PneumaticTire - HS 810 561
PneumaticTire - HS 810 561
PneumaticTire - HS 810 561
Preface
For many years, tire engineers relied on the monograph, “Mechanics of Pneumatic Tires”,
edited by S. K. Clark, for detailed information about the principles of tire design and use.
Published originally by the National Bureau of Standards, U.S. Department of Commerce
in 1971, and in a later (1981) edition by the National Highway Traffic Safety
Administration (NHTSA), U.S. Department of Transportation, it has long been out of
print. No textbook or monograph of comparable range and depth has appeared since.
While many chapters of the two editions contain authoritative reviews that are still rele-
vant today, they were prepared in an era when bias ply and belted-bias tires were in wide-
spread use in the U.S. and thus they did not deal in a comprehensive way with more recent
tire technology, notably the radial constructions now adopted nearly universally. In 2002,
therefore, Dr. H.K. Brewer and Dr. R. Owings of NHTSA proposed that NHTSA should
sponsor and publish electronically a new book on passenger car tires, under our editor-
ship, to meet the needs of a new generation of tire scientists, engineers, designers and
users. The present text is the outcome.
The chapter authors are recognized authorities in tire science and technology. They have
prepared scholarly and up-to-date reviews of the various aspects of passenger car tire
design, construction and use, and included test questions in many instances, so that the
book can be used for self-study or as a teaching text by engineers and others entering the
tire industry.
Contents
Chapter 1: An Overview of Tire Technology.............................................................. 1
by B. E. Lindemuth
13136 Doylestown Road, Rittman, OH 44270
by A. N. Gent
Polymer Science 3909, The University of Akron, Akron OH 44325-3909
by E. T. McDonel
10867 Fitzwater Road, Brecksville, OH 44141-1115
by S. M. Padula
Michelin North America, One Parkway South, P.O. Box 19001,
Greenville SC 29602
by M. J. Trinko
4426 Provens Drive, Green OH 44319
by M. G. Pottinger
1465 North Hametown Road, Akron OH 44333
by M. G. Pottinger
1465 North Hametown Road, Akron OH 44333
by D. M. Turner
Swithuns Gate, Ostlings Lane, Bathford, Bath BA1 7RW,
UNITED KINGDOM
by K. A. Grosch
Uelenbender Weg 22, 52159 Roetgen, GERMANY
by T. J. LaClair
Michelin North America, One Parkway South, P.O. Box 19001,
Greenville SC 29602
by K. A. Grosch
Uelenbender Weg 22, 52159 Roetgen, GERMANY
Chapter 14: Tire Properties That Affect Vehicle Steady-State Handling Behavior .. 594
by J. D. Walter
Civil and Mechanical Engineering Depts., The University of Akron,
Akron OH 44325-3905
Chapter 15: Introduction to Tire Safety, Durability and Failure Analysis .......... 612
by J. D. Walter
Civil and Mechanical Engineering Depts., The University of Akron,
Akron OH 44325-3905
by A. I. Isayev and J. S. Oh
Polymer Engineering, The University of Akron, Akron OH 44325-0301
Chapter 1
by B. E. Lindenmuth
Chapter 1
An overview of tire technology
by B. E. Lindenmuth
1. Introductory comments
Tires … round, black and expensive! That is the impression of most consumers who often
consider them a low-tech commodity and make purchasing decisions based solely on
price. Those with an opportunity to tour a tire production facility are surprised to learn that
there are 20 or more components, with 15 or more rubber compounds, assembled in a
typical radial passenger car tire and marvel at the massive amount of machinery and
processing involved to achieve the finished product. Tires are highly engineered
structural composites whose performance can be designed to meet the vehicle manufac
turers’ ride, handling, and traction criteria, plus the quality and performance expectations
of the customer. The tires of a mid-sized car roll about 800 revolutions for every mile.
Hence, in 50,000 miles, every tire component experiences more than 40 million loading-
unloading cycles, an impressive endurance requirement.
Historically, pneumatic tires began in Great Britain during the late 1800s as an upgrade
from solid rubber tires. They had small cross-sections and high pressures, principally for
bicycle applications. Larger “balloon” tires were introduced in the early 1920’s with
applications in the mushrooming motor vehicle industry. Tubeless tires were introduced
with improvements in rim design in the early 1950s. Belted bias tires (see Section 2.2,
Figure 1.1) became popular in the late 1960s. Radial tires, first introduced in Europe,
became popular in the USA starting in the early 1970s and now dominate the passenger
tire market.
This chapter serves as an introduction and overview of radial passenger tire
construction, performance, and testing typical of today’s product.
2. Tire basics
2.1 Function
Vehicle to road interface
The primary function of passenger car tires is to provide the interface between the
vehicle and the highway. The rubber contact area for all four tires for a typical mid-size
car is less than that of an 8½ x 11 inch sheet of paper; each tire has a footprint area of
about the size of an average man’s hand. Yet we expect those small patches of rubber to
guide us safely in a rain storm, or to allow us to turn fast at an exit ramp, or to negotiate
potholes without damage.
ferential direction, strengthening and stabilizing the tread region (see Figure 1.1).
Advantages: Improved wear and handling due to added stiffness in the tread area.
Disadvantages: Body ply shear during deflection generates heat; higher material and
manufacturing cost.
Radial tires
Radial tires have body ply cords that are laid radially from bead to bead, nominally at 90º
to the centerline of the tread. Two or more belts are laid diagonally in the tread region to
add strength and stability. Variations of this tire construction are used in modern passen
ger vehicle tire (see Figure 1.1).
Advantages: Radial body cords deflect more easily under load, thus they generate less
heat, give lower rolling resistance and better high-speed performance. Increased tread
stiffness from the belt significantly improves wear and handling.
Disadvantages: Complex construction increases material and manufacturing costs.
Load capacity
Tables of tire load ratings and load carrying capacity have been established by TRA. Their
purpose is to maintain a rational basis for choosing tire size, load and inflation. Details
will be covered in Chapter 5. TRA also coordinates its standards with other international
organizations such as ETRTO (European Tyre & Rim Technical Organization) and
JATMA (Japanese Automobile Tire Manufacturing Association). Note that ETRTO and
JATMA sizes can have different load-carrying capacities than like-sized P-metric tires.
3. Tire components
3.1 Rubber compounds
Purpose
Beyond the visible tread and sidewall compounds, there are more than a dozen specially
formulated compounds that are used in the interior of the tire. They will be discussed in
Section 3.3: Tire components.
Basic ingredients
Polymers are the backbone of rubber compounds. They consist of natural or synthetic
rubber. Properties of rubber and rubber compounds are described in more detail in chapter 2.
Fillers reinforce rubber compounds. The most common filler is carbon black although
other materials, such as silica, are used to give the compound unique properties.
Softeners: Petroleum oils, pine tar, resins and waxes are all softeners that are used in
compounds principally as processing aids and to improve tack or stickiness of
unvulcanized compounds.
Antidegradents: Waxes, antioxidants, and antiozonants are added to rubber
compounds to help protect tires against deterioration by ozone, oxygen and heat .
Curatives: During vulcanization or curing, the polymer chains become linked,
transforming the viscous compounds into strong, elastic materials. Sulfur along with
accelerators and activators help achieve the desired properties.
polymerization/spinning or melt spinning. The most common usage is in radial body plies
with some limited applications as belt plies.
Advantages: High strength with low shrinkage and low service growth; low heat set;
low cost.
Disadvantages: Not as heat resistant as nylon or rayon.
Rayon is a body ply cord or belt reinforcement made from cellulose produced by wet
spinning. It is often used in Europe and in some run-flat tires as body ply material.
Advantages: Stable dimensions; heat resistant; good handling characteristics.
Disadvantages: Expensive; more sensitive to moisture; environmental manufacturing
issues.
Aramid is a synthetic, high tenacity organic fiber produced by solvent spinning. It is
2 to 3 times stronger than polyester and nylon. It can be used for belt or stabilizer ply
material as a light weight alternative to steel cord.
Advantages: Very high strength and stiffness; heat resistant.
Disadvantages: Cost; processing constraints (difficult to cut).
Steel cord is carbon steel wire coated with brass that has been drawn, plated, twisted
and wound into multiple-filament bundles. It is the principal belt ply material used in
radial passenger tires.
Advantages: High belt strength and belt stiffness improves wear and handling.
Disadvantages: Requires special processing (see figure 1.16); more sensitive to
moisture.
Bead wire is carbon steel wire coated with bronze that has been produced by drawing
and plating. Filaments are wound into two hoops, one on each side of the tire, in various
configurations that serve to anchor the inflated tire to the rim.
Body plies
Body plies of cord and rubber skim wrap around the bead wire bundle, pass radially across
the tire and wrap around the bead bundle on the opposite side. They provide the strength
to contain the air pressure and provide for sidewall impact resistance. The tire example
shown has one body ply. In larger sizes, two body plies are typically used.
Bead bundles
Individual bronze plated bead wires are rubber coated and then wound into a bundle of
specified diameter and configuration prior to tire assembly. The bead bundles serve to
anchor the inflated tire to the wheel rim.
8 Chapter 1. An Overview of Tire Technology
Bead filler
Bead filler (also known as the apex) is applied on top of the bead bundles to fill the void
between the inner body plies and the turned-up body ply ends on the outside. Varying the
bead filler height and hardness affects tire ride and handling characteristics.
Sidewall
Tire sidewall rubber serves to protect the body plies from abrasion, impact and flex
fatigue. The sidewalls also carry decorative treatments, sometimes including white or
colored stripes or letters. The rubber compound is formulated to resist cracking due to
environmental hazards such as ozone, oxygen, UV radiation and heat.
Belt wedges
Small strips of belt skim or other fatigue resistant compounds are sometimes placed
between the belts near the edge of the top (number 2) belt. The purpose is to reduce the
interply shear at the belt edge as the tire rolls and deflects.
Shoulder inserts
Shoulder inserts are small, sometimes contoured strips of rubber placed on the body ply,
under the belt ends. They help maintain a smooth belt contour and insulate the body plies
from the belt edges.
Tread
The tread must provide the necessary grip or traction for driving, braking and cornering,
and the tread compound is specially formulated to provide a balance between wear,
traction, handling and rolling resistance.
A pattern is molded into the tread during vulcanization or curing. It is designed to
provide uniform wear, to channel water out of the footprint, and to minimize pattern noise
on a variety of road surfaces.
Both the tread compound and the tread design must perform effectively in a multitude
of driving conditions, including wet, dry or snow covered surfaces, while also meeting
customer expectations for acceptable wear resistance, low noise, and good ride quality.
For driving in severe winter conditions, snow tires with increased tread depth and
specially formulated tread compounds are recommended.
Subtread
The subtread, if used, is typically a lower hysteresis, cooler-running compound extruded
under the tread compound to improve rolling resistance in order to meet the OE vehicle
manufacturers’ goals for fuel economy. It also can be used to fine-tune ride quality, noise,
and handling.
Undertread
The undertread is a thin layer of rubber placed under the extruded tread/subtread package
to boost adhesion of the tread to the stablilizer plies during tire assembly and to cover the
ends of the cut belts.
10 Chapter 1. An Overview of Tire Technology
Construction selection
Body ply denier, cord style, EPD (Ends Per Decimeter) or EPI (Ends Per Inch), and
number of plies affect body strength and are chosen based on manufacturing, engineering,
and design criteria. Likewise steel cord construction (style) and EPD both affect belt
strength and are chosen typically based on tire size and application. Belt widths and belt
crown angle (see figure 1.8) also influence tire performance. Different crown angles
change the belt package stiffness, laterally and longitudinally, which can affect cornering
ability and ride. Belt widths can also be varied. If a high speed rating is required, the addi
tion of nylon cap strips at the belt edges or full-width nylon cap plies may be added. The
bead and sidewall areas can also contribute to subtle performance enhancements. The
bead filler volume and height, as well as the location of the end of the turned-up body ply
(see figure 1.8) all impact sidewall stiffness.
Chapter 1. An Overview of Tire Technology 11
Materials selection
Tread compounds are chosen to meet handling and traction requirements for wet, dry and
snow (if necessary), but must have suitable wear potential and resistance to gravel chips
and tearing. Subtread compounds and thickness are often determined by the rolling
Chapter 1. An Overview of Tire Technology 13
and guidelines and standards have been developed by individual tire manufacturers as
well. They serve as a starting point. In addition to experience, tire engineers use com
puter models and performance maps to help guide their selections and predict if per
formance targets will be met. Using an iterative process of design, construction and mate
rial choices, the engineer can reach a balance of compromises for each application. Figure
1.9 illustrates the impact of just one component change, wider belts, on selected tire per
formance parameters. In this so-called “spider diagram” a higher rating (outside the cir
cle) indicates improved performance. Handling and wet traction are improved but ride,
rolling resistance and weight have suffered. If the customer desires the handling improve
ments but is unwilling to accept the loss in ride quality and higher rolling resistance, the
tire engineer must look at other factors to balance the overall performance. This dilemma
is what drives new tire technology in design, materials and construction.
Irregular wear
Abnormal wear features, such as heel and toe, cupping, or shoulder wear (see figure 1.10)
can significantly shorten the service life or mileage potential of tires. While tread design
and tire construction are influential, many external factors such as vehicle mis-alignment,
vehicle suspension geometry and driving factors such as high speed cornering, rapid
acceleration or braking and underinflation of tires play significant roles in promoting
irregular wear patterns. Consumers who regularly check tire inflation pressure and
maintain a schedule for rotating tire positions and checking vehicle alignment will
maximize tire mileage.
Gravel chip/tear
Some tread compounds and tire designs can be sensitive to chipping and tearing of tread
elements during off-road or gravel road applications. Many outdoor wear and durability tests
include a small percentage of gravel roads to assure that this performance is acceptable.
Most test facilities can run the same test in both wet and dry conditions. Snow tests are
conducted at special facilities where the snow can be groomed and compacted to make a
consistent surface.
Ride comfort
A vehicle’s perceived ride comfort, whether “sporty” or “plush,” can be significantly
influenced by tires. Engineers’ evaluations go far beyond expectations of shake-free and
vibration-free ride on smooth highways. Tires are evaluated for impact harshness over
highway joints and railroad tracks, and for damping and bounce memory after road
disturbances. They are also graded for plushness (road isolation), nibble (steering wheel
oscillations), shake, vibration and other vehicle-specific features. Most tire manufacturers
have test facilities with dedicated lanes specifically designed for consistent evaluation of
tire and vehicle combinations by professional ride evaluators. Ride is one of the
compromises encountered in designing tires. For instance, wider belts may improve
vehicle handling but can contribute to increased ride impact harshness.
Noise
Significant time and effort goes into designing tire tread patterns and constructions to
minimize noise. Patterns have tread elements of varying pitch lengths to prevent tires from
generating identifiable tones. Multiple pitch lengths, typically 3 to 7 (see Figure 1.6), are
assembled in a computer-generated pattern around the tire circumference to dispel any
constant frequency noise as the tire rotates. Professional tire evaluators, using prepared
test areas, are able to sense not only airborne pattern noise but also structure-borne noise
as well. Structure-borne tire noise is transmitted through the tire carcass, the wheel and
suspension, sometimes aggravating resonances in a vehicle component. Evaluators rate
coarse road noise transmitted through the tire from textured highway surfaces. They
listen for growl, a low frequency noise noticed during low speed braking, and sizzle, a
hissing sound on ultra-smooth surfaces. These and other noise conditions add to the tire
engineers design dilemma. Increasing tread thickness and softening the bead filler reduces
coarse road noise but increases rolling resistance and affects ride and handling.
Drift/pull
A vehicle with drift/pull has a tendency to pull right or left while driving on a straight, flat,
level highway with minimal wind. Tires can contribute to this condition but vehicle
alignments and suspension geometry are also key factors. Drivers find the constant
steering correction annoying.
Endurance
Outdoor testing for tire endurance usually involves loading a vehicle to the maximum
specified load and inflation, or more, and driving on a closed road course at a specified
Chapter 1. An Overview of Tire Technology 17
schedule of speeds. With a three shift per day operation, and measurement/
inspection/maintenance delays, it takes approximately 45 working days to accumulate
40,000 mi. Each tire company has its own proprietary test protocol.
Endurance
In high speed testing, load is constant and speed is varied. In most indoor drum endurance
tests, the speed is constant and load is varied. Tests similar to DOT 109 and the new DOT
139 are as follows. The ambient temperature remains at 38°C. Pressure for a P-metric
standard load passenger radial is 180 kPa (26 psi) with a constant speed of 80 km/hr (50
mi/hr) for DOT 109 type tests. New DOT 139 regulations require 120 km/hr (75 mi/hr)
testing, becoming mandatory in June 2007. The test load begins at 85% of maximum load
for 4 hours and then becomes 90% for 6 hours and finally 100% for 24 hours. Tires com
pleting the initial 34 hour test must also complete an additional new DOT 139 low
pressure step at 140 kpa (20 psi) for an additional 90 minutes at the 100% load condition.
(See Chapter 17).
Rolling resistance
The force necessary to overcome hysteretic losses in a rolling tire is known as rolling
resistance. This parameter became important to USA vehicle manufacturers with
implementation of C.A.F.E. (Corporate Average Fuel Economy) standards for new cars. It
is measured by placing load cells in the wheel spindle and measuring the rolling
resistance force in the horizontal (longitudinal) direction. It requires precise instrumenta
tion, calibration, speed control and equipment alignment for repeatable results. Rolling
resistance is usually expressed as a coefficient: resistance force per 1000 units of load. OE
passenger car tires designed for fuel efficiency may have coefficients in the range 0.007
to 0.010 when fully inflated and evaluates at thermal equilibrium (see chap. 12). The test
load, speed and inflation pressure vary according to the vehicle manufacturers’
requirements. Rolling resistance is significantly influenced by inflation pressure, as
illustrated in Figure 1.11. Since tire rolling resistance can consume up to 25% of the
energy required to drive at highway speeds, it is economically wise to keep tires inflated
properly.
18 Chapter 1. An Overview of Tire Technology
Resistivity
Moving vehicles can generate static electricity which is aggravated by low temperature
and humidity. While rubber is usually thought to be an insulator, it is partially conductive,
and tire compounds influence the rate of static discharge. Test fixtures in humidity and
temperature-controlled laboratories are used to measure tire resistivity.
Uniformity
Due to material and assembly variations that occur during manufacturing and curing,
small deviations in tire cross section circumferentially can result in measurable spring rate
or dimensional changes, for example, an out-of-round condition. Tire Uniformity Grading
machines are used to measure the variations that occur around the circumference of the
tire. Inflated tires are loaded against an instrumented rotating drum. The radial and later
al force variations measured are compared to acceptance standards for smooth, vibration-
free ride.
Chapter 1. An Overview of Tire Technology 19
Flat spotting
Some tires, when parked, can develop a temporary “set” in the rubber compounds and
reinforcement cords, referred to as a “flat spot”. To test for this condition, tires are warmed
up or exercised at high speed, measured for uniformity and then loaded statically against
a flat plate for a prescribed time (usually days). Tires are then retested for uniformity,
exercised and the recovery time observed for the flat spot to disappear.
Air permeation
Innerliner compounds are formulated to minimize permeation of air through the tire
carcass. The permeation rate depends on the compound properties and gauge (thickness)
as well as the temperature and inflation pressure. Long term tests, taking months, require
regulated temperatures and leak-free tire mounting and test plumbing. Typical passenger
car tires lose approximately 1 psi per month due to air permeation.
High speed
DOT 139 requires testing conditions as described in Section 4.2. Tires must complete the
160 km/hr (100 mi/hr) step without failure to be in compliance.
Endurance
DOT 139 requires completion of all three steps at 120 km/hr (75 mi/hr) as described in
Section 4.2. All tires must complete the endurance portion, plus a 90-minute low inflation
pressure step, without failure to be in compliance.
Bead unseat
DOT 139 requires that tires retain air pressure and beads remain seated on the wheel in a
test where an anvil is pressed against the tire sidewall. Wheel, tire inflation and anvil
location are specified by rim diameter and tire type. Potential revisions to this test are
under study.
energy is calculated. Minimum energy requirements without tire rupture occurring must
be met at multiple locations around the tire circumference to be in compliance. Research
continues to determine if new or revised test procedures are needed to accommodate new,
lower aspect ratio tire sizes.
5. Tire manufacturing
5.1 Compound preparation (see Figure 1.12)
Raw materials
Approved vendors supply the basic ingredients including polymers, fillers (carbon black
and/or silica), softeners and antidegredants. Lab tests are run to sample, code and release
the materials for use in production.
Polymers
Carbon black/fillers
Sulfur/curatives
Oil/softeners
Other chemical agents
Mixing
The appropriate blend of polymers, fillers, oils and pigments for a specific compound
formula are combined in a closed mixer in batches of 180 kg (400 lbs) to 500 kg (1100
lbs). Batch temperatures are closely controlled, as are mixing power, cycle time and rotor
speed, in accordance with the compound specification. Each batch is flattened into slabs
or extruded and cut into pellets (not shown in figure 1.12) for storage and later blending
with other batches or materials.
Blend/feed mills
Large, closely spaced, water-cooled rollers squeeze and kneed a bank of compounds to
blend mixed batches and to warm up compounds prior to extrusion or calendering.
sheets are used in tire assembly for inner-liner, gum strips, or belt wedges, or in preparing
body ply or stabilizer ply material.
Body plies
In textile reinforcements of nylon, polyester, rayon, aramid, etc., individual filaments are
twisted and cabled together to form cords. The cords are woven , with pick cords to
maintain spacing, into a wide sheet of fabric prior to dipping in a latex adhesive to
enhance bonding to rubber, followed by a high-tension heat treatment (see Figure 1.13).
Body ply fabric is prepared in rolls approximately 57 inches wide by 3,000 yards long
having the appropriate denier (cord style) and EPD (ends per decimeter). The fabric is
then passed through a four-roll calender (see Figure 1.14) where a thin sheet of rubber
(body ply skim) is pressed onto both sides and squeezed between the cords of the fabric.
The calendered fabric is wound into 350-yard rolls, with a polypropylene liner inserted
to keep the fabric from sticking to itself, and then sent to a stock-cutting process.
cord sheet is cut, rotated, and spliced together again to form a continuous strip of a
specified width, with the cords at a specified angle. Since the number 1 and number 2 belts
have different widths and opposite angles, they must be prepared accordingly.
the bundle for tire building efficiency. Some tire manufacturers use a cable bead that
features a solid core with several bead wires cabled around it. This requires a different
manufacturing process than that shown in figure 1.16.
Figure 1.16: Tire manufacturing - bead bundle preparation
beads and rolled or stitched to adhere to the body ply lying flat on the drum. Strips of
sidewall compound are then placed on top of the turned-up ply on both edges. The drum
is then collapsed so the completed body carcass can be removed and taken to the second
stage machine.
Figure 1.18: Tire manufacturing - radial tire assembly
the diameter of the tire, most often in the middle of the center rib. The two mold halves
are attached to the top and bottom of the curing press and the green tire is inserted between
them. As the press closes, bringing the two halves together, the green tire is expanded by
inflating an internal rubber bladder that forces the outside of the green tire to conform to
the inner surface of the mold. After curing is completed, the mold parts separate and the
tire is removed. Note that the tread must deform to some degree as the mold opens, which
can be more difficult for some tire profiles, compounds, and tread designs.
Segmented molds are more complicated but are preferred for certain radial tires. This
is because the green tire diameter can more closely approach the mold diameter,
minimizing expansion of the belt. The tread region of the mold typically consists of 8 or
9 radially-divided pieces or segments that come together as the mold closes. They must
fit together precisely, both radially and with the top and bottom sidewall plates, when the
mold is completely closed. The advantage of segmented molds is that less expansion is
required of the green tire to fill the mold and the segments pull directly away from the tire
after curing. This is an important feature in manufacturing low-profile tires and tires that
use some tread compounds designed for low rolling resistance.
Curing (vulcanization)
As the mold closes, a bladder inside the tire expands and presses the green tire against the
mold. The high bladder pressure (several hundred psi) forces the uncured rubber into
every detail of the inner surface of the mold. Super-heated steam or hot water is then
circulated within the bladder and around the mold for about 12 to 15 minutes. This rise in
temperature causes a chemical reaction (curing, or vulcanization) to occur in the rubber
compounds whereby the long polymer molecules become crosslinked together by sulfur
or other curatives. The rubber compounds are transformed in this way into strong, elastic
materials in the finished, cured tire. Curing times, temperatures and pressure are
computer-controlled to give full cure of the chosen rubber compounds.
Visual inspection
All tires are inspected visually for any imperfections (e.g., plugged vents or trapped air)
before being transferred to a warehouse. If a minor imperfection cannot be buffed away
or repaired, the tire is scrapped.
Uniformity grading
Most passenger tires are screened for uniformity (Section 4.3) against limits established
by the OE vehicle manufacturer and/or the tire manufacturer. The test includes laser or
probe inspection of the sidewalls and tread.
Statistical sampling
Once in production, QA (Quality Assurance) protocols utilize statistical process control
and sampling methods to assure that production tires remain in compliance with their
original quality and performance standards.
6. Consumer care
6.1 Maintain proper inflation
By law, every vehicle sold in the United States has a placard that identifies the tire size(s)
and the inflation pressures recommended for front, rear, and spare tires by the OE vehicle
manufacturer. Beginning in the 2006 model year, the placard is to be located on the
driver’s door B-pillar. The location on older vehicles varies, sometimes being on the
driver’s door, the trunk lid, a door pillar or the glove box. In addition, the maximum load
and inflation information is stamped on the sidewall of every tire, usually in the bead area
just above the rim.
Tire pressures should be checked regularly. Radial passenger tires can be underinflated
by 12 psi or more and still look normal. As explained in Section 4.3, air permeates through
tires slowly, so that they typically lose about 1 psi per month, and even more in hot
climates. Moreover, a small puncture from an imbedded nail or screw can cause a tire to
be significantly underinflated. Underinflation contributes to rapid and uneven tread wear,
a loss in fuel economy, poor vehicle handling and excessive heat buildup which may lead
Chapter 1. An Overview of Tire Technology 27
to tire failure. It is recommended that tire pressures be checked at least monthly and before
long trips or when additional passengers and luggage are carried. Pressures should be
checked when the tires are cold, i.e., when the vehicle has not been driven for several
hours, and using an accurate gauge. It should be noted that every 10°F drop in ambient
temperature results in about one psi drop in tire inflation pressure. It is also recommended
that tire valve assemblies be replaced when a new tire is installed.
Acknowledgement
The writer would like to thank Dan Saurer, Division Vice-President, Consumer Tire
Development, Bridgestone/Firestone North American Tire, LLC, for providing the
resources and support to assemble the material contained in this chapter.
Bibliography
1. D. Beach and J. Schroeder, “An Overview of Tire Technology”, Rubber World, 222 (6),
44-53 (2000).
2. T. French, Tyre Technology, Hilger, New York, 1989.
3. F. J. Kovac and M. B. Rodgers, “Tire Engineering”, in Science and Technology of
Rubber, 2nd Ed., ed. by J. E. Mark, B. Erman and F. R. Eirich, Academic Press, New
York, 1994, pp. 675-718.
4. G.F. Morton and G.B. Quinton, “Manufacturing Techniques”, in Rubber Technology
and Manufacturing, 2nd Ed., ed. by C.M. Blow and C. Hepburn, Butterworth, London,
1982, pp. 405-431.
28 Chapter 2. Mechanical Properties of Rubber
Chapter 2
by A. N. Gent
4. Abrasion .................................................................................................................. 68
References ................................................................................................................ 76
Chapter 2
Mechanical Properties of Rubber
by A. N. Gent
A B C
Passenger Sidewall Steel belt
tire tread compound
Elastomers
Natural rubber - 50 100
Styrene-butadiene
copolymer (25/75) 75 - -
Cis-polybutadiene 25 50 -
Fillers
Fine carbon black (e.g., N220) 75 - -
Medium carbon black (e.g., N330) - - 60
Coarse carbon black (e.g., N660) - 50 -
Vulcanization agents
Zinc oxide 3 3 10
Stearic acid 3 1 1.2
Sulfur 1.55 2 5.5
Vulcanization accelerator
(sulfenamide type) 1.9 1 0.5
Secondary accelerator
(guanidine) 0.25
Processing aid
Processing oil 10 10 -
Protective additives
Antioxidant/antiozonant 1.5 3.5 1
Resin and adhesion promoter
Resin components - - 8
Adhesion promoter
(e.g.cobalt naphthenate) - - 2
Vulcanized for 10-15 minutes in a tire mold at a temperature of 170-175°C
small - only the entropy of the network is reduced. Thus, rubber is an entropic spring - it
returns to the undeformed state only because that is the most probable one. As a result, the
stiffness of rubber increases with temperature T because the contribution ∆S of entropy to the
free energy of deformation is given by the product: - T∆S. This characteristic feature of
rubber is in marked contrast to the behavior of springy crystalline materials, like metals,
where the stiffness decreases with rising temperature because, on deforming metals, the
atoms are forced out of an ordered state into a more random arrangement, with lower bond
energies and higher entropy.
An approximate statistical calculation gives the tensile modulus E (Young’s modulus)
of elasticity for an unfilled rubber vulcanizate under small tensile strains as
E = 3NkT (1.1)
The average network strand consists of about 100 repeat units (mers) of a particular
chemical group, typically with a molecular weight of around 60. Some repeat units in
rubbery materials commonly used in tires are:
-C4H6- : cis-1,4 polybutadiene
-C5H8- : natural rubber (cis-1,4-polyisoprene)
-CH2C3H6- : butyl rubber
(-CH2-C7H12-)n-(-C4 H6 -)m : SBR (a copolymer of styrene, n units, and butadiene, m units)
But why are only relatively few polymers rubbery? There are two main reasons why a
polymeric solid might not exhibit rubberlike elasticity. When the long molecules are highly
regular and able to pack closely together, they tend to arrange themselves in this way
under the weak van der Waals attractive forces that are present in all materials – they spon-
taneously crystallize. For example the commercial plastic – polyethylene - is as much as
60% crystalline at room temperature and shows little elasticity. Instead, it behaves as a
typical crystalline solid. Initially rather stiff, after the material is deformed by a few
percent the crystallites begin to yield and the deformation then becomes ductile and
plastic, rather than elastic.
The other reason why polymeric materials may fail to show rubbery behavior is that
the chemical repeat units are relatively heavy and bulky, so that thermal energy at normal
temperatures is not sufficient for molecular segments to move freely. Polymers in this
state are termed glassy solids. Typical examples are polystyrene and polymethylmethacry-
late that are glasslike at temperatures below about 100oC.
On raising the temperature, crystals melt and immobile molecules acquire more
energy, so that both crystalline and glassy polymers become rubbery at sufficiently high
temperatures, for example, above 150oC for polyethylene and above 100oC for
polystyrene and polymethylmethacrylate. And all rubbery polymers turn into glassy solids
at a characteristic low temperature, even if they have not spontaneously crystallized as
they are cooled down. The transition temperature (glass temperature, Tg) is one of the
most important characteristics of a polymer and, as shown later, it controls the
mechanical behavior at temperatures far above the actual glass temperature. Values of Tg
for selected rubbery polymers are given in Table 2.2.
32 Chapter 2. Mechanical Properties of Rubber
The other important physical feature of a rubbery solid is the average length of the
molecular strands comprising the network, generally denoted by the molecular weight Mc.
The original molecules have molecular weights in the range 100,000 to 500,000, not high
er than this because very high molecular weight materials are difficult to extrude and
shape. After crosslinking, the molecular weight Mc of the network strands ranges from
about 5,000 to 20,000 molecular weight units. Although this number may, in principle, be
varied widely by introducing more or fewer crosslinks, in practice if Mc is small then the
material is stiff and brittle and if Mc is large then some of the original long molecules are
not tied into the network completely, and contribute little or nothing to stiffness and
strength. Nevertheless, the practical range of crosslinking and hence of elastic modulus E
is from about 1 MPa to about 3 MPa, a greater range (a factor of 3) than for any other
structural solid. And by adding particulate fillers, the modulus can be raised still further
without serious loss in strength - indeed, often with a pronounced gain in strength, termed
reinforcement (see later).
IRHD E(MPa)
30 1.0
35 1.2
40 1.5
45 1.9
50 2.3
55 2.9
60 3.6
65 4.5
70 5.5
75 7.5
80 9.5
85 15
Note that when a solid rubber block is “compressed”, its volume does not decrease
significantly unless the pressures are extremely high. Instead, the block compresses
vertically and expands laterally - the volume remains virtually unchanged. Compression
of a block bonded between two plates is sketched in figure 2.3. Due to the restraint on lat
eral expansion, the effective modulus Ee can be quite high.
Rubber molecules move freely because the intermolecular forces are low. The
expansion in volume on heating is correspondingly high, similar to that for simple liquids.
The linear thermal expansion coefficient α for unfilled rubber compounds ranges from
about 1.5 to 2 x 10 -4/oC. Due to the low thermal expansion coefficients of filler particles,
values of α for typical filled compounds are somewhat lower, 1.2 to 1.5 x 10 -4/oC, but
still far higher than for metals, about 50 times greater. Thus, when a rubber article is
removed from the metal mold in which it has been shaped and crosslinked at a high
temperature (typically about 160oC) and then allowed to cool down to ambient
temperature, it shrinks in linear dimensions by about 2%.
(Note: λ1 = 1 + e1, etc., where e1 is the strain in the 1 direction.) Because rubber is
virtually incompressible, the product λ1λ 2λ3 = 1. Equation 1.2 is termed the neo-
Hookean constitutive law because it reduces to Hookean (linear ) elasticity, figure 2.2, in
the limiting case of small strains and gives reasonably accurate predictions at moderately
large strains.
34 Chapter 2. Mechanical Properties of Rubber
distinguish between the two usages, σ is sometimes termed engineering stress. For an
incompressible material, like most rubbery solids, the imposition of a stretch ratio λ leads
to a contraction in cross-sectional area by a factor 1/λ and thus there is a simple relation
between the two measures of stress (see figure 2.5):
σ = t/λ (1.6)
In rubber technology, the term “modulus” is often used to describe the tensile force per
unit of unstrained cross-sectional area at a stipulated tensile strain. For example, M100 denotes
the stress σ when e = 100%. Values for M100, M200, M300, etc., are often reported also - they
give an indication of the shape of the stress-strain curve, see figure 2.6.
The stress-strain relation in simple extension is obtained from Equation 1.5 as:
t = (E/3) (λ12 - 1/λ1) (1.7)
This is the large-deformation equivalent of the result: t = Ee, applicable at small strains.
Experimental stress-strain relations for soft rubber compounds are found to be in reasonable
agreement with equation 1.7 up to moderately large extensions, about 300%.
Note that stress-strain relations for rubber are usually non-linear, figures 2.7 - 2.9.
Equation 1.7, for example, predicts that the tensile stress-strain relation, after the expected lin
ear region at small strains, will be concave with respect to the strain axis, as is observed. At
higher strains a pronounced stiffening occurs as the molecular strands approach the fully-
stretched state, figure 2.8. Thus, non-linearity does not indicate that internal yielding or frac
ture has occurred. Instead, it follows directly from a consideration of the elastic behavior of
a network of long molecular chains, each of which is linearly-elastic over a wide range of
extension but because they are randomly arranged in space, the combined effect is usually
non-linear.
There is one outstanding exception to the general non-linear character of rubberlike
elasticity, however, and that is a simple shear deformation. Using equation 1.5 the calculated
stress-strain relation in simple shear is linear, and rubber compounds are found
to be approximately linear in shear up to moderately large strains, 100% or more, with a slope
(shear elastic modulus G) of E/3.
Many constitutive laws have been proposed to model the elastic behavior of rubber more
accurately than the simple neo-Hookean form for W given in Equation 1.2, for
example by taking into account the stiffening observed at high strains, Figures 2.6 and 2.8.
This feature is not predicted by Equation 1.2. It can be modeled by introducing a maximum
possible value Jm for the deformation term J1. Thus one modified version of Equation 1.2 is
the logarithmic form:
which reduces to equation 1.2 when the deformations are small compared to Jm, but gives
increasingly large values for W and thus for stresses when the deformation approaches its
maximum possible value Jm.
However, rubber compounds currently used in tires are seriously inelastic, as described
below, and attempting to describe their elastic properties with great precision is probably
unwarranted. Examples of typical tire formulations are given in Table 2.1. They illustrate
the complexity that results when compromises must be made to meet demanding but often
conflicting requirements: stiffness, strength, resilience, good traction, wear resistance,
durability, etc.
Chapter 2. Mechanical Properties of Rubber 37
This large effect of strain history on the stress-strain behavior is termed the “Mullins”
effect [1]. It appears to be due to slipping of rubber molecules over energy barriers on the
surface of filler particles or becoming detached from the filler surface. It renders computer
modeling of the stress-strain behavior of typical tire tread compounds extremely difficult.
For example, we know very little about the effect of pre-strains other than tensile strains,
or of complex strains as is, of course, the case for tire components, or about the rate of
recovery from prior straining.
To make matters even more complicated, a similar effect is found when quite small
strains are imposed, in the range, say, from 0.1% to 10%. Such measurements are usually
carried out dynamically, by imposing a small oscillatory tensile or shear strain at a
convenient frequency in the range 0.1 to 100 Hz, and reporting the results in terms of the
effective dynamic modulus E´ or G´, the ratio of peak stress to peak strain. E´ or G´ are
found to decrease dramatically over this range of dynamic strain amplitude, often to only
1/3 or less of the initial value, figure 2.10. Again, although partial recovery by 30 to 50%
occurs almost immediately on returning to small deformation amplitudes, complete
recovery takes months of rest unless the sample is heated to accelerate the recovery.
This small-strain softening phenomenon is often termed the “Payne” effect [2]. It
appears to be due to disruption of chain-like aggregations of filler particles within the
rubber matrix. Because the effect is so large in highly-filled compounds, they are seldom
used in rubber springs because the relevant “modulus” for calculations of spring rate and
resonant frequency is ill-defined. And, again, it is difficult to take the “Payne” effect into
account in modeling the stress-strain response of tire compounds in treads and sidewalls.
Figure 2.10: Stress-softening of filled
rubber at small strains (Payne effect)
40 Chapter 2. Mechanical Properties of Rubber
In summary, typical filled rubber compounds show two disturbing features: the
“Payne” effect”, a marked softening that starts at strain amplitudes of only 0.1 % or less,
attributed to breakdown of weak inter-particle bonds, and “Mullins” softening”, a further
substantial softening at higher strains, up to the breaking strain, that is attributed to
progressive rupture of bonds between rubber molecules and filler particles. Because these
two processes overlap to a considerable degree, they cannot be easily separated. However,
they appear to be a direct consequence of the high stiffening power of fillers of small
particle size and highly interactive surfaces when they are incorporated into rubber.
Note that, because filled rubber compounds are not elastic, i.e., they do not follow
reversible stress-strain relations, their behavior cannot be described successfully by any
elastic constitutive law relating stresses to strains. Instead, the tire designer must accept
quite approximate representations of the elastic properties of present-day tire compounds,
recognizing that they are subjected to complex strains and strain histories that have major,
and unfortunately ill-defined, effects on the corresponding stresses.
1.3 Visco-elasticity
Under repeated oscillations of shear strain γ, the stress-strain relation becomes an ellipse,
figure 2.11, provided that the strain amplitude is small. The slope of the line joining points
where tangents to the ellipse are vertical represents a dynamic shear modulus G´ (MPa)
and the area of the ellipse represents energy Ud dissipated in unit volume per cycle of
deformation, given by
Ud = π G″ γm2 (1.9)
where γm is the amplitude of shear strain and G″ is termed the dynamic shear loss modulus.
(For an oscillatory tensile deformation the corresponding moduli are denoted E′ and E″.)
The ratio G″ /G´ (or E″/E′) is the tangent of an angle δ, the phase angle by which the strain
lags behind the applied stress. When the ellipse axes lie in the horizontal and vertical
directions, δ is 90o (tan δ is infinitely large), and the rate of strain reaches its maximum value
when the applied stress is a maximum. This is the response of a viscous liquid. On the other
hand, when the ellipse degenerates into a straight line, tan δ = 0 and the material is a perfect
ly elastic solid. Values of tan δ for rubber compounds at room temperature range from about
0.03 for a highly resilient, “springy” material with low energy dissipation to about 0.2 for a
typical tread compound with relatively high dissipation.
Figure 2.11: Oscillatory deformation: dynamic effects
Chapter 2. Mechanical Properties of Rubber 41
where h1 is the drop height and h2 is the rebound height. If the impact is regarded as one
half-cycle of a steady oscillation, then
ln R = - π tan δ (1.11)
The loss modulus G″ increases with frequency to an even more marked degree than G´,
often becoming larger than G´ in the transition range of frequencies, because G″ is a direct
measure of viscous resistance to segmental motion. At sufficiently high frequencies, how
ever, the segments become unable to respond to the rapidly-alternating applied stresses,
and internal motion ceases. Energy dissipation that is associated with the motion of
Chapter 2. Mechanical Properties of Rubber 43
molecular segments in a viscous environment also ceases and G″ falls to the relatively low
value characteristic of polymeric glasses.
The rate ϕ of Brownian motion of molecular segments depends only upon the internal
viscosity and hence only upon temperature. Below Tg the polymer is glassy: above Tg it
is liquid (if low in molecular weight), rubbery (if high in molecular weight or crosslinked),
or crystalline (if the molecules are sufficiently regular), figure 2.13. The dependence of ϕ
upon temperature, or more precisely upon the temperature difference (T – Tg), follows a
characteristic law [3]:
log [(ϕ(T)/ϕ(Tg)] = A(T - Tg)/(B + T - Tg) (1.13)
where A and B are constants, having approximately the same values, 17.5 and 52oC, for
nearly all rubberlike substances, and Tg is a reference temperature, the glass transition
temperature, at which molecular segments move so slowly, about once in 10 seconds, that
for all practical purposes they do not move at all and the material becomes a rigid glass.
[By common consent, Tg is defined as that temperature at which the rate of Brownian
motion ϕ(Tg) has fallen to 0.1 Hz.] Values of Tg for some common elastomers are given
in Table 2. Equation 1.13 is represented graphically in figure 2.14.
In many applications molecular motion is required at frequencies much higher than 0.1
Hz. For example, for high resilience in a rebound experiment we require virtually
complete rubberlike response in a time of impact of the order of 1 ms. But molecular
segments will move in 1 ms only when the value of ϕ is about 1000 jumps per second.
Figure 2.14: Dependence of segmental mobility φ on temperature
[Williams, Landel & Ferry 1955]
From equation 1.13, that is only at a temperature about 16oC higher than Tg. Indeed,
for coordinated motion of entire molecular strands consisting of many segments to take
place within 1 ms, the segmental response frequency must be higher still, by a factor of
100 or so. This rate of response is achieved only at a temperature about 30oC above Tg.
Thus, fully rubberlike response will not be achieved until the temperature is Tg + 30oC,
or even higher. On the other hand, for sufficiently slow movements taking place over sev
eral hours or days, a material would still be able to respond at temperatures significantly
below the conventionally-defined glass transition temperature. This region is represented
by the lower portions of the curves in figure 2.14.
It is important to recognize that the conventional glass transition temperature is defined
in terms of relatively slow motions, taking place in about one minute, and requiring only
small-scale motions of individual molecular segments rather than motion of entire
molecular strands between crosslinks. The frequencies at which the entire rubber-to-glass
transition occurs depend on temperature in accordance with Equation 1.13.
The numerical coefficients, 17.5 and 52oC, in equation 1.13 are about the same for a
wide range of elastomers, because most elastomers have similar values of thermal
expansion coefficient and the molecular segments are similar in size. However, an
important exception is polyisobutylene and its derivatives, butyl rubber and halo-butyl.
For these materials, the coefficients appear to be about 17.5 and 100oC, the latter being
considerably different from the “universal” value that holds for other common elastomers.
Chapter 2. Mechanical Properties of Rubber 45
Thus, the rate of segmental motion increases much more slowly above Tg as shown by the
curve for butyl rubber in figure 2.14. Although the reason for this peculiarity is not well under
stood, it is probably associated with an unusually large size for the basic moving segment.
Two important consequences are that butyl rubber and halo-butyl exhibit unusually low
resilience, low gas permeability and good ozone resistance at temperatures well above Tg.
Equation 1.13 can be used to relate the dynamic behavior at one temperature T1 to that
at another, T2. For example, the dynamic modulus G′ and loss modulus G″ are found to
depend on the frequency of vibration as shown schematically in figure 2.12. When the
temperature is raised to T2, the curves are displaced laterally by a fixed distance, log aT,
on the logarithmic frequency axis, where log aT reflects the change in characteristic
response frequency of molecular segments when the temperature is changed from T1 to
T2. Thus log aT is given by:
log aT = log [ϕ(T2)/ϕ(T1)]
= 17.5 x 52(T2 – T1)/(52 + T2 - Tg)(52 + T1 - Tg) (1.14)
from Equation 1.13. In this way, measurements at one temperature can be transformed
into results at another. This is a powerful way of predicting viscoelastic response over
wide range of frequency from measurements over a limited range of frequency but at
many temperatures, using the general principle that a temperature shift is completely
equivalent to a change in frequency:
Log fT = Log fTg + Log aT (1.15)
The modulus of filled compounds is also decreased by raising the temperature, because
the rubber-particle bonds become weaker. This effect is not directly associated with
changes in molecular mobility and thus it cannot be represented by an equivalent change
in frequency. Hence, equation 1.14 does not apply to the modulus of filled compounds
without a correction for additional effects of temperature.
artificial cuts, and thus difficult to observe. Because the effective size deduced for all of
the rubber compounds studied to date, both filled and unfilled, is similar, they might be
the result of a common feature of rubber processing, for example a natural heterogeneity
of crosslinking, or they may be created in the course of stressing rubber up to the point of
rupture by a precursor mode of fracture. Local cavitation under tensile stress is one
possibility.
Further studies are highly desirable because initiation of fracture is an important aspect
of fracture mechanics. At present, we can account successfully for the strength of rubber
under a variety of conditions on the assumption that inherent flaws are present, equivalent
to sharp cuts about 50 µm deep.
Gc = 2P/w (2.1)
Similar considerations were applied by Griffith to tensile rupture of glass rods containing
a small edge crack of length c [4]. Here the crack grows at the expense of strain energy
W stored in the stretched sheet. When W is sufficiently high that the reduction -dW caused
by growth of the crack by a small amount dc is sufficient to meet the requirements for
fracture, Gcw.dc, then fracture ensues. The fracture criterion is:
Gc = - (1/w)(dW/dc) (2.2)
where the derivative is taken at constant length of the sample to avoid including
Figure 2.17: Relation between tear force P and fracture energy Gc
Chapter 2. Mechanical Properties of Rubber 49
further work input by the applied force. For a sheet of a linearly-elastic material (figure
2.18) the breaking stress σb and strain eb are obtained in this way as:
σb = 1. 63 (EGc/ho)1/2 (2.5)
where ho is the unstrained height of the strip, a measure of the amount of material from
which strain energy is released as the crack passes. Note that the breaking stress now
depends on the sample height ho and not on the crack length c.
On comparing Equations 2.3 and 2.5, the breaking stress is seen to be much lower in a
50 Chapter 2. Mechanical Properties of Rubber
constrained tensile test than in a regular tensile test because the initial height ho of the
specimen, usually several mm, is much greater than the depth c of an accidental edge flaw
in a simple tensile specimen, typically only about 50 µm. Thus the applied stress at which
rupture occurs is not a valid criterion for fracture. Moreover, the breaking stress at the
crack tip, which would presumably be a valid fracture criterion, is virtually impossible to
measure. On the other hand the fracture energy for a rubber compound can be measured
in several ways, including those sketched in Figures 2.17, 2.18 and 2.19, and proves to be
a simple, consistent and reproducible fracture criterion. It has therefore been widely
adopted. We now turn to consider theoretical and measured values of Gc for typical
rubber compounds.
Figure 2.19: Crack propagation in pure shear (constrained tension)
Indeed, using WLF shift factors calculated from Equation 1.13, as shown schematically
in figure 2.23, the results from figure 2.22 are accurately superposable; see figure 2.24.
This proves that the high tear strength exhibited at low temperatures is associated with
52 Chapter 2. Mechanical Properties of Rubber
reduced molecular mobility, and not with a change in the intrinsic strength of the rubber
molecule.
It was pointed out earlier that the molecular mobility of butyl rubber is anomalous - it
increases more slowly as the temperature is raised above Tg than for other elastomers. We
would therefore expect the tear strength of butyl rubber compounds to decrease more
slowly as the temperature is raised, and this turns out to be the case [8].
old strength [6]. Note that the threshold strength is inversely related to the degree of
crosslinking - it decreases as the degree of crosslinking is increased. [This raises an impor
tant practical issue. Because the threshold tear strength is reduced by crosslinking whereas
the elastic modulus is increased (equation 1.1), the appropriate level of crosslinking for a
particular rubber compound will depend on the anticipated service conditions.]
The tear strength under non-equilibrium conditions appears to be a product of two
terms, the inherent strength Go of the molecular network and a factor reflecting
dissipative processes. Indeed, when the relative value Gc/Go is plotted against tear speed
at a temperature of Tg, obtained by applying shift factors calculated from equation 1.13 to
experimental measurements of Gc at various temperatures, the same curve is obtained for
three elastomers having quite different chemical structures and different Tg’s, figure 2.24
[9]. Results are shown for polybutadiene (BR) with a Tg of -96oC, an ethylene-propylene
Figure 2.22: Fracture energy (tear strength) vs. rate
and temperature for HS-SBR vulcanizate (Tg = - 30°C)
BR (Tg= - 96oC)
EPR (Tg= - 60oC)
HS-SBR (Tg = - 25oC)
Chapter 2. Mechanical Properties of Rubber 55
2.3 Reinforcement
Internal dissipative processes obviously make a large contribution to the strength of
rubber. This raises two important questions:
1. Is it possible to create a strong material with low dissipative properties, because at
first sight these properties appear to be incompatible?
2. What other reinforcement mechanisms can be employed?
The apparent dilemma posed in the first question can be circumvented by recognizing
that high strength is only required in highly stressed regions, near a stress-raiser. If a
dissipative mechanism is activated only at high stresses, then both requirements can be
met: low dissipation under “normal” low-stress conditions that exist in the bulk of the
material, and high dissipation at high-stress sites, where needed.
One mechanism of this type is strain-induced crystallization. Certain elastomers,
notably natural rubber, have a regular molecular structure capable of being closely
packed, and they crystallize rapidly when stretched by more than about 300%. As a result,
they become transformed into tough partially-crystalline fibrous solids that break only
when crystallites in the tear path are pulled apart plastically, with considerable energy dis
sipation. At lower strains, before crystallization occurs, they are resilient, highly-elastic
solids with low energy losses.
Employing the same principle, we can postulate a desirable feature in the reinforce
ment of elastomers by incorporating particulate fillers, for example carbon black. We
require the rubber molecules to be bonded to filler particles, with bonds that do not fail at
low stresses and cause unwanted energy dissipation under normal operating conditions.
But the bonds must not be as strong as the elastomer itself, because then they would not
fail at all. They should be somewhat less strong, and break before the elastomer molecules
do, thus creating a dissipation mechanism at high-stress sites, where it is needed, without
rupture of the molecular network.
Although this would be a highly desirable mode of reinforcing rubber by fillers, at
present the bonds between rubber and particles appear to have a wide range of strengths,
so that some energy dissipation occurs even at low stresses, considerably more than in the
corresponding unfilled compound.
Rubber is usually crosslinked by sulfur, often by polysulfidic -S-S-S-S- crosslinks. The
resulting materials have greater strength than those with carbon-carbon crosslinks. We
note that sulfur crosslinks are weaker than C-C crosslinks and main-chain bonds, and
postulate that they serve a sacrificial function – by breaking first, they reduce the tension
in the elastomer molecule and permit it to survive. This function can also be interpreted
as providing a dissipation mechanism at points of high stress.
Other common reinforcing mechanisms that can be interpreted in the same way are :
Plastic yielding of inclusions
Friction at internal interfaces
Formation of internal cavities
Detachment from fibers
The conclusion that strength is directly related to energy dissipation was recognized
quite early [11]. A comparison was made of the work-to-break a rubber sample in tension
Ub with the energy lost in stretching a similar sample nearly to the breaking point Ud and
then relaxing it. The two quantities are in principle independent, but in practice an
excellent correlation was found to hold between them for a large number of rubber
56 Chapter 2. Mechanical Properties of Rubber
both quantities being measured in J/m3. As the amount of energy dissipated Ud cannot
possibly exceed the work-to-break Ub, Equation 2.6 suggests that the maximum possible
tensile breaking energy of a rubbery material is about 80 MJ/m3. However, Equation 2.6
is wholly empirical and, even though it applies successfully over an enormous range of
tensile strengths, any extrapolation of it must treated with caution.
direction of stretching, then deviation of a growing tear into that direction would be
inevitable and the material would be effectively self-reinforcing. Note that this mode of
reinforcement is not based on enhancing the strength of the material; it is attributed to
developing low strength in the strain direction, leading to ineffective tearing and blunting
of the tear tip. At least some of the reinforcing ability of particulate fillers appears to arise
from this mechanism.
Figure 2.26: Multiple tears at the tip of a cut in one edge
of a tensile test-piece: “Knotty” tearing
K = (EG)1/2 (2.7)
When energy G (< Gc ) is made available for fracture, even though large-scale tearing
does not occur, a tear tip is found to grow by a small distance dc each time the stress is
applied. The growth step dc depends strongly on G, following a simple empirical relation:
58 Chapter 2. Mechanical Properties of Rubber
where B and α are crack growth constants. Although an equation of this form is widely used
to predict growth of a crack under fatigue conditions, that is, under many applications of sub-
critical stresses, the actual behavior is more complex, see figures 2.27 and 2.28, and equation
2.8 is only a useful guide over a limited range of values of G, from somewhat higher than the
threshold strength Go to about one-tenth of the catastrophic tear strength Gc.
When the stress is relaxed to zero between each application, the value of the crack
growth constant B is found in all cases to be a molecular distance, about 0.05 nm. Thus,
under threshold stresses a crack advances by an extremely small step, less than the size of
a molecular strand, at each load application. On the other hand, the exponent α is quite
different for compounds based on different elastomers. For example, α is about 2 for nat
ural rubber compounds, 3 – 4 for butadiene-styrene (SBR) compounds, and 4 – 6 for
polybutadiene compounds, and appears to depend inversely on the dissipative properties
of materials. For highly-dissipative materials, α appears to approach a lower limit of 2,
whereas for perfectly elastic, non-dissipative materials α appears to become infinitely
large. Between these two limits, values of α for partially-dissipative materials can be
represented empirically by the relation:
α = 2/(1 – R) (2.9)
where R is the resilience and 1 – R is the fraction of input strain energy that is dissipated.
Filled rubber compounds are more dissipative than their unfilled counterparts and the
value of α is found to be correspondingly smaller. For typical filled natural rubber com
pounds α is about 1.5 instead of 2, and for typical filled SBR compounds α lies between
2 and 3 instead of between 3 and 4.
Why are values of α for natural rubber compounds, which are highly-resilient, in fact
more appropriate for highly-dissipative materials? It is thought that strain-induced
crystallization of natural rubber takes place at high stresses, especially at the tips of stress-
raising flaws, figure 2.29, and leads to marked dissipation of energy as a crack advances.
from equation 2.2, assuming for simplicity that the material is linearly-elastic. Using
Equation 2.9, the growth of a crack each time the strain is applied is:
Although approximate, equation 2.12 gives useful pointers to the effect of various
parameters on the tensile fatigue life of rubber specimens. It shows, for example, that the
fatigue life Nf decreases rapidly as the imposed strain e is increased. Taking α as 2 for
natural rubber compounds, when the strain is increased by a factor of 2, the fatigue life is
predicted to decrease by a factor of 16. For an SBR compound, the corresponding change
60 Chapter 2. Mechanical Properties of Rubber
in Nf is predicted to be much more severe, by a factor of 256. Thus, the fatigue life of SBR
and similar non-crystallizing elastomers falls sharply as the imposed strain is increased.
The dependence of fatigue life on the depth c of an initial defect or flaw is also large,
and different for different materials. For natural rubber compounds the fatigue life is seen
to be approximately proportional to 1/c, and thus is decreased by a factor of 2 if the depth
of an accidental flaw is increased by a factor of 2. On the other hand, under the same con
ditions the fatigue life of an SBR compound is decreased by a factor of 8. Thus, fatigue
failure of SBR and similar compounds is extremely sensitive to the severity of accidental
nicks or flaws.
When natural rubber compounds are subjected to repeatedly-applied strains, starting
from zero strain, some tearing takes place at the crack tip while strain-induced crystallites
are simultaneously being formed there. The crack growth steps are generally smaller than
for non-crystallizing elastomers, as discussed above. But natural rubber compounds show
a remarkable resistance to fatigue cracking if the imposed strain is not reduced to zero in
each strain cycle, figure 2.30. Apparently, the crystallites that develop at the crack tip,
even at modest overall strains, do not melt unless the strain is reduced to zero or close to
zero, and thus they persist and prevent further crack growth when the strain is increased
again. No comparable effect is expected for non-crystallizing elastomers such as SBR.
Some classic observations of the fatigue life of bonded cylinders of natural rubber sub
jected to repeated tensile or compressive strains are shown in figure 2.31 [13]. The fatigue
life Nf is seen to be a minimum when the applied strains were relaxed to zero during each
strain cycle, both for tensile and compressive strains. On the other hand, if the minimum
level of strain in each cycle was not reduced to zero, then the fatigue life was greatly
increased, by a factor of up to 100 times. Indeed, the mechanism of failure may change
altogether under non-relaxing conditions, from mechanical rupture of the elastomer
molecules, now protected by crystallization, to molecular scission by reaction with atmos
pheric ozone.
Figure 2.31: Fatigue life N for cylinders of NR cycled between a minimum strain
εmin and a maximum strain εmin + ∆ε
Ga = 2P/w (2.13)
where P is the peel force and w is the width of the bonded layer.
Unfortunately, even when Ga is lower than G, failure tends to occur within the rubber,
instead of at the interface, if the difference in fracture energy is not large enough. From
mechanical considerations, more energy is available for fracture at a plane somewhat
removed from the interface. Thus a bond may appear to be stronger than it really is. A
more stringent test is needed, in which failure is induced to occur at the interface. A test
of this kind, especially suitable for evaluating rubber-to-cord adhesion, is described later.
62 Chapter 2. Mechanical Properties of Rubber
In tire building, layers of different rubber compounds and rubber-cord laminates are
joined to construct the raw tire. These layers must adhere together during handling of the
raw tire and inserting it into the tire mold for final shaping and vulcanization. Thus, good
adhesion of unvulcanized rubber is an important requirement in tire manufacture. It is
commonly assessed using peel tests. In this case, energy is largely dissipated in viscous
processes as the unvulcanized rubber layer is pulled away. Thus, the apparent strength of
adhesion reflects viscous dissipation of energy in tensile flow. Note that for this to be the
major factor, the rubber must flow without breaking under a tensile flow stress that is
smaller than the strength of the interfacial bond itself. This principle of optimizing energy
dissipation under the constraint of a limiting stress is employed in the design of pressure-
sensitive adhesives.
For assessing the strength of adhesion of vulcanized rubber to cords, another test
arrangement, shown in figure 2.33, has been proposed. Using linear elastic fracture
mechanics, an estimate of the pull-out force P is obtained as:
for an inextensible rod or cord embedded partway in a rubber block, where r is the
radius of the rod or cord, A is the cross-sectional area of the rubber block in which it is
embedded, and E is the modulus of elasticity of the rubber. Because the pull-out force
increases with r, i.e., as the fracture path moves away from the interface, failure in this test
arrangement is induced to occur at or near the interface.
Note that equation 2.14 resembles Griffith’s result for a edge crack, Equation 2.3,
Chapter 2. Mechanical Properties of Rubber 63
because energy is again made available for fracture by stretching the lower part of the
block. (The upper part is rendered inextensible by the embedded rod.)
An interesting feature of this experimental arrangement is that the total force required
to pull out simultaneously n cords embedded in the same block increases in proportion to
n1/2. The reason for this surprising result is that the energy required to pull out n cords is
proportional to n, whereas the energy stored in a linearly-elastic device is proportional to
P2, where P is the applied force. Experiments on wire cords embedded in rubber blocks
have confirmed this result [14] which is clearly important in maximizing the strength of
cord-rubber laminates.
µ = F/N (3.1)
But for soft rubber sliding on a smooth surface, the frictional force is found to be more
or less constant, independent of the load N, or pressure P, as the applied load or pressure
increases from small initial values, probably because complete contact is achieved
between soft rubber and a smooth countersurface at quite small loads. Beyond this point,
further increases in normal load cannot increase the degree of interaction between the two
materials and so the frictional force no longer increases. Consequently the “coefficient” of
friction decreases continuously as the pressure increases, as shown schematically in figure
64 Chapter 2. Mechanical Properties of Rubber
2.34.
However, for harder rubber compounds sliding against a rough surface, the frictional
force does increase, approximately in proportion to the applied load or pressure, and thus
a “coefficient of friction” can be defined in this case that is largely independent of
pressure. Apparently, contact is incomplete, figure 2.35, for harder rubber compounds
such as those used in tire treads. An increase in pressure creates a larger true area of con
tact and hence a larger frictional force.
where G* [ = (G´2 + G´´2 )1/2] is the complex dynamic shear modulus and R is the cylinder
radius. (The term in parentheses is a measure of the length of the contact zone.) This sim
ple theory successfully accounts for rolling friction in terms of energy dissipated as
rubber is compressed and released in the contact patch. But tan δ must be measured at the
effective frequency of rolling (about 10 Hz at 30 mph) and at the service temperature
because rolling friction depends on speed (frequency) and temperature in the same way
that tan δ depends on frequency and temperature.
The characteristic length d (= vm/fm) is now only about 5 nm, a molecular distance.
From these model experiments Grosch was able to explain the more complex process
of sliding on a dry rough surface. In this case, both dissipative processes appear, at quite
different speeds corresponding to the different length scales of track asperities and
molecular strands, figure 2.40. And in the case of rubber sliding on a lubricated smooth
surface, sliding is frictionless because there is no mechanism for energy dissipation: both
molecular contact between rubber and countersurface and deformations by asperities are
now absent.
tered in service can be selected so that tires exhibit the desirable and yet apparently
contradictory features of low rolling resistance and high sliding friction.
δ = AWf (4.1)
where δ is the loss (m) in height of the block, Wf is the work expended in sliding per unit
of apparent contact area (J/m2), and A is a coefficient, termed “abradability”, (m3/J).
Typical values of A ranged from 0.1 - 0.5 mm3/J. They depended markedly on the test
speed and temperature, in accordance with the WLF rate-temperature dependence,
Equation 1.13. Minimum abrasion occurred at high speeds, near the glass transition, when
the tear strength is a maximum, figure 2.42. Indeed, the abradability was found to follow
an inverse proportionality to the strength of the rubber compound, represented by the
work Ub (J/m3) of tensile rupture,
A ≈ C/Ub (4.2)
where C is a fitting constant. Schallamach noted that the correlation shown in Equation
4.2 required that the tensile breaking energy Ub be measured at high rates of strain, about
104 strain units per sec.
The empirically-determined constant C, about 1 x 10-3, represents the ratio of the
volume of rubber abraded away to that volume which would be brought to rupture if an
amount of strain energy equal to the work expended in sliding friction were applied to
stretch the rubber uniformly up to the breaking point. It is clear that only a small fraction of
the frictional work causes rupture - the major part, over 99%, must be dissipated in other
ways.
Figure 2.41: Measurement of abrasion
Chapter 2. Mechanical Properties of Rubber 69
Later work showed that equation 4.1 was inadequate - the rate of abrasion is not
strictly proportional to the work expended in sliding but increases at a faster rate. A
modified relation was proposed [19]:
d = A′Wfα (4.3)
them to become progressively undercut. Eventually the tips fall off as large particles of
debris, 50 to 1000 µm in size.
Thus two abrasion processes occur: a small-scale intrinsic abrasion at the base of ridges
resulting in small particles, 1 – 5 µm in size, and detachment of relatively large fragments
from the ridge tips. Although much fewer in number, the large particles generally account
for most of the weight loss.
Because the ridges of a Schallamach abrasion pattern are unsymmetrical, leaning
towards the abrader, they can be used to deduce the direction of sliding. For example, if
the center rib of a tire exhibits abrasion ridges lying perpendicular to the circumference,
then abrasion must have taken place as a result of fore-and-aft sliding motions. If side ribs
have abrasion ridges lying at an angle to the tire circumference, then sliding occurred
primarily in that direction. Moreover, because an abrasion pattern is unidirectional, lean
ing forwards in the direction of motion of the rubber, one can deduce whether the ribs slid
outwards or inwards against the abrading surface - the road.
Both the ridge height and ridge spacing increase with increasing severity of wear, i.e.,
for sharp abrasives, with high frictional forces, and for soft rubber. When the frictional
force is increased to reach the tear strength of the rubber, then the rate of abrasion abruptly
increases and the mode changes to a gross gouging of the surface without a characteristic
pattern being formed.
The Schallamach abrasion pattern also does not form if the direction of sliding is
changed repeatedly. Abrasion then takes place more slowly and on a finer scale – a few
µm rather than the 50 µm to 1000 µm characteristic of pattern spacings – by an intrinsic
abrasion process in which small particles of rubber, only a few µm in size are plucked out
from the surface by frictional forces. This process is closely related to tensile rupture, as
described in section 4.1.
advance by a distance ∆c, given by equation 2.8. This leads to a loss in thickness of rub
ber of ∆c.sin θ. Thus,
The angle θ may be estimated by direct inspection of the way in which abrasion
patterns move over the surface during wear. It is found to be small, 5 – 10 degrees. Turner
has accounted for these small values by considering the severe tilt of the principal tension
directions in highly-sheared blocks [24]. All other terms in equation (4.4) can be
determined from tear-growth measurements. Thus, the theory does not involve any
arbitrary fitting constants. Figure 2.45 shows a comparison by Southern and Thomas of
rates of abrasion (points) with rates of crack growth (lines). Good agreement was found
in two cases: SBR and an isomerized NR (non-crystallizing). But NR (triangles), which
has excellent fatigue resistance but poor abrasion resistance, is anomalous.
Thus, although the theory is remarkably successful in accounting for the rate of
abrasion of two unfilled elastomers, the agreement is unsatisfactory for natural rubber
which abrades much more rapidly than crack-growth measurements would predict. It is
Figure 2.45: Crack growth vs fracture energy
and abrasion rate vs frictional energy
72 Chapter 2. Mechanical Properties of Rubber
possible that, under abrasive conditions, natural rubber does not undergo strain-induced
crystallisation and therefore lacks the fatigue resistance that it usually demonstrates, and
cracks grow rapidly. It should be noted also that filler-reinforced rubber compounds
abrade more slowly than would be predicted on the basis of crack-growth measurements.
Further work is needed to clarify these points, which are of great practical importance.
For a given side force S, the slip angle θ and hence the sliding distance xmax (= θL)
will be greater for tires that have greater compliance. Thus, abrasion will be greater for
tires that are more compliant for sideways deflections.
The amount of energy expended in deforming the tire is given approximately by
SθL2/2 = S2C/L. However, only a fraction R of this energy is expended in sliding, where
R is the resilience of the tire, because the tire will not return all of the input energy. Thus,
the expected rate of wear of a cornering tire is given by
d ∝ AS2CR/L (4.5)
and is clearly affected by several other factors besides the abradability A of the tread
compound. The lateral compliance C of the tire will contain a contribution from
deformation of the carcass in addition to the tread compliance. The contact patch length
L is of course governed by structural aspects of the tire and will be insensitive to
properties of the tread compound. And the most significant factor in tire wear is seen to
be the side force S, largely set up the driver. As far as the tread compound is concerned,
three physical properties are important: intrinsic abradability A, sideways compliance C,
and its contribution to deformational energy losses represented by the effective tire
“resilience” R. These properties are not necessarily related. For example, a harder, less
compliant compound may well be less resilient. Thus a successful tread formulation
requires a judicious balancing of sometimes conflicting requirements.
Although, for simplicity, the foregoing discussion dealt with the case of a cornering
tire, the same factors govern wear under driving and breaking forces. The compressive or
stretching deformation of the tire increases through the contact patch until slip takes place
when the local circumferential force at the interface exceeds the maximum that friction
can support. Strain energy built up in the tire by the circumferential deformation is then
released to provide the work of sliding against friction. Note, however, that in driving and
breaking the relevant tire compliance is in the circumferential direction. It may be consid
erably different from the lateral tire compliance (C in the previous paragraph), depending
on the geometry of cords and belts.
74 Chapter 2. Mechanical Properties of Rubber
5. Aging of rubber
Rubber undergoes profound changes on storage that are accelerated at higher tempera
tures. Deleterious changes occur in tire properties after storage at ambient temperatures
for five years or after use on cars for similar periods [29]. They are caused by a variety of
chemical reactions:
(i) Ozone attack. Although the concentration of ozone in the atmosphere is quite small,
typically only a few parts per 100 million, ozone reacts rapidly and efficiently with the
unsaturated elastomers commonly used in tire compounds, leading to molecular scission.
However ozone cannot penetrate deeply into the material – reaction takes place at the
exposed surface and produces a relatively-innocuous thin degraded surface layer, about 20
µm thick, which protects the interior. However, if a small tensile strain of the order of 10%
is present in the rubber surface, then the scission reaction with ozone causes characteris
tic sharp cracks to form in the surface and grow inwards, continuously exposing new
material to further attack. The cracks grow surprisingly rapidly. They become about 1mm
deep after only two weeks exposure of an unprotected rubber compound to normal
outdoor air with an ozone concentration of about 5 parts per hundred million. Thus ozone
cracking is potentially a serious problem in tire sidewalls where tensile stresses are
commonly present both in storage and in use. Special additives, termed antiozonants,
inhibit ozone cracking when added to the rubber compound in sufficiently large amounts,
about 3 %, probably by competing with rubber molecules for reaction with ozone. Butyl
rubber is much less susceptible to attack by ozone than other common elastomers, at least
at ambient temperatures, because it contains only a relatively small fraction of reactive
C=C bonds in the molecular backbone.
(ii) Oxidation. Another cause of aging is reaction with atmospheric oxygen. Oxidation
is slower than ozonolysis and oxygen therefore penetrates for some distance into the
material before reacting. Thus oxidation does not cause cracking directly although the
oxidized material is often brittle and cracks on flexing. Depending on the relative rates of
diffusion and reaction, the affected depth can range from several mm at ambient temper
atures, when the process takes years to reach a significant stage, or a fraction of 1 mm at
elevated temperatures when oxidation is rapid, taking only a few hours. Typical hydrocarbon
elastomers undergo an autocatalytic reaction that results in addition of oxygen groups to
the molecule and formation of new crosslinks by interaction with neighboring molecules.
As a result the material generally becomes harder and eventually brittle. However another,
generally minor, consequence of the complex oxidation reaction is occasional molecular
scission and hence softening. This provides a convenient way of characterizing the
sensitivity of a rubber compound to oxidation. Samples are stretched and aged in an oven
at various temperatures, usually in the range 70oC to 130oC, and the tensile stress is
monitored continuously over a period of several days. As oxidation proceeds and some
elastomer molecules break, the stress falls and gives an indication of the extent of
oxidation. The rupture reaction follows an Arrhenius dependence on temperature to a first
approximation, with an activation energy of about 25 kcal/mole. Thus an increase in
temperature of 10°C causes an increase in rate of oxidation by a factor of about 2x.
Another way of assessing the sensitivity of a rubber compound to oxidation is to
expose samples for various periods at elevated temperatures and then measure the
remaining strength and extensibility at room temperature. A typical specification for aging
resistance would require that the tensile strength does not change by more than a
specified fraction, say 20%, and the extensibility does not decrease by more than a
Chapter 2. Mechanical Properties of Rubber 75
specified fraction of the original value, say 30%, after a period of aging of 7 days at 70oC
or 22 h at 100oC.
(iii) Additional vulcanization. Vulcanization does not stop when the cured compound
is removed from the mold. Continued curing takes place subsequently but at much lower
rates, of course, depending on the temperature. As a result if tires are stored or used at
elevated temperatures the material hardens as more crosslinks are introduced, or softens
(a phenomenon termed reversion) as those crosslinks already formed gradually
decompose. These processes are a consequence of a series of complex reactions
involving elastomer molecules, existing crosslinks, residual sulfur, activators and
accelerators, and byproducts of the various intermediary steps in the crosslinking reaction.
In conventional aging measurements these processes are difficult to distinguish from the
effects of oxidation, but they can be studied separately by aging samples in an oxygen-
free environment. In thick rubber articles, material far from the surface, say over 10 mm
deep, may undergo solely anaerobic aging because oxygen reacts before it diffuses so
deeply. Passenger car tires, on the other hand, operate for long periods at moderate
temperatures, so that oxygen may diffuse extensively before reaction. Thus oxidation is
regarded as the normal mode of aging of tire components.
(iv) Weathering. This mode of aging is rather ill-defined. Insofar as new aging processes
occur, other than oxidation and ozone attack, they appear to be associated with irradiation
by UV and sunlight. Radiation causes free-radical reactions that can initiate or catalyze
oxidation and ozonolysis, as well as being itself a direct cause of crosslinking and/or
molecular scission.
6. Concluding remarks
Rubber compounds used in tires today are astonishingly effective and durable, as a result
of a long period of semi-empirical research and development. Even better materials could
presumably be developed with a better understanding of the mechanics and chemistry of
strength, fatigue, friction and wear. An outline has been given of our present understand
ing of basic rubber science but there are clearly substantial and serious deficiencies,
notably in the areas of filler reinforcement and chemical changes on aging, that call for
further study.
Bibliography
“The Vanderbilt Rubber Handbook”, ed. by R. O. Babbit, R. T. Vanderbilt Company,
Norwalk, 1978.
“Science and Technology of Rubber”, 2nd ed., J. E. Mark, B. Erman and F. R. Eirich
“Engineering with Rubber: How to Design Rubber Components”, 2nd. ed., A. N. Gent
References
1. L. Mullins, Rubber Chem. Technol. 42, 339 (1969).
2. A. R. Payne, J. Appl. Polymer Sci. 7, 873 (1963).
3. J. D. Ferry, “Viscoelastic Properties of Polymers”, 3rd. ed., John Wiley & Sons, New
York (1980).
4. A. A. Griffith, Phil. Trans. Roy. Soc. 221, 163 (1920).
5. J. P. Berry, Chap. 2 in “Fracture: An Advanced Treatise; Vol.7. Fracture of Non-
Metals and Composites”, ed. By H. Liebowitz, Academic Press, New York, 1972.
6. G. J. Lake and A. G. Thomas, Proc. Roy. Soc. (London) A300, 108 (1967).
7. T. L. Smith, J. Polymer Sci. 32, 99 (1958).
8. W.-J. Hung, Ph.D. Dissertation, Polymer Science, The University of Akron (2001).
9. A. N. Gent and S. M. Lai, J. Polymer Sci: Part B: Polymer Phys. 32, 1543 (1994).
10. W. G. Knauss, in “Deformation and Fracture of High Polymers”, ed. by H. H.
Kausch, J. A. Hassell and R. I. Jaffee, Plenum Press, New York, 1974, pp. 501-540.
11. K. A. Grosch, J. A. C. Harwood and A. R. Payne, Nature 212, 497 (1966).
12. A. N. Gent, M. Razzaghi-Kashani and G. R. Hamed, Rubber Chem. Technol. 76, 122
(2003).
13. S. M. Cadwell, R. A. Merrill, C. M. Sloman and F. L. Yost, Ind. Eng. Chem., Anal.
Ed. 12, 19 (1940).
14. A. N. Gent, G. S. Fielding-Russell, D. I. Livingston and D. W. Nicholson, J.
Materials Sci. 16, 949 (1981).
15. J. A. Greenwood, H. Minshall and D. Tabor, Proc. Roy. Soc. (London) A259, 480
(1961).
16. K. A. Grosch, Proc. Roy. Soc. Lond. A274, 21 (1963).
17. R. A. Schapery, Tire Sci. Technol. 6, 3 and 98 (1978).
18. A. Schallamach, J. Polymer Sci. 9, 385 (1952); K. A. Grosch and A. Schallamach,
Trans. Inst. Rubber Ind. 41, 80 (1965).
19. A. Schallamach, Rubber Chem. Technol. 43, 701 (1966).
20. D. H. Champ, E. Southern and A. G. Thomas, Org. Coat. Plast. Chem. 34(1), 237,
(April, 1974).
21. E. Southern and A. G. Thomas, Plast. Rubber: Mater. Appl. 3, 133 (1978).
22. A. N. Gent and C. Nah, Rubber Chem. Technol. 69, 819 (1996).
23. A. Schallamach, Rubber Chem. Technol. 41, 209 (1968).
24. D. M.Turner, unpublished results.
25. A. N. Gent and C. T. R. Pulford, J. Appl. Polymer Sci. 28, 943 (1981).
26. A. N. Gent and C. T. R. Pulford, J. Materials Sci. 14, 1301 (1979).
27. G. V. Vinogradov, V. A. Mustafaev and Y. Y. Podolsky, Wear 8, 358 (1965).
28. A. Schallamach and D. M. Turner, Wear 3, 1 (1960).
29. T. Kataoka, P. B. Zettterlund and B. Yamada, Rubber Chem. Technol. 76, 507
(2003).
Chapter 2. Mechanical Properties of Rubber 77
Test Questions
1. If a rubbery material has a small-strain elastic modulus G in shear of 1 MPa, what is small-
strain elastic modulus E in tension?
What is the value of the coefficient C1 in the neo-Hookean strain energy function, W= C1
(λ12 + λ22 + λ32 - 3)?
If a thin block of the same material with an area of 100 x 100 mm was subjected to a shear
force of 10 kN, what would be the angle of heel (the shear angle)?
2. A constrained tension test is one in which the sample is not free to contract laterally when
it is stretched. If a sample of the above material is stretched in constrained tension by 200%,
what are the values of the stretch ratios λ1, λ2, λ3?
5. Rebound resilience is used as a rough measure of energy dissipation and rolling resistance.
It is measured at a low temperature, say 0oC. The contact time is quite short, about 2 msec,
equivalent to a test frequency of about 250 Hz. Under these test conditions, are the results rel
evant to the rolling resistance of a tire in service, at say 50 mph and at a temperature of 60oC?
If the tread material has an effective Tg of – 50oC, what test temperature would make the
measurement of rebound resilience more relevant to the service conditions?
6. In a test for adhesion of cured sheets of rubber, two wide strips are bonded together over a
narrow section, 20 mm long and 6 mm wide. The peel force P is measured as the strips are
peeled (torn) apart. If P is 48 N, what is the strength G of adhesion?
If the speed of tearing was increased from 1 mm/sec to 100 mm/sec, what increase would you
Chapter 3
Tire Cord and Cord-to-Rubber Bonding
by E. T. McDonel
1. Textile cord
A number of references give detailed and comprehensive information on the chemistry,
production, and properties of today’s tire textiles (1-5). The chapter by Takeyama, Matsui,
and Hijiri in the 1981 edition of the “Mechanics of Pneumatic Tires” provides an in-depth
review of tire cord technology that is still current. The other references contain information
on all aspects of the chemistry, manufacture, processing and physical properties of industrial
cords for tires.
The present discussion will review current use and current trends for industrial textiles in
tires as well as the physical attributes of cords now used. Textiles developed for use in tires
are a small but exacting part of the huge textile industry. The average tire engineer, unless
specializing in this area, is not always familiar with textile vocabulary, the chemical
composition of tire textiles, the manufacturing process, the rationale for selecting certain
textiles for certain tires, possible deficiencies of textiles in some applications, and the very
important need for excellent cord-to- rubber adhesion in all applications. This chapter will
provide an overview of these topics.
would contain about 1 kilogram (2.2 pounds) of steel cord and 0.5 kilograms (1.1 pounds) of
polyester cord.
Table 3.1: Textile cord makeup of US tires - 2001
Belt Material Steel Aramid Nylon
Passenger/Light Truck
OE 97 2 1
Replacement 99 1 0
shows this construction, but on a larger scale. The filaments are twisted “Z” into yarns and
the yarns are back-twisted “S” to form a cord. The size of a tire filament, yarn, or cord is
measured by its weight per unit length - linear density or “denier” (denier is the weight in
grams of 9000 meters) or “decitex” (weight in grams of 10,000 meters). Textile cords are
identified by their yarn denier and their construction. Thus a 940/2 8x8 nylon cord is formed
from 2 - 940 decitex yarns twisted separately at 8 turns per inch and then back-twisted togeth
er at 8 turns per inch to form the cord. A 1650/3 10x10 rayon cord would comprise 3-1650
denier yarns twisted at 10 turns per inch separately and back-twisted together at 10 tpi. For a
given material, use of higher denier yarns or more yarns per cord result in a higher breaking
Table 3.4
Functions of tire cords
- Maintain durability against bruise and impact
- Support inertial load and contain inflating gas
- Provide tire rigidity for acceleration, cornering, braking
- Provide dimensional stability for uniformity, ride, handling.
Cord Requirements
- Large length to diameter ratio , eg, long filaments
- High axial orientation for axial stiffness and strength
- Good lateral flexibility (low bending stiffness)
- Twist to allow filaments to exert axial strength in concert with other filaments in the bundle
- Twist and tire design to prevent cord from operating in compression.
Textiles for the carcass and for the radial belt have different requirements. Kovac (4)
and Pomies (9) have succinctly summarized the ideal properties for cords in each of these
applications:
As with all tire components, choice of a textile cord for a given tire application may
require compromises involving cost, intended market segment and end-use application. The
tire engineer has a number of choices for a tire textile:
•Chemical composition of textile
•Cost per unit length and weight (cost in tire)
•Denier – filament size and strength
•Cord construction – number of yarn plies
•Cord twist
•Number of cords per unit length in ply
•Number of plies in the tire
These choices will naturally be predicated on the tire specifications for the particular
application and market, usually balancing cost against required performance.
Five materials currently make up the major tire textile usage – rayon, nylon, polyester,
aramid, and steel. Table 3.7 lists the physical properties of these materials. The high modulus
of steel and aramid find their major use in radial belts and in single-ply carcasses for large
radial tires. Rayon is used in both carcass and belt of passenger radial tires but lacks strength
for durable heavy-duty tires. Modern polyester cord is an excellent carcass textile for use with
steel belts in passenger and light truck tires and is becoming dominant worldwide. However,
it lacks the toughness and heat resistance required for large tires where nylon is the textile of
choice in large bias truck, earthmover and aircraft tires but nylon and polyester do not have
the high stiffness necessary for good performance in radial belts. Yarn and textile producers
are making continuing improvements in their products, therefore these data are for a general
comparison only. In particular, modifications in the last twenty years have resulted in
significant improvements in the dimensional stability of polyester and the tensile strength of
steel. Also, aramid modifications have improved its compression fatigue properties.
Physical properties of commercially-available tire cords are usually more meaningful to
tire engineers. The following table summarizes typical physical properties of some common
ly used cord sizes of organic textile fabrics (10). Data are for untreated yarn bundles lightly
twisted at 0.2 to 0.3 turns per inch.
Table 3.9 qualitatively ranks tire cord reinforcements for important tire performance proper
ties. It should be understood that the textile industry is continually upgrading their specific
products so that these rankings may change.
Again note that nylon and polyester do not have the requisite bending stiffness to make
useful radial tire belts. As discussed below, glass fiber and polyvinyl alcohol are potential belt
materials.
truck tires. Polyester must be used with carefully designed rubber adhesion systems and
carcass rubber compounds to prevent cord deterioration in use. Polyester cord is not
recommended for use in high-load/high-speed/ high-temperature applications, as in truck,
aircraft and racing tires, because of rapid loss in properties at tire temperatures above about
120C.
Rayon (12) - Tire cord strength has been improved 300% since its introduction by
improved coagulation and heat treatment. The low- shrink, high-modulus, good-adhesion
properties of rayon make it an excellent choice for use in passenger tires. However, rayon has
lost market share to polyester due to higher cost and environmental concerns with production
facilities. Rayon had historically been used in truck tires but has been displaced by nylon with
higher strength and impact resistance. Rayon is used for racing tires and has gained renewed
interest in the development of an extended-mobility self-supporting passenger tire.
Nylon- Nylon is a generic term for aliphatic polyamides. Two varieties are used in tires
cords Nylon 6 (polycaprolactam) and Nylon 66 (product of adipic acid/hexamethylene
diamine condensation). Both materials give similar properties with Nylon 6 being somewhat
less expensive, but more sensitive to moisture and subject to loss in tensile strength if mois
ture is present at tire curing temperatures. Nylon tire cord strength has been improved 25-50%
from early versions by processing modifications. Its low modulus and low glass transition
temperature make it unacceptable as a belt material or for applications where aesthetics, ride,
and handling are important, i.e., in passenger tires. Nylon is preferred in uses requiring
carcass toughness, bruise and impact resistance, high strength, and low heat generation, e.g.,
in tires for medium and heavy-duty trucks, off-road equipment, and aircraft. In these applica
tions nylon can be used in the bias-ply tire carcass or in radial tire carcasses with steel or
aramid belts.
Aramid (13 ) - Aramid is a wholly aromatic polyamide. The most common commercial
material is poly(p-phenylene terephthalamide), eg, Kevlar™ or Twaron™. Aramid cords have
very high strength, high modulus, and low elongation. The relatively high cost has slowed
adoption as a general radial belt material where steel cord is performing well. It is
particularly suited where weight is important, such as in the belts of radial aircraft tires or in
overlay plies for premium high-speed tires. As with steel cords, aramid can be used as
multiple plies in flat belts. However, in carcass applications aramid must be used as a single
ply. In a multiply carcass construction, aramid’s low elongation will prevent the outer ply
from adjusting to the average curvature, thus placing the inner plies into compression. This
reduces the contribution of the inner plies to the total strength, but, more seriously, early
failures of the inner ply are encountered due to the poor dynamic fatigue resistance of aramid
in compression. Work on aramid copoloymers to improve elongation and fatigue resistance
has been reported.(14).
gravity is only 2.54 compared to 7.85 for steel. The initial modulus is 2150 cN/tex compared
to 1500 for steel. Rubber adhesion is excellent with no problems with rusting due to water in
the belt. Each filament of fiber glass is coated with a latex dip before the filaments are
twisted into a yarn. It has been established that this latex must be formulated from a low glass
transition polymer to prevent the premature glass breakage seen in the early glass fiber
development.
Polyvinyl alcohol – PVA fibers have properties similar to both rayon and advanced
polyester, but with higher tenacity than rayon and lower shrinkage than polyester. Reported
properties (15) are – tenacity 14 g/den, elongation 8.7%, Tg = –90C, Tm = - 265C, Modulus
= 180 g/den, specific gravity = 1.3. High molecular weight, high crystallinity fibers are
stable in water at 115C. This textile has been used successfully in the carcass and belts of
radial passenger tires. A major drawback has been the lack of suppliers of multiple cord
material.
Polyethylene Naphthalate (PEN) - PEN is similar to the standard polyethylene
terephthalate (PET) polyester, being a copolymer of ethylene glycol and naphthalic acid. This
new textile has been developed by Allied-Signal (presently Honeywell High Performance
Fibers) and is being evaluated for tires. Its properties have been reported by Rim (16 ). It is
claimed to surpass DSP-PET for use in the carcass of passenger car tires, having lower
shrinkage, higher modulus, and higher Tg (120C vs. 80C). It also has potential as a
restrictive overlay belt for light truck and high-speed passenger tires, replacing nylon
overlays. A disadvantage is the high price, about 2.5 times that of polyester. Table 3.10
compares treated cord properties of PEN with other tire textiles.
Melt spinning
In the melt spinning process molten polymer is filtered and pumped through a spinnerette
containing a large number of very fine holes. A positive displacement pump is used to give
an extremely accurate and constant flow through the spinnerette. The extruded semi- molten
polymer is stretched to about 25 times its original length while solidifying in a cool air stream.
The solidified bundle is then treated with a spin finish to lubricate the filaments and
cold-drawn over a series of take-up rolls (“godets”). The drawing process elongates the
filaments by several hundred percent while the polymer is still above its glass transition
temperature. This procedure increases the strength and modulus, and reduces the breaking
elongation, by increasing the polymer crystallinity and molecular orientation. The final
Chapter 3. Tire Cord and Cord-to-Rubber Bonding 89
Solution spinning
Rayon (regenerated cellulose) and aramids have no defined melting temperature and must be
dissolved for extrusion as continuous filaments. A concentrated solution or slurry is pumped
through the spinnerette into a coagulating bath of non-solvent (wet spinning) or air-dried to
evaporate the solvent (dry spinning). Filaments are air-dried under tension and not
extensively drawn since crystallinity and orientation are already highly developed. Spin
finish is applied, as with melt-spun textiles, and the filament bundles are “beamed” for
downstream processing. Aramids emerge with their high-strength properties intact. Rayon
tenacities have been increased over three-fold by refinements in coagulation, modified
finishing procedures, and heat treatments to alter crystal size.
Cord assembly
In a conventional cord production, producer yarns (lightly twisted bundles of 0.2 to 2 turns
per inch) are removed from beams and ply twisted to a specified level, 6 to 12 turns per inch
(tpi), usually in the “Z” direction. Cords are formed by cable twisting when two or more yarn
plies are back twisted in the “S” direction to form a greige (untreated) cord. Tire cords are
usually balanced with equal twist levels in yarns and cords. Figure 3.2 shows a cord construc
tion with “Z” and “S” twists.
Tire cords can be constructed in a wide range of sizes and strengths. Yarn producers
generally offer standard denier sizes, for example, nylon might be offered as 840, 1260, and
1680 denier yarns, going from lightest to strongest. Cord constructions would be identified
as, for example, 840/2 or 1680/3 with the heaviest cords having the highest strength.
Twist effects
Twist levels are important for tire cord performance. Higher twists allow a cord to behave like
a spring which will not open up under compression, while lower twists allow a cord to behave
as a rod, maximizing the strength. Table 3.11 lists changes in general performance with
increasing twist level. As twist increases the tenacity decreases, fatigue in compression
improves (the main reason for higher twists), the cord cost per tire increases (because cords
become shorter as they are twisted), and shrinkage during processing and cure increases.
Tenacity and fatigue resistance are sometimes reduced with increasing twist. In a tire the
construction should be such that no cord will open up, chafe, or fret in a compression mode.
Cord fatigue in compression is a critical factor in tire sidewalls where bending stresses and
strains are high. In bias tires sidewall compressive stresses are more likely to occur and
typical twists are 12x12 tpi (turns per inch), while in radial tires carcass cord twists are
typically 6x6 tpi.
90 Chapter 3. Tire Cord and Cord-to-Rubber Bonding
Figure 3.2: An example of cord construction using “S” and “Z” twists
Twist multiplier
The amount of twist relates to both tenacity and compression fatigue resistance. To obtain
equivalent fatigue performance in a product when changing cord size, cords must be twisted
to the same helix angle using a “twist multiplier” relationship:
For the same material:
Cord A tpi x √Cord A denier = Cord B tpi x √Cord B denier.
[For example: Cord A of 2000/2 construction at 8 tpi would be equal to Cord B of 1000/2
at 11.3 tpi.]
For different materials specific gravity must also be considered:
Chapter 3. Tire Cord and Cord-to-Rubber Bonding 91
Weaving
After twisting yarns into cords, 1000 to 1500 cords are woven into a coherent sheet using a
very light “pick” fabric as the weft at a very low fill count of one to two picks per inch. Rolls
of this fabric (which is about 1.5 to 1.75 meters wide - the practical width of rubber-cord
calenders) are transferred for further operations. The function of the pick is to maintain a
uniform warp cord spacing during the downstream operations, such as, shipping, adhesive
dipping and heat treating, calendering, tire building and lifting. Uniform cord distribution in
the finished tire is essential for tire uniformity and performance. In bias-ply tires the pick
fabric is usually a weak cotton yarn which breaks readily during tire shaping. In radial tires it
consists of a highly extensible filament (undrawn nylon or polyester) in a cotton sheath. The
core ensures uniform cord distribution as the tire is shaped and the sheath holds the cord
spacing during adhesive treatment and calendering but breaks readily during tire lifting and
shaping.
Treatment: Greige Heat set Relax In mold Out of mold Post cure inflation
Temperature C 25 225 205 180 180 180 to 60
Crystallinity (%) 60 63 70 70 70 70
Orientation High Very high Moderate Low Low High
Postcure inflation
On release from the tire curing press, viscoelastic cords in the hot tire are almost completely
free to shrink. Postcure inflation (PCI) is therefore employed to stabilize tire size and
uniformity. All-steel or all-rayon tires do not require a postcure treatment. Because the
postcure inflating equipment is expensive to install and maintain, some companies have
minimized or eliminated PCI for their radial tires by predictive mold sizing, control of cord
properties, and controlled cooling. The use of PCI entails the following:
Passenger and light truck tires are automatically ejected from the press after the usual 12
to 24 minute cure time, depending on size, and then immediately loaded onto a postcure
inflator which re-inflates the tire to 200 to 400 kPa (30 - 60 psi). This loads and stretches the
hot cords. Post inflating controls the size, shape, uniformity, and growth of the finished tire.
However, the results depend on the time, temperature and load applied to the cords during the
Chapter 3. Tire Cord and Cord-to-Rubber Bonding 93
inflation process. Moreover, the cords should be cooled evenly to below their glass transition
temperature before release from the inflator. Lim (22) has reported on studies that simulate
PCI and non-PCI conditions by measuring EASL (elongation at specified load - the inverse
of modulus). For rayon the modulus is constant for cords heated to 177C under 0.06N/tex and
cooled to RT with or without tension. No modulus change took place. Under the same
conditions nylon showed a 20% increase in modulus and PET a 30% increase. Equilibrium
times for the cords were 30 minutes. A practical cooling time for factory tires is usually about
two cure cycles. Also, it should be noted that uneven cooling, e.g., from one side to the other,
can result in tire distortion, so that tires may be rotated during the PCI treatment.
Skolnik (23) has reported a “coefficient of retraction” for loaded cords in a simulated
postcure inflation study. Cords were loaded to 0.9 g/den., heated to 165C, and cooled to
various temperatures where the load was released. The coefficient of retraction (CR) is the
percent length change per degree C.
1680/2 Nylon 6,6 CR above Tg Tg (C) CR below Tg
1680/2 Polyester 0.050 50 0.022
(standard) 0.050 85 0.011
Cord-rubber adhesion
The adhesion of rayon, nylon, polyester, and aramids has been reviewed extensively.
Takeyama and Matsui (24) reviewed adhesives for rayon, nylon, and polyester. Solomon (25)
updated this work in 1985 to include aramid adhesion and the effects of environmental
exposure on degradation of adhesion. Chawla (17) summarized both cord processing and
adhesive dip treatments. Dipping and baking of the adhesive is intimately tied in with cord
stretching and relaxation procedures.
There are many variations in cord dipping procedures, e.g., one-step vs two-step dipping,
dipping in the tire-plant vs in the cord-plant, surface activation of the cord, etc. We consider
here only the goal of using adhesives, general operating procedures, and potential problem
areas.
The prime goal of the cord adhesive is to avoid separation at the cord-adhesive interface,
at the rubber-adhesive interface, or within the adhesive itself. This objective is achieved by
using proper dipping procedures. Tire carcass failures that were initially attributed to failure
at the adhesive interface were often shown on microscopic examination to be due either to
fatigue failure of rubber close to the cord, caused by high stresses resulting from improper
construction or irregular cord spacing, or to cord fatigue from excessive compressive stresses.
Basic requirements for a good cord adhesive are listed in table 3.13.
Table 3.13: Requirements for cord-to-rubber adhesives
Good bonding to both cord and rubber
The resorcinol-formaldehyde-rubber latex system (RFL) developed in the 1940s for use with
nylon and rayon is still used throughout the tire industry. A synthetic 2-vinyl pyridine-butadi
ene-styrene copolymer latex, developed for nylon, has replaced the natural rubber latex
originally used for rayon, in all modern dips. Resorcinol and formadehyde react in the dip to
give a strong polar polymer with good adhesion the polar tire cord, while the rubber
component of the latex provides good bonding to the rubber. Specific recipes are given in the
referenced literature for all textile cords and for both single and double dip treatments.
RFL is used for rayon and nylon exclusively and as the outer dip for polyester and aramid
cords. Typically, rescorcinol and formaldehyde are mixed and “matured” for up to 24 hours.
The latex is blended and the cord is dipped before tensioning. Dip formulations contain 2
5% total solids and dip pickup is controlled to about 6 - 8%. The cord is then tensioned and
baked. Complete total wetting of the cord is necessary to prevent spotty adhesion. Good dip
penetration is important for good adhesion and cord compaction. Dip penetration of 2 - 3
filament layers is optimal. Figure 3.3 shows a cross section of a 1680/3 rayon cord with high
penetration of the adhesive into the filaments, with resulting good adhesion and cord
compaction.
Polyester and aramid polymers are much less reactive to standard RFL and must be
pretreated to obtain good adhesion. A common practice is to employ a multistage dipping
process. The cord is first dipped in an aqueous solution of a reactive chemical, such as an
epoxide, e.g., the diglycidyl ether of glycerol or a blocked isocyanate, e.g., phenol-blocked
polyisocyanate [“Hylene MP”], along with a small amount of wetting agent to give uniform
dip pickup. After tensioning and baking the cord is again dipped in a standard RFL for final
baking and relaxation. Processing times through each of the steps is generally 30-60 seconds.
A general review of cord treatment temperatures is given in table 3.14.
The dip formulation, amount of dip pickup and the curing conditions can all affect adhe
sion and must be optimized. Strict quality control must be implemented once optimum con
ditions are established. Problems that must be avoided are: inadequate wetting of the cord,
inadequate dip pickup, excessive dip pickup (which can result in flaking off of the adhesive),
or overbaking during heat treatment. Any of these conditions can reduce adhesion. The fin
ished cord must be protected from nitrous oxides (if gas or oil heating ovens are used) and
from exposure to sunlight, humidity, or ozone if the cords are stored or shipped before being
calendered with rubber. The treated cords are generally protected by storing them in
polypropylene cloth liners and sealing them in black polyethylene film.
Chapter 3. Tire Cord and Cord-to-Rubber Bonding 95
Table 3.14: Drying, baking and heat setting temperatures for textile cords (°C)
Steel cord components: (1) filaments; (2) strands; (3) cord and (4) spiral wrap
differences. Steel filaments are much larger in diameter and the specific gravity is much
higher, so that “tenacity” is not particularly meaningful. Individual filament strength is
therefore reported in megaPascals (Mpa). Steel cords are described based on the way they are
constructed rather than in terms of denier and number of strands, as is the case with textile
cords. The generic steel cord, as depicted in figure 3.4, consists of filaments, strands, cord,
and wrap, as defined below (26):
Filament – the basic element of a steel cord is a single fine metallic wire, typically 0.15
to 0.38 mm in diameter.
Strand – two or more filaments are combined together.
Cord – a strand when it is used as the final product or, more usually, the result of cabling
strands or strands and filaments.
Wrap (Transfil) – a single filament usually 0.15 mm in diameter wrapped around a cord
package to maintain compactness.
The significant cord properties, in addition to the number and diameter of the filaments,
are listed in Table 3.15. The breaking load is the key property for the tire engineer.
to about 1-3 mm. The wire is then heat-treated (“austentized/patented”) to maximize the ten
sile strength by relieving internal strains and modifying the iron carbide structure. It is then
electroplated, first with copper and then with zinc, and heat treated to cause metallic interdif
fusion yielding a brass coating containing 63/70 Cu and 37/30 Zn. Brass plating facilitates the
multiple drawings needed to achieve the final diameters of 0.15-0.40 mm and is a critical
factor in cord – rubber adhesion. The brass plating thickness is ultimately about 0.1 to 0.7 µm.
Drawing dies must have no burrs that would scratch the brass layer, or corrosion problems
might later arise from galvanic action between the iron and copper when in contact with
moisture or rubber chemicals.
the structure of the cord, the length and direction of lay, and the product type. In defining the
structure the basic rule is that the description follows the manufacturing sequence, starting
with the innermost strand. A full description of the cord is given by the formula:
Strand 1{(NxF)xD} + Strand 2{(NxF)xD} + Strand 3{(NxF)xD} + ………
where N = number of strands
F = number of filaments
D = diameter of filaments (mm)
A simplified form is when N or F =1, then the 1 is not stated, and if D is the same for
several strands it is stated only at the end of the sequence. The wrap is always stated
separately. Examples of constructions of some commonly used cords are illustrated below
and in Figure 3.5.
3x0.20 + 6x0.35
2 + 2x0.28
7x7x0.22
example:
3 + 9 + 15x0.22 + 0.15 5/10/16/3.5 SSZS
5S: lay length and direction of strand 3x0.22
10S: lay length and direction of strand 9x0.22
16Z: lay length and direction of strand 15z0.22
3.5S lay length and direction of wrap.
Product type
Several types of cord products are available based on variations in the manufacturing and
twisting procedures.
Regular cord – standard cord production in which the lay direction in the strands is
opposite to the lay direction in closing the cord. This product is easy to produce, cost
effective, and processes well in the tire factory.
Lang’s lay cord (LL) – cord in which the lay direction of the strands is the same as the lay
direction in closing the cord. High elongation cord (HE) is a Lang’s lay cord in which the
strands are loosely associated and moveable with respect to each other. This allows the cord
to be stretched substantially and gives useful cut protection when used in the top belt of
radial truck tires and impact resistance in the rock penetration zone of earth mover and mine
tires.
Open cord (OC) - A cord in which the filaments are loosely associated and moveable
relative to each other. This permits rubber to penetrate into the cord to maximize adhesion to
the filaments and to prevent moisture wicking along the cord that could result in steel
corrosion. This cord is difficult to process with standard calendering equipment as excessive
tension during processing can close the cord resulting in void formation along the cord. Open
construction cord has been the subject of numerous patents for various production techniques
and cord designs. Bekaert offers a BETRU™ cord which is less sensitive to calender tensions
and cord wicking (28).
Compact cord (CC) – cords are produced in a single compact bundle in which the
filaments have mainly linear contact with each other. This construction is useful in
applications such as for the carcasses of heavy-duty radial tires where severe fretting fatigue
can occur at crossover points in a standard cord.
Table 3.17: Constructions and physical properties of steel radial tire cords
Construction Lay Lay Breaking Cord Linear
Length direction load diameter density
(mm) (N) (mm) (g/m)
Passenger tire belt
2+1x0.28 HT -16 -S 555 0.70 1.47
2+2x0.25 HT -14/14 -SS 605 0.65 1.55
LT carcass
3x0.20+9x0.175(CC) 10 S 855 0.75 2.49
2+7x0.20 HT 5.6/11.2 SS 915 0.76 2.26
LT belt
2+2x0.35 HT 16/16 -SS 1060 0.84 3.03
MT carcass
3+9x0.22+0.15 6.3/12.5/3.5 SSZ 1290 1.17 3.85
0.20+18x0.175 (CC) 12.5 Z 1300 0.90 3.71
MT belt
3x0.20+6x0.35 10/18 SZ 1660 1.13 5.34
HDT carcass
3+9+15x0.175 5/10/18 SSZ 1770 1.07 5.20
0.25+18x0.22 (CC) 16 Z 2050 1.13 5.85
HDT belt
3+9+15x0.22+0.15 6.3/12.5/18/3.5 SSZS 2750 1.62 8.50
(protector belt – 6.5% elongation)
3+7x0.22 (HE) 4.5/8 SS 1820 1.52 6.95
could cause a capillary channel to form down the cord where moisture may propagate and
cause corrosion. Such cords have shortcomings similar to those with standard twist construc
tions, for example: 1x4x0.25.
entering the tire through road cuts and penetrating to the belt cord. Poor performance in either
area can result in reduced tire service life. Thus, prevention of loss in adhesion in steel-
belted radial tires is a prime consideration for tire engineers.
The exact mechanism of brass-to-rubber adhesion has been the subject of much study and
conjecture, but lies beyond the scope of this chapter. Van Ooij (31,32) has reviewed the
subject in detail. Basically, it is conjectured that a bond is formed between the polar metals
and the non-polar rubber during vulcanization with sulfur by formation of a Cu-S- Rubber
bond, as idealized in figure 3.6. However, this bond can readily be converted to CuS with loss
of adhesion under some circumstances. There are also indications that optimum bonding
involves interfacial layers of oxides and sulfides of both copper and zinc.
Bond durability is tested by measuring bond strength after various aging times and under
various conditions: Dry heat aging, steam aging, aging in high humidity, and salt bath immer
sion. The rubber coverage of the wire after testing is regarded as equally important as the
retained bond strength of the rubber compound.
Corrosion (rust) of steel can destroy both the adhesive bond and the wire itself. Figure 3.
shows an idealized view of moisture attack on wire through galvanic action if the brass
coating is damaged or if water can wick into the cord interstices.
Cord construction, brass composition, brass plating thickness, rubber compound
composition, tire curing and storage conditions can all affect wire adhesion. Table 3.19
summarizes the best choice of these parameters.
Each tire manufacturer adopts specific belt coat compounds. Many generic versions may
be found in the literature.
102 Chapter 3. Tire Cord and Cord-to-Rubber Bonding
3. References
1. Mechanics of Pneumatic Tires, S. K Clark, ed., University of Michigan, US Department
of Transportation, National Highway Traffic Safety Administration, Washington, DC,
20590, 1891.
2. Handbook of Fiber Science and Technology, Lewin M., Sello S. B., eds, Marcel Dekker,
Inc., New York, NY, 1989.
3. Wellington Sears Handbook of Industrial Textiles, S. A. Adanur, ed, Technomic
Publishing Co., Inc., Lancaster, PA, 1995.
4. Kovac, F. J., “Tire Technology”, Goodyear Tire and Rubber Co., 1970.
5. Synthetic Fibre Materials, H. Brody, ed., Polymer Science and Technology Series,
Longman Scientific and Technical, John Wiley and Sons, New York, NY, 1994.
6. Smith, W., “Automotives – a Major Textile Market”, Textile World, September 1994.
7. Rubber and Plastics News, March 31, 2003.
8. Modern Tire Dealer, January 2002.
9. Pomies F., Burrows, J., Rubber World 217, #2, Nov. 1997, p. 23.
10. Courtesy of Accordis Industrial Fibers catalog 2003.
11. Yang, H. H., “Aromatic High Strength Fibers”, p.228, John Wiley and Sons, New York,
NY, 1989.
12. Elkink F., Steyn E., Uihlein K., ITEC 2002 paper 12C, Rubber and Plastics News, Sept.
2002.
13. Tanner, “3.1 Tires” High Technology Fibers (Part B), Handbook of Fiber Science and
Technology III, Lewin M, Preston J, eds., Marcel Dekker, Inc. New York, NY, 1989.
14. Ozawa , Matsuda “Aramid Coplymer Fibers”, ibid. , p. 22.
15. Sakuradi,I, Okaya,T, “Handbook of Fiber Chemistry, 2nd ed.”, p 296, Lewin, Pearce, eds.,
Marcel Dekker, Inc., New York, NY, 1989.
16. Rim, PB, Rubber World 213, #2 Nov. 1995, p. 23.
17. Skolnik L, “Tire Cords”, Kirk-Othmer, Encyclopedia of Chemical Technology, Vol 20,
2nd ed, p. 328, John Wiley and Sons New York, NY 1969.
18. Chalwa, SK, “Rubber Composites”, p. 203 , Synthetic Fibre Materials, Brody H., ed.,
Polymer Science and Technology Series, Longman Scientific and Technical, John Wiley
and Sons, New York, NY, 1994.
104 Chapter 3. Tire Cord and Cord-to-Rubber Bonding
19. Skolnik, L., Draves, C. Z., “Processing Rayon Tire Cord”, 2nd Pulp Conference, TAPPI,
New Orleans, 1968.
20. Skolnik, L., private communication.
21. Aitken, R. G., Griffith, R. L., Little, J. S., McLellan, J. W., Rubber World, 151(5),
p.58(1965).
22. Lim, W. W., Rubber Chem. Tech. 75 581, 2002.
23. Skolnik, L., “Tire Cords”, ibid., p. 341.
24. Takeyama, J, Matsui, J, Rubber Chem. Tech. 42, 159 (1969).
25. Solomon, T. S., Rubber Chem. Tech. 58, 561 (1985).
26. Riva, G., “Steel Cord Technology”, Educational Symposium #47, Basic Tire Technology:
Passenger and Light Truck, 157th Spring Technical Meeting, Rubber Division, ACS,
Dallas, Texas, April 2000.
27. Goodrich, J., “Steel Cord Technology”, Educational Symposium #48, Basic Tire
Technology: Medium and Heavy Duty Truck, 159th Spring Technical Meeting, Rubber
Division, ACS, Providence, RI, April 2001.
28. Arkins, O., Peterson, J. R., “ITEC 96 Select “, p. 183, Rubber and Plastic News,
September 1997.
29. Basaran, M., Rubber World 228, #6 Sept. 2003, p.28.
30. “Steel Cord Technology”, Ed, R. M. Shemenski, Wire Association International, Inc.,
1570 Boston Post Road, Guiliford , CN 06437
31. van Ooij, W. F., Rubber Chem. Tech. 57, 421 (1984).
32. van Ooij, W. F., Rubber Chem. Tech. 52, 605 (1979).
4. Review Questions
What are the main functions of cords in tires?
How does a carcass cord differ in physical properties from a belt cord?
What are the preferred cord materials for passenger tires? Why?
What are the two processes for producing textile cord filaments? How does steel filament
How does the processing of a thermoplastic textile cord differ from that of non-thermoplas
tic textiles?
What chemical materials are required as adhesives for good polyester or aramid adhesion to
What steel cord constructions are useful to maximize resistance to fretting and resistance to
corrosion?
Chapter 4
Materials
1The Goodyear Tire & Rubber Company, Corporate Tire Research, Akron, OH 44316
2 The University of Akron, Depts of Civil Engr. and Mechanical Engr., Akron, OH 44325
106 Chapter 4. Mechanics of Cord-Rubber Composite Materials
much stiffer than the stiffness in the 2-direction, transverse to the cords.
Figure 1.1: Diagram of a typical cord-rubber specimen
2
3
General orthotropic, or
anisotropic - normal stresses
cause normal and shear strains,
shear stresses cause shear and
normal strains.
To obtain the desired structural properties, different arrangements of cords and rubber
compounds are combined in single layers or lamina. Many layers are often stacked
together to get the desired properties in different directions. A structure with multiple
layers is called a laminate. It is common to combine many laminae with different cord
angles to obtain the desired strength and stiffness in various directions in the laminate. In
order to analyze and design the stiffness and strength of belt and ply layers in tires,
composite theory is often utilized. The simplest, linear, composite theory for multiple
layers bonded together is called laminate theory. The following sections will address the
mechanical analysis of composite laminae and laminates with some discussion of finite
element methods for cord-rubber composites.
Exercises
True or false
1. For most isotropic materials, a uniaxial normal stress causes extension in the direction
of the applied stress and a contraction in the perpendicular direction.
2. For orthotropic materials, like isotropic materials, normal stress in a principal material
direction results in extension in the direction of the applied stress and contraction perpendi
cular to the stress. However, due to different properties in the two principal material direc
tions, the contraction can be either more or less than the contraction of a similarly loaded
isotropic material with the same elastic modulus in the direction of the load.
3. For an anisotropic material, application of a normal stress can result in an extension in
the direction of the stress, contraction perpendicular to the applied stress, and a shearing
deformation.
4. Coupling between both loading modes (normal and shearing) and both deformation
modes (extension and distortion) is characteristic of orthotropic materials subjected to
normal stress in a non-principal material direction.
Chapter 4. Mechanics of Cord-Rubber Composite Materials 109
(2.1)
For a three dimensional stress state, such as that illustrated in figure 2.1, the generalized
Hooke’s law for a linear isotropic material is given by equation 2.2.
Figure 2.1: Three dimensional state of stress
110 Chapter 4. Mechanics of Cord-Rubber Composite Materials
(2.2)
where [S] is the compliance matrix. The “engineering” shear strains γij are used such that,
e.g., τ12 = Gγ12. The tensorial shear strains are half the engineering shear strains, and will
be used when coordinate system rotations are required.
The relationship in equation 2.2 is valid for any orientation. Material stability requires
that the material matrix [S] relating the state of stress to the state of strain be positive def
inite. This requirement is satisfied when E > 0, G > 0, and -1 < ν < 0.5.
(2.4)
where the nine material properties for linear orthotropic elasticity are the three moduli E1,
Chapter 4. Mechanics of Cord-Rubber Composite Materials 111
E2, E3; Poisson’s ratios , ν12, ν23, ν31, and the shear moduli G12, G23, and G31. The quan
tity νij is the Poisson’s ratio for a uniaxial normal stress applied in the i-direction, or,
(2.5)
21
(2.6)
Due to the symmetry of [S], the three Poisson’s ratios and three Young’s moduli are
related by
(2.7)
For analysis in the principal material directions (equation 2.4), there is no interaction
between normal stresses σ1, σ2, σ3 and shearing strains γ12, γ23, γ31, or between shearing
stresses τ12, τ23, τ31 and normal strains ε1, ε2, ε3, or between the shearing stresses and the
shearing strains on different planes.
Material stability for an orthotropic material also requires that [S] be positive-definite,
which results in the criteria:
(2.8)
When the left hand side of the last inequality is zero, the material is said to be incompress
ible, and a special set of equations for incompressible orthotropic elasticity are needed.
For the isotropic case, equation 2.8 ensures that ν < 1/2.
A special subclass of orthotropic materials is obtained if at every point of the material
there is one plane in which the mechanical properties are equal in all directions. This type
of material is termed transversely isotropic. If, for example, the 2-3 plane is the special
plane of isotropy, then the material properties in any direction in the 2-3 plane are the
same, and the 2 and 3 subscripts can be used interchangeably. The number of independent
elastic constants needed to describe the structural behavior of a transversely isotropic
material is reduced from nine for orthotropic, to five. The five constants can be listed as:
E1, E2(=E3), ν12(=ν13), ν23, G12(=G13), with
Oftentimes G23, rather than ν23, is specified. Steel and organic tire cords may be
considered transversely isotropic as shown in figure 2.2.
The most general type of linear elastic material is fully anisotropic where there are no
planes of symmetry for the material properties. To describe a fully anisotropic material,
112 Chapter 4. Mechanics of Cord-Rubber Composite Materials
Figure 2.2: Cords and wires are sometimes treated as transversely isotropic
3
Plane 2-3 is assumed plane of
material symmetry
(2.9)
To analyze a structure with a fully anisotropic elastic material, a finite element code would
require the input of all 21 independent constants. The restrictions imposed upon the
elastic constants by the stability requirements are too complex to list in terms of simple
equations. However, they can be satisfied if one ensures that the compliance matrix is
positive definite. This criterion is often checked automatically by commercially available
finite element computer codes [e.g., 2].
σ3=τ23=τ31=0
Chapter 4. Mechanics of Cord-Rubber Composite Materials 113
Figure 2.3 Portion of single belt or ply layer with plane stress assumptions
When these stresses are set to zero, it follows that the following shear strains are also
zero:
The normal strain ε3 can be calculated from σ1, σ2 and the orthotropic material proper
ties given in equation 2.4.
The orthotropic stress-strain relationship in equation 2.4 can be simplified for plane
stress and presented in a more compact form:
(2.10)
or
(2.11)
where [S] is now the plane stress compliance matrix for the lamina. The non-zero terms of
the [S] matrix retained for plane stress in equation 2.10 are S11, S12, S21, S22, and S66. For the
orthotropic plane stress problem, there are four independent elastic constants: the Young’s
moduli in the direction along the reinforcement and in the transverse direction, E1 and E2,
respectively, the in-plane shear modulus G12, and the major Poisson’s ratio ν12. The other
114 Chapter 4. Mechanics of Cord-Rubber Composite Materials
Poisson’s ratio, ν21 can be determined from the reciprocity relationship (equation 2.7). When
the lamina compliance matrix, [S], is inverted, the lamina stiffness matrix, [Q] is obtained
-1
(2.12)
Thus, the stresses in terms of the strains for the lamina in the principal material directions
are given as:
⎡σ 1 ⎤ ⎡Q11 Q12 0 ⎤ ⎡ε 1 ⎤
⎢σ ⎥ = ⎢Q Q 0 ⎥ ⎢ε ⎥ (2.14)
⎢ 2 ⎥ ⎢ 21 22 ⎥⎢ 2 ⎥
⎢⎣τ 12 ⎥⎦ ⎢⎣ 0 0 2Q66 ⎥⎦ ⎢⎣ε12 ⎥⎦
It is customarily acceptable to refer to the 3-3 entry in equations 2.13 and 2.14 as Q66. The
notation finds its origin in the general expression of the stress-strain relations for an
anisotropic material (equations 2.4, 2.9) before being simplified to plane stress in an
orthotropic material as expressed in equation 2.13. In equation 2.14 the shear strain has
been changed to ε12(=γ12/2), so that tensorial transformations are valid. The components
of the stiffness matrix, [Q], are given as follows:
E1 E2 E1
Q11 = , Q22 = , Q12 = ν 21 ,
(1− ν 12ν 21 ) (1−ν 12ν 21 ) (1− ν 12ν 21 ) (2.15)
E2
Q21 = ν 12 , Q66 = G12
(1 − ν 12ν 21 )
Table 2.1 shows the interrelationships among the different forms of the elastic constants.
Chapter 4. Mechanics of Cord-Rubber Composite Materials 115
The principal directions of material orthotropy (1,2) do not always coincide with the
reference directions used for the structure (x, y). An example is the angled belts of a
radial tire, where the direction of the belt wires is at an angle (e.g., 20 degrees) to the tire
axes x-y-z. The circumferential (x) and lateral (y) stiffnesses of the belt are desired for tire
design and analysis. When this occurs, a method of transforming the stress-strain relations
from one coordinate system to another is needed.
The angle of rotation from the x-axis to the principal 1-axis is θ as shown in figure 2.4.
Stress and strain are second order tensors and they transform with change of coordinate
reference in a specific manner.
Figure 2.4: x-y axes rotated θ degrees from principal material
axes 1-2. Positive rotation is in the counterclockwise direction
116 Chapter 4. Mechanics of Cord-Rubber Composite Materials
The transformation equations for expressing stress and strain in the reference
coordinate system (x, y) in terms of the stresses and strains in the material principal
directions (1, 2) are given as follows.
⎡σ 1 ⎤ ⎡σ x ⎤ ⎡ε1 ⎤ ⎡ε x ⎤
⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥
⎢σ 2 ⎥ = [R ]⎢σ y ⎥ , ⎢ε 2 ⎥ = [R ]⎢ε y ⎥ (2.16)
⎢⎣τ 12 ⎥⎦ ⎢τ ⎥ ⎢ ⎥
⎣ xy ⎦ ⎣⎢ε12 ⎦⎥ ⎣ε xy ⎦
where [R] is the transformation matrix given in terms of the angle θ as
(2.17)
Note that the inverse of the matrix [R] is obtained by changing the sign of the angle, i.e.,
The lamina principal stress-strain relations transformed to the lamina reference axes
(x, y) can be obtained by substituting Eqs (2.16) into (2.14) to give
Note that [Q] is symmetric. Note also that the engineering shear strain γxy, rather than the
tensorial shear strain εxy, is used in equation 2.19, so that the common definitions for the
Qij result. The Qij are related to the reduced stiffnesses Qij by the following
Q11 = Q11c 4 + 2(Q12 + 2Q66 )s 2 c 2 + Q22
s 4
Q 22 = Q11s 4 + 2(Q12 + 2Q66 )s 2 c 2 + Q22
c 4
(2.21)
Q16 = (Q11 − Q12 − 2Q66 )sc 3 + (Q12 − Q22 + 2Q66 ) s 3c
Q 26 = (Q11 − Q12 − 2Q66 )s 3c + (Q12 − Q22 + 2Q66
) sc 3
Q 66 = (Q11 + Q22 − 2Q12 − 2Q66 )s 2 c 2 + Q66 ( s 4 + c 4
)
Chapter 4. Mechanics of Cord-Rubber Composite Materials 117
The matrix [Q]is now populated with nine non-zero components. There are still only four
independent elastic constants, since the lamina is planar and orthotropic. In the new ref
erence coordinate system (x, y), there is coupling between shear strain and normal stress
es and between shear stress and normal strains. Thus, in the new reference coordinate sys
tem (x, y), the lamina behaves like a completely anisotropic material.
In a similar fashion, the strains can be expressed as a function of the stresses in the
arbitrary coordinate system (x, y) as
⎡ε x ⎤ ⎡ S 11 S 12 S 16 ⎤ ⎡σ x ⎤
⎢ ⎥ ⎢ ⎥⎢ ⎥ (2.22)
⎢ε y ⎥ = ⎢ S 12 S 22 S 26 ⎥ ⎢σ y ⎥
⎢γ ⎥ ⎢ ⎥⎢ ⎥
⎣ xy ⎦ ⎢⎣ S 16 S 26 S 66 ⎥⎦ ⎣τ xy ⎦
where the components of the lamina compliance matrix are given by
S 11 = S11c + (2S12 + S 66 )s c + S 22 s
4 2 2 4
S 22 = S11s 4 + 2(S12 + 2S 66 )s 2 c 2 + S 22 c 4
(2.23)
S 12 = (S11 + S 22 − S66 )s 2 c 2 + S12 (s 4 + c 4 )
S 16 = (2S11 − 2S12 − S 66 )sc 3 + (2S12 − 2S 22 − S 66 )s 3c
S 26 = (2S11 − 2S12 − S66 )s 3c + (2S12 − 2S 22 − S 66 )sc 3
S 66 = 2(2 S11 + 2S22 − 4S12 − S66 )s 2 c 2 + S66 (s 4 + c 4 )
All the elastic coefficients in equations (2.19), (2.21) and (2.23) for plane stress are func
tions of the four orthotropic elastic constants, E1 , E2 , G12, ν12, and the angle θ from the
1-direction of the orthotropic lamina. The stress-strain relations governing the off-axis
response of a single ply can be written as [3]
(2.24)
1 cos 4 θ sin 4 θ ⎛ 1 2ν 12 ⎞ 2
= + +⎜ − ⎟ sin θ cos θ
2
E x E1 E2 ⎝ G12 E1 ⎠
1 sin 4 θ cos 4 θ ⎛ 1 2ν 12 ⎞
2
= + +⎜ − ⎟ sin θ cos θ
2
E y E1 E2 G
⎝ 12 E1 ⎠
1 cos 2 2θ ⎡1+ ν 12 1+ ν 21 ⎤ 2
= +⎢ + ⎥ sin 2θ
Gxy G12 ⎣ E1 E2 ⎦
ν xy ν yx ν 12 1 ⎡1+ ν 12 1+ ν 21 1 ⎤ 2
= = − ⎢ + − ⎥ sin 2θ (2.25)
Ex E y E1 4 ⎣ E1 E2 G12 ⎦
⎡ sin 2 θ cos 2 θ 1 ⎛ 1 2ν 12 ⎞ ⎤
λx = sin 2θ ⎢ − + ⎜ − ⎟ cos 2θ ⎥
⎣ E2 E1 2 ⎝ G12 E1 ⎠ ⎦
⎡ cos 2 θ sin 2 θ 1 ⎛ 1 2ν 12 ⎞ ⎤
λy = sin 2θ ⎢ − − ⎜ − ⎟ cos 2θ ⎥
⎣ E2 E1 2 ⎝ G12 E1 ⎠ ⎦
In addition, the elastic constants are functions of the mechanical properties of the
constituents, e.g., the cord and the rubber modulus. In section 3, methods for determining
the orthotropic properties from the constituent properties will be addressed.
Figure 2.5 shows how the composite engineering constants vary with angle for a
typical tire cord-rubber ply. Figures 2.6 and 2.7 show the corresponding data for the
compliances and the reduced stiffnesses, respectively. These plots are based on data from
Patel et al. [4].
Often, it is easier experimentally to work with the compliances rather than the reduced
stiffnesses. For example, it is easier to apply σxx = σ0 (with all other stresses equal to zero)
Figure 2.5: Variation of Gxy, Ex and νxy with cord angle for a 1000/2 polyester ply [4]
Chapter 4. Mechanics of Cord-Rubber Composite Materials 119
90,000
Reduced Stiffness (psi)
80,000
Q11
70,000
60,000
50,000
40,000
Q16 Q12
30,000
Q66
20,000
10,000
0
0 10 20 30 40 50 60 70 80 90
Cord Angle (degrees)
to determine S11=1/E1, than it is to apply εxx=ε0 (with all other strains zero) to determine
Q11. The latter test requires bi-axial grips to apply a zero strain boundary condition in the
transverse direction.
One important consideration is the difference between Ex (determined with zero
transverse and shear stress boundary conditions) and Q11(determined with zero transverse
and shear strain boundary conditions). Figure 2.8 shows these two functions. Note that
120 Chapter 4. Mechanics of Cord-Rubber Composite Materials
there can be an order of magnitude difference in these two coefficients which both relate
normal stress to normal strain as in
σ x = Exε x , for σ y = τ xy = 0, or
(2.26)
σ x = Q11ε x , for ε y = γ xy = 0.
Figure 2.8: Comparison of modulus Ex with transformed reduced stiffness Q11
Second, determine S22 = 1/E2 and ν21 by running a simple tension test with loading in
the 2- direction, or 90 degrees from the cord direction as shown in figure 2.10. The trans
verse modulus E2 is usually dominated by the rubber response. For a 50% volume frac
tion of cords, an initial estimate of E2 would be twice the rubber modulus, however many
factors will affect this value. Stiffening due to the constraint imposed by the cords on the
rubber in the 1–direction, which minimizes the rubber’s lateral contraction, and edge
effects can be significant in this test. The cords can even debond from the rubber at the
free edges.
With E1, E2, ν12, and ν21 determined, reciprocity (Eq 2.7) can be checked.
Figure 2.9: Moire fringes on a cord-rubber composite. The displacement is constant
within each fringe. Fringes for x and y displacements are shown.
Figure 2.11: Torsion of a cord-rubber cylinder for G12 determination. Grid, moire
or DIC can be used for shear strain measurement away from the grips.
θ in degrees between the membrane local direction and the rebar (cord) direction. A
positive angle defines a rotation from local 1 direction toward local 2 direction.
Figure 2.12: Rebars defined relative to local coordinate system
(2.27)
or, the rate of heat flow per area is proportional to spatial temperature gradient. The
proportionality is the heat conductivity k in W/m°C. If the material is orthotropic, the
analysis along the principal material directions gives
(2.28)
The heat flow and temperature gradient are first rank tensors, or vectors. Consider the case
for rotations about the z-axis consistent with the planar lamina stress-strain analysis. The
transformation matrix for vectors in the x-y plane is simply
(2.29)
124 Chapter 4. Mechanics of Cord-Rubber Composite Materials
for angles θ measured from the 1 direction to the x axis. The transformed variables are
given as
(2.30)
The orthotropic conduction law in the arbitrary x-y reference plane is then
(2.31)
(2.32)
where, for rotations about the z-axis the components are given by
(2.33)
Often tire heat transfer FEA is performed on a 2D axisymmetric tire section where the
transformed properties in Eq 2.32 are input to the analysis. In a 3-D analysis the
properties could be input in the principal orientation together with the orientation to the
structural coordinate system.
principal material directions. For an orthotropic lamina analyzed with respect to the
principal material directions, a change in temperature that produces no stresses gives rise
to the following strains
⎡ε11 ⎤ ⎡α1∆T ⎤
⎢ε ⎥ ⎢ ⎥
⎢ 22 ⎥ ⎢α 2 ∆T ⎥
⎢ε 33 ⎥ ⎢α 3∆T ⎥
⎢ ⎥=⎢ ⎥ (2.34)
⎢γ 23 ⎥ ⎢0 ⎥
⎢γ ⎥ ⎢0 ⎥
⎢ 31 ⎥ ⎢ ⎥
⎣⎢γ 12 ⎦⎥ ⎣⎢0 ⎦⎥
where α1, α2, and α3 are the coefficients of thermal expansion in the three principal mate
rial directions. Note that for cord-rubber composites with organic cords with molecular
orientation, these coefficients can be negative. For linear analysis, the thermal strains and
those due to applied stresses are simply added together. Thus, the inclusion of thermal
strains changes equations such as 2.4 and 2.13, respectively to
⎡1 −ν 21 −ν 31 ⎤
⎢E 0 0 0⎥
E2 E3
⎢ 1 ⎥
⎢ −ν 12 1 −ν 32 ⎥
0 0 0⎥
⎡ε1 − α1∆T ⎤ ⎢ E1 E2 E3 ⎡σ ⎤ (2.35)
⎢ε − α ∆T ⎥ ⎢ ⎥⎢ 1 ⎥
⎥ ⎢ −ν 13 −ν 23 1 ⎥ σ 22
⎢ 22 0 0 0⎥⎢ ⎥
2
⎢ε 33 − α 3∆T ⎥ ⎢ E1 ⎢ ⎥
⎢ ⎥=⎢
E2 E3 ⎥ ⎢σ 33 ⎥
γ
⎢ 23 ⎥ ⎢0 0 0
1 ⎥ τ
0 0 ⎥ ⎢ 23 ⎥
⎢γ ⎥ ⎢ ⎢ ⎥
⎢ 31
⎥ ⎢
G23 ⎥ ⎢τ 31 ⎥
⎢⎣γ 12 ⎥⎦ ⎢ 0 1 ⎥ τ
0 ⎥ ⎣⎢ 12 ⎥⎦
⎢ 0 0 0
⎢ G31 ⎥
⎢ 1 ⎥
⎢0 0 0 0 0 ⎥
⎢⎣ G12 ⎥⎦
and
⎡ ⎤
σ
⎡ 11 ⎤ ⎡ 11 12
Q Q 0 ⎤ 1⎢ε − α1 ∆T ⎥
⎢σ ⎥ = ⎢Q Q 0 ⎥ ⎢ε − α ∆T ⎥
⎢ 22 ⎥ ⎢ 21 22 ⎥⎢ 2 2 ⎥
⎢⎣τ 12 ⎦⎥ ⎣⎢ 0 0 2Q66 ⎦⎥ ⎢ γ 12 ⎥ (2.36)
⎢ ⎥
⎣ 2 ⎦
A numerical analysis code requires the input of the three principal coefficients of thermal
expansion and the orientation to the principal material directions. Typical values of the
linear coefficient of thermal expansion for filled rubber and the most commonly used
reinforcing cords are listed in the following table:
126 Chapter 4. Mechanics of Cord-Rubber Composite Materials
Exercises
1. For the lamina shown below, find the stresses in the 1,2 directions and the strains in
the x,y directions. The lamina has the following elastic constants:
E1=14 GN/m2 70 MN/m2
E2=3.5 GN/m2
G12=4.2 GN/m2
ν12=0.4 y
1
ν21=0.1 2
-3.5 MN/m2
60°
x
-1.4 MN/m2
3. Use the derivatives of Ex to find its maxima and minima. Show that Ex is greater than
both E1 and E2 for some values of θ if:
4. Show that Ex is less than both E1 and E2 for some values of θ if:
Thus, an orthotropic material can have an apparent Young’s modulus that either exceeds
or is less than the Young’s modulus in both principal directions.
Chapter 4. Mechanics of Cord-Rubber Composite Materials 127
In general, the key feature shared by all these approaches is the mechanics of
materials reasoning that the strains in the fiber direction of a unidirectional fibrous
composite are the same in the fibers as in the matrix. In addition, sections normal to the
fibers remain plane before and after being stressed.
Some example properties of tire cords and rubber are shown in table 3.1.
Simple models
The rule of mixtures assumes no interation between the constituents. This method works
well for the modulus in the direction of the cords. It is equivalent to considering elastic
springs in a parallel configuation and summing the volume-weighted stiffnesses to get the
overall stiffness. The equation can be written as
(3.1)
where,
E1 is the Young’s modulus of the composite in the direction of the cords,
Ec is the Young’s modulus of a single cord,
Vc is the volume fraction of cords in the composite
Er is the Young’s modulus of the rubber, and
Vr is the volume fraction of rubber in the composite
Note that
where D is the diameter of the cord, t is the thickness of the ply layer, and “e” represents
the cord “ends per unit length” along the 2 direction of the ply layer.
In many cord-rubber composites, the modulus of the cords is 100-1000 times larger
than the modulus of the rubber, so the approximation
(3.2)
is sometimes used. The same rule of mixtures approach can provide a reasonable estimate
for the major Poisson’s ratio, ν12 for loading in the 1-direction:
(3.3)
i.e., the overall contraction in the transverse direction can be envisioned as the volume-
weighted sum of the contraction of each constituent.
Consider the idealized geometry in figure 3.2 with loading in the transverse direction.
By using the independent elastic spring analogy for the transverse stiffness, the model
becomes two springs in series as shown in figure 3.3.
Figure 3.2 - Idealized cord-rubber composite loading in the transverse direction
Chapter 4. Mechanics of Cord-Rubber Composite Materials 129
Figure 3.3 - Independent elastic springs in series model for transverse modulus
When accounting for the different volume fractions of cord and rubber, the simple
springs in series model gives
(3.4)
or
(3.5)
For the simple spring model of equations 3.4 and 3.5, the transverse modulus of the cord,
which may be much less than the axial modulus, could be used for Ec. Also, one could
argue that since the rubber is constrained from contracting in the 1-direction due to the
stiff cords, the “plane strain modulus” of the rubber Er/(1-νr) should be used in place of
Er. In any case, if Ec>>Er, then equation 3.4 reduces to
(3.6)
Consider figure 3.2 under shear rather than tensile loading. The springs in series model
can also be used to give an estimate for G12. For Gc >> Gr, an equation analogous to equa
tion 3.6 results
(3.7)
A common theme with most equations for the transverse or shear composite modulus is
that they have some factor times the modulus of rubber.
130 Chapter 4. Mechanics of Cord-Rubber Composite Materials
The minor Poisson’s ratio can be obtained from the reciprocity condition that arises
from symmetry of the stress-strain relationships
(3.8)
The simple expressions above often give a good first approximation to the orthotropic
elastic constants required for structural analysis of the belt and ply cord-rubber composites.
Many factors are neglected such as the Poisson’s ratio mismatch, and the potential for the
cord to be anisotropic, or the fact that the cord is actually a structure and not a homoge
neous material. Relationships which can add some accuracy for cord-rubber composites
are considered next.
Halpin-Tsai equations
The composite transverse (2 direction) and shear properties of the composite are the most
difficult to predict. The following equations are referred to as the Halpin-Tsai equations
[see e.g., 6,13] and are widely used with the rule of mixtures for establishing the five elas
tic constants for a single orthotropic ply.
(3.9)
(3.10)
where the ξ1, ξ2 are factors depending on the cord geometry and spacing. These equations
are semi-empirical and have shown good correlation with data. If Ec >> Er, and using
common values of ξ1=2and ξ2=1, Equations 3.9 and 3.10 reduce to
(3.11)
(3.12)
The rule of mixtures equations 3.1 and 3.3 are used for major modulus and Poisson’s ratio,
respectively. Figure 3.4 shows how E1 and E2 vary with cord volume fraction for rule of
mixtures and Halpin-Tsai equations.
Figure 3.4 - Dependence of composite moduli on
cord volume fraction for a 1000/2 polyester ply [25]
Moduli ratio (E1 or E2)/Er
E1 rule of mixtures
E2 Halpin - Tsai
E2 springs in series
Chapter 4. Mechanics of Cord-Rubber Composite Materials 131
Gough-Tangorra equations
Gough and Tangorra [7] have developed expressions specifically tailored to the properties
of cord reinforced rubber. The expression for the transverse modulus is:
(3.13)
They also used the simple approximation for the shear modulus in Eq 3.7 and assumed
major Poisson’s ratio ν12 =0.5.
Akasaka-Hirano equations
The Akasaka-Hirano equations [8] are a simplified version of the rule of mixtures and the
Gough-Tangorra equations with:
(3.14)
S. K. Clark equations
In this approach [9], an energy method is used to formulate expressions for the lamina
elastic constants without requiring detailed cord properties such as shear modulus and
Poisson’s ratio.
The theory uses a stiffening parameter φ indicating the degree of stiffening imposed
by the cord structure.
(3.15)
Then
(3.16)
(3.17)
with
132 Chapter 4. Mechanics of Cord-Rubber Composite Materials
prediction and measured data for a 1500/2 Kevlar - rubber ply [10]
The following tables from Clark [9] compare values calculated from the various
theories with measured ones.
To highlight the accuracy of the various approaches in predicting the effective properties
of a single lamina, two types of reinforcing materials were considered: steel and rayon
with the following properties:
Table 3.2 Cord and rubber properties [9]
Elastic properties Rayon Steel
Cord Young’s modulus, Ec (GPa) 3.41 50.50
Rubber Shear modulus, Gr (MPa) 2.94 5.30
Cord volume fraction, νc 0.23 0.11
Parameter φ 102.70 466.40
The theoretically predicted and experimentally measured elastic contants for a rayon-
rubber lamina are listed in Table 3.3.
Table 3.3 Elastic constants for a rayon-rubber lamina [9]
Halpin- Gough- Akasaka- Clark Experimental data
Tsai Tangorra Hirano Theory
E1 (GPa) 0.786 0.786 0.772 0.811 0.779
E2 MPa) 16.60 9.10 11.70 15.20 13.90
G12 (MPa) 3.28 2.28 2.94 3.57 3.82
ν12 0.54 0.50 0.50 0.50 0.49
Thermal conductivity
When performing a heat transfer analysis for predicting tire temperatures, the thermal
conductivities of the materials are required. If the cord-rubber layers are represented by
effective composite properties, the conductivity will depend on direction. With reference
to figure 3.1, the heat typically travels more efficiently along the cords (1–direction) than
transverse to the cords (the 2- or 3 - direction). Consider a representative volume element
as shown in figure 3.7. In the 1-direction, the effective composite conductivity is repre
sented by the rule of mixtures as
(3.18)
134 Chapter 4. Mechanics of Cord-Rubber Composite Materials
As with the moduli, the transverse properties are more difficult. As an example, consider
the idealized geometry in figure 3.8. The square cord is a representation of ¼ of the effec
tive cross-sectional area of the cord, and the remaining area comprises the rivet between
cords and the rubber treatment above or below the cords (assumed symmetric). A model
with thermal resistors for heat flow in the 3–direction is shown in figure 3.9. The thermal
resistance is in general
(3.19)
where L is the length of the heat flow, A is the cross sectional area, and k is thermal
conductivity. For this example consider the resistance per unit depth in the 1 – direction.
Figure 3.8 - Unit cell for transverse
cord
Chapter 4. Mechanics of Cord-Rubber Composite Materials 135
Figure 3.9 - Equivalent thermal resistance model for conduction in the 3-direction
(3.20)
(3.21)
and
(3.22)
First, combine Rc and Rr1 in series. Note that thermal resistances in series add directly,
and thermal resistances in parallel add reciprocally,
(3.23)
(3.24)
(3.25)
The effective conductivity for conduction in the through-thickness direction for the cord-
rubber composite, keff, can be determined from equation 3.25. For the more simple case
136 Chapter 4. Mechanics of Cord-Rubber Composite Materials
without an extra rubber layer above and below the cords, Eq 3.25 simplifies to
k r b + kc t
keff = (3.26)
(b + t )
where the rule of mixtures for the conductivity is evident. A similar analysis can be
performed for the conductivity in the 2-direction. A more precise estimation of the
equivalent conductivities could be obtained by performing a heat transfer finite element
analysis of the geometry in figure 3.7.
Exercises
1. A composite specimen has dimensions of 25.4 cm x 25.4 cm x 0.3 cm and a weight of
218 g. The fibers weigh 186 g. The densities of the fibers and matrix are 1.0 g/cm3 and
1.2 g/ cm3, respectively. Determine the volume fractions of fibers and matrix in the
specimen.
2. The constituent materials in the composite described in the previous exercise have the
properties Ef1=32.0E+06 psi (220 GPa), Ef2=2.0E+06 psi (13.79 GPa), and Em=0.5E+03
psi (3.45 MPa). Estimate the longitudinal and transverse moduli of the composite by
where k2, kf, km are the thermal conductivity for the composite in the transverse direction,
What will be the thermal conductivity perpendicular to a ply with Vm=50% with the
kf
−1) (
k2 1 + ζηV f km 1
= ,η = , and ζ =
km 1− ηV f k 4 − 3Vm
( f +ζ )
km
Chapter 4. Mechanics of Cord-Rubber Composite Materials 137
LAMINATE
LAMINATE
z
x
y
governing constitutive equation of the laminate is derived in terms of the coupled bending
and stretching stiffnesses caused by the different principal directions among the layers of
the laminate.
(4.1)
where d is the distance from the neutral surface to the outer surface. The normal strain
εx can be expressed as
Chapter 4. Mechanics of Cord-Rubber Composite Materials 139
(4.2)
(4.3)
(4.4)
where κ is the curvature and I is the area moment of inertia of the cross section about the
neutral surface. Also note in figure 4.3b that the assumption of “plane sections remain
plane” has been used, since the planes AE and BF are assumed to remain planar after
deformation.
This is an analysis of bending of a beam about one axis. Composite laminate analysis
includes bending of the laminate “plate” about two axes as shown in figure 4.4. Note the
convention that Mx causes stresses σx, rather than being a moment about the x-axis.
Strain-displacement relationships
The kinematics of strain and displacement including bending deformations follow those
of plate theory developed originally for isotropic materials [12]. The treatment here
follows that in the text by Jones [13]. Note that the kinematic relations for strain and
deformation are determined independent of the material properties (isotropic, anisotropic,
heterogeneous, etc).
To establish a relation between the displacement components and the corresponding
strain field, a section of a laminate is considered as shown in figure 4.5.
The laminate is deformed due to transverse (z-direction) loading and also in-plane
loading. The equations relating the strain at any point C in the laminate will be established
in terms of the displacements at the geometrical midplane (i.e., the x-y plane passing
through B) of the laminate (u0,v0) and the displacement in the z direction (w).
140 Chapter 4. Mechanics of Cord-Rubber Composite Materials
Figure 4.5 - Deformed shape of the laminate in the x-z plane [13].
The “plane sections remain plane” assumption means that there are no shearing
deformations in the x-z and y-z planes. This is to say that the layers that make the
laminate cross-section DBA do not deform over one another: the plane represented by the
line DBA remains plane – i.e., straight and normal to the deformed midplane. With this
assumption, known as the Kirchhoff-Love hypothesis, the displacement at any point C on
the line DBA is given by the following linear relationship:
(4.5)
(4.6)
Note that the Kirchhoff-Love hypothesis may not be accurate for many cord-rubber
applications, since it does not allow for any interply deformations. The laminate with
Kirchoff-Love is stiffer than a laminate that allows interply shear deformations.
Since the line DBA is assumed to remain straight with negligible stretching or
shortening, it is consistent to assume that the normal strain in the z direction is nearly zero.
Therefore, the normal deflection at any point in the laminate is considered to be the same
deflection (w) as the geometric midplane.
At this stage, the definitions of the normal strain as the ratio of the change in length
divided by the original length, and the shear strain as the total angle change, are used to
derive the expression for the strain at any point in a laminate in terms of the displacements
u0,v0, and w. The strains can be defined in terms of the derivative of the displacements
with respect to spatial position as:
Chapter 4. Mechanics of Cord-Rubber Composite Materials 141
(4.7)
Using equations 4.6 and 4.7, the normal and shear strains can be written in terms of the
midplane strains and laminate curvatures as follows:
⎡ε x ⎤ ⎡ε 0 x ⎤ ⎡κ x ⎤
⎢ ⎥ ⎢ 0 ⎥ ⎢ ⎥ (4.8)
⎢ε y ⎥ = ⎢ε y ⎥ + z ⎢κ y ⎥
⎢γ ⎥ ⎢ 0 ⎥ ⎢κ ⎥
⎣ xy ⎦ ⎢⎣γ xy ⎥⎦ ⎣ xy ⎦
where the geometric midplane strains are related to the displacements of the midplane as:
⎡ ∂u0 ⎤
⎢ ⎥
⎡ε 0 x ⎤ ⎢ ∂x ⎥ (4.9)
⎢ 0 ⎥ ⎢ ∂v0 ⎥
⎢ε y ⎥ = ⎢ ⎥
⎢ 0 ⎥ ⎢ ∂y ⎥
⎣⎢γ xy
⎦⎥ ⎢ ∂u0 ∂v0 ⎥
⎢ +
⎥
⎣⎢ ∂y ∂x ⎦⎥
and the midplane curvatures are related to the midplane displacement w in the z direction
as:
(4.10)
Using matrix notation, the above equations can be written in a more compact format:
[ε ] = ⎣⎡ε 0 ⎦⎤ + z [κ ] (4.11)
Thus, the midplane strains are functions of the midplane displacements u0,v0 and the
curvatures are functions of the midplane deflection w.
or,
(4.13)
(4.14)
By substitution of the strain from Eq 4.8 into the above stress-strain relation, the
stresses in the kth layer can be expressed in terms of the laminate midplane strains and
curvatures as
[σ ]k = ⎣⎡Q ⎦⎤ ⎡⎣ε 0 ⎤⎦ + z ⎣⎡ Q ⎦⎤ [κ ]
k k
(4.16)
This is equivalent to computing the stresses due to axial and bending deformations in a
simple beam. The expression (4.16) can be used to compute the stress in a lamina when
the laminate midplane strains and curvatures are known.
Since can be different for each layer of the laminate, the stress variation through
the laminate thickness is not necessarily linear, even though the strain variation is assumed
linear. A hypothetical variation of stress and strain is depicted in figure 4.6.
Figure 4.6 - Hypothetical variation of strain, stiffness and stress in a laminate [13]
es in the z direction.
These stress resultants are positive in the same sense as the corresponding stresses, and
since they are stresses times length (dz), they have the dimension of force per length.
(4.17)
In addition, the moment resultants are given as the sum of the stresses multiplied by the
moment arm with respect to the midplane.
(4.18)
With the definitions of equations 4.17 and 4.18, a system of three stress resultants and
three moment resultants has been defined which is equivalent in their actions to the
actual stress distribution across the thickness of the laminate.
The stress resultants can be written in a vector form in terms of the stress components
as follows:
(4.19)
If this load system is applied to a laminate composed of n layers, stress resultants can be
expressed as the sum of n simple integrals. Furthermore, if the stress in each layer is
written in terms of the midplane strains, the plate curvature, the z coordinate, and the plate
elastic stiffness properties, the following expression can be established:
Each of these integrals can be easily evaluated because [ε0] and [κ]are not functions of
z, and within any layer [hk-1, hk], [Q]k is, is not a function of z.
Since [ε0] and [κ] are independent of the layer number k, equation 4.21 can be written in
the simpler form:
⎡ N x ⎤ ⎡ A11 A12 A16 ⎤ ⎡ε 0 x ⎤ ⎡ B11 B12 B16 ⎤ ⎡ κx ⎤
⎢ ⎥ ⎢ ⎥ ⎢ 0 ⎥ ⎢ ⎥⎢ ⎥
⎢ N y ⎥ = ⎢ A12 A22 A26 ⎥ ⎢ε y ⎥ + ⎢ B12 B22 B26 ⎥ ⎢κy ⎥ (4.22)
⎢ ⎥ ⎢ ⎥ ⎢ 0 ⎥ ⎢ ⎥⎢ ⎥
⎣ N xy ⎦ ⎣ A13 A26 A66 ⎦ ⎣⎢γ xy ⎥⎦ ⎣ B16 B26 B66 ⎦ ⎣ κxy ⎦
or
where
n
Aij = ∑ (Qij ) hk − hk −1
k
( )
k =1
(4.24)
n
1
Bij =
2
∑ (Q ) (h
k =1
ij k
2
k − h 2k −1 )
Chapter 4. Mechanics of Cord-Rubber Composite Materials 145
Equation 4.23 indicates that for a general laminated plate, the midplane stress resultants
are given in terms of the midplane strains and the plate curvatures. Or, conversely, that
coupling exists between extensional forces and bending, or twisting, deformations.
The moment resultants can also be defined in terms of the stresses as follows:
(4.25)
Similarly, if this load system is applied to a laminate of n layers, the moment resultants
can be expressed as the sum of n simple integrals.
(4.26)
Following the same procedure as for the stress resultants, the matrices can be removed
from the integral in the summation:
(4.27)
The constitutive relationship for the moment resultants can be expressed as:
(4.28)
(4.29)
where
(4.30)
(4.31)
(4.32)
In this equation, the Aij are called extensional stiffnesses, the Bij are called the coupling
stiffnesses, and the Dij are called the bending stiffnesses. The presence of the Bij implies
coupling between bending and extension of a laminate. Thus, pulling on a laminate that
has non-zero Bij will cause both extension and bending or twisting of the laminate. Also,
such a laminate will extend or contract as well as bend when loaded by a moment.
Examples of the terms in equation 4.31 calculated for different tire ply/belt systems are
given in [14,25,73].
Figure 4.8 shows the extension-twist coupling for a simple 2-ply, +/-θ cord-rubber lam
inate vs. the extension-shear deformations of the single lamina. Note that each lamina is
described by the Aij stiffnesses derived from the which have extension-shear coupling,
but no out of plane bending. It is only when two +/-θ plies are bonded together that non
zero Bij terms appear and cause the out-of plane twisting.
Figure 4.8 - Deformations that occur in angled ply laminates. a) individual plies at +
and - cord angles; b) the two ply laminate [25] [13].
Example
A simple example is presented to provide a baseline for subsequent discussion. The
primary purpose of this example is to illustrate the calculation of the constitutive matrix
for the laminate.
Chapter 4. Mechanics of Cord-Rubber Composite Materials 147
Consider a specific two-ply 0/+45 degree laminate. The bottom lamina is a 0 degree
layer with 0.20 inch thickness and the following properties:
Since this is a zero degree ply, the x-y coordinate system coincides with the 1-2 prin
cipal material coordinate system.
The second lamina is a +45 degree layer with the same material properties as the first
lamina ( in the 1-2 coordinate system), and it is 0.10 inch thick. The Qij terms are found
in the x-y coordinate system by using the transformation equations described in Eqs 2.21:
Therefore,
The geometric configuration of the laminate and the location of the geometric midplane
are defined in figure 4.9.
Figure 4.9 - Notation for lamina thickness for the example [0,+45] laminate
The laminate constitutive equation can then be obtained by calculating the extensional
stiffnesses Aij, the coupling stiffnesses Bij, and the bending stiffnesses Dij.
148 Chapter 4. Mechanics of Cord-Rubber Composite Materials
n
Aij = ∑ (Q ) (h
k =1
ij K k − hK −1 )= (Qij ) [0.05 + 0.15] + (Qij ) [0.15 − 0.05]
1 2
⎦
1 n 1 1
Bij = ∑ (Qij ) (h 2 k − h 2 k −1 )= (Qij ) ⎡0.05
⎣
2
+ 0.152 ⎤⎦ + (Qij ) ⎡⎣0.152 − 0.052 ⎦⎤
2 k =1 k 2 1 2 2
⎦
1 n 1 1
Dij = ∑
3 k =1
( k
( 3
)
Qij ) h3 k − h3 k −1 = (Qij ) ⎡0.05
1 ⎣
3
+ 0.153 ⎤⎦ + (Qij ) ⎡0.15
3 2 ⎣
3
− 0.053 ⎤⎦
Combining the above results, the total set of constitutive equations for this two-ply
laminate can be written:
6.75 ⎤ ⎡ε x ⎤
0
⎡ Nx ⎤ ⎡ 697.5 97.5 67.5 − 20.25 6.75
⎢N ⎥ ⎢ 0 ⎥
⎢ 97.5 157.5 6.75 6.75 6.75 6.75 ⎥ ⎢ε y ⎥
⎢ y ⎥ ⎢ ⎥⎢
⎢N ⎥ ⎥
⎢ 67.5 67.5 97.5 6.75 6.75 6.75 ⎥ ⎢γ 0 xy ⎥
⎢ xy
⎥ =104 ⎢ ⎥
⎢M x ⎥ ⎢ −20.25 6.75 6.75 4.56 0.956 0.731 ⎥ ⎢κ x ⎥
⎢ ⎥ ⎢ ⎥
⎢ 6.75 6.75 6.75 0.956 1.406 0.731 ⎥ ⎢κ y ⎥
⎢M y ⎥ ⎢ ⎥
⎢ ⎥ ⎣⎢ 6.75 6.75 6.75 0.731 0.731 0.956 ⎥⎦ ⎢κ ⎥
⎣ M xy ⎦ ⎣ xy ⎦
where the units of Ni are (lb/in) and of the Mi (moment per length) are (lb).
Case1, [B] = 0: The terms in the [B] matrix are obtained as a sum of terms involving
the[Q]matrices and squares of the z coordinates of the top and bottom of each ply. Since
the Bij are, ( hk − hk −1 ) they are zero for laminates which are symmetrical with respect to
2 2
z. That is, each term in Bij is zero if, for each lamina above the mid-plane there is an iden
tical lamina (in properties and orientation) located at the same distance below the mid-
plane. Such mid-plane symmetric laminates are an important class of laminates. They are
commonly constructed because the extension-bending or twist coupling of non- sym
metric laminates causes warping due to in-plane loads or thermal contractions. The gov
erning constitutive equations for symmetric laminates are also considerably simplified
compared to non-symmetric laminates. Bending and extensional portions of the problem
can be considered separately. Although mid-plane symmetric laminates exhibit no
extension-bending coupling, they do in general exhibit both in-plane and bending
anisotropy with extension or bending and shear coupling due to the A16, A26, and D16,
D26 terms.
Case 2, A16=A26=0: This class of laminates behaves as an orthotropic plate with
respect to in-plane forces and strains. This occurs if, for every lamina of a plus θ
orientation there is another lamina of the same orthotropic properties and thickness with
a negative θ orientation, regardless of the stacking sequence. The laminate then behaves
as an orthotopic lamina loaded in the principal directions with respect to in-plane forces
and strains. However, bending and twisting can still occur. Laminates with this property
are called balanced.
Case 3, D16 =D26=0: The simplification of the bending matrix [D] is also possible. The
terms D16 and D26 can equal zero, if all the laminae are oriented at 0o or 90o, or if, for
every lamina oriented at a positive θ orientation above the mid-plane there is an identi
cal lamina placed at an equal distance below the mid-plane but oriented at a negative θ
orientation. These laminates are balanced laminates with the additional condition on the
distance of the +/- pair from the mid-plane.
Case 4, [B] =0, and D16=D26=0: For this special case, the laminate must have all the
restrictions listed in cases 1-3—i.e., the laminate is balanced and symmetric. The laminate
acts like an orthotropic lamina loaded along its principal axes. Four independent elastic
constants can be identified – Ex, Ey, Gxy and νxy – in the same way as for a single
orthotropic lamina. A simple example is a [+θ,-θ,-θ,+θ] lay-up. To avoid out-of plane
deformations, Lee and co-workers [47, 74] often use a [+θ,-θ,-θ,+θ ] composite for
fatigue testing rather than the 2-ply specimen illustrated in Fig 4.8.
With the limits of Case 4, the elastic constants can be computed from the laminate stiff
ness matrix as
Some examples of single lamina vs. balanced and symmetric laminate elastic constants
are shown in figure 4.10 a) and b). The laminate has cord lay-up [+θ,-θ,-θ,+θ].
150 Chapter 4. Mechanics of Cord-Rubber Composite Materials
single ply and 4 ply [+θ, -θ, -θ, +θ ] laminates of 840/2 nylon cord-rubber
For a general laminate under a general loading condition, equation 4.31 describes the
governing behavior. By inverting the stiffness matrix, the mid-plane strains and plate
curvatures are determined as a function of the applied loads as:
(4.33)
where [A’], [B’], and [D’] are related to [A], [B], and [D]. Once the mid-plane strains and
plate curvatures are determined, the strains in any laminae can be calculated from
(4.34)
Consider a lamina at a distance z from the geometric mid-plane, and transform the above
strains into the lamina principal material axes 1-2:
(4.35)
(4.36)
The stresses in equation 4.36 and/or the strains in equation 4.35 could be used in the
appropriate failure criterion to determine if the loads are acceptable. If the strength crite
rion is based on a maximum strain theory, a comparison is made using the state of strain.
If it is based on a maximum stress theory then a comparison will be made using the state
of stress. The topic of laminate strength analysis will not be dealt with in toto. However,
fatigue and durability of composite laminates is covered in section 6.
(a)
(b)
In the tire footprint the belt and ply flatten in the crown region and the following
coordinate directions are nearly equivalent
In the following discussions we will use the equations developed previously in the
x-y-z Cartesian system with the assumption that they approximately represent the actual
toroidal structure.
Plysteer
Plysteer is the tendency for a rolling loaded tire under zero camber and zero slip angle to
track at an angle to its centerline. It is usually observed as a net lateral force under zero
steer angle and zero camber conditions in a laboratory test. Consider the simple case of a
passenger tire belt package with a lay-up of [+20, -20]. This is not symmetric and the
coupling terms Bij are not zero. As the belt package is bent with a change in curvature κx
going into the footprint, the B16 term can create a force Nxy which could result in a steer
ing torque, and the B12 term can create a cornering force Ny. Both of these forces can con
tribute to plysteer. Note that whether the tire rotates clockwise or counterclockwise, the
plysteer force still acts in the same direction.
Obstacle envelopment
This tire performance parameter is related to harshness. To envelop objects such as joints
in the road, a low circumferential bending stiffness is desired. For moderate sized rocks,
a low meridional bending stiffness may also help reduce harshness. Of course, these
requirements need to be balanced with other requirements for belt package stiffness.
For circumferential bending and extension, with other strains and curvatures unchanged,
equation 4.31 gives
(4.37)
If the resultant force in the x-direction is assumed to be zero in the footprint, then
Chapter 4. Mechanics of Cord-Rubber Composite Materials 153
(4.38)
(4.39)
which is the same moment-curvature relation as equation 4.4, except that euqation 4.39 is
for a wide, composite laminate.
Figure 4.12: Tread centerline deflection in tire footprint due to cornering [25]
(4.40)
where the constants Ci have units 1/length and relate to the footprint dimensions and
lateral deformation characteristics of the tire construction.
Interlaminar deformations
The theory of composite laminates presented in the previous sections does not allow for
shear deformation between layers. Laminate theory may be useful for overall stiffnesses,
154 Chapter 4. Mechanics of Cord-Rubber Composite Materials
If the loading direction is the x-direction and the z-direction is normal to the laminate
surface, then the shear indicated by the change in angle of the white line is the γzx com
ponent. The tendency for the composite to twist is also evident in the loaded picture. A
distribution of interlaminar shear across the width of a 2-ply specimen is shown in figure
4.14. The maximum value of interlaminar shear strain occurs at the edges of the specimen.
With the stress riser caused by the ends of the cords, this makes the edge a likely location
for cracks to initiate. Note that zero, or nearly zero interlaminar shear strain develops for
cord angles 0o, 90o, 75o, and 60o.
Figure 4.14 - Interlaminar shear strain ( xz )
It is interesting to note that, as the rubber layer between the cord layers decreases in
thickness, the shear strain will often increase for the same extensional strain, since the
“gauge” over which the shear deformation acts is decreasing. Other effects such as the
change of stiffness distribution with thickness makes the relation of shear strain vs.
Chapter 4. Mechanics of Cord-Rubber Composite Materials 155
rubber gauge less than linear [17]. This decrease in interlaminar shear strain with
increasing gauge is one of the reasons that a belt edge wedge or rubber layer is often found
between, and at the edge of, the working belts in radial tires. Another interesting
behavior is interlaminar shear as a function of cord angle for a fixed extension as shown
in figure 4.15. The maximum interlaminar shear strain occurs at around a 20 degree cord
angle, which is close to the cord angle (measured with respect to the circumferential
direction) used in many tire belts. However, this experiment was performed with constant
extensional strain, whereas tire belts need to resist an inflation load and periodically a
bending deformation in the footprint. Belt angles around 20 degrees usually work well,
Figure 4.15 - Interlaminar shear vs. cord angle for [+θ ,- θ] laminate [20].
combined with the 90 degree ply to provide sufficient Gough stiffness and stability for a
radial tire.
A method to illustrate interlaminar shear strain was described by Turner and Ford [67].
In this method, a row of pins is inserted across the width of a 2-ply specimen. The pins
tend to rotate as the load is applied; see figures 4.16a and 4.16b. The rotation angle of the
pins is proportional to the interlaminar shear strain.
Figure 4.16a - Pin experiment showing mechanism to demonstrate
interlaminar shear strain in 2-ply composite specimen [67]
156 Chapter 4. Mechanics of Cord-Rubber Composite Materials
The interlaminar shear strain can be many times the applied axial strain. Figure 4.17
shows that for cord angles of 22 degrees, the interlaminar shear strain at the specimen
edge is about 4 times the applied axial strain. This plot is for a specimen with geometry
shown in figure 4.13, with a large rubber gauge between the cord-rubber layers. As the
rubber gauge between the plies decreases, the interlaminar shear strain will increase fur
ther for the same axial strain. A parametric study on interlaminar shear strain is given by
DeEskinazi and Cembrola [75].
Macroscopic modeling
With macroscopic modeling, the cords are not explicitly represented, but their stiffness
contribution is made through the material constants. The composite analysis outlined in
section 3 is typically used to define the homogenized composite properties. Many different
approaches for the finite element analysis are available. Three approaches, using different
types of structural elements are described in the following subsections.
Care should be taken when smearing properties over a standard brick element, since
bending stiffnesses and deformations are not well represented. The nodes at the corners of
the brick have only displacement degrees of freedom (u, v, w). In contrast, shell elements
have both displacement and rotational degrees of freedom, and are often preferred for
representing layered composites with smeared properties.
Figure 4.19: Layered shell element with six degrees of freedom per node
Membrane elements can also be used to represent a single cord-rubber layer. They do
not have rotational degrees of freedom, so in order to model the bending behavior of a
cord-rubber laminate, a layer of membrane elements is used to represent a single cord-
rubber layer. Solid rubber elements are then used to separate the membrane layers as
shown in figure 4.21. An early analysis using this approach is found in Turner and Ford
[67]. Each membrane element can be assigned orthotropic properties, or alternatively,
rebars can be used to represent the cord stiffness and the membrane material properties
can represent the rubber behavior.
The rebar/membrane (or rebar/shell) approach offers flexibility in using material laws,
since properties for the rubber and cords are input separately. For example, a hyper
elastic/softening law to represent the Mullins effect in rubber can be assigned to the mem
brane, while a nonlinear viscoelastic law to represent creep in a nylon cord can be
assigned to the rebar. Note that the resulting stiffnesses distributed to the nodes are
homogenized stiffnesses that include effects of both the rubber and the cords. This
approach also provides shear deformation between layers to achieve a shear deformation
similar to the pattern in figure 4.20b.
Figure 4.21 - 2-D view of a 3-D cord-rubber modeling scheme using
membrane and brick elements. Each node has 3 degrees of freedom.
Chapter 4. Mechanics of Cord-Rubber Composite Materials 159
Macroscopic modeling, using brick, shell and/or membrane elements provides the
overall behavior of the composite structure. These elements are typically used to represent
the cord-rubber layers in a tire model. [e.g., 57, 58]. For a more detailed analysis, the cord
geometry needs to be represented explicitly, but this requires larger models and more
computation time. The next subsection describes two methods for detailed cord-rubber
modeling.
plastic, and damage effects. Interfaces are typically provided for user specific models.
Figure 4.24 shows results for strain energy density around a cord.
The Method of Cells does not satisfy all the governing equations of the mechanics
problem as a finite element approach would. What MOC gives up in consistency, it gains
in speed and the ability to look at very fine detail.
Software for the MOC can be coupled to finite element codes to provide a multi-level
(i.e., global-local) solution capability. For example, the MOC could be used to perform
the local analysis of the cord-rubber detail and provide equivalent anisotropic material
properties to macroscopic brick or shell elements used in a global tire analysis. Damage
could be evaluated by the MOC and the resulting softened stiffnesses provided back to the
Figure 4.23 -- Representative volume elements for
continuous fiber reinforced composite
Exercises
1. A symmetric laminate has both geometric and material property symmetry about the
mid-surface. Show that the laminate coupling stiffnesses Bij are zero for a [+45/-45/
45/+45] symmetric angle-ply laminate consisting of 0.25 mm thick unidirectional lami
nae with the following engineering constants: E1=138 GPa, E2=9.0 GPa, G12=6.9 GPa,
and ν12=0.3.
2. An antisymmetric laminate has plies of identical material and thickness at equal posi
tive and negative distances from the middle surface, but the ply orientations are anti-
symmetric with respect to the middle surface. Show that the laminate coupling stiffness
es Bij are not zero for a [-45/+45/-45/+45] antisymmetric angle-ply laminate consisting
of the same 0.25 mm thick unidirectional laminae used in the previous symmetric lami
nate.
volume) as the specimen is cycled from point A to point B and returned. This means that
when a cord-rubber composite material is cyclically stressed, a net energy loss occurs in
the form of heat, which either raises the temperature of the composite or dissipates to the
surroundings.
Figure 5.1 -- Typical stress-strain hysteresis loop
The consideration of viscoelasticity for the composite is important for accurate stress-
strain analysis of the tire structure, as well as the computation of energy loss for tire rolling
resistance, and the corresponding heat production which affects the tire temperature.
Hysteresis of cord-rubber composites is typically characterized by either sinusoidal
oscillatory testing, or by step-strain tests giving rise to quantities such as
which are the complex modulus in the 1 direction, the storage modulus in the 2 direction,
the loss modulus in the 2 direction, the relaxation modulus in the 2 direction, and the shear
relaxation modulus, respectively. As for the elastic properties, the 1-direction response is
typically dominated by the cords, and the 2-direction response is typically dominated by
the rubber.
The treatment of hysteresis in this chapter is limited to linear viscoelasticity and
orthotropic response. Often the moduli of cord-rubber composites turn out to be functions
of strain, or strain amplitude as well as time, temperature and frequency. The strain
dependence is beyond linear theory and not discussed here.
Stress-strain relationships
The mathematical basis for viscoelastic, or “lossy” behavior is due to Boltzmann [27] who
postulated that for all real solids the stress-strain relations were functions not only of
instantaneous values but also of the complete stress and deformation history of the
material. For linear viscoelasticity, one may use the principle of superposition; namely, the
stress at any time resulting from a sequence of separate strain histories applied at earlier
Chapter 4. Mechanics of Cord-Rubber Composite Materials 163
times is a sum of the stresses that would have been produced had the separate strain
applications occurred individually.
This may be illustrated by assuming that uniaxial strain ε1 was applied at the time t1
and then held constant, i.e., a step-strain history. At some later time t this strain has caused
a stress σ1 (t – t 1). Similarly a second step strain ε2 beginning at time t2 and held con
stant, will produce a stress σ2(t – t2) at time t . This is illustrated in figure 5.2.
Superposition postulates that if the strain history consisted of these two step-strains
acting together, then the total stress induced in the specimen at time t would be
(5.1)
The time dependent relationship between stress and step strain history is called the stress
relaxation function and is often denoted by the symbol E(t) for extension (and compres
sion), or by G(t) for shear. For example, if a step strain εο is applied at the time t = 0, then
stress at any time t is given by εο E(t). The stress relaxation function has the general time
decaying form shown in figure 5.2a. For a solid material, it decays to some non-zero
value. The stress relaxation function is a material property for viscoelastic materials just
as Young’s modulus is for elastic materials.
When the strain history consists of a series of N step-strain increments εn, individual
ly applied at times t = t n the stress σ(t) observed at a time t > tn is given by
(5.2)
An arbitrary strain history can be regarded as the superposition of infinitely many step
strain history increments. In this case equation 5.2 extends to equation 5.3
(5.3)
or,
(5.4)
This is known as Boltzmann’s superposition principle. For small strain analysis of cord
rubber composites, these equations can simply be used to compute the uniaxial extensional
response in, say, the 1-direction by using E(t) = E1(t), or similarly for E2(t) in the direction
164 Chapter 4. Mechanics of Cord-Rubber Composite Materials
σ 1 (t ) = E1 (t )ε 0 (5.5)
⎢⎣ 0 0 S66 (t ) ⎦⎥
⎡ Q11 (t ) Q12 (t ) 0 ⎤
[Q(t )] = ⎢⎢Q12 (t ) Q22 (t ) 0 ⎥
⎥ (5.9)
⎢⎣ 0 0 Q66 (t ) ⎥
⎦
So , the viscoelastic analysis of an orthotropic lamina or special laminate, there are four
independent material functions analogous to the four independent constants for orthotrop
ic elasticity. The strain and stress components in the principal material directions are given
by
Chapter 4. Mechanics of Cord-Rubber Composite Materials 165
(5.10)
(5.11)
where τ12,γ12 are τ6,ε6 respectively, in the contracted notation. As for the elastic case, it
is often easier to measure the compliances Sij(t) than the corresponding stiffnesses or mod
uli Qij(t), since it is easier experimentally to apply a single component of stress.
For a cord-rubber composite where the cord direction is much stiffer, and the cord
response can be considered elastic (e.g., for steel wires), then the following approxima
tions are often made [33]
(5.12)
With the approximations in Eq 5.12, only S22 and S66 remain to be determined as a func
tion of time. Applying uniaxial loading such as creep or constant stress rate in the 2-direc
tion can be used with Eq 5.10 to define S22(t), and similarly in shear to define S66(t). For
example,
(5.13)
(5.14)
Figure 5.3: Stress components resolved along non-principal material axes x-y
166 Chapter 4. Mechanics of Cord-Rubber Composite Materials
(5.15)
Tests in non-principal material directions are sometimes used to determine the shear
compliance S66(t); for example, S66(t) appears in equation 5.14 for S 11 (t ) . To obtain all
four compliance functions, the following creep tests and analysis can be performed anal
ogous to the elastic case[32]. First, a creep test in the direction of the cords gives
ε1 (t )
S11 (t ) = (5.16)
σ 10
and measuring the transverse strain vs. time during the same test gives
ε 2 (t )
S12 (t ) = (5.17)
σ 10
Second, another creep test can be performed in the 2-direction, transverse to the cords to
give
ε 2 (t )
S22 (t ) = (5.18)
σ 20
with a check on symmetry via
ε 1 (t)
S 21 (t) = 0 (5.19)
σ2
which should also equal S12(t).
Determining transverse strain ε1(t) with the creep stress applied in the 2-direction is
often difficult for cord-rubber composites, and it may be very non-uniform.
Thirdly, an off-axis test is performed at some angle θ to the cord direction. Then
ε 6 (t ) (5.20)
S66 =
σ6
where ε6 (i.e., γ12) and σ6 (i.e., τ12) are calculated from
∫S
(5.23)
66 (t − s)Q66 ( s ) ds = t
is used with specific functional forms, such as Prony series for S66 and Q66.
The most common representation of the orthotropic viscoelastic functions is with a
Prony series, similar to the case for isotropic viscoelasticity. For the modulus functions
Chapter 4. Mechanics of Cord-Rubber Composite Materials 167
(5.24)
and
(5.25)
for the compliance functions. In equation 5.24, the τp are the relaxation times, and in 5.25
the λp are termed the retardation times.
In the simplest forms shown in equations 5.24, 5.25, the τp are the same for all the Qij
and the λp are the same for all the Sij. In general, this may not provide the best represen
tation. For example if both the cords and the rubber are viscoelastic, then the cord domi
nated 1-direction and the rubber dominated 2-direction will likely need different relax
ation/retardation times.
ber. In this manner, the measured results can be fit more precisely.
One relatively simple combination of rebar elements for cords and solid or membrane ele
ments for rubber would be the following: linear viscoelastic properties for the cord—e.g.,
Prony series with initial and long-time modulus, and visco-hyperelastic properties for the
rubber. The visco-hyperelastic option available in most major FEA codes provides for the use
of a typical hyperelastic law coupled with a Prony series representation for viscoelastic
portion of the response [2].
Another advantage for specifying the properties of cord and rubber separately is that
the cords and rubber compounds can be tested separately and combined in the FEA code.
In this case, the numerous possible combinations of different cords and rubber compounds
do not need to be tested as composites. One disadvantage is that the potentially nonlinear
stiffness due to the elevated strain experienced in the rivet, or rubber between the cords,
will not be represented by the rebar, or any other “smeared” approach.
Energy loss
Consider the calculation of energy loss where the strain cycles are known from a FEA
calculation. For this discussion, 3-D solid elements are used for the belts and plies and the
energy loss is to be calculated. In general, the energy loss density is given by
(5.26)
where W″ is the energy loss per unit volume. With the strain cycles for one revolution for
a given element assumed known from the FEA solution, the stresses can be calculated
from Eq 5.11 and used with Eq 5.26 and then multiplied by the appropriate volume to
calculate the energy loss. As cords or wires do not dissipate much energy, the loss from
the cords is often neglected in comparison with the loss from the rubber [e.g., 34].
response, stiffness, and thermal properties have been covered. These composite properties
together with similar properties for the rubber compounds can be used as input to a tire
analysis to predict performance such as rolling resistance, or handling. Tire durability
analysis requires additional properties to describe the fatigue behavior of the materials
and/or components.
Fatigue and fracture of cord-rubber composites can provide information for tire
durability assessment in different ways. With some limitations due to the complexity and
multiple factors involved with tire durability performance, the following investigations
can often provide some insight: 1) fatigue properties of tire composites combined with tire
finite element analysis for a tire life estimation, 2) fatigue of belt and ply composites to
assess their relative performance, 3) failure mechanisms of laboratory composites to
provide understanding of failure mechanisms in tires, and 4) crack growth properties of
rubber compounds combined with detailed fracture mechanics analysis of the cord-rubber
layers in tires. Some of the factors involved are: aging, temperature, deformation modes,
minimum stress or strain, strain rates, etc.
The scope of this section is limited to the fatigue and failure mechanics of cord-rubber
composite specimens produced in the laboratory. The range of stress, strain and
temperature, and the degree of cure and aging of the laboratory composites should mimic
the actual conditions in the tire of interest for meaningful results.
The choice of load or strain control depends on the nature of the deformation cycles in the
tire component of interest. Deformation Index analysis [46,65] can be used to ascertain
stress control, strain control, or energy control cycles. A typical radial tire might have strain
controlled (bending) cycles in the sidewall, and “energy” controlled cycles in the belts [e.g.,
46, 60]. Energy control is difficult to achieve in the laboratory, so load control is often used.
Figure 6.2: Load or strain cycles for fatigue of cord-rubber composities, sine and pulsed
Many investigators have studied the fatigue response of cord-rubber composite speci
mens. The results and interpretations given here follow the work of Lee and co-workers [47
49,74], Causa and co-workers [50-51], Kawamoto and Mandell [52], and Huang and Yeoh
[53]. Figure 6.3 shows specimen axial strain vs. time for a fatigue test under load control.
Note the curve looks much like a typical static creep curve, except here the phenomenon is
cyclic creep. After the initial or elastic deformation at time t = 0, the maximum (and also the
minimum) strain in the cycle continually increases until failure. For discussion purposes, the
curve is often divided into three regions: primary, secondary, and tertiary. Although the exact
mechanisms of damage may differ slightly depending on the type of cords (e.g., nylon vs.
steel wire) and rubber, some general observations can be cited. During the primary stage, vis
coelastic response and softening due to temperature rise and Mullins effect in the high strain
regions account for most of the increased strain. Very small cracks near the cord-rubber inter
face are also possible in this stage. A significant portion of these strains are recoverable over
time, upon release of the load. A typical recovery period might be on the order of hours, or
days to recover 99% of the creep strain.
In the secondary stage, further viscoelastic creep, cracking around the cords, cord-rubber
debonding, and small matrix cracks form to further soften the overall structure.
In the tertiary stage, crack coalescence and progressive delamination leads to specimen
failure. Figure 6.4 shows a schematic of this process with the socketing and delamination
cracks depicted.
Figure 6.3 - Maximum strain vs. time or cycles for load controlled composite fatigue test.
Chapter 4. Mechanics of Cord-Rubber Composite Materials 171
Figure 6.4 - Cord-rubber creep and crack growth under load control
Figure 6.7 - Delamination cracking towards the end of the specimen's life.
One common method of comparing fatigue data from different composite specimens is
by plotting the applied stress range (load range divided by the original cross sectional
area) vs. the number of cycles to failure (Nf). An example is given as the schematic plot
in figure 6.8. This plot is called an S-N diagram, fatigue-life curve, or Woehler diagram.
172 Chapter 4. Mechanics of Cord-Rubber Composite Materials
Multiple specimens are typically run at each stress range, since data are often scattered.
Plotting either: 1) all the data, 2) data ranges (as in figure 6.8) and/or 3) confidence
intervals, aids in interpreting the statistical nature of the results. Tests with no overall
failure, or “run-outs” are also possible at lower load ranges as indicated by the arrow in
the figure. Different cord angles, cord spacings, rubber compounds and other design
features can be investigated by comparing their composite fatigue results. An actual plot
of an S-N diagram for a cord-rubber composite is shown in figure 6.9. This plot shows the
effect of minimum stress on fatigue life.
Figure 6.8 - Typical stress-life fatigue plot
When the minimum stress in a fatigue test of a rubber compound is greater than zero,
the fatigue life is generally greater than the same test run with the same stress cycle and
zero minimum stress – as long as the maximum stress is well below the upper strength
limit. This effect is the opposite of that typically exhibited in fatigue of other materials.
This is sometimes termed the R-ratio effect, since R is defined as
(6.1)
The stress in equation 6.1 can be replaced with other loading parameters, and for rubber
it is usually the energy release rate G (see equation 6.2). A discussion of this effect can be
found in Mars and Fatemi [61]. The mechanism of increased fatigue life for R>0 in rub
ber articles is often reported to be linked to strain-induced crystallization. A more gener
al picture is to envision molecular orientation near a crack tip in the direction of maximum
principal strain. This orientation can act to inhibit “self-similar” crack growth and can
cause crack branching along weaker directions. A branched crack will propagate more
slowly than a single sharp crack. These combined effects contribute to the R-ratio effect
in many rubber compounds. Since cord-rubber composites generally fail due to cracking
in the rubber matrix, the same phenomenon can be observed in composite fatigue tests.
The R-ratio effect for cord-rubber composites was examined by Ku et al [49].
Chapter 4. Mechanics of Cord-Rubber Composite Materials 173
Figure 6.9 - Stress-life fatigue plot with dependence on minimum stress [49].
Figure 6.10 - Specimen geometries for evaluating crack growth in the rubber
Figure 6.11 - Edge delaminated cord-rubber specimen with initial cracks [52].
rubber compound to the cord. Figure 6.13 shows some examples of pull-out surfaces from
an aramid-rubber test, and figure 6.14 shows the same for a steel wire-rubber test. Good
wire or cord adhesion is obtained if all the failure occurs in the rubber, thus indicating that
the cord-rubber interface is stronger than the rubber itself. In figure 6.13, the outline of
Chapter 4. Mechanics of Cord-Rubber Composite Materials 175
the cord and filament surfaces is clearly visible indicating that the failure is at least close
to the interface. Close inspection revealed some fracture in the cord dip and also some in
the dip-fiber interface. In figure 6.14, there is no indication of the wire geometry and all
the failure is in the rubber compound indicating good adhesion.
Figure 6.13 - Aramid cord-rubber surface from pull-out test, 20x and 50x. Some
failure occurs in cord dip and cord-dip interface [55].
(6.2)
where U is the strain energy in the structure and A is the increase in crack area. Griffith’s
criterion states that a crack will grow when G in Eq 6.2 reaches a critical value, Gc.
Consider the cord pull out geometry in figure 6.12 under “fixed grip” conditions. For
an increase in the cylindrical crack around the cord of ∆c, Griffith’s criterion gives [76]
176 Chapter 4. Mechanics of Cord-Rubber Composite Materials
(6.3)
where Gc is the critical energy release rate (also called tear energy or fracture energy)
related to the cracking mechanism—e.g., cracking in the rubber adjacent to the cord or
cracking in the cord-rubber interface. The change in strain energy in the structure is
denoted by ∆U. The change in strain energy is deduced by considering the change in
volume for a portion of the rubber cylinder, away from the cord, that has a homogeneous
uniaxial strain field. The volume of this region increases by A∆c = πr2∆c. Also, the strain
energy density in the homogenous region is given by
1 1P 2 (6.4)
W = σε =
2 2A2 E
Thus, the load at crack growth initiation is
1/ 2
P = ⎡⎣ 4π 2 R 2 rEGc ⎤⎦ (6.5)
Simple energy release rate approximations can also be derived for two mechanisms of
cracking in the 2-ply composite specimen. The following discussion follows Breidenbach
and Lake [36,37] and Huang and Yeoh [53]. Consider first the case for delamination
starting from the edges of the specimen as illustrated in figure 6.11.
In a manner similar to the energy release rate analysis for the pure shear geometry [56],
consider the change in strain energy in the structure as the edge cracks grow by a small
amount ∆c. The plies in the cracked region are assumed to have negligible strain energy
compared to the central region. The volume of material that has significant strain energy
is reduced by an amount: (length)(thickness)(∆c). If there is a region of homogeneous
strain energy density Wh (e.g., the central region of the specimen away from the cracks)
that is also reduced by this volume, then the energy released as the crack grows by an
amount ∆c is
∆G = W ht ∆c (6.6)
where the thickness t represents the rubber gage between the cord layers. This is the same
expression used for the pure shear geometry. Given the distribution of interlaminar shear
strain in figure 4.14 for a typical cord-rubber specimen, it may be difficult to define the
quantity Wh, but nonetheless, Eq 6.6 can be useful in an approximate sense.
If the mechanism of cracking is socketing, as in figure 6.15, rather than delamination,
then the strain energy analysis remains the same, but the calculation of new crack area
changes to
(6.7)
where D is the diameter of the cords and e is the number of cords per unit length. Then,
(6.8)
Chapter 4. Mechanics of Cord-Rubber Composite Materials 177
Finite element analysis (FEA) can provide more accurate estimates for energy release
rate than the simple closed form solutions given above. Many studies with finite element
analysis have been performed for cord-rubber composites, some by Pidaparti and
co-workers [17,38-41]. With FEA, a more representative material model and the actual
geometry of the structure and the observed crack(s) can be included. However, for cracks
such as those in figures 6.6 and 6.7, the actual geometry may be difficult to represent, and
even if the actual crack was represented, the analysis might provide only a specific result
that is not useful in general. Thus, finite element analysis of the more general situations,
such as those treated above, may be the best approach. Examples of analysis of cord-
rubber composite fatigue with fracture mechanics methodology is provided in the
references [53, 60].
in many complex structures without a priori knowledge of how the structure will fail.
In general, fatigue damage is the progressive loss of mechanical integrity of the
material and/or structure. It is often a complex phenomenon that can be driven by many
factors. In the critical regions of highly localized stress, strain, and/or temperature, failure
could be caused by any one, or a combination of these quantities. In addition, other
non-critical damage mechanisms such as local loss of cord-rubber adhesion or chemical
degradation due to reversion or oxidation could become the primary cause of failure, or
they could accentuate the state of damage. Some typical damage in the form of cracking
mechanisms were shown in Figures 6.5-6.7. Damage mechanisms due to micro-cracking
and chemical aging are not as easily seen and must be evaluated through their effects on
mechanical properties.
Damage often starts in the rubber compound at the micro level by developing
microcracks or cavitations in the neighborhood of micro defects, or at the interface of
inclusions such as reinforcing particles or long reinforcing cords. Other failure modes
could also exist such as rupture of the reinforcements under extreme loading conditions,
delamination by macrocracks between layers, or creep rupture under applied cyclic stress.
The underlying physics of each one of these failure modes might differ in the initiation
stage, but the subsequent damage could be similar in the accumulation stage. Therefore,
different approaches are typically employed. For example, one such combination of
approaches is to use CDM to predict initiation of a macro-crack, and to use crack growth
analysis to predict propagation of the crack.
One simple expression of continuum damage is the Palmgren-Miner approach [42]
which is based on the S-N diagram. The damage estimate is made from the following
(6.9)
where the damage di associated with the ith set of cycles at a given load or stress is equal
to the number of cycles ni at the associated stress amplitude, divided by the number of
cycles to failure at that stress amplitude, Nfi. Then failure is defined when
ni
D=∑ =1 (6.10)
i N fi
This is called a linear damage law since each cycle at a given amplitude contributes the
same fraction to the damage. Equation 6.10 allows for no load sequence effects or depend
ence on the current state of damage. In actual cases, load sequence and aging effects
probably exist [e.g., 62].
A more general approach parallels the CDM method advocated by Kachanov [43] and
Rabotnov [44]. It is a phenomenological model which depends on laboratory testing to
describe the evolution of the damage and contains a scalar damage parameter to describe
the collective effect of material degradation, similar to di in equation 6.9.
The method will be illustrated for a 2-ply composite [+θ,-θ] specimen under cyclic
tensile loading as in figure 6.16. In equation 6.9, the damage is attributed to stress cycles,
since the data is given by an S-N diagram.
Chapter 4. Mechanics of Cord-Rubber Composite Materials 179
The following analysis is based on the premise that the cyclic interlaminar shear strain (γzx
cyclic) is the primary cause of damage. The model constants are derived from fatigue data at
room temperature, and the numerical simulation of the test was conducted using ABAQUS.
The stress-life data from the S-N curve was transformed to interlaminar shear strain cycle vs.
life data for use in this example.
The damage law is expressed in a rate form as
Cγ a1−n
ω& = (6.11)
(1− ω )
m
where ω · is the damage rate expressed in terms of the driving force, γa, and the current state
of damage, ω. Material constants C, n, and m are to be determined from the Ya vs. life
information. Note that the rate in equation 6.11 can be expressed in terms of time or
fatigue cycles with time = #cycles/frequency. Also note that with m = 0 in equation 6.11,
the linear damage rule of Equation 6.10 is obtained.
Integrating Equation 6.11 using the conditions that ω=0 at t=0, and ω=1 at t=tf, where
tf is the failure time yields:
γ −n
tf = (6.12)
C (1+ m )
The instantaneous damage state, , can be then derived to be:
1
⎡ t ⎤ 1+m
ω (t )= 1 − ⎢1 − ⎥
⎢⎣ t f ⎥⎦ (6.13)
The effect of the current state of damaged material can be incorporated into the constitu
tive law of the material using an expression such as
U = (1 − ω (t ))U 0
(6.14)
where U0 is the undamaged strain energy function. This creates a softening of the mate
rial as damage progresses.
The following is an example to demonstrate the potential utility of this technique to
predict the number of cycles to failure of a 2 ply composite laminate. The increase of the
damage parameter ω was monitored throughout the analysis, and failure is predicted when
ω reaches one in one of the elements. Note that this is a simple definition of failure, and
it is inherently mesh size dependent, but it is used here for simple illustration. Localized
softening can actually drive the damage parameter towards one at a rate depending on the
size of the mesh [e.g.,78].
180 Chapter 4. Mechanics of Cord-Rubber Composite Materials
Figure 6.17 shows the finite element model used for the calculations. Membrane
elements with rebars were used to represent the cord-rubber layers. The location and the
magnitude of the damage parameter at different loading conditions is indicated.
Figure 6.17 - Evolution of the damage as predicted
by FE implementation of the CDM model
Temperature prediction
Since viscoelastic and fatigue properties of cord-rubber composites depend on temperature,
and elevated temperatures can accelerate aging processes, the distribution of temperature
within a specimen or structure should be determined for a more accurate durability assess
ment. The temperature prediction is accomplished by a method similar to [34] where three
modules: 1) an elastic structural solution, 2) a viscoelastic calculation of energy dissipation,
and 3) a heat transfer model for temperature prediction, are linked together. The FE geome
try shown in figure 6.17 is also used for this example. The Mooney-Rivlin law is used for the
rubber elements with slight compressibility
(6.14)
The nonlinear elastic behavior of the reinforcing cords is captured by using the
Hypoelastic option in ABAQUS where the rate of change of stress is defined as a tangent
modulus matrix multiplying the rate of change of the elastic strain:
(6.15)
where dσ is the rate of change of the Cauchy (true) stress dε, is the rate of change of
the elastic strain, and Del is the tangent elasticity matrix with entries provided by giving
the Young’s modulus E, and Poisson’s ratio ν of the cord as functions of the strain invari
ants. Rebar elements embedded in membrane elements are used to represent the cord-rub
ber layers.
To simulate the cyclic effect of the loading conditions, the sample was statically
subjected to the maximum and minimum loads in the cycle. The cyclic load, coupled with
Chapter 4. Mechanics of Cord-Rubber Composite Materials 181
The volumetric heat flux is calculated from the following approximate relationship
(6.16)
where ∆H is the heat flux in (BTU/hr.in3), V is the volume of the element, f is the frequency
or the number of deformation cycles per second, and tan δ is the loss tangent for the material.
The effective orthotropic conduction in a membrane is calculated using the rule of mix
tures with thermal conductivity of rubber kr=0.014 (BTU/ hr.in.oF), and kc =2.016
(BTU/hr.in.oF) for the cords. The heat transfer film coefficient used is relatively small
because of the low speed of the test that causes a pseudo laminar flow around the sample. A
value of h=0.055 (BTU/ hr.in2.oF) is used, which is slightly larger than free convection.
By subjecting the structural model to a sinusoidal load range, a temperature distribution is
developed throughout the sample. In this case, the concentration of the stresses and strains at
the free edges creates a large strain energy density and consequently the volumetric heat flux
at the same location. The temperature is also maximum near the free edges as shown in
figure 6.18.
Figure 6.18: Predicted temperature distribution in a 2-ply cord-rubber composite
Experimental infrared (IR) camera measurements of the 2-ply composite specimen are
shown in figure 6.19, which depicts the temperature at the edge of the sample during the
early stage of a fatigue test (a) and the latter stage of a fatigue test (b). The average
measured temperature around a cord in the early stage of damage evolution where
socketing is the predominant mode of failure is on the order of 64.5°C or 135.2°F. This
temperature agrees well with the temperature calculated (129 - 130 °F) by the finite ele
ment analysis. In the later stage of the failure mode where delamination is more dominant,
182 Chapter 4. Mechanics of Cord-Rubber Composite Materials
the experimental data shows a shift in the location of the maximum temperature and a
reduction of its magnitude (56.5°C or 122.4°F). The maximum temperature is lower on
the surface, since socketing and delamination have caused the region of high stress/strain
concentration to occur in the interior of the specimen, while the IR camera is seeing the
surface where the material has become largely unloaded.
Figure 6.19 - Thermal images (°C) of the edge of a 2-ply composite at an early
stage of damage (a), and a later stage (b).
(a) (b)
Acknowledgments
The authors would like to thank Drs. Ming Du and Yao-Min Huang for providing figures
and data for the composite testing and analysis.
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186 Chapter 5.Tire Load Capacity
Chapter 5
by Stephen M. Padula
Chapter 5
Tire Load Capacity
by Stephen M. Padula
1. Introduction
A primary responsibility of The Tire and Rim Association, Inc. (TRA) for over 100 years
has been the establishment of interchangeability standards for tires. This chapter presents
the evolution of the TRA load formula for passenger car tires from the early years to its
current application. The load formula is then compared to one based upon constant rela
tive deflection. This alternative method provides a possible methodology for future research
and review. Although the focus of this work is passenger tires, the same model and
methodology can be used for other pneumatic tires including light truck tires.
TRA was founded in 1903 to establish and promulgate interchangeability standards for
tires, rims, valves and allied parts. These standards include tire loads and dimensions, and
rim contour and valve dimensions. The formulae used to calculate tire loads are empiri
cally based, are derived from information and field experience from member companies,
and are fundamentally similar in format for all types of tires, except aircraft. In the interest
of brevity, only the formula for calculating passenger car tire loads will be analyzed. The
load limits calculated and shown in the TRA publications are considered maximum for the
pressure shown and, conversely, the pressures shown are considered minimum for the corre
sponding loads shown. Higher pressures for high speed and other special circumstances are
often recommended by vehicle and/or tire manufacturers and are acceptable as long as
they do not exceed the maximum pressure marked on the passenger car tire. TRA and
other standardizing bodies provide guidelines for adjusting the tire load/pressure relation
ship as a function of speed.
The intent of the formula from a standards perspective is to:
a) Determine a load rating that allows a tire manufacturer to design and produce a tire
that can perform satisfactorily to the tire manufacturer’s individual design
requirements and still be interchangeable with the same tire size produced by other
manufacturers;
b) Provide rational increments of load carrying capacities over the range of tire sizes of a
given type and series;
c) Take into account the requirements of the vehicle manufacturer and service conditions.
A load capacity formula ideally should be:
a) Scientifically based and contain the physics that allow accurate predictions;
b) Calculated from obvious, easily measured tire characteristics.
For lesser loads the ratings were determined by taking a direct proportion of the maximum
inflation pressure as illustrated by the formula below:
L=L0 (P/P0) (1)
In 1928 tire load ratings were adjusted to take into account the rim diameter, but this again
was done by agreement and not by calculation.
The first formula adopted by TRA was developed in the mid 1930’s by C. G. Hoover,
a mathematician who later served as the staff director of TRA. The formula was
empirically derived, based on the effects of inflation pressure, tire section width (or
section diameter as it was called for the early circular section tires) and rim diameter. It is
thought that this formula was based on maintaining a uniform degree of deflection in the
tires at their assigned loads and inflation pressures. The formula for what were called
“Low Pressure” passenger car tires was:
L=6.65xP0.585xS1.702x[(DR+S)/(19+S)] (2)
where:
L = tire load carrying capacity at pressure P
P = tire inflation pressure
S = tire section width (on rim width = 62.5% of tire section width 1
DR = nominal rim diameter
The structure of this formula, with input for tire dimensions in the form of section width S
and rim diameter DR, and inflation pressure P, has essentially not changed since its inception
for most ground vehicle tires. However, it has been updated by adjustments to the coefficient
K and the exponents for P and S to accommodate changing tire service requirements and tire
sizing as well as a change from U.S. customary system of units to S.I. units.
3. Basic formula
As stated above, the origins of the load formula are not well documented. However, based
on the available information, Hoover related the tire load carrying capacity to the tire
volume in developing the formula. Assuming this to be the case, Schuring, in an unpub
lished paper written in 1985, provided the following plausible speculation of how and why
the various terms were selected:
b) The new relationship indicates that for each tire design, a constant value exists for
the ratio L / Pn = L0 / P0n.
c) The tire load carrying capacity was assumed to be directly proportional to the air
volume V. Thus, L/Pn would depend linearly on air volume V:
L/Pn = const. V.
As the cross-section of a tire was approximately circular in the 1930’s, the volume V was
given by
V = const. S 2 (DR + S),
where DR is rim diameter, and S is section diameter. Combining the above two equa
tions yields:
L = const. Pn S 2 (DR + S).
However the volume of a circular annulus is not exactly proportional to S2. Thus it
must have soon become apparent that a load formula based on this approximation would
produce overloads for larger tires. Hence, the exponent of 2 for S was reduced to 1.39 –
probably based on field experience of tire performance at that time – so that the basic tire
load formula became:
L = const. Pn S 1.39 (DR + S).
4. Constant
The “constant” is a very significant part of the load formula. It is referred to as the “K”
or “service” factor. Different values are assigned for different types of tire (passenger
car, truck and bus, agricultural, etc.) and for different service applications. The addition
al numerical factor of 0.425 in equation (3) was included simply to keep the value of “K”
at approximately 1.00.
In the first application of the formula, K was made equal to 1.00 for truck and bus tires,
and to 1.10 for passenger car tires. When P-Type (metric) tires were introduced in the mid
1970s, conversion factors to allow input of millimeters and kilopascals were included in
the formula and the factor of 0.425 was incorporated into the new value of K.
5. Pressure exponent
The exponent of inflation pressure gives the tire load over the range of pressures
standardized for a specific tire. The reason for selecting an exponent of 0.585 was that as
the inflation pressure increases in a given tire size, the stress in the carcass and tread area
will increase. Consequently introducing a positive exponent for P will tend to limit the
deflection of the tire, and therefore the stress, as the inflation pressure increases. A value
of 0.585 was adopted for passenger car tires in the 1950s and 1960s (alpha-numeric series)
and a value of 0.50 was adopted for P-Type tires in the 1970s based on service
considerations and tire performance.
When P-Type tires were developed, tires were selected for a vehicle based on two
criteria [both given in a TRA guideline and as part of the National Highway Traffic
Administration (NHTSA) regulations]. The “normal” operating vehicle load must be
equal to or less than a Design Load, and the maximum vehicle load had to be equal to or
less than a Tire Maximum Load. Passenger car tires existing at that time used the
190 Chapter 5.Tire Load Capacity
alpha-numeric system for Standard Load (Load Range B) tires with a maximum inflation
pressure of 32 psi and a Design Load based on a lower pressure of 26 psi. The TRA
committee that developed the P-Type tires thought that higher inflation pressures should
be required for Design Load applications as well as for maximum load conditions. The
Design Load pressure for P-Type tires was therefore increased to 28 psi and the pressure
associated with the maximum load was set at 35 psi. An exponent of 0.5 for pressure kept
the Design Load at 28 psi for P-Type tires equal to that for alpha-numeric tires at 26 psi.
Similarly, the maximum load for P Type tires was only slightly increased above that for
alpha-numeric tires of the same dimensions, but at 35 psi instead of 32 psi. No differen
tiation was made between bias and radial tires other than in the tire designation: symbols
D, B and R were introduced for diagonal (bias), belted bias, and radial tires, respectively.
departure occurred from the “round” cross section (Aspect Ratio = 1) upon which the
formula was originally based. Even larger rim widths and section widths are under
consideration. A review of the load formula is therefore necessary, especially as new rules
are introduced in connection with Tire Pressure Monitoring Systems (TPMS). Under the
proposed TPMS rules, tires could be allowed to operate at reduced pressures (thus at
increased deflection) for extended periods of time before a warning light would be
activated.
For decades, the basis for the maximum load was to first consider the ‘normal load’
from the regulations and then select a pressure coefficient that had the effect of limiting
the deflection at maximum load. It is believed the intent of the ‘normal’ load as defined
by the federal regulations (FMVSS110) was to take into account how vehicles are
‘normally’ used in service, that is, at less than full occupancy and not fully loaded.
Moreover, as defined by this same regulation, the normal and maximum load on the tire
is based upon a passenger weight of 68kg (150 lbs.). In contrast, certain FAA regulations
use 170 lbs. for occupant weight and one might argue that even this number is rather low.
What basis should be used today for weight of occupants in vehicle load calculations?
8. A new approach
Over the decades, much work has been done in the tire industry and in the academic world
to study the effect on tire performance of stresses, temperatures, load distribution and
many other factors. Much of the fundamental work, that still applies, was described in the
NHTSA-funded monograph, “Mechanics of Pneumatic Tires”, edited by S. K. Clark and
published in the early 1980s. However, it is now an opportune time to explore changes
that might be made to the tire load formula, for the following reasons:
Tire sizes continue to evolve towards larger rim diameters and wider section widths.
The historic differentiation between passenger cars and ‘light truck’ has changed with
the wide usage of passenger car tires on SUVs. This fact has been recognized by NHTSA
with the recent promulgation of FMVSS139 which essentially combines the regulations
for these two types of tires for use on vehicles with gross vehicle weight ratings (GVWR)
of 10,000 pounds or less.
There is a need for a formula that is more closely related to the physical characteristics
of today’s tires.
The following section develops a methodology and presents relevant data for this pur
pose. An equation is developed, based upon constant relative deflection, and compared to
the current formula. Guidance is given as to further verification of the result, with sug
gestions for future work.
2 V.E. Gough, "Chapter 4, Structure of the Pneumatic Tire", Mechanics of Pneumatic Tires, S.K. Clark, Editor,
3Koutny, F., "A Method for Computing the Radial Deformation Characteristics of Belted Tires," Tire Science
in the industry generally concur that the stiffness of pneumatic tires is controlled primari
ly by the inflation pressure and the tire dimensions, principally the tread width and out
side diameter. The tire structure itself only accounts for about 10 to 15% of the tire’s load
carrying capacity at typical operating pressures.
In a more recent work, Rhyne5 extended the ring model used by Koutny and applied it
directly to the question of vertical stiffness. We can adopt Rhyne’s stiffness model to cal
culate the deflection for any given tire size and operating pressure and thus develop a new
load formula.
First, one must look at typical load-deflection curves at various operational inflation
pressures, as shown in Figure 1. The tire stiffness at a given pressure is derived from the
slope (the tangent vertical stiffness) of the individual curves, which appear to be quite
linear over normal ranges of operating load. As tires become lower in aspect ratio and
develop a square footprint, the value of the tangent stiffness approaches that of the secant
stiffness.6
Figure 5.1
Secondly, as a result of Rhyne’s work7, we know that the tangent stiffness, KZ, is a
function of tire pressure, footprint width and outside diameter and may be expressed as
follows:
(4)
where:
OD = Outside diameter - mm
4Koutny, F., "Air Volume Energy Method in Theory of Tyres," Plasty a Kaucuk, 15, No. 4, 1978, pp. 100-105,
(English translation in International Polymer Science and Technology, Vol. 5, No. 8, 1978).
5Rhyne, T.B., "Development of a Vertical Stiffness Relationship for Belted Radial Tires," Presented at the 23rd
For initial verification, the predicted KZ was calculated for 50 different tires of 34 sizes
for which tangential stiffness had been measured at various loads and inflations pressures.
Figure 5.2 shows the very strong correlation between the measured and predicted values8.
In order to make the stiffness equation more practical for use by standardizing bodies
it is desirable to have the equation written in terms of parameters that these bodies
normally deal with. First, a large sample of tires was evaluated to determine the
relationship between footprint width and nominal section width. The following
relationship results:
(5)
where:
W = Footprint width - mm
AR = Aspect Ratio
SN = Nominal Section Width – mm
a = Factor from Table 5.2
Thus the above value for W may be introduced into equation (4) as follows:
The effect of equation (5) is that as the Aspect Ratio decreases, the factor ‘a’ by which the
nominal section width, SN, is multiplied, increases, as shown in Table 5.2 below. Thus,
for lower aspect ratio tires, the footprint width as a percentage of the width of the tire sec
tion, is inversely proportional to the aspect ratio.
9Availableby subscription from: The Tire and Rim Association, Inc., 175 Montrose West Ave., Suite 150,
(7)
where:
H = Design Section Height - mm
DR = Rim Diam Code – mm
and
(8)
where:
SN = Nominal Section Width – mm
AR = Aspect Ratio
Thus OD may be expressed in terms of section width, aspect ratio and rim code as follows:
(9)
and equation (6) may be expressed in terms useful to the engineer and standardizing
bodies as shown below:
(10)
Figure 5.3
196 Chapter 5.Tire Load Capacity
11. Methodology
Tire designers and vehicle engineers generally speak in terms of tire load and inflation
pressure. In fact, what they are really concerned with is the deflection of the tire at a given
load and pressure. As the deflection increases, the tire is strained more severely and there
fore more heat is generated. Consequently the operating temperature increases. The
energy expended in rolling also increases. Thus any review of load capacity should
consider the corresponding deflection.
Figure 5.4 shows a schematic of a tire mounted on a rim at a given inflation pressure.
As a load is applied, the tire will deflect by the amount “d”.
Using equation (10) for tangential stiffness, an equivalent static deflection may be
calculated for any combination of load and pressure in the linear range of operation, for
any given size tire:
(11)
where:
d = deflection – mm
L = load – kg
KZ = tangential stiffness – kg/mm
The deflection under load is considered to be the main determinant of tire durability
and for different types of tire the relative deflection is the appropriate measure. Thus, to
compare a variety of tire diameters, aspect ratios and rim diameters, it is desirable to
express the deflection as a percent of the section height (SH) as defined in Figure 5.4.
Thus:
(12)
where:
FD = rim flange diameter – mm
SH = section height (above rim flange) – mm
OD = outside diameter - mm
Figure 5.4
Using equations (10) through (12), deflections for the entire range of tire sizes may
easily be calculated.
(13)
Combining equations (10) and (13) results in a deflection-based general equation for
load for any size of passenger car tire.
(14)
In order to calculate maximum loads, the pressure may be assumed to be 240 kPa13.
Using a coefficient chosen from a table of coefficients in terms of aspect ratio, the percent
deflection can be calculated for comparison with the current formula.
Solving equation (14) for a range of standard load tires with an aspect ratio of 70 we
can select a percent deflection and generate loads for comparison with the existing TRA
formula as shown in Figure 5.11.
Figure 5.11: % calculated load1/TRA load (70 aspect ratio)
The penalizing effect of increased rim codes with the current formula is again clearly seen.
13Reference pressure associated with maximum load of today's standard load P-metric tires
Chapter 5.Tire Load Capacity 201
15. Summary
The basic load formula for tires has been in existence for many decades and should be
reviewed to bring it into accord with today’s requirements. The methodology presented
here can be used to evaluate the maximum loads that are currently accepted. Test pro
grams could determine optimum deflections and provide guidelines for future tire load
calculations.
Acknowledgements
The author wishes to thank all those who contributed directly and indirectly to this work.
Special thanks go to the following:
Mr. Joe Pacuit of the Tire and Rim Association, Inc. (TRA) who gathered and summa
rized documents from TRA archives, including previous work by Louis F. Michelson
(Goodyear, Retired)
Mr. Frank S. Vukan (BF Goodrich, Retired)
Dr. Dieter J. Schuring (Firestone, Retired).
Dr. Tim Rhyne (Michelin Americas Research and Development Corporation) who
initiated the proposed new method of calculating tire loads.
Appendix 1
Notes from The Tire & Rim Association, Inc.
Figure 5.1.1: Relation between section width/rim width ratio – Part 1
With tire of circular section
S1 = Section width
W1 = Rim width
α = Angle subtended by rim W1
Θ = α/2
SIN Θ = (W1/2)/(S1/2) = W1/S1
Θ = 38.67º
180 - Θ = 141.33º
202 Chapter 5.Tire Load Capacity
Let:
∴
204 Chapter 5.Tire Load Capacity
The perimeter of a tire having a depressed crown and a section S1 on a rim width W1=
(2)
Appendix 2
Secant stiffness
Load
Tangent stiffness
Deflection
206 Chapter 6. Tire Stress Analysis
Chapter 6
by M. J. Trinko
Chapter 6
Tire Stress Analysis
by M. J. Trinko
1. Introduction
The pneumatic tire presents a challenging stress analysis problem. It is a pressurized mem
brane structure of revolution reinforced by a network of cords laid at an angle to the circum
ferential direction. The cords are folded around a rigid hoop of steel wires, which anchors the
tire to the rim.
The bias-ply pneumatic tire was in commercial use for decades before engineers were able
to solve any but the most basic problems. Calculations of the initial inflated shape were devel
oped to assist the tire designer determine fundamental relations between the shape of the mold
and width of the building drum, but they required substantial manual calculation. A by-prod
uct of the shape calculation was the prediction of cord force distribution in the mounted and
inflated tire. Virtually nothing was known about the cord loads and rubber stresses in the
loaded tire. In the early 1970s, the combination of the development of the finite element
method and the availability of reasonably high-speed computers made it possible to analyze
the pneumatic tire in more detail. Initially, only axisymmetrical analysis was possible, but
within a few years solutions were obtained for the statically loaded tire. More advanced prob
lems were tackled using the static solution as a starting point. Several years passed before
usable solutions for the rolling tire problem appeared.
At about the same time that the computer-based tools became available to study the bias-
ply tire, the shift to radial tires had begun in the U.S. Application of the new analysis tech
niques to radial tires became critical as the shape of the mold and the location of components
in the cross section of the radial tire are much more important than those same quantities in
a bias-ply construction. The transition from bias analysis to radial analysis was relatively
smooth as the input information for the structural analysis programs only needs to describe
the dimensions, material properties and loadings. In a way, the analysis of the radial construc
tion is less complicated, as the radial cords do not change orientations significantly as the tire
deforms. The cord angles in the more flexible bias tire are directly related to the distance from
the axis of rotation when the tire is under inflation loading. As the tire is loaded, the cords
behave like a netting structure and change their orientations in response to the relative loads
in the circumferential and meridional direction. At the time, the commercially-available finite
element codes were not able to take these angle changes into account. Radial tires do not
exhibit this behavior, so problems introduced because of orientation angle changes were
small or zero.
The discussion given in this section is based on a combination of analysis and application
of the basic principles of mechanics. The results of finite element analyses (FEA) provide
insight into the behavior of the tire under various loading conditions, that would otherwise be
difficult to understand. By evaluating the results, we can determine the appropriate design
parameters that affect the behavior.
208 Chapter 6. Tire Stress Analysis
By applying equilibrium conditions in the axial and hoop directions, along with a
moment balance, Purdy derived a single integral equation for the natural shape, in the
form of a hyper-elliptic integral. Details of this calculation are given in “Mathematics
Underlying the Design of Pneumatic Tires”1. Note that Purdy’s solution is valid for any
cord path. However, for each combination of cord angle, drum width and diameter, a
numerical integration was required, involving extensive calculations that were initially
done manually, using only a slide rule.
Another approach to the solution of this problem was developed by Bidermann,
Hofferberth and Walter using a combination of “thin shell” theory and netting theory.
Their contribution is included in the approach described here. These equations describe
various features of tire behavior. The first uses the relation between cord angle and radius
that was described previously. The second is the relation between in-plane stress result
ants and cord angle. By calculating the values of the circumferential (Nθ) and meridion
al ( Nφ ) stress resultants for a network of cords with cord tension t and at an angle α, typ
ically measured from the circumferential direction, it follows that
tan2 α = Nφ / Nθ .
This can also be derived by considering the equilibrium of a free body diagram of a sec
tion of a doubly-curved membrane.
The third equation is derived from the equilibrium of a section of the tire lying between
the radius of the widest section ( ρm ) of the tire and the centerline radius. The force bal
ance in the axial direction includes the pressure term and the meridional stress resultant.
Nφ = p θ ( r2 - rm2) / 2 r 2
The fourth equation is the equilibrium equation for a doubly-curved surface. It is an exten
sion of the equilibrium equation for a circular cylinder, which is derived in introductory
210 Chapter 6. Tire Stress Analysis
statics textbooks. For a cylinder of radius R and thickness t, under internal pressure p, the
hoop stress σ is given by
σ=pR/t
This equation can be transformed into an integral equation for z by reducing the order to
z′, decomposing using partial fractions and applying the boundary condition at the center-
line. The integral equation can then be solved numerically, yielding the inflated shape of
the cross-section.
This is the ‘natural’ shape of the tire that it would take if it behaved as a membrane with
no resistance to bending. If the mold shape deviates from the natural shape, the inflated
cross section will not attain a true natural shape because of resisting forces generated in
the rubber.
ear fashion. These materials are also temperature and rate dependent, exhibit creep and
have different stiffnesses in tension and compression. Two common cord materials, nylon
and polyester, undergo shrinkage during the curing process. Finally, the cord reinforce
ment, the primary contributor to the tire stiffness, is placed at bias angles giving the struc
ture orthotropic characteristics. Moreover, the cords change their orientation somewhat
during loading. All of these features contribute to the tire analysis problem and must be
addressed in order to obtain useful results.
side of the vertical diameter must be set equal to the vertical displacement of the corre
sponding node on the other side of the vertical diameter and the other two components of
the displacement on one side of the vertical diameter must be set equal and opposite of
those on the corresponding point on the other side. A problem with bandwidth arises
because these constraints connect nodes, which would not normally be connected geomet
rically. This results in a few rows in the matrix that are substantially wider than the aver
age. The wider rows cause symmetry considerations in tire problems to give less compu
tational benefit that in conventional problems.
For a typical linear static analysis, the stiffness matrix must be solved once to obtain the
final solution. But tire analysis is not typical. There are a number of non-linearities in the
analysis, which require a more advanced approach. The dominant problem is that the final
loaded shape is substantially different from the initial shape. This means that the stiffness of
the structure, the loading locations and directions all change as the structure responds to the
external loads and constraints. To successfully solve this problem, the loading is typically
applied in increments which yield a stable equilibrium configuration at the end of each incre
ment, that serves as the starting point for the next increment. The initial stiffness of the
unloaded structure is used as a starting point. The first increment of load is applied and the
system is solved. The process is repeated using the same load until there is no change in the
response from successive iterations. The stiffness is re-calculated using this new configura
tion, the next increment of load is applied, and the next solution is obtained. This process is
repeated until the loading sequence is completed. The size of the allowable load steps depends
on the severity of the non-linearity at that loading level. If there are other types of non-lin
earity in the problem, they can usually be addressed concurrently with the large deformation
non-linearity using the incremental loading approach. Since this method requires the creation
and solution of a matrix, whose size is determined by the number of degrees of freedom, the
required run time and storage requirement grow at least in proportion to the square of the
number of degrees of freedom.
Mounting
As of this writing, no publications have reported a successful simulation of the actual
214 Chapter 6. Tire Stress Analysis
process of mounting a tire on a wheel. The problem is neither axisymmetric nor static. It
resembles a buttonholing action where the bead slides over the safety hump at one point
on the circumference and then continues the seating process around the circumference
until the whole bead is seated against the flange. This is a dynamic process, which requires
using a dynamic solution procedure along with simulation of the frictional resistance at
the bead/rim interface. The explicit finite element approach would appear to be the appro
priate choice for solving this problem. A successful analysis would give an increased
understanding of the mounting process and predict the inflation pressure necessary to
fully seat the tire bead on the rim.
Rolling resistance
Among other performance requirements, automobile manufacturers stipulate that tires do
not exceed certain levels of rolling resistance, as this quantity has a significant effect on
fuel consumption. Rolling resistance is principally the total effect of the hysteretic losses
that occur as a tire rolls. Using the results of a static deflection analysis, the engineer can
extract stress and strain values for every element in the model. By assuming that the stat
ic stresses and strains are the same in a slow rolling tire, we can calculate a ‘cycle’ repre
senting a complete revolution by examining the stresses and strains in a ‘ring’ of elements.
Each ring is the set of elements that are traced around the circumference by a single ele
ment. These results can be combined with material energy loss data to predict the energy
loss in each ring during one complete revolution. Combining this result for all rings in the
model yields a total loss for the tire for one revolution. The tire engineer can use these
results to modify the design. Options include choosing less hysteretic materials in critical
regions or adjusting the location of components within the tire cross-section to reduce the
energy loss.
Chapter 6. Tire Stress Analysis 215
Modal analysis
Automobile manufacturers also require information describing the vibration characteris
tics of tires so that the tire and vehicle system can be ‘tuned’ to provide the best possible
ride. To meet this requirement, a model that represents the inflated and deflected tire is
required as a starting point. The model must provide the natural vibration frequencies and
the corresponding mode shapes for frequencies up to 200 Hz. Since the extraction of
modal information over the range of interest requires substantial computing resources and
the calculation of internal quantities such as local stresses and strains are not of direct
interest, a simplified model made up of layered elements is the usual choice. With this
simplification, we are able to obtain the required results quickly and accurately.
The typical modal tire model uses shell elements in which all of the reinforcing mate
rial layers and rubber layers are ‘smeared’ into a single layer. In the tread area, an addi
tional layer of 8-noded bricks is added to represent the rubber tread. This layer of rubber
could include circumferential grooves if desired. The contact patch points are fixed to the
road while the rim may be either fixed or allowed to move as a rigid body. The tire is
mounted on the rim, inflated, and deflected against the road surface. The mode shapes
and natural frequencies are extracted using the stiffness matrix from the static deflection
loading. A description of this method was given by Sundaram et al2.
which comes with this approach, does not come without some penalties. The model must
be precisely axisymmetric. For tires with smooth treads or straight circumferential
grooves, such as aircraft tires and some farm and racing tires, this provides a workable
approach for analysis. Estimates of force and moment quantities, contact patch pressures
and traction forces are obtained. However, although tires with conventional tread patterns
may be analyzed this way, details of the tread pattern can not be represented exactly.
The third approach is to use the explicit finite element method described previously.
This method lends itself to the calculation of global force and moment quantities generat
ed by a rolling tire, and the response of a rolling tire to an impact with a road irregulari
ty. Koishi, Kabe, and Shiratori 6 reported results for a passenger tire operating at various
slip angles. The results of lateral force vs. time were shown for slip angles of 0º, 1º, 2º and
3º. In these analyses, ‘time’ refers to the time for the dynamic event and not the computer
time needed to complete the analysis.
The results exhibit two characteristics. The first is an oscillation from step to step of
0.6 kN for all the loading conditions. For comparison, the measured lateral forces for the
four conditions were approximately 1, 5, 9 and 13 kN. The second feature is that the aver
age result for a sequence of steps also shows an oscillation of approximately 1 kN. These
oscillations do not negate the validity of the solutions but the analyst must take them into
account in assessing the precision of the results.
Kamolaukus and Kao 7 analysed the problem of a rolling tire impacting a full width
cleat on a rotating test wheel using the explicit method. Results showed the vertical and
tangential reaction force traces immediately after the tire strikes the cleat, along with the
frequency content of the vibrations following impact. During the static inflation and load
ing step, they carefully applied nodal damping and observed the results as the structure
reached static equilibrium. To assure that all of the kinetic energy had been dissipated,
they allowed the process to continue through a number of additional time iterations. The
tire was then rotated against the load wheel until equilibrium was reached at the desired
speed. The steady-state dynamic equilibrium was evaluated for stability. Finally, the cleat
effect was introduced, causing horizontal and vertical reaction forces and the harmonic
response. To demonstrate the stability of the solution, they allowed the analysis to contin
ue until the bump had made a total of six passes through the contact region. The predict
ed vertical and lateral peaks were larger than those measured experimentally, but the fre
quency content and shape of the response curves were similar to experimental curves. The
authors suggested a number of causes for the discrepancy. This analysis was the first
attempt at a difficult problem, and the objective was to show that the explicit approach
could be applied successfully.
These examples show the types of problems that can be addressed with finite element
analysis. Other applications could be included. The list will expand as FE programs add
capabilities, and as faster and more powerful computers become available.
At equilibrium, the effect of air pressure acting in the axial direction is balanced by the
meridional force at the tire equator:
2 π ρ0 Nφ = p π (ρ02 – ρm2).
This equation yields Nφ in terms of known quantities, where Nφ is the sum of the stress
resultants in the belt, Nφb , and body ply, Nφp:
Nφ = Nφb + Nφp
Chapter 6. Tire Stress Analysis 219
Nθ = ρθ (p + Nφ / ρφ )
yields the hoop stress resultant, Nθ. The value of the curvatures ρφ and ρθ can be taken
directly from the tire mold or from an actual measurement on the tire. The circumferen
tial curvature ρθ is the radius to the belt package and is equivalent to ρ0. Note that only
the belt cord tension contributes to the circumferential stress resultant Nθ as there is no
circumferential contribution from the radially-directed ply cords.
Using the value for Nθ, the belt cord tension tb can be calculated from
tb = Nθ / ( 2 nb cos2 α )
or in terms of initially-known quantities,
tb = p ρθ [ 1 – (ρ02 – ρm2) / ( 2 ρφ ρ0 ) ] / ( 2 nb cos2 α ).
Note that as ρφ becomes larger, i.e. as the belt becomes flatter, tb approaches the value
obtained using that for a simple cylindrical hoop. To obtain the ply cord tension, we apply
the netting relation to the belt cords to calculate that portion of Nφ , denoted Nφb , that is
contributed by the belt cords:
Nφb = Nθ tan2 α
We now calculate that portion of Nφ which comes from the body ply cord tensions Nφp ,
by subtracting the portion that comes from the belt from Nφ. This gives
N φ p = Nφ - N φ b
With Nφp, we can calculate the belt and ply cord tension using
tp = Nφp / (np nl)
where np is the number of ply layers. Substituting the value of Nφp , the ply cord tension
in terms of the given quantities is
This calculation, using the additional information of the meridional curvature and sidewall
shape, gives an improved estimate of the belt cord tension compared to that assuming a
simple cylindrical shape and also gives an estimate for the centerline body ply cord ten
sion under inflation loading.
at the belt edge is reduced. If the force is small, as with a more curved sidewall, the stress
concentration is increased. On the other hand, the handling performance typically
improves with a flatter sidewall. Achieving the proper balance is a major challenge for
the tire designer.
Bead tension
The bead tension is a result of the interference fit between the tire bead and the rim and
the tension in the cords that are anchored around the bead due to the inflation pressure.
The interference is required to retain the tire firmly on the rim so that the tire does not
rotate relative the to wheel under driving and braking maneuvers nor unseat from the rim
under extreme cornering maneuvers. The load in the bead due to the interference fit may
be estimated by combining known information (bead bundle diameter, rim diameter,
thickness of material underneath bead, etc.). For the radial tire, the load due to the cord
tension is estimated by the equation
Bt = ts ρb cos φ
where ts is the sidewall cord tension, ρb is the radius of the bead, and φ is the angle of
approach of the cords to the bead. However, care must be used when combining these two
results as the interference load is reduced as the inflation load is increased. It is important
that the interference between the bead and the rim is sufficient to hold the bead firmly
under all of the conditions mentioned above. There appears to be no published data on
bead bundle cord tensions.
This equation shows that the total belt tension can be increased by increasing the total
width of the tire and/or by increasing the area under the belt but above the line through the
widest point on the tire cross section. It also shows that the shape of the sidewall from the
widest point on the cross section to the bead area has no influence on the total belt tension.
5. Related topics
Stored energy in an inflated tire
The work that is expended in forcing air into a tire is stored as potential energy in the com
pressed air in the tire cavity. The amount of energy stored is substantial and a sudden
release of this energy due to a catastrophic failure of the carcass can be damaging. In the
interest of safety, it is important that any person working in the vicinity of tires be aware
of this danger and take appropriate precautions to avoid injury.
Tire manufacturers routinely test their product for ultimate pressure to assure the car
cass integrity under normal operating loads. The burst pressure is usually 6 to 10 times
greater than the recommended operating pressure. To assure the safety of laboratory per
sonnel, water is used to fill and pressurize the tire, because water is relatively incompress
ible and when the tire bursts, only very little stored energy is released, and the carcass
remains relatively intact, except in the immediate area surrounding the failure. If air were
to be used as the inflation medium instead, sudden release of the much greater amount of
stored energy would cause substantially more damage to the tire and the surrounding area.
The equation for calculating the stored energy W in the compressed air within a tire is
given by
W = ( p2V2 - p1V1 ) / ( 1 - k )
where k is the gas constant for air. The pressure is absolute pressure. State 1 refers to the
compressed pressure and volume and state 2 refers to the free air pressure and volume.
Table 6.1 lists the amount of energy stored in different types of tire.
222 Chapter 6. Tire Stress Analysis
Load transfer
Many structures carry their intended load by creating tensile or compressive forces with
in their volume. These structures are stress free in the unloaded state and become stressed
in the loaded state. Tires, on the other hand, actually transfer a portion of their intended
load by relieving the pre-stress caused by inflation pressure. At the interface between side
wall and belt, the contact patch flattens the belt package and displaces it towards the rim.
This causes the sidewalls to bulge outward, reducing the radius of curvature in the side
wall, which results in a reduced cord tension. For an inflated tire under no load, the side
wall tire cords are anchored to the beads and they pull the beads radially outward. When
the tire is loaded at the contact patch, the reduced tension in the sidewall cords in the con
tact region reduces the loading by the cords on the bead, causing the bead to be pushed
against the rim. This increased load on the rim is the end of the load path from the road to
the rim.
Table 6.1
Type of tire Inflation Stored energy Height that a bowling ball
pressure (ft-lbs) would be thrown (feet)
Automobile 35 10000 625
Automobile 250 68000 4250
Truck 105 80000 5000
Farm tractor 45 136000 8500
F-4J jet fighter 480 394000 24625
Eathmover 75 597000 37313
C-130 military transport 570 2889000 180563
However, the load follows a second path from the contact patch to the rim. When the
belt hoop deforms, it flattens in the contact patch area and is also displaced vertically by
a small amount. For a typical passenger tire, the section of the tread opposite the contact
patch displaces away from the rim by approximately 10% of the displacement at the con
tact patch. This motion causes two related changes, which aid in the load transfer. First,
in the area of the tire directly opposite the contact patch, the sidewall curvature increases
slightly which causes the bead to pull away from the rim slightly, in the opposite manner
to that at the contact patch. Second, in regions that are about 90o from the contact patch,
cords that were directed radially are now angled slightly upward. This helps to lift the rim
and transfer the load.
So we see that, although the load from the contact patch is applied in one portion of the
tire tread, the load path into the rim involves the complete circumference of the tire.
6. Experimental approaches
Interest in radial tire construction began in the domestic tire market in the early 1970s. At
that time, FEM was only beginning to be developed as an engineering analysis tool.
Analysis of an inflated tire was available in the mid 1970s but the capability to analyze an
inflated and deflected tire did not become available until the late 1970s. Because there
were no analytical techniques available to understand radial tire behavior in the beginning,
new experimental methods were developed to measure properties related to the perform
ance of radial (and bias-belted) tires.
Chapter 6. Tire Stress Analysis 223
Rubber strains
The level of strain in the rubber in various regions of the tire is an important quantity for
tire designers and compounders. If the strains approach the limit that the compound is
capable of withstanding, failure may occur. Key areas of interest include the outer surface
at the mid-sidewall and regions near the belt and ply endings. In addition, strains in the
inner surface and the crown area were of general interest to better understand the basic
behavior of the tire.
Because strain is a tensor quantity, it is necessary to measure strain in three directions
on the surface. Typically, three gauges are placed at 00, 450 and 900 to the radial direction,
and offset slightly around the circumferential direction. The values of maximum and min
imum elongational strain, and their directions, along with the maximum shear strain, may
be calculated from the three measurements.
One of the challenges in measuring rubber strains on the surface of a tire arises from
their large magnitude. Typical commercial strain gages are designed for use on metal,
where strains are of the order of 0.5% or less. But typical operating strains on the surface
of a tire are of the order of 20% to 50%. A number of techniques have been developed to
measure them. One, first suggested by Kern 8, is a simple thin metal strip formed into a
“U” shape (figure 6.5) with a conventional strain gage attached to the base of the “U”.
The pointed tips of the “U” are pressed into the surface of the tire. Relative movement of
the tips creates a bending deformation in the base of the “U” that can be measured by the
attached strain gage. This device typically has a gage length of 10 to 20 mm, so it is not
capable of making the local strain measurements that can be done with smaller gages. This
limitation is common to all of the gages described in this section. Figure 6.5 shows vari
ous gages used for measuring tire surface strains.
A second type of gage consists of a rubber cylinder with a fine wire wrapped in a hel
ical pattern around the diameter. This combination is attached with an adhesive to the tire
surface. The level of strain can be measured by the change in resistance of the helically
wound wire.
A third type of gage is made by molding a rubber block with a small cavity running
along the long axis of the block. The dimensions of the block are approximately 3 x 3 x
30 mm. The cavity is filled with a continuous column of liquid mercury and the device is
attached with an adhesive to the tire surface. Small lead wires are inserted into the ends
of the cavity and the small changes in resistance of the mercury column that occur when
the block is stretched or compressed can be calibrated to yield the magnitude of the
strains.
Walter and Jannsen 9 applied a number of these gages to the internal and external sur
faces of bias, bias-belted and radial tires. They examined the strains at approximately six
locations on the outer surfaces and five locations on the inner surfaces. The results
showed that the patterns of surface strains are different for each type of tire.
Cord loads
Until the finite element method was developed to the level where it could be applied to
mounted, inflated, loaded tires, it was not possible to determine cord loads. The tire indus
try had gained sufficient experience to design bias-ply tires successfully, but with the
advent of belted tires an understanding of cord tensions in these constructions became
important. Experimental methods were therefore developed to measure cord tensions
directly. Walter10,11 applied a technique using a small metal billet with holes at each end,
and with strain gages attached to the billet between the holes. Cords were threaded
through the holes in the billets. At this point, the assembly can be calibrated in a test
machine equipped with a standard load cell to determine the relation between strain gage
output and cord tension. Cords were then removed from a “green” uncured tire and
replaced by the test assembly, with the billet at the site of interest. The tire build was then
completed, with the leads from the strain gage being carefully led out of the tire carcass.
The tire was then cured in the mold and the finished tire tested in the laboratory under con
trolled conditions. Results are shown in the following figures for cord loads at the center-
line for bias and radial tire constructions.
Another type of gage was developed by Clark and Dodge12. It consisted of a small
beryllium copper cylinder, sized to slip snugly around the cord. Small strain gages were
attached to opposite sides of the cylinder, thus minimizing the effect of bending in meas
uring tensile forces. The cylinder was attached to a single cord in the “green” uncured tire
the cord at the point of interest, using an epoxy adhesive. [An earlier version was clamped
between the two ends of a cut cord.] The tire building process was then completed and
the tire cured, as with the billet type gage.
Walter and Kiminecz13 developed a small pressure transducer that could be installed in an
axial hole in the end of a standard 8/32 inch bolt, as shown in figure 6.6.
Using a 8/32 inch nut welded on the outside surface of the rim, relative to the tire cav
ity, the bolt was inserted so that the end of the transducer was flush with the inside sur
face of the rim. Measurements of rim pressure could be made in this way while the tire
was rolled under various combinations of load, pressure, camber and steer.
between the tire circumference and the distance traveled in one revolution. Under brak
ing, accelerating, camber and cornering loads, the slip patterns are more severe and more
complex (see chapter 7).
In order to study forces and displacements in the contact region, a number of test meth
ods were developed. For example, a cylindrical rod carrying strain gages, and calibrated
to measure normal and lateral forces, was mounted in the surface of a model track, with
the end flush with the track surface as shown in figure 6.7.
A tire was rolled over the track with desired steer and camber settings and also with
applied drive or a braking torques. It was rolled at least one complete rotation to allow the
shear forces to reach stable values representing continuous rolling. For a tire with a tread
pattern, multiple passes were necessary to obtain data for the complete pattern. But with
a single transducer, a substantial amount of time was required to test a tire under a single
loading condition.
Figure 6.7: Footprint pressure transducer
the tire surface. Using image recognition software, the motion of each mark was record
ed and the slip path plotted. This information was combined with data from a force trans
ducer similar to those described earlier, and the values of slip energy were calculated.
Although the glass surface does not have the same characteristics as an actual road, this
approach gives a visual record of slip behavior with no restriction on the magnitude of
slip. Contact patch and footprint phenomena are fully described in chapter 7.
The process of designing and building the special transducers required to measure the
surface and internal stresses, the tedious process of installing the transducers in tires, and
the careful handling required during testing, made testing a time-consuming and expen
sive process. In addition, a transducer was a single use item and many were required to
obtain a complete picture of the stress pattern in a complete tire. As a result, after labora
tory experiments were found to validate FEM results, tire designers tended to rely on FEM
calculations, rather than on experimental measurements .
Figure 6.8: Footprint pressure and motion transducer
228 Chapter 6. Tire Stress Analysis
7. Problems
Tire Description
Size: P235/75R15
Inflation Pressure 30 psi
Diameter 28.25 inch
Section Width 9.50 inch
Belt Cord Angle 220
End Count for Belt 18 epi (ends/inch)
Tread Rubber Thickness 0.60 inch
Belt Width 7.75 inch
Number of Body Plies 2
End count for Body Plies at Bead 28 epi
Bead Construction 5 x 5 ( 25 strands )
Rim Diameter 15 inch
Rim Width 6 inch
Tread Radius 42 inch
1. Calculate an approximation for the centerline belt cord tension in the tire described
above, using only the first six lines of information.
2. Repeat the calculation using all of the information available for a more accurate
approximation.
3. Estimate the contained air volume in the tire. Using this volume, calculate the energy
stored in the air of the inflated tire.
8. References
1. Purdy, J. F., Mathematics Underlying the Design of the Pneumatic Tire, Hiney
Printing Company, Akron, Ohio, 1963.
2. Richards,T., Sundaram, S. V., Brown, J.E., and R. L. Hohman, Modal Analysis of
Tires Relevant to Vehicle System Dynamics, International Modal Analysis Conference
III, Orlando, FL, January 28 – 31, 1985.
3. Rothert, H., and R. Gall, On the Three Dimensional Computation of Steel Belted
Radial Tires, Tire Science and Technology TSTCA, Vol. 14, No. 2, Apr-Jun, 1986.
4. Faria, L., Bass, J., Oden, J.T., and E. Becker, Three Dimensional Rolling Contact
Model for a Reinforced Rubber Tire, Tire Science and Technology TSTCA, Vol. 17, No.
3, Jul.-Sep., 1989.
5. Kennedy, R. and J. Padovan, FEA of a Steady Rotating Tire Subjected to Point Load
or Ground Contact, Tire Science and Technology TSTCA, Vol. 15, No. 4, Oct-Dec.,
1987.
6. Koishi, M., Kabe, K., and M. Shiratori, Tire Cornering Simulation Using an Explicit
Finite Element Analysis Code, Tire Science and Technology TSTCA, Vol. 26, No. 2,
Apr-Jun, 1997, pp. 109-119.
7. Kamoulakos, A. and B. G. Kao, Transient Dynamics of a Tire Rolling over Small
Obstacles — A Finite Element Approach with Pam-Shock, Tire Science and Technology
TSTCA, Vol. 22, No. 2, Apr-Jun, 1998 pp. 84-108.
8. Kern, W. F., Strain Measurements on Tires by Means of Strain Gauges, Revue
Generale du Caoutchouc, Vol 36, October 1959.
9. Janssen, M. and J. D. Walter, Stresses and Strains in Tires, Akron Rubber Group, Fall
Meeting, 1972.
10. Walter, J. D., A Tirecord Tension Transducer, Textile Research Journal, Vol 39,
February 1969.
11. Walter, J.D. and G.L. Hall, Cord Load Characteristics in Bias and Belted-Bias Tires,
Society of Automotive Engineers, 690522, 1969.
12. Clark, S. K. and R. N. Dodge, Development of a Textile Cord Load Transducer,
ORA Report 01193-1-T, The University of Michigan, May 1968.
13. Walter, J.D., and R.K. Kiminecz, Bead Contact Pressure Measurements at the Tire-
Rim Interface, Society of Automotive Engineers. 750458, 1975.
14. Howell, W., Tanner, S., & Vogler, W., Static Footprint Local Forces, Areas, and
Aspect Ratios for Three Type VII Aircraft Tires, NASA Technical Paper #2983, Feb.
1991.
15. Lazeration, J., An Investigation of the Slip of a Tire Tread, Tire Science and
Technology TSTCA, Vol. 25, No. 2, Apr-Jun, 1997, pp. 78-95.
230 Chapter 6. Tire Stress Analysis
Symbols
ρ0 , rc — Centerline radius
θ0 — Cord angle at the centerline,
measured from the circumferential direction
ρ, r — Radius to general point on carcass line
∆s — Element of length along carcass line
Nθ — Stress resultant in circumferential direction
Nφ — Stress resultant in meridional direction
α — Cord angle
ρm, rw — Radius to the widest point on tire cross section
p — Inflation pressure
R — Radius of cylinder
T — Thickness of cylinder
σ — Stress in cylinder wall
ρθ — Radius of curvature in circumferential direction
ρφ — Radius of curvature in meridional direction
z — Axial distance to point on tire carcass line
t — Cord tension
th — Cord tension in belt cord
t — Cord tension in carcass cord
nb — Number of cords / length — belt
np — Number of carcass ply layers
Bt — Bead tension
ts — Ply cord tension
φ — Carcass tangent angle
ρb — Radius of bead
B — Total load in belt package
W — Energy stored in air contained in tire
p1, p2 — Pressure at conditions 1 & 2
V1 ,V2 — Volume of air at conditions 1 & 2
k — Adiabatic gas constant
Chapter 7. Contact Patch (Footprint) Phenomena 231
Chapter 7
by M. G. Pottinger
4.1.4 Relationship of footprint stresses and tire forces and moments .............. 272
4.2.1 The solid tire as surrogate for a tread pattern ........................................ 272
4.2.2 Deviations in footprint stresses and motions due to tread pattern .......... 274
Chapter 7
Contact Patch (Footprint) Phenomena
by M. G. Pottinger
1. Introduction
The tire has three boundary regions. One is the zone of contact with the road or test sur
face and the other two are the zones of contact between the tire bead areas and the rim.
This chapter is concerned with the contact between the tire and a road or test surface.1 The
road surface is assumed to be dry and rigid, with limited texture unless otherwise noted.
The term footprint is employed in place of the commonly-used alternative: contact patch.
If a term is not defined in the text, the definition found in SAE J2047 [1]2 applies.
The study of the tire footprint is a very complex matter for several reasons.
The tire is a doubly-curved surface. It is curved both circumferentially and transversely.
A doubly-curved surface is not a developable surface and cannot be made to conform to
a flat or round surface simply by bending. Conformance requires that the tire structure be
stretched and compressed as well as bent.
The tire is a relatively soft or flexible, pneumatically pre-stressed structure whose
behavior depends on applied loads and operating conditions.
The friction of the road or test surface affects the deformation of the tire and, hence,
the tire’s footprint.3
In spite of these difficulties the obvious association between the tire footprint and other
properties of the tire, including traction, tire/pavement interaction noise, ride over road
irregularities, and wear, motivated engineers to study tire/road contact.
It is uncertain when the first studies of the tire footprint were conducted, but at least
prints of the footprint area were commonly made by the early 1930s [2]. This work was
preceded by mathematical studies of the contact of high-modulus elastic bodies from the
time of Hertz [3], with practical applications to gears, rolling bearings, and contact
between railroad wheels and rails. Certainly, contact studies related to the endurance of
roads and rutting were well underway before the dawn of the twentieth century.
In the two editions of the book, “The Mechanics of the Pneumatic Tire”, edited by
Clark, footprint literature is reviewed up to 1981 [4,5]. In this chapter the discussion is
primarily based on sources published since 1981. The earlier literature is discussed only
when it provides the single or best source of information on a given subject.
This chapter primarily covers footprint physics and experimental techniques.
Analytical techniques such as finite element analysis are mentioned as appropriate, but no
attempt is made to discuss these techniques thoroughly. In order, we discuss the terminol
ogy used for describing the tire footprint, equipment and methodologies that have been
used for studying the footprint, and footprint physics, followed by a few concluding
remarks.
3Plainly, a deformable road surface is going to affect the tire footprint. In the case of a wet surface, which is
discussed later, water can alter the effective road surface topography preventing physical contact between the
tire and road. This can completely alter the mechanics occurring in the tire footprint.
Chapter 7. Contact Patch (Footprint) Phenomena 235
tation for measuring gross tire forces and moments has a wheel based orientation. Thus,
it makes sense to discuss footprint behavior in terms of an axis system which is road sur
face fixed, but which can be readily related to the Tire Axis System.
The FAS, figure 7.2, is a right-handed, three axis, orthogonal Cartesian coordinate system
with its origin at the contact center in the road plane. The road plane is the plane tangent to
the road surface at the contact center. In experimental footprint studies it is the active plane
for footprint sensing instrumentation. The contact center is the point in the road plane where
the line defined by the wheel plane (the plane halfway between the rim flanges) to road plane
intersection is cut by the projection of the spin axis onto the road plane. Thus, the contact cen
ter is defined by the wheel, not the tire5, and is a common point for both the FAS and the SAE
tire axis system. The FAS X-axis is in the road plane and is coincident with the tire’s trajec
tory velocity. Positive X is in the direction of the tire contact center’s velocity V over the road
plane. The FAS Y-axis lies in the road plane perpendicular to the FAS X-axis. Positive Y is
to the right when the system is viewed from the rear looking in the positive X-direction. The
FAS Z-axis is defined by the cross product of + X into + Y. It is perpendicular to the road
plane with its positive sense into the road plane.
Figure 7.2: Footprint axis system (reprinted from tire science and technology,
vol. 20, no. 1, 1992 with Tire Society permission) [6]
Data in the FAS system may be transformed into data in the tire axis system or vice
versa by a simple rotational transformation about the Z or Z’ axes involving the sines and
cosines of the slip angle, α.
4The FAS is the author's suggestion to fill a gap in tire technology and has not so far been adopted by any
standards organization.
5This avoids a definition in terms of the tire center of contact, which has no fixed location.
σX+ pushes the tire forward (tending to accelerate the vehicle) and σX- pushes the tire
rearward (tending to cause deceleration). σX has both positive and negative values.
σY+ pushes the tire to the right and σY- pushes the tire to the left. σY has both posi
tive and negative values.
σZ- supports the tire - it prevents the tire from sinking into the road. Thus σZ has only
negative values.
Figure 7.3: Positioning definitions for the footprint (based on a figure from Tire
Science and Technology, Vol. 27, No. 3, 1999 with Tire Society permission) [7]
(a)
To
e
Y
He
el
(b)
Chapter 7. Contact Patch (Footprint) Phenomena 237
X
Leading Edge of Footprint
S
t1
u
t0 v
To
e
Y
He
el
6For a surface with texture, like a real road surface, the actual area of tire/road contact is smaller because the
tire's surface within the footprint cannot conform perfectly to road surface irregularities.
7The axes shown in Figure 7.5 are the SAE Tire Axes, as discussed in Chapter 8, not the FAS X and Y axes
It is also possible to adapt glass plate photography to determine simultaneously both nor
mal stress in the footprint and to image the footprint, see section 3.2.4.
Imaging of tire footprints operating at speed over flooded glass plates has long been
important in hydroplaning studies [9]. Indeed, it is possible to apply shadow Moiré to
ascertain the depth of water existing between various portions of the tread in the footprint
and the surface of the glass plate. In recent years, combination of finite element methods
with computational fluid dynamics has allowed the prediction of the contact area existing
under wet conditions [10,11] as illustrated in figure 7.6.8
8Only the right half of the footprint is shown since symmetry could be used to reduce the amount of computa
tion required.
9For a general discussion of the global forces and kinematic conditions existing during operation, see Chapter 8.
10Partial
coverage is discussed in detail in section 7.2.2.2.
11The dimensional resolution problem can be regarded as an experimental analog to the problem of mesh
The gap must be small so that the stresses derived from the measured forces are given
correctly by equation 7.1.
Figure 7.8 shows schematically the effect of a transducer installation that is not flush
with the test surface. If the transducer is installed with its top flush with the test surface,
the forces sensed are representative of those over the rest of the plane surface. However,
if it is installed with the top above the road surface, it gives readings that are erroneously
high, because it is supporting stress from a larger area than the top surface of the trans
ducer. If it is installed with its top below the road surface, then the readings are erroneous
ly low.12
The necessity for the gap, figure 7.7, has consequences with regard to cleanliness of
the test facility. Particles that become lodged in the gap can short-circuit the force path.
Thus, it is important to be able to clean the gap and avoid production or introduction of
particles. Thus the best work is done with tires that have a clean surface13 and when abra-
Figure 7.8: Miniature force transducer
installation heights: correct and incorrect
Transducer
Transducer
Transducer
Further, equation 7.1 yields three stresses which are assumed to be uniform on the
12An irregular tread surface can produce effects like those caused by incorrect installation of the transducer.
Thus it is important to remove mold stubble and other local surface irregularities from new tires prior to test
ing. When testing worn tires, dirt adhering to the surface should be removed, but the surface irregularities due
to wear are a real part of the sample and should not be removed.
13Particles are not the only troublesome form of surface contamination. Chemical contamination will change
the surface friction. Thus, it is necessary to clean surfaces periodically with a solvent and to try to maintain a
constant coating of diffusion products on the tire surface. For example, after surface cleaning, 8 to 10 passes
of the tire may be necessary to approximately stabilize surface friction.
242 Chapter 7. Contact Patch (Footprint) Phenomena
transducer top. Given that the contact stress field on a tire tread can change rapidly with
distance over the tread surface, as will be seen in section 4, and that many tread elements
are not large, the transducer top must be reasonably small if a reliable approximation to
the stress field is going to be determined. Ideally, the transducer top should be vanishing
ly small, but in practice it is typically from one to five millimeters square or in diameter.
In regions of the footprint where the stresses are changing very rapidly, as illustrated
in figure 7.9, appreciable moments can exist on the top of the transducer. The transducer
should be designed so it rejects the effect of these moments, otherwise the resulting forces
and, hence, stresses will be in error. It would be possible to use the moment information,
if available, to generate couple stress components in the analysis, but this has not been
done in systems designed to date.
Figure 7.9: Moments on a footprint transducer with a
finite top surface area in a rapidly changing stress field
σZ
σ
Y
σ X
Y
Z
larly encoding the tire’s circumferential position, both synchronized to a mutual registra
tion location, for example, the point where the FAS y-axis and the projection of the spin
dle center onto the road plane coincide. The lateral position of the transducer must be
indexed so that it is known with respect to a known lateral position on the test surface, for
example, the origin of the SAE Tire Axis System. This combination allows the coverage
state of the transducer to be determined for all other positions of interest in the region of
the tire under test. The result is an intelligent form of a data acquisition grid like the one
superimposed on a footprint image in figure 7.11.
Figure 7.10: Partial coverage illustration
Partial Coverage Complete
Coverage
Tread Element
Obviously, testing at every point in the grid shown in figure 7.11 would be impossible
within a reasonable period of time using a single transducer. Thus, there is a need to take
data at multiple tread points every time the transducer system is within the footprint. This
requires an array of transducers. Given the physical constraints on sizing and arranging
transducers into an array, 3-D force transducer arrays have been used as line arrays.
Figure 7.11: Tread pattern with a superimposed data acquisition
road. However, in the literature the lateral line array has been applied most often in a
low-speed plank type machine [7,15] or in a real road. The example shown in figure 7.12
is arranged in the test surface of the test machine as indicated in figure 7.17[7]. The
longitudinal line has been most commonly applied in roadwheel machines. An example,
taken from reference 16, is shown schematically in figure 7.18.
Technology, Vol. 27, No. 3, 1999 with Tire Society permission) [7]
X
Transducer 1
Transducer 2
Transducer 3
Transducer 4
14The specific transducer spacing shown in Figure 7.12 is a particular designer's concept. There are many
other feasible spacings depending on the detailed mechanical design of the transducers and the array. Indeed,
the patent [13] for the array shown in Figure 7.12 shows another possible transducer spacing, as does Figure 4
in Reference 15.
15The example is actually a combined stress and slip displacement transducer. It will be discussed in Section
7.2.6.1
16The first commercially produced line arrays appear to have been made by the Precision Measurement
Company in the 1980s.
Chapter 7. Contact Patch (Footprint) Phenomena 245
Figure 7.18: Drawing of the drum test stand at Continental AG for "stochastic
footprint measurement technology" (reprinted from Tire Science and Technology,
Vol. 31, No. 3, 2003 with Tire Society permission) [16]
The efficient use of a lateral line array to map the tread surface, as indicated by the cir
cles in figure 7.11, requires that the array have a carefully controlled lateral position,
which can be adjusted from footprint pass to footprint pass. This is combined with circum
ferential position tracking/setting, discussed earlier in connection with rejecting data from
partially covered transducers. The inability to precisely control and track their position
makes experiments involving force transducer arrays in real road surfaces much less sat
isfactory than those conducted using testing machines.
The efficient use of a longitudinal array requires that the tire position also be carefully
controlled and tracked, as noted above. Reference 16 provides an interesting solution to
the problem of tire lateral position through precise stepwise change of tire position over
the course of an experiment.
In the force transducer form of rolling footprint experiments engineers have historically
made a tacit assumption that the force transducer is in contact with a given sample of the
tread surface throughout contact. This tread surface sample is inherently thought of as a
point on the tread surface. Thus, the results from a transducer are ascribed to a point.
Indeed, it is common to say that the stresses were thus and so at that point during contact.
In reality, slip during contact causes the portion of the tread sampled by a transducer to
change throughout contact. If slip is small, this deviation from assumed behavior is not
important. If slip becomes large enough, however, the simple concept no longer applies
and the test engineer cannot decide which point on the tread is associated with the results.
For that matter, with appreciable slip, partial coverage has a different definition from loca
tion to location throughout the footprint. The exact slip conditions at which these complex
changes have to be taken into account depend on the physical size of the transducers
among other things. For 5 mm diameter transducers, as used in Reference 13, the problem
248 Chapter 7. Contact Patch (Footprint) Phenomena
A grille of conductive, but force insensitive, leads is deposited on a flexible Mylar sheet.
A second layer of Mylar acting as a mask is laminated over the sheet. An ink with pres
sure sensitive conductivity is deposited over the mask and the excess removed leaving an
array of ink dots through the thickness of the mask. Finally, a second grille of leads running
at right angles to the initial grille is bonded to the sandwich to form a flexible circuit card.
By rapidly querying the leads, one pair after another, the state of the sensitive element
at each crossing point can be obtained, allowing production of a 2-D matrix of pressure
measurements. It is possible with current technology to obtain pressure images under slow
rolling conditions as shown in Section 4.1.1.17
At this time, mats have very limited durability under conditions of applied global shear
such as those imposed by slip angle or torque. Thus, mats are basically limited to either
static loading or straight-ahead free rolling.
17The pressure mat results in Section 7.3.1.1 were obtained using a Tekscan, Inc, resistive mat [17].
Chapter 7. Contact Patch (Footprint) Phenomena 249
lengths of light of the glass surface). The angle at the glass/air interface exceeds the crit
ical angle determined by Snell’s law, see figure 7.20.
This situation can be changed if a third medium with a high optical density is brought
PLASTIC SHEET
LIGHT ZOOM
SOURCE
TIRE
TREAD
PVC
GLASS PLATE
* GLASS
LIGHT
CAMERA
SOURCE
The camera sees a
bright spot.
close to the interface. Light is then partially transmitted across the interface, the total inter
nal reflectance is frustrated, and an illuminated spot appears in the field. In figure 7.20 a
thin vinyl sheet with small dimples is introduced between the test tire and the glass.18 The
brightness of the illuminated spot at each dimple depends on the intimacy of contact at the
dimple location, which depends on how firmly the sheet is pressed against the glass. Thus
the field, interpreted in terms of brightness, is a measure of the applied pressure. Using a
video camera and appropriate software, a pressure image like that in figure 7.21 can be
produced. The method works statically or in slow free-rolling, but is not suitable for test
ing when the tire is subject to significant shearing forces such as those encountered in cor
nering or when spindle torque is applied.
A related procedure using absorption of light at a rubber/glass interface is described in
reference 19. This method, which works with a matte rubber surface, does not require any
intermediate sheet when used with a properly conditioned tire. Thus it will work in the
presence of global shear if improved optics can be developed. It has been used only stat
ically. A variant, based on enhanced optics, might offer results in rolling operation.
18Fortuitously,a very inexpensive textured vinyl table cloth with a tiny diamond pattern repeating at a small
fraction of a millimeter is an adequate sheet.
250 Chapter 7. Contact Patch (Footprint) Phenomena
19Correlationwith Moiré studies carried out on a smooth glass plate confirm that properly designed needle
systems [7,13] can give reasonable slip displacement results.
Chapter 7. Contact Patch (Footprint) Phenomena 251
ern radial passenger tires, the system described in reference 13 only works up to about 1.5
slip angle.
Needle type transducers adapt well as line arrays. Figure 7.12 shows the system used
Figure 7.22: Schematic of a slip displacement needle apparatus
Tread
Test Surface
Instrumented
Compliant
Needle
Base
Machine Structure
in reference 7. A line array of displacement transducers is mounted coaxially within a line
array of force transducers. Figure 7.23 shows the system used in reference 15. In this sys
tem separate line arrays of stress and motion transducers are used. The coordinated use
of stress and displacement transducer results is discussed in section 3.6.1.
To date, the displacement needle method has only been applied at low speeds due to trans-
Figure 7.23: An array produced by Precision Measurement Company
Stress
Array
Line
Displacement
Array Line
252 Chapter 7. Contact Patch (Footprint) Phenomena
ducer dynamics. It would be possible to apply this method at appreciably higher speeds, if
high-stiffness, low-mass, composite materials were applied in transducer fabrication.
dition. Thus, studies of behavior at high slip and inclination angles or under appreciable
torque are possible. It does, however, require special tire preparation and the attention of
a skilled operator.
Like the needle method it works best at modest rolling speeds. This is due to constraints
on frame rates while maintaining good optical resolution.
20Displacements internal to the tread blocks are not available using this method.
21The drum of this machine has no side plate on its top, thus, instrumentation can be easily mounted inside the
drum.
(point a). At successive times the line scan camera sees a different image as the tread ele
ment slips across the window. This is illustrated in figure 7.26 where the evolution of the
image from time 1 to time 2 to time 3 is illustrated. The rectangle surrounded with the
heavy line in the figure is the block location at the indicated instant.
The example shown in this section is for slip displacement in one direction. To consid-
Figure 7.25: Line scan camera slip displacement measuring equipment schematic
Reflection No Contact: b)
b)
Tire
Reflection Contact: a)
Glass plate
Camera
Laser
Figure 7.26: Evolution of the line scanned image (reprinted from Tire Science and
Technology, Vol. 32, No. 2, 2004 with Tire Society permission) [21]
er displacement of the block edges in both directions, it is necessary to have two systems,
one providing u-displacement and the other providing v-displacement. Again, tracking
which point on the tread surface is associated with the data is very important.
254 Chapter 7. Contact Patch (Footprint) Phenomena
This arrangement does not have severe dynamic limitations and functions in an auto
mated way at reasonable test speeds up to 40 km/hr.
3.4.1 Thermocouples
Miniature thermocouples, figure 7.27, have been applied to measure the change in tread
surface temperature during passage through the footprint [22]. In reference 22, the exam
ple data are for a 6 slip angle, which is a severe condition in terms of wear, and for a par
ticular test surface texture which produces a contact area of about one-tenth of that meas
ured on a smooth surface. Fujikawa and his associates [22] provide experimental data to
support the use of a one-tenth factor. Whether this factor is precisely correct or not, the
idea that the real tread contact area on a textured surface is less than that measured on a
smooth surface is definitely correct. Figure 7.28 [22], based on computations known to
correlate with experimental results, shows the effect of the thermal properties of the test
surface on the tread surface temperature in the footprint. This is briefly discussed in
footnote 22.
Science and Technology, Vol. 22, No. 1, 2004 with Tire Society permission) [22]
22Obviously, the effect of heat transfer with the test surface is important. If the test surface is highly conduc
tive and not at a temperature reasonably near the mean tread surface temperature, the results of tread surface
temperature studies will be markedly affected.
Chapter 7. Contact Patch (Footprint) Phenomena 255
150
Pavement:
Temperature rise (K)
Concrete
100
Alumel
50
Steel
0 50 100 150
Distance into Patch (mm)
Walters [24] observed, as noted at the outset of Section 3.4, that it is possible to attrib
ute the temperature rise to the shear energy developed as the tire rolls through the foot
print. It is necessary to start with the tire and an insulated test surface at the same temper
ature, and gather data for only a short period after rolling begins at a given set of test con
ditions. For example, one might test at a given load, slip angle and speed, from the start
of rolling for a period of 90 seconds to two minutes, and then take data. The tire would
then be removed from the test surface, allowed to cool, and test using the next set of con
ditions initiated after about 15 minutes or so. This procedure could be repeated as neces
sary to gain related data at many test conditions in a reasonably short period of time.
23Today,almost all customer dissatisfaction with tire wear performance arises from uneven wear, which either
produces unsatisfactory vibration and noise performance or leads to locally severe wear in an otherwise good
tread.
Chapter 7. Contact Patch (Footprint) Phenomena 257
wear rate to the magnitude of the shear force. Different drivers generate shear forces that
can vary enough on a given course so that the wear rate varies by a factor of six [28].
Different courses can lead to wear variations by as much as a factor of 10 [28].
Reference 29 demonstrates that the exact way the forces are generated, the combina
tion of slip and inclination angles existing due to suspension characteristics, has a dramat
ic influence on the uneven wear pattern developed even if the gross tire forces are nearly
unchanged. Plainly, it is not appropriate to study footprint behavior under conditions that
are unrelated to real operation. For example, straight free-rolling does not occur at zero
slip and camber angles as noted in Chapter 8, Section 4.1.4.1. Therefore, a proper under
standing of actual tire operating conditions is necessary if rolling footprint studies are
going to be relevant.
R = KFn (7.2)
Different vehicles, with different tires, operating according to the acceleration and
velocity description of a course will develop different force and alignment details. Thus,
it is necessary to extend the course description into a force description applicable to the
individual tires operating on the test vehicle24: longitudinal force (FX), lateral force (FY),
inclination angle (g)25, normal force (FZ), and velocity (VX), if footprint studies are to
yield realistic results.
The force description can be determined experimentally, but in practice it is more effi
cient to derive the individual tire force description from the acceleration and velocity
imposed by the course, by applying vehicle dynamics. This can be done by complex mod
eling [32] or by a blend of modeling and experiment [30].
The direct use of the force descriptions as time functions is briefly discussed in Section
4.3.3.
24The operational conditions of a tire on a single vehicle depend on the position in which the tire is mounted.
Front and rear are not the same. Left and right are not the same.
25Since lateral force is a nonlinear function of slip and inclination angles, it is necessary to allow the angles to
7.32. This approach has been applied in many publications such as [7], [15], [21], and
[33]. Then, the force descriptions appropriate to each bin are computed.
Figure 7.32: Route spectra for a truck tire public road wear course
0.250
F
r 0.200
a J
c o
t u 0.150
i r Ay ( g
o n 0.100 )
)
(g
n e
x
y
A
o 0.050
f 0.2
0
0.000 0.0
8
0.28
Da
0.20
ta
0.12
-0.
0.04
16
-0.04
-0.12
-0.
-0.20
28
-0.28
The engineer conducting the footprint study must decide which of the bins to examine.
If the intent is to produce a description, the route spectra can be used as discussed in
Section 4.3.3.
(Eq. 7.3)
Where:
S = Path of integration. Physically, the path of motion of a point ij on the tread relative
26In Finite Element Analysis the shear stresses and displacements should be computed so that the computation
27Energy also crosses the boundary in the footprint due to vertical deflection of the tire and heat transfer.
Equation 7.4 is the differential form of Equation 7.3. In Equation 7.4 the footprint forces
and differential displacement vectors at the points of the tread pattern-sampling grid have
been decomposed into their X and Y components at a specific location on the tread.
(Eq 7.4)
Where:
dWij = Differential shear work at point ij.
Fxij = X component of the footprint force at point ij.
Fyij = Y component of the footprint force at point ij.
duij = X component of the differential displacement vector at point ij.
dvij = Y component of the differential displacement vector at point ij.
As the points on the tire tread follow path S through the footprint the differential shear
work varies point-by-point. It can be approximated as indicated in Equation 7.5 for the
transition from point k - 1 to point k on path S.28
(Eq 7.5)
Where:
k = The index of points along the integration path S. Data begins with k = 0, but
the first k for which Equation 7.5 is defined is k = 1.
Shear work has a sign. At times the road will do work on the tread and at other times the
tread will do work on the road. But from the standpoint of the tread material at the inter
face between the tire and road, the sign of shear work is not important. What is important
is the total amount of shear energy that is generated at the interface per unit area of the
tread surface during passage through contact. This energy per unit area is the shear ener
gy intensity. Equation 7.5 can be rearranged to yield differential energy intensity by
dividing the forces at the tread pattern-sampling points by the area of the transducer used
to sense the shear forces and taking the absolute value. The result is Equation 7.6.
(Eq 7.6)
Where:
dIij,k = Differential energy intensity for the kth step.
28More complex differential formulations could be used, but in practice, given data spacing, this has proven
adequate.
Chapter 7. Contact Patch (Footprint) Phenomena 261
Summing over k from one to n, equation 7.7, through the length of the footprint, gener
ates the energy intensity at point ij applicable to passage of the tread through contact for
the current operational condition. When this is done at all points of contact between the
tread pattern-sampling grid and the tread, intensity maps are obtained like those in Section
4.3.
(Eq 7.7)
Where:
Iij = Energy intensity at point ij.
Ixij = Energy intensity due to X stresses and displacements at point ij.
Iyij = Energy intensity due to Y stresses and displacements at point ij.
4. Footprint physics
The footprint is a kinematically driven boundary. As noted at the outset of this chapter, the
double curvature of the tire structure requires that the tire structure be stretched, com
pressed and bent to conform to the road surface. These forced displacements give rise to
the footprint stress field. They are combined with the effect of the tire’s pneumatically pre
stressed structure, the complex displacement field due to compression loading of nearly-
incompressible treads cut into complex patterns, and the soft response of the tire structure
to shearing displacements.
Figure 7.33 is an example of the stresses and slip displacements at a single point, with
a truck tire rolling straight ahead. A similar set of data is found at every other point in the
footprint. Each point is unique and experiences a unique set of operational conditions.
Further, the precise data depends on the tire’s structural design and architecture (shape,
tread pattern, etc.). Thus, the presentation in this section is necessarily generic. In trying
to use it, never forget that the footprint physics or mechanics applicable to a particular tire
design and operational condition are unique.
262 Chapter 7. Contact Patch (Footprint) Phenomena
ó Z N o rm a l S re s s (k P
-1 2 0 0
-8 0 0
-4 0 0
0
150 100 50 0 50 100 150
X Ax is (m m )
X Axis (mm)
1 50 100 50 0 50 1 00 150
0.04 160
u Long tudinal D isplacem ent (m
0.04 80
0.08 40
óX
0.12 0
0.16 40
X Axis (mm)
0 80
Lateral Stress (kP
0 .0 2 60
0 .0 4 40
ó
0 .0 6 20
0 .0 8 0
For ordinary passenger tires, as in figure 7.34, it is typical to find the largest
normal stress magnitudes on the shoulders with lower magnitudes in the crown.
Figure 7.34: Typical lateral, longitudinal, and normal stress isometrics for a
passenger tire rolling at zero slip and inclination angles
-200
40
-150 30
-100
PSI
20
PSI
-50 10
0 0
-10
-20
-30
-40
40
30 NOTE:
20 •There are no grooves.
10
PSI
0 •Red is negative.
-10
-20 •Blue is positive.
-30
-40
Figure 7.35: Cord loads for the tire used to produce the data presented in figure
7.34 (reprinted from Tire Science and Technology, Vol. 20, No. 1, 1992 with Tire
Society permission) [6]
Figure 7.37: Normal stress isometric for a radial medium duty (TBR) truck tire
rolling at zero slip and inclination angles (reprinted from Tire Science and
Technology, Vol. 27, No. 3, 1999 with Tire Society permission) [7]
-1600
-1400
99
48
80
+X
61
23
42
+Y
23
-3
4
-15
Longitudinal Distance from -28
-34
-53
-79
-110
Center of Contact
(mm) Fig 7.36
Thus, if the normal stress profile of an ordinary passenger tire is plotted in the Y-Z plane
it has a bowl shape with the magnitudes on the shoulders representing the sides of the
bowl and the magnitudes in the crown representing the bottom of the bowl. Figure 7.38
shows Y-Z plane normal stress profiles for a 50-Series Performance tire and a medium
duty truck tire, both at typical service loads. The profiles in figure 7.38 are very different
from the typical passenger profile. Why?
The 50-series tire is operating at about 60-percent of its load rating. The footprint,
Figure 7.3, is not at all square, as is that one for the tire in figure 7.34. The pneumatic pre-
strains in the belt cords have not been fully relieved, as was the case in figure 7.35.
Figure 7.38: Comparative normal stress on the Y-Z plane for a performance tire
and a radial medium duty (TBR) truck tire rolling at zero slip and inclination
400
600
Normal Stress (kPa)
800
1000
1200
1400
1600
150 100 50 0 50 100 150
Lateral Distance from Center of Contact (mm) +Y
Passenger Tire Raw Data Truck Tire Raw Data
Truck Tire Polynomial Fit Passenger Tire Polynomial Fit
Chapter 7. Contact Patch (Footprint) Phenomena 265
On the other hand, in the case of the medium duty truck tire, the inflation pressure is
sufficiently high that very substantial pneumatic prestrains exist in the cords in the foot
print during loading.
In figure 7.34, the longitudinal stress is positive at the front of the contact and negative
at the rear. This is exactly what is expected based on figure 7.36, and is driven by the cord
load changes portrayed in figure 7.35.
Another interesting feature can be noted in the longitudinal stress data. The longitudi
nal stress in the shoulders is more positive than in the crown. Now, by definition the slip
ratio of the free-rolling tire is zero. For the longitudinal stress in the shoulder to be more
positive than that in the crown, the shoulders must travel further than the crown during
each tire revolution. This means that the shoulders have a larger effective rolling radius
than the crown.30 Looking at figure 7.35 this is expected since belt tension will affect belt
length. The crown area of the belt, where pneumatic tension is reduced during transit of
the footprint, will act as if it is shorter. This is a good example of kinematic forces and of
the tire fighting with itself in the footprint region.
The lateral stresses in figure 7.34 also show the effect of cord load relaxation with the
right shoulder showing positive stresses and the left shoulder showing negative stresses.31
The outward shear direction is maintained. The very low level of a positive lateral stress
bias is a result of plysteer.
30The point whose data are shown in Figure 7.33 is on the shoulder, thus, its longitudinal stresses are also
shifted positive.
31The lateral stresses in Figure 7.33 are also positive as would be expected since the example tread location is
32The footprint shape shown in Figure 7.39 applies to Figures 7.40 and 7.41 as well.
Figure 7.40: Isometric of footprint lateral stresses of standard radial passenger tire
loaded statically and in slow rolling (reprinted from Tire Science and Technology,
Vol. 20, No. 1, 1992 with Tire Society permission) [6]
The normal stresses in figure 7.39 are highest in magnitude in the shoulder and lowest in
the crown, as seen in figure 7.34, regardless of whether the tire is stationary or rolling. The
change is in relative magnitude. The difference from shoulders to crown is appreciably
larger for the rolling tire. The normal stresses on the shoulder are higher in magnitude and
the ones on the crown are lower. Basically, the shoulders are more heavily loaded and the
crown is more lightly loaded during rolling.
The change in the normal stress pattern is associated with changes in lateral stress from
static to rolling. As shown in figure 7.40, the outwardly-directed lateral stresses are great
ly reduced in rolling. Fundamentally, the motion occurring in rolling frees the shoulders
to move inward as a result of the relaxation of belt cord tension in the footprint, which is
shown in figure 7.35 and was discussed earlier. Relatively, the shoulders move inward and
roll under, while the crown tends to deform away from the road, reducing the magnitude
of crown normal stress. An idealized sketch of the process is shown in figure 7.42.
Under static loading the longitudinal stress pattern in the contact region is symmetrical
with respect to X, regardless of the Y position of the X - Z section examined. Rolling caus
es the shoulder stresses to become more positive and those in the crown region to become
Chapter 7. Contact Patch (Footprint) Phenomena 267
more negative. This is due to the non-constant rolling radius effect noted in the discussion
of longitudinal stress, based on data in figure 7.34.
Figure 7.42: Normal and lateral stress association schematic
Technology, Vol. 20, No. 1, 1992 with Tire Society permission) [6]
Figure 7.43 shows how the change from static to rolling can cause gross changes in the
apparent footprint shape. In this case the tire actually buckled out of contact in much of
the crown region when rolling.
Figure 7.43: Comparison of static (ink block) and slow rolling (pressure mat) foot-
prints of a self-supporting tire (data courtesy of Smithers Scientific Services, Inc.)
Inflated
Flat
Flat
X Y
Slip Angle
34It should be noted that existing flat surface machines are economically not feasible for studies of abrasive
wear studies, because of first cost per test position and high maintenance costs. Thus, these tests have been
The lateral stresses in turning are very much larger than for a tire rolling straight ahead.
For a left turn, at a positive slip angle, the lateral stresses are essentially totally negative,
and in the right turn they are essentially totally positive. The deformations discussed in
section 8.4 are responsible for these lateral stresses. A careful look at the lateral stress
fields in figure 7.44 reveals that the stress develops differently according to lateral posi
tion. Figure 7.45 gives a simplified view of the lateral stresses and associated slip dis
placements developed by the performance tire when turning right, at -1° slip angle.35 Rib
1 (the left-most rib) is lengthened and subject to higher normal stresses. The lateral slip
displacement occurs primarily in the rear of contact and the lateral stress distribution is
definitely shifted to the rear. Rib 3, which is subject to a moderate level of normal stress
reaches a maximum lateral stress at about halfway through contact and then the lateral
stress ceases to grow (sliding is occurring) in spite of increasing slip displacement. Rib 5
is lightly loaded and its contact length is short. Again, sliding begins about halfway
through contact. The basic lateral slip displacement field portrayed is in accord with that
shown in reference 20.36
Figure 7.45: Rib by rib lateral stress and slip displacement for a cornering per-
formance tire (reprinted from Tire Science and Technology,
V SAE X’
-1°á
Rib 1
Rib 5
Rib 3
500 01
00
400
01
Lateral Displacement (mm)
02
300
Lateral Stress (kPa)
03
200 04
05
100
06
07
0
08
100 09
60 40 20 0 20 40 60 60 40 20 0 20 40 60
X X
Distance hrough Contact (mm) Distance hrough Contact (mm)
As shown in figure 7.44, longitudinal stress is warped by the effect on the cord loads
of the slip angle induced deformation. The heavily loaded shoulder shows an exaggera
tion of the reduced rolling radius effect seen in the crown and discussed in the previous
section. The lightly loaded shoulder shows an exaggeration of the increased rolling radius
effect seen in the shoulder and discussed in the previous section.
36In reading Reference 20, take care about the meaning of terms and the senses shown in the diagrams, as they
are not the same as those used in this chapter.
270 Chapter 7. Contact Patch (Footprint) Phenomena
The data of figures 7.44 and 7.45 come from low slip angles. The distortion of the foot
print increases as slip angle increases, as shown in figure 7.5. Thus, the general behavior
observed should grow in magnitude with increasing slip angle.
Figure 7.46 shows that the effect of small inclination angles is much less that of small
slip angles. This is expected, given the response of passenger tire forces and moments to
inclination angle, section 8.4. The primary results are what would be expected when one
shoulder is loaded more heavily and the load on the other is simultaneously reduced.
Figure 7.46: Effect of inclination angle on
the stress field of a smooth treaded tire
Y
X
-200
PSI
-100
0
-40
-20
PSI
-20
-40
γ = 0.0° γ = 1.0°
In all cases, the slip displacement due to an operational parameter has increased toward
the rear of the footprint. The same is true in the case of torque application. Addition of
braking slip to the free-rolling state leads to a change of the longitudinal stress seen at zero
slip and inclination angle in figures 7.34 and 7.46, into the pattern seen in figure 7.47. This
indicates that it is reasonable to assume as a first approximation that the effect of longitu
dinal slip is a linear change in stress from the front to rear of the footprint, as represented
schematically in figure 7.48. Line 1 in figure 7.48 is the total component of longitudinal
stress due to driving torque. Line 2 is the total component of longitudinal stress due to
braking torque. Line 4 is the free-rolling longitudinal stress in the crown zone of the tire.
Chapter 7. Contact Patch (Footprint) Phenomena 271
Adding lines 1 and 4 produces line 3, which gives a first approximation to the longitudi
nal stress in a driven tire. Adding lines 2 and 4 produces line 5, which gives a first approx
imation to the longitudinal stress in a braked tire.37
Figure 7.47: Longitudinal shear stresses for a braking smooth treaded tire [5]
X
Y
Longitudinal Stress (kPa)
10
0
-10
-20
-30
37The discussion of deformation given in Reference 5 is not presented here as it is thought to be inadequate.
272 Chapter 7. Contact Patch (Footprint) Phenomena
Equation 7.13 makes a very important point. Aligning moment is a function of not
only lateral stress, but also of longitudinal stress. It is common to think in terms of later
al stress only and in terms of pneumatic trail.38 This is an incomplete model that can lead
to misconceptions in the presence of inclination angle and applied torque.
Figure 7.49: Isometric of lateral stresses in the footprint of a static and slow rolling
solid tire (reprinted from Tire Science and Technology,
Vol. 20, No. 1, 1992 with Tire Society permission) [6]
Figure 7.50 shows the longitudinal stress field for a solid tire, both static and rolling.
Again, the sense of the stresses is reversed in comparison with that seen in the pneumat
ic tire, figure 7.41. But the rolling radius is larger on the shoulders and smaller in the
crown, just as in the pneumatic case. The effect of rolling is very clear in the crown where
negative shear stresses are absolutely dominant.
Figure 7.50: Isometric of longitudinal stresses in the footprint of a static and slow
The result is a solid tire shear field like that in figure 7.51. Addition of this field to the
global pneumatic field conceptually presented in figure 7.36 causes significant and com
plex changes.
274 Chapter 7. Contact Patch (Footprint) Phenomena
Figure 7.51: Fundamental sense of shear stresses acting in the footprint of a solid
tire (reprinted from Tire Science and Technology,
Vol. 20, No. 1, 1992 with Tire Society permission) [6]
Groove 3
Groove 1
Groove 4
- Smooth
Groove 4
- Grooved
Strip Lateral Force (N)
Groove 3
30
Groove 2
Groove 1
0
-30
-60
is a sharp change at each groove edge. The sense is exactly as expected based on figures
7.49 and 7.51. Consider groove 2 for example. The lateral force on the left edge is sharply
negative compared to the value prior to having the groove cut. The right edge of the sec
ond rib is trying to slide into groove 2, inducing a leftward, negative, shear force. The
effect on the right edge of groove 2 is the exact reverse, and this alteration continues
across the tread.
Figures 7.53 and 7.54 show the rib/groove effect in terms of lateral stresses along the
y-axis for a performance tire and a medium duty truck tire. The effect is the same regard
less of tire type. However, it is worth noting that the underlying stress pattern in the truck
tire does not show the effect of cord load reduction, as mentioned earlier in the discussion
of figures 7.37 and 7.38. This is because the inflation preload of the cords in the truck tire
is high enough to prevent a serious loss in cord load during contact deformation.
Figure 7.53: Effect of longitudinal grooves on the lateral stress field
across the crown of a performance tire (reprinted from Tire Science and
Technology, Vol. 27, No. 3, 1999 with Tire Society permission) [7]
Figure 7.54: Effect of longitudinal grooves on the lateral stress field across the
crown of a medium duty (TBR) truck tire (reprinted from Tire Science and
Technology, Vol. 27, No. 3, 1999 with Tire Society permission) [7]
276 Chapter 7. Contact Patch (Footprint) Phenomena
Adding cross-cut grooves to the longitudinal groove pattern shown in figure 7.52 alters
the longitudinal stress pattern as shown in figure 7.56. In this case the net force on a series
of 5mm square areas along the leading edge of the blocks is positive while that on a series
of 5mm square areas along the trailing edge of the blocks is negative. This is what would
be expected based on Poisson bulging under compression.
Figure 7.56: Effect of lateral grooves on footprint forces
Figure 7.57 shows the effect of the edges of a lateral block element in the tread of a
performance tire. The heel, i.e., the block leading edge, is subject to positive stresses as
expected, and the toe, trailing edge, shows a definite negative bias, also as expected.
The Poisson effect in this case, when mixed with some braking, will produce heel /toe
wear because the stresses on the toe are larger than on the heel. This is a common occurrence.
Figure 7.57: Effect of lateral grooves on the longitudinal stress field of a tread ele-
ment in the crown of a performance tire (reprinted from Tire Science and
Technology, Vol. 27, No. 3, 1999 with Tire Society permission) [7]
Chapter 7. Contact Patch (Footprint) Phenomena 277
Where:
A = Abradability (mm removed / (J/m2/rev))
Rijlmn = Tread (mm) removed at point (i,j) in the lth revolution under the conditions
existing in the (m,n)th route spectra bin.
Knowing the route spectra as represented in figure 7.32 it is possible to restate
Equation 7.14 to represent the whole route, Equation 7.15.
RTij = A ITij = A lT (ΣΣ Iijmn(lmn/lT)) (Eq. 7.15)
mn
Where:
ITij = Route Intensity, Intensity indicative of the entire route (J/m2)
lmn/IT = The fraction of the route spectra represented by bin m,n.
lmn = The number of revolutions in bin m,n.
lT = Total number of tire revolutions.
RTij = Total Rubber Removed by driving the entire route (mm).
Obviously, a repeated course could be characterized by determining its route spectra
then computing tread loss on the basis of the total operational distance.
Plainly, the ideal situation would be one in which ITij at every point (i,j) would be iden
tical. In this case, RTij would be the same at every point and wear would be even. This is
nearly impossible to achieve, but great effort goes into trying to achieve a reasonable
approximation to this ideal state.
Figure 7.58: Slip angle and shear energy intensity (partially reprinted from Tire
Science and Technology, Vol. 27, No. 3, 1999 with Tire Society permission) [7]
Figure 7.59: Tread pattern effects on shear energy intensity (partially reprinted
behavior, figure 7.61. Zheng simulated the entire wear process, including stepwise tread
removal, using abradability based on over-the-road experiments. This is a comprehensive
application of the concept expressed in equation 7.15.
Figure 7.60: Tire-by-tire agreement between wear rate and shear energy intensity
Figure 7.61: Cross section tread wear profiles for the right-front tire (reprinted
4.3.4 Indoor laboratory simulations based on duplicating the energy history in an outdoor
wear test
Over the last decade it has become common to use route characterizations of the type dis
cussed in Section 3.6.1, to do indoor wear tests with machines like that portrayed in fig
ure 7.16, equipped with dust applicators and collectors as portrayed in figure 7.62. The
results give quite a reasonable simulation of outdoor wear results, figure 7.63 [32, 34].
Figure 7.62: Features of indoor wear test equipment (reprinted from Tire Science
and Technology, Vol. 30, No. 2, 2002 with Tire Society permission) [30]
Figure 7.63: Comparison of fleet- and indoor-tested tread loss profiles for tire
5. Concluding Remarks
An orderly framework for discussing footprint behavior has been provided and the wide
range of experimental methods that are used for analyzing footprint behavior has been
described. Example results have been presented and the underlying physics and mechan
ics have been discussed. However, it should be obvious from the presentation that this
subject is far from completely worked out - much remains to be elucidated. Although
space limitations prevented the inclusion of many details of our present understanding of
footprint behavior, it is hoped that the references provided will help the interested reader
pursue the subject further.
Chapter 7. Contact Patch (Footprint) Phenomena 281
6. References
1. “Tire Performance Terminology,” Society of Automotive Engineers, SAE J2047,
Warrendale, PA, 1998.
2. Michael, F., “Zur Frage der Abmessungen von Luftreifen für Flugzeuglaufräder,”
Jahrbuch 1932 der D. V. L., 3 p.17. Available in English Translation as NACA TM 689
(1932).
3. Hertz, H., “Gessammelte Werke,” Vol. I, Leipzig, 1885.
4. “Mechanics of Pneumatic Tires,” S. K. Clark Ed., Monograph 122, National B u r e a u
of Standards, Washington, D.C., November 1971.
5. “Mechanics of Pneumatic Tires,” S. K. Clark Ed., DOT HS 805 952, National
Highway Traffic Safety Administration, Washington, D. C., August 1981.
6. Pottinger, M. G., “The Three-Dimensional Contact Patch Stress Field of Solid and
Pneumatic Tires”, Tire Science and Technology, TSTCA, Vol. 20, No. 1, January-March
1992, pp. 3-32.
7. Pottinger, M. G., “Effect of Suspension Alignment and Modest Cornering on the
Footprint Behavior of Performance Tires and Heavy Duty Radial Tires,” Tire Science and
Technology, TSTCA, Vol. 27, No. 3, July – September 1999, pp. 128-160.
8. “Standard Practice for Tread Footprints of Passenger Car Tires Groove Area Fraction
and Dimensional Measurements,” F870-84, Annual Book of ASTM Standards, Vol. 09.02,
ASTM, W. Conshohocken, PA, 2005.
9. Yeager, R. W., “Tire Hydroplaning: Testing, Analysis, and Design,” The Physics of Tire
Traction, D. F. Hays and A. L. Browne, Eds., Plenum Press, New York, N.Y., 1974.
10. Grogger, H., and Weiss, M., “Calculation of the Hydroplaning of a Deformable
Smooth-Shaped and Longitudinally-Grooved Tire,” Tire Science and Technology,
TSTCA, Vol. 25, No. 4, October-December 1997, pp. 265-287.
11. Seta, E., Nakajima, Y., Kamegawa, T., and Ogawa, H., “Hydroplaning Analysis by
FEM and FVM: Effect of Tire Rolling and Tire Pattern on Hydroplaning,” Tire Science
and Technology, TSTCA, Vol. 28, No. 3, July-September 2000, pp. 140-156.
12. Ginn, J. L., and Marlowe, R. L., “Road Contact Forces of Truck Tires as Measured in
the Laboratory,” Society of Automotive Engineers, SAE 670493, Warrendale, PA, 1967.
13. Pottinger, M. G., “Apparatus for Measuring Tire Tread Force and Motion,” U. S.
Patent No. 4,986,118.
14. Marshall, K. D., Phelps, R. L., Pottinger, M. G., and Pelz, W., “The Effect of Tire
Break-In on Force and Moment Properties,” Society of Automotive Engineers, S A E
770870, Warrendale, PA, 1977.
15. Knisley, S., “A Correlation Between Rolling Tire Contact Friction Energy and Indoor
Tread Wear,” Tire Science and Technology, TSTCA, Vol. 30, No. 2, April- June 2002, pp.
83-99.
16. Koehne, S. H., Matute, B., and Mundl, R., “Evaluation of Tire Tread and Body
Interactions in the Contact Patch,” Tire Science and Technology, TSTCA, Vol. 31, No. 3,
July-September 2003, pp. 159-172.
17. Malacaria, C., “Making no Mark,” Tire Technology International, June 2003,
pp. 51.
18. Gentle, C. R., “Optical Mapping of Pressures in the Tyre Contact Patch,” Optics and
Lasers in Engineering, No. 4, 1983, pp. 167-176.
19. Sakai, E. H., “Measurement and Visualization of the Contact Pressure Distribution of
Rubber Disks and Tires,” Tire Science and Technology, TSTCA, Vol. 23, No.4, October
282 Chapter 7. Contact Patch (Footprint) Phenomena
Problems
1. Figure 7.2 presents the Footprint Axis System. Figure 8.1 presents the SAE Tire Axis
System. Derive the transformation from the FAS System to the SAE Tire Axis System.
2. Figure 7.1 presents an ink block plot of a footprint. The area enclosed by a line circum
scribing the footprint periphery and touching image is referred to as the gross footprint
area. The portion of the gross area that is printed in black is referred to as the net footprint
area. The groove area is defined as the gross area minus the net area and the groove void
fraction as the groove area divided by the net area times 100. Using the Problem 7.2
PowerPoint file as raw data, determine the gross footprint area, the net footprint area and
the groove void fraction. NOTE: For purposes of this problem all parts of the grooves
have been assumed to be of equal depth and the tread thickness has been assumed to be
equal to the groove depth.
Problem 7.2:
3. Examining the half footprints shown in figure 7.6 it is obvious that the area in contact
is shrinking with rising velocity. Given the concept that full hydroplaning will occur when
the contact area is reduced to zero, at what speed should full hydroplaning occur?
4. Assuming that the footprint system in your laboratory is composed of transducers like
that sketched in figure 7.7 with a 5 mm X 5 mm top and a 0.5 mm wide gap, compute the
footprint stresses imposed on the tire when the transducer reports the following forces
applied to it. What is the apparent coefficient of friction?
FX = 3 N; FY = - 4 N; FZ = 5 N
5. Why is it important to reject data arising from partially covered footprint stress trans
ducers? If the decision were to use data from partially covered transducers, what must be
known besides the transducer loading in order to make the data valid?
6. What tare relationship must be known between a displacement needle transducer and
applied normal stress in order to insure that the displacement data are accurate? NOTE
that this question that must be considered to insure that slight needle height discrepancies
do not introduce errors. Surface lapping can take care of stress transducer height discrep-
ancies.
7. Assuming a footprint length of 10 cm, what would be the largest slip angle at which a
284 Chapter 7. Contact Patch (Footprint) Phenomena
± 1.75 mm range displacement needle would likely be able to fully measure lateral dis
placement? Assume that the full lateral displacement occurs from beginning to the end of
contact. Figure 8.3 provides a conceptual sketch of the expected displacement.
8. (Figure 7.33 data.XLS – worksheet, Osc Plot) is the data set used to draw figure 7.33.
Cross plot sX vs. u and sY vs. v. There are large changes in sX and sY in the absence of
appreciable changes in u and v. What feature of the tire’s structure makes this possible?
Problem 8.5
Tire manufacturer
Tire construction number
Tire size 295/75R22.5
Tread pattern
Load 6175 lbs.
Deflection n/a
Inflation 125 psi
Rim width 8.25 in.
Slip angle 0 degrees
Chamber angle 0 degrees
Operator CHP
Test date 3/10/1998
Comments 11 in data window
Slip factor 5
Torque value 0
Speed 3
9. The tire in figure 7.43 is a self-supporting tire. It will roll as much as 80 km at 80 km/hr
when uninflated, before self-destructing. Given the situation portrayed in the flat rolling
data, what would be your expectation with regard to wet road behavior?
10. (Figure 7.45 data.XLS – worksheet, lateral summary) is the data set used to draw fig
ure 7.45. Turning to the Sz worksheet, add the normal stresses to the data in the Lateral
Summary. What is the relationship of the stresses in each rib?
Chapter 7. Contact Patch (Footprint) Phenomena 285
11. Using the data included for problem 10, how does the lateral intensity accumulate on
Ribs 1, 3, and 5? What relative wear relationship would be expected if the tire continued
to operate at a -1° slip angle, as it is in the example data?
12. If braking torque were applied to the tire portrayed in figure 7.56, what would be the
expected worn block profile? What if driving torque had been applied instead?
Chapter 8
by M. G. Pottinger
1. Introduction .......................................................................................................... 288
2.3 Tire usage variables or inputs that lead to development of tire forces and ..........
4.4 Advantages and disadvantages of indoor and over-the-road testers ............ 303
5.1.3 Steady state response to combined slip and inclination angles .............. 313
5.1.4 Special cases of steady state free-rolling and cornering ........................ 315
7.5 The special problem of modeling down to zero speed .................................... 353
Chapter 8
Forces and Moments
by M. G. Pottinger
When pneumatic tires were first produced commercially, after John Boyd Dunlop invented
them for the second time [1]1, they were viewed as vibration isolation devices with reduced
rolling resistance. A bicyclist could readily perceive these characteristics. There was no
immediate recognition that pneumatic tires had an improved ability to produce control forces.
The importance of the tire as the source of vehicle control forces was recognized when
the automobile became the important tire market. Recognition began with the need for
safety when braking. Understanding of the importance of lateral force came later.
Analytical vehicle dynamics began when tire force and moment data first became
available. During the 1930s, R. D. Evans [2] of Goodyear and then A. W. Bull [3] of U.
S. Rubber measured tire force and moment properties. Evans’ work was on a smooth steel
drum. Bull examined the question of testing on flat versus curved surfaces and was
already concerned with test surface friction.
Maurice Olley, who can be viewed as the father of vehicle dynamics in the United
States, noted the importance of this force and moment measurement work in a 1961
speech. Olley said, “With the introduction of independent front suspension… in this coun
try and the first tire tests on smooth drums, by Goodyear in 1931 (by Cap Evans)…, the
real study of the steering and handling of cars began.”
World War II interrupted this early automotive progress and focused attention on other
problems, particularly aircraft shimmy. This dynamic problem was of concern to all air
forces, particularly the Luftwaffe, which held a meeting on shimmy at Stuttgart in 1941 [4].
Some of the first effective tire force and moment mathematical modeling was presented at
that urgent meeting.
At the end of World War II, engineers expert in aeronautical stability theory began to apply
modeling to the problem of automotive control and stability. William F. Milliken and Leonard
Segel at the Cornell Aeronautical Laboratory, working under the sponsorship of General
Motors, were leaders in this effort. Today, all tire and vehicle manufacturers apply modeling
to the problem of vehicle control. The tire force and moment data requirements of vehicle
modeling sparked the development of the current methods for determining, characterizing,
and modeling tire force and moment behavior. These methods continue to evolve.
The data requirements of the existing virtual prototyping system for vehicle stability
and handling design have largely been met by tire force and moment data developed on
smooth road surfaces, without vertical undulations or bumps. Road-surface-excited tire
vibration, which is discussed in chapter 9, has usually been treated as a separate subject.
The split between subjects occurred because of the practical difficulty of solving the gen
eral problem.
In parallel with the major thrust to describe tire force and moment properties on a
smooth dry road, substantial effort has been expended on the effects of contaminants such
as water, snow, and ice. These effects have been studied extensively, along with the effects
of pavement micro-texture and macro-texture, as noted in chapter 11.
This chapter treats tire force and moment behavior classically with an emphasis on dry
road behavior. It then looks at modifications due to surface texture and surface contaminants.
2.3 Tire usage variables or inputs that lead to development of tire forces and moments
In use, tires generate the forces and moments we have just defined. Tire force generation
2The reason for describing the forces and moments in terms of the road's action on the tire is that these are the
forces that move the vehicle and the engineer's interest is in vehicle behavior.
3Complete mathematical correctness would say the F [5] should be F ' [6]. While this correction was made
X X
in Reference 6, common usage has continued to follow Reference 5. The author has chosen common usage in
this chapter.
4If the tire is mounted on a steered axle, F has a moment about the Kingpin Axis [5] and can cause lateral
X
movement of the automobile by steering the tires. Also, FX can steer the vehicle by modulating lateral force.
This is discussed in Section 8.4.3. These steering effects can usually be considered secondary in normal driv
ing.
5Obviously, if F can modulate F as noted in footnote 4, F can modulate F . Thus, cornering affects the
X Y Y X
ability to accelerate or decelerate a vehicle. This is discussed in Section 8.4.3.
Chapter 8. Forces and Moments 291
comes about kinematically in response to a number of usage variables, and inputs such as
pressure and driving speed. The effects of these parameters are considered in Section 8.5.
The usage variables are defined in this section.
Tire load, which is characterized in terms of Normal Force (FZ), is the most important
tire usage variable. It largely determines the tire structural deformations. Normal force is
a crucial player in the generation of frictional forces and, hence, in this whole discussion.
Rl, Loaded Radius, is the distance from the spin axis to the contact center in the wheel
plane. This is an important geometric variable in tire force and moment studies. It is
dependent on normal force and the tire’s structure.
A tire generates lateral force in response to two principal angular variables shown in
figure 8.1.
Slip angle, α, is measured from the X′-axis to the direction of wheel travel, trajectory
velocity, VT. VT, the velocity of the contact center across the road, lies in the road plane.
α is positive clockwise around the positive branch of the Z′-axis. A positive slip angle is
associated with a left turn.
Inclination angle, γ, measures the tilt or camber of the wheel plane with respect to the
Z′-axis. Inclination angle is measured from the Z′-axis to the wheel plane within the Y’
Z′ plane. It is positive clockwise around the positive branch of the X′-axis. The top of a
tire showing a positive inclination angle is moved to the right with respect to its contact
center, as seen from the rear.
Tire engineers use inclination angle instead of camber angle to maintain sign consis
tency at all tire positions on a vehicle. Inclination angle is positive when the top of a tire
leans to the right. Camber angle is defined as positive, if the tire is leaning outward on a
vehicle. Thus, on the right side of a vehicle camber angle is equal to inclination angle, but
on the left side of a vehicle camber angle is the negative of inclination angle.
If the path followed by the tire has very large curvature, i.e., very small turning radius,
the path curvature itself measurably affects lateral force and aligning torque generation.
For automotive and truck tires, path curvature at normal operating speeds is small and is
ignored. However, in the case of motorcycle weave, path curvature should be considered.
Longitudinal force generation depends on angular velocity about the spin axis and in
particular on the relationship of the instantaneous angular velocity to the angular velocity
existing when the tire is free-rolling in a straight line.6 The moment or torque that is
applied from the axle to the tire is what determines the tire angular velocity. Thus, the
torque applied about the spin axis is also important in the generation of longitudinal force.
Spin angular velocity, Ω, is the angular velocity of the tire about the spin axis.
Slip ratio, SR, equation 8.1, characterizes spin angular velocity at a given time relative
to the spin angular velocity of the straight free-rolling tire, Ω0. When Ω is greater than Ω0,
SR > 0, the tire generates a driving force. When Ω is less than Ω0, SR<0, the tire gener
ates a braking force.7
SR = (Ω – Ω0)/Ω0 (8.1)
Wheel torque, T, is the external moment applied to the tire about the spin axis, see fig
ure 8.1. T causes the tire to operate in either a driven or braked state. When T is greater
than zero, it is called Driving Torque. When T is less than zero, it is called braking torque.
6A free-rolling tire is subject only to tire rolling resistance and bearing drag.
7The definition given above is the common definition. There are other definitions sometimes used in the liter
ature because they lend themselves to particular mathematical formulations [7].
(Reprinted with permission from SAE J2047 1998 SAE International) [6]
senting the tread centerline. This will be done as the simplified tire is rolled against the
road with a positive slip angle, yaw angle, existing between the trace of the wheel plane
on the road surface and the local velocity vector of the contact center, the trajectory vec
tor. This left turn situation is portrayed in figure 8.3 where the string is referred to as the
equatorial line.
When a block on the surface of the string first touches the road it tends to adhere to the
Figure 8.3: String tire at a slip angle in contact with the road
road surface. Then, as the tire rolls forward the block moves further into the contact patch,
initially following the direction of the trajectory velocity. This induces frictional forces
between the block and the road because the string’s elastic foundation is trying to return
the string to the wheel plane. At some point in its path through contact the block can no
longer provide the friction needed to perfectly follow the path of the trajectory velocity
and it begins to slide over the road surface. Given long enough, or really a large enough
rolling distance for the situation to equilibrate, the string takes up a configuration like that
represented in figure 8.3.
The integral of the lateral stresses, lateral force, is negative for a positive slip angle and
has a line of action located behind the center of contact for slip angles that occur in ordi
nary driving. Thus, the lateral force has a moment, aligning moment, about the Z′-axis,
which tends to return the tire to its unsteered state. The distance from the contact center
to the line of action of lateral force is called the pneumatic trail, lTY. The example illus
trates that aligning moment is inherently positive for a small positive slip angle, which is
what we encounter in normal driving.
Assuming that the normal stresses act on the line of the string, then it is plain that their
integral, normal force, has a positive moment, overturning moment, about the X′-axis.8
Real tires have treads of finite width, and belt packages. The tread and belt package,
which is a ring, also has bending stiffness. Thus, if the package is deformed laterally, as
in the string model, simple continuity considerations force it to twist out of the road plane.
This causes the footprint to go from a roughly symmetric shape laterally, as shown in
figure 8.4, into a trapezoidal shape like that in figure 8.5. The heavily loaded, long side of
the footprint lies on the more forward lateral edge of the footprint. In the example case,
8In this discussion sidewall compliance effects are assumed to have been accounted for in the characteristics
of the elastic foundation supporting the string.
294 Chapter 8. Forces and Moments
positive slip angle – left turn, the long edge is the right edge. This tends to cause the nor
mal force to lie to the right of the X′-axis, induce a negative overturning moment.
Figure 8.4: Glass plate photo of footprint shape for a straight free-rolling tire
Figure 8.5: Glass plate photo of footprint shape at a positive slip angle
Considering that there are two mechanisms driving the development of overturning
moment, each with a different relationship to slip angle, overturning moment can be either
positive or negative dependent on the tire’s design, on load, and on which mechanism pre
dominates under a given operating condition.
Chapter 8. Forces and Moments 295
Fortunately, the mechanics for lateral force and aligning moment are only modulated
by out-of-plane twisting, not changed completely. Thus, the string model presents a sim
ple conceptual model for them.
In the case of a real tire, the approximately symmetric straight free-rolling footprint as
shown in figure 8.4 is again distorted into a trapezoidal shape due to inclination angle, fig
ure 8.7. The lateral deformation of a longitudinal slice through contact becomes more
severe as the slice’s location moves from the lightly loaded shoulder toward the heavily
loaded shoulder. Due to the ideally symmetric character of the lateral stresses, sliding, if
it occurs, will be in front of the contact center. This will cause the lateral force resultant
to lie in front of the center of contact. Hence, one would expect a positive aligning
moment to be associated with positive inclination angle. Furthermore, the local rolling
radius of each longitudinal slice is inherently smaller as the slice’s location moves from
the lightly loaded to the heavily loaded shoulder. At some lateral location the local slice’s
rolling radius would be the same as the tire’s rolling radius. To the left of this section, the
tread surface would be driven, and to its right, it would be braked. The resultant differen
tial longitudinal stress distribution would also produce a positive aligning moment com
ponent in response to a positive inclination angle.
296 Chapter 8. Forces and Moments
Roadway curvature causes complex deviations of the tire force and moment properties
from the ones existing on a flat surface. Figures 8.10 and 8.11 [8] show ratios of force and
moment results from an external drum and a flat surface. These ratios are results from the
drum divided by results from the flat surface with both sets of data being taken at the same
test conditions. They were obtained on the TIRF machine at the Tire Research Facility,
part of the General Dynamics Company facility in Buffalo, New York. 9 That machine has
the unique characteristic of allowing measurements to be made on both a drum and a flat
surface using the same measuring head.
Unfortunately, the relationship between results on curved and flat surfaces is not only
complex, but depends on the tire diameter with respect to the diameter of the test surface
and on the individual tire construction. No simple rule of thumb exists for converting
force and moment data taken on a curved surface to data taken on a flat surface.
9TheTire Research Facility was originally associated with CALSPAN and in 2004 is owned by General
Dynamics.
298 Chapter 8. Forces and Moments
Figure 8.10: TIRF 1.7m (67 in.) roadwheel / TIRF Flat belt - slip angle
(Reprinted with permission from SAE 760029 1976 SAE International) [8]
Figure 8.11: TIRF 1.7m (67 in.) roadwheel / TIRF flat belt - inclination angle data
(Reprinted with permission from SAE 760029 1976 SAE International) [8]
If there is such a problem converting data from round to flat, why would anyone use a
round test surface? Historically, round surfaces were used for studying tire endurance and
high speed performance at the time that Evans [2] and Bull [3] began to study tire force
and moment properties. They began with what existed. Further, though it is easy to create
a low speed flat surface machine using either a slowly moving table like that in a large
milling machine or by moving the tire manipulator over a stationary table; creating a high
Chapter 8. Forces and Moments 299
speed flat-surface test machine is not easy. Such machines are much more complex and
expensive than drum machines, and not as durable. Internal drum machines allow the
ready introduction of real pavement surfaces, and adding water or ice. In the end, howev
er, the high-speed flat-surface test machine has become the preferred device for determin
ing dry surface forces and moments. It typically consists of a steel belt running between
two drums with a flat surface in the central contact zone maintained by using a special
bearing under the belt, figure 8.12. A major machine control task is to retain the lateral
position of the steel belt in the presence of lateral force. This is a dynamic process requir
ing a belt tracking system. Practically speaking, the TIRF Machine (where the belt is sup
ported by a hydrostatic air bearing) [9], figure 8.13, was the first truly viable high-speed
flat-surface machine. The MTS Flat-Trac® (where the belt is supported by hydrodynamic
water bearing) [10, 11, 12], figure 8.14, came later. It is the most commonly used test
machine in commercial service. 10
(Reprinted with
permission from SAE
730582 1973 SAE
International) [9]
The photograph is
personal.
10Since the introduction of the Flat-Trac®, it has passed through a number of models and in 2004 exists in a
light truck size.
300 Chapter 8. Forces and Moments
(Reprinted with
permission from SAE
800245 1980 SAE
International) [10]
The photograph is
personal.
(Reprinted from Rubber World Magazine 1996 with Rubber World permission) [13]
Chapter 8. Forces and Moments 301
Ignoring the question of surface contaminants like water, snow, ice, mud, etc., it is still not
easy to deal with the question of surface friction. Indoors, the common practice is to coat the
belt with sandpaper11. Sandpaper surfaces are believed to be both more realistic than smooth
steel and to give relatively stable friction.12 Does the friction of sandpaper correlate with that
of a real road? Often, not as well as would be desired. Figure 8.15 from Reference 13 gives
an example of the inherent problem. In this case, the indoor surface on the TIRF machine was
a 120-grit previous-generation THREE-M-ITE ® with a cloth backing. The outdoor surface
used by UMTRI (University of Michigan Transportation Research Institute) was a cross-
brushed concrete. Figure 8.15 shows the relationship between the data. It is not a single val
ued function relating forces on one surface to forces on the other. A true function exists only
at low slip ratios where one set of data is linearly related to the other. It is plain that the two
machines measured considerably different values even where the force data were functional
ly related. Some will say that there is a simple solution: test over-the-road on a real surface.
The question is which real road surface? And when?. Whitehurst and Neuhardt [14] showed
that real road surfaces vary in friction from day-to-day, season-to-season, and year-to-year.
Surface friction is a thorny issue that demands further work.
12To insure relative frictional stability, a statistical process control procedure must be applied to the abrasive
surface or experiments defining wear along with usage tracking must be applied.
(Reprinted from Tire Science and Technology, Vol. 20, No. 2, 1992 with Tire Society permission) [11]
Fixed systems may be configured with either unitary or distributed measuring heads.
Unitary heads gather all the load cells into a compact arrangement directly supporting the
spindle bearings, figure 8.16. A typical arrangement has a spindle supported by two bear
ings, one thrust and one free-floating. This arrangement allows direct determination of FX,
FY, FZ, MX, and MZ. Distributed heads have the load cells strategically placed over a con
siderable volume of space perhaps even surrounding the frames that form the tire manip
ulator as in the flat surface tire dynamics machine [15]. In either case, the geometric
arrangement of the cells is carefully defined so that the load cell outputs can be trans
formed into the forces and moments defined in Section 8.1.2. For unitary heads it is also
necessary to measure Rl and γ in order to account for the geometry so that the results can
be reported in terms of the tire or wheel axis systems.
Today, the unitary head is the common arrangement due to its compactness and high
stiffness resulting in a potential high-frequency response. When used in a free-rolling con
dition, one end of the spindle supports the tire and the other end is unconstrained with
respect to rotation. If braking is the only desired torque input, a brake rotor can be installed
on the end of the spindle shaft opposite to the tire. However, there is one drawback to using
a brake; it is difficult to control the slip ratio after the peak braking force is reached. If both
braking and driving torque are desired, it is common to place a coupling between the end
of the spindle shaft opposite to the tire and a hydraulic motor. The slip rate control prob
lem is non-trivial, but reasonable control can be achieved using a torque motor.
Load cell systems not only measure forces in the design direction of a given load cell,
but also measure a small amount of the force applied in other directions. For example, no
matter how well a measuring head is designed and constructed; applying a normal force
will cause small readings in the FX, FY, etc. load cells. These small but spurious compo
nents can be eliminated from linear systems through a calibration procedure that allows
computation of an interaction matrix. This matrix describes the output of all force and
moment channels arising from each separately-applied force or moment so that the inter
actions can be taken into account. Several SAE J-Documents, of which reference 16 is an
Chapter 8. Forces and Moments 303
Disadvantages
Does not use a real road surface
Does not use real weather conditions
Advantages
Real road surfaces
Real weather conditions
Disadvantages
Low tire testing rate (tires tested per unit time)
Less precise control of usage cariables (α ,γ , FZ, SR)
Some tests endanger operators
Reduce dynamic responses
Real weather conditions13
Real road surfaces14
moves to the case of combined inputs. The example results are expressed in the SAE Tire
Axis System. One or two problems provide an understanding of how the same data appear
when expressed in the ISO Wheel Axis System.
The example results shown should not be considered correct in detail for every tire.
Further, the magnitude of responses for various tire specifications15 can be quite different
one from another.
15A tirespecification is a unique design expressed in terms of mold profile, tread pattern, compounding, cord
characteristics, details of the tire composite lay-up, etc.
Chapter 8. Forces and Moments 305
Lateral force as a function of slip angle, figure 8.17, is approximately reflected with
respect to the point at which FY = 0, reference 17. As shown in figure 8.17, lateral force
is not typically zero at the point that α = 0.16 This is an important characteristic of tire
behavior related to on-center behavior and is discussed in section 8.4.1.4.1.
A negative lateral force is associated with a positive slip angle and a positive lateral
force is associated with a negative slip angle. In the normal driving range the slope of lat
eral force as a function of slip angle is negative.
The absolute value of slope of the lateral force curve at zero slip angle is commonly
Figure 8.17: Lateral force at a single normal force as a function of slip angle
16Itis possible to design such that FY = 0 when = 0, but this has a cost and has only been done as an R&D
exercise [18, 19].
306 Chapter 8. Forces and Moments
called the Cornering Stiffness [6]. This is a very important parameter in determining the
linear range behavior of vehicles, the area in which most driving is done. Cornering stiff
ness depends on load (normal force), figure 8.18. It typically increases up to some frac
tion of the tire rated load [6] then gradually falls off in magnitude as load continues to
increase. The data in figure 8.18 extend only to the peak of cornering stiffness with load.
Where the cornering stiffness peaks depends on the tire design. For good handling, it is
desirable that the peak occurs at or above tire rated load, as it does for the example tire.
This produces a positive value of load sensitivity [20, 21], which is the slope of the later
al force curve from 80 percent of tire rated load to 100 percent of tire rated load.
The increase in lateral force with slip angle is a complex non-linear function. The mag
nitude of FY eventually peaks and then declines in a similar way to that for a tire subject
to longitudinal braking. Think back to section 8.2.1 and the deformation of the string tire,
figure 8.3. As slip angle becomes large, more and more of the available contact area is
involved in sliding, thus, a maximum or peak amount of lateral force will be generated at
some slip angle. Beyond the slip angle associated with the peak, increasing sliding sys
tematically decreases the lateral force.
Aligning moment as a function of slip angle, figure 8.19, is also approximately sym
metrical with respect to the point at which MZ = 0, reference 17. However, in this case,
the slope in the normal driving range is positive. As shown in figure 8.19, the aligning
moment is not typically zero at the point when α = 0 17 nor is it typically zero at the slip
angle for which the lateral force is zero. This is an important characteristic of tire behav
ior related to on-center behavior and is discussed in section 8.5.1.4.1.
Figure 8.19: Aligning moment at a single normal force as a function of slip angle
The absolute value of the slope of the aligning moment curve at zero slip angle is com
monly called the aligning stiffness [6]. This is an important parameter in determining the
linear range understeer behavior of vehicles, chapter 14. Note that the aligning stiffness
17Itis possible to design for a particular value of MZ when = 0. Though this is a complicating detail, the
required design modifications are made to insure proper on-center vehicle performance [18, 22].
Chapter 8. Forces and Moments 307
As noted above, lateral force has the basic form illustrated in figure 8.17. A change in
load produces detailed adjustment in the shape of the FY data, figure 8.21. The cornering
stiffness generally increases with load as was illustrated in figure 8.18, but the rate of
increase declines as load increases. The peak of the lateral force curve occurs at higher
and higher slip angles as the normal force increases. However, if a pseudo coefficient of
friction is computed by dividing the peak value of FY at each load by the load itself, one
discovers that the frictional capability of the tire declines with increasing load. Thus, high
performance vehicles on a dry road will exhibit their maximum cornering ability using
large tires operating at relatively light loads.
In a practical sense, the initial region for lateral force up to the slip angle associated
with the peak in FY is the range where a driver can maintain vehicle control. The proba
bility of loss of control in cornering becomes higher as the operating slip angle approach
es the angle associated with the peak lateral force for the normal load, FZ. In general, low
ering the tire aspect ratio causes the lateral force data and data for the other forces and
moments to exhibit higher initial slopes and a more abrupt transition from the initial
slopes to the behavior at high slip angles. Thus, the width of the transition zone between
308 Chapter 8. Forces and Moments
well controlled behavior and a possible loss of control is smaller in terms of slip angle for
lower aspect ratio tires.
For higher aspect ratio passenger tires like the one that provided the data in figures
8.21, 8.22, 8.24, and 8.25, the overturning moment, figure 8.22, is basically generated by
a lateral shifting of the tire contact patch, which can be approximately modeled by the
deformation of a string on an elastic foundation, section 8.2.1, figure 8.3. The trend in MX
with slip angle has a positive slope.
Figure 8.21: Lateral force vs. slip angle and load for a 75 aspect ratio tire
Figure 8.22: Overturning moment vs. slip angle and load for a 75 aspect ratio tire
Chapter 8. Forces and Moments 309
The situation for a low aspect ratio tire like that used to obtain the data shown in fig
ure 8.23 is different. The out-of-plane twisting that causes the footprint shape to become
trapezoidal, see figure 8.5, becomes the predominant effect. The result is a basically neg
ative trend of MX with increasing slip angle. As load increases, the lateral displacement
of the contact patch, as shown in figure 8.3, also becomes a significant source of overturn
ing moment. This is seen at the rated load and above in figure 8.23.
Figure 8.23: Overturning moment vs. slip angle and load for a 45 aspect ratio tire
Typically, results for MX can be very complex. Depending on the tire aspect ratio and
inflation pressure, almost any relationship can be found to hold.
Like lateral force, curves of aligning moment versus slip angle retain the same charac
teristic shape as load changes, figure 8.24, but change in amplitude.
Figure 8.24: Aligning moment vs. slip angle and load for a 75 aspect ratio tire
310 Chapter 8. Forces and Moments
Curves of loaded radius versus slip angle are bell-shaped, and the shape becomes more
pronounced as load increases, figure 8.25.
Figure 8.25: Loaded radius vs. slip angle and load for a 75 aspect ratio tire
The data in all figures representing pure slip angle behavior have been shown as a func
tion of both positive and negative slip angles. Sometimes when there is no interest in the
offsets near zero shown in figures 8.17 and 8.19, engineers will compute “mirrored” data
using the method illustrated in Eq. 8.2.18
f(α) = (f(α+) – f(α-))/2 (8.2)19
Data obtained in this way are used to produce carpet plots like the one in figure 8.26,
where both slip angle and normal force are shown as negative numbers. Carpet plots are pro
duced by plotting both normal force and slip angle on the horizontal axis. For each succes
sively higher slip angle, the origin of the normal force axis is moved to the right. The result
overlays a group of FY vs. FZ graphs for single slip angles, one on top of the other, where the
origins have been moved to the right by an amount proportional to the slip angle. The loads
at each slip angle are then connected. The constant slip angle - variable load curves together
with the constant load - variable slip angle curves form a comprehensive data set.
18Reference 17 provides a detailed discussion of different ways to look at slip angle only data.
19 + signifies positive slip angle. - signifies negative slip angle.
Chapter 8. Forces and Moments 311
The response of lateral force to inclination angle, figure 8.27, is much weaker than the
response to slip angle, figure 8.17. The inclination stiffness, the inclination equivalent to
cornering stiffness, is 10 percent or less of the cornering stiffness in terms of force per
degree. In the example case used here, the inclination stiffness is less than 5 percent of the
cornering stiffness. Note that the slope of lateral force with inclination is positive; a pos
itive change in inclination angle is associated with a positive change in lateral force. This
is the reverse of the situation with slip angle where a negative lateral force is associated
with a positive slip angle. As expected, the lateral force at zero inclination angle is not
zero, due to pull forces. One of the pull forces (plysteer) is associated with a structurally
induced self-steering and the other (conicity) is associated with a structurally induced self
312 Chapter 8. Forces and Moments
In the examples, the tire is inclined positively, such that the lateral force generated by
inclination is positively directed. Thus, the inclination-generated lateral force adds to the
lateral force generated by a negative slip angle in the Tire Axis System, see section 8.2.1.
If the tire is inclined at a negative angle, it is evident that the reverse applies.
The lateral force, figure 8.31, shows three characteristics that are due to the asymmet
ric behavior that occurs in the presence of combined slip and inclination angles. Note that
as slip angle becomes quite large, inclination angle does not have much effect on the mag
nitude of the lateral force. This is because the tire is operating beyond the peak of the fric
314 Chapter 8. Forces and Moments
tion curve in lateral slip. It is moving closer and closer to full lateral sliding as the slip
angle continues to increase.
Inclination angle offsets the data in the vicinity of zero slip angle, as is shown in
figure 8.27. This apparent simple addition of the effects of slip and inclination angles first
noted by Evans [2], is only valid at low slip and inclination angles. As noted below, behav
ior at higher angles is more complex.
When the lateral forces due to slip angle and inclination angle have the same sense and
act in the same direction, the peak lateral force is higher and sharper than in the case of
slip angle only. When the inclination angle is positive, this occurs for a negative slip angle.
Both lateral forces are then positive, to the right.
Figure 8.32: Effect of inclination angle on
overturning moment as a function of slip angle
When the lateral forces due to slip angle and inclination angle have opposite senses,
the peak in lateral force is lower and broader than in the case of slip angle only. When the
inclination angle is positive, this occurs for a positive slip angle. The lateral force due to
the inclination angle is positive, to the right, but the lateral force due to the slip angle is
negative, to the left.
Figure 8.33: Effect of inclination angle on aligning moment as a function of slip angle
Chapter 8. Forces and Moments 315
Overturning moment, figure 8.32, is offset negatively, as would be expected for a pos
itive inclination angle on the basis of the results for inclination alone, shown in figure
8.28. It is important to note that the offset is larger when the direction of lateral forces
induced by the slip angle and the inclination angle are the same, and the offset is smaller
when they are in opposite directions.
Aligning moment, figure 8.33, is warped into asymmetry, much as is the case for lat
eral force, except that in this case the direction of the aligning moments is important.
When the aligning moments have different directions, the magnitude of the aligning
moment is reduced in comparison with that for a pure slip angle. This occurs when incli
nation angle is positive and slip angle is negative.
When the aligning moments are in the same direction, the magnitude of the aligning
moment is increased with respect to that observed in the case of pure slip angle. In the
example, this occurs when the inclination angle and the slip angle are both positive.
Figure 8.30 showed that the vertical stiffness of the tire increases with increasing incli
nation angle. This appears in the combined case, figure 8.34, with the effect being aug
mented when the components of lateral force due to inclination and slip angles act in the
same direction. The lateral distortions of the tire carcass then reinforce each other.
Figure 8.34: Effect of inclination angle on loaded radius as a function of slip angle
enters free-control (see [5]) and the aligning moment will become zero, i.e., self-steering,
but typically the lateral force is not zero. The front of the vehicle will be pushed to the
side and a yaw moment will be imposed on the car. In this case the car will only travel
straight along a straight road if the driver exerts an aligning torque to maintain zero later
al force on the front of the car to maintain direction (see [5]). The driver is aware that the
car is not tracking properly and becomes fatigued by the process of simply driving straight
ahead. If the driver is inattentive or ceases to maintain trim, the car may change lanes,
cross the centerline of the road, or drift off the road altogether. Also, the driver will bring
his car in for wheel alignment service which will continue to be unsatisfactory as long as
the properties of the tires are unsatisfactory, as discussed later in this section.
Figure 8.35: Small slip angle lateral force and
aligning torque exerted on the front axle of a car
The first authors to demonstrate that tire-induced steering pull had its source in the
facts mentioned in the last paragraph were Gough et al. in 1961 [25]. Unfortunately, their
discussion was not as clear as it might have been. Topping [26] gave a clearer explanation
in 1975. He showed that tire-induced steering pull was a direct result of the presence of
residual lateral force on the front axle when the steering wheel was allowed to seek its free
position. Topping showed that the rear axle could basically be ignored as a source of tire-
induced vehicle pull, but is a source of vehicle side-slip, denoted dog-tracking, when trav
eling straight. The conclusion that the rear axle did not contribute to the tire-induced pull
problem was based on experiments and simple, but unpublished, analysis. In 2000, Lee
[27] published a model equivalent to the one used by Topping.
There are two features of tires that give rise to a residual lateral force.20 The first is
generally termed conicity. Conicity is usually the result of the tire belt being applied
slightly off-center, as a result of manufacturing variance. This constrains the rolling radius
on the side of the tire toward which the belt is shifted to be slightly smaller than the rolling
radius on the other side of the tire. At the belt level, where rolling radius is determined,
the tire acts as if it had a conical cross-section, and develops lateral force and aligning
20Residual lateral force is the free control expression of tire pull. In fixed control, the case where the driver
is maintaining trim, tire-induced pull expresses itself as a residual aligning moment (torque).
Chapter 8. Forces and Moments 317
torque as if it was cambered toward the more constrained side of the tire. The distribution
of conicity values for a group of tires is very flat, as shown schematically in figure 8.36.
The mean value is near zero, but may not be precisely zero due to asymmetries in the man
ufacturing process. Other effects are termed plysteer. Plysteer arises when the tire struc
ture is anisotropic, causing the tread band to undergo in-plane shear when it is forced to
become flat in the footprint [18], figure 8.37. Additional in-plane shearing occurs due to
the change in belt tension in the contact zone, and causes footprint curvature and a
plysteer residual aligning moment or torque. Pottinger [28] has discussed these effects in
detail. Plysteer effects are due to tire design.21 The distribution of values is narrow, as
shown schematically in figure 8.36, centered on the design value, which can have a con
siderable range as illustrated in figure 8.38. Indeed, the research used in producing figure
8.38 could have yielded a target plysteer anywhere in the range from –370 N to + 370 N
with an associated residual aligning torque from –11 Nm to +11 Nm.22
Figure 8.36: Schematic of distribution of conicity and plysteer
21[18], [19], and [22] present basics on designing to eliminate or modify plysteer components.
22Note that practical considerations would not have allowed the full range of values indicated to be present in
production tires.
(Reprinted with permission from SAE 760731 1976 SAE International) [18]
The importance of conicity and plysteer effects has led to an SAE recommended prac
tice covering their determination [29].
As pointed out in reference 28 and in Yamazaki et al. [30], the resulting steering pull
depends on tire properties, suspension alignment, and road crown. The consequence for
the tire industry is that automobile manufacturers set limits on components of pull force
for their tire suppliers. These limits make tire-induced pull effects manageable from the
standpoint of the automobile manufacturers. All restrict conicity to a narrow band about
zero so that they can ignore its effects. The situation with regard to plysteer residual
aligning torque (moment) is more complex and less satisfactory. Each OEM establishes
an allowable band of values for residual aligning moment or torque, depending on their
own design philosophy. Some automobile manufacturers use tire plysteer residual align
ing torque (PRAT) to counteract road crown. Thus, their tire specifications require that
tires supplied to them have particular values of plysteer residual aligning torque. Other
automobile manufacturers demand zero PRAT, within a narrow tolerance, and counteract
road crown effects solely through the suspension alignment. This can lead to vexing dif
ferences in tire design that tire manufacturers must introduce in nominally identical tires
in order to meet the requirements of different automobile manufacturers. These differ
ences are not obvious to tire stores or purchasers. The result can be pull problems when
OE tires are replaced because the available tires in the aftermarket may not match the
properties specified by the car manufacturer. The car may not travel straight without driv
er input, even when the suspension is aligned in accordance with OEM specifications.
is expelled be broken so that intimate contact is achieved between the tire tread and the
road. Expelling the gross water layer depends on having both an appreciable groove vol
ume in the tire tread and having voids present on the road surface, termed macro-texture.
Often the road surface macro-texture will be worn away over time leaving a road sur
face with a low void content. In this situation the chance of a wet traction accident is sig
nificantly increased, particularly at high speeds when the lack of drainage contributes to
hydroplaning during a heavy rainfall. To counter this problem, highway departments treat
Figure 8.39: Tire contact overlaid over rain grooves
23A similar sensation exists when driving on a bridge deck surfaced with a steel grating.
24M. Pottinger and J. McIntyre.
320 Chapter 8. Forces and Moments
rolling tire because small lateral movements made the stress field oscillate in a repeatable
fashion each time the lateral motion of the car took it across a groove spacing. Peters
applied finite element analysis to the stress field for tires with a variety of tread patterns
and predicted that the lateral force waveform would look like the one portrayed in figure
8.41. This waveform is quite similar to the one obtained from the indoor experiments.
Recently, Nakajima [34], applied the same concept used in Peter’s work and obtained
an excellent correlation to subjective ratings by drivers, figure 8.42. This figure shows
subjective ratings of groove wander by the test driver as a function of the computed pea
to-peak amplitude of the lateral force waveform resulting from tire interaction with longi
tudinal grooving in the pavement.
Figure 8.40: Illustration of coincidence concept
of tread pattern to rain groove interaction
(Reprinted from Tire Science and Technology, Vol. 29, No. 4, 2001 with Tire Society permis
sion) [33]
Figure 8.41: Example tread pattern to rain groove interaction lateral force waveform
(Reprinted from Tire Science and Technology, Vol. 29, No. 4, 2001 with Tire Society permis
sion) [33]
Chapter 8. Forces and Moments 321
(Reprinted from Vehicle Systems Dynamics, Vol. 6, No. 2, Taylor & Francis Ltd.,
http://www.tandf,co.uk/journals) [34]
be seen in the discussion of figure 8.44, the peak in driving is not usually equal to the peak
in braking.
Figure 8.43: A typical longitudinal force response to braking torque application
Slide longitudinal force is the value of FX that occurs when the slip ratio in braking
becomes -1.00 (the wheel is locked). When wheel spin occurs in driving, SR can exceed
1.00. Examples of cases where spin occurs are the launch of a dragster or when excess
driving torque is applied on a slick surface. In this particular example case, FX SLIDE was
obtained by extrapolation of the end of the response curve.25
Figure 8.44 illustrates FX (SR) over a range of slip ratios encompassing both driving
and braking. It shows the response asymmetries just mentioned.
The magnitudes of the driving and braking data are not the same. For example, at the
highest load the magnitude of the peak longitudinal force in driving is over 4500N where
as the magnitude of the peak longitudinal force in braking does not quite reach 4000N.
25In the example case the experiment was terminated before slide in order to prevent tire flat-spotting which
will lead to vibrations that can cause test load control problems.
Chapter 8. Forces and Moments 323
The location of the peaks is also different. The driving peak occurs at a lower slip ratio
than the braking peak.
In general, as noted a few paragraphs back, the driving and braking longitudinal force
curves are generically similar, but have different values of the descriptive parameters. The
braking response surface is warped differently than the driving response surface because
of these parametric differences which arise from footprint stress distributions that do not
mirror each other as the tire changes from braking to driving, see chapter 7.
The rate at which slip ratio is applied affects the response obtained. Application of
torque and release of torque, the two sides of a half triangular wave that can be imposed
using test machines equipped with controlled-rate hydraulic systems, do not typically lead
to the same FX (SR) response curve. This effect, which appears to be thermally induced,
is separate from the tire dynamics and relaxation effects noted in section 8.4.4. Figure 8.45
[11] is an illustration of this effect. As the slip ratio ramp rate is reduced, the total exper
iment becomes longer, and the difference increases between the longitudinal force curves
for application and release. Also, if the tread is severely overheated, the surface will
become tacky and the apparent coefficient of friction will change because of altered adhe
sion, see chapter 11. Wear during testing should also be monitored in order to avoid data
which are, unknowingly, dependent on the wear state.
Always bear in mind the importance of test surface friction. As illustrated in figure 8.15
and discussed in Section 8.3.1, even different dry surfaces can give quite different results.
Figure 8.45: Longitudinal force in response to both
braking torque application and release
26Longitudinal force in straight ahead operation is usually thought of as producing only vehicle acceleration
or deceleration.
324 Chapter 8. Forces and Moments
In the design of vehicles, particularly with front wheel drive, considerable engineering
effort is expended to insure against inadvertent introduction of yaw moments or steering
system inputs due to longitudinal forces occurring during acceleration or deceleration.
Anti-lock braking systems and traction control systems help in limiting yaw moments due
to variable friction across the roadway. Stability control systems deliberately induce a sta
bilizing yaw moment by inducing a desirable longitudinal force differential.
In straight line operation there are effects due to differences between the tires mounted on
an axle, particularly on the drive axle of front wheel drive cars, that can contribute to unde
sirable yaw moments. These effects are lumped together under the name torque steer [28].27
If a torque steer problem arises, the engineer should check three potential tire sources of force
differences: rolling radius, torsional spring rate, and inconsistent frictional properties. The
most common source is a side-to-side rolling radius difference. If the difference in rolling
radius between the left and right tires is too small to cause the car’s differential to function,
but still significant, the vehicle will follow a radius defined by the rolling radius difference.
On a straight road, relatively minor differences can be significant.
27Always remember that the car design itself may contribute to or be the torque steer problem. Some vehicle
design sources are described in Reference 28, which also outlines a road test procedure to distinguish between
the various sources of tire induced vehicle pull.
28The version shown in Figure 8.46 is the all slip angle version. It is a modification of the figure shown in
reference 35 to include the form of Eqs. 8.3 - 8.6, which appear in an Appendix to the Reference 35.
29F(s,β ) is usually assumed to be an ellipse, but as pointed out in the discussion associated with Figure 8.52,
this is not always a valid assumption.
Chapter 8. Forces and Moments 325
At least three ways are conventionally employed for functionally or graphically visu
alizing combined data. The first, figure 8.47, represents lateral force, FY, at a fixed slip
angle as a function of slip ratio. Longitudinal force, FX, for driving is also included in fig
ure 8.47 to illustrate a point discussed in the next paragraph. The second way, figure 8.48,
represents lateral force at fixed slip angles as a function of longitudinal force. This repre
sentation is the one that is mainly used in this section. The third, figure 8.49, represents
Figure 8.46: Slip Circle
The effect of combined slip on longitudinal force is to make the peak longitudinal force
occur at lower and lower magnitudes and at ever higher slip ratios, figure 8.49.30
Unfortunately, actual data, like these, do not always perfectly match the ideal as represent
ed in figure 2.18 in reference 7. Although the expected trend is generally present, at low
Figure 8.49: Longitudinal force as a function of braking slip ratio at fixed slip angles
30The data are the average of tests of four tires and show some of the difficulties that can occur in experimentally
characterizing real force and moment data. These data were developed in the course of preparing Reference 37.
Chapter 8. Forces and Moments 327
Figure 8.50: Illustration of the return of the FY(FZ) curves at high slip ratios
(Reprinted with permission from SAE 960180 1996 SAE International) [35]
slip angles the behavior is often not in good accord.
If the results from cycles of torque application and release at numerous slip angles are
plotted together, as in figure 8.51 drawn from reference 36, it is evident that the shape of
F(s,β) changes with slip angle. Also, as shown by Pottinger et al [37] the shape of the
response depends on normal force. Thus, many diagrams like figure 8.51 are needed to
represent the combined response in actual operation. Also, as noted in Reference 36, there
Figure 8.51: Lateral force as a function of slip angle and longitudinal force
(Reprinted with permission from SAE 960180 1996 SAE International) [35]
approaches 1.00.
Aligning moment as a function of cornering and driving or braking is a more difficult
problem mathematically than lateral force. MZ versus FX, figure 8.54, is just not a simple
function. This is because anything that affects the lateral location of the resultant longitu
dinal force vector is going to affect the aligning moment. Inclination can have enormous
effects on MZ in the presence of torque. This is illustrated in figure 8.55, where the incli
nation angle is minus three degrees. Indeed, in the case of pure inclination combined with
longitudinal force the resulting diagram (unpublished) bears a striking resemblance to a
line drawing of the petals of a Spider Chrysanthemum blossom.
As noted by Pottinger [36], overturning moment is also not an orderly function of longi
tudinal force.
Figure 8.54: Aligning moment as a function of slip angle and longitudinal force
F = A (1 – e-Bt) (8.7)
Where:
A = the steady state value of F.
B = a constant with the form 1/τ.32
F = a particular force or moment.
t = time.
Assuming a constant test velocity, v, equation 8.7 can be restated in terms of distance
rolled.
F = A (1 –e-(X/l ) x ) (8.8)
Where:
l = vτ is relaxation length.33
x = vt
The value of relaxation length is different for the different inputs, for example, a torque
input will produce a different result than a slip angle input. These differences are because
31This book summarizes a professional lifetime's experience on tire transient effects and tire force and moment
modeling.
32 τ is the length of time require for e-Bt to change from 1 to 1/e.
33In the context of this discussion, relaxation length is defined as the distance a tire subject to a step input in slip
angle, load, torque, or other usage variable must roll in order to reach 63.2 percent of the steady state value that
the force and moment in question will attain as a result of the step input in the pertinent usage variable. In certain
cases, for example, step slip angle, there is a corresponding physical definition as noted in Reference 6.
Chapter 8. Forces and Moments 331
the relevant deformations are not the same. The value of the relaxation length also varies
with testing parameters such as load and with tire design features.
Relaxation effects with respect to slip angle have been studied with considerable thor
oughness over many years. Figure 8.56 is a sample response. In this case, the step was
imposed by rapidly imposing a fixed slip angle on a slowly rolling tire. Data were taken
after a period of steady state rolling sufficient to characterize the pull force and uniformi
ty signals produced by the tire, which were then subtracted to produce the data plotted in
figure 8.56. It is also common to simulate a step steering response by loading a non-mov
ing tire onto a plank type roadway and then to start the roadway to generate an equivalent
to a step steer response [40].
Figure 8.56: Example data from a relaxation length experiment (uniformity and
pull force removed in data processing)
It is important to note that lateral relaxation length in response to slip angle is depend
ent on slip angle magnitude [41], figure 8.57. Darnell, Mousseau, and Hulbert [42] pro
vide force development curves at various slip angles and loads.
Figure 8.57: Distance from the start of motion to
equilibration at steady state for radial tires
(Reprinted with permission from SAE 760031, 1976 SAE International) [41]
Relaxation in transient inclination response has been studied much less than in the case
of relaxation due to imposed steer. Higuchi and Pacejka [40] provide results from both
332 Chapter 8. Forces and Moments
experimentation and modeling. An experimental complication in this case is that the only
practical way to simulate a step, given existing equipment limitations, is some form of
static preloading followed by rolling from the preloaded condition. Preloading at an incli
nation induces a static lateral force response, which is not the same as the normal response
to inclination. The result is that it is necessary to assume that the relaxation length repre
sented in the change from the static response to the normal rolling response is indeed the
correct relaxation length for inclination.
Relaxation length during torque application is important to the designers of anti-lock
braking systems. Extensive information does not exist. The literature [38, 43, 44] indi
cates that the relaxation length in braking is appreciably less than in the case of cornering
and grows shorter as the applied longitudinal force becomes closer to producing a com
pletely locked condition. In the step to a locked wheel case the behavior of the tire,
excluding stick-slip oscillations, would be very like the behavior observed in a static lon
gitudinal spring rate test.
(Reprinted from Tire Science and Technology, Vol. 15, No.3, 1987 with Tire Society permis
sion) [46]
Chapter 8. Forces and Moments 333
ωp = ω/v (8.11)
This ability to express the results in terms of path frequencies, i.e., radians steered per
meter of travel, allows the data to be exhibited as in figure 8.59. These data illustrate the com
plexity of the actual response as path frequency increases, especially for aligning moment.
Figure 8.59: Sinusoidal steer
(Reprinted from Tire Science and Technology, Vol. 20, No.3, 1992 with Tire Society permis
sion) [11]]
334 Chapter 8. Forces and Moments
(Reprinted from Tire Science and Technology, Vol. 20, No.3, 1992 with Tire Society permis
sion) [11]
34Under dynamic conditions, it is always well to remember that most force measuring devices are to some degree
also accelerometers.
Chapter 8. Forces and Moments 335
As observed by Schuring [45], transients are important for limit handling, primarily in
terms of the effect on phase lag, not in terms of amplitude effects. For example, consider
a violent maneuver and a vehicle with a 20:1 steering ratio. The upper end of human per
formance is reached when about 800 degrees per second is applied at the steering wheel.
This would produce 40 degrees per second at the road, or about 0.7 radians per second.
Assuming a 100 km/hr speed, the path frequency would be approximately 0.025, which
would have little effect on lateral force magnitude, but would yield a measurable phase
shift or lag. The lag effect could be accounted for with the relaxation length model as
noted in Section 8.4.4.1. Thus, the amount of detail inherent in sinusoidal excitation data
is not really necessary for handling studies. Certainly more important than the pure later
al effect would be the combination of tire lateral dynamics with load variation effects
occurring during the vehicle maneuver.
Schroeder and Chung [47] demonstrated that the effect of the relaxation length associ
ated with slip angle on transient handling at modest lateral accelerations is discernable,
but not nearly as important as the effect of cornering stiffness under normal driving con
ditions.
increased further.
Increasing inflation reduces the size of the footprint, and thus the amount of tread rub
ber in contact with the road. At some point, the reduction in cornering stiffness due to a
lower area of contact outweighs the increase in carcass stiffness due to increased pressure
and the cornering stiffness begins to fall. A reduction of aligning torque is associated with
an increase in inflation pressure because the shorter footprint also reduces pneumatic trail,
section 8.2.1. Figure 8.61 is a schematic representation of the phenomena just described.
Figure 8.61: Aligning moment corrected and uncorrected for inertia
Additionally, increasing unit loading on the tread rubber reduces friction as shown by
Ervin and associates [49]. Thus, if the inflation pressure is adequate to prevent buckling
under the expected lateral and longitudinal forces, increasing the inflation pressure will
likely lead to lower apparent friction.
Figure 8.63: |FXpeak| vs. water depth and speed for a new tire
(Reprinted with permission from SAE 962153 1996 SAE International) [50]
338 Chapter 8. Forces and Moments
The changes just discussed will warp tire lateral and longitudinal force performance
with speed to generate a sharper peak as sketched in figure 8.65. It is evident that the con
sequence of warping the tire’s response for the pure slip cases, which are the axes in the
slip circle, figure 8.46, will be to change the shape of the friction ellipse as speed rises.
Thus, the combined behavior is speed dependent.
Figure 8.65: Expected warping of either lateral or longitudinal force with speed
35The discussion in this reference is extensive and includes the actual aging changes in the tire compound stiff
nesses, a discussion of aging in service, and a never-explained observation that two different aging mechanisms
are discernable from the force and moment data.
Chapter 8. Forces and Moments 339
increase of cornering and aligning stiffnesses with the age of the tire. Using cornering
stiffness as an example, the changes can be appreciable, see figure 8.66, and the rate of
change is strongly dependent on storage temperature. Eq. 8.12 shows how the rates of
change at different temperatures are related, for temperatures below about 70°C (158°F).
In a practical sense, whenever tire forces and moments are compared for different tire
specifications or even for different samples drawn from the same tire specification, it is
important to know the age of each sample and how each one has been stored prior to testing.
Samples drawn from the same tire specification, but stored under different conditions can
yield very different force and moment properties. And old samples are not the same as new
ones. “Control” tires change just due to storage times and conditions. Machine correlations
can be rendered invalid by aging effects. The rule is: be careful and never ignore aging.
reduction was dependent on distance and slip angle magnitude, i.e., lateral force level, as
illustrated in figure 8.69. It appears that after traveling some distance at a given slip angle
the cornering stiffness stabilizes, and ceases to change. Interestingly, operation at a slip
angle of one degree did not appear to induce a break-in. Thus, there must be a threshold
strain level for the break-in effect.
Figure 8.67: Matrix experiment to determine the
(Reprinted with permission from SAE 810066 1981 SAE International) [52]
(Reprinted with permission from SAE 770870 1977 SAE International) [53]
Chapter 8. Forces and Moments 341
(Reprinted with permission from SAE 770870 1977 SAE International) [53]
The break-in effect did not depend on removal of a skim coat by wear. Thus, the source
of the break-in effect is internal to a tire’s structure.
The reduction in cornering stiffness was independent of the tread depth so the source
of the break-in effect is within the carcass and belt package, not in the tread.
The implication of these results is that strain cycling of the tire carcass, induced by
applying a pure slip angle, will alter a tire’s force and moment properties by a magnitude
that is dependent on the slip angle, applied normal force, distance traveled, and time since
the end of operation at a higher slip angle. Practically, this means two things.
A force and moment test protocol should be designed so as to expose a tire to approx
imately the same strain state an engineer would expect to occur on the vehicle.
In a real emergency maneuver, tire forces and moments will change throughout the
actual maneuver.36
Wear itself leads to increased cornering stiffness.37 This is due to the effect on tread
element shear stiffness of tread element height or groove depth [54]. The effect is most
marked in tires with appreciable tread depth, like snow tires or truck tires. By way of
example, the change from a 15 mm tread depth to a 3.75 mm tread depth as a 295/75R22.5
steer axle tire wore down by 75 percent increased its cornering stiffness by 48 percent
[50]. In the case of blocky tread patterns, wear will also increase longitudinal stiffness as
the tread depth is reduced.
36In the latter part of Reference 52, a limited experiment tracing small samples of tires over thousands of kilome
ters of highway use indicates that normal driving does not lead to appreciable changes in tire lateral forces from
those which exist in new not-broken-in tires. Apparently, aging, break-in, and wear combined to produce compen
sating effects. Thus, a sudden exposure to high slip angles will lead to changing force and moment properties in a
real, one-time only, emergency maneuver.
37Racing tire engineers minimize tread depth to achieve the maximum possible cornering stiffness.
342 Chapter 8. Forces and Moments
neers are very concerned with tread temperatures because special racing compounds vary
strongly in adhesion depending on tread temperature. Chapter 11 considers both the adhe
sion and viscoelastic components of friction.
The stiffness of the tire structure declines as its temperature increases primarily
because of a decrease in the stiffness of the compounds that form the matrices of the tire
composites. To a first approximation, for normal environmental conditions the tempera
ture of a tire increases by a given temperature increment for a defined operational state
regardless of the tire’s cold temperature. Figure 8.70 shows that cornering stiffness
decreases with the logarithm of the tire’s cold temperature for a tire soaked at a constant
temperature and then tested. Viewing the soaked results as being the effect of varying the
ambient temperature, we can deduce the effect of variations in ambient temperature on
force and moment data. It is clearly important to maintain good control of ambient tem
perature during force and moment testing.
Figure 8.70: Cornering stiffness coefficient vs. logarithmic temperature
38Micro-texture is the small sharp texture that feels rough on sandpaper [55].
Chapter 8. Forces and Moments 343
(Reprinted with permission from STP 929 1986 ASTM International) [14]
The question of the effect of surface contamination (water, snow, and ice) is covered
thoroughly in chapter 11 from the standpoint of friction. In general the effect of contami
nates is to grossly alter tire force and moment properties.
The figures in reference 50, of which figure 8.63 is an example, illustrate characteris
tic tire responses to water depth, test speed, tread depth, and load. Typically, force levels
in the wet decline with respect to those measured in the dry as:
- Tread depth declines due to wear40,
- Water depth increases,
- Speed increases,
and
- Load decreases.
Effectively the force and moment response surfaces become warped.
39These data were acquired over a limited period in the autumns of 1996 and 1997.
40Interestingly, Williams and Evans [57] have shown that tread wear unevenness decreases tire performance on a
wet pavement.
Warping of all features of the tire forces and moments, both stiffnesses and friction lev
els, on contaminated surfaces is illustrated in figure 8.72, which compares behavior on a
dry road, a snow covered surface, and ice. Plainly, using a simple scaling factor to account
for surface differences is not an adequate representation of real data.
Figure 8.72: Lateral force dry, on snow, and on ice for two tire specifications
eter models can be used as substructure elements, but that requires determining some
structurally-related parameters. Empirical models with prudent insertion of some lumped
parameter aspects (making them semi-empirical) are the current models of choice for
evaluating tire forces and moments in vehicle modeling.
This approach began with the work of von Shlippe and Dietrich [66] where the infi
nitely long transverse element was a string. Later von Shlippe and Dietrich [67, 68]
included the effect of contact width by representing the tire by two parallel strings. Over
41The emphasis in this chapter is on forces and moments. Lumped parameter models are widely used in studying
dynamics in the wheel plane for purposes of ride, and impact with road irregularities. They are also important in
trying to deal simultaneously with ride factors, and forces and moments.
346 Chapter 8. Forces and Moments
the years the string on an elastic foundation approach has been elaborated to include mul
tiple parallel strings and an elastic tread, figure 8.74. Chapter 5 of Pacejka [38] discusses
string models in great detail with a particular emphasis on transients.
Figure 8.74: Tire model with single-stretched string and model extended
with more parallel strings provided with tread elements with longitudinal
flexibility from "tyre and vehicle mechanics"
(ISBN 0750651415), H. B. Pacejka, page No. 220, copyright 2002, with permission of Elsevier) [38]
Gough [69, 70] replaced the string with a composite beam, a more appropriate structure
to represent a radial tire. This concept led to a number of design insights about forces and
moments. Sakai [71] used the beam on the elastic foundation concept, but with the beam
formed into a hoop. Then to simplify the situation, he assumed that the beam was rigid later
ally, but could bend in the tire-wheel plane, to establish the footprint. The flexibility of the
tread was used to generate longitudinal and lateral forces along with the relevant moments.
Carrying the idea of a flexible tread to its logical conclusion, one can simply allow the
tread to possess all of the required flexibilities. The carcass can be considered rigid. This
generates the so-called brush model, figure 8.75. It is capable of simulating pure and com
bined slip forces and moments, given the right selection of properties for the bristles.
However, simulation of transient effects requires a flexible carcass so that it is now com
mon to attach a brush tread to a flexible carcass. A large number of authors have used ver
sions of the brush model. Again, Pacejka [38] contains extensive discussions.
42The precise form of the "standardized" models in use often varies from corporation to corporation and from one
software package to another, even though they often share a common origin, because the engineers and program
mers who devised the particular version had divergent ideas in detail.
Chapter 8. Forces and Moments 347
vehicle dynamicists who use commercial software like DELFT TYRETM can concentrate
on vehicle simulation without being intimately concerned with understanding tire struc
tural behavior or becoming an expert on tire data.43
Figure 8.75: The brush tire model
(Reprinted from "Tyre and Vehicle Mechanics" ((ISBN 0750651415), H. B. Pacejka, Page No.
94, Copyright 2002, with permission of Elsevier) [38]
In this section, we focus on modeling where, by one method or another, the parameters
in the model can be related to actual tire behaviors like cornering stiffness, peak coeffi
cient of friction, etc. Neural network modeling is not considered because it gives no
insight on how to deal with what-if questions. For example, a vehicle dynamicist may
wish to know what happens if the tire has the cornering stiffness increased by 10 percent.
This is an appropriate question that a tire modeling routine should be able to answer.
Perhaps the simplest way to deal with empirical data is to apply interpolation to tables of
tire force and moment values. This has been done in the past and can be done efficiently today
for simple cases where the forces and moments are functions of two, or at most three, tire
usage variables. However, interpolation quickly becomes an unwieldy procedure in the face
of combined operations and transients, where large amounts of experimental data are
required. Simple interpolation has therefore become relatively uncommon.
Pure polynomial fits can work well for a limited range of tire data and for data con
fined to restricted operational ranges. Polynomials can be made useful for fitting data
accurately over a wide range if used in structured sets. For example, it is possible to divide
the total data range into small sections, then fit each section by a power series, and pro
vide good continuity at the section joints [72]. Another way to ensure continuity and tight
fits is to apply spline techniques [73]. For the case of pure slip angle and normal force,
regionalized bi-cubic spline models work effectively over the normal range of operating
slip angles. But regionalized fitting methods can lead to large numbers of coefficients and
43Given that the subjects of tire modeling and tire properties are not yet completely understood, development of
force and moment testing and modeling technology is continuing to take place.
348 Chapter 8. Forces and Moments
If two transforms are applied (equations 8.15 and 8.16), the location of the center point
of the curve (x = y = 0) can be defined in terms of real physical variables X and Y. As
shown in figure 8.76, the curve then has a form commonly seen in tire data.
x = X + Sh (8.15)
Y(X) = y(x) + Sv (8.16)
Six important properties of the “Magic Formula” are noted in a subsequent paper by
Bakker, Pacejka, and Lidner [76]. The two that seem most important are: a) the formula
can match the experimentally observed characteristics of lateral force, aligning torque,
and longitudinal force quite well and b) the coefficients can be related to actual physical
tire data in a recognizable way.
The discussion in Schuring, Pelz, and Pottinger [77] is very helpful for understanding
the mathematical features of the “Magic Formula” and sheds light on both the interpreta
tion of the coefficients B, C, D, E, Sh and Sv and the consequences of the values chosen
for C and E. [In that paper, Sh and Sv are denoted Sx and Sy respectively, and the x to X
transformation formula is expressed in a slightly different way.]
Sv and Sh simply locate the center point of the Magic Formula curve with respect to
the origin when the traditional representation of tire data is used. For example, in the case
of lateral force Sv and Sh arise from plysteer and conicity, Section 8.4.1.4.1.
B•C•D is the initial slope of the curve. In practice, this product is the cornering stiffness,
aligning stiffness, or longitudinal stiffness depending on which force or moment is being
modeled.
B and D are size factors. D can be estimated from the maximum value of y when the curve
has a definite peak. The fact that y cannot exceed D leads immediately to this relationship.
C and E are shape factors. Graphic examples of the effect of various choices for the
values of C and E are given in Reference 77. For E < 1, the region in which “Magic
Formula” curves are shaped like typical force and moment curves, the value of C controls
the initial slope, final level, extreme values and passage through zero (where y goes from
plus to minus or vice versa). For typical curves 1 < C < 4 and indeed in almost all cases
1 < C < 3.
Chapter 8. Forces and Moments 349
Following the methods outlined in reference 78 all the “Magic Formula” constants can
be determined from tire force and moment characteristics. The resultant individual fits as
functions of only lateral or only longitudinal slip angles are quite good, figure 8.77
through 8.81.
(Reprinted with permission from SAE 931909 1993 SAE International) [77]
350 Chapter 8. Forces and Moments
(Reprinted with Permission from SAE 931909 1993 SAE International) [77]
(Reprinted with permission from SAE 931909 1993 SAE International) [77]
Chapter 8. Forces and Moments 351
(Reprinted with permission from SAE 931909 1993 SAE International) [77]
(Reprinted with permission from SAE 931909 1993 SAE International) [77]
sent B, C, D, etc., adequately is acceptable. Figures 8.82 and 8.83 were drawn from an
implementation that used simple polynomials in two variables as representations.
Figure 8.82: Lateral force at three loads
(Reprinted with permission from SAE 931909 1993 SAE International) [77]
(Reprinted with permission from SAE 931909 1993 SAE International) [77]
Chapter 8. Forces and Moments 353
It is worth mentioning that the asymmetry induced by inclination, see figures 8.31
through 8.34, renders the raw curves unfit for “Magic Formula” modeling. This problem
can be dealt with by making E dependent on the sign of the base usage variable. In this
case, E at positive slip angle takes one value and E at negative slip angle takes another.
Basically, two curves are derived by splitting the original curve into two pieces at its cen
ter point and treating each piece as one-half of a complete and symmetrical “Magic
Formula”. It is just an application of the relationship given in Eq. 8.14.
8. Concluding remarks
The discussion in this chapter has covered a very broad range of tire force and moment
behavior and effects. The literature reviewed suggests that research and development is
still needed in these areas of tire mechanics.
Tire force and moment modeling needs improvement with respect to clarity, overall
standardization, and economic efficiency. At this time, the complexity of models often
makes them the domain of specialists. Thus, the engineering value potentially present is
not realized because use by the broader community is inhibited. A drive for standardiza
tion could eliminate considerable duplication of effort. Also, it is rather rare to see a dis
cussion of the economic merits of a modeling approach, including the data acquisition
needs and the engineering man-hours required for routine use.
Where to get certain tire test data is a vexing question. The ability to acquire force and
moment data is limited for larger tires, light truck sizes and above. The existing infrastruc
ture for testing these tires is definitely inadequate.
354 Chapter 8. Forces and Moments
9. References
1. Dunlop, J. B., “An Improvement in Tyres of Wheels for Bicycles, Tricycles, and Other
Road Cars,” Provisional Specification, 20 July 1888.
2. Evans, R. D., “Properties of Tires Affecting Riding, Steering, and Handling,” Society
of Automotive Engineers, SAE Transactions, Vol. 30, February 1935, pp. 41-49.
3. Bull, A. W., “Tire Behavior in Steering,” Society of Automotive Engineers, SAE
Transactions, Vol. 45, No. 2, 1939, pp. 344-350.
4. “Papers on Shimmy and Rolling Behavior of Landing Gears Presented at Stuttgart
Conference Oct. 16-17, 1941,” NACA Technical Memorandum 1365, W a s h i n g t o n ,
August 1954.
5. “Vehicle Dynamics Terminology,” Society of Automotive Engineers, SAE J670e,
Warrendale, PA, 1978.
6. “Tire Performance Terminology,” Society of Automotive Engineers, SAE J2047,
Warrendale, PA, 1998.
7. Milliken, W. F. and Milliken, D. L., “Race Car Vehicle Dynamics,” Society of
Automotive Engineers, SAE J670e, Warrendale, PA, 1995, pp. 39-40.
8. Pottinger, M. G., Marshall, K. D., and Arnold, G. A., “Effects of Test Speed and Surface
Curvature on Cornering Properties of Tires,” Society of Automotive Engineers, SAE
760029, Warrendale, PA, 1976.
9. Bird, K. D., and Martin, J. F., “The CALSPAN Tire Research Facility: Design,
Development, and Initial Test Results,” Society of Automotive Engineers, SAE 730582,
Warrendale, PA, 1973.
10. Langer, W. J., and Potts, G. R., “Development of a Flat Surface Tire Testing
Machine,” Society of Automotive Engineers, SAE 800245, Warrendale, PA, 1980.
11. Pottinger, M. G., “The Flat-Trac II® Machine, the State-of-the-Art in Tire Force and
Moment Measurements,” Tire Science and Technology, TSTCA, Vol. 20, No. 3, July-
September 1992, pp. 132-153.
12. Jenniges, R. L., Zenk, J. E., and Maki, A. E., “A New System for Force and Moment
Testing of Light Truck Tires,” Society of Automotive Engineers, SAE 2003-01-1272,
Warrendale, PA, 2003.
13. Pottinger, M. G., Tapia, G. A., Winkler, C. B., and Pelz, W., “A Straight-Line
Braking Test for Truck Tires,” Rubber World, September 1996, pp. 29–36.
14. Whitehurst, E. A. and Neuhardt, J. B., “Time-History Performance of Reference
Surfaces,” The Tire Pavement Interface, ASTM STP 929, M. G. Pottinger and T. J. Yager,
Eds., American Society for Testing Materials, W. Conshohocken, PA, pp. 61-71.
15. Ginn, J. L., and Marlowe, R. L., “Road Contact Forces of Truck Tires as Measured in
the Laboratory,” Society of Automotive Engineers, SAE 670493, Warrendale, PA, 1967.
16. “Combined Cornering and Braking Test for Truck and Bus Tires,” Society of
Automotive Engineers, SAE J2675, Warrendale, PA, 2004.
17. Pottinger, M. G., & Fairlie, A. M., “Characteristics of Tire Force and Moment Data,”
Tire Science and Technology, TSTCA, Vol. 17, No. 1, January-March 1989, pp. 15-51.
Chapter 8. Forces and Moments 355
18. Pottinger, M. G., “Plysteer in Radial Carcass Tires,” Society of Automotive Engineers,
SAE 760731, Warrendale, PA, 1976.
19. Kabe, K., and Morikawa, T., “A New Tire Construction Which Reduces Plysteer,”
Tire Science and Technology, TSTCA, Vol. 19, No. 1, January-March 1991,
pp. 37-65.
20. Nordeen, D. L., “Analysis of Tire Lateral Forces and Interpretation of Experimental
Tire Data,” Society of Automotive Engineers, SAE 670173, Warrendale, PA, 1967.
21. Nordeen, D. L., “Application of Tire Characterizing Functions to Tire Development,”
Society of Automotive Engineers, SAE 680409, Warrendale, PA, 1968.
22. Matjya, F. E., “Steering Pull and Residual Aligning Torque,” Tire Science and
Technology, TSTCA, Vol. 15, No. 3, July-September 1987, pp. 207-240.
23. Gillespie, T. D., “Fundamentals of Vehicle Dynamics,” Society of Automotive
Engineers, Warrendale, PA, 1992, pp. 351.
24. Marshall, K. D., Pottinger, M. G., and Gibson, G. E., “Nibbling – the Force and
25. Gough, V. F., Barson, C. W., Gough, S. W., and Bennett, W. D., “Tire Uniformity
PA, 1961.
26. Topping, R. W., “Tire Induced Steering Pull,” Society of Automotive Engineers,
SAE 750406, Warrendale, PA, 1975.
27. Lee, J-H., “Analysis of Tire Effect on the Simulation of Vehicle Straight Line Motion,”
Vehicle Systems Dynamics, Vol. 33, No. 6, Swets & Zeitlinger, Lisse, The Netherlands,
June, 2000, pp. 373-390.
28. Pottinger, M. G., “Tire/Vehicle Pull: An Introduction Emphasizing Plysteer Effects,”
Tire Science and Technology, TSTCA, Vol. 18, No. 3, July-September 1990, pp. 170
190.
29. “Residual Aligning Moment Test,” Society of Automotive Engineers, SAE J1988,
Warrendale, PA, 1994.
30. Yamazaki, S., Fujikawa, T., Suzuki, T., and Yamaguchi, I, “Influence of Wheel
Alignment and Tire Characteristics on Vehicle Drift,” Tire Science and T e c h n o l o g y ,
TSTCA, Vol. 26, No. 3, July-September 1998, pp. 186 -205.
31. Tarpinian, H. D. and Culp, E. H., “The Effect of Pavement Grooves on the Ride of
Passenger Cars – The Role of Tires,” Society of Automotive Engineers, SAE 770869,
Warrendale, PA, 1977.
32. Doi, T. and Ikeda, K., “Effect of Tire Tread Pattern Design on Groove Wander of
Motorcycles,” Tire Science and Technology, TSTCA, Vol. 13, No. 3, July-September
1985, pp. 147 -153.
33. Peters, J. M., “Application of the Lateral Stress Theory for Groove Wander Prediction
Using Finite Element Analysis,” Tire Science and Technology, TSTCA, Vol. 29, No. 4,
October-December 2001, pp. 244 -257.
34. Nakajima, Y., “Prediction of Rain Groove Wandering,” Vehicle Systems Dynamics,
Vol. 40, No. 6, Swets & Zeitlinger, Lisse, The Netherlands, 2003, pp. 401-418.
35. Schuring, D. J., Pelz, W., and Pottinger, M. G., “A Model for Combined Tire Cornering
and Braking Forces,” Society of Automotive Engineers, SAE 960180, Warrendale, PA,
1996.
36. Pottinger, M. G., “Tire Force and Moment in the Torqued State an Application of the
356 Chapter 8. Forces and Moments
54. Akasaka, T., Kabe., K., Koishi, M., and Kuwashima, M., “Analysis of the Contact
Deformation of Tread Blocks,” Tire Science and Technology, TSTCA, Vol. 20, No. 4,
55. Bond, R., Lees, G., and Williams, A. R., “An Approach Towards the Understanding
Performance of the Tire,” The Physics of Tire Traction; Theory and Experiment, D. F.
Hays and A. L. Browne, Eds., Plenum Press, New York – London, 1974, pp. 339-360.
56. Pottinger, M. G., Pelz, W., Winkler, C. B., Pottinger, D. M., and Tapia, G. A.,
Steer, and Trailer with Evolution from New to Naturally Worn-Out to Retreaded
57. Williams, A. R. and Evans, M. S., “Influence of Tread Wear Irregularity on Wet
STP 793, W. E. Meyer and J. D. Walter, Eds., American Society of Testing and
58. Pottinger, M. G., McIntyre, J. E. III, Kempainen, A. J., and Pelz, W., “Truck Tire
2000.
59. Grogger, H., and Weiss, M., “Calculation of the Three-dimensional Free Surface
Flow Around an Automobile Tire,” Tire Science and Technology, TSTCA, Vol. 24,
60. Grogger, H., and Weiss, M., “Calculation of the Hydroplaning of a Deformable
61. Seta, E., Nakajima, Y., Kamegawa, T., and Ogawa, H., ”Hydroplaning Analysis
by FEM and FVM: Effect of Tire Rolling and Tire Pattern on Hydroplaning,” Tire
Science and Technology, TSTCA, Vol. 28, No. 3, July-September, 2000, pp. 140
156.
62. Okano, T., and Koishi, M., “A New Computational Procedure to Predict Transient
Hydroplaning of a Tire,” Tire Science and Technology, TSTCA, Vol. 29, No. 1,
63. Darnell, I., Mousseau, R., and Hulbert, G., “Analysis of Tire Force and Moment
Response During Side Slip Using an Efficient Finite Element Model,” Tire Science and
Technology, TSTCA, Vol. 30, No. 2, April – June 2002, pp. 66-82.
64. Ohishi, K., Suita, H., and Ishihara, K., “The Finite Element Approach to Predict the
Plysteer Residual Cornering Force of Tires”, Tire Science and Technology, TSTCA, Vol.
65. Rao, K., Kumar, R., and Bohara, P., “Transient Finite Element Analysis of Tire
Dynamic Behavior”, Tire Science and Technology, TSTCA, Vol. 31, No. 2, April – June
66. Schlippe, B. von and Dietrich, R., “Das Flattern eines Bepneuten Rades,” Papers on
Shimmy and Rolling Behavior of Landing Gears Presented at Stuttgart Conference Oct.
16-17, 1941, NACA Technical Memorandum 1365, Washington, August 1954. “”
67. Schlippe, B. von and Dietrich, R., “Zur Mechanik des Luftreifens, ” Zentrale fur
Wissensvhaftliches Berichtwesen, Berlin-Adlershof, 1942.
68. Schlippe, B. von and Dietrich, R., “Das Flattern eines mit Luftreifen versehenen
Rades,” Jahrbuch der Deutsche Luftfahrforschung, 1943.
69. Gough, V. E., “Nondestructive Estimation of Resistance of Tire Construction to Tread
Wear,” Society of Automotive Engineers, SAE 667A, Warrendale, PA, 1963.
70. Gough, V. E., Kautschuk and Gummi, Vol. 20, p469, 1967.
71. Sakai, H., “Study on Cornering Properties of Tires and Vehicles,” Tire Science and
Technology, TSTCA, Vol. 18, No. 3, July – August 1990, pp. 136-169.
72. Sitchin, A., “Acquisition of Transient Tire Force and Moment Data for Dynamic
Vehicle Handling Simulations,” Society of Automotive Engineers, SAE 831790,
Warrendale, PA, 1983.
73. DeBoor, C., “Bicubic Spline Interpolation,” J. Mathematics & Physics, Vol. 41, 1962,
pp. 212-218.
74. Radt, H. S., and Glemming, D. A., “Normalization of Tire Force and Moment Data,”
Tire Science and Technology, TSTCA, Vol. 21, No. 2, April – June 1993, pp. 91-119.
75. Bakker, E., Nyborg, L., and Pacejka, H.B., “Tyre Modeling for Use in Vehicle
Dynamics Studies,” Society of Automotive Engineers, SAE 870432, Warrendale, PA,
1987.
76. Bakker, E., Pacejka, H. B., and Lidner, L., “A New Tire Model with an Application in
Vehicle Dynamics Studies,” Society of Automotive Engineers, SAE 890087,
Warrendale, PA, 1989.
77. Schuring, D. J., Pelz, W., and Pottinger, M. G., “The BNPS Model – An Automated
Implementation of the ‘Magic Formula’ Concept,” Society of Automotive Engineers, SAE
931909, Warrendale, PA, 1993.
78. Schuring, D. J., Pelz, W., and Pottinger, M. G., “The Paper-Tire Concept: A Way to
Optimize Tire Force and Moment Properties,” Society of Automotive Engineers, SAE
970557, Warrendale, PA, 1997.
79. Bayle, P., Forissier, J. F., and Lafon, S., “A New Tyre Model for Vehicle Dynamics
Simulations,” Automotive Technology International, 1993, pp. 193-198.
80. Svendenius, J., and Gafvert, M., “A Brush-Model Based Semi-Empirical Tire-Model for
Combined Slips,” Society of Automotive Engineers, SAE 2004-01-1064, Warrendale, PA,
2004.
81. Bernard, J. E., and Clover, C. L., “Tire Modeling for Low-Speed and High-Speed
Calculations,” Society of Automotive Engineers, SAE 950311, Warrendale, PA, 1995.
Chapter 8. Forces and Moments 359
10. Problems
1. Define the transformation matrices required to convert data from the SAE Tire Axis
System to the ISO Wheel Axis System.
2. What is the sense of the aligning moment and lateral force associated with a positive
inclination angle in the ISO Wheel Axis System?
3. Due to the width of the tire to be tested and clearance problems with respect to the load
cell system, a special adapter was manufactured to move the test wheel plane off-center
with respect to the design wheel plane location for the force and moment machine. The
assembly wheel plane is now 5 cm outboard (Y′+) of the location that the machine’s pro
gramming assumes to be the origin of the Tire Axis System. Derive the equations that
properly express the data in the actual SAE Tire Axis System using the machine reported
data as input. Be sure to assume that inclination angle is present. Review the definition of
the origin of the Tire Axis System given in Section 8.1.1 before beginning this problem.
4. Transform the FY and MZ data used to generate figures 8.17 and 8.19 from the SAE Tire
Axis System to the ISO Wheel Axis System. These data are for a constant FZ. The data
are in the problem 4 data set.
Problem 4: Data set
SA (deg) Fy (N) Mz (Nm)
-10 5689 -25
-9 5602 -34
-8 5440 -40
-7 5306 -52
-6 5155 -67
-5 4917 -83
-4 4595 -97
-3 4124 -117
-2 3275 -116
-1 2006 -79
0 242 -10
1 -1588 64
2 -2966 105
3 -3893 115
4 -4397 103
5 -4744 88
6 -5000 74
7 -5101 57
8 -5248 49
9 -5357 44
10 -5455 35
5. Sketch revised versions of figures 8.31 - 8.34 showing the expected effect of a
negative inclination angle.
360 Chapter 8. Forces and Moments
6. Beginning with reference 29 as a guide, determine the plysteer, conicity, plysteer resid
ual aligning moment, conicity residual aligning moment, and plysteer residual lateral
force for the data in the problem 6 data set. For the sake of simplicity assume that linear
models of FY(α) and MZ(α) are adequate. FZ is a constant.
Problem 6: Data set
Right rotation (operation as if on right side of car.)
SA Fy Mz
Deg N N-m
-1.00 1897 -74.7
-0.80 1601 -64.3
-0.60 1275 -52.5
-0.40 968 -40.8
-0.20 637 -27.6
0.00 268 -13.0
0.20 -83 1.1
0.40 -436 15.3
0.60 -784 29.2
0.80 -1121 42.4
1.00 -1425 54.5
7. Under appreciable torque a particular differential is known to not function until the
difference in angular velocity between the left and right front tires reaches 0.265 percent
of the angular velocity of the larger tire. As an engineer at a company that is considering
using the differential design in question, you are asked to estimate the potential magnitude
of the lateral movement for a new car design with a 1520 mm track width. You decide to
characterize the problem in terms of how far the vehicle must travel forward in order to
move 3.66 m laterally on a dead level road. Further, you decide the characterization will
assume the smaller tire is on the right side of the front axle. Based on data in your com
pany’s files, the larger tire probably produces 470 revs/km, thus it is logical to consider
the smaller tire in the worst case would produce 471.25 revs/km. How far forward does a
worst-case equipped car roll under torque before it moves 3.66 m laterally?
Chapter 8. Forces and Moments 361
8. Based on a qualitative examination of figures 8.47 through 8.51, what is your expectation
of the change in effectiveness of anti-lock braking as a means of avoiding lateral instability
due to braking as cornering severity, slip angle, increases?
9. Based on the problem 9 data set, estimate the relaxation length associated with figure 8.56.
The data provided are with the uniformity removed. NOTE: If uniformity were present, the
data would be appreciably more difficult to analyze. Uniformity is discussed in chapter 9.
Problem 9: Data set
Time Distance rolled Fy
sec (m) N
0.00 0.000 -3
0.02 0.038 -2
0.04 0.076 2
0.06 0.114 1
10. If you were the driver of a car that was going to be operated over a large speed range,
what are the handling implications inherent in concepts expressed within figure 8.65? Give
special consideration to how the vehicle will behave in the vicinity of the peak lateral force.
11. A sample of tires stored in an air conditioned warehouse is split into two sub-samples.
One sub-sample remains in the original warehouse; the other is shipped to a warehouse
without air conditioning located in a southwestern state. On average the air-conditioned
warehouse is maintained at 24°C for the next 150 days whereas the average temperature
in the desert warehouse is 42°C. When the sample was split, the cornering stiffness coef
ficient was 0.175. After the 150 days of differential storage, what would you expect the
two cornering stiffness coefficients to be?
12. Test drivers are given four sets of tires. They are told that the interest is in low accel
eration, ordinary driving maneuvers so the tires are not to be “broken-in”. The tires are
really the same except for pretreatment. One set is not broken-in, the second has been bro
ken-in at 2° slip angle, the third has been broken-in at 4° slip angle, and the fourth at 8°
slip angle, but are marked as if they were different constructions. Given that low lateral
acceleration behavior is proportional to cornering stiffness, how would you expect the
drivers to rank the “constructions”? Also, given that the drivers would consider a 5 per
cent change as significant, which “constructions” would they consider to be significantly
different?
13. Handling tests are performed on several different days over a period of weeks, but
someone did not keep the control data so the raw results and the ambient temperature on
the test days are all the data you have. The ambient temperatures are 13°C, 22°, and 29°.
You have to decide how a control tire would probably have ranked on the three days in
question. Assume that the cornering stiffness coefficient of the control is 0.175 at 22°C.
Further, would the drivers have judged significant differences to exist between control tire
results from the different days based on the 5 percent rule expressed in problem 12?
14. A vehicle is driven on a dry road, a snowy road, and an icy road. Assuming figure 8.72
represents the tires, what differences in perceived behavior will occur based on the
changes in tire force and moment properties with surface?
364 Chapter 9. Tire Noise and Vibration
Chapter 9
by K. D. Marshall
3.1.1 Blank tire on SW surface (smooth surface) - 74.8 dB(A) ........................ 388
3.1.2 Blank pattern on HRA surface (rough surface) - 88.4 dB(A) .................. 388
4.3 Finite element analysis and boundary element methods .............................. 393
5.3.3 Road roughness (or road roar, 25 to 300 hertz) ...................................... 397
Chapter 9
Tire Noise and Vibration
by K. D. Marshall
Introduction
The tire is a complex dynamic system. When it vibrates it can generate noise and
vibration signals observers may find objectionable. These problems are typically referred
to as “ride disturbances,” and over the years many tactile (feel) and acoustic (hearing) ride
disturbances have been identified, figure 9.1 [1]1. The definitions of these terms, and
several others, are provided in the Glossary and closely follow the SAE Standards [2]. As
figure 9.1 shows, both the road and the tire may be the source of ride disturbances, and
these problems can occur over the frequency range of near zero to several thousand Hertz
(cycles per second).
Figure 9.1: Noise and vibration ride disturbances [1]
Figure 9.2 illustrates the relationship between the road, the tire, the vehicle and the
observer. Ride disturbances can be excited by the road or from mechanisms internal to the
tire. The ride disturbances can propagate either through the air (airborne noise) or through
the structure of the tire and the vehicle (structural borne vibrations). Airborne noise can
travel away from tire and arrive at the location of observers near a passing vehicle. This
is often referred to as Passby Noise. Airborne noise can also enter the vehicle by passing
though openings in the body or by acoustically coupling with the vehicle and re-radiated
inside the vehicle by panel vibrations. The structure borne vibrations travel through the
tire, the suspension and other components of the vehicle, and appear inside the vehicle as
either tactile or acoustic signals. Finally, the tire and the vehicle can exhibit individual and
joint characteristics that can amplify and/or attenuate the noise and vibration signals.
The emphasis of this chapter is on what the tire or vehicle engineer needs to know
about tire noise and vibration. However, it is impossible to comprehend why certain
aspects of tire performance are important without some consideration of how the tire and
the vehicle interact, and how human beings respond to noise and vibration disturbances.
Figure 9.2: Airborne and structure borne transmission paths [1]
Throughout this chapter we will provide references to the topic being discussed. These
references are not exhaustive, however. They are merely a roadmap to what can be found
in the literature.
1. Tire vibrations
Like many dynamic systems, the behavior of a tire is dependent on the operating
conditions. At low frequency a tire can be approximated by lumped parameters and a
single, linear second-order system. At higher frequencies the tire behaves like a continu
um of linear second-order systems, although some non-linearities can appear at times.
2Several other techniques also exist, and they are discussed in any good vibration text.
368 Chapter 9. Tire Noise and Vibration
8.25-14
Most investigators believe these changes in spring rate and damping are the result of the
tire footprint constraint being relaxed as the tire begins to rotate. The spring rate and damp
ing of a tire are also generally piecewise linear about any given set of operating conditions.
Figure 9.5 illustrates the effect of various testing conditions and tire materials on the
rolling spring rate of a radial tire. The spring rate demonstrates a slight increase with load
and speed, a large increase with inflation pressure, and is unchanged for many carcass/belt
combinations. Table 9.1 provides additional spring rate data in the vertical, lateral and
longitudinal directions, and about the steering axis of the tire.
Chapter 9. Tire Noise and Vibration 369
Figure 9.5: Effect of load, speed, belt material and inflation pressure changes on
radial tire vertical spring rate behavior, V = 80.5 km/h [5]
3The term "free tire" implies that the tire is suspended above the ground using rubber bands or by some other
soft supporting method that does not affect the vibrational motion of the tire. The low frequency "bouncing" of
the tire on the rubber bands is not considered a vibrational mode of the tire.
370 Chapter 9. Tire Noise and Vibration
The nomenclature identifies the mode shape number and the resonant frequency. For
example, “n1=63 Hz” indicates that the first vertical mode of vibration has a frequency of
63 Hertz. As the mode number increases the deformed shape of the tire becomes increas
ingly complex, and the number of nodal points for any mode of vibration is equal to twice
the mode number. That is, the 3rd mode of vibration has 6 nodal points. Although only
Chapter 9. Tire Noise and Vibration 371
four modes of vibration are shown in figure 9.6, in reality there is no limit to the numbers
of modes of vibration that can exist in any direction.
Investigators have also considered the vibrational behavior of a tire for a variety of
different boundary and operating conditions, Table 9.2 [9]. In this experiment the wheel
was free to move in all three axial directions, or in six directions if one considers the
torsional modes. Earlier it was mentioned that the tire was axisymmetric in the plane of
the tire, but this is not the case once the tire is in contact with the ground. The modal
frequencies in the vertical and longitudinal directions are significantly different. Table 9.2
also shows that an increase in inflation pressure generally causes an increase in the
resonant frequency, the tire load has a relative minor effect on the resonant frequencies,
and an increase in speed from 0 to 8 km/h lowers the resonant frequencies. It should be
mentioned, however, that the speed effect is limited to low speeds. Increasing the speed
above 20 km/h has little effect on the resonant frequencies. This can be seen in figure 9.7.
Four of the lower order modes in the vertical and longitudinal directions deserve
special mention, figure 9.8 [11] and Table 9.2. As will be seen later, these modes have a
critical effect on the ride performance of the tire/vehicle system. The lowest mode in the
vertical direction is essentially a rigid body motion of the tire/wheel assembly at 18 Hertz.
In this case the tire and wheel move as a single unit and deflect the carcass of the tire in
the vicinity of the tire footprint. The longitudinal mode at 42 Hertz is a torsional mode
where the wheel rotates clockwise and the tread rotates counterclockwise (or vice versa).
The next higher vertical and longitudinal modes, 80 and 88 Hertz, respectively, are
flexural modes where there is an out-of-phase motion between the tire and the wheel. For
convenience these vertical and longitudinal modes will be designated as VR1, VR2, LR1
and LR2. Note that VR1 and VR2 are prominently visible in figure 9.7.
In relative terms, for ride performance the higher order vertical and longitudinal modes
are usually less important (3rd mode and above). This is principally because the amount
of rim motion becomes progressively smaller at the higher order modes. In fact, the even-
order modes exhibit no wheel motion whatsoever for a free tire, whereas the odd-order
modes demonstrate a small amount of wheel motion, and will thus have some effect on
the ride performance of the tire.
There are a number of construction and operational factors that can influence the resonant
frequencies of the tire. Some of the more important effects are shown in Table 9.3.
Table 9.3: Factors that can increase radial direction resonances in radial ply tires [12]
r = effective radius,
4Similar factors in the other rotating components can also be a problem, but will not be discussed here.
374 Chapter 9. Tire Noise and Vibration
Figure 9.10: Radial force variations for one tire revolution [1]
Chapter 9. Tire Noise and Vibration 375
If the rotational rate of the tire is known, it is possible to determine the frequency of
any harmonic order at any test speed by using the following formula.
f = N x RPS (2)
where f = frequency (Hertz)
N = harmonic order
RPS = revolutions per second of the tire
Since the rotational speed of a passenger tire is approximately 1 revolution for every 5
mph of test speed, at 50 mph the frequency of 1st harmonic of radial force variation (RH1)
will be approximately 10 Hertz, the 2nd harmonic, 20 Hertz, etc. As the speed of the tire
increases, the frequencies of the harmonic orders will also increase. At some point the
frequency of a harmonic will coincide with and excite a resonant frequency of the tire, and
the level of the force variation will increase dramatically. Figure 9.11 illustrates how an
aligning moment resonance of a HR70-15 tire can be excited by a number of different
harmonics. Note that each harmonic order increases in amplitude until it excites the tire
resonance at a certain speed. At the resonance speed the force variation will exhibit a peak
value, after which the force variation value will decrease as speed further increases.
Any force and moment variation can potentially cause a ride disturbance, but
experience has shown the radial (vertical) and longitudinal (fore/aft, or drag) directions
are the most important contributors [13]. In addition, the aligning moment resonance
shown in figure 9.11 can cause a torsional vibrational disturbance of the steering system
of the vehicle.
Figure 9.12 shows the normal trend for the 1st harmonics of radial and longitudinal
force variation as a function of speed. Typically RH1 will show a gradual increase with
test speed, whereas LH1 is very small at low speed and increases dramatically as the speed
increases. The behavior of LH1 was explained by Walker [14]. Equation 3 shows that the
longitudinal force variation is proportional to the radial runout of the tire and the square
of the velocity.
F ∝ v 2 ∆R [K ] (3)
where ∆R = radial runout
v = velocity
K = constant
There are a number of potential causes of tire nonuniformities, many of which have
been reported in the literature. There is also a general tendency for certain material or
construction anomalies to preferentially affect either the radial or longitudinal force
variation. Anomalies that affect mass variation around the tire are very important to
radial force variation, and anomalies that affect stiffness or runout variation are more
important to longitudinal force variation. Table 9.4 provides a comparison of a number of
potential sources of force variation.
Table 9.4: Sources of tire nonuniformity force variation
Source type Radial FV Longitudinal FV
Heavy or light splices ** *
Spread or bunched cords ** *
Tread thickness variation ** *
Non-symmetric bead setting * **
Building drum irregularities * **
Mold irregularities * **
Carcass run-out * **
For many years tires have been “ground” on the factory uniformity machines to reduce
radial force variations [15]. This process is generally most effective for the 1st harmonic
of radial force, which is critical to improving a ride problem that will be discussed later.
Grinding is accomplished by removing a small amount of rubber from the shoulder of the
tire, or across the entire tread surface, at the high point of the RH1 term. Figure 9.13
shows that grinding can have a beneficial effect on the radial force variation of the tire.
In recent years, however, grinding has lost favor and become less acceptable to automo
tive manufacturers and tire consumers.
Figure 9.13: Effect of tread grinding on radial first harmonic force variation [15]
Match mounting is another technique that has been used to minimize RH1. Match
mounting is usually applied at the end of the vehicle assembly line where the tires and the
wheels come together, figure 9.14 [16]. In this process the tire and wheel are positioned
such that the high point of RH1 for the tire is aligned with the low point of the 1st har
monic of the bead-seat runout of the wheel. This process minimizes the overall 1st har
monic of radial force variation for the tire/wheel assembly.
Figure 9.14: Effect of match mounting on radial first harmonic force variation [16]
378 Chapter 9. Tire Noise and Vibration
(5)
[ ]
SPLTotal = 10 x log 10 (92 /10 ) +10(94 /10 )
= 96.12 dB
There are a number of potential sources of tire noise. Several of these sources are dis
cussed below [1, 17, 18, 19].
6. Test speed has a major influence on tire noise. Passenger car tire noise increases at
a rate of approximately 40 log V, and truck tire noise increases at a rate between 30
log V and 40 log V.
7. Truck tires are typically 3-5 dB noisier than passenger tires, principally due to the
larger loads they carry.
8. Among a class of tire, such as passenger tires, tire to tire differences are generally in
the range of 4-6 dB.
The degree of acceptability of the sound presented to the passengers in a vehicle, or
near a passing vehicle, is the result of both the amplitude and the spectral content of the
noise signal. The best tread patterns are those which minimize the overall amplitude of
the noise, spread the noise energy over as wide a frequency range as possible, and avoid
narrow frequency peaks that will cause the tire to be viewed as tonal. One of the funda
mental ways to control tire noise is by the design of the tread. There are a number of fac
tors involved in tread pitch design of a tire.
a. The total number of pitches around a tire
b. The size of the pitch lengths
c. The pitch ratio (the relative size of the different pitch lengths)
d. The sequence of pitches around the tire
e. The number of pitches of each size
f. The circumference of the tire
Figure 9.15(a) shows a conceptual half-tire representation of a pitch length (or pitch
380 Chapter 9. Tire Noise and Vibration
segment) where the black areas designate the grooves in the tread. The pitch length is the
fundamental design unit of the tread pattern. If the same pitch length was repeated around
the circumference of the tire the result would be a Mono-Pitch design, figure 9.15(b).
Mono-pitch treads are still found in off-the-road, farm service and a few other types of
tires, but they are never used in highway passenger tires. The reason is the frequency spec
trum of a mono-pitch design is principally confined to the rotational order of the tire and
its harmonics. That is, if the tire has 56 equal pitch lengths around the circumference, the
frequency spectra would exhibit a 56th order and probably the 112th, the 168nd, etc. Tread
designs that exhibit this type of “peaky” spectrum will result in a tonal sound that is very
disturbing to the observer.
To avoid tonality, tires are normally designed using a number of different pitch lengths
around the circumference, often three or more, and a carefully designed pitch sequence.
figure 9.15(c) illustrates a situation where the fundamental pitch length has been modified
to accommodate three pitch lengths of different sizes. In this case the scaled sizes of the
pitch lengths are 0.80, 1.00 and 1.20 units. Since the largest pitch length is 50% greater
than the smallest pitch length, the pitch ratio of the tire would be 1.50. Note also that the
pitch sequence is different on the left and right sides of the tire. Although there are aes
thetic and practical limits on what is possible in a real tire, techniques that have proven
useful in spreading out the noise energy over a wide frequency band are listed below.
a. Use as many different pitch lengths as possible
b. Use as large a pitch ratio as possible, but avoid ratios of integers such as 3/2, 4/3, etc.
c. Use as many pitches as possible around the tire
d. Use a randomized pitch sequence around the tire
e. Use a different pitch sequence on the left and right side of the tire
The first detailed analysis of pitch sequence theory was provided by Varterasian [20].
In this classic study he discussed the importance of utilizing “mechanical frequency mod
ulation” to eliminate tonal situations. This technique is analogous to electrical frequency
modulation where a modulating signal is used to transfer energy from a dominant frequen
cy peak into its sidebands, thereby making the peak amplitude and its sidebands nearly
equivalent and smoothing out the spectra. It should be noted that this technique flattens
the spectrum but the total energy contained in the spectrum is normally not reduced.
Figure 9.15: Tread design pitch length examples
Chapter 9. Tire Noise and Vibration 381
Over the years a number of U. S. Patents have been issued to tire companies that cover
a variety of pitch sequence design methods. Most of the approaches use some variation of
an up-down methodology where the pitch sequence sinusoidally increases and then
decreases one or more times around the tire. There are, however, a couple of exceptions.
Williams [21] revealed a technique where three pitch lengths are used, but in a significant
percentage of the time the smallest pitch length is immediately followed by the largest
pitch length, or vice versa. This produces a much less sinusoidal translation between the
pitch lengths. In a related vein, Bandel [22] patented a technique that utilizes numerical
methods to generate random sequences of at most three pitch lengths, and preferably only
two pitch lengths.
Figure 9.16 illustrates that a properly designed tread design can markedly improve the
noise performance of a tire [22]. The curve labeled “poor randomization” is a very tonal
tire with large frequency peaks. The “optimized randomization” tire has spread out the
noise energy and greatly lowered the distinct frequency peaks.
Pitch sequencing is not the only technique that is useful in improving tire noise.
Shown in Table 9.5 are a number of “design rules” that have proven useful. However, it
is important to remember that, depending on the design of the tire and the intended mar
ket, some of these suggestions may not be feasible because they may negatively affect
other desirable tire properties such as wet traction, handling, power loss, etc.
Figure 9.16: Effect of randomizing pitch sequence [19, revised Fig 10.29]
382 Chapter 9. Tire Noise and Vibration
Table 9.5: Methods of reducing tire noise and smoothing the spectral content*
Technique Reasoning
1. Use narrower, less deep grooves. Smaller transverse grooves produce less vibra
tion and pump less air.
4. Grooves should be well ventilated. Avoids enclosed pockets of air, and minimizes
the number of partically closed grooves that
5. Randomize/Optimize the Tread Design and might resonate.
Pitch Sequence.
Stagger the lateral grooves so multiple grooves Insures that large excitations do not occur at
do not intersect the leading or trailing edge of discrete places around the tire, and maintains a
contact at the same time, and the shape of the fairly constant amount of void area intercepted
grooves should not match the shape of the lead by the edges of contact around the tire.
ing and trailing edges of contact.
The pitch sequence on the left side of the tire Further randomizes the pitch sequence.
should be different than that on the right side of
tire. This is often called "phasing" or "offset."
Use a high Pitch Ratio, but avoid integer ratios Generally aids in spreading the noise energy,
such as 3/2. 4/3, etc. but integer ratios can cause harmonic orders
that are independent of pitch sequencing.
Use as many different pitch lengths as possible. Helps smoothes out the energy spectrum.
A larger number of pitches is generally better Reduces the block size and the resulting tread
than a smaller number of pitches. impacts.
The Fourier analysis of the pitch sequence Generally helps smooth the frequency
should avoid a 1st harmonic and should contain spectrum.
2nd, 3rd and 4th harmonics with approximately
the same amplitude.
6. Reducing the modulus of the tread or side Soften impacts of tread elements with the road.
wall generally reduces the noise level.
7. Decreasing the load on the tire or increasing It is usually beneficial to have a smaller
the inflation pressure often reduces the noise amount of the tread in contact with the road,
level. and hence fewer active noise sources.
* It is important to remember that many of these suggestions can affect other tire properties, such
as wet traction, handling, power loss, etc.
Chapter 9. Tire Noise and Vibration 383
Sakata [24] reported that the tire can exhibit higher order modes that are harmonically
related to the fundamental modes, but only the first vertical and longitudinal modes will
interact with the wheel and transfer energy to the vehicle. Investigators have also found
that these cavity modes are fairly independent of inflation pressure, tire load and tire
aspect ratio.
Scavuzzo, et al. [23] reported that the cavity modes can also couple with a bending
mode of the wheel. The bending of the wheel disk (see figure 9.18) will result in the
wheel rotating about the longitudinal axis and this, in turn, will produce a vertical force
on the vehicle spindle. This structural/acoustic coupling will augment the energy transfer
to the vehicle by the tire cavity modes and produce a higher noise level in the vehicle.
Figure 9.19 illustrates the sound pressure level inside a vehicle. In the upper graph, the
78 dB SPL peak in the vicinity of 230 Hertz is the result of the interaction of the tire cavity
and the wheel resonance. The middle chart shows that if the tire is filled with helium the large
SPL peak is reduced to 54 dB, or a change of about 24 dB. This is because the speed of sound
is much higher in helium than it is in air and the tire cavity modes are shifted upwards and
away from wheel resonance. The bottom chart shows that if an aluminum wheel is used in
place of the steel wheel, the SPL peak is reduced to about 66 dB. This occurs because the
aluminum wheel resonance is significantly higher than the steel wheel, and the
structural/acoustic coupling of the cavity/wheel resonances is virtually eliminated.
Figure 9.18: Structural wheel resonance,
strips to the fine surface texture of the pavement. The pavement’s characteristics of inter
est are changes in the elevation of the pavement as the vehicle traverses the road. The ele
vation changes are normally classified according to wavelength.
a. Undulations – wavelength greater than 50 cm
b. Road Roughness – wavelengths between 10 cm and 50cm
c. Macrotexture – wavelength between 0.5 mm and 10 cm
d. Microtexture – wavelengths less than 0.5 mm
The temporal frequency of the vibrations created by a vehicle traveling along the
road is given by the following equation.
ft=V/λ=Vk
(7)
where ft = temporal frequency (cycles per second)
V = vehicle velocity (m/sec)
λ = wavelength (m/cycle)
k = wave number (cycles/m)
Suppose a vehicle is traveling at 27.8 m/s (100 km/h) and the tire encounters a sinu
soidal pavement undulation with a wavelength of 1.39 m. In this case the tire will experi
ence a vibrational input of 20.0 Hertz. Since tactile inputs as low as 1.0 Hertz and acoustic
inputs as high as 20,000 Hertz are of interest, highway characteristics that fall into groups
‘b’ and ‘c’ can impact the ride quality of a vehicle.
Although every highway is different, highway engineers have measured hundreds of
roads over the years and it is possible to define an “average” road. figure 9.20(a) shows
some representative highway elevation curves, the “average” for Portland Cement
Concrete roads (PCC), and PCC overlaid with a smooth asphalt coating (Bituminous).
Road irregularities with a long wavelength are on the left side of the chart and those with
a short wavelength are on the right. The amplitude measurement is power spectrum den
sity, a convenient statistical measure of pavement roughness.
Assuming a vehicle is traveling on the road at 80 km/h, figure 9.20(b) shows the dis
placement input to the tire, and by differentiation the velocity (9.20(c)) and acceleration
(9.20(d)) inputs to the tire. Elevation (displacement) inputs are large at low frequencies
and acceleration inputs are large at high frequencies.
Chapter 9. Tire Noise and Vibration 387
(a) Typical road elevation power spectral density (PSD) versus wave number
(b) Elevation PSD versus frequency
(c) Velocity PSD versus frequency
(d) Acceleration PSD versus frequency
388 Chapter 9. Tire Noise and Vibration
replicated surface also contributes to the tire noise. The increase in tire noise is 13.6
dB(A).
Barone [28] extended envelopment studies to highway speeds and found that the input
forces excite vibrational signals that correspond to the resonant modes of the tire in the
vertical and longitudinal directions. The frequency ranges were 80-100 Hertz and 30-50
Hertz, respectively. The envelopment process and the resulting forces at the axle are
therefore extremely important to the ride performance of the vehicle in the frequency
range of 20 to 100 Hertz.
Bandel and Monguzzi [29] developed a semi-empirical approach to the envelopment
process that modeled the in-plane motion of the tire as second-order systems in the verti
cal and longitudinal directions. The envelopment of the bump was treated as a low speed
process where a “basic curve” was determined by the size and shape of the bump. The
basic curve and its mirror-image were offset by an amount that depended on the deflec
tion of the bump, and the two curves were added together to determine the input from the
road. The predicted forces at the tire/vehicle axle in the vertical and horizontal directions
correlated very well with the measured results, figure 9.23. This tire model was designed
to serve as a “black box” that could be interfaced with vehicle models and solved with
minimal computational effort.
Mancuso et al. [30] proposed a methodology that extended the Bandel model and elim
inated the need to provide experimental tire data, thus allowing “virtual” testing of the tire
during the design process. They used a 3D FEA model of the tire and extracted a mathe
matical-physical model of the tire that could be interfaced with vehicle models to model
the in-plane ride performance from dc to 100 Hertz. The tire contact information, includ
ing envelopment, is determined from a separate FEA model of the tread pattern of the tire.
Efforts are also underway by investigators to extend the frequency range of the analy
sis for tire durability modeling. Tire durability requires modeling the response of the tire
as it undergoes severe impacts and other extreme conditions that might stress the tire
and/or suspension to the point of failure [31]. FEA and multi-body system models are
being used to investigate time domain transient results as the tire encounters very large
obstacles.
4. Tire modeling
Numerous techniques have been used over the years to model the vibrational behavior of
tires. A few of these approaches are discussed in this section.
Figure 9.24: Driving point FRF for a P225/60R15 freely suspended tire [32]
392 Chapter 9. Tire Noise and Vibration
The benefit of using a modal representation to model the tire is a very small number of
parameters can accurately describe the vibrational behavior of the tire. All one needs are
the modal frequencies, masses and damping values and the mode shapes at the locations
of interest, which are usually the tire footprint and the tire/wheel spindle. The disadvan
tage of modal analysis is that data must be gathered for every tire and all operating con
ditions of interest. This can be very time consuming. There is no direct way to determine
the modal properties from the design parameters of the tire such as the belt angle, carcass,
compound modulus, etc.
Finally, it has previously been noted that the boundary conditions at the ground and the
wheel have an important effect on the vibrational behavior of the tire. W. Soedel [33]
reported that the “preferred” or “baseline” set of experimental (or analytical) modal prop
erties should be obtained for a free, axisymmetric tire. That is, a tire should be measured
without the imposition of any constraints at the tire/wheel spindle or the ground plane.
The reason for this is constraints can be easily added to the axisymmetric data to obtain
the non-axisymmetric results. However, removing constraints from non-axisymmetric
data to obtain the axisymmetric results is a more difficult process and requires knowledge
of the forces of constraint.
The equations of motion are developed using Hamilton’s principle and the assumption
that the tread ring is inextensible in the circumferential direction. A solution is sought by
assuming the deflected motion of the ring can be expanded as a complex Fourier series.
This is essentially a modal extraction procedure, and the solution to the characteristic
equation of the system is the modal properties of the tire. Since the model is developed
without any constraints being imposed, it is possible to utilize this type of model to pre
dict the vibrational behavior of the tire and the transmission of vibrational energy from the
ground to the vehicle.
There are certain limitations to this type of model. First, there is not a provision to
account for the vibration of the tire in the lateral direction. Second, since the stiffness of
the ring and the sidewall are lumped together it is not possible to easily incorporate tire
design changes into the model without performing additional calculations.
frequencies is desired. SEA is a statistical tool that allows the average response of a system
to be estimated by grouping the vibrational modes into subsystems that are represented by
idealized components such as plates, beams, acoustic cavities, etc. As figure 9.26 shows,
SEA provides a good estimate of the system response through several thousand Hertz.
Figure 9.26: Accelerance for tire on steel wheel using a SEA model [after 35]
5. Tire/vehicle systems
In the previous sections various aspects of tire noise and vibration behavior were dis
cussed. How the tire and the vehicle behave as a dynamic system will be considered in
this section.
as the “ride rate,” which is the effective spring rate of the suspension and the tire
RR = (KtKs)/(Kt+Ks) , (8)
where Kt and Ks are the spring rates of the tire and suspension respectively.
The undamped resonant frequency of the “quarter car model” is given by,
f = 0.159 RR / M (9)
The other three rigid body motions of the vehicle body are longitudinal and lateral
translations (x & y axes) and yaw about the z-axis. These motions generally occur in the
frequency range of 3-5 Hertz and are usually less important than the vertical bounce and
pitch and roll motions.
5A hemi-anechoic test room has a hard floor but absorptive material is attached to the other five surfaces.
396 Chapter 9. Tire Noise and Vibration
and a Wheel Hop resonance between 10 and 20 Hertz. The damping ratio is between 0.2
and 0.4. [25]
Tire uniformity is the most important excitation source for wheel hop, although under
some circumstances road undulations may excite this mode of vibration. The prime con
tributor is the first harmonic of radial force variation (RH1) and the critical driving speed
is 80 km/h and above. When the wheel hop resonance is excited the ride disturbance is
typically referred to as “Shake.” Historically, Shake has been the leading cause of ride
adjustments for tire manufacturers. Shake can be controlled in two ways.
RH1 should be small throughout the operating speed range. The required force level
depends on the sensitivity of the vehicle and the tolerance level of the observer.
Shift the Wheel Hop resonance upward as much as possible. There are two reasons for
doing this. First, to insure that the speed at which wheel hop is most strongly excited is
above the normal driving speed range. Second, the human body is less sensitive to a 20
Hertz vibration than to a 10 Hertz vibration, so a higher wheel hop frequency is less dis
turbing to an observer (also see Section 6 and figure 9.32).
5.3.4 Boom
Like most enclosed spaces, it is possible for the passenger compartment of a vehicle to
support the development of standing waves. There are a number of causes of boom
including Road Roughness, tire uniformity (4th or 5th harmonics of radial and longitudi
nal force variation), and rotational inputs from the engine and driveline. The frequency
range is generally between 30 and 100 Hertz. FEA techniques are often used to address
this problem.
5 A hemi-anechoic test room has a hard floor but absorptive material is attached to the other five surfaces.
Chapter 9. Tire Noise and Vibration 399
Figure 9.32: Equal comfort zones for seated passengers, vertical excitation [41]
observer can be very different in both amplitude and phase. This is called an interaural
difference, and binaural measurements are required to account for these effects.
References
1. Pottinger, M. G. and Yeager, T. J., Ed., The Tire/Pavement Interface, American Society
for Testing and Materials, STP 929, Baltimore, MD, 1986.
2. Vehicle Dynamics Terminology , Society of Automotive Engineers, SAE J670E, 1976.
3. Rasmussen, R. E. and Cortese, A. D., “Dynamic Spring Rate Performance of Rolling
Tires,” Society of Automotive Engineers, SAE 680408, 1968.
4. Melvin, J. W., Klein, R. G. and Marshall, K. D., B. F. Goodrich file data, 1968.
5. Pottinger, M. G., Thomas, R. A. and Naghshineh, K., “Stiffness Properties of
Agricultural Tires,” International Conference on Soil Dynamics, Auburn, AL, June, 1985.
6. Marshall, K. D., Ohio Dynamics, Inc., unpublished file data, 1997.
7. Jianmin, G., Gall, R. and Zuomin, W. “Dynamic Damping and Stiffness Characteristics
of Rolling Tires,” Tire Science and Technology, TSTCA, Vol.29, No. 4, Oct-Dec, 1997.
8. Potts, G. R., Bell, C. A., Charek, L.T. and Roy, T.K., “Tire Vibrations,” Tire Science and
Technology, TSTCA, Vol. 5 No. 4, Nov. 1977.
9. Scavuzzo, R.W., Richards, T. R. and Charek, L. T., “Tire Vibration Modes and Effects
Chapter 9. Tire Noise and Vibration 403
on Vehicle Ride Quality,” Tire Science and Technology, TSTCA, Vol. 21, No. 1, Jan-
March, 1993.
10. Mills, B. and Dunn, J. W., “The Mechanical Mobility of Rolling Tires,” Vibration and
Noise in Motor Vehicles, Institute of Mechanical Engineers, London, England, pp 90-101,
1972.
11. Richards, T. R., Charek, L. T. and Scavuzzo, R. W., “The Effect of Spindle and Patch
Boundary Conditions on Tire Vibration Modes,” Society of Automotive Engineers, SAE
860243, 1986.
12. Kung, L. E., “Radial Vibrations of Pneumatic Radial Tires,” Society of Automotive
Engineers, SAE 900759, 1990.
13. Marshall, K. D., Wik, T. R., Miller, R. F. and Iden, R. W., “Tire Roughness – Which
Tire Nonuniformities are Responsible,” Society of Automotive Engineers, SAE 740066,
Detroit, MI, 1974.
14. Walker, J. C. and Reeves, N. H., “Uniformity of Tire at Operating Speeds,” American
Society for Testing and Materials Committee F-9, Symposium on Tire Uniformity and
Vibrations, Akron, OH, November, 1973.
15. Nedley, Q. L. and Gearig, D. M., “Radial Improvements in Tire and Wheel
Manufacture – Their Effects upon Radial Force Variation of the Assembly,” Society of
Automotive Engineers, SAE 700089, Detroit, MI, 1970.
16. Hofelt, C., Jr., “Uniformity Control of Cured Tries,” Society of Automotive Engineers,
SAE 690076, 1969.
17. Hayden, R. E., “Roadside Noise from the Interaction of a Rolling Tire and the Road
Surface,” Technical paper presented at the 81st meeting of the Acoustical Society of
America, Washington, D. C., April, 1971.
18. Plotkin, K. Fuller, W. and Montroll, M., “Identification of Tire Noise Generation
Mechanisms Using a Roadwheel Facility,” Proceedings –International Tire Noise
Conference, Stockholm, Sweden, August, 1979.
19. Sandberg, U. and Ejsmont, J. A., “Tyre/Road Noise Reference Book,” Informex, SE
59040, Kisa, Sweden, 2002.
20. Varterasian, J. H., “Quieting Noise Mathematically – Its Application to Snow Tires,”
Society of Automotive Engineers, SAE 690520, Detroit, MI, 1969.
21. Williams, T. A., ”Multiple Pitch Sequence Optimization,” U.S. Patent Document
5,309,965 (1994) The General Tire and Rubber Company, May 10, 1994.
22. Bandel, P. et al.,”Low Noise Sequence of Tread elements for Vehicle Tire and Related
Generation Method,” U.S. Patent Document 5,371,685 (1994) The Pirelli Tire Company,
December 6, 1994.
23. Scavuzzo, R. W., Charek, L. T., Sandy, P. M. and Shteinhauz, G. G., “Influence of
Wheel Resonance on Tire Cavity Noise,” Society of Automotive Engineers, SAE 940533,
Detroit, MI, 1994.
24. Sakata, T., Morimura, H. and Ide, H., “Effects of Tire Cavity Resonance on Vehicle
Road Noise,” Tire Science and Technology, TSTCA, Vol. 18, No. 2, April-June, 1990, pp
68-79.
25. Gillespie, T. D., Fundamentals of Vehicle Dynamics, Society of Automotive
Engineers, Warrendale, PA, 1992.
26. Walker, J. C., “The Reduction of Noise Generated by Tyre/Road interaction,” The
German Rubber Conference, Wiesbaden, Germany, June, 1993.
27. Julien, M. A. and Paulsen, J. F., “The Absorptive Power of the Pneumatic Tire,
404 Chapter 9. Tire Noise and Vibration
33.Soedel, W. and Prasad, M.G., “Calculation of Natural Frequencies and Modes of Tries
34. Gong, S., “A Study of In-Plane Dynamics of Tires,” Delft University of Technology,
MS Thesis, 1993.
35. Lee, J.J., Pham, H.Q. and Moore, J.A., “Structure-Borne Vibration Transmission in a
Tire and Wheel Assembly,” Tire Science and Technology, TSTCA, Vol. 26, No. 3, July-
September, 1998, pp 173-185.
36. Healey, A. J., Nathman, E. and Smith, C. C., “An Analytical and Experimental Study
of Automobile Dynamics with Random Roadway Inputs,” Journal of Dynamic Systems
Measurement and Control, Trans ASME, December, 1977.
37. Yu, H. J. and Aboutorabi, H., “Dynamics of Tire, Wheel and Suspension Assembly,”
Tire Science and Technology, TSTCA, Vol. 29, No. 2, April-June, 2001, pp 66-78.
38. Marshall, K. D. and St.John, N.W., “Roughness in Steel Belted Radial Tires –
Measurement and Analysis,” Society of Automotive Engineers, SAE 750456, Detroit, MI,
1975.
39. Marshall, K. D., Ohio Dynamics, Inc., unpublished file data, 1994.
40. Robinson, D. W. and Dadson, R. S., “A Re-determination of the Equal Loudness
Relations for Pure Tones,” British Journal of Applied Physics, Vol. 7, 1956, pp. 166-181.
41. Smith, C. C. and McGee, D. Y., “The Prediction of Passenger Riding Comfort from
Acceleration Data,” ASME paper 77-WA/Aut-6, American Society of Mechanical
Engineers, New York, 1977.
42. Leatherwood, J. D., Dempsey, T. K. and Clevenson, S. A., “A Design Tool for
Estimating Passenger Ride Discomfort within Complex Ride Environments,” Human
Factors, Vol. 22, No. 3, 1980, pp 291-312.
Chapter 9. Tire Noise and Vibration 405
Glossary
Acoustic:
Boom
A high intensity vibration (25 – 100 Hz.) perceived audibly and characterized as a sensation
of pressure by the ear.
Noise
Unwanted sound is typically referred to as noise.
Road roar
A high intensity vibration (100 – 300 Hz.) perceived audibly and excited by the pavement tex
ture such as pebbly surfaces.
Thump
A periodic vibration and/or audible sound generated by the tire and producing a pounding
sensation which is synchronous with wheel rotation.
Tactile:
Harshness
Vibrations (15 to 100 Hz.) perceived tactily and/or audibly, produced by the interaction of
the tire and road irregularities.
Tire roughness
Vibration (15 to 100 Hz.) perceived tactily and/or audibly, generated by a rolling non-uni
form tire on a smooth road, and producing the sensation of driving on a coarse surface.
Shake
The intermediate frequency (5-25 Hz.) vibration of the sprung mass as a flexible body, gen
erated by the first harmonic of radial (normal) force variation.
General:
Nodal point
A nodal point is a location on a vibrating structure where the amplitude of motion for some
particular mode of vibration is equal to zero. Other vibrational modes may or may not
exhibit zero motion at the same location.
Sprung mass
Sprung mass is all the mass that is supported by the suspension of the vehicle, including the
mass of a portion of the suspension itself.
Unsprung mass
Unsprung mass is the mass of the tire and the wheel plus the portion of the mass of the sus
pension that is not part of the Sprung Mass.
406 Chapter 9. Tire Noise and Vibration
1. What are the frequency ranges for tactile and acoustic ride disturbances?
a. Tactile 1 – 60 Hertz, Acoustic dc – 10000 Hertz.
b. Tactile 20 – 300 Hertz, Acoustic dc – 10000 Hertz.
c. Tactile 1 – 300 Hertz, Acoustic 20 – 20000 Hertz.
d. Tactile 20 – 300 Hertz, Acoustic 20 – 20000 Hertz.
Answer: c
2. What are representative values for a passenger tire spring and percent critical damping?
a. Spring rate 600 kN/m, Damping 1.5 percent.
b. Spring rate 200 kN/m, Damping 1.5 percent.
c. Spring rate 600 kN/m, Damping 5.0 percent.
d. Spring rate 200 kN/m, Damping 5.0 percent.
Answer: b
3. What are the most important vibrational frequency ranges for a loaded passenger
tire?
a. Vertical 10 to 25 Hertz and 60 to 120 Hertz,
Answer: a
4. Assume a passenger vehicle has a boom frequency of 42 Hertz. What is the speed of
the vehicle when the 3rd, 4th and 5th harmonics of force variation excite the peak value
of this ride disturbance? Assume the tire rotational rate is 1 rps per 5 mph.
Answer: 3rd harmonic = (42 Hz / 3rd) x (5 mph) = 70.0 mph
4th harmonic = (42/4) x (5) = 52.5 mph
Answer:
Hertz).
b. Air pumping from the tread pattern (frequencies higher than 2000 Hertz).
c. Both ‘a’ and ‘b’ are important between 1000 and 2000 Hertz.
Answer: b.
9. Rank the following tire/road interaction noise combinations from quietest to loudness.
Answer: d, a, b, c
Answer: d
11. What is the damped suspension frequency for a ¼ car model for a vehicle with a mass
of 1,400 kg, and tire and suspension spring rates of 200 kN/m and 30 kN/m, respectively.
Assume the weight distribution is 50/50 front/rear, and the damping ratio is 0.25.
12. If the unsprung mass of the suspension in problem 11 is 50 kg, what is the
13. Assume a noise signal is composed of two 1/3 octave bands of acoustic energy; B400
14. What is the A-weighted sound pressure level for the noise signal in problem 13?
Chapter 10
by D.M. Turner
Chapter 10
Waves in Rotating Tires
by D. M. Turner
1. Introduction
This chapter describes the phenomenon of “standing waves” observed in tires at high
speeds, causing increased energy consumption and overheating. A simple theoretical
treatment is proposed that accounts for the main features in bias-ply tires: the existence of
a critical speed C at which the waves first appear, and the increase in energy consumption
at speeds above C. The theory is then modified to apply to radial tires and more complex
tire constructions.
In the 1950s in Europe there was intense competition between motorcycle manufactur
ers in Grand Prix Racing. At least four tire manufacturers were also competing. The
500cc bikes were achieving speeds of over 150 mph and occasionally a tyre lost its tread.
Tests on drum dynamometers showed that at speeds above 70 to 80 mph there was a steep
increase of power consumption with increasing speed. Close examination of the tyre
profile as it parted from the surface of the drum showed the presence of a wave. Gardner
and Worswick [1] carried out an experimental investigation and found that the angle of
tire cords relative to the circumference of the tire (crown angle) was a vital factor - the
lower the angle the higher is the speed at which the wave effects arise. Hysteresis in the
rubber causes the wave to be heavily damped so that it extends for only a short distance
round the tire. This implies that there will be no reflection of the waves in the circumfer
ential direction.
To some extent the waves are analogous to the wake behind a ship. These appear to
travel through the water at a phase velocity that is the same as that of the ship. However
the energy of the wave travels at a lower velocity, the group velocity, and consequently
energy is carried away at a rate proportional to the difference between the two velocities.
2. Wave mechanics
A tire is remarkably complex in its structure and geometry. A model embracing too many
of the features becomes over-complicated and the basic understanding is lost. Here we
consider three basic cases of wake formation in a wheel/tire, selected to demonstrate the
principles that are involved.
In figure 10.2 m represents the mass of a tire segment and l is the pitch length. The lin
ear density is given by ρ = m / l. We imagine that the spring-mass system is arranged
radially around a wheel of radius R that is compressed afgainst a flat surface so that an
angle φ is subtended between the center and the end of the contact patch. Vt is the veloc
ity of the tire. The vertical component of the velocity of the tire as it leaves the contact
patch is Vt sin f but as φ is generally small, replacing sin φ by φ is a satisfactory approx
imation for present purposes.
Thus tire segments leave the ground at a vertical velocity of φVt with energy ½ m (φ
Vt)2. All of this energy is eventually dissipated by damping processes in the tire. The
rate of energy loss is ½ m (φ Vt )2 Vt / l = ½ ρ φ2 Vt3.
If E is the elastic constant of each spring, the resonant frequency of each mass will be
F = √ (E /m ) and the wavelength will be λ = Vt /F. Note that this model does not predict
a critical velocity that must be reached before waves can be seen or before they contribute
to the power consumption.
Chapter 10. Waves in Rotating Tires 411
Figure 10.3
Figure 10.3: The Y axis is in the vertical direction and the X axis is horizontal. A sin
gle cycle is illustrated and further cycles would occur along the X axis. It is seen that the
displacements are greatest at the centre line and zero at the beads.
In one cycle the distance travelled by the wave front at velocity C is OQ. The distance
between peaks on the centre line is the wavelength λ = OP. The phase velocity Vp is the
velocity of propagation of the peak in the X direction:
Vp = (OP/OQ) C = C /cos θ. Therefore cos θ = C/ Vp.
The group velocity Vg is the velocity of flow of energy in the direction of the X axis:
Vg = (OR/OQ) C = C cos θ. The wavelength λ = OP = 4 B tan θ, where tan θ = √((1 –
cos2θ)/cos2θ). Thus λ = 4B Vp /C√(1 – C2/ Vp 2).
So far the diaphragm has been considered to be static. However the waves are observed
to be stationary relative to the axis of the tire. This can only happen if the velocity of the
tire Vt is the same as the phase velocity Vp but in the opposite direction. This means that
that wave is actually travelling through the tire towards the contact point with the road.
This is illustrated in Figure 10.4.
Chapter 10. Waves in Rotating Tires 413
Figure 10.4
Figure 10.4: The path of the center line of the tire running on a drum at a velocity Vt
that exceeds the critical velocity C for wave formation. The wave is travelling at velocity
Vp where Vp is equal to the tire velocity Vt but in the opposite direction.
The speed of flow of energy away from the contact area is Vt - Vg, where
Vt - Vg = Vt – Vt cos2θ = Vt (1 – C2/ Vt 2). The energy per unit length of a single diag
onal wave is ½ Β ρ( ( φ Vt)2. The power consumption of a pair of waves is thus 2 (Vt -
Vg) ½ Β ρ ( ( φ Vt)2 = φ2 ρ Β Vt (Vt 2 – C2 ). Note that there is no power consumption
due to the waves until the velocity of the tire Vt exceeds the critical velocity C.
3. Cross-ply tires.
A tensioned diaphragm of uniform density is a reasonable simplification for a light treaded
racing motorcycle tire. The inflation pressure and the radius of curvature prevailing in the
cross-section determine the lateral tension. However it is the circumferential tension that
controls the critical velocity.
the angle of the cord to the centre line is θ. Then Tc = Tt / tan2 θ. Also at the crown, P =
The critical velocities in the two directions are Cc = √( Tc /ρ) and Ct = √( Tt /ρ).
This difference will cause anisotropy in the propagation of diagonal waves (not con
sidered in the earlier treatment [3]. The main effect will be equivalent to a proportional
increase in the value of the distance B from the centre line to the bead and consequently in
the wavelength.
414 Chapter 10. Waves in Rotating Tires
At the crown the pressure is supplemented by centrifugal force ρ Vt2 / R acting radially.
Then Tc/Rc + Tt/Rt = P + ρ Vt2/Rc, where
Tc = (P Rc + ρ Vt2) /(1 + tan2 θ Rc/Rt).
Thus to maximise the tension Tc, Rt must be large. This is the case in the crown of
wide-section low-profile tires.
Chapter 10.4 refers to networking of the cords and shows how the equilibrium shape
of the casing depends on the crown angle. The lower the crown angle the squatter the equi
librium shape. Motorcycles using tires with low crown angles, inflation pressures of
around 30 psi and light nylon casings won all of the Grand Prix races in the late 1950’s.
Thus a tire that is moulded in a squat shape with a low crown angle can be very effective.
Standard 80 series low profile cross-ply tires were used very successfully for Formula
Ford Racing in Europe in the 1960’s. Tires for Indianapolis car and F1 racing continued
the trend for low aspect ratios.
In radial tires with low profiles and very low belt angles virtually all of the centrifugal
force will be contained by the circumferential tension. The additional contribution to Tc
is: Tcirc / Rc = ρVt2/ Rc. Then C = √(Tcirc /ρ)= Vt. Thus centrifugal force should always
prevent waves from forming in radial tires. However, waves are observed near the belt
edges (see figure 10.1) as discussed in section 5. Centrifugal force also reduces the angle
of deflection φ in accordance with an effective increase in pressure.
Figure 10.4
Figure 10.5: Measured (exp) and theoretical power consumptions for a 3.50-18
motorcycle racing tire [3]. An amount of 0.018 bhp per mph has been added to the
theoretical power consumption as an allowance for non-wave contributions.
The motorcycle tire with a relatively light tread, a well rounded section and a low
overall tire to wheel diameter ratio gives the best chance of obtaining a good match
between experiment and predictions from the simple model. The present results show at
least a satisfactory first approximation. A comparison between the different loading
conditions is more valid. The tire with the lowest pressure has the lowest critical velocity
but it also benefits most from the contribution of the centrifugal force.
Units
Lx = length of segment in circumferential direction m
Ly = length of segment in transverse direction m
e(y)= mass per unit area kg/m2
M(y) = ρ(y) Lx Ly = mass of a segment kg
Tx(y) = circumferential tension N/m
Ty = Lateral tension N/m
Sx = Tx(y) Ly /Lx = elasticity in the circumferential direction N/m
Sy = Ty Lx /Ly = elasticity in the transverse direction N/m
ηx = viscous constant in the circumferential direction Nsec/m
ηy = viscous constant in the transverse direction Nsec/m
The last four constants relate to the forces acting on an individual segment. The viscosity
constants are empirical and are chosen so that the distance the wave extends round the tire
is in accord with observations.
Where there is an array (y), values for the parameter can change according to the
transverse position. The derived parameters minimise the number of repeated calcula
tions. The time step is set so that the results are stable and further reductions in the time
step do not affect the result. Typically there will be 100 x steps, 10 y steps and 2000 time
steps, which take only a few minutes to run on a PC.
Figure 10.6: Mass and tension network for the time domain model
Fx(x,y)and Fy(x,y) are two of the four forces acting on the segment of mass m(x,y).
Fx(x-1,y ) and Fy(x,y- 1) are the other two. Z(x,y) is the displacement and V(x,y) is the
velocity of the segment (x,y) perpendicular to the plane of the casing.
Fx(x,y) = ( Z(x+1,y) - Z(x,y)) Sx + (V(x+1,y) - V(x,y)) hx
Figure 10.7
418 Chapter 10. Waves in Rotating Tires
5. Radial tires
The time domain model allows the question of waves in radial tires to be tackled. In
general, tires with fabric belts and folded edges or steel belted tires with cord fabric over
lapping the belt do not exhibit waves. However a radial tire with a steel belt can exhibit
waves that are confined to the belt edges. The photograph in Figure 1 of a 205/55R16 tire
running on a drum at 320 km/h illustrates such a case. If the upper steel ply terminates
before the lower steel ply at each side the tread will not be fully supported by the belt.
Consequently the circumferential tension for this band will be reduced. There is no
circumferential tension due to the cords in the sidewall but the shear stiffness of the
sidewall rubber provides some interconnection in that direction.
The program was used to calculate vertical displacements of the 13 mm wide bands of
another tire. The densities of the bands are as follows:
Tread 20.2 kg/m2
Belt edge 16.5 – 11.8 kg/m2
Sidewall 5.9 kg/m2
The tread edge wave is a diagonal wave. On one side a reflection is caused by the highly
tensioned tread area. On the other side there will be a partial reflection due to the
impedance mismatch at the interface with the sidewall and a full reflection at the bead.
The wavelength of the sidewall wave differs from the tread edge wave and although
initiated by the latter it will have its own regime. At present the sidewall tensions and
stiffnesses to be employed are only guesses. Fitting the results to some substantial
experimental data is necessary to carry the modeling further.
The energy consumed by the wave can be calculated by multiplying the force due to
viscous damping by the relative velocity at each face of a segment and summing along the
length for each band. Figure 9 shows the result of this operation for the case illustrated in
figure 10.8.
Figure 10.9: Rate of consumption of energy in watts predicted by the time domain
model for a 155/80 R13 tire on a drum at 45, 50 and 55m/sec. The dark colours are for an
unsupported tread edge and the light colours for a fully supported edge.
Figure 10.9
6. Discussion
There is an indication of a second wave affecting the tread bands. It is estimated that the
wavelength is about 800 mm giving a frequency of 63 Hz at a speed of 50m/sec, while the
frequency at the tread edge is in the region of 550 Hz. Runs on the model over a range of
speeds suggest that the frequency of the former may be constant indicating that the mech
anism is of a mass, i.e. the tread and belt as a single band, supported on a spring, i.e. the
sidewall, as described in section 2.2. The model covers this case but it should be noted
that the spring rate due to the cord tension in the sidewall needs to be that when the side
wall is curved rather than a plane.
Concern has been expressed that the load on the tire will reduce the circumferential
tension. Rough calculations on high speed tires which have low loads, high inflation
pressures and high centrifugal forces indicate that they should not lose more than 10% of
the tension. Dedicated numerical or FE programs are needed to provide accurate results
420 Chapter 10. Waves in Rotating Tires
7. Conclusions
A proper study of standing waves in tires requires an approach that considers the tire as a
surface. The mechanical properties will vary very considerably from the tread centre line
to the bead. Thus a numerical approach is required. It has been shown that a model using
extremely simple concepts and mathematics will cope automatically with some quite
complex physics. There is considerable scope for further development of the model in
response to needs arising from the operation of tires at high speeds.
It is unfortunate that the term “standing wave” has been accepted for this phenomenon.
The wave is in fact a travelling wave, progressing with a phase velocity equal and
opposite to that of the tire perimeter. A standing wave would only be formed if there were
a reflection of the travelling wave at the far end of its path. No waves have been reported
that have travelled completely around the tire and thus the resonance that causes a
standing wave has not been encountered.
Acknowledgement
I thank Mike Hinds and John Luchini of Cooper Tires for helpful discussions.
References
1. E. R. Gardner and T. Worswick, Trans. Instn. Rubb. Industr. 27, 127 (1951).
2. A. Chatterjee, J. P. Cusumano and J. D. Zolock, J. Sound Vibration 227 (5),
1049-1081 (1999).
3. D. M. Turner, Proc. 3rd. Internatl. Rubber Conf., London, 1954, pp. 735-748.
4. T. Akazaka and K. Yamagishi, Trans. Jap. Soc. Aerospace Sci. 11 (18),
12-22 (1968).
Chapter 11. Rubber Friction and Tire Traction 421
Chapter 11
by K. A. Grosch
1.1.1 The master curve on smooth and rough surfaces .................................. 424
1.1.3 Effect of filler and oil extension on the shape of the master curve ........ 427
2. Use of slipping wheels to determine the friction of rubber compounds .......... 443
2.1 Slip and load dependence of the side force .................................................. 443
2.2 Speed and temperature dependence of the side force coefficient ................ 446
2.4 The friction/side force master curve over a reduced range of log aTv values .. 451
2.5 Effect of the temperature rise and lubrication on the side force .................. 451
3.2 Speed dependence of peak and slide braking coefficients ............................ 456
3.3.1 The braking force slip curve under controlled slip ................................ 458
3.3.2 The relation between braking force and slip for ................................................
4.2 Comparison of laboratory measurements with road test ratings ................ 462
4.3 Road test correlation with side force measurement at a constant speed,
4.5 The influence of the track surface structure on the correlation between
4.6 Data evaluation treating speed and temperature as independent variables .... 468
4.7 Correlation between road test ratings and simulated road tests using
Chapter 11
Rubber Friction and Tire Traction
by K. A. Grosch
Introduction
Tire traction describes the force transmitted between tire and road under all circum
stances. It is a prerequisite for controlled steering, acceleration and braking of self-pro
pelled vehicles. It finds its upper limit in the frictional force when total sliding occurs.
Two aspects have therefore to be considered: The mechanics of force transmission of elas
tic wheels under limited slip and rubber friction.
1. Rubber friction
Some aspects of rubber friction are dealt with in the chapter on tire wear and abrasion.
Since traction is rigorously associated with friction it is necessary to discuss the important
aspects that bear on tire traction in more detail. Rubber friction differs from friction
between hard solids in that it depends on the load and strongly on speed and temperature,
while in hard solids it is virtually independent of these parameters [1]. The load depend
ence is discussed in connection with abrasion in Chapter 13, where it is shown that
although the effect of load is important for soft rubbers on smooth surfaces [2,3] it is not
so important for tire compounds on roads which are always sufficiently rough for the load
dependence to be small [4,5]. The temperature and speed dependence of friction, howev
er, have such a large impact on tire traction that they require a much more detailed treat
ment than is given in Chapter 13.
measuring equipment, the sliding speed and the magnitude of the friction coefficient. The
highest attainable friction coefficient during stick-slip is taken as representative of poly
mer friction in order to establish the decreasing branch of the master curve.
Figure 11.1: Experimental friction data (left) as function of log speed at different of
temperatures and master curve (right) of an ABR gum compound on a clean dry
silicon carbide 180 track surface referred to room temperature [from ref 11].
coefficient of friction
0 coefficient of friction
2.5
^-5.2
4.5
2.0 10
-10
20
30
40
1.5 50
60
80.5
90.5 -15
1.0 100
-17.5
70
0.5
log (aTv 10-2m/s)
log (v 10-2m/s) log (v 10-2m/s)
0
-5 -4 -3 -2 -1 -5 -4 -3 -2 -8 -6 -4 -2 0 2 4 6 8 10
The corresponding log aT curve is then superposed on the WLF function by vertical
and horizontal shifts. Figure 11.2 shows the WLF equation with experimental points
obtained from friction measurements for four different gum rubber compounds. From the
difference between the reference temperature and the zero value of the WLF equation the
Standard Reference Temperature is obtained. This is approximately Tg + 50oC, where Tg
is the glass transition temperature of the elastomer. Table 11.1 compares the actual stan
dard reference temperatures obtained from friction measurements with values of Tg + 50
oC. The deviation for the friction experiments is generally less than 5°C. Hence the mas
ter curve can also be constructed using the WLF equation with a Standard Reference
Temperature of Ts = Tg + 50 to calculate the required shift factors. This may not be quite
as accurate as the actual shifting procedure but is much less tedious and generally leads to
very good results.
Figure 11.2: Shift factors fitted to the WLF equation of four different gum com
pounds on a silicon carbide 180 track [from ref 11].
10
SBR
8
ABR
6 Butyl
isom. NR
4
-2
-4
(T-Ts)°C
-6
-60 -40 -20 O 20 40 60 80 100 120 140
424 Chapter 11. Rubber Friction and Tire Traction
Master curves of the friction coefficient have been obtained for a wide range of rubber
compounds on different types of track and for dry and wet surfaces. The technique would
undoubtedly also function for sliding on ice. However, because ice is generally close to
its melting point it displays special properties that influence the friction of rubber in con
tact with it. In addition the temperature in the contact area cannot rise above 0°C.
The shape of the master curve and its position on the log aTv axis depends both on the
rubber compound and on the structure of the track surface. If it can be obtained, it con
tains all the friction information for a rubber on a particular surface. Slight deviations have
been observed recently for complex tire tread compounds. These will be discussed
towards the end of this chapter.
ly the same log aTv position as for the same compound on glass. However, it then rises
further to reach a maximum similar in height to that on the glass surface but at a much
higher value of log aTv. If the rough track is dusted with a fine powder, preferably mag
nesium oxide, the hump disappears but the larger maximum remains virtually unchanged,
at the same height and in the same position as on the clean surface. The presence of fine
powder has impaired the molecular adhesion between rubber and track. This is confirmed
if the friction of a dusted sample is measured on glass. In this case the friction coefficient
remains low at a value of about 0.2 for all temperatures and speeds.
Figure 11.3: Master curve of the friction coefficient of an ABR gum compound on
smooth clean dry glass, referred to room temperature. [from ref 10]
2.5
2.0
1.5
1.0
0.5
log aTv
0
-8 -6 -4 -2 0 2 4 6 8
The value of log aTv at which friction is a maximum for sliding on silicon carbide bears
a constant relation to the frequency of the maximum loss angle of the material, indicating
that this friction process is based on a classic cyclic hysteresis loss mechanism. The con
stant length of 5x10-4 m that relates sliding speed and deformation frequency is consis
tent with the spacing of the track asperities. A number of authors have published theoret
ical treatments of this mechanism of friction due to deformational hysteresis [13, 14, 15].
It is important to bear in mind that adhesion also plays a major role. Purely vertical defor
mations lead to friction values that are much too small [16]. The fact that friction on a
dusted track is still high at very low log aTv values, a range where the adhesion is also
low, suggests that quite a small adhesion component is sufficient to produce large local
tangential stresses in the rubber at the contact points with track asperities, as is also borne
out by the occurrence of abrasion. This is discussed in more detail in Chapter 13. These
tangential deformations cause elastic energy to be stored that cannot be returned to the
rubber on release and is therefore dissipated as heat, akin to an adiabatic process in ther
mo-dynamics, leading to the high friction observed in the region of low log aTv values
where hysteretic and adhesion losses are low.
426 Chapter 11. Rubber Friction and Tire Traction
1.5 1.5
1.0 1.0
0.5 0.5
0 0
-8 -6 -4 -2 0 2 4 6 8 -8 -6 -4 -2 0 2 4 6 8
Log(aTv 10-2m/s) Log(aTv 10-2m/s)
The adhesion and deformation mechanisms of friction are related to the relaxation
spectrum and loss factor, respectively. Thus the shape of the friction master curve on
rough surfaces depends on the difference between the position of maximum loss modulus
and maximum loss factor. The smaller this difference is, the broader the friction master
curve on rough surfaces. For smooth surfaces the following relation holds for all polymers
between log aTv for maximum friction and log aTf for maximum loss modulus
[log(a v )] −[log(a f )] = −9.22 [1a]
T smooth T E"
From this it follows that, if the difference between the two log frequencies of the dynam
ic properties is small, the difference between the two log speeds of the friction master
curve must be large, which is indeed the case. For ABR, for example, the difference
between the two log frequencies is small and hence the difference between the two log
speeds of maximum adhesion friction and deformation friction is large. For butyl rubber
exactly the opposite is true and the difference between adhesion and deformation friction
is so small that it cannot be distinguished. The shape of these two master curves are com
pared in figure 11.5 where results are shown for a gum ABR and butyl rubber sliding on
a silicon carbide 180 track.
Chapter 11. Rubber Friction and Tire Traction 427
Figure 11.5: Comparison of the master curves of an ABR gum compound (upper)
with a Butyl gum compound (lower) both on 180 clean silicon carbide.
2.5
friction coefficient
2.0
1.5
A
1
0.5
log aTv
0
-8 -6 -4 -2 0 2 4 6 8 10
2.5
friction coefficient
2.0
1.5
B
0.5
log aTv
0
-8 -6 -4 -2 0 2 4 6 8 10
1.1.3 Effect of filler and oil extension on the shape of the master curve.
The unfilled compounds used above to demonstrate the influence of polymer and track
surfaces are too soft to be used in tires. Tire compounds invariably contain a considerable
amount of filler, predominantly carbon black but increasingly also silica. Addition of filler
does not affect the WLF speed - temperature equivalence but changes the shape of the
friction master curve considerably.
Figure 11.6 shows the master curves for an ABR compound (a) unfilled, (b) filled with
20 parts of carbon black per hundred of rubber by weight (pphr) and (c) with 50 pphr of
carbon black, sliding on smooth glass and on a dusted and a clean silicon carbide track.
On glass the curves for gum and filled rubbers have a similar shape, the single maximum,
however, is reduced progressively as the filler content is increased. It also shifts slightly
towards lower speeds. On the two rough tracks the deformation friction maximum also
decreases with increasing amount of black. On the clean rough track the deformation peak
has decreased to the same level as the vestiges of the adhesion maximum so that a broad
plateau appears between the maxima for the adhesion and deformation friction compo
nents maxima when 50 parts of black have been added.
On the dusted track the adhesion friction component is clearly lower for all three com
pounds when comparing the results with the master curves obtained on a clean track
The friction plateau observed for the compounds filled with 50 parts of black (which
428 Chapter 11. Rubber Friction and Tire Traction
is typical for tire tread compounds) is found for most elastomers, as shown in figure 11.7.
The width of the plateau depends on the difference between the frequencies of maximum
loss modulus and loss factor as explained above. If this difference is small, the plateau is
broad; if the difference is large, the plateau becomes narrow. For butyl it is non-existent
and only a single distinct maximum is displayed.
Figure 11.6: Master curves on smooth, wavy glass, on a silicon carbide track dusted
with magnesium oxide and on a clean silicon carbide track of three ABR compounds:
unfilled, filled with 20 pphr carbon black and 50 pphr respectively [from ref 11].
Figure 11.7: Master curves of different types of polymer all filled with 50 parts of car
bon black referred to 20 °C (from ref 10]
3.0
coefficient of friction
Butyl
2.0
SBR
ABR
1.0
NR
log aTv
0
-8 -6 -4 -2 O -2 -4 -6 -8 -10
If oil is added to the compound during mixing and the carbon black level is adjusted to
maintain the same hardness, the master curve on glass displays a maximum that is only a
little lower than for the gum rubber, as shown in figure 11.8 for an oil-extended black
filled NR. The dotted line indicates the master curve of the gum rubber. It is a general
experience in compounding tire treads that incorporating oil with raised black levels to
maintain the same hardness causes the friction coefficient to be increased significantly.
Chapter 11. Rubber Friction and Tire Traction 429
2.0
gum NR
oil-extended NR+black
1.0
dry silicon carbide dusted with magnesium oxide powder (d) an alumina track wetted with
distilled water and (e) wetted with water + 5% detergent.
with distilled water and (b) with 5% detergent added to the water [from ref. 17]
2.0
coefficient of friction
1.5
distilled water
1.0
water+5%detergent
0.5
log(aTv 10-2m/s
O
-8 -6 -4 -2 0 2 4
glass plate when (left) wetted with distilled water and (right) with added
detergent. The distilled water collects into globules leaving virtually dry
regions while a thin continuous film forms when detergent is added [from ref. 18]
(a) dry glass, (b) dry clean silicon carbide 180, (c) dry silicon carbide
dusted with magnesium oxide powder (d) Alumina 180 wetted with
2.0
1.0
log(aTv 10-2m/s)
0
-8 -6 -4 -2 0 2 4
allel to each other, because the rubber will deform under the hydrodynamic pressure.
Roberts [18] has demonstrated this convincingly by letting a rubber sphere slide over a
smooth glass surface lubricated with a silicon oil film. His results are shown in figure
11.12. The Newtonian fringes represent lines of equal height; the central contour line is
shown at the bottom left. For the case of a rubber cylinder sliding on a flat hard surface,
the average film thickness is given by Archard and Kirk [20] as
[3]
Figure 11.12: Newton's fringes used to measure the film thickness and deformation of
a rubber sphere sliding at different speeds over a smooth glass plate lubricated with
silicon oil. The graph at bottom left shows the center contour line of the rubber
sphere (ordinate) along the center contact length (abscissa) [from ref.18]
where R is the radius of the rubber cylinder and E is the modulus of the rubber. From
dimensional considerations a similar relation should also hold when a hard solid sphere
slides on a soft rubber surface. A typical water film thickness map under a smooth tire
sliding over a wet hard surface is shown in figure 11.13 [21].
Chapter 11. Rubber Friction and Tire Traction 433
Figure 11.13: Water film thickness in the contact area of a smooth tire sliding over a
hard surface; v = 68 km/h; inflation pressure = 1.3 bar [from ref 21]
where n is the number of spheres in the contact area. Using equation 2 for the film thick
ness, the friction coefficient m can be written as
[6]
or with equation 3
[7]
[8]
where µd is the coefficient of friction on a dry track and vcrit is the speed at which the fric
tion coefficient becomes zero, i.e. when perfect lubrication is achieved. The value of vcrit
depends on which model is used to calculate it. And, in practice, the friction does not
434 Chapter 11. Rubber Friction and Tire Traction
become zero. A small frictional force remains because of the resistance to displacing the
bulge of excess water that is created in front of the tire or sample. Also, of course, the
model simplifies the road surface structure. In reality the road surface contains particles
of different size and shape having different contact radii and different height. In addition
the asperities themselves often have a micro roughness. [22-25].
The theoretical considerations outlined above suggest that in lubricated sliding the fric
tion coefficient should decrease with the square root of the speed. Practically all friction
measurements on wet tracks in the speed range where hydro-dynamic lubrication occurs
are in the form of tire skid measurements. They will be discussed in greater detail in sec
tion 2.2. Only one instance will be discussed here. Figure 11.14 shows the results of a
braking test for a smooth tire on wet finely-structured concrete. The friction coefficient
was measured as function of the speed after the wheels had been locked. Plotting the data
as a function of the square root of the speed results in a straight line graph with an extrap
olated value of the abscissa at zero friction coefficient of 10.2, corresponding to a critical
speed of 104 km/h. The intercept at zero gives a dry friction coefficient of 1.35, which is
very close to what would be expected from laboratory measurements of tread compounds
on rough surfaces. The critical speed can be related to the roughness of the track. Using
equation 2 with a = 0.12 m, b = 0.15 m, L = 3500 N, vcrit = 28.9 m/s and η = 1.01x10-3
N.s/m2, the radius of the track asperities is obtained as r = 0.13 mm. If the calculation is
based on equation 3, using R = 0.31 m as the radius of the tire, E = 24 N/mm2 and the vis
cosity as given above, the asperity radius r is obtained as r = 0.21 mm. Although the two
values differ by almost a factor of 2, both are close to reality. In any case equation 6 suc
cessfully describes water lubrication between rubber and a rough track using two measur
able quantities: the dry friction coefficient and a critical speed at which the friction
becomes zero, two very useful quantities to judge the wet traction behavior of tires on a
particular road surface. Notice that in this model the water film thickness does not enter
directly into the calculations – it appears indirectly in the critical sliding speed.
Figure 11.14: Locked wheel braking coefficients of a tire with a
root of the speed. (Tire size 175 R 14, load 350 kg, inflation pressure 1.9 bar)
0.8
friction coefficient
0.7
0.6
0.5
0.4
0.3
0.2
0.1
V1/2 (V in km/h)
0
0 1 2 3 4 5 6 7 8 9 10
Chapter 11. Rubber Friction and Tire Traction 435
Consider now the effect of the inertia of the water. This plays no part in laboratory
experiments with test samples but it always manifests itself in tire use. It will be discussed
here to complete the discussion of models that predict the effect of water on friction. The
inertia effect is associated with aquaplaning. It has been discussed extensively by R. W.
Yeager [26]. In aquaplaning the thickness of the water film is greater than the height of
the asperities and water has to be squeezed out of the contact area. As a point on the rub
ber surface moves through the contact area it sinks through the water film until it makes
contact with the tips of the asperities at a distance x along the contact patch (see figure
11.15). This length x is proportional to the sink time ta which is itself a function of the
water height above the asperities and to the speed vf of the vehicle
Figure 11.15: Sketch of a water wedge forming in the
front part of the contact area of a tire rolling on a wet road with a
water film thickness rising above the road asperities.
contact length l
x=vfta [9]
With increasing speed the point at which dry contact is made moves further along the
contact patch reducing thereby the extent of dry contact and hence the friction. When the
speed is so high or the sink time is so long that even at the end of the contact patch (of
length l) the water cannot be removed, then all frictional contact is lost and with it the
steering and braking/accelerating capability of the vehicle.
The first experimental and theoretical attempts to describe this phenomenon were
made by Saal [27]. A number of other authors have described experimental procedures
and other theoretical models [28,29]. The very brief and simple approach to be presented
here is based on Barthelt’s theory [30]. Under a given set of conditions determined by road
surface structure, state and design of the tire, in particular the tread pattern, and the water
level on the road, let the length of the dry contact region be l - x. Part of the load on the
tire, proportional to x, will be carried by the water film and will not contribute to the fric
tional force required to drive, brake or steer the tire. The frictional force Fw will then be
given by
FW=µdryRd [10]
where Rd is the reaction force of the “dry” contact area. This can be deduced from the
wedge length x and contact length l as
[11]
where
1=vaqtc [12]
Combining these equations leads to
[13]
where vaq is the critical speed at which contact is lost. This depends on the road surface
conditions including the height of the water level above the road asperities and tire prop
erties such as tread pattern design and the condition of the tire, primarily the available
void volume of the tire tread.
Using the Barthelt model and neglecting secondary terms like the influence of water
outside the contact area, the aquaplaning speed vaq can be written as
v aq=f(tire)⋅g(road)[14]
where pm is the mean ground pressure which is about equal to the inflation pressure of the
tire, ρ is the density of the water, lm and bm are the contact length and width respectively.
The function f is itself a characteristic speed of the tire. For a smooth passenger tire of
width 205 mm and an inflation pressure of 2.5 bar, the contact width and length are both
about 0.18 m. Hence the characteristic speed in this case is 55 m/s, i.e. 198 km/h.
The road function is characterized by two heights, hr which is a measure of the struc
ture of the road surface and ho, which is the water film thickness above the height hr
g(road)=(1n(ks))-1 [16]
with
[17]
Note that equation 17 is valid only for films of water with thickness ho > hr . If hr = 0.5
mm and ho = 2.0 mm, corresponding to quite a large film thickness on a relatively smooth
surface, then g(road) = 0.48 and the aquaplaning speed for the smooth tire would be 96
km/h. With a tread pattern, the contact width is reduced to the rib width and the aquaplan
ing speed increases dramatically, provided the water level is such that the void volume of
the tread pattern is not completely filled.
The sink time of equation [12] is made up of two additive terms. One acts until the
voids of the tire have been filled with water and the second term is that for a smooth tire
to which the tread pattern degenerates when the voids are filled. During free rolling the
available frictional force is determined by the forward and the aquaplaning speed. If
forces are transmitted, sliding lubrication sets in and the friction coefficient, modified by
sliding lubrication, limits the maximum force transmission further. The factor µo of equa
tion [13] contains the temperature and speed dependence of the friction coefficient and the
effect due to sliding lubrication. Which of the two lubrication contributions dominates
depends on the external conditions. If the water level is small, sliding lubrication is the
Chapter 11. Rubber Friction and Tire Traction 437
Figure 11.16: Sliding braking coefficients on wet concrete and asphalt as function of
vehicle speed (left) and of the square root of the vehicle speed (right) [from ref 31]
0.7 0.7
06 0.6 asphalt
asphalt
sliding braking coeff.
05 0.5
0.4 0.4
concrete concrete
03 0.3
02 0.2
0.1 0.1
0 0
20 40 60 80 100 120 5 6 7 8 9 10 11
speed (km/h) sqr(speed (km/h))
friction coefficient
2.0
KCl
NaCl NH4Cl
1.5
Mg(NO3)2
1.0
0.5
LiCl
0
-60 -50 -40 -30 -20 -10 0
Ice temperature °C
0.05 m/s
0.005 m/s
3.0
0.0005 m/s
2.0
1.0
their different glass transition temperatures they occupy different ranges on the log aTv
axis. If all four master curve segments are plotted as a function of log aTv on one graph,
they form the shape of a single master curve with a single maximum which has the same
position as on glass, as shown in figure 11.20 (compare with figure 11.4, right). This
shows clearly that the friction process on ice is the same as on smooth glass. A peak for
deformational friction is absent demonstrating clearly that the friction on ice is due to
adhesion.
Figure 11.19: Friction coefficient of four different gum compounds having
different glass transition temperatures as a function of the ice track
temperature at a constant sliding speed of 5 mm/s [from ref 34]
4.0
coefficient of friction
BR
3.0
NR
2.0
SBR
1.0
ABR
0
0 -10 -20 -30
ice track temperature °C
3.0
Coeff.of friction
2.0
BR SBR
1.0
ABR
0.0
-4 -2 0 2 4 6
log aTv
440 Chapter 11. Rubber Friction and Tire Traction
Friction coefficients were measured recently on the ice track disk of an LAT 100 trac
tion and abrasion tester, comparing several compounds at different ice surface tempera
tures, right up to – 4.5 °C, a temperature which is of great practical importance for winter
conditions in many northern countries with a moderate winter climate [35]. Table 11.2
shows a summary of the results at this critical ice surface temperature. A is a Standard 100
OSBR summer tire compound, B is an NR + N234 carbon black traditional winter com
pound with a low glass transition temperature and C is a 70 BR/30 SBR blend + N375,
i.e., a modern black filled winter tread also with a low glass transition temperature. D has
the same polymer composition but is filled with silica with silane coupling. The difference
between the two black filled compounds with low glass transition temperatures and the
summer compound is striking. Addition of silica improves the performance still further.
Because of the closeness to the melting point of the ice at – 4.5°C the absolute values of
the friction coefficient are very low, a dangerous situation in any case, so that the advan
tage conferred by a superior tread compound is even more important.
Table 11.2: Friction coefficients and relative rating at an ice track temperature of
- 4.5°C and a sliding speed of 0.6 km/h for four tire tread compounds of different
composition: A: OESBR + N375 black, B: NR+N220 black, C: 70 BR/30 SBR + N375
black, D: 70 BR/30 SBR + silica
Measurement average rating
Compound 1 2 3 4
A 0.0528 0.0770 0.0642 0.0718 0.0665 54.7
B 0.1208 0.1279 0.1188 0.1182 0.1214 100.0
C 0.1390 0.1360 0.1370 0.1364 0.1371 112.9
D 0.1402 0.1375 0.1479 0.1566 0.1455 119.8
µ=fice(ϕ)⋅glub(v,µd)⋅µd(v,ϕ)[18]
where fice represents the ice properties and is a function of the ice temperature ϕ, and glub
describes the influence of lubrication. Finally, µd is the friction master curve for a dry
track surface. As shown in figure 11.20, the shape and position of this curve are those for
adhesional friction.
The function fice is determined by two boundary conditions. First, at zero temperature
the function is zero if sufficient ice is present that no ground contact is made, implying
that the ice/water mixture has no shear resistance. Second, at a particular temperature the
full strength of the ice is reached and the ice surface behaves like a normal hard surface.
This temperature can be determined by experiment. It lies between –10 and –15 °C. At
this temperature the function fice = 1 and its slope is zero. A function that fulfills these
conditions is the integral of the normal distribution function, normalized to give 1 when a
given temperature is reached
[19]
Chapter 11. Rubber Friction and Tire Traction 441
where
[20]
In these equations, ϕ is the ice surface temperature, ϕhard is the temperature at which it is
assumed that the ice has reached its full strength (from a friction point of view), and σ is
the width of the normal distribution function. To cover the complete area the integration
of the norm should go from -∞ to +∞. However, since a range of 6σ covers 99.9 %, this
is taken as giving sufficient accuracy to the integration. Hence the integration covers the
range from - 3σ at 0°C to 6σ at ϕhard with ϕm=ϕhard/2.
Frictional heating in the contact area melts some ice, forming a thin water film. The
function glub defines the state of lubrication through this melting.
[21]
where h is the water film thickness and hcrit is the water film thickness when contact is
completely lost. For this to happen, enough water has to be melted to fill all voids of the
two contacting surfaces. The water film thickness is obtained from power consumption in
the contact area
F⋅v=bw⋅h⋅v⋅ρ⋅{ch(0-ϕ)+Mh}[22]
where F is the frictional force, ϕ is the contact temperature, v is the sliding speed, bw
is the contact width, which is generally the block or groove width of the tire tread pattern,
h is the film thickness, ρ is the water density, ch is the specific heat of ice, and Mh is the
latent heat of melting. The frictional force F can be written as
F=µ⋅p⋅bw⋅1b1 [23]
where p is the ground pressure, taken to be equal to the inflation pressure, and lbl is the
contact length. Combining these two equations and estimating the melted film thickness
it is easily seen that it is only a few nanometers thick. Zero temperature is reached very
quickly as the friction coefficient rises with decreasing ice temperature. The high latent
heat of melting is responsible for the fact that only a very thin film of melted water can
form.
On the other hand, even for rather smooth surfaces of ice and rubber, the surface rough
ness is at least two orders of magnitude larger than the melted film thickness. Hence melt
ed ice water plays no significant role in braking on ice at any speed once the ice temper
ature is a few degrees below its melting point. Thus the critical water film thickness does
not need to be known accurately.
Combining the above equations leads to
⎛ µplc ⎞
g lub = ⎜⎜1− ) ⎟⎟ [24]
⎝ ρ (cw (0 − ϕ )+ M h )hcrit ⎠
Writing
[25]
and
442 Chapter 11. Rubber Friction and Tire Traction
glub=(1-k2µ) [26]
the friction coefficient becomes
[27]
Figure 11.21 shows the friction coefficient of two model tread compounds as function of
the ice temperature and the sliding speed. Both have the same shape of the master curve,
a normal distribution function to represent the friction master curve on a smooth surface,
but they have different glass transition temperatures, - 68°C for the winter tread and –
46°C for the summer compound. As expected the friction coefficient is very low around
0°C and rises to a maximum near the point where the ice reaches its full hardness. Once
friction has reached a value high enough to create sufficient heat to bring the contact tem
perature to zero, melting of the ice sets in and keeps the temperature at zero. Hence the
friction coefficient is virtually independent of the ice temperature, but drops with increas
ing speed because of the increasing value of log aTv.
Figure 11.21: Friction coefficient of two tread compounds (master curve
simulated by a normal distribution function with the following parameters:
µmin = 0.2, µmax = 1.45, log(aTv)µmax = - 1.5,σ = 2), as a function of sliding speed
and ice surface temperature ϕ, with ϕhard = - 15°C . The only difference
between the compounds is their glass transition temperature: For the winter
compound Tg = - 68°C and for the summer compound Tg = - 46°C
g
1.4 1.0
1.2
0.8
10
fric. coeff.
fric. coeff.
0.8 0.6
06 0.4
0.4
02
0.2
00 -26 0.0 -26
-18 -18
5
5
-13 -13
25
25
-9 (°) -9 . (°
)
45
p.
45
sp e -5 spe -5 p
em tem
65
65
ed ed
( km -1
ic et (km
/h
-1 ice
/h ) )
The rating of these two model compounds is shown in Figure 11.22 as a function of the
ice track temperature and speed. Clearly, the winter tread compound has a much higher
rating over the whole range of variables, in particular when the ice temperature is very
close to its melting point. This is important since the absolute value of the friction coeffi
cient is very low in this region. These conclusions are in good agreement with the results
of laboratory experiments shown in Table 11.2.
The reason for this behavior is the high values of log aTv which occur because the ice
temperature cannot rise above zero. This is demonstrated in Figure 11.23 which shows
the model master curve and the operating log aTv values for the dry friction coefficient of
the two compounds. Because they are on the right-hand branch of the curve the compound
with the higher glass transition temperature also has higher log aTv values and hence a
lower friction coefficient. A broader treatment of this is given in section 3.3 below.
Chapter 11. Rubber Friction and Tire Traction 443
225
200
relative rating
175
150
125
100
-1
-5
-9
65
-13
45
ice /h )
tem -18
25
°C -26 (km
ed
spe
5
Figure 11.23: Model friction master curve for a smooth surface represented by a
normal distribution function as for figure 11.21, showing the experimental ranges of
the friction simulation for the winter and summer tire tread compounds.
angles while the coefficient of friction determines the side force at large slip angles. (The
same conclusions hold for longitudinal slip.) These results were obtained using a simple
model of a tire - the so-called “brush” model proposed by Schallamach and Turner [refer
ence 25 of Chapter 13] - in which the tire is treated as a large number of independent radi
al elements (“bristles”) that deform elastically under side and/or longitudinal forces. A
fuller account is given in Section 3.1.2 of Chapter 13. Although it is extremely simple, the
model describes the main features of slipping tires remarkably well. Figure 11.24 shows
a curve of the side force coefficient vs. slip angle for two different friction coefficients cal
culated using the brush model with the same stiffness and applied load. It is seen that
above 10° the influence of the friction coefficient is clearly apparent. This offers a very
useful practical method to evaluate traction properties of rubber compounds in the labo
ratory. In this case, too, it is convenient to describe the data by means of the side force
coefficient, defined in the usual way as the ratio of the side force to the normal load. This
coefficient depends on the load. The side force is shown in Figure 11.25 for four different
tread compounds at a constant slip angle and speed. The lines drawn through the experi
mental points represent the curve obtained with the brush model of slipping wheels with
the two parameters, friction coefficient and cornering stiffness, adjusted to give the high
est linear correlation coefficient between the calculated and measured values [36]. These
best-fit parameters are shown in Table 11.3 for measurements of the four compounds on
three different surfaces. A possible load dependence of the friction coefficient was taken
into consideration, using a power function, but in all cases the power index n was either
zero or very close to it. It is clear that the load dependence of the side force coefficient is
entirely due to the properties of the slipping wheel and not due to a load dependence of
the friction coefficient. Also, while there is some variation of the cornering stiffness, it
appears to be random. The ranking of the friction coefficient of the four compounds, how
ever, is very clear and independent of the type of surface. The discrimination between
compounds increases as the surfaces become blunter. This is particularly true for the SBR
and 3,4 IR compounds. Finally the friction coefficients are lower the smoother the surface,
indicating that water lubrication is more effective on blunter surfaces, as expected.
of the slip angle for friction coefficients of 1.2, and 0.9, using the
1.2
a
Side force coefficient
1.0
0.8
b
0.6
0.4
0.2
0.0
0 5 10 15 20
slip angle (°)
Chapter 11. Rubber Friction and Tire Traction 445
angle of 13° and a speed of 0.06 km/h on a wet, blunt alumina 180 surface
for four tread compounds based on four different polymers, all filled
with N 234 black. Full lines were determined with the brush model
Table11.3: The parameters for the curves of Figure 25: cornering stiffness Ks,
friction coefficient µo, power index n to allow for a possible load dependence
of the friction coefficient, and correlation coefficient r between the calculated
values using the brush model and measured values
Slip angle: 13.1°
Speed: 0.15 m/s
Compound Ks µo n r
based on (N/rad)
Surface: 180 sharp wet
OE-SBR 413 1.735 0 0.997
BR 448 1.133 0 0.996
3,4 IR 549 1.736 0 0.998
NR 408 1.39 0 0.995
Surface: 180 blunt wet
OE-SBR 418 1.562 0 0.998
BR 474 1.039 0 0.996
3,4 IR 548 1.58 0 0.999
NR 447 1.18 0 0.998
Surface: ground glass wet
OE-SBR 427 0.803 -0.01 0.997
BR 402 0.56 -0.01 0.997
3,4 IR 502 0.925 0 0.997
NR 377 0.671 0 0.997
446 Chapter 11. Rubber Friction and Tire Traction
Figure 11.27: The data of figure 26 plotted as function of log aTv with
is the friction master curve for the same compound on the same surface.
1.5
force coefficient
side force
1.3
friction
1.1
0.9
0.7
Log(aTv 10.2m/s)
0.5
-6 -5 -4 -3 -2 -1 0 1 2 3
Figure 11.28: Master curve of the side force coefficient of two tread compounds with
different glass transition temperatures on wet alumina 180: 3,4 IR (Tg = - 21°C) and
OESBR (Tg = - 46°C) [from ref 36]. Two spot measurements at different water
temperatures show that the ranking of the two compounds reverses.
1.5
side force coefficient
3,4 IR+black
1.3
OESBR+black
1.1
0.9
0.7
0.5 -5 -3 4
-6 -4 -2 -1 0 1 2 3
Log(aTv 10-2m/s)
influence of temperature rises and lubrication effects in the contact area. These effects
profoundly influence the traction capabilities of tires. In the chapter on abrasion and tire
wear, it is shown that the temperature rise in the contact area can be considerable as the
sliding speed is increased and this influences abrasion by oxidation and thermal degrada
tion. Friction is primarily influenced because the temperature rise affects the operating
range on the log aTv axis of the master curve.
In order to calculate the relevant log aTv values a relation is required between the tem
perature rise in the contact area and the sliding speed. Carslaw and Jaeger [38] have dealt
in detail with the problem of moving heat sources over plane surfaces of semi-infinite
bodies. They predict that the maximum temperature rise, occurring for rubber near the
end of the contact area, is given by the following relation:
[28]
where Q is the heat generated per unit area and per unit time, lr is the contact length of the
heat source, ρ is the density, c the specific heat and K its heat conductivity. In this case it
is assumed that all the heat flows into the rubber. If the heat is produced by friction, part
of it will flow into the rubber and the other part will flow into the track surface. Assuming
a large heat capacity for both and neglecting heat losses to the sides, the amount of heat
flowing into the rubber is given by
[29]
where the subscripts t and r stand for track and rubber respectively, k is the heat diffusiv
ity, and qr is the amount of heat which flows into the rubber per unit area and time. The
temperature rise near the end of the slider will then become
[30]
where the constant ξc contains the constants listed above. Unfortunately, it is hardly ever
possible to deduce ξc from this formula. The value has to be determined by experiment.
For practical road traction tests on wet surfaces a method will be described in section 5.7.
If the temperature is a function of speed, then the factor log aT is also a function of
speed, changing the operating point on the master curve. Log aTv increases with speed and
decreases with increasing temperature. At very low speeds log v wins, but as the speed
increases the influence of log aT becomes larger, eventually overtaking log v and the term
log aTv passes through a maximum. Figure 11.29 shows log aTv values as a function of
the sliding speed for an NR and an SBR tread compound under different track surface con
ditions at a constant ambient temperature. To obtain these curves the constant ξc had to be
estimated. For dry track conditions this was derived from figure 11.36 of the chapter on
abrasion and tire wear. For wet conditions its value cannot exceed 100°C and was reduced
accordingly. For icy conditions the value cannot exceed the melting point of ice. In each
Chapter 11. Rubber Friction and Tire Traction 449
case the range of log aTv values obtainable is limited. On dry and wet surfaces log aTv
goes through a maximum. In addition, on wet surfaces lubrication leads to a decrease of
the temperature at high speeds so that the log aTv values rise again. And because of the
temperature limit of 0°C on ice, the curve rises continuously.
Figure 11.29: Log aTv values as a function of speed for an
The range of log aTv values obtainable in practical tire tests is therefore limited to
about 2 decades and only a small part of the master curve is covered. However this por
tion depends strongly on the ambient conditions.
Because of the different glass transition temperatures for different polymers the oper
ative values of log aTv also differ considerably for different compounds since they are
determined by the difference between the contact temperature and the standard reference
temperature Ts, or more fundamentally, the glass transition temperature Tg. Figure 11.30
indicates the relevant regions of log aTv for sliding speeds between 1 and 10 km/h (cor
responding to about 10 to 100 km/h forward speed with ABS braking) on the two master
curves for unfilled NR and SBR compounds sliding on glass (see upper part of the dia
gram) under wet and icy environmental conditions respectively. On wet surfaces the SBR
compound is better than the NR compound because the operative log aTv range is on the
rising part of the master curve. On ice it is on the decreasing branch, and the ranking is
reversed, as already remarked when discussing rubber friction on ice. This is the reason
for using separate sets of tires for winter and summer.
450 Chapter 11. Rubber Friction and Tire Traction
Figure 11.30: The log aTv speed function of the previous figure is combined with the
friction master curves for an NR and an SBR gum compound on glass showing the
limited range of friction values (and their position on the log aTv axis for different
testing conditions) that are obtained when the sliding speed is increased.
ice NR
wet SBR
wet NR
ice SBR
The maximum in the log aTv curve also produces a maximum in the curve relating fric
tion to speed. But this has nothing to do with the maximum in the friction master curve.
On dry or wet surfaces the operating points lie generally on the rising part of the master
curve for friction. Thus, the maximum friction is determined by the maximum value of log
aTv. This is demonstrated in figure 11.31 which shows friction measurements on an
unfilled NR compound with a slider which had a thermo-couple embedded in it. Both
friction and the corresponding contact temperature were measured, allowing transforma
tion of the data into a friction vs. (log aTv) curve. The three experimental curves were
obtained under different loads producing different temperature rises. It is seen that a sin
gle rising curve results from the transformation. The load dependence is revealed in this
case as a disguised temperature effect.
Figure 11.31: Coefficient of friction (A) and temperature rise (B) in the contact area
between an NR gum compound and a thermo-couple slider as a function of speed at
different loads. Also shown are the data plotted against log aTv (C), indicating a rising
portion of a single master curve which absorbs also the apparent load effect.
coefficient of fricition
A
1.5 1.5 150
C
B
0.5 0.5 50
-2 -1 0 -2 -1 0 1
log aTv log (v m/s)
Chapter 11. Rubber Friction and Tire Traction 451
2.4 The friction/side force master curve over a reduced range of log aTv values
Since only a limited region of the master curve is relevant for a particular environmental con
dition, for routine practical compound development a limited number of experimental points
are often sufficient. Figure 11.32 shows a comparison of three tread compounds over a range
of log aTv obtained using five temperatures at a constant speed. A quadratic curve was fitted
to the data by the least squares method to represent the portion of the master curve.
Compound A is a classic summer OESBR black filled compound, compound B is a blend of
solution SBR and BR with a silica filler and silane coupling. It is seen that compound B is
better than the standard compound A at high values of log aTv and worse at low ones. This
corresponds to practical experience. With modern ABS braking systems contact temperatures
are kept low, particularly on wet surfaces, so that the values of log aTv are relatively high.
The silica filled compound is then superior to the standard black filled one. With locked wheel
braking the contact temperatures are high and the log aTv values are low: the ranking of the
braking performance of these two compounds is then reversed. Compound C is a classical
winter tread compound which is poorer over the whole experimental range of log aTv values.
It would show its merits at temperatures below zero.
Figure 11.32: Side force coefficient at a constant load, slip angle and speed as a func
tion of log aTv obtained by using five different temperatures as measuring points and
assuming that the WLF transformation applies. To obtain a continuous curve a quad
ratic equation was fitted to the data using the least squares method
1.2
B 1.1
side force coefficient
0.9
A
0.8
C
0.7
0.6
-5 -4 -3 -2 -1 0
log aTv
2.5 Effect of the temperature rise and lubrication on the side force coefficient as a func-
tion of slip angle on wet tracks
The temperature rise in the contact area also influences the side force coefficient during
partial sliding. The brush model assumes a constant friction coefficient. However, the slip
speed vs will affect the temperature in the contact area and hence the effective speed aTvs
and thus the friction coefficient. This also affects the relation between side force and slip
452 Chapter 11. Rubber Friction and Tire Traction
vs=vfsin θ [32]
[33]
where ξc is a constant, F sin θ is the energy dissipated by the slipping wheel, and ta is the
ambient temperature. To obtain the appropriate side force coefficient from the friction
master curve for the relevant compound the log aTv value is calculated for the above tem
perature and slip speed using the WLF equation. The side force F is then calculated using
the brush model with the relevant friction coefficient µ. If the friction master curve for the
compound is available the friction coefficient can be assigned for the calculated values of
log aTv. If only a limited number of points are available, a curve has to be fitted to them
and the appropriate friction coefficient is read from that. Usually a quadratic equation is
sufficient as seen from Figure 11.32. In general the operative values of the friction coef
ficient are on the left branch of the master curve. As the temperature in the contact area
rises, the log aTv value drops and the friction coefficient, and with it the side force,
decrease with increasing slip angle.
In addition to the influences of temperature and speed on the log aTv values, the fric
tion coefficient is reduced on wet tracks as the sliding speed increases because of the
increasing hydrodynamic lift. For a constant forward speed the hydrodynamic lift increas
es with the slip angle in cornering and with the circumferential slip when braking or accel
erating. The side force coefficient can be calculated using the brush model with a friction
coefficient modified by the temperature rise in the contact area and by water lubrication
as described by equation [8], but replacing the speed v by the slip speed vf sin θ
[33]
An example is shown in Figure 11.33 using the master curves of Figure 11.32 for com
pounds A and B. Also shown is the calculated contact temperature, the values of log aTv
obtained from the temperatures and sliding speeds and the rating of compound B relative
to A. In all cases the curves were obtained for a friction coefficient (a) without consider
ing the effect of water lubrication and (b) taking it into account, using the same tempera
ture calibration constant for both cases. The reduction of the friction coefficient through
water lubrication leads also to lower temperatures in the contact area. The higher temper
ature dependence of compound B results in a cross-over at higher slip angles as is already
apparent from the master curve of Figure 11.32. This is a general experience in tire tests.
In fact compound B has primarily been developed for use in conjunction with the ABS
braking system which limits the slip to about 10%, a range in which this compound is
clearly superior to the standard black filled OESBR both on dry and lubricated tracks.
Chapter 11. Rubber Friction and Tire Traction 453
Figure 11.33: Side force coefficient as a function of slip angle (upper left), calculated
by using the data from figure 32 for compounds A and B, the brush model and a vari
able friction coefficient taking into account temperature rises and lubrication effects
on wet tracks. Curves were constructed allowing temperature rises without lubrica
tion effects and with a hydrodynamic lift due to water lubrication. Also shown are the
operating log aTv values (top right) the calculated temperature rises (bottom left) and
the rating of compound B relative to A (bottom right)
side force coefficient log aTv
1 0.5
0
0.8
side force coeff.
-0.5 0 10 20 30 40 50 60 70 80 90
(b) without lubrication
0.6 -1 With
log aTv
-1.5
0.4 (a) With lubrication -2
-2.5
0.2
-3 Without
0 -3.5
0 10 20 30 40 50 60 70 80 90 -4
slip angle (°) slip angle (°)
3. Tire traction
3.1 Braking test procedures
In a typical braking test deceleration of the test car is measured over a short time starting
from a set initial speed. A wide range of speeds can be covered. The rear brakes are dis
connected so that the side forces on the rear wheels keep the vehicle stable up to the high
est approach speeds. Because the drop in speed is small during the time of braking the
speed dependence of the braking coefficient can be obtained over a wide range. A typical
deceleration time diagram for a locked wheel braking test is shown in Figure 11.34. The
speed is also recorded. Before the brakes are applied the vehicle is allowed to roll freely
for approximately one second to establish a base line. The deceleration increases as the
brakes are applied, passes through a maximum, and then levels off at a lower value that
remains approximately constant with time, provided that the time interval of braking is
short so that the decrease in speed is small. To obtain the actual braking coefficient from
the measured deceleration, two corrections due to load transfer during braking have to be
applied: (a) the normal reaction on the front wheels increases and (b) the vehicle tilts for
ward, decreasing the measured deceleration in the absence of an inertia platform. Both
effects are vehicle dependent but can be obtained from the vehicle data. The actual brak
ing coefficient bc can be written as
[34]
454 Chapter 11. Rubber Friction and Tire Traction
where F is the frictional force, R is the normal reaction on the front wheels of the testing
vehicle, c1 and c2 are constants depending on the testing vehicle and b′ is the measured
braking coefficient.
Figure 11.34: Braking deceleration time record for a short-time braking test
time (s)
The abrupt drop from the peak to the sliding value of the braking force is associated
with spontaneous locking of the wheels that occurs because of the negative force-slip rela
tion, which itself is due to the rising temperature and increasing lubrication, as shown in
Figure 11.32.
The sliding value of the braking coefficient is not always constant. Often it rises with the
time of braking, as shown in Figure 11.35. This is expected from decreasing lubrication
and falling temperature in the contact area as the speed decreases. However, if experi
ments are carried out with different initial speeds and the instantaneous speed is also
recorded, as shown in the time records above, a single curve of the sliding coefficient
should be obtained over a wide range of instantaneous speeds, independent of changes
with time during the test. But this is often not the case, as shown in Figure 11.36 where
the slide coefficient is shown as a function of tire speed on a wet asphalt track. The dif
ferent symbols refer to different initial speeds. The tire which has skidded for a long time
has a higher slide coefficient than the one that has skidded for only a short time, both com
pared at the same instantaneous speed. Thus conditions in the contact area change not only
Figure 11.35: Braking deceleration time record for a short-time braking test on wet
asphalt at a lower initial speed showing a sliding coefficient rising with time
Chapter 11. Rubber Friction and Tire Traction 455
with speed but also with time. There is no obvious explanation for this. The tire surface
in contact with the track appears to be modified during braking, most likely by abrasion,
perhaps even through some thermal degradation, thereby influencing the friction coeffi
cient. For example, NR tread compounds tend to become sticky in friction experiments on
wet tracks at quite moderately elevated water temperatures, showing an increase in fric
tion at log aTv values where it would not be expected on visco-elastic grounds.
In a second type of braking experiment a trailer is pulled by a car and the tow bar pull
is measured while the brakes of the trailer wheels are applied. Either one or two wheeled
trailers are used. In both case there are a number of disadvantages over the car skid test.
The tow bar pull measures the force acting between car and trailer and any acceleration
or deceleration of the car during the experiment causes an error. This can be overcome to
some extent by allowing the car to roll freely during the experiment and determining its
own rolling resistance in a separate experiment. The range of speeds that can be covered
is also much smaller since it is difficult to maintain the trailer in a stable condition at high
er speeds. In this case, too, load transfer occurs, but in the opposite sense - the normal
reaction to the tire load is reduced and transferred to the connecting pin of the tow bar.
The formula for the braking coefficient is then given by
bc = c1b′(1+c2b′) [35]
where the constants c1 and c2 are obtained from force transducer calibration and trailer
dimensions. b′ is the apparent friction coefficient - the frictional force divided by the nor
mal load on the tires.
In recent times the simple measurement of stopping distance has become popular
because it is less sensitive to variations in road surface structure and also, in many cases,
to slight variations in water depth. This method, however, produces only a single reading.
Some indication of the effect of speed can be obtained by using different initial speeds.
More sophisticated equipment uses measuring hubs either on one wheel of a testing vehi
cle or on a fifth wheel mounted on a large mobile testing platform. In most cases five com
456 Chapter 11. Rubber Friction and Tire Traction
ponents can be measured: three forces and two moments. Simpler versions can still meas
ure the three force components. Both measurements can be made under controlled longi
tudinal slip and cornering.
Speed1/2(km/h)
rising branch of the friction master curve, the braking coefficient decreases with increas
ing contact temperature. In addition, the braking coefficient drops because the water lubri
cation effect increases with increasing sliding speed. There is not sufficient information
available to separate the two effects which both work in the same direction. The fact that
straight line graphs are obtained when plotting the data in this way suggests that the water
lubrication effect dominates.
From the two straight-line graphs for the peak and slide values (the latter is also shown
in Figure 11.14) two parameters can be deduced. The “dry” braking coefficient at zero
speed is nearly the same for both: 1.045 for the slide value and 1.085 for the peak brak
Chapter 11. Rubber Friction and Tire Traction 457
ing coefficient. From the intercepts on the square root of speed axis, the critical sliding
speeds vc are 118 and 388 km/h respectively. Writing equation [8] as
[36]
where bcd is the braking coefficient at zero speed, sl is the slip, (sl = 1 in sliding), and vf
is the vehicle speed at the time of measurement. The ratio of [vc]slide to [vc]peak should
give the slip sl. The value of 0.3 is somewhat larger than the value expected from direct
measurements, about 0.2.
Figure 11.38: Slide values of the braking coefficient as a function of (speed)1/2 for
a radial ply tire on wet concrete, comparing two basic patterns with a smooth tire.
(Tire size
g 175 R 14, load 350 kg, inflation pressure 1.9 bar)
braking coefficient
Sped1/2 (km/h)
458 Chapter 11. Rubber Friction and Tire Traction
where pm is the mean ground pressure, which is about the same as the inflation pressure,
and ρ is the density of water. For a normal passenger car this flow speed is about 80 km/h.
Water is mainly removed towards the sides. Hence longitudinal ribs with short lateral
channels are required to provide short drainage distances towards the sides and thus to
increase the critical aquaplaning speed of the tire. To optimize the removal of water, diag
onal ribbing and directional tread patterns are employed. A modern high performance tire
can no longer be mounted in either direction on the rim. Attention has to be paid to mount
ing the correct side-wall towards the outside in order to achieve maximum water drainage.
Diagonal ribbing also helps to reduce tire noise. Pattern design has become very sophis
ticated, to combine maximum safety with the lowest noise level.
Figure 11.39: Braking force coefficient as a function of slip at zero slip angle on
wet asphalt at a constant speed of 30 mph, obtained with the Mobile Traction
Laboratory of NHTSA. The curve was fitted using the brush model and a variable
friction coefficient taking account of temperature and lubrication effects.
braking coefficient
Slip
as indicated above. The data presented here were obtained with the NHTSA Mobile
Traction Laboratory (MTL). The curve was fitted to the data using the brush wheel model
and a variable friction coefficient as defined in equation [8]. Regression coefficients
between calculated and measured values were used to find the best-fit parameters.
Chapter 11. Rubber Friction and Tire Traction 459
Theoretically, it is possible to take into account separately the temperature and speed
dependence of the friction coefficient µd. However, because they tend to be additive, both
effects are included in the power index and µd appears as a constant. It is seen that the
braking coefficient decreases steadily with increasing slip. The abrupt drop between peak
and slide values in the time record of a braking test is due to the negative slope of the
curve relating braking coefficient to slip.
3.3.2The relation between braking force and slip for composite slip on wet roads
The force-slip relation when both cornering and longitudinal acceleration/braking forces
are acting on a tire simultaneously was discussed under abrasive tire wear for the brush
model with a constant friction coefficient (see chapter 13). With the MTL it was possible
to establish braking force curves under a set slip angle as shown in Figure 11.40. It is seen
that the side force coefficient decreases as the braking force increases. The reason is that
the total friction force is limited to a maximum value of µL, where L is the normal load.
The brush model describes this behavior very well as seen by the curves that have been
fitted to the data using the model with a friction coefficient which takes into account the
lubrication effect as a function of the slip speed. Figure 11.41 shows calculated curves for
different set slip angles using the brush model and a variable friction coefficient.
These data can also be plotted with the side force coefficient portrayed as a function of
the braking/acceleration coefficient, producing a semi-ellipse for a constant friction coef
ficient. For a variable friction coefficient these ellipses turn inward at high braking forces,
indicating that the braking coefficient decreases, as shown in Figure 11.42. Similarly, the
maximum side force coefficient decreases for large slip angles.
Figure 11.40: Braking and side force coefficients as a function of the longitudinal slip
for a set slip angle of 8° on wet asphalt at a constant speed of 30 mph, obtained with
the Mobile Traction Laboratory of NHTSA. The curves were fitted using the brush
model for composite slip with a variable friction coefficient.
Slip
460 Chapter 11. Rubber Friction and Tire Traction
Figure 11.41: Braking coefficient as a function of slip for different slip angles,
calculated with the brush tire model and a variable friction coefficient.
Segel [39] obtained similar graphs on dry tracks again using traction equipment that
could measure all of the force components and moments under controlled slip at set slip
angles. At zero slip angle (and at all other slip angle settings) the braking force coefficient
increased with increasing slip and passed through a shallow maximum at about 20% slip
as shown in Figure 11.43 A. The experimental points have been taken from one of Segel’s
curves (Figure 11.7 of his chapter). They agree with the hypothesis that the friction coef
ficient decreases with increasing temperature brought about by increasing slip speed. The
solid line was drawn using the brush model and a temperature and speed dependent fric
tion coefficient. The temperature rise in the contact area required to obtain the curve fit is
Chapter 11. Rubber Friction and Tire Traction 461
shown in Figure 11.43 B and the resulting log aTv values are shown in Figure 11.43 C.
The friction coefficient was calculated using a model master curve represented by a nor
mal distribution function with parameters given in Figure 23 and with Tg = - 46°C.
Finally, with the above data, the side force coefficient was plotted vs. the braking coeffi
cient for two slip angles of 4 and 8°. The turn-in at high braking force coefficients is due
to the increasing temperature in the contact area. These curves agree with the correspon
ding curves that Segel shows in Figure 11.17 of reference 39.
Figure 11.43: Braking force coefficient as a function of slip (A). The marked points
are from figure 7 of Segel's paper [ref 39]. The line was obtained from the brush
model with a variable friction coefficient and taking into account temperature rise
with increasing slip. Also shown: the contact temperature rise (B) and log aTv as a
function of slip. D shows the side force vs. braking force coefficient for set slip angles
of 4° and 8° (compare with fig. 17 of ref 39)
1.2 120
1.0 100
contact temperature (°)
B
braking coefficient
0.8 80
0.6 60
0.4
A 40
0.2 20
0.0 0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.6
-1.4
-1.6 0 0 01 02 03 04 05 06 07 08 09 D 05
-1.8
side force coefficient
-2 04
-2.2
log aTv
03
-2.4
-2.6 C 0.2
-2.8
01
-3
-3.2 00
-3.4 -1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2
The following laboratory surfaces have given high correlations with a wide range of
road test data:
Alumina of several grades of coarseness and sharpness
Ground glass
Stainless steel with finely cut grooves.
Because of the limited range of log aTv values, a range of temperatures between 1 and
50°C and a sliding speed range of three decades from 2 km/h downward is sufficient to
cover most of the slip speeds and log aTv values encountered in the contact area. Most of
the laboratory evaluations are carried out on wet surfaces, where the water provides the
required temperatures. In order to obtain the best correlation between road test ratings at
one testing condition and laboratory ratings over a given range of log aTv it is useful to fit
a simple mathematical function to the experimental points of the master curve. As already
pointed out, a quadratic equation usually gives good results. A more sophisticated
approach, that is better when minimum and maximum values of the master curve are to
be included, is to use the normal distribution function but this is rarely necessary. Figure
Figure 11.44: Side force coefficient of compound A (an OE-SBR black filled tire tread
compound) on wet blunt alumina 180 as a function of log aTv obtained at three speeds
and five temperatures (open squares) with a quadratic equation fitted to the data
(black solid line). The red marked points were obtained at one speed for five tempera
tures with the dotted red line the best-fit quadratic equation, showing the risk of
extrapolating from a limited set of data.
1.5
1.4
1.3
side force coefficient
1.2
1.1
1.0
0.9
0.8
0.7
-5 0 -4.0 -3.0 -2 0 -1.0 0.0 10
log aTv
Table 11.4: Parameters for fitting a quadratic equation to the side force coefficients as
a function of log aTv for six tread compounds on wet, blunt alumina 180 at different
speeds and temperatures.
Parameter of quadratic equation Correlation coefficient
Compound a b1 b2 r
A 1.262 0.136 0.013 0.962
B 1.074 0.113 0.009 0.961
C 1.604 0.188 0.006 0.976
D 1.513 0.226 0.014 0.992
E 1.365 0.288 0.035 0.942
F 1.520 0.226 0.016 0.969
Chapter 11. Rubber Friction and Tire Traction 463
Figure 11.45: Ratings of the six compounds of table IV as a function of log aTv using
compound A as reference
140
130
120
C
rel rating
110
A
100
F D E
90 B
80
70
-5 -4 -3 -2 -1 0 1 2
log aTv
11.44 shows data for an OE-SBR black-filled passenger tire tread compound, obtained at
different temperatures and speeds (black open squares and solid black line) and trans
formed into a master curve using the WLF equation, fitted with a quadratic function using
the least-squares method. Table 11.4 shows the coefficients of such a function, together
with the correlation coefficients, applied to six tire tread compounds for which road test
results were also available. Figure 11.45 shows the compound ratings referred to com
pound A obtained from the fitted quadratic functions. It appears that most of the other
compounds show improvements of ratings in relation to the control only over part of the
experimental range.
Asphalt - Alumina 60
1
correlation coefficient
0.8
0.6
correla ion
0.4
regression
0.2
0
-5 -4 -3 -2 -1 0 1
log aTv
correlation
correlation/regression coeff.
08
06
0.4
regression
02
00
-5 -4 -3 -2 -1 0 1
-0 2
log aTv
0.8
correlation
0.6
regression coefficient
0.4
0.2
0
-5 -4 -3 -2 -1 0 1
log aTv
compared. In the present case the laboratory rating was used as the independent variable
and the road rating as the dependent one. This means that the discrimination of the labo
ratory measurement is higher than that of the road if the regression coefficient is smaller
than 1, and the reverse is true if it is larger than 1. It is seen that the regression coefficient
becomes smaller the blunter and finer the surface. On the other hand, the repeatability of
the laboratory measurements is higher the sharper the surface, being often better than 1%
while on the very blunt ground glass it is about 3 - 5%. The finer surfaces are useful for
screening compounds in the early stages of development because of their high discrimi
nating power despite their greater variability.
Chapter 11. Rubber Friction and Tire Traction 465
4.3 Road test correlation with side force measurements at a constant speed, load and slip
angle
Since the range of the master curve values of log aTv needed for a high correlation
between road test ratings and laboratory measurements is only about two to three decades
it may be sufficient to limit the experiments to a small number of temperatures at a con
stant speed. This saves a considerable amount of time in the early stages of compound
development. To interpolate and extrapolate the curve to some extent, again a square-law
relation is fitted to the curve relating side force coefficient to log aTv. In Figure 11.44 the
red points are such measurements taken at five different temperatures at a constant speed
of 0.6 km/h. The dotted red line is the square relation obtained by the least squares method
based on the five measured points. It is seen that good agreement is obtained within the
experimental range but that extrapolation in both directions leads to considerable devia
tions from the curve obtained from a wider range of measurements. It is important to be
always conscious of this risk.
If road test ratings are available the same procedure can be followed.
4.4 Road test correlation between laboratory side force measurements at a single test
condition.
The high correlation obtained over a range of log aTv values suggests that a high correla
tion may also be obtained at a single temperature and speed test condition if this condi
tion is chosen with reasonable care. Figure 11.47 shows such a correlation obtained with
10 different tire tread compounds [37]. The road test ratings were obtained from the times
it took to negotiate a wet slalom test track between two electronic measuring gates. The
laboratory tests were carried out on wet ground glass at a water temperature of 2°C and a
speed of 0.6 km/h. The road test rating times were converted into force-based ratings
using the argument that the time taken is a measure of the maximum cornering capability
of the tread compound.
If the slalom course is described by a sine function the acceleration of a mass traveling
along the sinusoidal path is given by
[37]
where vf is the forward speed of the mass. The maximum acceleration is given by the
maximum cornering force that can be sustained without complete sliding
[38]
where µp is the peak friction coefficient on a wet track and W is the weight. Since this
coefficient is proportional to (maximum speed)2 the rating is given by
[39]
Figure 11.48: Correlation between road test ratings supplied by different tire
manufacturers and average laboratory ratings on a wet, blunt alumina 180
surface from a testing program using a range of temperatures and speeds.
130
laboratory rating (side force)
125
120
115
110
105
100
95
90
90 95 100 105 110 115 120 125 130
road test rating
Chapter 11. Rubber Friction and Tire Traction 467
Figure 11.48 shows laboratory ratings that were obtained from the average values of
tests on an LAT 100 tester at Degussa in cooperation with several tire companies who used
different road test evaluation methods for determining the wet traction properties of the
same compounds [40, 41]. In this case, too, a high correlation is obtained between road
and laboratory test results. Such screening tests give obviously only limited information,
as discussed above. It is known that traction performance often depends critically on the
test conditions and a more extensive laboratory testing program will reveal regions of bet
ter and worse traction performance.
4.5 The influence of the track surface structure on the correlation between laboratory and
road test ratings.
Considerable efforts have been expended to find the best possible laboratory surface for
evaluating the traction of tread compounds. The simplest way would be to bring road sur
faces into the laboratory. This is not possible for two reasons. First, there are no well-
defined road surfaces. Although they are primarily made up of either concrete or asphalt,
they differ drastically in coarseness and sharpness. Second, none of them is durable
enough to withstand continuous use in the laboratory.
In the present context, with the emphasis on wet friction, only surfaces with some
roughness are considered. Grindstones of different grain size have been used but also
glass surfaces either dulled chemically or mechanically and in one case stainless steel with
fine interlacing groves.
Table 11.5: Correlation (upper right) and regression coefficients (lower left)
obtained with a single test condition using six different laboratory surfaces and
road test ratings obtained with one testing condition on three different proving
ground surfaces. Laboratory test condition: side force measurements at 13° slip
angle, 76 N load, and 2°C water temperature. Road test condition: stopping
distance from 90 km/h to 10 km/h, ABS braking, ambient temperature 5°C.
Stle. steel
Al 60 Al400 Al400 Al180 Ground finely
sharp sharp blunt blunt glass lathed Concrete Asphalt-1 Asphalt-2
Al 60 0.860 0.999 0.946 0.965 0.954 0.968 0.991 0.986
Al 400 sharp 0.999 0.859 0.951 0.850 0.961 0.962 0.951 0.993
Al 400 blunt 1.077 1.075 0.953 0.853 0.870 0.968 0.980 0.941
Al 180 blunt 0.884 0.959 1.116 0.967 0.877 0.884 0.970 0.942
Ground glass 2.362 1.946 2.106 2.772 0.993 0.896 0.864 0.970
St. steel 0.965 2.380 2.016 2.189 2.881 0.954 0.876 0.908
Concrete 0.374 0.377 0.928 0.792 0.866 1.071 0.943 0.922
Asphalt-1 0.879 0.326 0.340 0.824 0.687 0.751 0.939 0.921
Asphalt-2 0.732 0.654 0.240 0.243 0.613 0.536 0.587 0.681
Experiments with the six different compounds referred to above on six laboratory sur
faces of widely differing sharpness and coarseness and tire test ratings on three different
road surfaces showed that a good correlation existed in all cases when the surfaces were
468 Chapter 11. Rubber Friction and Tire Traction
compared with each other at a single, carefully chosen testing condition, as shown in
Table 11.5. The correlation coefficients are given in the upper right and the regression
coefficients in the lower left part. If the coefficient is larger than 1, then the surface listed
in the left column is more discriminating than the surface listed in the corresponding row.
The results in this table confirm again that the surface structure is not a major variable
when considering friction and traction under a limited range of testing conditions on wet
surfaces.
1.3
1.3
1.2
side force coefficient
1.2
side force coeff.
1.1
1.1 comp A
1 comp C
1
0.9
09
0.8
0.8
0.7
1.3
0.7
2.0
0.2
0.6
0.6
-0.8
06
-1.9
-0.8
-1
-2.9
-1
-0.25
-2.2
aTv
-4.0
-0.25
0.5
v
-3.6
aT log
0.5
-5.0
log log
log
-5.0
v v
Chapter 11. Rubber Friction and Tire Traction 469
Figure 11.50: Correlation coefficient between road test ratings on a wet concrete
track and laboratory measurements on a wet blunt alumina 180 disk. Upper part
as a function of log aTv and lower part as a function of log aTv and log v.
four constants by the least-squares method. The results have to be presented as a table or
as a three dimensional graph. As an example, the side force coefficients are shown in
Figure 11.49 for compounds A and C of Table 11.4. It is seen that the major variation is
due to the log aTv terms and only a small part is due to the additional log v term. The cor
relation with road test ratings of the six compounds of Table 11.4 is shown in Figure
11.50, comparing the results using log aTv only (upper part) and the extended version with
the added log v term (lower part). Both laboratory data sets were obtained on wet blunt
alumina 180. The road test ratings were obtained on a concrete track from stopping dis
tance measurements for speeds between 90 and 10 km/h using ABS braking. It appears
that the point of highest correlation shifts slightly with the speed and is also improved.
If the WLF transform is not obeyed rigorously, a more empirical analysis is possible,
treating temperature and speed as separate variables. The temperature dependence can be
described by a square law, and the speed dependence by a linear relation between the fric
tion or side force coefficient and log (speed). The addition of an interaction term, i.e. a
dependence of friction or side force on both temperature and speed, would amount to rec
ognizing some kind of equivalence between them, i.e. invoking a treatment similar to the
WLF equation described above.
Figure 11.51 shows the side force coefficient (upper part) of compound C of Table 11.4
and its rating (lower part) relative to compound A as a function of temperature and log
(speed). The side force coefficient depends on temperature and log (speed). Indeed, the
rating of compound C relative to A depends on temperature so strongly that a reversal in
ranking occurs. The dependence on speed, however, is small. Figure 11.52 compares the
correlation coefficient between a road test on wet concrete and laboratory side force meas
urements for the six compounds of Table 11.4 on blunt, wet alumina 180 (a) with a log
(aTv) - log v evaluation and (b) with a temperature - log v evaluation. It appears that the
correlation coefficient for the temperature - log v evaluation depends strongly on both
temperature and speed, i.e. on the laboratory testing condition, while in the log (aTv) - log
v evaluation the correlation depends on log (aTv) but not on log v. Both methods: log aTv
- log v and temperature - log v give equivalent results and similar correlations with road
test data and both are slightly more precise than the evaluation based on log (aTv alone).
Clearly, the direct temperature-log v evaluation is easier to handle and probably also more
easily understood in routine evaluation and comparison of compounds. The log (aTv)
470 Chapter 11. Rubber Friction and Tire Traction
Figure 11.51: Side force coefficient of compound C of table IV and its rating
relative to compound A as a function of temperature and log v.
method, however, has a clear physical interpretation and if deviations from the master
curves are observed repeatedly then it would be worthwhile to spent some research effort
to identify the cause. In particular, if deviations are compound-specific, it may be very
useful and of practical benefit to understand their origin.
4.7 Correlation between road test ratings and simulated road tests using laboratory
measurements.
It was pointed out above that when practical traction tests are to be simulated using labo
ratory data the heat transfer constant and hence the operating temperature in the contact
area cannot be determined explicitly. However, if road test ratings are available they can
be used to determine this constant.
For this purpose a road test simulation program is necessary using the test conditions
under which the actual road test ratings were obtained and a set of laboratory measure
ments to determine the parameters with which laboratory ratings can be calculated for a
range of log (aTv) and log v values.
Starting with the initial vehicle speed, the braking slip condition and an initial value for
the constant ξc, a temperature in the contact area and hence a value of log aTv and a provi
Chapter 11. Rubber Friction and Tire Traction 471
sional value of the braking coefficient for the first compound is obtained. Because the brak
ing coefficient is employed in the temperature rise equation an iteration process is required to
find the actual temperature, the log aTv value and the corresponding braking coefficient.
For braking distance measurements the distance and speed attained after a small time
interval are then calculated. This procedure is repeated and the incremental distances are
added up until the final (set) speed is reached, giving the total braking distance. The whole
procedure is repeated for all compounds in the testing program. From the braking dis
tances ratings are calculated for all compounds relative to the chosen reference tire. A lin
ear regression equation and correlation coefficient between these ratings and road test rat
ings are obtained. The procedure is then repeated with different ξc values until the high
est correlation coefficient is reached.
Such an evaluation is shown in Table 11.6, which lists the road test conditions, the road
test ratings for the six compounds on wet concrete and ratings calculated from laboratory
data of side force coefficients at five temperatures and four logarithmically-spaced speeds
on wet alumina 60. It is seen that a high correlation is achieved using a heat transfer con
stant of 0.79. This is not a universal constant, however; it depends strongly on the type of
braking test, in particular whether it is ABS or locked-wheel braking, and also on road
conditions such as surface structure and water level and ambient temperature.
Table 11.6: Comparison of a road test simulation using laboratory data obtained on
wet alumina 60 with six compounds (parameters from table IV) with a road test on
wet concrete to give the calibration constant of the contact temperature rise with the
highest correlation
Conclusions
The visco-elastic properties of an elastomer have been shown to play a dominating role in
the friction of tire tread compounds. Two processes contribute to the total friction: adhe
472 Chapter 11. Rubber Friction and Tire Traction
sion and internal energy losses though cyclic deformation. Both are strongly influenced
by visco-elasticity. Adhesion is an activated process akin to the cohesion of liquids. This
is demonstrated by the applicability of the WLF speed-temperature equivalence.
Moreover, the conversion constant between the speed of maximum friction and the time
of the maximum in the relaxation spectrum of the polymer coincides with the frequency
of the maximum loss modulus or the maximum of the first derivative of the real part of
the modulus. On optically smooth surfaces this molecular adhesion process is the domi
nating factor in elastomer friction. On rough surfaces a cyclic local deformation process
causes internal energy losses. In this case the conversion constant is a length directly
linked to the asperity spacing of the track. But adhesion also plays a major role on rough
tracks. Purely normal deformation contributes only a very limited fraction of the total fric
tion. Tangential stresses are required to account for the high friction on rough surfaces
and these are made possible by the presence of adhesion. Even a small adhesion compo
nent, as for contaminated rubber surfaces, produces large tangential stresses in the rubber.
Indeed, friction on rough surfaces is not only due to internal cyclic energy losses, but
rather the total energy stored through the presence of tangential stresses is lost. The
process can be likened to an adiabatic thermodynamic cycle.
This basic rubber friction process is present on all surfaces, dry, wet or icy, being mod
ified only by the external conditions. On wet surfaces this is primarily water lubrication
which is itself influenced by the water depth, roughness of the road surface and the state
of the tire tread pattern. For given conditions the lubrication depends on the slip speed
between tire and road and modifies the dry friction coefficient. An additional effect is pro
duced by the inertia of the water that must be removed from the contact area. The lubri
cation effect becomes operative only when forces are transmitted between tire and road.
This occurs also during free rolling but at considerably higher speeds if sufficient tread
pattern is available to allow good water drainage from the contact area.
The low friction on ice near its melting point is mainly due to the properties of the ice.
A special winter tread compound is required because the contact temperature cannot
exceed 0°C. Because of the large latent heat of melting, the film of water produced
through heat generated in the contact area is so thin that it does not affect the friction coef
ficient significantly. Hence high friction coefficients on ice are possible at temperatures
below about –10°C. The limitation is set by the compound properties, expressed through
its glass transition temperature.
The friction master curve of a polymer on a particular surface contains practically all
of the frictional information. However, recent experiments suggest that there may be small
deviations from this, particularly for modern blends where the glass transition tempera
ture is not well defined. There are also possibly effects due to use of new filler systems
that may affect rubber friction separately from the visco-elastic relation to some extent.
This could become an interesting area of research since every possibility of increasing
friction on the road without simultaneously raising the internal friction and hence the
rolling resistance of the tire is of great practical value.
Basically, a road traction test differs from a laboratory test only in that the temperature
in the contact area is allowed to rise and is not really measurable, while in the laboratory
the speed is kept so low that the temperature rise may usually be neglected. The different
surface structures of the road and laboratory test surfaces appear to play a minor role. This
is not surprising since at a given speed the coarseness of a track (the average spacing of
the asperities) influences the friction only on a logarithmic scale. Also the observed
Chapter 11. Rubber Friction and Tire Traction 473
dependence of the friction coefficient on load for soft rubber compounds on smooth sur
faces disappears for harder black or silica filled tread compounds on rough surfaces.
A good correlation between road tests and laboratory measurements is usually obtained
only over a limited range of the master variable log (aTv). Comparisons of the friction
coefficients of different compounds over a wider range of log (aTv) values with road test
ratings identify the useful range and give a detailed picture of its capabilities in practical
tire use. Further developments of compounds can be limited to laboratory assessments
over a limited range of log (aTv), i.e. of temperatures and speeds with a strong assurance
that the road performance will reflect the laboratory findings. Winter compounds are still
best developed on an ice track because this is the easiest way to produce the necessary low
temperature range.
Since modern compounds are highly sophisticated compositions of polymer types and
fillers, a detailed understanding of the exact shape of a master curve, its relation to the
WLF transform, and possible deviations from it, still requires further research.
References
1. F. B. Bowden and A. D. Tabor (1954), Friction and Lubrication of Solids, Oxford
University Press (London)
2. P. Thirion, Gen. Caout. 23, (1946), 101
3. A. Schallamach, Proc. Phys. Soc. B65, (1952), 657
4. A. Schallamach, Wear 1,(1958) 384
5. K. A. Grosch, Rubber Chem. Technol. Rubber Reviews, 69, (1996), 3
6. M. L. Williams, R. F. Landel, and J. D. Ferry, J. Am. Chem. Soc 77 (1955) 3701
7. See literature on visco-elasticity, for example J. D. Ferry (1961) The Visco-elastic
Properties of Polymers
8. L.Mullins, Trans I.R.I., 35, (1959), 213
9. T. L. Smith, J. Polym. Sci. 32, (1958), 99
10. K. A. Grosch, Proc. Royal Soc. A 274 (1963) 21
11. K. A. Grosch, Ph.D. Thesis “Sliding Friction and Abrasion of Rubbers”, University of
London, 1963
12. A. Schallamach, Wear 6 (1963) 375
13. F. A. Greenwood and D. Tabor, Proc. Phys. Soc 71 (1958), 989.
14. F. A. Greenwood and J. B. P Williams, Proc. Roy. Soc, A 295 (1966), 300
15. M. Klüppel, A. Müller, A. Le Gal and G. Heinrich ACS, Rubber Division Meeting San
Francisco, Spring 2003
16. H. Rieger, Dissertation TH Munich 1968
17. K. A. Grosch, The Speed and Temperature Dependence of Rubber Friction and its
Bearing on the Skid Resistance of Tires, in “The Physics of Tire Traction, Theory and
Experiment”, ed. by D. L. Hayes and A. L. Browne, Plenum Press New York, 1974, 143
18. A. D. Roberts, Lubrication studies of smooth rubber contacts, in “The Physics of Tire
Traction, Theory and Experiment”, ed. by D. L. Hayes and A. L. Browne, Plenum Press New
York, 1974, 143
19. A. G. M. Mitchell, “Viscosity and Lubrication, The Mechanical Properties of Fluids”,
Blackie and Sons Ltd, London, 1944
20. T. Kirk and T. F. Archard, Proc. Roy. Soc: London, 261, (196), 532
21. A. L. Browne, Tire Sci. Technol. 3, No. 1 (1975) 6
474 Chapter 11. Rubber Friction and Tire Traction
22. K. A. Grosch, Kautschuk und Gummi, Kunststoffe, 49, Nr. 6 (1996), 132
23. G. Heinrich, ACS Rubber Division Spring Meeting Montreal 1996
24. B. Persson, Surface Science, 401, (1998) 445
25. M. Klüppel, G. Heinrich, ACS Rubber Division Meeting Spring 1999, paper 43
26. R. B. Yeager, Tire Hydroplaning,.Testing, Analysis and Design, in “The Physics of Tire
Traction, Theory and Experiment”, ed. by D. L. Hayes and A. L. Browne, Plenum Press New
York-London 1974, 143
27. R. N. J. Saal, J. Soc. Chem. Ind., 55, (1936), 3
28. A. L. Browne, Tire Sci. Technol. 1, (1977)
29. W. B. Horne and U.T. Joyner, SAE report 870 C (1965)
30. H. Barthelt, Automobil Technische Zeitung, 10, (1973), 368
31. G. Maycock, Proc. Inst. Mech Eng. 80, 2A (1965-66), 122; Rubber Chem. Technol. 41,
(1968) 780
32. K. A. Grosch and G. Maycock, Trans I.R.I., 43, (1966), 280; Rubber Chem.
Techn.41,(1968) 477
33. A. D. Roberts and J. D. Lane, J. Phys. D, 16 (1983) 275
34. W. Gnörich and K. A. Grosch, J. I R I., 6 (1972) 192
35. K. A. Grosch, to be published
36. K. A. Grosch Kautschuk, Gummi, Kunststoffe, 6 (1996),432
37. K. A. Grosch, International Tire Technology Conference, RAPRA Technology Ltd
London (1998)
38. H. S. Carslaw and J. C. Jaeger, Oxford University Press (1959), p. 279
39. I. Segel, Tire Traction on Dry, Uncontaminated Surfaces, in “The Physics of Tire Traction,
Theory and Experiment”, ed. by D. L. Hayes and A. L. Browne, Plenum Press New York-
London 1974, 65
40. M. Heinz; Conference on Time Dependent Materials 2000; 17-20 Sept. 2000, Erlangen,
Germany
41. K. A. Grosch and M. Heinz; International Rubber Conference 2000; Helsinki, Finland
Chapter 12. Rolling Resistance 475
Chapter 12
Rolling Resistance
by T. J. LaClair
1.2 Rolling resistance relations for rolling under various conditions .................... 478
2.5 Basics of rubber compounding for low rolling resistance .............................. 502
2.6 Rolling resistance and the first law of thermodynamics .................................. 505
2.6.1 First law considerations for transient speed operation .......................... 507
2.7 Modeling of rolling resistance and temperature using finite element analysis
4.1 Simple analysis of the effect of rolling resistance on fuel consumption .......... 524
4.2 Relationship between rolling resistance and fuel economy .............................. 528
Chapter 12
Rolling Resistance
by T. J. LaClair
1. Introduction
When a tire rolls on the road, mechanical energy is converted to heat as a result of the phe
nomenon referred to as rolling resistance. Effectively, the tire consumes a portion of the
power transmitted to the wheels, thus leaving less energy available for moving the vehi
cle forward. Rolling resistance therefore plays an important part in increasing vehicle fuel
consumption.
Since the oil embargo in 1973, there has been increasing pressure to reduce fuel con
sumption. The Corporate Average Fuel Economy (CAFE) program was established by the
United States federal government as part of the Energy Policy and Conservation Act of
1975. The CAFE standards were mandated in an effort to minimize the country’s depend
ence on foreign oil. The regulations require each automobile manufacturer to achieve an
annual sales-weighted average fuel economy that meets or exceeds the specified CAFE
standards. Otherwise, the manufacturer is subject to fines, which are currently set at $5.50
per vehicle sold for each 0.1 mpg by which the average fuel economy is below the stan
dard. Separate standards exist for passenger cars and light trucks, and the average fuel
economies are calculated separately for the two categories of vehicles.
Vehicle manufacturers have therefore continued to push tire manufacturers for reduc
tions in rolling resistance. While the CAFE standards have forced tire manufacturers to
focus heavily on tires for new vehicles, it is likely in the future that they will also be
required to provide replacement tires with low rolling resistance. The State of California
recently passed legislation that will require replacement tires to have a level of rolling
resistance that is, on average, equivalent to or better than original-equipment tires [1], and
the U.S. Senate is also proposing to establish tire efficiency standards and labeling for
replacement tires. The pressure for attaining improvements in rolling resistance is likely
to continue.
This chapter is intended to provide the reader with an introduction to the subject of tire
rolling resistance. The following topics are discussed: physical causes and effects of
rolling resistance, design and use factors that impact rolling resistance, prediction and
measurement methods and test standards, and the effect that rolling resistance has on vehi
cle fuel consumption. While the author has attempted to provide a comprehensive
account, the technical literature should be reviewed for details on specific topics that are
covered only briefly here. Several thorough reviews exist [2-4], and these should serve as
excellent starting points for more detailed studies.
their car tires - that additional energy is needed to drive with under-inflated tires - proba
bly because cars have sufficient power so that the reduction in performance is not so obvi
ous.)
Rolling resistance is the effort required to keep a given tire rolling. Its magnitude
depends on the tire used, the nature of the surface on which it rolls, and the operating con
ditions — inflation pressure, load and speed. Rolling resistance has historically been treat
ed as a force opposing the direction of travel, like a frictional force. While the force con
cept is appropriate for a free-rolling tire on a flat surface, it proves unsatisfactory in other
cases. A more general concept is in terms of the energy consumed by a rolling tire.
Although several authors [5-8] recognized the importance of energy consumption, the
concept of rolling resistance as a retarding force has persisted for many years. Schuring
[9] provided the following definition of rolling resistance as a loss in mechanical energy:
“Rolling [resistance] is the mechanical energy converted into heat by a tire moving for
a unit distance on the roadway.”
He proposed the term “rolling loss” instead of “rolling resistance” so that the long-
standing idea of a force would be avoided. Although the term “rolling loss” has been used
regularly since then, at present the generally accepted term, and that appearing in current
test standards (for example the SAE J1269 rolling resistance standard) is “rolling resist
ance,” which is also used throughout this chapter [10].
Schuring pointed out that although rolling resistance — defined as energy per unit dis
tance — has the same units as force (J/m = N), it is a scalar quantity with no direction
associated with it. This distinction is important, once we understand the causes of rolling
resistance, and the idea of rolling resistance as a retarding force should be disregarded.
Even though the normal symbol used to denote rolling resistance is FR, it is emphasized
that, in many instances, the value of FR is calculated from operating and geometrical
parameters – it does not correspond to an actual physical force. Rolling resistance includes
mechanical energy losses due to aerodynamic drag associated with rolling, friction
between the tire and road and between the tire and rim, and energy losses taking place
within the structure of the tire. Bearing losses at the tire spindle, however, are excluded,
since they result in a reduction in the energy transmitted to the tire.
Schuring defined rolling resistance as “mechanical energy converted into heat.” For
any device where the output (generation) or transmission of mechanical energy is a pri
mary concern — as for the automobile and, ultimately, the tire — thermodynamics
becomes a useful tool for evaluating losses in mechanical energy (work). Detailed discus
sion of this view is postponed until Section 2.6, but it is appropriate at this point to men
tion the importance of heat with respect to rolling resistance. Thermodynamics tells us
that energy transfer across system boundaries takes only two forms, work and heat trans
fer. As a result of energy transfer, the internal energy often changes within a system,
resulting in changes in temperature. In a tire, much of the energy conversion occurs inter
nally, resulting in heat dissipation within the volume of tire, causing its temperature to
increase. This heat dissipation is a result of mechanical hysteresis of the materials in the
tire. Additionally, friction between the tire and road, and between the tire and rim, causes
heat to be generated, and the aerodynamic drag between the tire and surrounding air leads
to thermal energy generation as a result of the dissipative action of the air’s viscosity.
Thus, all of the mechanical energy losses associated with the rolling tire are, indeed, con
verted into heat.
478 Chapter 12. Rolling Resistance
Holt and Wormeley [5] presented an analysis of the energy losses associated with a
rolling tire (although they considered the energy loss to be different from the rolling resist
ance). They evaluated the power losses for tires using a test apparatus consisting of a tire
driven by an electric motor (i.e., operating under torque) rolling against a test drum that
was connected to an electrical generator, which provided a braking force to maintain the
tire at a constant speed for a given torque input. They performed an energy balance result
ing in the following expression:
Ptire=Pin-Pout (1.1)
The term Ptire denotes the rate of energy converted to heat by the tire, while Pin and Pout
are the input and output mechanical energies, which were directly measured through the
electrical power input and output from the motor and generator, respectively. Schuring
noted that the energy balance approach, as applied in this specific case, is entirely gener
al and claimed that it could be used in any rolling situation to determine the rolling resist
ance. Equation 1.1 applies to steady-state operating conditions and can be easily modified,
as shown in section 2.6, for transient conditions.
During a time interval, dt, the tire travels a distance ds = ν dt with respect to the sur
face on which it rolls. The road speed, v, is the speed at which the road or test surface
moves relative to the tire, whether the tire physically moves, as in the case of a vehicle
driving on the road, or the tire rolls in place, as on a test wheel or flat track machine that
moves under the tire. The mechanical energy loss of the tire during the interval dt is given
by
δWtire=FRds=FRvdt, (1.2)
where FR is the rolling resistance. Since power, in general, is given by force times
velocity, the rolling resistance can be calculated from the input and output powers, as
FR=(Pin-Pout)/v. (1.3)
Schuring regarded this equation as the most general expression for calculating rolling.
resistance. Note that Pin-Pout is the net mechanical power input to the tire, denoted W in
standard thermodynamic terms.
net input power applied to the tire is (Tω-Fxv), where ω is the angular velocity of the
Figure 12.1: (a) Tire rolling under applied torque on a flat surface.
(b) Free-body diagram of the tire.
(a)
wheel. The rolling resistance in this case is therefore obtained from Eq. 1.3 as
(1.4)
It is seen that in the case of a free-rolling tire (ω = 0) the rolling resistance is equal in mag
nitude to the force Fx opposing the motion of the tire. However, in the general situation
this is clearly not the case.
The rolling resistance for other operating conditions can similarly be determined by
application of Eq. 1.3. Table 12.1 provides formulas for the calculation of rolling resist
ance for steady operating conditions on either a flat surface or a cylindrical drum for dif
ferent combinations of camber, slip angle and torque, as derived by Schuring [9]. The
rolling resistance is calculated from the various forces and moments acting on the tire, as
given by the equation
FR=AT+BFx+CFy+DMx+EMz. (1.5)
In Eq. 1.5, Fy is the lateral force, Mx is the overturning moment, and Mz is the aligning
torque, as defined in the SAE standard terminology [10].
480 Chapter 12. Rolling Resistance
0 0
γ(t)=γ0sinωt, (1.6)
Figure 12.3: (a) A rubber sample subjected to pure shear. (b) Imposed sinusoidal
strain γ and the corresponding stress response σ, illustrating the phase lag δ.
(a)
(b)
482 Chapter 12. Rolling Resistance
is applied to the sample, it is found that the stress response is also sinusoidal. However,
the stress exhibits a phase lag relative to the input shear strain:
σ(t)=σ0cos(ωt+δ). (1.7)
The strain input and stress output are plotted as a function of time in figure 12.3(b), and
the stress and strain cross-plotted over a cycle give the hysteresis loop shown in Fig 1-2.
Eq. 1.7 can be expressed in the form
σ(t)=γ0(G′sinωt+G′′cosωt)=σst+σ1, (1.8)
where the constants G´ and G″ are defined as the elastic shear storage and loss moduli,
respectively, and σst and σl are the corresponding storage and loss contributions to the
total stress. It is easily shown that the ratio G″/G´ is related to the phase lag δ, as
, (1.9)
, (1.10)
where γ(t)=wγ0cosωt is the strain rate. During each period of oscillation, the net work
input per unit volume of the material can be calculated as
, (1.11)
Calculation of the integral after substituting and the relation for from Eq. 1.8 into Eq. 1.11,
yields
. (1.12)
Over a complete cycle, the energy that is stored elastically is fully recovered, but the ener
gy associated with G″ is simply converted to heat in the sample. G″ is termed the “loss
modulus” since all of the mechanical energy loss is associated with it. Although the above
analysis is based on an imposed strain, similar results are obtained for an imposed stress
or other modes of operation. In short, any deformation of a polymer results in energy dis
sipation as a result of viscoelasticity. It is precisely the viscoelastic behavior of rubber that
makes it ideal in tires for damping vibrations and absorbing large shocks, and it is also
responsible for the tire’s grip. However, although viscoelasticity of rubber provides the
tire with some of its most desirable — and critical — qualities, it is also the cause of the
energy dissipation that comprises a major part of rolling resistance.
Chapter 12. Rolling Resistance 483
Figure 12.5: Longitudinal bending of the crown: (a) overall deformations of the
tire, (b) deformations at different circumferential locations in the crown region.
The tire crown can be likened to a three-layered composite structure where the central
layer is the belt with reinforcing cords that make it both inextensible and incompressible
and the two outer layers – the tread and the liner – are soft visco-elastic materials. When
the crown bends, the outside layer (tread) is stretched and the inside layer (liner) is com
pressed. Conversely, when it flattens, the outside layer is compressed and the inside layer
stretched. Similar compression/stretching phenomena occur across the width of the crown
due to transverse bending. Such repeated deformation leads to energy dissipation and con
tributes substantially to the rolling resistance.
Compression of the tread also occurs in the contact patch due to the load that the tire
carries. Typical compression strain levels are around 5% for passenger car tires and 14%
for truck tires. The actual level depends on the local stress, which in turn is dependent on
inflation pressure and load. In the contact region of the tire, the normal forces resulting
from the load must be balanced locally by the force generated by the tire’s internal pres
sure, hence the vertical stresses at the carcass are effectively limited by the tire pressure.
As a result, the contact patch length adjusts in order to carry the load and the mean stress
and strain levels of the tread are nearly independent of the load itself. When a higher load
is applied the longer contact patch results in a longer period of compression for each mate
rial point passing through the contact patch but the transient portion of the stress-strain
cycle is similar. Since the energy dissipated by hysteresis is determined by the integral
∫ σ dε ,
adding a period in which the strain remains constant in the contact patch will not result in
additional energy loss. This suggests that the energy dissipation might be unchanged in
the case of a higher applied load, resulting in a reduced coefficient of rolling resistance,
which is simply the ratio of rolling resistance to applied load. However, the increased flat
Chapter 12. Rolling Resistance 485
tening in the longer contact patch and additional bending of the sidewall tend to counter
act this effect, and the rolling resistance coefficient in fact tends to remain nearly constant
with load.
The void ratio of the tread design and the tread block size have an important effect on
the compressive stress and strain in the tread blocks. If the tread design employs a large
void ratio, the stresses in the tread must be larger in order to transmit the same load to the
carcass. The average pressure over the contact patch (the total area of the road surface
below the tire surface including the area of voids) is close to that of the tire pressure, i.e.
around 2 bars for a passenger car tire and 8 bars for a truck tire. However, the tread pat
tern includes grooves and other features, i.e. voids, which make up about 30% of the tread
for most tires. Only the other 70% is in contact with the road surface. The pressure exert
ed on the tread blocks in the contact patch (Pcontact) depends on the void ratio (Void) and
tire inflation pressure (Ptire) in the following manner:
Pcontact=Ptire/(1-Void) (2.1)
Equation 2.1 indicates that the average pressure exerted on tread blocks in contact with
the road surface is about 45 % higher for a typical value of the void ratio, i.e. about 3 bars
for a passenger car tire and 11 bars for a truck tire.
When a tread block is compressed vertically it expands sideways because rubber com
pounds are essentially incompressible in volume. Thus, the compression stiffness of a
tread block increases as the compression strain is increased. The effect depends on the
ratio of height to width. A short, wide block increases in compression stiffness more than
a tall, narrow block. The compression strain may be estimated using a formula such as the
following:
, (2.2)
where σ is the pressure, M10 is the tensile stress measured at 10 % strain, and F is the
aspect ratio, equal to S/S′, where S and S′ are defined in figure 12.6.
Figure 12.6: Tread block geometry.
In addition to bending and compression strains that occur in the crown of the tire, shear
stresses and strains are also generated due to driving and braking stresses that occur as a
natural consequence of the rolling process. When a tread block enters the contact patch,
because the tire is round, the block does not make contact with the road surface vertical
ly but at an angle, as illustrated in figure 12.7. At this instant, it appears to lean back
wards. Since the tire belt is largely inextensible (because of the cords) but the tread block
is deformable, the tread block’s progression is dictated by the belt. In the absence of slip
page between the road surface and tread (good grip on a dry road), the angle of the tread
block is determined by the relative position between its point of impact on the road sur
486 Chapter 12. Rolling Resistance
face (blue dot in figure 12.7) and its point of attachment to the belt (green dot). To “keep
up” with the belt as it approaches the center of the contact patch, the tread block gradual
ly becomes upright again. Finally, the shearing force exerted on it just before leaving the
contact patch makes it appear to lean forwards.
Figure 12.7: Movement of a tread block through the contact
patch for a -free-rolling tire with no slip angle.
Shear stresses are also generated in the lateral direction within the contact patch as a
result of the lateral bending of the crown; force coupling interactions in the tread construc
tion, which is orthotropic; and as a result of the transmission of forces through the tread
that are generated as a result of interactions during bending of the belt plies, which are laid
at opposing angles and thus form an orthotropic composite structure. While it is beyond
the scope of the current chapter to discuss the mechanics of stresses within the contact
patch in detail, it should be clear that these stresses and strains will affect the rolling resist
ance. It is fortunate that a minimization of these stresses is generally desirable for most
aspects of tire performance; hence it is desirable to optimize the crown profile both to
minimize these stresses and simultaneously to reduce the rolling resistance.
region. Hence it is important to understand the primary deformations that take place in the
bead and sidewall regions.
For both the bead and sidewall, flexure of the region directly above the ground contact
area is significant. Loading of the tire and accompanying flattening of the contact patch
force the lower sidewall and bead to bulge and flex as shown in figure 12.4(a). Flexure
in this region results in compressive and tensile stresses in the inner and outer regions of
the bending zone, in a similar manner to that discussed above for the crown of the tire. In
the sidewall the cord or cords typically define the neutral bending axis. Flexure therefore
results primarily in tension of the rubber material on the outer surface of the tire and com
pression of the inner liner. Out of the contact region, in the upper half of the tire the cords
in the sidewall support the applied load and the sidewall region will straighten somewhat
relative to the unloaded shape of the tire. This can cause bending stresses in the tire oppo
site to those occurring in the lower region, although the bending resulting from this
“counter-deflection” is less significant than that occurring above the contact patch. The
flexing of the sidewall and the cyclically varying compressive and tensile stresses and
strains as each point travels through the contact region is responsible for most of the ener
gy dissipation in the sidewall.
Another deformation in the sidewall and bead region that contributes to the rolling
resistance is termed deradialization. As a result of the inextensibility of the belt plies,
when the crown of the tire flattens as it enters the contact patch, the position of the body
plies in the crown region deviates from its initially radial position in the unloaded tire, as
shown in figure 12.8. This deradialization extends through the sidewall and into the bead
region. As a result, the rubber surrounding the cords of the body plies is stretched and
sheared, causing energy dissipation. Note that, while it may be possible to limit deradial
ization in some regions, the deformation is inherent in the operation of the radial tire, and
hence cannot be completely eliminated.
Figure 12.8: Shearing of sidewalls due to deradialization.
During acceleration and braking, and to overcome the drag forces acting on a vehicle
during steady state operation, a torque must be applied to the wheel. This torque is trans
mitted through the tire to generate a force at the road surface. When torque is applied, the
bead and sidewall regions transfer the forces from the rim to the crown by means of r-θ
shearing of the materials present. This generates torsional shearing in the materials and
488 Chapter 12. Rolling Resistance
also results in deradialization of the body plies, although in a different manner than that
resulting from tire loading. It should be pointed out that steady state shearing of the side
wall would not result in energy dissipation if the shear levels were constant around the
tire, but instead simply shift the operating point to a different shear level. Of course, the
shear strains due to any applied torque are superimposed on the circumferentially varying
shear strains present in the sidewall due to rolling.
As discussed above, many of the deformations that take place in the sidewall also occur
in the bead region of the tire to some degree. Additionally in the bead, however, at boundaries
between materials of significantly different stiffness, shearing of the softer material can be
important and the associated energy dissipation may contribute significantly to the rolling
resistance. Of particular importance is the shearing of the rubber in the calendered body ply.
In the direction of the textile cord, the body ply provides a high stiffness that resists exten
sion. Hence, any flexing of the tire in a direction that attempts to extend the ply cords will
instead cause the surrounding material to shear. The relatively high modulus of the bead filler
results in a similar effect, although the stiffness of the rubber material is less than that of the
cords, so the effect is reduced somewhat. Additionally, the bead filler is typically sandwiched
between the inner and outer wraps of the body ply and is thus itself forced to shear under the
stresses imposed by the tire bending. In general, designs that permit the bead to flex more eas
ily, thereby minimizing the stresses in the bead, tend to result in a lower contribution to the
rolling resistance. The contributions of the various stresses and strains, however, must be
evaluated carefully using analytical methods, for example Finite Element Analysis (FEA),
which will be discussed in section 2.7.
Figure 12.9: Comparison of tire deformations. δf and δd are the total deflections
when rolling on a flat surface and a drum, respectively.
the presence of standing waves, which are discussed in more detail in Section 2.2.2,
results in significant stress-strain cycling of the crown of the tire, the energy dissipated
increases substantially. This increased amplitude of waves on a drum can significantly
impact the rolling resistance at high speeds, even at speeds below the point at which stand
ing waves are visible.
The effects due to surface curvature discussed above primarily affect the crown region
of the tire. While the bending of the sidewall and bead will also be different when rolling
on surfaces of different curvature, the differences in energy dissipation in these regions
tends to be smaller and the relative contribution to the total rolling resistance from these
areas of the tire is usually reduced. In conclusion, the contribution from the crown of a tire
to its rolling resistance is typically amplified during road wheel testing, while improve
ments in the sidewall and the bead area may be masked somewhat.
Frequently, when only road wheel data is available and a flat surface value is desired,
a correction is made using an expression developed by Clark [14]:
FRd=FRf(1+r/R)1/2, (2.3)
where FR,d is the rolling resistance measured on a drum of radius R, FR,f is the rolling
resistance on a flat surface, and r is the tire radius. This equation was based on a number
of simplifications and assumptions related to the tire deflection and stresses in various
regions in the tire. While the theoretical model does not encompass all of the curvature
effects influencing rolling resistance, the result has gained wide acceptance and is fre
quently used to estimate rolling resistance on a flat surface.
road surface has also been suggested as a possible mechanism. Surface roughness has
been found to increase rolling resistance more significantly in tires with high-hysteresis
tread compounds, suggesting that deformational hysteresis is the cause of the observed
differences in rolling resistance. Although deformations due to surface asperities are rela
tively small compared to the bulk deformations of loading, the local stresses may be large.
Additionally, the high deformation frequency causes the hysteretic losses to be increased
(see section 2.5).
Among typical road surfaces, the rolling resistance may vary by up to about 10%, but
considerably higher values have been reported in some cases [15]. Descornet [16] meas
ured energy losses due to surface variations while pulling an instrumented trailer and
reported that larger scale unevenness in road surfaces can result in even greater energy
losses. This may be a result of losses due to suspension interactions with the road surface
and not a true rolling resistance effect, however.
For rolling resistance measurements, an 80-grit surface is specified in the SAE rolling
resistance test standards (Safety Walk and 3M-ite for the SAE J1269 and J2452 standards,
respectively) and is permitted in the ISO standard. The grit surface is intended to mimic
a road surface and has been reported to increase tire rolling resistance by 2-11% compared
to a smooth steel surface. Luchini [17] found that rolling resistance measurements were
more reproducible when a smooth steel roadwheel was coated with a grit surface, and the
effect of the age of the test surface, if any, was small [18].
where k = 0.006 °C -1. Other values of k are sometimes used, but its value is typically in
the range of 0.005 to 0.008 °C -1.
Chapter 12. Rolling Resistance 491
CR=FR/Fz, (2.5)
where Fz is the vertical load applied to the tire. While the coefficient is a number, without
units, it is frequently expressed either as a percentage or by using units of kilogram force
for the rolling resistance (a kilogram-force = 9.81 N) and metric tons for the load (1 ton-
force = 1000 kgf). The resultant unit is kg/t, stated as “kg’s per ton”. Thus a coefficient
CR of 0.008 may also be expressed as 0.8% or 8 kg/t. A typical range of CR for modern
passenger car tires is 7-12 kg/t and for heavy truck tires it is 5-7 kg/t. The lower values
for truck tires are a result of their higher stiffness, due to increased operating pressures and
the use of a steel cord carcass.
Although the load effect on tire rolling resistance is approximately linear, the increase
in energy dissipation that accompanies an increased load causes the temperature of the tire
to rise. This results in a lower hysteretic loss coefficient for the elastomer materials, and
as a result the coefficient of rolling resistance often decreases somewhat with increasing
load, as shown in figure 12.11.
492 Chapter 12. Rolling Resistance
Figure 12.11: Effect of load on rolling resistance for passenger car tires.
Rolling resistance plotted as a percentage of that measured at 80% of
the tire's maximum load capacity.
A reduction in pressure affects the global deformations of the tire in a manner similar
to that of an increased load. The increased bending and shear stresses result in a larger
rolling resistance. However, in some cases the reduction of compressive stresses in the
tread can offset the bending deformations to such an extent that a pressure reduction has
little effect on the rolling resistance, and in some instances the rolling resistance may even
decrease. Truck tires having a block structure in the tread pattern are most likely to exhib
it this effect, since the tread blocks tend to deform more under compression than in a rib
sculpture. For solid ribbed tread patterns, the added stiffness provided by the incompress
ibility of rubber reduces the compressive strains due to load in the tread, as was discussed
in section 2.1.1. This reduces the contribution to the tire’s rolling resistance resulting from
compression. Due to the same effect, some of the stresses associated with bending may
actually be higher in the case of a solid ribbed tire, which tends to increase the rolling
resistance when the pressure decreases. For passenger car tires, the inflation pressure is
not usually high enough for compression to be a significant source of the total energy dis
sipation. Figure 12.12 shows the typical effect of pressure on rolling resistance for both
passenger car tires and truck tires.
With increased speed, several effects influence the tire rolling resistance. In general,
rolling resistance increases as a result of centrifugal forces and, at higher speeds, the for
mation of standing waves. Also, increased speed causes the frequency of deformation to
increase, which tends to increase the loss tangent, tan δ, of the rubber materials. On the
other hand, the higher heat generation rate,q , causes the tire temperature to rise and the
increased temperature causes tan δ, and therefore the rolling resistance, to decrease, as
discussed in section 2.2.1. The combined influence of these effects is typically an increase
in rolling resistance with speed, but there are some cases where the rolling resistance may
decrease as a result of the temperature dependence of tan δ or due to tire construction.
While changing the load or pressure by a given percentage results in differences in rolling
resistance that are of the same order of magnitude as the parameter change, the variation
in rolling resistance with speed is generally much smaller, at least for speeds below that
at which standing waves begin to form.
Chapter 12. Rolling Resistance 493
(a) passenger car tires and (b) truck tires. Rolling resistance plotted as
(a)
(b)
On a flat surface at speeds below 120 kph, standing wave formation is not typically
observed and the increase in rolling resistance is fairly small. Nonetheless, this speed
range is of particular interest in the United States, where legal speed limits are everywhere
below 75 mph (121 kph). Centrifugal forces cause the tire’s radius to increase in the upper
region opposite ground contact, while the deformations associated with flattening of the
contact patch remain nearly constant at various speeds. The result is that the relative dis
placements through each revolution are larger at higher speed, resulting in higher energy
dissipation for each cycle. In tires with a nylon cap, particularly tires with H speed rating
or above, the added crown reinforcement tends to minimize the effects of centrifugation
and the change in rolling resistance with speed tends to be reduced. Figure 12.13 shows
the typical effect of speed on rolling resistance for two passenger car tires, both with and
without a nylon cap ply. In some instances the rolling resistance may even decrease with
increasing speed due to the improved centrifugation performance.
494 Chapter 12. Rolling Resistance
Because of the influence of temperature on rolling resistance, the degree to which a tire
is allowed to reach thermal equilibrium at a given speed plays an important role in the
speed dependence. When a passenger car tire begins rolling at a steady speed, it takes
more than 30 minutes to reach a fully steady-state operating temperature. If the rolling
resistance is measured for a tire immediately following a speed increase, a higher value
will be observed than if the measurement is made after allowing the tire to achieve a con
stant temperature. If the variation of rolling resistance due to speed is not very large, the
temperature effect can even reverse the direction of the sensitivity between stabilized
measurements and those made immediately after changing test conditions, i.e. the rolling
resistance may increase with speed for non-stabilized measurements but decrease with
speed for measurements in which the tire is at thermal equilibrium. Except in highway
driving, a tire rarely operates for extended periods of time at a constant speed, hence
defining an appropriate measurement depends on the objective of the test. For making
comparisons of rolling resistance among different tires, however, it is generally accepted
that rolling resistance measurements should be made after allowing the temperature to sta
bilize at a given speed. This approach is incorporated into all of the major test standards
for rolling resistance measurements made at a single speed.
The formation of standing waves causes rolling resistance to increase very rapidly.
Even before they can be observed visually, the rolling resistance begins to increase at an
accelerated rate, which may be an indication of the onset of standing waves. As described
in Chapter 10, standing waves occur as a result of the large momentum of deformed por
tions of the tire exiting the contact region at high speed. This momentum carries the
deformed contact region, including both the crown and sidewall, through the exit of con
tact and beyond the normal deformation path in the tire; i.e. the tire cannot recover from
the deformation imposed by ground contact to its equilibrium loaded shape at high speed
[19]. Beyond the contact region, the over - deformed elements of the tire are acted upon
by forces which attempt to restore the tire to its equilibrium loaded shape. This interaction
of the momentum and restoring forces results in a circumferentially traveling vibratory
motion of each element of the tire. Although each element experiences an oscillatory
Chapter 12. Rolling Resistance 495
motion during its trajectory about the axis of rotation of the tire, when viewed from a fixed
location adjacent to the rolling tire the deformation path is stationary relative to the con
tact patch, which is where the standing waves are initiated.
It should be apparent that the large strain cycles associated with the standing wave phe
nomenon will cause significant energy dissipation due to hysteresis. Ultimately, hystere
sis dampens the standing waves and prevents them from continuing around the full cir
cumference of the tire. Increasing the tire’s speed provides greater momentum and kinet
ic energy to the tire, which is converted to vibrational energy in the standing waves and
subsequently to rolling resistance. Therefore, the rolling resistance continues to increase
with increasing speed as a result of the standing wave phenomenon.
To characterize the rolling resistance of a tire, it is important to determine the effects
of load, pressure and speed, including interactions between them. Empirically, it has been
found that these effects can be correlated quite well using various mathematical relations
when taken independently. Schuring [2] includes an excellent review of correlations that
have been used in the past. The SAE J1269 measurement standard calls for a regression
fit of the following form:
This is based on measurements conducted at a single speed of 80 kph since speed has a
minor influence, as discussed above. However, it is desirable to use a more refined model
that includes the speed effect since evaluations for a specific application will frequently
consider only a single operating load and pressure combination. A single speed data point
clearly does not provide a complete picture in this case. A simple quadratic form incorpo
rating interactions between the effects is
(2.7)
The reciprocal pressure is used since the rolling resistance is approximately hyperbolic
when plotted against pressure. Although this form can provide a good correlation for a
suitably large set of data, it provides little insight into the functional dependencies. A more
recent method is provided in the SAE J2452 standard for passenger car and light truck
tires. In this test standard, discussed in more detail in section 3.2, rolling resistance is
measured at several load/pressure combinations during a simulated coastdown. Data are
then regressed to obtain coefficients for the relation
(2.8)
This form was proposed by Grover [20] after evaluating a variety of functional forms, and
it fits data quite well for a broad range of tire designs and load, pressure and speed com
binations. Values of α are typically in the range of -0.3 to -0.5, and for β, the value is close
to unity. A typical plot of rolling resistance measurements from the SAE J2452 standard
and the curve fit to the data is presented in figure 12.14. The functional form given by
Eq. 2.8 is quite flexible and can be used to represent data obtained by measurements other
than the simulated coastdown, provided that sufficient data is available to perform the
regression.
496 Chapter 12. Rolling Resistance
Figure 12.14: Measured rolling resistance data for a P265/75R16 tire following the
SAE J2452 test standard, along with curves showing the regression to the form
FR=PαFB(a+bv+cv2) .
Equation 2.8 shows that rolling resistance is rather strongly dependent on inflation
pressure. While load and speed are largely fixed for a given vehicle and application, tire
pressure is easily adjusted by the user. When a tire is under-inflated, its rolling resistance
increases by a factor of, on average, about (P/P0)-0.4, where P0 is the specified inflation
pressure. Since fairly substantial under-inflation is rather common in the United States,
there is an opportunity to improve the average level of rolling resistance even without
changing tire designs. In a study conducted by NHTSA [21], it was discovered that 27%
of passenger cars and 32% of light trucks operating with P-metric tires had at least one
tire under-inflated by at least 8 psi (0.55 bar), and more than 10% of vehicles had at least
one tire under-inflated by more than 10 psi (0.69 bar). These pressure levels represent, on
average, under-inflation by more than 25%, with a corresponding rolling resistance
increase, relative to proper inflation, of about 12%.
Schuring [22] evaluated the torque effect and claimed that the minimum rolling resist
ance does not occur under freely rolling conditions, but rather under a small driving
torque. As shown in section 1.2, the rolling resistance for a tire rolling under an applied
torque is given by
FR=(Tω/v)-Fx, (2.9)
where Τ is the torque, ω is the angular velocity of the rotating wheel and v the speed of
travel. Under free-rolling conditions, the torque is zero and the value of Fx is negative, i.e.
the force acting on the tire in the contact patch opposes the forward motion. When a pos
itive torque is applied, this force becomes less negative and eventually becomes positive,
so that the rolling resistance is reduced by the force Fx acting on the contact patch. If Fx
increases more quickly than the term Tω/v, then rolling resistance will decrease. Based on
the relationships for the slip ratio — which is directly related to ω/v — and the longitudi
nal force as a function of the applied torque, Schuring showed that indeed there is a min
imum of the rolling resistance at positive torque for the tires he considered. He argued that
the physical mechanism responsible for this surprising result is that a small positive torque
may reduce slippage losses occurring during free rolling more than it increases hysteretic
dissipation in the tire. In his comprehensive review, Schuring [2] comments that in some
cases a minimum in rolling resistance is observed under braking conditions, and this has
been observed also by Bäumler [23]. It may be that some factor of tire design dictates the
498 Chapter 12. Rolling Resistance
relative rates of increase of the terms Tω/v and Fx, and thus determines the torque level
at which the rolling resistance is a minimum.
where Rl and R are the dynamic loaded radius of the tire and the surface radius of curva
ture, respectively, and Mx is the so-called overturning moment, which is the moment act
ing in the contact patch about the direction of travel. On a flat surface, this reduces to
Since the slip angles for both toe and steering are generally quite small, typically less than
3°, the value of cos α is close to unity and sin α ≅ α. The value of Fx does not change sig
nificantly with slip angle, and the lateral force, Fy, is approximately a linear function of
slip angle, so
Fy=Cαα, (2.12)
where Cα is the cornering stiffness. Substituting these simplifications into Eq. 2.11, the
rolling resistance at small slip angles reduces to
FR=Fx+Cαα2=FR0+Cαα2 (2.13)
where FR0is the rolling resistance without a slip angle applied. It should be noted that the
cornering stiffness is roughly proportional to the applied load, hence the coefficient of
rolling resistance is largely independent of load for normal operating conditions.
Taking as an example a 225/60R16 tire with a rolling resistance of 47 N at a load of
620 kg, inflation pressure 2.2 bars and speed of 80 kph, the cornering stiffness is about
1.5 kN/deg (86 N/mrad). For a total toe angle of 0.3° (0.15°/tire), the rolling resistance
increases by about 0.6 N, or 1.3%, while at 1.0° of total toe (0.5°/tire), the increase is 6.5
N, or about 14%. Clearly, if the vehicle is configured with a significant toe angle, the
resulting increase in rolling resistance can quickly counteract any improvements made in
tire design. This result also indicates the importance of precise tire alignment during
rolling resistance measurements.
Chapter 12. Rolling Resistance 499
, (2.14)
where m is the tire mass, cp (1300 J/kg-K) is an average specific heat for the tire, h0 (100
W/m2-K) is the convection coefficient corresponding to a reference speed v0 (80 kph), A
is the tire surface area, p (0.5) is an exponent characterizing the heat transfer coefficient
as a function of speed (a power law in velocity, h=h0(v/v0)P, is frequently used for con
vective heat transfer correlations), T∞ is the ambient temperature, FR* is the stabilized
rolling resistance corresponding to the instantaneous speed condition, and k (0.0061 /°C)
is the temperature sensitivity of rolling resistance. The temperature calculation must be
solved numerically for a given velocity cycle. Subsequently the rolling resistance is cal
culated from the following:
. (2.15)
Mars and Luchini used the values given in parentheses above for the parameters of the
model and found quite good agreement with experimental measurements. Due to the
model’s simplicity, it is quite valuable for prediction purposes.
500 Chapter 12. Rolling Resistance
Figure 12.16: Rolling resistance as a function of (a) distance and (b) mass for a
215/65R15 tire run on-vehicle for 15,000 miles.
(a) (b)
Chapter 12. Rolling Resistance 501
ly to be quite variable. Tire rotation (to different positions on the vehicle) will result in the
crown profile changing to a different shape. As a result, this phase may be repeated, at
least in part, following each rotation. It should be noted that artificially induced changes
in the crown profile (buffing) can have a positive or negative impact on rolling resistance,
and in some cases, changes to the profile may increase rolling resistance even when the
tire mass decreases [26].
Continued operation after the crown profile and the material properties are essentially
fully stabilized results in the rate of decrease of the rolling resistance reaching an approx
imately constant value that continues throughout the remainder of the tire tread life
(assuming that tires are not rotated). The rolling resistance is plotted against the tire mass
in figure 12.16(b): it is clear that mass reduction is the main cause of continued reduction
in rolling resistance during this stage of tire life.
Reduction of mass from other parts of the tire also improves rolling resistance, but tends
to be less effective than crown modifications. Nonetheless, it is worthwhile for rolling
resistance, not to mention cost, to remove as much mass as possible from a tire if the addi
tional rubber is not necessary for a specific function.
The curvature of the crown has an important effect on the stresses generated in the con
tact patch during rolling, and also on stresses in the belts. Although a flatter crown
(decreased curvature) is not always better, this is generally the case. It results in reduced
lateral bending since achieving good contact with the road requires less flattening of the
summit. Lateral stresses in the contact patch are also reduced, so that shear stresses in the
tread are generally lower. However, the tire shoulders can be excessively loaded if the
crown radius becomes too large. This causes the stress levels there, and thus the energy
dissipation, to increase. Additionally, if the carcass shape in the meridional plane remains
the same, a flatter crown will result in an increased thickness of tread in the shoulder
regions of the tire. In this case the mass effect may counteract any reductions in energy
dissipation obtained through flattening, and the rolling resistance may increase as a result.
The addition of a nylon cap ply tends to increase the rolling resistance by several per
cent, and this difference can be even larger if the tire that was not initially designed for
use with a cap ply. Although a nylon cap helps reduce rolling resistance at high speeds, as
discussed in section 2.2.2, this is not the case at more typical operating speeds. Below
about 120 kph the rolling resistance is larger when a nylon cap is present due to several
effects. First, adding the cap ply simply adds mass to the tire. Second, the added rubber
and the cords dissipate energy. Although energy loss in the cords is not as significant as
that in a similar volume of rubber, it may contribute up to 2% to the rolling resistance due
to hysteretic energy dissipation as the cords are cyclically stretched and relaxed.
Additionally, the cap ply stiffens the crown region circumferentially, which influences the
stresses both in the contact patch and the belts, and it restricts growth in the shoulders
when the tire is inflated thus making the crown profile more round than in a tire without
a cap ply. These effects require that the tire design be adjusted when a cap ply is used.
Otherwise, the rolling resistance may be impacted negatively.
Various studies have been conducted on the effects of tire size (see Schuring [2] for a
rather detailed review). The results have been quite mixed, probably because of interac
tions among the many variables of tire design. One parameter that appears to have a clear
effect is the tire outer diameter: a larger diameter tends to reduce the coefficient of rolling
resistance. Pillai and Fielding-Russell [27] found that the coefficient of rolling resistance
is approximately proportional to the outer diameter raised to the -1/3 power,
(2.16)
for tires of similar construction. This relationship holds for a wide range of tire sizes within
the same tire line, using the same construction and materials. Pillai and Fielding-Russell used
this relation to predict the effect of aspect ratio using different combinations of other dimen
sional variables. They considered section height, section width and seat diameter, which were
interrelated with the tire outer diameter to predict their effects using Eq. 2.16.
placement of an appropriate material in each location of the tire is critical for the overall
performance. The purpose of this section is to provide an overview of some issues in rub
ber compounding that have a significant impact on energy dissipation and hence on the
tire’s rolling resistance. Additionally, compounding approaches are discussed that reduce
energy dissipation while still maintaining satisfactory wear and grip performance. A
detailed description of rubber compounding is given elsewhere [see, for example, refer
ences 28 and 29].
Figure 12.17: Elastomer molecular structure: (a) Pure polymers, and (b) reinforced
polymer structure, showing sulfur bridges that form following vulcanization.
(a) (b)
low traction. Changing the polymer used in a tire will just cause a shift in the hysteretic
properties, both for rolling resistance and grip. Figure 12.18 illustrates the effect of the
choice of polymer on the dependence of hysteresis on temperature and strain magnitude.
New approaches to resolving the dilemma involve changing the bonding of reinforcing
fillers within the rubber compound.
Figure 12.18: Effect of the polymers used in a rubber
compound on its propensity to dissipate energy.
Propensity to dissipate
energy
Carbon black-reinforced compounds
based on high-hysteresis polymers
Modulus in MPa
Strain
in %
Stretching/compression
Shear
(2.17)
In this equation, W is the rate of work input (power) into the system, Q· is the rate of heat
transfer to the system, and dE/dt is the rate of change of the total energy of the system.
Equation 2.17 states simply that the combined power inputs (and/or outputs in the case of
negative values of W or Q · ) due to work and heat transfer, into a closed system, result in
506 Chapter 12. Rolling Resistance
, (2.18)
where KE is the kinetic energy of the tire, including rotational and translational forms. If
the kinetic energy increases during a transient, Eq. 2.18 requires that additional power be
supplied to the tire, and this must be in the form of mechanical power for a traditional tire.
Therefore, this portion of the mechanical energy input is not converted to heat, and, in
fact, it can be converted back into mechanical energy. Based on the definition of rolling
resistance, then, this term does not contribute to the rolling resistance. Similarly, a nega
tive rate of change of kinetic energy does not directly affect the heat-related terms, Q and
dU/dt, hence it does not reduce the rolling resistance. Under transient speed conditions, it
is possible to have a net power output from a tire simply by decelerating the tire sufficient
ly quickly. If the rolling resistance were defined strictly in terms of the net work or power
input to the tire, then with a sufficiently negative rate of kinetic energy change, the rolling
resistance would itself become negative, which is undesirable since it is not consistent
with the concept of mechanical energy consumption, or a conversion of mechanical ener
gy to heat. This of course still does take place, through material hysteresis, when the tire
is rapidly decelerated. Therefore, for transient speed operations, the mathematical descrip
tion of the rolling resistance must be modified to account for mechanical energy inputs,
or outputs, into the system that do not result in an increase in thermal energy. Instead of
relying on the net mechanical power input for the general formulation, as was done for the
constant speed case in Eq. 1.3, it is appropriate to define the rolling resistance directly in
terms of changes in the thermal energy of the tire, as follows:
. (2.19)
For the case discussed above, this reduces to the desired result of a reduction of the net
power input by the rate of change of kinetic energy of the system,
. (2.20)
It should be noted that other changes in operating conditions may affect other aspects of
the total energy of the system in Eq. 2.17. For example, transient changes in load will
modify the stored elastic energy of the tire. Equation 2.19 generalizes Eq. 1.3, and allows
508 Chapter 12. Rolling Resistance
the rolling resistance to be determined in cases where operating conditions are transient
ly varying.
, (2.21)
which is valid at any instant in time. Consider what happens to the tire as it begins to roll.
A positive velocity and rolling resistance are attained but there is no heat transfer since
heat transfer cannot occur without a difference in temperature. Equation 2.21 thus requires
that the internal energy increases, which results in the tire temperature beginning to rise.
As the temperature increases, heat is transferred from the tire to the environment, and the
heat transfer rate becomes more and more significant as the tire temperature rises. Based
on Eq. 2.21, this results in a reduction in the rate of increase of the internal energy of the
tire. After some period of time, the tire temperature increases to a point at which the heat
transfer rate exactly balances the rolling resistance and a steady state condition results. At
this point, the rate of change of internal energy becomes zero and thermal equilibrium
again exists between the tire and its surroundings. This is the condition described as “ther
mally stabilized”. It is apparent, therefore, that under the conditions described above the
energy dissipation rate due to the rolling resistance initially contributes entirely to a rise
in the tire’s internal energy (when there is no heat transfer from the tire) and later is redis
tributed so that all of this energy is removed through heat transfer, with intermediate dis
tributions of the energy between these endpoints.
Due to transient variations in speed and other operating conditions, it is also possible
that the tire temperature can be above the thermally stabilized temperature corresponding
to the instantaneous operating speed. In this situation, the heat transfer rate from the tire
will exceed the power input due to the rolling resistance, and the rate of internal energy
change, and hence the tire’s temperature, must decrease. Alternatively, if ambient temper
atures increase, the tire temperature may be below the ambient temperature and heat trans
fer will occur from the environment to the tire instead of in the other direction. These sce
narios show that it is possible to have several combinations of positive or negative values
of heat transfer and internal energy changes for the tire, depending on the history of oper
ating conditions and, ultimately, the current state of the tire and ambient conditions.
In most situations an increase in rolling resistance will be accompanied by a rise in tire
temperature. If two tires of similar size and construction but with different values of
rolling resistance operate under the same conditions, the tire with the higher rolling resist
ance will generally experience higher temperatures. However, that the heat transfer from
Chapter 12. Rolling Resistance 509
the tire to its environment is a function of the tire’s boundary temperature. For the same
level of rolling resistance, two tires can have different internal stabilized temperatures if
the heat transfer inside the two tires is not similar. For example, a tire with a thicker crown
region but identical rolling resistance will reach higher internal temperatures since the
added thickness results in a larger thermal resistance. Additionally, one should note that
the heat transfer rate appearing in Eq. 2.21 is a net heat transfer rate for the complete tire.
Different distributions of temperature may exist for two tires with the same total heat
transfer rate if the boundary conditions are not identical, for example in tires with very
different tread patterns, or even as a result of sidewall lettering [31]. These features can
influence air flow patterns over the tire, which may alter the local convective heat trans
fer coefficient and result in differing heat transfer efficiencies among different tires. In
addition to these effects, different temperature distributions will occur when different
regions of the tire contribute to the rolling resistance in differing proportions. If one tire
has a very hysteretic bead region but uses an efficient tread compound, for example, the
temperature distribution will be quite different than for a tire employing a more hysteretic
tread and a low hysteresis bead filler.
As mentioned in previous sections, and as shown in Eq. 2.21, the power input from
rolling resistance is not simply the rolling resistance itself, but the product of rolling
resistance and speed. This is an important point that affects the tire temperature at high
speeds. As the speed is increased the tire temperature continues to rise, although not typ
ically in direct proportion to the speed. Since heat transfer coefficients, and thus the heat
transfer rate, usually increase with speed, the increase in temperature with speed tends to
be sub-linear if rolling resistance remains relatively constant. As discussed in section
2.2.2, at speeds above 120 kph, the rolling resistance can increase quite rapidly, and the
temperature can rise sharply as a result.
2.7 Modeling of rolling resistance and temperature using finite element analysis (FEA)
Several approaches have been taken for prediction of tire rolling resistance, with various
degrees of complexity. Schuring [2] and Schuring and Futamura [4] provide excellent
reviews of different models developed over the years for predicting rolling resistance.
Many early models considered specific effects of operational and/or design variables, and
were successful for providing physical insight into some of the mechanisms responsible
for rolling resistance. However, the development of many of these models, often purely
empirical or semi-empirical in nature, frequently required that significant quantities of
experimental data be used to derive the model parameters, and the applicability was often
limited to tires of a specific construction, tread pattern, size, etc.
Computer simulation is a powerful tool that has become standard practice in tire
design. With the availability of fast desktop computers and workstation clusters, tire mod
eling is now primarily based on numerical simulations using well-established methods.
Finite element analysis (FEA), in particular, has gained widespread acceptance due to its
flexibility, the level of development and availability of both commercial and proprietary
software, and the level of detail and precision that FEA modeling can provide. Complete
three-dimensional tire models used for loaded analyses frequently contain more than
100,000 degrees of freedom, and very detailed models can easily grow to be an order of
magnitude larger. Typical simulations can be run in a period of a few hours or less, to eval
uate the effects of specific design changes on tire performance and to make comparisons
510 Chapter 12. Rolling Resistance
between several designs in order to optimize the design. Furthermore, very detailed inves
tigations of tire physics are possible by interpreting FEA-predicted stresses and strains
along with other calculated physical quantities at precise locations in the tire.
The starting point for FEA modeling of tires is a mathematical formulation of the
physics of tire behavior, consisting of the differential equations governing the physical
phenomena to be studied, appropriate boundary and initial conditions for the particular
problem, and material behavior laws. The tire geometry is first subdivided, or discretized,
into sub-domains, or elements. An approximate solution for the dependent variables is
then calculated simultaneously for each of the elements of the model. The solution proce
dure in FEA consists of a minimization of weighted residuals for the differential equations
resulting from the approximating functions [32,33]. This process is complicated by many
factors in tire modeling, including non-linearities due to material behavior and large
deformations, discontinuities in material properties with large differences in stiffness
(metal, textiles, rubber), the dependence of properties on previous history, difficulties in
determining pre-strains and stresses of the tire prior to inflation, and complex boundary
conditions associated with tire rolling. Many assumptions and approximations, along with
other auxiliary data are necessary to specify a rolling tire problem sufficiently and solve
it completely, and the details of FEA methodologies employed by tire manufacturers are
generally proprietary as a result.
For standard tire mechanical analyses, the FEA procedure is typically applied in a
sequence of mechanical simulations employing non-linear elastic material models to esti
mate the complete field of deformations for an inflated and loaded tire. The calculated
deformations are subsequently used to determine local stresses, strains, and reactive
forces. For calculating rolling resistance, various methods can be employed. It is not prac
tical to routinely solve a complete viscoelastic model of a three-dimensional tire [34].
However, the deformation field obtained from a standard elastic analysis of a loaded,
rolling tire may be used to calculate heat generation rates (dissipated energy) in the tire
using an independent viscoelastic model on an element by element basis. The heat gener
ation rates can subsequently be applied to the elements in a thermal model for tempera
ture prediction and the sum of all of the energy losses represents the tire rolling loss. This
approach has been referred to as “semicoupled” or a “one-way coupling” of the mechan
ical, viscoelastic and thermal solutions. A more direct coupling that better represents the
physics of the material behavior is through the use of a true viscoelastic model [35,36].
Although a purely elastic analysis neglects the dynamic variations of stiffness that result
from the viscoelastic behavior of the rubber, the deformation results are similar to those
obtained in a viscoelastic simulation, and the basic physics of the hysteretic contribution
to rolling resistance can be captured in subsequent analysis with this approach.
Furthermore, through iteration, converged thermo-mechanical results can be obtained. A
typical semi-coupled thermo-mechanical analysis procedure is depicted in the flow chart
of figure 12.21. The details of such a modeling approach are discussed in greater detail
below.
imposed loads and deformations. Rubber, on the other hand, is frequently used in situa
tions where large levels of strain are imposed, and the linear theory breaks down. A more
general theory is necessary to account for the non-linear mechanical behavior. For cal
culating the stress and strain relationship of any elastic material, it is convenient to use a
strain energy function (see Chapter 2). The elastic properties of rubber are frequently
modeled assuming incompressibility, and in this case the strain energy function reduces to
a function of just two measures of strain,
W=f(I1,I2). (2.23)
where I1 and I2 are the strain invariants defined in Chapter 2, Section 2.1. Among mate
rial models, the Mooney-Rivlin model is frequently cited. Their strain energy function is
W=C10(I1-3)+C01(I2-3). (2.24)
It results in a linear relationship between stress and strain in the case of simple shear [39]
and hence it cannot capture non-linearities observed in shear experiments at high strains.
Furthermore, the Mooney-Rivlin equation does not provide a satisfactory prediction of
rubber behavior under compression [40]. In general, a strain energy function that includes
more terms in the strain invariants is necessary to describe adequately the stress-strain
relations of rubbery materials.
3D strain cycles
Heat generation
rate Q(x,y,θ)
2D axisymmetric thermal
model
2D temperature
T(x,y)
Converged
temperatures?
No
Yes
End
512 Chapter 12. Rolling Resistance
. (2.25)
The FEA simulation represents the tire relative to a reference frame attached to, but not
rotating with, the axle of the tire. However, the nodes and elements of the model do not
move to simulate the rolling of the tire. Instead, one can consider that the material points
of the rolling tire pass through each point in the stationary finite element model but fol
low the path determined by the calculated deformation field. Stated in the terms of con
tinuum mechanics, the loaded tire analysis used for FEA is based on an Eulerian formu
lation of rolling. Since Eq. 2.25 applies to each material point in the tire, the derivative is
actually the substantial derivative,
. (2.26)
The position of each material point of the tire can be determined, at a time t, by apply
ing a rotation about the axle of the wheel (due to the rotational velocity) and subsequent
ly adding the deformation calculated in the FEA model at the corresponding position.
Mathematically, the position of a material point can be expressed as
. (2.27)
In Eq. 2.27, represents the reference configuration of the tire, which also corresponds to
the undeformed finite element geometry. The vector function gives the initial position of
a material point whose coordinates are given by (r, θ, z), and it is convenient to use an
Chapter 12. Rolling Resistance 513
identity mapping. The velocity vector is then determined by direct calculation from these
relations, and the strain rate is calculated from Eq. 2.26, where the strain gradient in each
element is calculated from the displacement field. From this, the energy dissipation is cal
culated from Eq. 2.25. The sum of the energy dissipation for all elements in the tire for
one revolution, divided by the distance that the tire travels in one revolution, is the rolling
resistance, while the local heat dissipation rate, calculated on a unit volume basis, is the
internal “heat source” term, q, that appears in the heat transfer equation, which for steady
state conditions, is
, (2.28)
cases. The methods described in the test standards include the following: measurement of
the resistive force at the tire spindle while rolling at constant speed (spindle force
method), measurement of the resistive torque on the drum hub at constant speed (torque
method), measurement of the electrical power used by the motor to keep the drum rotat
ing at a constant speed (power method), and measurement of deceleration when the driv
ing force at the drum is discontinued (deceleration method). Each of these methods is
described in more detail below.
where Faero=τaero/Rt is the energy loss per distance traveled due to aerodynamic drag. The
hysteretic loss, , itself is often referred to as “rolling resistance”, even in the test standards,
and hence it is important to ascertain whether it includes aerodynamic drag or not when
reviewing test data.
This confusion about including or removing aerodynamic drag has arisen due to sev
eral factors. One important reason is that the magnitude of the aerodynamic drag on the
tire when operating on a vehicle can be very different than in laboratory measurements
made on a drum. In the former case, there is a net velocity of the vehicle, accompanied by
a bulk flow of air over the tire. This results in a greater overall air speed than for rolling
on a drum, where the only air flow is that resulting from the tire’s rotation. Also, air flow
516 Chapter 12. Rolling Resistance
around the tires is quite variable among different vehicles, depending on wheel well siz
ing, height of the vehicle above the ground, etc., so that the total aerodynamic drag
depends on the vehicle design. As a result of the inability to include aero drag that is fully
representative of actual use, it has been decided that rolling resistance measurements
should simply not include the aerodynamic drag at all, i.e. it should be subtracted from the
rolling resistance measurement.
In addition, there is a practical limitation to the measurement itself for some of the test
methods. They include techniques to remove systematic (bias) errors associated with
machine offset and parasitic losses due to bearing losses, aerodynamic drag of the road
wheel, etc. from the rolling resistance measurement. In order to do this, a reference meas
urement that includes the parasitic losses must be made and subtracted from the test meas
urement. In several test methods, such a measurement can be made only while including
the effect of aerodynamic drag on the tire. Friction between the tire and roadwheel is also
subtracted in many of the measurement methods in common use.
It should be noted that in adjusting for parasitic losses, the bearing losses are usually
assumed to be independent of load. If the bearing loss is characterized as a function of
load (as specified by the bearing supplier, for example), this effect can be taken into
account in the rolling resistance calculation. However, with modern, low friction bearings,
the bearing loss is a rather small percentage of the total losses measured in rolling resist
ance measurements. Thus, compensating for load rarely has a significant effect. For nor
mal speed operations, the ratios between hysteretic, aerodynamic and bearing losses are
approximately on the order 100:10:1, although the ratios depend significantly on the load,
pressure and speed conditions. Figure 12.24 compares hysteretic, aerodynamic and spin
dle bearing losses for a typical rolling resistance measurement. In this figure, the bearing
loss was calculated based on the manufacturer-supplied bearing properties and the aero
dynamic component was measured from a skim measurement (see below) reduced by the
calculated bearing friction component.
FR=-Fx(1+Rl/Rd) (3.2)
where Fx is the measured spindle force, Rl is the dynamic loaded radius of the tire (the
distance, along the line connecting the centers of the roadwheel and tire spindle, from the
center of the spindle to the surface of the test drum) and Rd is the radius of the test drum.
The negative sign for Fx is based on the fact that the force opposes the direction of trav
el of the wheel, which is taken as the positive direction.
For the spindle force method, a measurement of aerodynamic drag on the tire —
referred to as the “skim measurement” — is made by repeating the spindle force measure
ment while a very light load is applied to the tire. The load should be large enough for the
tire to remain in contact with the test drum at the test speed, but small enough that hys
Chapter 12. Rolling Resistance 517
teretic dissipation in the tire is negligible. For passenger car tires, a load of about 100 N
is sufficient. It is assumed that the aerodynamic drag acting on the tire during the skim
measurement is equal to that during the loaded tire measurement. For the spindle force
method, the standards specify that Fx should be replaced by the difference Fx-Fx0 in the
calculation appearing in Eq. 3.2, where Fx0is the skim measurement.
The primary advantage of the spindle force method is that the only significant poten
tial source of error is the spindle bearing loss. Bearing losses in the roadwheel have no
effect on the calculated rolling resistance. However, load misalignment and interactions
between load and spindle force can result in significant errors. These effects should be
quantified and either removed or compensated for in the measurements [47].
τd=-Fx(Rd+Rl), (3.3)
where Fx is the spindle force. Therefore, from Eq. 3.2, the rolling resistance is
FR=τd/Rd (3.4)
As in the case of the spindle force method, a skim force measurement is made, and the
torque from the skim measurement is subtracted from that in the fully loaded condition.
The standards also permit a machine offset reading instead of a skim measurement. In this
case the torque is measured at the measurement speed with no tire contacting the drum.
518 Chapter 12. Rolling Resistance
This provides a measure of the bearing and aerodynamic losses of the roadwheel alone. It
should be noted that if the machine offset method of parasitic loss subtraction is used, then
the calculated rolling resistance includes the aerodynamic drag in addition to hysteretic
losses in the tire.
The main advantage of the torque method is that it is a nearly direct measurement. Only
the net torque applied to the road wheel must be measured to obtain the rolling resistance.
However, the measurement includes frictional losses in the spindle bearing in addition to
the roadwheel bearing and aerodynamic losses. These can be effectively removed by
applying the parasitic loss correction. It is important that the speed is held constant.
Otherwise, the torque measurement can include large errors associated with transient
accelerations of the roadwheel. Such errors can be minimized by averaging the measured
torque over a suitable time period.
FR=P/v. (3.5)
The standards specify that either a skim measurement or machine offset reading, as in the
spindle force or torque methods, should be performed to account for parasitic losses. As
in the previous methods, the measured power in the parasitic loss measurement is sub
tracted from that measured under the specified test condition before calculating the rolling
resistance.
The power method is extremely simple since force and torque transducers are not need
ed to determine the rolling resistance. However, this method does not account for electri
cal losses in the motor or other transmission losses. A careful calibration of the motor is
necessary to ensure accurate results, and it may be necessary to determine motor efficien
cy as a function of speed and power output. In addition, special controls may be needed
to prevent power fluctuations.
down periods are necessary. The measurement consists of two distinct periods, during
which three separate decelerations are measured. Following the break-in and warm-up
periods, the first step is to measure the deceleration of the test drum against which the tire
is being held (ω · d) while the tire and drum roll together at the same speed. (Although the
term ω · is referred to as a deceleration, its value is negative, following normal conven
d
tion for a change in angular velocity, as are other ω · terms discussed below.) The deceler
ation is measured when the speed of the tire and roadwheel pass through the test speed.
The angular deceleration ω · tire of the tire is calculated based on the ratio of the drum and
tire radii.
In the second step, the tire and drum are brought back to a speed slightly above the
measurement speed. The tire is then moved away from the test drum, and the deceleration
of the unloaded tire ω · tire,u, due to tire spindle friction and aerodynamic drag, is measured
while the tire slows through the measurement speed. Simultaneously, the deceleration is
measured of the unloaded test drum alone, ω · d,u, as the result of aerodynamic and bearing
frictional losses acting on it. This step is analogous to the skim measurement in the other
test methods, although it should be noted that in this method, there is no contact friction
present in this step. (It is possible that leaving out contact friction in the measurement of
parasitic losses is the reason for differences observed between the results of deceleration
and other test methods. If this is so, then the deceleration method should be a better meas
ure of rolling resistance, i.e. more representative of road operations, since road friction is
included.)
As is standard, the assumption is made that the parasitic losses acting in each compo
nent of the system are identical in the loaded and unloaded cases. Appropriate substitu
tions are made to calculate the rolling resistance. From angular momentum considera
tions, the rolling resistance is calculated as
, (3.6)
where Rr is the rolling radius of the tire, Id and It are the moments of inertia for the drum
and tire, respectively, and the other variables are as previously defined.
The main advantage of this method is its relative simplicity. Only the tire deceleration
needs to be measured, along with the moments of inertia. Also, since the motor does not
drive the roadwheel during the measurements, errors due to transients in the speed con
troller that can occur in the other test methods are absent. If the deceleration of the tire is
calculated as a finite difference in velocity or from a measurement of the slope of the
velocity - time curve, it is important to make precise measurements of velocity and to con
sider the variability and smoothness of the test data to obtain accurate results.
The different rolling resistance standards are very similar in terms of the methods of
measurement, but the test conditions and equipment, warm-up times, etc. for each stan
dard are somewhat different. The main differences between the ISO 8767 and SAE J1269
standards are the measurement conditions, but they both consist, essentially, of stabilized
rolling resistance measurement(s) at a single speed of 80 kph. The SAE J1269 standard
expressly includes measurements at several different load and pressure combinations to
characterize the rolling resistance sensitivity to these effects, while the ISO 8767 standard
specifies that only one standard test condition be measured (other operating conditions are
optional).
The SAE J2452 standard is unique in that the speed dependence of rolling resistance is
measured during a simulated coastdown from 115 to 15 kph at multiple load and pressure
conditions. An example of the measurement cycle for one load and pressure condition is
shown in figure 12.25. The specified set of load, pressure and speed test conditions pro
vides a complete characterization of the rolling resistance with respect to these parame
ters. The resulting data are fitted to the following empirical model
(2.8)
which was discussed somewhat in section 2.2.2. This approach was developed largely in
response to changes in test methods for vehicle emissions and fuel economy in North
America. Vehicles are tested on a dynamometer, with load forces based on vehicle coast-
down tests performed over the same 115 to 15 kph range [48,49]. Vehicle manufacturers
are only required to test a portion of their fleets; the remaining fuel economy values can
be calculated based on modeling. Since the rolling resistance of tires is one of the inputs
to these models, the speed dependence of rolling resistance is a required input.
With rolling resistance data available for different speeds, vehicle fuel economy can be
modeled for different use scenarios. Additionally, the average rolling resistance corre
sponding to a particular drive cycle can be calculated using Eq. 2.8. This procedure is
described in the SAE J2452 standard, in which the mean equivalent rolling force (MERF)
is defined as
, (3.7)
where the integration is performed for velocities corresponding to the desired drive cycle
(The term denotes a function of velocity at time t, not a product.) The form of the rolling
resistance regression given in Eq. 2.8 is convenient for calculating the MERF since the
pressure and load terms are constant with respect to the integration. Therefore, Eq. 3.7
simplifies to
, (3.8)
where C1, and C2 are the average values of the velocity and square of velocity over the
given drive cycle, respectively. These terms only need to be calculated once for each drive
Chapter 12. Rolling Resistance 521
cycle. The MERF depends on the specific load and pressure conditions, but a standard
mean equivalent rolling force, SMERF, is defined for a standard reference load and pres
sure test condition, which is defined in the test standard. The SMERF is intended to serve
as a single measure of rolling resistance for quantitative comparisons between tires. The
combined EPA urban and highway drive cycles are normally used to calculate MERF and
SMERF values.
Table 12.2: Summary of Rolling Resistance Testing Standards
for Passenger/Light Truck (LT) Tires
SAE J1269 1,2 SAE J2452 1,3 ISO 8767 1
Effective November 1979 June 1999 August 1992
Date of
standard
Test methods Force, torque or power Force or torque method Force, torque, power or
method deceleration method
Test wheel No specification given. Minimum diameter of 1.5 to 3.0 m diameter.
diameter Standard states that the 1.219 m (48 in).
most common test wheel is
1.708 m diameter.
Test wheel Medium-coarse (80 grit) 80 grit textured surface. A Smooth steel surface is nor
(drum) surface. A surface condi- surface conditioning proce- mal, but an 80 grit textured
surface tioning procedure is dure is required to ensure surface is permitted.
required to ensure consis- consistent results.
tent results.
Test rims Normal rim for testing is Normal rim for testing is Standard measuring rim, or,
the design rim for the tire, the measuring rim for the if not available, the next
although other approved particular dimension, wider rim.
rims may be used. although other approved
rims may be used.
Test Equilibrium (thermally sta- An array of speed/load Equilibrium (thermally sta
conditions bilized) rolling resistance /pressure conditions are bilized) rolling resistance
measured at several test tested during a stepwise measured at several test
conditions: coastdown (not equilibrium conditions:
A single test speed of 80 conditions) from 115 to 15
kph is used. kph.
522 Chapter 12. Rolling Resistance
Test cell Held between 20 and 28°C. Held between 20 and 28°C. Must be between 20 and
temperature Rolling resistance values Measurements corrected to 30°C. Measurement cor
must be corrected to an ambient reference tempera- rected to 25°C if made at
ambient reference tempera- ture of 24°C. different temperature.
ture of 24°C.
Chapter 12. Rolling Resistance 523
.
3. The SAE J2452 standard specifies that rolling resistance data will be regressed to provide a rela
tion of the following form (Eq. 2.8):
FR = Pα Fzβ (a + bv + cv 2 )
In addition, the Standard Mean Effective Rolling Force (SMERF) is typically reported, which is
defined by
where the velocity profile in the integration typically corresponds to the EPA combined Urban and
Highway cycle.
4. Temperature corrections are made using Eq. 2.4,
,
with k=0.006 °C -1 for the SAE standards and k=0.01 °C -1 for the ISO 8767 standard.
524 Chapter 12. Rolling Resistance
, (4.1)
where Etract is the total tractive energy delivered over the drive cycle, ε is the drivetrain
efficiency, Pacc is the average accessory power requirement, and T is the total time of
travel.
The fuel consumed by the engine is directly proportional to the thermal energy Efuel
released through combustion of the fuel. Efuel is the product of the average fuel consump
tion per distance traveled, Fc, the distance traveled over the drive cycle, D, and the lower
volumetric heating value of the fuel, H0:
Chapter 12. Rolling Resistance 525
E fuel=FcDH0. (4.2)
The purpose of the engine is to generate mechanical power, but the thermal to mechani
cal energy conversion is limited by the engine’s thermal efficiency, and frictional losses
within the engine cause the power output to be further reduced. The total mechanical ener
gy output from the engine can be related to the fuel energy by the following relation [51]:
E fuel=aN+Eengine/η. (4.3)
In Eq. 4.3, a is a frictional loss coefficient, N is the number of revolutions of the engine
over the drive cycle, η is an engine thermal efficiency, and Eengine is the total mechani
cal energy output by the engine over the drive cycle. Ross and An [53] showed that this
equation works quite well for engine operations that are typical of normal driving condi
tions, specifically for power levels that are below two-thirds of the power generated at
wide-open throttle. They evaluated the parameters a and b=1/η for a wide array of engines
and found that the values were quite consistent. By combining Eqs. 4.1 through 4.3, we
obtain an equation relating the fuel consumption of the vehicle over a drive cycle to the
required tractive energy input,
, (4.4)
The tractive energy Etract is the total mechanical energy provided to the drive wheels
to overcome all of the forces opposing the vehicle’s motion during a specific drive cycle.
These forces include aerodynamic drag, gravitational forces (when the vehicle is climb
ing a slope), inertial forces, and of course, the force required to overcome the rolling
resistance of all of the tires on the vehicle. The tractive energy is therefore given by
Figure 12.26: Use (and loss) of fuel energy in a vehicle. The size of the
arrow for each term is intended to indicate the relative proportion of the
total fuel energy consumed. (Reprinted from [50]).
526 Chapter 12. Rolling Resistance
Etract=Eaero+Egrav+Einertia+Etire. (4.5)
Each of the terms Eaero, Egrav, and Einertia is calculated by integrating the correspon
ding force over the distance traveled during the drive cycle. It can be shown that these
forces, and the rolling resistance, do not contribute to the total tractive energy during peri
ods of braking for standard vehicles that do not use regenerative braking, but this gener
ally has a rather small impact on the fuel consumption for most drive cycles and is not
included in the current analysis for simplicity.
Substitution of Eq. 4.5 into Eq. 4.4, along with the basic definition of rolling resistance
(energy dissipated per unit distance traveled), provides the functional dependence of fuel
consumption on rolling resistance as
,(4.6)
where FR,tot is the sum of the average rolling resistance over the drive cycle for all tires
on the vehicle. The appropriate average is that calculated by integrating over the drive
cycle with respect to the distance traveled, which can also be written as a time integral
through a change of the integration variable:
. (4.7)
This is similar, but not equivalent, to the mean equivalent rolling force (MERF) that is
specified in the SAE J2452 rolling resistance standard. (FR,avg can be calculated from the
coefficients determined from the J2452 standards, however, in a manner similar to that
described in section 3.2.)
Equation 4.6 can be used to obtain an estimate of the fraction of total vehicle fuel con
sumption attributable to rolling resistance for a drive cycle. Without evaluating the con
tributions of the other loss terms, this is calculated from a measured fuel consumption rate
by
, (4.8)
where Fc,tire is the consumption per unit distance due to the tires. Values for representa
tive vehicle classes, based on typical values of the various parameters, are presented in
Table 12.3.
We now consider the effect of a variation in rolling resistance for the drive cycle, and
assume that the other loss terms do not change when rolling resistance is modified. Based
on Eq. 4.6, the variation of the energy consumption per distance traveled based on a
change in tire rolling resistance is given by
, (4.9)
where Wvehicle=mvehicle g is the weight of the vehicle. The second equality in Eq. 4.9
assumes that the coefficient of rolling resistance for all tires on the vehicle is uniformly
decreased by ∆CR. Since the engine thermal efficiency η and drivetrain efficiency ε are
Chapter 12. Rolling Resistance 527
both less than unity, this relation shows that a reduction in rolling resistance results in an
amplified fuel energy savings. As mentioned above, the value of the thermal efficiency is
rather consistent among modern engines, and an average value of 0.408 is used. The value
of ε is taken as 0.90. With a value of H0=32 kJ/cm3 for gasoline, Eq. 4.9 predicts that for
every 1 kg/ton reduction in the coefficient of rolling resistance, the gasoline consumption
rate decreases by 0.835 cm3/km for each metric ton of vehicle weight. The general trend
that fuel efficiency varies linearly with changes in rolling resistance for a given vehicle
and drive cycle has been confirmed by numerous measurements of fuel economy, as
shown in figure 12.27. The precise ratio of ∆FC/∆FR, of course, is somewhat variable, but
measured values are relatively well predicted by this simple approximation. It is interest
ing that the impact of rolling resistance on the fuel consumption is largely independent of
the vehicle on which tires operate, except for the effect of the vehicle weight, since rolling
Table 12.3: Representative values of fuel economy, vehicle weight, and ratio of tire
contribution to total fuel consumption by vehicle class. Fuel economy (FE) and fuel
consumption are representative values based on the EPA combined urban/highway
drive cycles. Values of Fc,tire/Fc are based on a CR range of 7-11 kg/ton.
Vehicle class FE, mpg/Fc, L/km Vehicle weight, lbs. Calculated Fc,tire/Fc
Compact 33/0.071 2800 0.105-0.165
Midsize sedan 25/0.094 3500 0.099-0.155
Large sedan 22/0.107 4200 0.104-0.164
Small SUV 20/0.117 3500 0.079-0.125
Large SUV 17/0.138 5300 0.102-0.160
Minivan 22/0.107 4500 0.112-0.176
Small pickup 18/0.130 4100 0.084-0.132
Large pickup 15/0.156 6000 0.102-0.161
We now discuss the impact that a global reduction in rolling resistance can have on fuel
consumption. It was calculated above that a one kg/ton improvement in rolling resistance
(about 10-12% of current average values for passenger cars) results in a fuel savings of
about 0.835 cm3/km traveled per ton of vehicle mass. If the average vehicle mass is
assumed (conservatively) to be 1600 kg (3520 lbs.), then the estimated fuel consumption
savings possible nationwide by a reduction in the coefficient of rolling resistance of 1
kg/ton is 1.5 billion gallons of gasoline per year. This figure is based on year 2000 data
from the Federal Highway Administration (FHWA) of 2.7 trillion vehicle miles traveled
(VMT) annually in the U.S. Since the significantly higher mass of trucks results in even
larger energy consumption associated with the tires, the total potential savings due to the
same level of improvement in rolling resistance is even larger when all vehicles are con
sidered.
, (4.10)
where FE is the fuel economy. This has frequently been stated as a ratio in the form of
Chapter 12. Rolling Resistance 529
It is assumed, further, that the consumption per distance traveled associated with other
losses is not affected by changes in the rolling resistance. The relative rate of change of
fuel economy due to a relative change in the tire contribution is then obtained by taking a
differential in Eq. 4.11.
The result can be written
as
,(4.12)
where the last equality is based on the definition of Fc,tire from the previous section. Also,
it is again assumed that the thermal efficiency η and drivetrain efficiency ε do not change
with rolling resistance. Eq. 4.12 can be rewritten in the same form as Eq. 4.10, yielding
the return factor
, (4.13)
Values of the ratio of the tire contribution to the total fuel consumption rate over a drive
cycle were provided in Table 12.3. More accurate values are easily calculated from Eq.
4.8 if both the rolling resistance of the tires on the vehicle and the actual fuel economy are
known. Of course, the values of η and ε are also necessary for a detailed calculation, so
estimates can only be expected to be accurate within, roughly, 10-20% in the absence of
specific information regarding these parameters. Nonetheless, a comparison of the
Fc,tire/Fc values in Table 12.3 to the typical ranges provides support for this approach.
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13. Aklonis, J.J. and MacKnight, W.J., 1983, Introduction to Polymer Viscoelasticity (2nd
Edition), John Wiley and Sons, New York.
14. Clark, S.K., 1976, “Rolling Resistance Forces in Pneumatic Tires,” U.S. Dept. of
Transportation, TSC Rep. No. DOT-TSC-76-1.
15. Deraad, L.W., 1978, “The Influence of Road Surface Texture on Tire Rolling
Resistance,” SAE Paper 780257, SAE International, Warrendale, PA.
16. Descornet, G., 1990, “Road-Surface Influence on Tire Rolling Resistance,” in Surface
Characteristics of Roadways: International Research and Technologies, ASTM STP
1031, Ed. by W.E. Meyer and J. Rechert, American Society for Testing and Materials,
Philadelphia, 401-415.
17. Luchini, J.R., 1983, “Rolling Resistance Test Methods,” Tire Rolling Resistance,
Rubber Division Symposia Vol. 1, Edited by D.J. Schuring, Rubber Division, American
Chemical Society, Akron, Ohio.
18. Fuller, D.L., Hall, G.L., Conant, F.S., 1984, “Effect of Testing Conditions on Rolling
Resistance of Automobile Tires,” SAE Paper 840068, SAE International, Warrendale, PA.
19. Wong, J.Y., 1993, Theory of Ground Vehicles, Chapter 1, 2nd edition, John Wiley and
Sons, New York.
20. Grover, P.S., 1998, “Modeling of Rolling Resistance Test Data,” SAE Paper #980251,
SAE International, Warrendale, PA.
21. Thiriez, K. and Bondy, N., 2001, NHTSA’S Tire Pressure Special Study, February
2001,” US DOT/National Highway Traffic Safety Administration, Paper Number 256.
22. Schuring, D.J., 1976, “Energy Loss of Pneumatic Tires Under Freely Rolling,
Braking, and Driving Conditions,” Tire Science and Technology, TSTCA, v. 4(1), 3-15.
23. Bäumler, M., 1987, “Development and application of a measurement procedure for
investigating the power loss of passenger car tires,” Ph.D. Thesis, University of Karlsruhe.
24. Schuring, D.J., Siegfried, J.F. and Hall, G.L., 1985, “Transient Speed and Temperature
Effects on Rolling Loss of Passenger Car Tires,” SAE Paper #850463, SAE International,
Warrendale, PA.
25. Mars, W.V. and Luchini, J.R., 1999, “An Analytical Model for the Transient Rolling
Resistance Behavior of Tires,” Tire Science and Technology, TSTCA, v. 27(3), 161-175.
Chapter 12. Rolling Resistance 531
26. Luchini, J.R., Motil, M.M. and Mars, W.V., 2001, “Tread Depth Effects on Tire
Rolling Resistance,” Tire Science and Technology, TSTCA, v. 29(3), 134-154.
27. Pillai, P.S. and Fielding-Russell, G. S., 1991, “Effect of Aspect Ratio on Tire Rolling
Resistance,” Rubber Chemistry and Technology, v. 64(4), 641-647.
28. Mark, J.E., Erman, B., and Eirich, F.R. (Eds.), 1994, Science and Technology of
Rubber, 2nd edition, Academic Press, San Diego.
29. Barlow, F.W., 1993, Rubber Compounding: Principles, Materials, and Techniques,
2nd edition, Marcel-Dekker, New York.
30. Wark, K., 1988, Thermodynamics, 5th Edition, McGraw-Hill, New York.
31. Oswald, L.J. and Browne, A.L., 1981, “The Airflow Field Around An Operating Tire
and Its Effect on Tire Power Loss,” SAE Technical Paper 810166, SAE International,
Warrendale, PA.
32. Reddy, J.N., 1993, The Finite Element Method, 2nd Edition, McGraw-Hill, New York.
33. Stasa, F.L., 1995, Applied Finite Element Analysis for Engineers, Oxford University
Press.
34. Ebbott, T.G., Hohman, R.L., Jeusette, J.-P., Kerghman, V., 1999, “Tire Temperature
and Rolling Resistance Prediction with Finite Element Analysis,” Tire Science and
Technology, TSTCA, v. 27(1), 2-21.
35. LeTallec, P. and Rahier, C., 1994, “Numerical Models of Steady Rolling for Non-lin
ear Viscoelastic Structures in Finite Deformations,” Int. J. Numer. Meth. Eng., v. 37,
1159-1186.
36. Becker, A., Dorsch V., Kaliske M., and Rothert H., 1998, “Material model for simula
tion of the hysteretic behaviour of filled rubber for rolling tyres,” Tire Science and
Technology; v. 26(3), 132-48.
37. Hall, D.E. and Moreland, J.C., 2001, “Fundamentals of Rolling Resistance,” Rubber
Chemistry and Technology, v. 74(3), 525-539.
38. Rivlin, R.S., 1956, “Large Elastic Deformations,” in Rheology: Theory and
Applications, Vol.1, Ed. by F.R. Eirich, Academic Press, New York, Chapter 10.
39. Yeoh, O.H., 1993, “Some Forms of the Strain Energy Function for Rubber,” Rubber
Chemistry and Technology, v. 64, 754-771.
40. Erman, B. and Mark, J.E., 1994, “The Molecular Basis of Rubberlike Elasticity,” in
Science and Technology of Rubber, 2nd edition, Edited by J.E. Mark, B. Erman, and F.R.
Eirich, Academic Press, San Diego, Chapter 4.
41. Dehnert, J. and Volk, H., 1991, “An Approach to Predict Temperature Distributions in
Rolling Tires Using Finite Element Methods,” presented at the 10th meeting of the Tire
Society, Akron, Ohio.
42. Browne, A.L. and Arambages, A., 1981, SAE Technical Paper 810163, SAE
International, Warrendale, PA.
43. SAE Surface Vehicle Recommended Practice, 1979, “Rolling Resistance
Measurement Procedure for Passenger Car, Light Truck, and Highway Truck and Bus
Tires,” SAE J1269, SAE International, Warrendale, PA.
44. SAE Surface Vehicle Recommended Practice, 1999, “Stepwise Coastdown
Methodology for Measuring Tire Rolling Resistance,” SAE J2452, SAE International,
Warrendale, PA.
45. ISO International Standard, 1992, “Passenger Car Tyres—Methods of Measuring
Rolling Resistance,” ISO 8767, ISO, Geneva.
532 Chapter 12. Rolling Resistance
46. ISO International Standard, 1992, “Truck and Bus Tyres—Methods of Measuring
Rolling Resistance,” ISO 9948, ISO, Geneva.
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Light Truck, and Highway Truck and Bus Tire Rolling Resistance,” SAE J1270, SAE
International, Warrendale, PA.
48. SAE Recommended Practice, 1996, “Road Load Measurement Using Onboard
Anemometry andCoastdown Techniques,” SAE J2263, SAE International, Warrendale,
PA.
49. SAE Recommended Practice, 1995, “Chassis Dynamometer Simulation of Road Load
Using Coastdown Techniques,” SAE J2264, SAE International, Warrendale, PA.
50. National Research Council, 1992, Automotive Fuel Economy: How Far Should We
Go?, National Academy Press, Washington, D.C.
51. An, F. and Ross, M., 1993, “A Model of Fuel Economy and Driving Patterns,” SAE
Paper #930328, SAE International, Warrendale, PA.
52. Adler, U., Bauer, H., Bazlen, W., Dinkler, F., Herwerth, M. (Eds.), 1986, Bosch
Automotive Handbook, 2nd Edition, Robert Bosch GmbH, Stuttgart.
53. Ross, M. and An, F., 1993, “The Use of Fuel by Spark Ignition Engines,” SAE Paper
#930329. SAE International, Warrendale, PA.
54. Schuring, D.J., 1988, “Tire Rolling Loss—An Overview,” Presented at the Seventh
Annual Meeting and Conference on Tire Science and Technology, Akron, Ohio.
Chapter 13. Rubber Abrasion and Tire Wear 533
Chapter 13
by K. A. Grosch
2.3 The dependence of abrasion on energy consumption in the contact area ...... 538
2.4 The relation between rate of cut growth and tearing energy .......................... 539
2.7 The effect of smearing and abrasion in an inert atmosphere .......................... 551
2.8 A brief summary of the basic factors contributing to abrasion ...................... 554
3.1.2 The relation between the force acting on the wheel and simple slip ...... 556
3.5 Combining the energy dependence and speed dependence of abrasion .......... 570
4.2 Conditions affecting tire wear in road tests and normal usage ...................... 579
4.3.4 Influence of the shear stiffness of the tire tread compound .................... 586
5. Correlation between laboratory road test simulation and road wear test results 587
5.2 Correlation with truck tire road test ratings .................................................... 589
5.3 Energy consumption and slip speeds in road wear .......................................... 590
Chapter 13
Rubber Abrasion and Tire Wear
by K. A. Grosch
1. Introduction
Abrasion or wear occurs whenever two bodies slide against each other under friction.
Material is transferred from one body to the other and this process can go in both direc
tions. Wear is therefore associated with friction. Friction is present in every aspect of life
and indeed life would not be possible without it. Although it is a dissipative process in
which mechanical energy is turned into heat, high friction is often useful, and even essen
tial, and the resulting wear has to be tolerated. Research tries to find ways to retain high
friction while minimizing wear.
High friction is required when forces must be transmitted across two surfaces in con
tact. The most common case is whenever human beings or animals want to walk or run.
The case that concerns us here is the movement of self-propelled vehicles, i.e., automo
biles, under controlled conditions over road surfaces of all kinds. In all these cases some
wear occurs: both surfaces lose some material.
This chapter deals with the forces that are transmitted between tire and road and the
wear of the tire that occurs and affects its useful life. Although the main emphasis is on
wear, some remarks are included on rubber friction since abrasion and tire wear is close
ly associated with friction.
First, the factors will be considered which govern the abrasion of rubber in controlled
laboratory experiments, then the major contributing factors to tire wear will be discussed
and a link established between laboratory abrasion and tire wear.
2. Sliding abrasion
2.1 The load dependence of friction
The frictional force is a function of the load, the force that acts normal to the contact sur
faces and presses the two bodies together. For friction between two hard bodies the fric
tional force is generally proportional to the load and can therefore be written as follows
where F is the frictional force and µ is the friction coefficient, which is independent of the
load. For rubber this is not always the case. For soft rubbers on smooth surfaces the fric
tion coefficient decreases with increasing load. Assuming that the frictional force is pro
portional to the real area of contact A [1,2,3], which is usually much smaller than the
apparent one, and that the rubber asperities are much larger than those of the hard smooth
contact surface and may be represented by hemispheres, the real area of contact increas
es with the load as follows:
2
⎧ L ⎫3 [2]
A = Ao ⋅ ⎨ ⎬
⎩ Lo ⎭
where Ao and Lo are reference values of the real contact area and load. This leads to the
following relation for the friction coefficient
Chapter 13. Rubber Abrasion and Tire Wear 535
⎛ 1⎞
⎜− ⎟
⎧ L ⎫⎝ 3⎠
µ = µo ⋅ ⎨ ⎬ [3]
⎩ Lo ⎭
Figure 13.1 shows the friction coefficient as function of pressure for three unfilled natu
ral rubber compounds with different elastic moduli [3].
Figure 13.1: Friction coefficient as function of pressure for three NR unfilled
compounds of different moduli on a smooth surface [from ref. 3]
Schallamach has shown that the exponent in Equation 3 changes to –1/9 if both surfaces
have hemi-spherical asperities [4]. The author has carried out extensive experiments with
different tread compounds on tracks of different asperity shape and coarseness and has
found that for these hard tread compounds the exponent of the friction coefficient ranged
between - 0.1 to + 0.1 but was in many cases very small so that the load dependence may
be neglected [5]. In figure 13.2 and table 13.1 the friction coefficients and values of the
exponent at a load of 100 N are given for several elastomer compounds sliding under var
ious surface conditions.
536 Chapter 13. Rubber Abrasion and Tire Wear
Table 13.1: Parameter of the friction load relationship under different surface con
ditions
Compound Ground glass Ground glass Corundum 180 Corundum 180
dry wet Sharp wet Blunt wet
µo n µo n µo n µo n
BR 1.080 0.111 0.613 0.098 0.815 -0.096 0.841 0.023
NR 1.519 0.049 0.775 -0.009 1.041 -0.097 1.042 -0.042
SBR 0.956 0.086 0.660 -0.033 1.043 -0.103 0.975 0.003
3.4 IR 0.803 0.080 0.844 0.038 1.180 -0.054 0.998 0.002
mean over all rubber compounds and conditions: 0.004
abr = abrref ⋅ ⎨ ⎬
⎩ po ⎭
Chapter 13. Rubber Abrasion and Tire Wear 537
with a positive exponent n, greater than 1. For a constant apparent area of contact, a sim
ilar relation results for the load dependence. The reference abrasion loss Abrref at the ref
erence pressure po and the exponent n depend both on the type of track and the rubber
compound. If the abrasion is referred to the amount of frictional energy dissipated in slid
ing, usually referred to as the abradability of the rubber compound, then the exponent
would vary if the friction coefficient depended on the load and would remain the same if
it did not.
If the track is smooth the abrasive loss may be so small that it eludes normal measure
ment, although the frictional force may be very high, while on rough sharp surfaces the
abrasive loss may be pronounced even at a moderate frictional force. It follows that stress
concentrations in the contact area enhance the abrasion process for similar external forces.
Figure 13.3 shows the abrasion loss of a BR tread compound on four surfaces of different
sharpness, as a function of the pressure using logarithmic scales for both axes. The
straight-line graphs differed both for their reference loss as well as for their exponent n
[6]. The smallest value for n was obtained for abrasion on the flat side of a silicon carbide
grinding wheel. The others were for a tarmac road surface and two concrete floor surfaces.
Note that n was always significantly greater than 1. It is generally observed that the index
becomes larger the blunter the abrading surface. On very sharp tracks, on the other hand,
the index n may be reduced to 1, as shown in figure 13.4 [7].
0-01
0.02 .1 .2 .6 1 2 6
pressure (kg/cm 2
)
538 Chapter 13. Rubber Abrasion and Tire Wear
Figure 13.4: Abrasion loss as function of pressure for four different compounds
[from ref .7] A, SBR+50 HAF black; B, NR +50 HAF black;
C, NR+50 thermal black; D, NR+50 activated CaCO3
Pressure(kg/cm2)
where s is the distance covered. Hence the energy dissipation W per unit distance becomes
equal to the frictional force. On very sharp tracks the abrasion loss is nearly proportional
to the frictional force; more generally, however, it is a power function of the energy dis
sipation, whereby the exponent depends both on the sharpness of the track and on the rub
ber compound in question. When referred to the energy dissipation, the abrasion loss is
generally called the abradability of the rubber compound on the track on which it was
measured.
This behavior is similar to the cut growth and fatigue behavior of rubber compounds.
The rate of the growth of a cut is a function of the tearing energy [8,9] which itself is pro
Chapter 13. Rubber Abrasion and Tire Wear 539
portional to the stored elastic energy density in the test piece. The exact value depends on
the geometry of the test piece.
2.4 The relation between rate of cut growth and tearing energy
Since this relation appears to be the basis of the abrasion process it is necessary to study
its characteristic shape. Figure 13.5 shows the relation for an unfilled natural rubber com
pound [10,11]. It can be divided into four regions.
Figure 13.5: Rate of cut growth dc/dn as function of tearing energy
Below a limiting tearing energy To there is no mechanical tearing. Cut growth is then due
to attack of ozone, which cleaves elastomer molecules. The rate of growth of cracks due
to ozone is independent of the tearing energy and depends only on the ozone concentra
tion.
Above To there is a small region for which the cut growth rate dc/dn is proportional to the
tearing energy T
dc
= Acr ⋅ {T − To } [6]
dn
where Acr is a material property that is influenced by temperature and oxygen.
In a third region, the cut growth rate can be described by a power law
dc
= BcrT β [7]
dn
540 Chapter 13. Rubber Abrasion and Tire Wear
In this region the growth rate constant Bcr is also influenced by temperature and the pres
ence of oxygen. The exponent β depends primarily on the polymer.
When the tearing energy approaches a critical value Tc the rate of cut growth becomes
suddenly very large, and spontaneous rupture occurs. The critical tearing energy Tc is pro
portional to the energy density at break. Figure 13.6 shows the cut growth rate for six
polymers as gum rubbers and filled with two levels of reinforcing carbon black [10]. They
can all be represented by a power function over a considerable range of tearing energies.
Note that addition of carbon black filler reduces the rate of cut growth but has only a small
effect on the exponent.
Figure 13.6: Rate of cut growth for six different rubbers ( ) gum rubber, (+) loaded
with 20 pphr reinforcing black and (") with 50 pphr reinforcing black.
[from ref. 10]
ABR Syntheitc BR
cis-polyisoprene
Generally it appears that abrasion occurs mainly in the third region, except when the
abrasive track is very sharp. In this case the number of cycles to detach a small piece of
rubber becomes small and the rate of abrasion is proportional to the reciprocal of the ener
gy density at break of the rubber compound. This becomes apparent when the temperature
dependence of sliding abrasion on a sharp silicon carbide track is compared with that of
the inverse of the energy density at break [12]. Both curves look very similar as shown in
Chapter 13. Rubber Abrasion and Tire Wear 541
figure 13.7. In order to superpose the two curves, the rate of extension employed in meas
uring the energy density at break had to be about 100/sec for a sliding speed of 10 mm/s.
The difference between the origins of the two ordinate scales represents the log of the
coefficient in the relation between abrasion resistance, defined by the energy required to
remove unit volume of rubber (J/mm3) and the energy density at break in the same units.
Figure 13.7: Abrasion loss per unit energy (abradability)(---) as function
Aresis tan ce
k=
U break [8]
where Aresistance is the inverse of the abradability, defined as the abrasion loss per unit
energy and Ubreak is the energy density at break. The value of k is about 2000, indicating
that the energy required to remove unit volume of rubber, even by a sharp abrasive track,
is used quite inefficiently. It is sufficient to break 2000 times the amount of rubber that is
actually abraded away.
icant temperature rise occurs in the contact area. Multiplying the experimental sliding
speed by the factor aT of the WLF equation and plotting the friction coefficient as a func
tion of log aTv, a single continuous curve results from measurements at various tempera
tures and speeds. This so-called master curve describes the frictional properties of the
compound on the track surface completely. Figure 13.8 shows the master curves of an
ABR gum rubber on polished glass (A), on a clean silicone carbide track (B) and on the
same track lightly covered with magnesium oxide powder (C) [14]. On glass the friction
is very low at low values of log aTv, then rises to a maximum and falls again at high log
aTv values. On a silicon carbide track, the friction at low values of log aTv is considerably
higher than on glass. It also rises with increasing log aTv and reaches a small plateau at
the point of maximum friction on glass. For further increase of log aTv, however, it does
not fall but increases further to a sharp peak at values of aTv several decades above that
for maximum friction on glass. Beyond that point the friction coefficient falls rapidly to
values expected for hard solids.
Figure 13.8: Master curve of the friction coefficient as function of log aTv for an
ABR gum compound on polished glass (A), clean (B) and dusted (C)
silicon carbide respectively [from ref. 14]. Dust used: MgO powder
When powder is present at the interface, the curve on silicon carbide still shows a max
imum at the same position as on the clean track, but the hump has disappeared. If powder
is used with a smooth glass track, the friction coefficient remains very low over the whole
range of log aTv, indicating that the hump on clean silicon carbide is due to the same fric
tion process as occurs on clean glass.
It is useful to compare the speed of maximum friction with the frequency dependence
of visco-elastic properties for different polymers. It turns out that there is a direct propor
tionality between the speed for maximum friction on smooth glass and the frequency of
maximum loss modulus E″. The factor relating them is a distance: λ =6 x 10-9 m, which
is of molecular dimensions and is the same for all elastomers examined. The fact that it is
Chapter 13. Rubber Abrasion and Tire Wear 543
associated with the maximum of the loss modulus suggests that friction on such smooth
surfaces is a molecular relaxation process since the loss modulus curve is a rough approx
imation of the relaxation spectrum of the material. Indeed, an identical length could also
be derived from the first derivative d(log E′)/d(log f) of the real part of the modulus, which
gives an improved approximation to the relaxation spectrum.
Schallamach has developed a theory based on this assumption, which predicts the
essential features of the observed curve [15]. Because it is based on a single relaxation
time the predicted dependence on log aTv is narrower than the experimentally-observed
one. Its greatest shortcoming, however, is that it predicts zero friction both at very small
and at large values of log aTv; i.e. it does not incorporate a “static” friction coefficient.
Thus, on smooth surfaces friction is due to an adhesion relaxation process. On such
tracks there is also no measurable material transfer, i.e. no abrasion loss.
For friction on the rough silicon carbide track the speed of maximum friction is relat
ed to the frequency of the maximum loss factor and the associated length is 1.5 x 10-4 m,
which is close to the spacing of abrasive particles on the track. It is thus obvious that this
type of friction is associated with energy losses under a cyclic deformation. However,
these losses are not only due to the deformation caused by the normal load, they are rein
forced through tangential stresses produced by adhesion and magnified by the shape of the
abrasive particles. Schallamach has demonstrated in a two-dimensional transparent model
[16], shown in figure 13.9, how stresses are distributed in rubber due to indentation by a
wedge under a normal force and when, in addition, a frictional force is applied. The
results agree closely with the calculated stress distribution due to a line force acting at an
angle to the surface of a semi-infinite solid (figure 13.10) [17]. Figure 13.11 (upper)
shows the elastic stored energy of two line forces acting at an angle a to the surface of the
semi-infinite body simultaneously and figure 13.11 (lower) shows the horizontal stress
component. As expected, both energy and stress show strong peaks at the points of con
tact. The stress is compressive in front of the contact, i.e. the friction component pushes.
At the end of the first contact the stress is strongly tensile, decreasing rapidly away from
the contact and it becomes again compressive at a point midway between the two contacts.
If more contact points are considered, this pattern is repeated. The stress is also limited to
a small range of depths. If the distance between the two contact points is 1 mm then at a
depth of 0.04 mm the horizontal stress component has dropped to about 20% of that at
0.01 mm depth. The stress at the surface itself cannot be calculated because a point has an
infinitesimally-small contact area and hence the stress becomes infinitely large. In reali
ty, the force due to an asperity with a small but finite tip radius produces a small but well-
defined contact area, particularly between a hard asperity and a soft rubber compound,
and keeps the contact stress finite. Nevertheless, the smaller the contact radius, the high
er are the stresses at the contact point. The contact radius defines the sharpness of the
wedge and in a three-dimensional case the sharpness of an asperity.
The stored energy between the line forces does not become zero and a tensile stress
acts at the end of the line forces and a compressive stress at the front. At each pass the
energy peaks at the contact points causing crack growth to start from small surface flaws.
This is the mechanical contribution to the abrasion process. In any case a large amount of
elastic stored energy is lost and turned into heat. This is not a hysteretic process in the nor-
mally-understood sense of energy lost in a cyclic process. It is a thermodynamic process
in which rubber is extended and energy is stored but it cannot be retrieved on retraction
544 Chapter 13. Rubber Abrasion and Tire Wear
and is therefore turned into heat. This is the reason for the high friction observed at low
values of log aTv in sliding on silicon carbide, figure 13.8.
Generally the stresses are not large enough to tear the rubber away in one pass but
repeated passes over a rough surface will detach particles. This process is aided by the
temperature rise that occurs under normal wear conditions and causes thermal and oxida
tive degradation of the rubber. Since the energy is stored in a very small depth and the heat
conductivity of rubber is low, the temperature rise within a small surface layer may
become quite high, promoting both modes of degradation.
Figure 13.9: Two-dimensional stress pattern in a transparent rubber block under a
line force [from ref. 16]
normal pressure
sliding direction
sliding direction
Chapter 13. Rubber Abrasion and Tire Wear 545
Figure 13.10: Calculated lines of equal stress for a line force made up of normal
load and frictional force acting on the surface of a semi-infinite body.
Figure 13.11: Calculated stored elastic energy and the horizontal stress component
due to two line forces at an angle to the plane of the rubber surface and a fixed
distance x apart for different depths from the surface of a semi-infinite body.
“master” curves as shown in figure 13.12 [12]. The abrasion loss decreases with decreas
ing temperature reaching a minimum value. At lower temperatures, it begins to rise
sharply. This is accompanied by a drastic change in appearance of the abraded surface, as
discussed in the next section. It appears that the lowest abrasion loss occurs when the rub
ber has a maximum extensibility. This also corresponds to the highest energy density at
break, as already discussed in section 2.3
Figure 13.12: Sliding abradability as function of the variable log aTv for four
non-crystallizing rubber compounds on dusted silicon carbide 180
Butyl Log aT
abradab ty (mm3/mkg)
T-To (°C)
SBR
ABR
Log aTv
The WLF transform only works well for abrasion of non-crystallizing gum rubbers.
For tire tread compounds the temperature dependence is smaller. However, in this case,
too, the abrasion loss, measured at a constant speed, reaches a minimum at a particular
temperature as shown in figure 13.7. A surprising result is that tread compounds have a
much higher abrasion loss than the corresponding unfilled compounds, as shown in figure
13.13 where the rate of abrasion is plotted as a function of temperature for three elas
tomers: (A) SBR, (B) ABR and (C) NR. The solid lines are for the unfilled compounds
and the dotted ones are for the same elastomers filled with 50 parts of HAF black. The
reason for the differences becomes apparent on examining the appearance of the abraded
surfaces.
Chapter 13. Rubber Abrasion and Tire Wear 547
of temperature at a sliding speed of 0.01 m/s (a) SBR, (b) ABR, (c) NR
Figure 13.14: Abrasion pattern: (a) appearance for different rubbers under
different testing conditions (b) cross section through abraded samples.
548 Chapter 13. Rubber Abrasion and Tire Wear
away. Thus, during sliding abrasion the pattern moves slowly in the direction of abrasion.
This abrasion mechanism prevails for soft rubbers with high extensibility. In fact, the
higher the extensibility, the coarser the abrasion pattern becomes, as seen in figure 13.15
which was obtained during the abrasion - temperature experiments described above.
Although the samples were rotated through 90° at regular intervals in order to suppress
their formation, abrasion patterns were formed. However, because of the frequent rota
tion of the sample they appear as nipples rather than ridges at low values of log aTv (high
temperatures, low sliding speeds) as shown in the upper photograph of figure 13.15. As
the minimum rate of abrasion is approached, very pronounced ridges formed despite the
rotation (middle photo of figure 13.15) and the rate of abrasion increased forming a small
hump in the abrasion vs. log aTv curves (figure 13.12 above). At still higher values of log
aTv, scoring marks appeared (lower photo, figure 13.15). The rubber now acts like a hard
solid, in this case like a glassy plastic. However, because of the high rate of extension
both in the abrasion process and also in the energy to break experiments, this change
occurs well before the conventional glass transition temperature is reached. The rate of
abrasion increases drastically, as seen in figure 13.12, in accordance with the decreasing
Figure 13.15: Abrasion pattern of ABR gum rubber from the sliding abrasion
experiment at different temperatures
90 °C
-10°C
-45°C
Chapter 13. Rubber Abrasion and Tire Wear 549
+95 °C
+20 °C
-15 °C
ing rubber compounds. Under their abrasion conditions strong abrasion patterns developed.
If the increase in the crack length at the base of an abrasion pattern ridge is dc the
increase in abrasion depth per pass, i.e. per cycle, is dc.sin β where β is the angle between
the cut growth direction and the surface of the sample. Assuming that crack growth in
abrasion resembles crack growth in a trouser test piece the tearing energy is given by
T = F / b ⋅ (1 + cos β ) [9]
where b is the width of the sample wheel and F is the tangential force on the blade. By
using different loads, different tearing energies are produced. The rate of crack growth
can be measured independently as a function of the tearing energy.
Figure 13.17 shows the results of their experiments for three unfilled compounds. The
solid lines represent the amount of abrasion per cycle and the dashed lines the rate of cut
growth. The agreement between the two measurements is very satisfactory. However, cut
growth rates of rubber compounds depend on the applied tear energy by a power law, with
the exponent α of tear energy being 2 to 4, depending on the elastomer and compound.
Whilst the agreement between the cut growth rates and the rate of abrasion was good for
the non-crystallizing elastomers, with α about 3, it was unsatisfactory for unfilled natural
rubber, which shows good resistance to crack growth by fatigue (α about 2) but poor
resistance to abrasion (α about 3). Strain-induced crystallization reduces the rate of crack
Figure 13.17: Log abrasion loss by a blade (solid lines) and log cut growth rate
(dashed lines) of non-crystallizing rubber compounds as function of log frictional
and log tearing energy respectively. 1, isomerized NR; 2, SBR; 3, ABR [from ref. 21]
Chapter 13. Rubber Abrasion and Tire Wear 551
growth significantly but in the abrasion experiment it appears to have less effect.
Pulford and Gent [22] have extended these experiments. They obtained straight-line
graphs on plotting log abrasion against log frictional energy dissipation, as shown in fig
ure 13.18. The slopes of the lines were compared to those obtained from cut growth exper
iments. For the black-filled BR compound the slope was much smaller than for cut
growth, whilst for the unfilled compound it was similar (compare with figure 13.6 above).
Figure 13.18: Log abrasion by a blade as function of log frictional energy
for three unfilled and a black filled rubber [from ref. 22]
NR
BR
SBR
BR+50HAF black
drops and abrasion may stop altogether. In road wear this phenomenon is only observed
under extreme conditions like car racing. Normally there is sufficient dust on the road to
be adsorbed by the debris and cause the rolls to disintegrate or not to form at all. Hence,
a suitable powder is often applied to the track in laboratory abrasion experiments in order
to absorb the smeary abrasion debris that would otherwise falsify the abrasion results.
Magnesium oxide has proved to be one of the most effective agents for absorbing abra
sion debris and thus preventing smearing. Schallamach carried out experiments on blunt
abrasive tracks using MgO and a mixture of Fuller’s earth and alumina powder [23]. The
results are shown in figure 13.19 for two blunt surfaces, one of knurled aluminum, and the
other of knurled steel. Aluminum produces a lower abrasion loss than steel. On both sur
faces, however, the rate of abrasion was much higher when MgO was used as an absorb
ing powder than with the mixture of Fuller’s earth and alumina. The two compounds
examined were an NR compound without antioxidant and one with 2 parts of Nonox ZA.
There is a very clear difference between the two compounds indicating that oxygen plays
Figure 13.19: Abrasion of two NR tread compounds, one unprotected, the other with
2 parts Nonox ZA on a knurled aluminum and a knurled steel surface in the presence
of (a) Magnesium oxide powder and (b) a dust mix of Fuller's earth and alumina
powder. [deduced from ref. 23]
60
NR
Abrasion loss (mm^3/200 rev)
50
NR+2 parts Nonox ZA
40
30
20
10
0
MgO dust Mix MgO dust mix
Al wheel AL wheel Steel wheel Steel wheel
an important role.
Abrasion experiments in nitrogen, first carried out in Russia [24], showed that abrasion
in practically all cases was higher in air than in nitrogen (with the interesting exception of
butyl rubber), and the effect was larger with blunt abrasives than with sharp ones. Figure
13.20 is an abrasion time record that Schallamach obtained on an Akron grinding wheel
when switching from air to nitrogen and back again. A mixture of alumina powder and
Chapter 13. Rubber Abrasion and Tire Wear 553
Fuller’s earth was fed into the nip between the sample wheel and the abrasive wheel to
counteract smearing. The sample was first run in air until a constant rate of abrasion had
been reached. On introducing nitrogen the abrasion loss dropped rapidly and significant
ly more so for the unprotected compound. After air was re-introduced the rate of abrasion
rose again to approximately the former value. Each time the atmosphere was changed it
took a few readings before a steady state was reached, indicating that a certain time was
Figure 13.20: Time record of the abrasion loss on a standard Akron grinding
wheel in nitrogen and in air of an NR tread compound (a) unprotected and
(b) protected with an antioxidant [from ref.23]
No anti-oxidant
2 parts Nonox ZA
needed before the effect of oxygen on the surface layer was complete.
In figure 13.21 the difference between abrasion in air and in nitrogen is plotted as a
function of the abrasion in air for different compounds abraded on several surfaces of dif
ferent sharpness and using either magnesium oxide or a mixture of Fuller’s earth and alu
mina powder as agents to counteract smearing. Introducing the powder has two opposing
effects on the abrasion loss, but to differing degrees in air and in nitrogen. It prevents
smearing and helps in this way to increase the abrasion loss. On the other hand it can lubri
cate the track and protect the rubber from abrasion. Thus if smearing is small in air and
the lubrication effect is significant, abrasion in nitrogen may be greater than in air. MgO
powder produced very large differences between abrasion in air and nitrogen when the
losses in air were high i.e. when the oxygen-induced smearing effect was large (unprotect
ed NR) suggesting it is highly effective in eliminating smearing. If the compound was
well protected by antioxidant, the rate of abrasion in air was similar to that in nitrogen,
suggesting that the effect of lubrication was also small. Thus MgO can be seen to be a very
effective agent to eliminate smearing without interfering in the abrasion process itself. On
the other hand the mixture of powders produced less abrasion in air and in some cases for
554 Chapter 13. Rubber Abrasion and Tire Wear
well-protected compounds the rate of abrasion was higher in nitrogen than in air. If it is
assumed that the lubrication effect is about the same in both air and nitrogen, then clear
ly the mixture of powders is not very effective in preventing smearing.
Gent and Pulford [55] noticed that black-filled compounds tend to smear more than
unfilled ones, especially for NR, SBR and EPR, while BR and TPPR showed no smearing at
all. Instead, the latter elastomers produced abrasion debris that was dry and powdery. This
was also the case for NR in nitrogen or in vacuum, showing clearly that oxidation is a major
cause of smearing in air. But thermal degradation may also play a role since it is known that
BR, for instance, has a very high decomposition temperature (> 500 °C) so that smearing
would be expected to be less than for NR which has a much lower thermal decomposition
temperature (about 260°C). Gent and Pulford suggested that there are three chemical contri
butions to the abrasion and smearing process: thermal degradation due to local frictional heat
ing, oxidation which is also aided by the raised temperatures, and formation of free radicals
in the rupture process which then also participate in oxidation reactions. In all experiments
BR has the smallest smearing effect, i.e. it has the highest resistance to thermal and oxidative
degradation. It is suggested that this is the reason for its well-known high resistance to abra
sion, rather than any special strength properties.
growth phenomena starting from small flaws in the rubber matrix. Cut growth, in turn, is
governed by the tearing energy available and hence it is not surprising that rubber abrasion
depends above all on the frictional energy supplied. Since the cut growth process is influ
enced by the presence of oxygen, abrasion is also affected by the surrounding atmosphere.
In addition, dissipation of frictional energy is likely to raise the temperature in the con
tact area, increasing the rate of oxidation and/or thermal decomposition and at the same
time weakening the resistance of the rubber to cut growth, thus further increasing the rate
of abrasion. Hence abrasion is not purely a physical phenomenon but is strongly influ
enced by chemical processes that depend on the chemical structure of the polymer and any
protective agents added to the compound. The cut growth process and the associated loss
of material by abrasion depend on the sharpness of the abrasive surface. This also influ
ences the oxidation. Blunt surfaces produce less abrasion per unit distance so that oxygen
has more opportunity to react. The coarseness of an abrasive is less important. Its influ
ence is limited to an interaction between asperity spacing and sliding speed, and the fre
quency dependence of the visco-elastic properties of the rubber compound.
For braking, vf > vc , s is positive with a maximum value of 1. For acceleration, vf < vc, s
is negative and becomes infinite when the forward speed vc is zero, i.e. when the driven
wheel spins.
Slip can be considered as the ratio of energy that is turned into heat by friction against
the road surface to the kinetic energy stored in the mass carried by the wheel. When the
wheels are locked, this ratio becomes 1. Under normal braking conditions the slip is
smaller than 1 and the kinetic energy of motion is mainly dissipated as heat in the brakes.
During cornering also, energy is lost and the forward speed is smaller than the circumfer
ential speed. When accelerating, energy is supplied to increase the forward speed. Hence
the negative sign of the slip. If the vehicle cannot gain any kinetic energy because the
wheel spins, then the slip becomes infinite.
3.1.2 The relation between the force acting on the wheel and simple slip
When forces are transmitted, slip occurs because the wheel is being deformed. The rela
tion between force and slip is one of the most important laws in tire mechanics, because
it influences the all-important properties of traction, durability and – important in the pres
ent context – tire wear. (This is not only true for tires but for all forces transmitted by
adhesional friction).
Even for a homogeneous elastic wheel the distortion is complex and requires sophisti
cated methods to arrive at a precise relation between force and slip. For tires this is even
more difficult because of their complex internal structure. Nevertheless, even the simplest
model produces answers that are reasonably close to reality in describing the force - slip
relation in terms of measurable quantities. This model, called the brush model – or often
the Schallamach model [25] when it is associated with tire wear and abrasion – is based
on the following assumptions:
The wheel consists of a large number of equally-spaced deformable elements, like the
fibers of a brush. They project radially outwards and carry the compressive load. Under
lateral forces they deform sideways in accordance with a linear elastic relation:
f = kf ⋅ y [10]
where f is the lateral force applied to an element, y is the lateral deformation and kf is the
spring constant of the element. This assumption of linear elasticity has the consequence
that no distortion of the wheel occurs outside the contact area.
Consider first the case of cornering, when the wheel runs at a slip angle. Lateral dis
tortion of the wheel creates a force normal to the plane of the wheel referred to as side
force or cornering force. In tires it serves to steer the vehicle.
The load on the wheel creates a contact area of finite length a caused by compression
of the air within the inflated tire. Note that lateral distortion of the wheel caused by the
load is ignored, whereas solid rubber wheels would bulge out. Thus, the above relation
for lateral force is really a shear relation. The fibers are assumed to have high compres
sive stiffness and low shear stiffness, which in fact is a good approximation for rubber.
Within the contact region, as the distance x from the point of contact increases, the lat
eral deflection increases proportionally. For unit area, the stiffness k is given by k = n kf
where n is the number of fibers per unit area. Hence the distorting stress t is given by t =
k x tan θ. This condition is maintained until a limiting stress, set by the available friction,
Chapter 13. Rubber Abrasion and Tire Wear 557
is reached. If it is assumed that the pressure distribution along the x-axis is elliptical and
the friction coefficient is a constant, sliding occurs along the curve f(µp). A diagrammat
ic view of this is given in figure 13.22. The total side force is proportional to the area
under the curve. A similar diagram is also obtained for the braking or accelerating force,
except that in these cases the force acts in the plane of the wheel instead of sideways. The
force-slip relation for the brush model is given by
[11a]
where S, B, A stand for either the side-force, braking-force, or accelerating force. µ is the
friction coefficient and the function q is given by
2c
q= [11b]
1 + c2
where c depends on the type of force considered. For a side force
π ksa 2
c= ⋅ ⋅ tan θ [11c]
8 µL
where L is the applied load, a is the length of the contact area and ks is the stiffness nor
mal to the plane of the wheel, µ is the friction coefficient and θ is the slip angle, the angle
which the plane of the wheel makes with the forward velocity of the wheel.
At small slip angles the force - slip angle relation reduces to
k a2
S = s ⋅θ [11 d]
2
This means that at small slip angles the force is independent of load (except for the
dependence of the length a of the contact patch on load) and independent of friction coef
ficient. The slope of the relation between side force and slip angle, termed the cornering
stiffness Ks of the wheel, is given by
k a2
Ks = s [11e]
2
Thus, Ks is a measurable quantity.
Similarly for circumferential slip
π ka 2 s l
c= ⋅ ⋅ [12]
8 µL 1 − s l
where Kc is the circumferential slip stiffness of the wheel. Note that Kc has a different
value from the cornering stiffness, but can also be measured directly.
F/µL is plotted as a function of the quantity c in Figure 13.23, together with the adhe
sion(A) and sliding (B) contributions, making up the total force. Up to a value of c of
about 0.82, the adhesion region dominates. At larger values the sliding region becomes
more and more important. At large values of c, i.e. at a high stiffness or large slip, or at
low friction coefficient or low load L, the force F tends asymptotically to µL.
Figure 13.22: Diagrammatic view of the contact area for the brush wheel model
under cornering.
µp
sliding region
adhesion region
plane of wheel
Length of contact area X
direction of travel
Figure 13.23: Side force coefficient (dimensionless quantity S/µL) as function of the
parameter c (Equation 11c) showing the two components due to adhesion and sliding
1.0
S/µL
0.8 C
0.6
B
0.4
A
0.2
C
0
0 0.4 0.8 1.2 1.6 2
Chapter 13. Rubber Abrasion and Tire Wear 559
where q is given by equation [11b] and a is the length of the contact area.
At small slip angles the self-aligning torque becomes
1 1
T= k s a 3θ = K s aθ
12 6 [13b]
Thus, the self-aligning torque is proportional to the cornering stiffness, the length of the
contact area and the slip angle. It is an important aid to the steering control of a vehicle.
A car driver controls the direction of his vehicle as much by sight as by the force on the
steering wheel supplied by the self-aligning torque. As long as the side force is propor
tional to the self-aligning torque the driver feels safe and in control (see figure 13.24). At
large slip angles, however, the torque decreases with further increase of side force, mis
leading the driver. This becomes particularly important when the friction coefficient is
low, as is the case on wet roads and even more so on snow and ice.
Figure 13.24: Side force coefficient (S/µL) as a function of
the self-aligning torque coefficient (T/aµL)
1.2
1
side force coeff.
0.8
0.6
0.4
0.2
0
0 2 4 6 8 10 12 14
self aligning torque coeff.
560 Chapter 13. Rubber Abrasion and Tire Wear
µp
lateral
traction
(A)
total traction
lateral traction shaded area
gives total
µp side force
(B)
[14a]
where Ks and Kc are the cornering stiffness and the circumferential slip stiffness respec
tively and
2σ
q com = π (K s2 sin 2 θ + K c2s co
2
)
1 + σ 2 and σ = ⋅
4µL cos θ − s co
with sco = 1 − s
l
The model is also capable of explaining the self- aligning torque that occurs during cor
Chapter 13. Rubber Abrasion and Tire Wear 561
nering of a braking or accelerating wheel, but this is not of concern when considering
abrasion. Abrasion is to be expected whenever energy is being dissipated in the contact
area of slipping wheels. The general expression for the energy dissipation W is
W = force x slip
For cornering this is given by
Ws = S ⋅ sin θ [15a]
The total energy loss in a slipping wheel under composite slip is then the sum of these two
components.
S/µL
Figure 13.27 shows results for a radial ply tire, obtained using a trailer equipped with
a force-measuring hub. Different load and slip angle settings were used on two different
wet road surfaces to give different friction coefficients. The self-aligning torque is also
shown. At low friction coefficients the experimental data for the side force deviate from
the model, but otherwise the agreement is very satisfactory.
562 Chapter 13. Rubber Abrasion and Tire Wear
Figure 13.27: Side force coefficient and self aligning torque of a radial ply tire
175 R14 on two wet road surfaces of different friction coefficient, at three slip angles
and loads as a function of the parameter c (equation [11c]) all on log scales. The solid
log S/µL
log R/µL
log c
Finally, side force measurements were carried out at different slip angles on a labora
tory abrasion tester at constant load and speed. The results are shown in figure 13.28. The
cornering stiffness and friction coefficient were obtained by a curve fitting method using
a computer program. Since the sample runs on the flat side of a circular disk, the side force
is not exactly zero when the slip angle is zero. The program also determined the devia
tion from zero. The calculated curve agreed excellently with the data.
A similar diagram was obtained for different loads as shown in figure 13.29. In this
case the friction coefficient depended on the load according to the relation
−0.1
⎡ L⎤
µ = µo ⎢ ⎥ [16]
⎣ Lo ⎦
After adjusting for this effect, all data could be fitted to the tire model curve. Thus, good
agreement is reached in all cases with the simple brush model. It is very useful for
describing the relation between force and slip using measurable properties of the tire: cor
Chapter 13. Rubber Abrasion and Tire Wear 563
nering stiffness, longitudinal slip stiffness, and friction coefficient. But, as it stands, it is
of little use to the tire construction engineer since it does not predict the important quan
tities: tire stiffnesses and friction coefficient.
Figure 13.28: Side force coefficient S/L as function of slip angle at a constant load
for a tire tread compound measured in the laboratory on an LAT 100
abrasion tester. Surface: alumina 60. Speed: 6 km/h. Load: 75 N
1.2
1.0
ks = 419 N/rad
0.8
µ = 1.141 0.6
θο = 0.55°
side forcd coefficient
0.4
0.2
0.0
-30 -25 -20 -15 -10 -5 0 5 10 15 20 25 30
-0.2
-0.4
-0.6
0.8
1.0
-1.2
Slip angle (°)
Figure 13.29: Side force coefficient S/L as a function of slip angle for
different loads. To fit the brush model relation the friction
coefficient was adjusted for load dependence using µ = µ′ L-0.1.
Surface: Alumina 180. Speed: 2 km/h.
564 Chapter 13. Rubber Abrasion and Tire Wear
the standard Akron abrader or the Lambourn abrader. In the Akron abrader the sample
wheel runs under a given slip angle against the abrasive surface of an alumina grindstone.
Speed and load are fixed; the side force is not measured.
The author [28, 29] developed an extension of the Akron abrader principle by incorporat
ing the following features:
3.5 130
3
16° slip angle 120
4° slip angle
110
2.5
log (abrasion (mg/km))
12° 100
2
rel rating
90 8°
1.5
8°
80 12°
1
4° 70
16°
0.5 60
0 50
20 40 60 80 100 120 20 40 60 80 100 120
load(N) load(N)
Chapter 13. Rubber Abrasion and Tire Wear 565
Also shown is the relative rating between an OE-SBR compound and an NR + carbon
black tire tread compound. At the smallest slip angle the rating of the NR compound is
better than the OE-SBR compound but the rating of the NR compound decreased with
increasing load. As the slip angle is increased the ratings are reversed.
Figure 13.31 shows the abrasion loss as a function of slip angle for an OE-SBR tread
compound and an 80NR/20 BR compound, measured at constant load. Clearly the abra
sion depends very strongly on the slip condition. Careful examination of the graph shows
that the abrasion loss for the NR/BR blend is higher at large slip angles but it becomes
lower at low slip angles. This is more apparent when the data are plotted on log scales, see
figure 13.32. Straight-line graphs are obtained with slightly different slopes for the NR
compound and the SBR compound. Plotting the relative rating of the NR compound with
respect to the OE-SBR compound as a function of log slip angle shows that the small dif
ference between the slopes is, in fact, highly significant when considering the relative rat
ing, i.e. in a practical comparison of the two compounds.
Figure 13.31: Abrasion of OESBR and an 80 NR/20 BR blend tire compound as a
function of slip angle at a load of 76 N and a speed of 19.2 km/h
60
80NR/20BR
Abrasion (mg/km)
40
SBR 20
Both load and slip angle can be combined if the abrasion is treated as a function of the
energy dissipated, as already discussed in some detail in the section on sliding abrasion.
If the side force is measured and the slip angle is known the data can be evaluated direct
ly as a function of energy. A comparison is shown in figure 13.33 between NR and SBR
abraded against a sharp alumina 60 and a blunted alumina 180 surface [30]. Again,
straight line graphs are obtained if the two variables are plotted on logarithmic scales. The
lines are seen to cross over at low energy levels for both compounds and for both abrad
ing surfaces. The slopes of the lines and the intercepts (the average absolute abrasion loss)
differ considerably for the two surfaces. Moreover, the slopes are larger than 1, indicat
566 Chapter 13. Rubber Abrasion and Tire Wear
ing that the process is more related to a fatigue failure by crack growth than to a direct
tearing process. This is particularly so for the blunt surface. Although the abrasion loss
is distinctly lower on the blunt surface at low energies, the slopes of the lines are much
higher and the lines eventually cross so that the abrasion losses at high energies are high
er on the blunt surface.
Figure 13.32: The abrasion data of figure 31 plotted on logarithmic scales and the
rating of the NR/BR blend relative to OESBR as a function of log slip angle
1.5
1
log abrasion
0.5
0
0 0.25 0.5 0.75 1 1.25 1.5
-0.5
-1
log slip angle
150
125
rel rating
100
75
50
0 0.2 0.4 0.6 0.8 1 1.2 1.4
log. slip angle
The difference in the slopes for NR and SBR is also larger on the blunt surface than on
the sharp one. In fact, this difference is much larger than would be expected from obser
vations of tire wear, suggesting that road surfaces are not blunt but act rather like a sharp
abrading surface.
Chapter 13. Rubber Abrasion and Tire Wear 567
Figure 13.33: Log (abrasion) for two tread compounds NR + black and SBR + black
on two surfaces of different sharpness: alumina 60 and alumina 180 (blunt) as a
function of log (energy dissipation) [from ref. 31]
Figure 13.35: Log (abrasion) as a function of log (speed) for three different tire tread
Surface: alumina 60
Abrasion rises with increasing speed but the slope of the line is considerably different
for the three compounds. The SBR compound has the smallest dependence on speed and
the NR compound the largest. This is most easily explained if it is assumed that the tem
perature in the contact area rises with speed and that both speed and temperature togeth
er influence the abrasion. In this temperature range it would be expected that the rate of
abrasion rises with increasing temperature, from considerations of both loss in strength
and thermo-oxidative degradation, whilst it would be expected to fall with increasing
speed if the temperature could be kept constant, because of viscoelastic strengthening.
Clearly the thermo-oxidative effect outweighs the viscoelastic effect. This is in accord
with the finding that the effect of speed is much smaller for SBR than for NR. The
SBR/BR blend filled with silica lies between the two. However, the present author does
not know of any measurements of the thermal and oxidative resistance and the visco-elas
tic properties for this type of compound, for comparison with abrasion rates.
decreases rapidly as soon as it leaves the contact area. Because the heat conductivity of
rubber is low the temperature also decreases rapidly with increasing depth.
Figure 13.36: Temperature rise in the contact area of a small steel
temperature (°c)
On rough surfaces the contact time between an asperity and the rubber is generally very
short and hence the temperature rise is small. However, since the next asperity arrives
before the temperature rise has died away the temperature continues to rise.
Figure 13.37: Theoretical temperature rise in the contact area of a
pad sliding over a semi-infinite solid at different depths from the surface.
temperature (°c)
where A is the abrasion volume loss per km, W is the energy dissipation per km and v is
the forward speed of the abrasive disk in the contact area relative to the sample wheel.
In order to determine the coefficients of this equation, at least four different testing
conditions are required: two energy levels, given by two combinations of slip angle and
load, and two speeds. The slip can be circumferential instead of at a slip angle – the
important point is that the resulting slip force is measured in order to obtain a measure of
the energy dissipation. In practice more test conditions are useful, say three energy and
three speed levels and for each condition repeat measurements are advisable because abra
sion is always subject to variation. The four coefficients are then calculated from the abra
sion results using the statistical method of least square deviations from the mean [31, 35].
Table 13.2 shows the coefficients obtained in this way for four passenger tire tread com
Chapter 13. Rubber Abrasion and Tire Wear 571
pounds. The coefficient a gives the expected abrasion loss at an energy dissipation of 1
kJ/km and a speed of 1 km/h. The coefficient b1 is the power index for the abrasion-ener
gy relation at constant speed and temperature. It is positive and larger than 1 for all com
pounds indicating that a cut-growth process is dominant. However, it is smaller than
would be expected purely from cut-growth results but larger than from simple tearing.
The coefficient b2 is related to the abrasion-speed relation at constant energy and temper
ature. It is negative, showing that at a constant temperature the rate of abrasion would
decrease with increasing speed, emphasizing the visco-elastic nature of abrasion resist
ance. The coefficient b3 is positive and can be taken as the effect of temperature due to
both energy and speed.
Table 13.2: Coefficients of the abrasion equation [18] for four passenger tire tread
compounds together with the correlation coefficient between calculated and meas
ured data.
Coefficients Correlation
Compound a b1 b2 b3 r
OESBR 0.683 1.640 -0.339 0.223 0.994
pol blend+silica l 0.583 1.706 -0.115 0.152 0.996
pol blend+ silica ll 0.908 1.502 -0.465 0.388 0.994
NR+black 0.690 1.636 -0.345 0.871 0.995
These coefficients can be used to calculate the best estimate of abrasion loss over a
wide range of energy and speed levels on the particular surface on which they were
obtained. Figure 13.38 shows a program of testing conditions designed to cover a wide
field of severities. The range suggested for log energy and log speed, from 0 to 1.6 respec
tively, has proved to be very useful. To fill it, some extrapolation from actual data is
required, which has to be carried out with some care. To extend the experimental condi
tions down to the left upper corner of the table is not practicable because under these very
mild conditions a long time would be required to obtain a measurable abrasion loss and
the results would be rather variable as environmental fluctuations become important. The
four conditions at the corners of the test scheme are mandatory. Two further conditions
at moderate speed and energy are desirable because they support the mildest condition,
which takes the longest time and is subject to the largest variations. A complete program
using nine conditions gives very reliable results.
Figure 13.38: Proposed testing scheme for evaluating the
coefficient of the abrasion equation [18]
speed (km/h
energy log 1.00 1.58 2 51 3.98 6.31 10.00 15.85 25.12 39.81
(kJ/km) energy 0 0.2 0.4 0.6 0.8 1 1.2 1,4 1.6
1.00 0
1.58 0.2
2.51 0.4
3.98 0.6
6.31 0.8
10.00 1
15.85 1.2
25.12 1.4
39.81 1.6
572 Chapter 13. Rubber Abrasion and Tire Wear
Abrasion loss as log(abrasion) vs. log(energy) and log(speed) is best presented either
in tabular form, figure 13.38, or as a three dimensional graph [30], as shown in figure
13.39. Notice that the rate of abrasion between the mildest condition (upper left) and the
most severe condition (lower right) differs by a factor of about 1000.
Figure 13.39: Log abrasion as a function of log (energy) and log (speed) for a tire
tread compound.
3
log abrasion
(mg/km)
0
0
1.2
0.6
0.6
log
1.2
spe ..
W.
0
ed .
.. log
More important for practical use is the relative rating of an experimental compound
with respect to a standard reference compound, usually a well-proven compound for
which a substantial set of road data are available. The relative rating is defined as
abrasionof standardcompound
rel. rating = ⋅100 [19]
abrasionof experimental comp.
Note that there is no single rating because the value depends on the severity of the test. In
fact, reversals in ranking occur. An example is shown in figure 13.40 as a three dimen
sional diagram. This compound (Compound 2 of table 13.2 compared to Compound 1) is
better than the control at low speeds and low energies. In this case the presentation of the
ratings in a table is more useful because it gives a detailed quantitative view of the behav
ior of each compound. Table 13.3 shows the ratings for Compound 2 and Compound 4
for which the abrasion coefficients are listed in table 13.2, using Compound 1 as the ref
erence. Both compounds show reversals in ranking with changes in severity. This agrees
closely with road wear experience. The upper compound (Compound 2) is better at low
energies and speeds than the reference and is known to be better under driving conditions
typical for the USA, i.e. under speed restrictions and on long straight roads that lead to
low and moderate rates of abrasion. Under European traffic conditions this type of com
pound is appreciated for its high wet grip but it is also known to be somewhat inferior in
wear resistance compared to the reference.
Chapter 13. Rubber Abrasion and Tire Wear 573
140
120
rel. rating
100
80
60
40
1.6
0.0
0.4
1.2
0.8
0.8
lo g )
1.2
km
0.4
s pee (J/
1.6
d
0.0
(km g W
/h) lo
The NR compound is known to be better under low temperature conditions than the
control but worse under high temperature conditions. Chemically, NR has the lowest ther
mal stability of the elastomers used for tread compounds in tire technology and it has
therefore the highest temperature dependence of abrasion and wear. Thus, it is generally
accepted that NR has better wear resistance in a moderate climate than, for instance, SBR
but is much worse in hot climates. This will be thoroughly documented below under tire
wear.
In table 13.3, NR was better whenever the sample surface temperature was near room
temperature, i.e. at low speeds for all energies and at low energies for all speeds. When
both energy and speed were raised the sample surface temperature rose considerably and
the rating of the NR decreased correspondingly.
Table 13.3a: Ratings of compound 2 relative to compound 1 of
table 13.2 as a function of log (energy) and log (speed)
SBR/BR+SI vs. OESBR black
0.2 122.1 110.9 100.7 91.4 83.0 75.3 68.4 62.1 56.4
0.4 118.5 108.3 98.9 90.4 82.6 75.5 69.0 63.1 57.6
0.6 114.9 105.7 97.2 89.4 82.3 75.7 69.6 64.0 58.9
0.8 111.5 103.2 95.6 88.5 81.9 75.9 70.2 65.0 60.2
1.0 108.1 100.8 93.9 87.5 81.6 76.0 70.9 66.0 61.5
1.2 104.9 98.4 92.3 86.6 81.2 76.2 71.5 67.1 62.9
1.4 101.8 96.1 90.7 85.7 80.9 76.4 72.1 68.1 64.3
1.6 98.7 93.8 89.2 84.8 80.6 76.6 72.8 69.2 65.7
574 Chapter 13. Rubber Abrasion and Tire Wear
0.0 113.2 113.6 113.9 114.2 114.5 114.8 115.1 115.5 115.8
0.2 113.4 111.2 109.0 106.8 104.7 102.6 100.6 98.6 96.6
0.4 113.7 108.9 104.3 99.9 95.7 91.7 87.9 84.2 80.6
0.6 113.9 106.6 99.8 93.5 87.5 82.0 76.7 71.9 67.3
0.8 114.1 104.4 95.6 87.5 80.0 73.2 67.0 61.4 56.2
1.0 114.3 102.2 91.5 81.8 73.2 65.5 58.6 52.4 46.9
1.2 114.5 100.1 87.5 76.5 66.9 58.5 51.2 44.7 39.1
1.4 114.7 98.0 83.8 71.6 61.2 52.3 44.7 38.2 32.6
1.6 114.9 96.0 80.2 67.0 55.9 46.7 39.0 32.6 27.2
A further example is shown in table 13.4. This gives the laboratory rating as a func
tion of log energy and log speed for four passenger tire tread compounds for which road
test ratings were available. They are shown on the left of each table. For compound 1,
practically all the laboratory ratings were less than 100 and in the tire road test the com
pound was also distinctly poorer than the control. Compound 3 was slightly better than
the control but the ratings were higher under some conditions and lower under others.
Hence, it is not certain whether the compound is better in general, under a variety of serv
ice conditions. On the other hand, compound 4 was much better than the control under
most testing conditions and this is reflected in the high road test rating. Note that the com
pound developer gets a much broader view of the wear potential of a compound with this,
admittedly more time consuming, laboratory abrasion method than with a single point test
result.
The cells marked in red in the laboratory abrasion table show by inspection only two
testing conditions for which a high correlation exists between road ratings and a single
laboratory abrasion test condition (speed, slip and load). It would have been very diffi
cult to pick one of these laboratory test conditions without any background knowledge.
Moreover they are only applicable to the particular road test; if the road test conditions
change, then the relevant laboratory test conditions also change.
By using regression analysis between the road test ratings and the ratings obtained for
any one cell, i.e. for any particular testing condition, the correlation coefficient, the regres
sion coefficient and the intercept of a linear regression equation can be calculated for each
cell, i.e. for each laboratory condition.
This is shown in table 13.5 for the data of table 13.4. If the criterion is used that the
regression coefficient should be nearly 1 and the intercept nearly zero, besides a high cor
relation coefficient, the two best testing conditions differ slightly from those obtained by
inspection. The range over which the correlation coefficient is high is quite large, but for
a perfect 1:1 correlation it is very limited.
Similar data were also obtained for truck tire compounds during an extensive sur
Chapter 13. Rubber Abrasion and Tire Wear 575
compound 3
log v
log W 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
0 91.9 105.0 120.1 137.3 157.0 179.5 205.2 234.7 268.3
0.2 97.9 106.7 116.3 126.7 138.1 150.4 163.9 178.6 194.6
0.4 104.4 108.4 112.6 116.9 121.4 126.1 130.9 135.9 141.1
107 0.6 111.3 110.2 109.0 107.9 106.8 105.6 104.5 103.5 102.4
0.8 118.7 111.9 105.6 99.5 93.9 88.5 83.5 78.7 74.2
1 126.6 113.7 102.2 91.9 82.6 74.2 66.7 59.9 53.9
1.2 134.9 115.6 99.0 84.8 72.6 62.2 53.2 45.6 39.1
1.4 143.9 117.4 95.8 78.2 63.8 52.1 42.5 34.7 28.3
1.6 153.4 119.3 92.8 72.2 56.1 43.7 34.0 26.4 20.5
compound 4
log v
log W 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
0 68.1 88.0 113.7 147.0 190.0 245.5 317.3 410.1 530.0
0.2 75.1 92.5 114.0 140.4 172.8 212.9 262.2 322.8 397.6
0.4 82.9 97.3 114.2 134.0 157.3 184.6 216.6 254.2 298.3
170 0.6 91.5 102.3 114.4 127.9 143.1 160.0 178.9 200.1 223.8
0.8 101.0 107.6 114.6 122.2 130.2 138.7 147.8 157.5 167.9
1 111.4 113.1 114.9 116.6 118.4 120.3 122.1 124.0 125.9
1.2 122.9 118.9 115.1 111.4 107.8 104.3 100.9 97.6 94.5
1.4 135.6 125.1 115.3 106.3 98.0 90.4 83.4 76.9 70.9
1.6 149.7 131.5 115.5 101.5 89.2 78.4 68.9 60.5 53.2
vey by the European Union of the durability and wear of re-treaded tires. An abrasion pro
gram carried out with three compounds using different polymer blends with the same car
bon black gave a high correlation with road test ratings on driven axles for different use
at high energy levels. Two such correlations are shown in figure 13.41. For seven com
pounds containing different types of filler with the same polymer formulation (80 SBR/20
NR) the correlation between road wear and laboratory ratings was narrower as shown in
figure 13.42. Notice that the regression coefficient was only about 0.5. This means that
the laboratory experiments were twice as discriminating between compounds as the road
wear data.
The laboratory data were obtained on an alumina abrasive track. The question arises as to
the degree to which this surface corresponds to real road surfaces.
576 Chapter 13. Rubber Abrasion and Tire Wear
Figure 13.41: Correlation coefficient between road test ratings of three truck tire
compounds on re-treaded tires at two different axles and laboratory ratings as func
tions of log (energy) and log (speed).
Tractor Drive Axle Rigids Drive Axle
1.0
1.0
0.8
0.8
correlation coefficient
0.6
0.6
correlation coefficient
0.4 0.4
0.2 0.2
0.0
0.0
-0.2
-0.4 -0.2
-0.6 -0.4
-0.8
-0.6
-1.0
1.6
0
1.6
1.2
0.4
0
1.2
m)
0.4
0.8
0.8
) 7k
0.8
0.8
log
1.2
/ km kJ
0.4
v( km
W(
1.2
lo g /kJ
1.6
0.4
v (k /h)
l og
0
gW
16
m/h
0
) lo
Chapter 13. Rubber Abrasion and Tire Wear 577
Figure 13.42: Correlation and regression coefficients between road test ratings of
seven truck tire compounds based on 80SBR/20NR but differing in type and amount
of filler and laboratory abrasion ratings obtained on LAT 100 testing equipment.
Correlation Coefficiernt Regression Coefficiernt
rmax 0.96
0.4
1.0
0.9
0.8 0.3
correlation coefficient
regression coefficient
0.7
0.6
0.2
0.5
0.4
0.3 0.1
0.2
0.1
0.0
0.0
-0.1
-0.1
1.6
0
1.2
0.4
1.6
0
0.8
0.8
1.2
)
0.4
/m
1.2
0.4
(kJ
0.8
0.8
log m)
1.6
gW
0
v( km log
1.2
0.4
/h) lo v( km 7k
( kJ
1.6
/h)
0
g W
lo
4. Tire wear
4.1 Tire wear under controlled slip conditions
One of the simplest instruments to measure actual tire wear under controlled load and slip
is a towed two-wheeled trailer with its wheels set at a given slip angle. Both wheels must
have tires mounted of the same construction, tread pattern and compound, so that the tires
run at the same slip angle. A trailer of this type was employed in abrasion studies by
Schallamach [34]. The rate of wear as a function of slip angle is shown in figure 13.43
for three different black filled tire tread compounds. Plotting wear and slip angle on log
scales produces straight line graphs indicating that the wear loss can be described by a
power law, in agreement with the model described previously. The straight-line graphs
cross, indicating reversals in ranking. The NR and SBR compounds cross at a low slip
angle. With a blend of 50 NR/50 BR the crossing point shifts to higher slip angles. All
tires developed a characteristic Schallamach abrasion pattern. The most important vari
able, however, was the tire surface temperature. It increased dramatically with increasing
slip angle.
If the rating of NR relative to SBR is plotted as function of the tire surface tempera
ture, as shown in figure 13.44, a clear relation is seen to hold with NR better at low tire
surface temperatures, but worsening rapidly as the temperature rises. The tire surface
temperature may also be affected by environmental conditions. Wear measurements with
tires set at a constant slip angle but with changing ambient temperature produced a larger
temperature dependence of the wear rate for NR than for SBR, as shown in figure 13.45.
The straight-line graphs showed a crossover at a surface temperature of about 48 °C. Such
temperatures are also produced in ordinary tire wear. Measurements on the trailer tires
themselves, where the slip angles were much smaller, gave much lower wear rates, as
expected, but the relative wear rating of NR to SBR corresponded to that expected from
the measured tire surface temperature, see figure 13.44 (open circles). The shift of the
crossover point towards higher slip angles (i.e. higher surface temperatures) for a NR/BR
blend indicates that this blend has a higher thermal stability.
578 Chapter 13. Rubber Abrasion and Tire Wear
All these observations are qualitatively in agreement with the laboratory investigations
described above. They underline the important role that temperature plays in the wear
process. Although temperature affects the dynamic properties of rubber its main effect is
to accelerate the thermal oxidative processes that occur in abrasion and wear, and lower
the resistance of compounds to them.
Figure 13.43: Log (tire wear) as a function of log (slip angle) for three tire tread
compounds. Results obtained with the MRPRA trailer
Figure 13.44: Relative compound rating as a function of the tire surface temperature.
Data for three compounds obtained with the MRPRA test trailer and including data
from the towing vehicle
Trailer
Car
re at ve rat ng
4.2 Conditions affecting tire wear in road tests and normal usage.
4.2.1 Influence of the road surface
Consider first a test car that is driven for an eight-hour shift over a prescribed route with
a length of about 600 km. The route could be selected to have special features like pre
dominantly interstate highway (motorway, turnpike or freeway) or passing through windy
and hilly country. In any case there will inevitably be a number of changes of road sur
face resulting in the need to characterize the average road surface structure, i.e. the sharp
ness and coarseness of the road surface, that are likely to have a significant influence on
the abrasion loss. The average structure may be different for different test routes but the
difference between the two averages are already likely to be smaller than the differences
encountered along each route separately. Moreover, the average of a test route is likely to
reflect the average of a whole geographical region. Differences between the average road
surfaces in highly developed countries are likely to be small while larger differences exist
between such countries and less developed ones. But even the structures of a chosen test
route vary with the season. It is well known that road surfaces in areas of moderate cli
mate are much sharper in winter than in summer. Even over much shorter periods the
sharpness often changes drastically as a result of weather conditions.
Since wear results are generally obtained over a span of time they are inevitably aver
ages. This is not too serious if the ranking of test compounds is not strongly influenced by
the road surface structure and in particular by the changes that occur. Nevertheless it is
obvious that repeatability is limited and as the few examples taken from laboratory exper
iments have shown, reversals in ranking are rather the rule than the exception. Hence a
single road test result can have only limited validity.
580 Chapter 13. Rubber Abrasion and Tire Wear
(a) With one tire of each type mounted on a trailer axle (equal force comparison).
(b) With two similar tires mounted on the trailer axle (equal slip)
Bias
equal force
radial
equal slip
Chapter 13. Rubber Abrasion and Tire Wear 581
equipped with a test group of four identical tires. If this is not possible, at least one axle
must have identical tires. Otherwise an average slip angle will be set up that brings into
balance the side forces acting on the axle. This force will be larger for the stiffer tire than
would be required if both tires had the same stiffness and it will be smaller for the softer
tire. Hence the wear result is falsified with an advantage being conferred on the softer tire.
The same argument holds for multi-section tires.
Figure 13.47: Measured distribution of (a) cornering and (b) fore and aft
accelerations in a controlled road wear test for passenger car tires.
0.45
0.4
0.35
0.3
Frequency
0.25
0.2
0.15
road Simul.
0.1
0.05
0
25
25
25
25
25
5
5
.7
.7
.7
.7
.7
0.
1.
2.
3.
4.
-4
-3
-2
-1
-0
0.5
0.45
0.4
0.35
Frequency
0.3
0.25
0.2
0.15
0
-3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5
Figure 13.48: Speed distribution in a controlled road test for passenger car tires.
Also shown is a distribution made up of two super-posed normal ones
30
road
20
Frequency
15
10
0
0-25 25-50 50-75 75-100 100-125 125-150 150-175 175-200 200-225 225-250
s pe e d ( k m / h)
0.030
frequency
0.025
0.020
0.015
0.010
0.005
0.000
-0.3 -0.2 -0.1 0 0.1 0.2 0.3
cornering acceleration (g)
0.06
0.05
frequency
0.04
0.03
0.02
0.01
0.00
0 10 20 30 40 50 60 70 80 90 100 110
speed (km/h)
With the assumption that the frequencies of the different distributions determine one
event, the frequency of each event is defined by the product of the individual frequencies.
Having calculated the force for a particular event the slip is calculated using the brush
model and hence the energy dissipation is obtained.
The forces are different for a driven and a non-driven axle and accordingly different
abrasion rates will result. The load transfer that occurs during cornering averages out and
is therefore not included. With the tire data, i.e. dimensions, tread width, net/gross area
ratio for the tread pattern, and tread depth, expected tire lives are calculated under the
above specified conditions. This approach is highly empirical (although supported by
some distribution measurements). It has the merit that, with an existing set of laboratory
abrasion data, a large number of road test simulations with different boundary conditions
can be run in a very short time and their effect on tire life and compound rating can be
estimated.
The tire construction influences the stiffness for both cornering and longitudinal slip.
This includes the tire carcass, breaker construction, inflation pressure, and tread pattern
design. However, since the two stiffness components can be measured, knowledge of the
construction details is not necessary. The vehicle geometry influences tire wear through
the air resistance it creates and through the load distribution between front and rear axle.
The driving parameters are determined by the maximum values of their distribution func
tions and in case of load and speed by the ratios between low, medium and high loads and
town, country road and highway traffic speeds, respectively. The tread pattern influences
tire life through its effect on the abraded rubber volume.
Figure 13.51 shows the major influences on tire life. The most important are clearly the
586 Chapter 13. Rubber Abrasion and Tire Wear
driving style, determined by accelerations and maximum speed, and the tire construction
as determined by the cornering and longitudinal slip stiffnesses. The figure compares two
representative modern passenger tire tread compounds: an OE-SBR black filled com
pound and a solution SBR/BR blend filled with silica. The two upper graphs show the
influence on tire life of the relevant distribution functions of the maximum acceleration
components at a constant maximum speed. It is clear that tire life decreases rapidly with
increasing severity. The solid lines refer to the cornering stiffness components of table
13.5 while for the dotted lines the stiffness components have been reduced by a factor of
0.75, keeping all other values the same. This reduces the tire life considerably. However,
not only are the tire lives affected but the relative ratings of the two compounds are also
changed. At low accelerations the OE-SBR/BR blend + silica is better than the OE-SBR
black compound but the rating reverses with increasing acceleration. The same is
observed if the maximum speed of the speed distribution function is varied and the accel
eration is kept constant. As the speed is increased the tire life is drastically reduced and
the rating of the silica compound reverses with respect to the OE-SBR black compound.
80 110
tire life (1000 km)
60 105
rel rating
40 100
20 95
0 90
0 0.1 0.2 0.3 0 0.1 0.2 0.3
max. acceleration (g) max. acceleration (g)
160 130
140
120
120
tire life (km)
rel rating
100 110
80
60 100
40 90
20
0 80
120 140 160 180 200 220 120 140 160 180 200 220
max. Speed (km/h) speed (km/h)
ferences between the moduli of different tread compounds are also small and hence the
results are reasonably correct. But ideally the cornering and longitudinal slip stiffnesses
should be measured. If this is not possible the two stiffness components ought to be cor
rected for differences in shear modulus of the tread.
Considering the tire as a composite short beam under shear the following correction
may be applied to the basic stiffness of the control
ρ (1+ ϕ )
K x = Ko [20]
(1+ ϕρ )
where Ko is the slip stiffness component of the control tire, ϕ is the ratio between the tread
and carcass stiffness and ρ is the ratio of stiffness of the experimental tread to that of the
control. The latter quantity can be estimated from the side force coefficients obtained dur
ing abrasion experiments at a small slip angle and a high speed. They reflect directly the
compound stiffness and since the dimensions of test wheels are the same, the same ratio
holds for the shear modulus.
For a balanced construction the ratio ϕ should be 1. If this is assumed to be the case the
correction becomes
2ρ [20a]
K x = Ko
(1+ ρ )
if ϕ<1 then the correction is larger, approaching ρ in the extreme case. If ϕ is larger than 1
the correction becomes smaller than given by equation [20a], so that Kx approaches Ko. If
multi-section tires or experimental and control tires are mounted on the same axle, as is often
the case for tests with truck tires on commercial fleets, the inverse of the above relation should
be applied because the tires are now running under an imposed common slip.
5. Correlation between laboratory road test simulation and road wear test results
5.1 Correlation with a set of passenger car tires
It has already been shown how a correlation between road test ratings and laboratory abra
sion can be obtained over a range of energies and speeds. Usually a good correlation, with a
high correlation coefficient and a regression coefficient near 1, is obtained only over a limit
ed range. This can be taken a step further by comparing road test ratings with a simulated
laboratory test rating. This reduces the comparison again to a single number, a result usual
ly desired by managers and compounders. However, the number is now based on the range
of boundary conditions defined by the road test. This means that the appropriate road test
conditions must be selected. Since today it is an easy matter for tire and vehicle manufactur
ers to carry out the required measurements, a representative set of boundary conditions and a
comprehensive set of laboratory abrasion data could save an enormous amount of compound
development time [37].
For the available road test data, such information was not available. However because a
road test simulation takes only a very short time and reasonable conditions are not hard to
guess, good correlations can be shown to exist between laboratory road test simulations and
actual abrasion road test ratings without exact knowledge of the boundary conditions.
Table 13.7 shows three road test simulations of increasing severity by raising the maximum-
acceleration components in three steps. The remainder of the boundary conditions were only
loosely known and were therefore guessed at. The calculated ratings are compared with the
588 Chapter 13. Rubber Abrasion and Tire Wear
road test ratings. A very good correlation is achieved at a maximum acceleration for both
components of 0.35g. This corresponds to a hard driving style, but one which is common for
tire test drivers. The calculated tire lives are accordingly short, in agreement with this style
of driving, and seem quite realistic.
Table 13.7: Correlation between road test ratings and laboratory road test
simulations for a group of four passenger tires, discussed above for their
laboratory abrasion and correlation to road test ratings
ratings for three truck tire re-tread compounds mounted on the axles of
several trucks, either as multi-section tires or as whole tires but with two
groups on the same axle together with the simulation and actual ratings when
compared (a) under equal force (b) under equal energy (assumes the same
slip stiffness for all groups and (c) under equal slip conditions
Since no data apart from a description of the general use of the truck (mostly short dis
tance haulage) and the axle position on which the tires were mounted were available,
again reasonable assumptions had to be made on maximum accelerations and speeds as
well as load distributions. They are listed in table 13.8 together with the three compar
isons. The boundary conditions chosen are reasonable for the kind of application under
which the tires operated and the achieved mileages agree well with those obtained from
the simulation. It is seen that good agreement is only reached if equal slip conditions are
assumed, with a correlation coefficient of 0.977.
25
laboratory energy
testing range
20
frequency
15
10
0
-1.1
-0.9
-0.7
-0.5
-0.3
-0.1
0.1
0.3
0.5
0.7
0.9
1.1
1.3
1.5
1.7
1.9
2.1
2.3
passenger truck
log energy (kJ/km)
In addition, the laboratory range of energies, marked by the blue box, is also close to
the range of energies in practical tire use. The reason for this is that much larger slip
angles are used but at much smaller forces. A similar situation exists for the slip speeds.
They are lower for trucks than for passenger tires but both are within the range of slip
Chapter 13. Rubber Abrasion and Tire Wear 591
6. Conclusions
During the last fifty years our understanding of the very complex phenomena of abrasion
and tire wear has progressed steadily. This chapter has traced the main strands of the
story. Some details may still be missing but the main points are now clear. Starting from
the underlying basic mechanisms, knowledge of the wear of tires has progressed so far
that it is possible to predict compound ratings from laboratory experiments with sufficient
accuracy that tire tread compounds can be developed with confidence without the neces
sity of tire road testing.
Basically abrasion is a cut growth process. This process is understood to be governed
by the tearing energy as defined by Rivlin and Thomas 50 years ago. Schallamach was
the first to conceive of abrasion as a function of energy dissipated in the contact area of
rubber and track. Basic experiments have demonstrated that the relation between cut
growth and tearing energy on the one hand and the relation between abrasion and energy
consumption on the other are closely connected. A power law relation between rate of
abrasion and energy consumption is now firmly established and holds over a wide range
of energies. The power index and constant depend on both the compound and the abra
sive track. The primary effect of the track is due to its sharpness rather than its coarse
ness. This influences both the level of abrasion as well as the value of the power index in
the relation between abrasion and sliding energy.
Both processes, abrasion and cut-growth, are not purely physical phenomenon because
chemical processes, primarily oxidation and thermal decomposition, play an important
role. Indeed, in abrasion they can become dominant; for example, highly-degraded rub
ber is a sticky material, pulled off the tire by adhesion to the road. It is important to rec
ognize that energy consumed in abrasion raises the temperature at the interface between
rubber and track and thereby modifies the abrasion process. Because the temperature in
the contact patch is a function of the power consumption, it depends also on the sliding
speed. The physical and the chemical processes are so significant that they lead to differ
ent behavior of compounds under different service conditions: reversals in ranking of
compounds for abrasion resistance are common. It is therefore in principle not possible
to design a single laboratory abrasion test that can reflect the practical experience of serv
ice conditions. A range of energies and speeds are a minimum requirement. Computer
simulation of road test conditions has shown that both these quantities can be applied in
laboratory experiments to cover the same ranges that occur in service, increasing the like
lihood of producing compound ratings that accurately reflect practical experience.
The question of the influence of the road surface on tire wear cannot be answered
unequivocally because of the large effect on their abrasive power of different composi
tions, state of use and effect of weather. Road surfaces are also not durable enough for
laboratory use. Hence reliance has to be placed on the correlation between laboratory
results on a laboratory abrasive surface and road test experience. Alumina of different
grain size (but primarily 60) has proved to be the most useful. Even so its sharpness
592 Chapter 13. Rubber Abrasion and Tire Wear
changes with use and test disks have only a limited useful life.
The link between laboratory abrasion test methods and road tests is now well estab
lished so that further research can concentrate on elucidating the basic underlying process
es. For example, the interaction between filler systems and elastomers is continually
changing through introduction of both new polymers and new fillers. But now the link
does not need to extend from basic experiments to road testing of actual tires. It can be
limited to some well-defined laboratory abrasion tests supplemented by measurements of
the basic physical properties of the compound.
References
1. F. B. Bowden and D. A. Tabor (1954), “Friction and Lubrication of Solids”, Oxford
University Press, London, 1954
2. P. Thirion, Rev. Gen. Caout. 23, (1946), 101
3. A. Schallamach, Proc. Phys. Soc. B65, (1952), 657
4. A. Schallamach, Wear 1 (1958) 384
5. K. A. Grosch, Rubber Chem. Technol. Rubber Reviews, 69 (1996) 495
6. K. A. Grosch and A. Schallamach, Kautschuk, Gummi und Kunstst. 22, (1969) 288
7. A. Schallamach and K.A. Grosch, “The Mechanics of Pneumatic Tires” ed. S. K. Clark,
US Dept of Transportation, National Highway Traffic Safety Administration, Washington
DC 20950, Chap. 6, p. 408
8. R. S. Rivlin and A. G. Thomas, J. Polymer Sci. 10 (1953) 291
9. A. G. Thomas, J. Appl. Polymer Sci. 3 (1960) 168
10. G. J. Lake and P. B. Lindley, Rubber J. 146 (10) (1964) 10.
11. P. B. Lindley and A.G. Thomas, 4th Rubber Technology Conference London, 1962
12. K. A. Grosch and A. Schallamach, Trans. IRI 41 (1965) T 80; Rubber Chem. Technol.
39 (1966) 287
13. M. L. Williams, R. F. Landel, and J. D. Ferry, J. Am. Chem. Soc. 77 (1955) 3701
14. K. A. Grosch, Proc. Roy. Soc. A 274 (1963) 21
15. A. Schallamach, Wear 6 (1963) 375
16. A. Schallmach, Wear 13 (1969) 13
17. S. P Timoshenko and J. N. Goodier, “Theory of Elasticity”, McGraw-Hill
International Book Co., 19th ed., 1983.
18. A. Schallamach, Trans. Inst. Rubb. Ind. 28 (1952) 256
19. K. A. Grosch and A. Schallamach, Trans. I.R.I, 41, T80; Rubber Chem. Technol. 30
(1966) 287
20. A. Veith, Polymer Test. 7 (1987) 177
21. D. H. Champ, E. Southern, and A. G. Thomas, “Advances in Polymer Friction and
Wear”, ed. by L. H. Lee, Plenum, New York (1974) p. 134
22. A. N. Gent and C. T. R. Pulford, J. Appl. Polymer. Sci. 28 (1983) 943
23. A. Schallamach, J. Appl. Polymer Sci. 12 (1968) 281
24. G. J. Brodskii, Sakhuoskii, M.M. Reznikovskii and V. F. Estratov, Soviet Rubber
Technol. 19 (1960) B. 22
25. A. Schallamach and D. Turner, Wear 3 (1960) 1
26. D. L. Nordeen and A. D. Cortese, Trans. S.A.E. 72 (1964) 325
27. K. A. Grosch, Conference Proc. IRC Kobe (1995), p.155
28. K. A. Grosch, Kautschuk Gummi und Kunststoffe 49 No. 6 (1998) 432
Chapter 13. Rubber Abrasion and Tire Wear 593
Chapter 14
by Joseph D. Walter
Chapter 14. Tire Properties That Affect Vehicle Steady-State Handling Behavior 595
Chapter 14
Tire Properties That Affect Vehicle
Steady-State Handling Behavior
by Joseph D. Walter
14.1 Introduction
Tires generate the control forces – characterized as driving and braking in the longitudi-
nal direction and cornering in the lateral direction – that affect and effect vehicle motion
in general and vehicle handling in particular. Handling is one of the important subjects of
vehicle dynamics along with ride and straight line tractive performance. Vehicle response
to steering wheel inputs (i.e., handling) is governed principally by cornering forces gen-
erated at the tire-road interface. For some vehicle classes cornering forces can also act as
a fulcrum if tire-road friction is sufficiently large precipitating vehicle rollover.
In the present chapter, we make a variety of simplifying assumptions that allow us to
focus solely on the influence of tires in controlling vehicle motion in a turn which allows
for an understanding of handling and rollover in elementary terms. It will be shown that
handling is strongly influenced by the understeer behavior of a vehicle, which quantita-
tively is dominated by differences in the cornering stiffnesses of the front and rear tires.
On the other hand, the rollover propensity of a road car is governed principally by vehi-
cle geometry with tires being of secondary influence.
Detailed derivations of the equations used herein and further discussion of vehicle
dynamics can be found in the texts of Wong[1], Gillespie[2] and Pacejka[3].
1 The terminology "wagon steer" is also used to describe the steering of articulated, tracked vehicles
(Wong [1], p. 425).
596 Chapter 14. Tire Properties That Affect Vehicle Steady-State Handling Behavior
controls the turn radius R for a given wheel base L. When R >> L, which is the usual case,
we can assume
δ f = L/R (1)
Figure 14.1: Wagon steering geometry
The steer angle is nominally the angle between the wheel plane and the longitudinal axis
of the vehicle. It is apparent that as steer angle increases, the turn radius decreases for a
vehicle of given wheelbase. It is, of course, not feasible to have a solid front axle pivot-
ing as shown on a road car.
However, if the front axle remains parallel to the rear axle in a low speed turn with the
front tires remaining parallel to each other as they pivot vertically, the yawing motion of
the front axle is eliminated. Figure 14.2 illustrates this concept for so-called parallel steer-
ing geometry. But the motion described occurs at the expense of a non-common turn cen-
ter with concomitant tire scrub.
Figure 14.2: Parallel steering geometry
The Ackermann steering geometry shown in Figure 14.3 eliminates the problems of
both wagon and parallel steer. For this lay-out, the front axle remains parallel to the rear
axle for a vehicle in a turn while the wheels rotate individually about vertical pins (known
Chapter 14. Tire Properties That Affect Vehicle Steady-State Handling Behavior 597
as king-pins) relative to the front axle.2 The concept is illustrated for a simple co-planar
steering mechanism; sub-axles AL and BR about which the wheels rotate are part of the
steering knuckles CAL and DBR, respectively. These knuckles are connected to the front
axle by king-pins at A and B which allow pivoting motion. The two steering knuckles in
turn are connected to the tie rod CD at C and D. Because the tie rod is shorter than the dis-
tance between the king-pins, the inside wheel will turn through a greater steer angle δi
than the angle δo of the outside wheel. The steering gear transmits the driver’s steering
wheel motion to the front axle steering mechanism. This ingenious arrangement, dating
from the era of horse-drawn carriages in the early 1800s, allows the front axle to remain
parallel to the rear axle with each tire-wheel assembly rolling about a common turn center.
There are two distinct equations that can be obtained from the Ackermann steering concept:
cotδo - cotδi = t/L (2)
and
⋅
(δi+δo)/2 = δf = L/R (3)
Equation 2, based solely on geometry, establishes the relationship between the steer
angles δi and δo with vehicle wheel base L and track t that minimizes tire scrub. Equation
3 defines an average front steer angle δf that forms the basis of subsequent analysis
(Section 14.3) used to describe the steady state handling characteristics of an automobile.
However, Equation 3 needs to be extended to include laterally compliant tires – i.e., tires
operating at a slip angle – in order to quantify the concept of understeer.
2While present day automobiles are generally produced with independent front and rear suspensions,
"axle" is used in the text and shown in Figures 14.1-14.3 for conceptual purposes.
598 Chapter 14. Tire Properties That Affect Vehicle Steady-State Handling Behavior
Fy = cornering force3
α = slip angle
Cα = cornering stiffness (initial slope of curve)
Figure 14.4: Representative trend of tire cornering force vs. slip angle
3 In the SAE coordinate system, the coordinate "y" denotes the lateral direction; other lateral forces
acting on a tire include camber thrust, ply steer and conicity - but these are generally smaller in
magnitude than cornering forces. (See Chapter 8).
Chapter 14. Tire Properties That Affect Vehicle Steady-State Handling Behavior 599
Note that the force Fy remains perpendicular to the wheel plane in a cornering maneuver
and is considered to be zero when slip angle α is zero. The travel direction is coincident
with the direction of the velocity vector of the tire-wheel assembly.
In the normal range of use parameters, the cornering stiffness Cα increases with
increasing wheel load, inflation pressure, rim width, tread wear, and other factors, and is
markedly influenced by tire cross-sectional geometry and internal construction features.
Surprisingly, Cα is relatively insensitive to vehicle velocity (at normal highway speeds)
and tread compounding ingredients (with the latter becoming important at higher slip
angles where the “Fy vs. α” relationship becomes non-linear). The complete set and nature
of tire force and moment properties, and especially cornering force, is discussed in
Chapter 8.
The units of cornering stiffness are commonly reported as lb/deg or N/radian.
Cornering stiffness is occasionally and mistakenly referred to as cornering power.
Cornering stiffness divided by wheel load is known as the cornering coefficient. The cor-
nering coefficient is sometimes cited as normalized lateral force. Note that as wheel load
increases, the cornering stiffness of a tire increases while the cornering coefficient
decreases (i.e., lateral force builds-up more slowly in magnitude than wheel load). Values
of the load and pressure dependent cornering stiffness may range from a low of 100 lb/deg
for a slightly underinflated bias ply passenger car tire to magnitudes in excess of 400
lb/deg for a radial performance tire.
For purposes of this analysis, we consider only the linear tire regime for which
Fy = Cαα (4)
The cornering force-slip angle curve for most passenger car tires can be considered to
remain linear to approximately four degrees of slip angle (which is generally about 0.3g’s
of lateral acceleration).
Note, however, for a vehicle in a turn with four nominally identical tires, that slip
angles will vary left-to-right as well as front-to-rear depending on the level of lateral load
transfer and front/rear axle loads (Figure 14.5). Knowing the tire cornering characteristics,
vehicle geometry, and operational conditions allows each of the four slip angles to be esti-
mated for a given level of lateral acceleration.
Figure 14.5: Variation of tire slip angles (left-to-right and front-to-rear)
for a vehicle turning at speed
600 Chapter 14. Tire Properties That Affect Vehicle Steady-State Handling Behavior
For purposes of subsequent analysis, left and right tire cornering stiffnesses are com-
bined at the front and rear axles, respectively, to produce the so-called “bicycle model” as
shown schematically in Figure 14.6. Note that the forward vehicle velocity (travel direc-
tion) at any point is perpendicular to the turn radius at that point and, again, cornering
forces front and rear are perpendicular to the wheel plane. Slip angles are measured
between the wheel plane and the velocity vector. The turn center is located at the intersec-
tion of line segments perpendicular to the velocity vectors at the front and rear tire-wheel
assemblies. With the turn radius much larger than the wheel base we can reasonably
assume small differences in the inside and outside steer angles and represent the average
angle (δi + δo)/2 as δf – the front wheel steer angle.
Figure 14.6: Bicycle model
Note that on a cornering vehicle, steer angle is not equal to tire slip angle while in the
laboratory environment of the tire test dynamometer the two angles are identical. Also
observe that slip angles on the rear tires of a vehicle also produce forces needed to maneu-
ver – just as the front tires – but produce such lateral forces without the benefit of a rear
steer angle.4
From Figure 14.6 the following geometrical relation can be obtained
L ⎛⎜ W f W ⎞ v2
δf = + − r ⎟ (6)
R ⎜⎝ 2Cαf 2Cαr ⎟⎠ gR
where linear tire properties have been employed (Equation 4), and
Wf, Wr = load on front and rear axles,
Cαf, Cαr = cornering stiffness of individual front and rear tires,
with other terms as previously defined. The local value of acceleration due to gravity is
denoted by “g” and the vehicle forward velocity by v.
The expression in parentheses in Equation 6 is often denoted by the symbol Kus and is
referred to as the understeer coefficient by Wong [1] due to vehicle weight distribution and
tires – i.e.,
Wf Wr (7)
K us = −
2Cαf 2Cαr
Note that the understeer coefficient as defined by Equation 7 is dimensionless but is often
expressed in degrees rather than radians. Each term on the right hand side of this equation
– with dimensions of load divided by cornering stiffness – is often referred to as corner-
ing compliance.
Equations 5 and 6 can now be re-written in the forms:
L K us v 2
δf = + (8)
R gR
or
δ f = L R + K us (a y g) (9)
Modifying the front and rear static axle loads in Equation 7 to accommodate longitu-
dinal load transfer due to braking or accelerating, or to account for aerodynamic lift or
down force, is inappropriate. However, the load transfer that occurs, e.g., to the front axle
while backing off the throttle upon entering a curve increases the cornering stiffness of the
front tires (due to the increased load) and reduces same in the rear. This action shifts the
handling balance of the vehicle toward oversteer. Further, if longitudinal forces are pres-
ent due to braking or driving, the ability of the tire to generate lateral (cornering) forces
is diminished and the cornering stiffness of the tire, Cα, is accordingly reduced.
From analysis or measurement, Equations 8 and 9, are often plotted to graphically
explain the condition of understeer, oversteer and neutral steer. Steer angle (δf) is plotted
schematically as a function of vehicle velocity (v) in Figure 14.7 and as a function of lat-
eral acceleration (ay/g) in Figure 14.8 for constant radius turns. As shown in these figures,
the steer angle δf increases as velocity and acceleration increase for understeering vehi-
cles; for oversteering vehicles, the steer angle decreases for the same conditions. Also,
understeering vehicles may tend to oversteer at higher lateral accelerations (dotted line
shown in Figure 14.8).
Figure 14.7: Relation between steer angle and vehicle velocity (equation 14.8)
Qualitative example
Consider the effect of rapid air loss due to foreign object penetration in one of the front
tires on the change in the steady state directional behavior of an automobile. Since the cor-
nering stiffness of the tires on the front axle is reduced, the understeer coefficient Kus
Chapter 14. Tire Properties That Affect Vehicle Steady-State Handling Behavior 605
(Equation 7) becomes more positive. While the driver may experience a pull in the steer-
ing wheel (in the direction of the deflated tire), vehicle response will tend to be more slug-
gish, but predictable.
On the other hand, for the loss of air in one of the rear tires, the vehicle will be less
understeering – and even perhaps oversteering – depending on the total amount of under-
steer present in the vehicle at the time of air loss. If the vehicle becomes oversteering, its
handling behavior is unstable.
One of the limitations of the “bicycle model” is that lateral load transfer and vehicle
roll effects cannot be accommodated in the analysis. It is known from on-road tests or
more advanced models than the reduction in the understeer coefficient (or gradient) due
to a rear tire disablement is position sensitive depending on the turning motion of the vehi-
cle. If the outside rear tire (left or right depending on the turn direction) is disabled, over-
steer is more likely to occur. The effect is exacerbated in vehicles with a solid rear axle.
Quantitative example
Consider an automobile with the following characteristics:
Weight = 4200 lb
Wheel Base = 110 in
Distance to c.g. =50 in (from front axle)
that is equipped with two radial tires (Cα=300 lb/deg) and two bias tires (Cα=200 lb/deg)
per axle.
For the case of bias tires at the front and radial tires at the rear:
Wf Wr 2290 1910
K us = − = − = 5.73° − 3.18° = +2.54° (0.0443 rad).
2Cαf 2Cαr (2)200 (2)300
This tire fitment tends to shift the control balance of the automobile to more understeer.
For the case of radial tires front and bias tires rear:
Wf Wr 2290 1910
K us = − = − = 3.82° − 4.75° = −0.93° (-0.0162 rad),
2Cαf 2Cαr (2)300 (2)200
and the vehicle becomes less understeering – and perhaps oversteering.
While it is not recommended to mix tire constructions on the same vehicle, if for some rea-
son this needs to be done, radials should be placed on the rear axle with bias tires on the front.
It may be useful to end this discussion of vehicle oversteer and understeer by listing
the assumptions used in developing the algebraic equations employed herein:
a) constant vehicle velocity;
b) path of constant curvature;
c) turn radius much larger than wheel base of vehicle;
d) tire cornering forces vary linearly with slip angle;
e) bicycle model.
606 Chapter 14. Tire Properties That Affect Vehicle Steady-State Handling Behavior
These assumptions are far removed from vehicle behavior in cornering maneuvers at
lateral accelerations in excess of 0.3-0.4 g’s, but the equations developed offer useful
insights into the handling behavior in the linear domain where the vast majority on-road
driving occurs.
Although the understeer coefficient was derived for the case of a vehicle in a turn, it
can be shown that it also applies to vehicle response in straight-ahead driving. For exam-
ple, oversteering vehicles have a critical speed at which forward motion is unstable [1].
Non-linear, steady state, tire behavior and its influence on vehicle directional control is
discussed in detail by Pacejka [3]. References to transient vehicle handling behavior can
be found in Wong [1] and Gillespie [2]. On-road tests allow one to evaluate non-linear tire
characteristics and transient vehicle behavior where elementary theories of ‘dynamic
understeer” have yet to be developed. Complex computer models are often employed to
analyze the limit behavior and skid recovery of complete tire-vehicle systems in non-
steady state conditions.
14.4 Rollover
If lateral forces produce a sufficiently large roll moment about the c.g. of a vehicle, it will
rollover about a longitudinal axis. When the lateral forces are due to cornering during
highway maneuvers, the rollover event is termed “untripped;” if the lateral forces arise
due to a roadway obstruction (such as a curb or soft shoulder on the edge of the highway),
the rollover is referred to as “tripped.” The majority of rollover accidents involve a trip-
ping mechanism – i.e., impact or engagement with an obstacle – but we consider only the
simpler case of untripped rollover because the results obtained correlate well with rollover
accident statistics whether the event is tripped or untripped [6].
Figure 14.9 shows the forces acting on a roll plane of a vehicle in a steady state turn
with a reversed inertia vector directed away from the turn center. There is no recognition
of front and rear axles or of a compliant suspension in this rigid body, planar analysis. As
velocity is increased or turn radius is decreased, lateral load transfer occurs; that is, the
normal load Fzi on the inside tires decreases while the normal load Fzo on the outside tires
increases. The accompanying tire cornering forces behave in a similar manner. If the coef-
ficient of friction between tire and road is sufficiently large, the cornering forces on the
front and rear outside tires act as a pivot axis about which the inertial (centrifugal) force
tends to rotate the vehicle in the roll plane away from the turn center. In other words, steer-
ing left into a turn causes the vehicle to roll to the right, and vice-versa.
Figure 14.9: Forces acting on a vehicle in a turn prior to rollover
Chapter 14. Tire Properties That Affect Vehicle Steady-State Handling Behavior 607
At the instant of incipient inside tire lift-off, Fzi = 0, and (roll) moment equilibrium
requires that
a g = t 2h (11)
y
where t is the track width of the vehicle and h is the height of its center of gravity (c.g.),
the other terms being previously defined. Equation 11 is termed the rollover threshold by
vehicle dynamicists or the static stability factor (SSF) by NHTSA. The terms are used
synonymously herein. Equation 11 predicts the lateral acceleration level (g’s) necessary to
cause inside tire lift-off prior to possible vehicle rollover. It involves only two parameters
related to vehicle geometry (track width and c.g. height) and no tire properties. Static sta-
bility factors predicted from this simple mechanics model range from 0.4 for heavy trucks
to 1.7 for sports cars [2]. The lower the SSF, the greater is the propensity for rollover to
occur – i.e., vehicles having higher values of SSF are safer, ceteris paribus.
The same free body diagram (Figure 14.9) can be used to show that
ay g = µ
where µ is the coefficient of friction between the tire and road surface . Thus, high µ sur-
faces are required to effect untripped rollover while low µ surfaces promote lateral slid-
ing of the vehicle.
Representative values of the static stability factor or rollover threshold for various
vehicles are given in Table 14.2.
The understeer gradients for these same vehicles are given in Table 14.1. It is apparent that
vans, pickup trucks and SUVs (so-called light trucks for fuel economy rating purposes)
have lower rollover thresholds than passenger cars principally due to their higher centers
of gravity. The problem is exacerbated in light trucks as passengers, cargo, and/or luggage
are added because the vehicle c.g. tends to move upward and rearward with added weight.
Both of these movements have a potentially destabilizing effect on stability: the upward
movement of the c.g. decreases the rollover threshold (Equation 11) and its rearward
movement decreases the understeer coefficient (Equation 7).
Since this elementary, quasi-static5 model lacks a suspension and does not consider
time-dependent steering inputs, it overpredicts the lateral g’s necessary to produce
rollover in controlled experiments. Stability factors (static and dynamic) predicted from a
variety of more complex models incorporating a generalized vehicle suspension and time
varying steering inputs are given in Table 14.3 for a generic small car having an SSF of 1.25.
Although there is a large reduction in the stability factors when dynamic steering inputs
are considered, the ability of the advanced models to properly discriminate among the dif-
fering rollover levels obtained from measurements made on vehicles of a given class
(such as SUVs) is no better than that given by the rollover threshold or SSF defined by
Equation 11. Further, it is again appropriate to point out that tire properties do not appear
explicitly in either the elementary or more complex models – though the compliance of
tires subjected to a vehicle roll moment, not a big effect in any case, could be accounted
for in the suspension parameters used in the more advanced models. Those interested in
those issues from a vehicle dynamics perspective should consult the chapter references,
namely [2 and 7].
Table 14.3: Prediction of small car rollover propensities from various models
[7, p. 342]
Static or Dynamic
Time Dependence Vehicle Suspension
Stability Factor
Quasi-static No 1.256
Quasi-static Yes 1.19
Step-steer Input Yes 0.92
Sinusoidal Steer Yes 0.80
While tire properties do not appear explicitly in any of the rollover equations, some
implicit factors are present. For example:
a) slight changes in inflation pressure produce very small changes in vehicle c.g. height
and possibly tire-road friction coefficient and cannot be effectively employed to
greatly change the rollover propensity of a given vehicle;
b) large reductions in inflation pressure ( ~ 25%) will tend to reduce the cornering force
available for untripped rollover as well as the cornering stiffness of the tire
(affecting understeer behavior); however, such large pressure reductions adversely
impact tire durability, rolling resistance, hydroplaning and high speed behavior.
c) as tires wear, the c.g. height of the vehicle is slightly lowered, thereby marginally
increasing the SSF;
d) in wet weather driving, the road surface friction coefficient is reduced, thereby
increasing the chance of vehicle slide-out rather than an untripped rollover in
driving maneuvers. Tripped rollovers, however, remain a possibility if the sliding
vehicle encounters an obstacle;
e) by installing wheels with negative offset7 on front and rear axles, vehicle track width
Table 14.4: Sales-weighted average SSF trends for different vehicle classes [8]
Model Year
Vehicle Class 1980 1985 1990 1995 2000 2003
Passenger Cars 1.36 1.36 1.37 1.41 1.42 1.41
Minivans N/A 1.11 1.16 1.19 1.24 1.24
Pickup Trucks N/A 1.18 1.17 1.18 1.18 1.18
Sport Utility Vehicles 1.07 1.08 1.07 1.09 1.11 1.17
Full Size Vans N/A 1.09 1.09 1.11 1.12 1.12
The positive trend for some of the more recent production vehicles is partially due to
changes in geometry (t/2h ratio), and the rising penetration of advanced electronic vehi-
cle systems that preferentially control driving and braking forces on all four tires of an
automobile which reduces the development of potentially de-stabilizing yawing motions.8
As electronic stability control (ESC) systems are increasingly being offered to purchasers
of new cars, it is expected that rollover thresholds, as predicted by Equation 11 supple-
mented by a government mandated dynamic on-road test sensitive to the presence (or lack
thereof) of ESC, should continue to show evolutionary improvements over time.
8Large yaw and roll motions occur prior to vehicle rollover; 3-D models incorporating both yaw
and roll response are discussed by Gillespie [2].
610 Chapter 14. Tire Properties That Affect Vehicle Steady-State Handling Behavior
References
1. J.Y. Wong, Theory of Ground Vehicles, Third Edition, Wiley, New York, 2001.
2. T.D. Gillespie, Fundamentals of Vehicles Dynamics, Society of Automotive Engineers,
Warrendale, PA, 1992.
3. H.B. Pacejka, Tyre and Vehicle Dynamics, Butterworth-Heinemann, Oxford, 2002.
4. SAE Recommended Practice J266, Steady-State Directional Control Test Procedures
for Passenger Cars and Light Trucks, Society of Automotive Engineers, Warrendale,
PA, 1996.
5. W.R. Garrott, et al., Experimental Examination of Selected Maneuvers That May
Induce On- Road Untripped, Light Vehicle Rollover. Phase 2 of NHTSA’s 1997-1998
Vehicle Rollover Research Program, National Highway Traffic Safety Administration,
DOT HS 808 977, East Liberty, OH, 1999.
6. Rating System for Rollover Resistance, An Assessment, Transportation Research Board
Special Report 265, National Academy Press, Washington D.C., 2002.
7. R. Stone and J.K. Ball, Automotive Engineering Fundamentals, Society of Automotive
Engineers, Warrendale, PA, 2004.
8. M.C. Walz, Trends in the Static Stability Factor of Passenger Cars, Light Trucks, and
Vans, National Highway Traffic Safety Administration, DOT HS 809 868, Washington
D.C., 2005.
Test Questions
1. On-road tests show that a sports car designed for responsive handling has a measured
understeer gradient of 1.1 deg/g. It weighs 3200 lb with the driver and has a 55/45
static split in front-to- rear axle weight distribution. The wheel base is 103 in. and the
cornering stiffness of the tires varies with load as shown:
Tire Load Cornering Stiffness
(lb) (lb/deg)
450 180
675 230
900 260
1125 280
a) Quantify and discuss the changes in vehicle directional control that can occur if a
tire puncture happens in a front vs. a rear position.
b) If 400 lb of “cargo” are placed in the vehicle with the resultant of the added weight
acting over the rear axle, compare and contrast quantitatively the expected handling
behavior vs. the original unladen condition (with four inflated tires).
c) Determine the steer angle required to negotiate:
- a 300 ft radius curve at 50 mph;
- a 1000 ft radius curve at 100 mph.
Chapter 14. Tire Properties That Affect Vehicle Steady-State Handling Behavior 611
2. Derive an equation for the rollover threshold for a vehicle traveling around a
superelevated curved highway with a transverse slope angle? Discuss the implicit role
that tire properties may have, if any, on the results.
612 Chapter 15. Introduction to Tire Safety, Durability and Failure Analysis
Chapter 15
Failure Analysis
2.7 Personal injury and property damage claim/lawsuit data ................................ 616
Chapter 15. Introduction to Tire Safety, Durability and Failure Analysis 613
Chapter 15
Introduction to Tire Safety, Durability and
Failure Analysis
by J.D. Gardner and B.J. Queiser
1. Introduction
Over time, as science, engineering, and manufacturing technology have advanced, tires
have offered an increasingly higher degree of utility, reliability, and value. However, tires
can become damaged, or when pushed beyond their limits, they will fail. Pneumatic tires
have always had a certain failure rate, and for this reason, vehicles have typically carried
spare tires—in the early days, more than one. Over time, failure rates have decreased, but
not vanished.
Tires are subjected to a wide variety of use and operating conditions, depending on the
vehicles they are attached to, the use and maintenance habits of their operators, the envi
ronment in which they operate, and random events that can inflict damage or affect their
operation. While great strides have been made in design and manufacturing, with rare
exception, it is what a tire experiences or is exposed to in the field that determines whether
it will fail in service.
Still more complicated is the issue of safety. Tire failure during operation, for instance
on a highway at speeds safe for conditions, is the precursor to an inconvenience, i.e.
pulling over and applying the spare, or, it is the first in a series of events that may lead to
loss of control, and in some cases an accident. However, the vast majority of the time
tire disablements do not result in an accident [1].
Performing a failure analysis on a tire is a scientific process that begins with an under
standing of the product, including its materials, mechanics, and operational characteris
tics. These items are evaluated as evidence that tells a story through testing, inspection,
and evaluation.
The goal of this chapter is to outline elements of tire safety, basic mechanics and fac
tors pertinent to tire durability, including common tire failure modes and analysis. These
subjects can be very broad, thus it is not practical to deal with all of them. The tire pho
tographs and examples shown in this chapter are intended to illustrate the concepts being
addressed and do not include all of the possible variants that can occur in testing or result
from use in the field.
2. Tire safety
2.1 Definition
The issue of tire safety is more complex than assessing the mere occurrence of a tire fail
ure; because a tire has failed does not necessarily mean that it was unsafe. If that were
true, all tires could be deemed unsafe because all tires can be made to fail under some con
ditions. Tire safety is usually part of a complex relationship involving the tire itself,
human beings, a vehicle or service equipment, and other external elements. There are two
primary areas of tire safety that pertain to failure:
614 Chapter 15. Introduction to Tire Safety, Durability and Failure Analysis
the ultimate result of a tire failure. A flow chart of the occurrences and influences is shown
in Figure 15.1.
Whether a tire failure in service is a safety issue depends upon numerous factors.
Therefore, assessing tire safety in this context can be difficult.
(NHTSA) have relied upon warranty or adjustment data as a measurement of tire perform
ance and customer satisfaction in the field. An adjustment refers to the credit a customer
receives at a retail outlet when a tire is exchanged under the terms of a warranty, or for
goodwill. Therefore, the data may include adjustments made on tires that are not under
warranty or for conditions not covered by a warranty, simply to maintain or foster cus
tomer satisfaction.
Adjustment data is prolific enough to provide statistical significance. It streams in contin
ually to the tire manufacturer where it is coded to identify the reason for the adjustment,
which can be for a variety of reasons, including durability, ride, comfort, and cosmetics. By
compiling and monitoring the data, a tire manufacturer can identify product performance
and/or customer satisfaction trends. This feedback can be used to implement countermeasures
on a timely basis, to enhance business relationships, and for future product development.
Because of its commercial value, tire manufacturers keep adjustment data confidential.
There is no standard within the tire industry or government regarding the evaluation of
adjustment statistics; for instance, there is no level which automatically triggers attention.
In addition, adjustment data for one tire may not necessarily be meaningfully compared
with data for other tires. For instance, performance trends that are apparent from a review
of data for a large, conglomerate population of tires may not apply to a particular tire
because of differences in such parameters as design or application. Essentially, to evalu
ate the performance of a particular tire, it is necessary to examine that tire’s unique per
formance history.
inspection for months, even years. In some cases alleging tire failure, the tire may no
longer be available, or may not be provided for inspection, making it practically impossi
ble to determine the cause of the alleged failure.
Another fundamental difference between a claim/lawsuit and an adjustment is that an
adjustment related to tire durability is commonly made before the tire has sustained a
complete failure. This is accomplished through routine inspection of tires by the consumer
or tire service professional or because of other warning signs, such as loss of inflation,
vibration, and irregular wear. Almost all claims and lawsuits could have been an adjust
ment if the tire was removed from the vehicle sooner. Failed tire adjustments are also
examples of a failure that does not result in an accident, contrary to some allegations
involving claims and lawsuits.
the tire in service. The rate of wear of tread rubber may constitute a durability parameter
in some cases, but it is not addressed here.
Tire engineers and chemists optimize the tire structure for performance factors relating
to traction, wear, ride comfort, handling, durability, rolling resistance, and more. Cost or
economy is a factor as well. Within the limits of science and engineering, certain param
eters may conflict with one another, or require engineering trade-offs. For this reason, the
marketplace contains a wide range of tire products designed with different operating and
performance parameters in mind, such as those necessary for off-road versus highway,
high speed use. From a durability perspective, the tire is designed to provide a balance of
desired attributes, while relying on the end-user for basic care and maintenance.
Because a tire has failed does not necessarily mean that there was anything deficient in
its design or manufacture. Structural failure of a tire can occur at virtually any point in its
expected lifetime, whether it is brand new with full tread depth, or nearly worn out after
years of use. Any tire can fail. The objective is to design a product that meets internal
company, industry, and government standards, and performs according to the consumer’s
needs until the tire is worn out and replaced. However, as previously noted, external fac
tors such as severe service conditions or damaging events can affect the tire’s ability to
endure and shorten or end its service life.
3.3 Deflection
The most basic factor that affects tire durability performance is the vertical deflection.
Using the simple model shown in Figure 15.2, when a tire is bearing a load P the deflec
tion d is the difference between the unloaded radius ro and the loaded distance rL between
the wheel center and the ground:
d = r o – rL
Considering the pressurized tire as a vertical spring, and applying Hooke’s Law:
P=Kd
where the vertical stiffness, K, is primarily attributable to the internal inflation pressure.
(It should be noted that the precise load/deflection behavior is more complex than this
model; see chapter 5.) The work W done in compressing a linear spring is
W = ½ K d2
As the deflection increases, more work energy is imparted to the tire, whether from situ
ations involving a decrease in K (such as from reduced inflation pressure) or an increase
Chapter 15. Introduction to Tire Safety, Durability and Failure Analysis 619
in load. As the tire rolls, a mechanical energy loss mechanism manifests itself through the
generation of heat [19 - 21]; see chapters 2, 11 and 12. As the tire deflection and/or load
increases, the energy loss increases as well, as shown in Figure 15.3 for a steel belted radial
tire.
Figure 15.2: Unloaded and loaded tires
The effect of the tire deflection can be seen by observing the size, shape, and pressure
of the contact patch—also known as footprint—of a tire on a flat surface, as shown in
Figure 15.4. The length of the footprint in the circumferential direction increases with an
620 Chapter 15. Introduction to Tire Safety, Durability and Failure Analysis
Figure 15.5: Example of FEA solid element cyclic strain energy density
3.4 Heat
As previously mentioned, tires develop heat as they are operated. As a tire is cyclically
deflected, it reaches thermal equilibrium in terms of its internal component temperatures, and
in terms of the contained air temperature within the tire cavity. Changing operating conditions
and physical changes to the tire structure itself can change the state of equilibrium.
Elevated and extended heat generation is a primary factor in the breakdown of a tire.
Increased heat decreases rubber tear resistance which promotes crack initiation and prop
agation [18]; see chapter 2. Permanent degradation of material properties from exposure
to elevated temperatures occurs as well, depending on the exposure history.
The shoulder areas of a radial tire are generally the highest in heat generation. Figure
15.6 illustrates the increasing temperatures that develop in the shoulder area when a tire
is operated with increasing deflection. Rubber deterioration (reversion) from excessive
heat build-up results in a decrease in tensile strength and a general softening. This break
622 Chapter 15. Introduction to Tire Safety, Durability and Failure Analysis
down from heat is an additive effect that can drive the temperature higher still. Ultimately,
a component or portion of the tire can reach a critical temperature range where the dete
rioration of the rubber can cause detachment from the tire of pieces or whole sections of
the tread. Reverted rubber has a porous appearance, such as the pieces of reverted tread
rubber shown in Figure 15.7. Other components may melt, such as the polyester body cord
shown in Figure 15.8.
Figure 15.6: Thermographic images of a tire operated with decreasing inflation [23]
Chapter 15. Introduction to Tire Safety, Durability and Failure Analysis 623
3.5 Speed
Changing the speed of tire rotation affects the centrifugal force and frequency of the
deflection cycle. Corresponding changes in the stresses and strains developed in the tire
components affect the tire’s heat build-up characteristics. Additional deflections occur
from “standing waves”, treated in chapter 10. This phenomenon depends upon the size,
construction, stiffness, and especially the rotational speed of the tire.
The fundamental effect of speed is its influence on the frequency of cyclic deforma
tion. With each rotation, a given radial section of the tire undergoes a stress-strain cycle
as it passes through the contact patch. Increasing the cyclic frequency increases the heat
that develops, and hence affects the performance of the tire as a whole, particularly param
eters that relate to durability.
The centrifugal force acting on a body is directly proportional to the mass, the distance
from the axis of rotation, and the square of the angular speed of rotation. For a tire, the
tread and belt structure is located the greatest distance from the center of rotation and
comprises a substantial portion of the tire mass. It is thus evident that centrifugal force
can affect the tread area, particularly the shoulders, which are concurrently undergoing a
stress/strain cycle with each rotation.
Testing has verified that increasing speed causes an increase in tire temperature, par
ticularly in the shoulder area [23, 24]. In Figure 15.9, results are shown for a tire tested
at speeds up to 120 km/h (75 mph) with numerous inflation pressure settings at constant
load. The results indicate a near-linear relationship between speed and temperature, with
increasing slope as the inflation pressure was decreased.
Figure 15.9: Belt edge temperature of a tire operated with
increasing speed at various inflation pressures
Chapter 15. Introduction to Tire Safety, Durability and Failure Analysis 625
Cracking of rubber can begin at or near an interface with a tire cord, as in the case of
the polyester cord shown in Figure 15.12. Such cracking may progress around the cord
itself, creating a condition known as socketing. Continued crack growth can result in
smaller cracks meeting one another and progressing further. Figure 15.13 shows a typical
separation between the steel belt edges. Fracture of cords or filaments can lead to struc
tural weakness that causes a whole-tire rupture or leads to more slowly developing inter
nal ply separation, depending on the extent of the damage. Tensile fracture of a steel cord
filament is shown in Figure 15.14.
Chapter 15. Introduction to Tire Safety, Durability and Failure Analysis 627
4.6 Overdeflection
As noted previously, the deflection of a tire is a dominant parameter in tire performance
because it determines stress, strain, and heat generation. When a tire is designed and test
ed for a rated load and inflation pressure, it has a specific deflection and will perform sat
isfactorily at that deflection within a reasonable tolerance. Overdeflection is defined as a
deflection that is greater than that intended for the rated load and inflation pressure.
Overdeflection occurs when the load is excessively high, or the inflation pressure is too
low, or when a combination of load and inflation pressure creates an excessively high
deflection. A moderate degree of overdeflection, depending on speed and other operating
parameters, can initiate and propagate steel belt edge separation and tread/belt detach
ments. Severe, gross overdeflection, generally at nearly flat-tire conditions, commonly
causes sidewall flex break failures.
If overdeflection occurs for a short period of time, the tire failure itself may be the only
evidence of overdeflection. If the overdeflection occurs over a longer period of time, it
can be evidenced by conditions such as rim grooving [23], balance weight impressions,
and/or tread wear patterns.
the air cavity to the atmosphere. First, they must pass through the innerliner which is
specifically formulated to resist air permeation, typically through the use of halogenated
butyl rubber. Any air molecules that pass through the innerliner then continue through the
remainder of the tire until they eventually reach the atmosphere. The overall rate of air
permeation through the tire depends on many factors. Loss in inflation pressure is gener
ally 1-2 psi per month for steel belted radial passenger and light truck tires.
During the process of air permeation a pressure gradient exists within the tire structure.
Over time, this pressure gradient attempts to reach equilibrium. If the interior of the tire
is breached by a puncture, impact, tear to the bead, etc., pressurized air within the tire cav
ity can pass directly into the tire structure. This creates two problems. First, the air brings
moisture and excess oxygen into the tire structure, which can break down chemical bonds
and consequently the adhesion/cohesion of internal components. Secondly, since the tire
cavity pressure is higher than the equilibrium gradient pressure within the tire structure,
the pressurized air will tend to force the structural components of the tire apart, particu
larly at high stress/strain areas of the tire such as the belt edges. These two conditions can
lead to or aggravate separations, including in some cases tread/belt detachment.
nering and other loads are superimposed on the inflation loads [30]. When a tire encoun
ters an object or irregularity in the roadway such as a pothole, rock, or other debris, the
part of the tire (i.e. tread, sidewall) that comes in contact with the object attempts to take
the shape of the object by deforming around it. Hence, the initial orientation of the rein
forcing cords and rubber must change to accommodate or envelope the object. Moving a
reinforcing cord relative to its rubber matrix has been studied in the laboratory by pulling
cords (steel, polyester, etc.) out of rubber pads or other configurations. The results of such
testing depend on the rate and degree of strain applied to the cord and its surrounding rub
ber material. At lower strains, the cord rubber matrix changes shape and no damage
occurs. At higher strain levels , the cord or the rubber matrix fractures. For a group of
cords, a combination can occur. Figure 15.21 shows the results of a steel cord pull test
from a rubber block matrix; in this case the cord fractured.
Figure 15.21: Steel cord tensile pull test, resulting in cord fracture
The same phenomenon can occur in a tire during an impact [31]. The additional
stress/strain loading imparted by an impact is different for different cords within the same
tire. Thus, after an impact injury, a unique damage pattern will exist in the tire. In some
cases the damage pattern will be so extensive that a hole will be present and an immedi
ate loss of air will result, as in Figure 15.17. In other cases, the tire will be damaged inter
nally, but capable of continued operation until it fails sometime after the impact.
If, when, and how the tire fails depends on the initial damage pattern and the
type and severity of the subsequent operation. The failure modes can be as diverse as the
damage pattern. They can be localized to the impact area or spread circumferentially
around the tire. If the impact causes a tear between the belts, broken cords, or a combina
tion of these, the damage can propagate and result in a tread/belt separation or detachment.
Chapter 15. Introduction to Tire Safety, Durability and Failure Analysis 637
In some cases, an opening or radial split of the tire cavity, particularly the innerliner, will
occur at the time of the impact. Over time, air will migrate through the split into the tire
causing intracarcass pressurization (previously addressed). The tire can subsequently fail
from a tread/belt detachment or other related failure mode.
Rarely is the object that the tire impacted available for inspection or the cord loading and
orientation at the time of the impact known. Impact failures are analyzed by examining
any damage the impact caused, such as rubber tears, fractured steel cord as in Figure
15.22, broken body ply cords, radial splits, propagation patterns, surface damage to the
tire, and wheel deformations.
Figure 15.22: Broken steel cords of a stabilizer (steel belt) ply
Design and manufacturing defects generally fall into two categories: those that affect
the performance of the tire and those that affect other things such as the appearance of the
tire. The categories that affect the performance of the tire include those that are safety
related and those that are not. Missing or misplaced components, improper vulcanization,
and foreign material are examples of manufacturing defects. However, whether a defect
such as a misplaced component is a performance or safety related issue depends on where
and how the components are misplaced.
Because the major tire manufacturers have been in business for decades and have
extensive research, design, development, manufacturing, and quality control activities and
procedures—and employ thousands of specially trained scientists, engineers, and produc
tion personnel—design and manufacturing defects in tires are extremely rare. Because of
this, defects tend to be one-of-a-kind occurrences and are typically evaluated accordingly.
References
1. Baker, J.S. and McIlraith, G.D., Tire Disablements and Accidents on High-Speed
Roads, Highway Research Record, Number 272, Highway Research Board,
Washington, DC, 1969.
2. Seiffert, U. and Wech, L., Automotive Safety Handbook, SAE International,
Warrendale, PA, 2003.
3. Burton, W.E., The Story of Beads and Tires, McGraw-Hill Book Company, Inc., New
York, 1954.
4. Care and Service of Automobile and Light Truck Tires Including Recreational Vehicle
Applications, Publication RMA ALT 9/95, Rubber Manufacturers Association,
Washington, DC, 1995.
5. Care and Service of Truck and Light Truck Tires Including Tires for Buses, Trailers
and Other Applications, Publication HTM-2-98, Rubber Manufacturers Association,
Washington, DC, 1998.
6. Demounting and Mounting Procedures for Automobile and Light Truck (LT) Tires
That Are Used on Single-Piece Rims, Wall Chart DMPAT-7/95, Rubber
Manufacturers Association, Washington, DC, 1995.
7. Demounting and Mounting Procedures for Truck/Bus Tires, Wall Chart TTMP-7/95,
Rubber Manufacturers Association, Washington, DC, 1995.
8. Multipiece Rim Matching Chart, Wall Chart RMC-7/93, Rubber Manufacturers
Association, Washington, DC, 1993.
9. Warning Zipper Rupture, Wall Chart RMA Z295, Rubber Manufacturers Association,
Washington, DC, 1995.
10. Ranney, T.A., Heydinger, G., Watson, G., Salaani, K., Mazzae, E.N., and Grygier, P.,
Investigation of Driver Reactions to Tread Separation Scenarios in the National
Advanced Driving Simulator (NADS), DOT HS 809 523, U.S. Department of
Transportation, Washington, DC, 2003.
11. Traffic Safety Facts 2003, DOT HS 809 775, U.S. Department of Transportation,
Washington, DC, 2005.
12. Shankar, U. and Longthorne, A., Analysis of FARS and Exposure Data Between 1982
and 2002; Why Are We Not Seeing Reduction in Highway Fatalities?, U.S.
Department of Transportation, Washington, DC., 2004.
13. Characteristics of Fatal Rollover Crashes, DOT HS 809 438, U.S. Department of
Transportation, Washington, DC, 2002.
14. US Tire Industry Facts, Rubber Manufacturers Association, Washington, DC, 2004.
15. Nashworthy, M., “Car Care Month: Tire Technology is on a Roll,” AAA Today, Vol.
22, No. 5, September/October 2003, p. 10.
16. “Federal Motor Vehicle Safety Standards; Tires; Final Rule; 49 CFR Part 571, Docket
No. NHTSA-03-15400, RIN 2127-AI54,” Federal Register, Vol. 68, No. 123, June 26,
2003, pp. 38115-38152.
17. Williams, J.G., Fracture Mechanics of Polymers, Ellis Horwood Limited,
Chichester, 1984.
18. Gent, A.N., “Strength of Elastomers,” Science and Technology of Rubber, Second
Edition, ed. by Mark, J.E., Erman, B., and Eirich, F.R., Academic Press, San Diego,
1994, pp. 471-512.
640 Chapter 15. Introduction to Tire Safety, Durability and Failure Analysis
19. Gehman, S.D, “Rubber Structure and Properties,” Mechanics of Pneumatic Tires, ed.
by Clark, S.K., DOT HS 805 952, U.S. Department of Transportation, Washington,
DC, 1981, pp. 1-36.
20. Rolling Resistance Measurement Procedure for Passenger Car, Light Truck, and
Highway Truck and Bus Tires, SAE J1269 Surface Vehicle Recommended Practice,
Society of Automotive Engineers, Inc., Warrendale, 2000.
21. Measurement of Passenger Car, Light Truck, and Highway Truck and Bus Tire
Rolling Resistance, SAE J1270 Surface Vehicle Recommended Practice, Society of
Automotive Engineers, Inc., Warrendale, 2000.
22. Walter, J.D. and Clark, S.K., “Cord Reinforced Rubber,” Mechanics of Pneumatic
Tires, ed. by Clark, S.K., DOT HS 805 952, U.S. Department of Transportation,
Washington, DC, 1981, pp. 123-202.
23. Grant, J.L. Rim Line Grooves as an Indicator of Underinflated or Overloaded Tire
Operation in Radial Tires, Paper 45, Presented at the International Tire Exhibition and
Conference, Akron, Ohio, September 21-23, 2004.
24. Song, T.S., Lee, J.W., and Yu, H.J. Rolling Resistance of Tires—An Analysis of Heat
Generation, SAE Technical Paper 980255, Society of Automotive Engineers, Inc.,
Warrendale, 1998.
25. Puncture Repair Procedures for Passenger and Light Truck Tires, Wall Chart PRP
PLTT-1004, Rubber Manufacturers Association, Washington, DC, 2004.
26. Puncture Repair Procedures for Truck/Bus Tires, Wall Chart PRP-TBT-1004, Rubber
Manufacturers Association, Washington, DC, 2004.
27. Herzlich, H.J., The Effect of Snaked Belt Anomalies on Tire Durability, Paper 15C,
Presented at the International Tire Exhibition and Conference, Akron, Ohio,
September 12-14, 2000.
28. Forbes, G. and Robinson, J., The Safety Impact of Vehicle-Related Road Debris, AAA
Foundation for Traffic Safety, Washingto n, D.C., June 2004.
29. French, R.W. and Gardner, J.D., The Distribution of the Inflation Forces on the
Structural Members of a Radial Tire, SAE Technical Paper 820814, Society of
Automotive Engineers, Inc., Warrendale, 1982.
30. Walter, J.D., “Cord Force Measurements in Radial Tires,” Paper No. 71-RP-A,
Journal of Engineering for Industry, American Society of Mechanical Engineers, New
York, May 1972, pp. 678-682.
31. Bolden, G.C., Smith, J.M., and Flood, T.R., “Impact Simulations II,” (Pending
Publication), 2005.
Chapter 16. Non-Destructive Tests and Inspections 641
Chapter 16
Chapter 16
Non-Destructive Tests and Inspections
by J.A. Popio and T. M. Dodson
1. Overview
In 1985 Trivisonno published a technical review of nondestructive evaluation (NDE) tech-
niques used on tires.1 He reviewed radiography, in particular x-ray; and optical methods
including holography and shearography; ultrasound; infrared; and electrical methods such
as eddy current. The same methods are used today. Although some NDE equipment has
become commercially available, the techniques and procedures often remain proprietary
and confidential.
In this section, we review some current techniques and the underlying fundamental
principles, and refer to commercially available equipment if it is available.
2. Introduction
Pneumatic tires are complex structures manufactured from many different materials using
complex processes in an attempt to meet various performance requirements. Often a tire
development or quality engineer would like to answer the question: “How did I do?” Or
an automotive engineer may want to know “How do these tires compare internally?” with-
out actually destroying the structure or tire. These questions and the desire to make con-
sistent and reproducible products have promoted the use of non-destructive evaluation
(NDE) of tires. Four commonly-used NDE techniques are X-ray, shearography, ultra-
sound, and eddy current analysis. They aid in evaluating the internal structure of the tire
without modifying or changing it. For example, the following conditions or areas of the
tire may be studied and analyzed2:
- Foreign material in the “green” or uncured tire
- Rust or corrosion of the steel belt
- Sidewall blisters
- Abnormal cord spacing or cord diameter
- Crossed and/or broken body ply cords
- Cord separation
- Steel belt placement (wandering, snaking, scalloping, necking, flaring)
- Loose wires
- Open and heavy splices
- Voids and porosity
- Bead concentricity
- Component thicknesses
3. X-ray examination
X-ray examination of tires started in the 1950s.3 It is particularly useful when there are
gross differences in the density of components, as in tires, and the technique has been
widely adopted for nondestructive evaluation. Cost savings can be realized through early
identification of tires that do not meet specified construction details or component place-
ment. This allows timely corrections to be made during the production cycle, and reduces
Chapter 16. Non-Destructive Tests and Inspections 643
X-ray
detector
X-ray
source
Figure 16.2: Measurement error caused by the conical shape of the x-ray beam.
644 Chapter 16. Non-Destructive Tests and Inspections
Panoramic tube
Conventional tube
X-ray methods are most helpful in studying steel components or other materials that
have significantly higher density than rubber compounds. For example, broken, missing,
loose or overlapping steel belt cords can readily be detected. Belt placement, belt step, for-
eign material, and significant corrosion of the steel belt filaments may also be detected.
Figure 16.4 shows a variation in belt step. The long white lines directed up and to the right
at an oblique angle are from one set of steel belts and the lines directed up and to the left
are from another steel belt package. The overlap is not symmetric about the centerline of
the tire. Figure 16.5 shows a white object, a stone that is embedded in the tread of the tire.
Figure 16.6 shows another piece of foreign material, in this case a nail. Figure 16.7 shows
variation in steel cord spacing, and Figure 16.8 shows a loose wire filament.
Figure 16.4: Variation in the belt step. One belt package is offset from
the other, whereas the belt step should have been symmetrical.
Belt step
variation
side to side
Belt step
variation
side to side
Chapter 16. Non-Destructive Tests and Inspections 645
tread
Nail in tire
Figure 16.7: A gap detected in the steel belt that runs up and to the right.
646 Chapter 16. Non-Destructive Tests and Inspections
Broken bead wires, body ply cord condition, spacing, and turn up heights are easily
detected. However it is more difficult to evaluate the condition of non-steel reinforcement
due to the similarity in densities of polyester, rayon, nylon, and rubber. The capability of
X-ray equipment to distinguish between these similar densities is commonly evaluated
using the ASTM F1035 standard.9 Figure 16.9 shows an image of the standard “Rubber
– Cord Pie Disk”.10 This disk has six different types of reinforcing materials embedded in
it: steel, nylon, polyester, fiberglass, Kevlar and rayon. The image shows the density dif-
ferences between these materials and the relative clarity and ease with which they can be
observed.
Figure 16.9: An X-ray image of the ASTM rubber - Cord pie disk that is used to
examine the discernment capability of a tire x-ray imaging system.
Steel
Rayon
Nylon
Kevlar
Polyester
Fiberglass
Chapter 16. Non-Destructive Tests and Inspections 647
Another application of x-rays is to evaluate bead fitment. Tangential x-rays are used to
study the relative placement or fitment of the bead grommet and the bead seat relative to
the wheel flange. Bead fitment is important when evaluating tire uniformity for improv-
ing vehicle ride quality. Figure 16.10 shows a schematic of a tangential x-ray device.11
The x-ray beam passes longitudinally through the tire and wheel, and the image is collect-
ed on the other side. Figure 16.11 shows the resulting radiograph. In this case a gap can
be seen between the tire bead and the rim flange.
X-ray analysis is the most firmly established and widely used of all of NDE techniques,
with different techniques being used in development and production.
Figure 16.10: The tangential x-ray technique.
X-ray
detector
X-ray
source
648 Chapter 16. Non-Destructive Tests and Inspections
Rim
flange
Air
gap
Tire
bead
4. Shearography
Shearography is an interferometric method that permits full-field observation of surface
strains in tires.12 Holography, another interferometric method, is also used. The differ-
ence between the two is that shearography measures displacement gradients directly while
holography measures actual displacements. It has been shown that it is easier to correlate
voids with variations in displacement gradient rather than in actual displacements.13 As a
result, shearography is now more popular. It can detect internal voids when corresponding
surface displacements can be excited. Voids in belts and sidewalls, corrosion damage of
steel cables, blisters between components, broken belts and cord socketing may all be
observed with this technique.
The basic shearography method entails illuminating the tire surface with a single sta-
ble-frequency light source. Laser light is the preferred choice. A baseline photograph is
then taken using a Shearography camera This camera is equipped with a shearing device,
i.e., a thin glass angle prism (shear crystal) located at the iris plane of the lens and cover-
ing one-half of the lens aperture.14 The prism allows the light rays from each point P0
(x,y) to be mapped into two points P1′ and P2′ in the image plane. Thus, the shearing
camera brings light rays scattered from one point on the object surface to meet rays scat-
tered from a neighboring point in the image plane.15 As a result the rays interfere with
each other, causing a random speckle pattern. Figure 16.12 shows two points on the
object:
P(x,y) and P(x + dx,y , y).16
Chapter 16. Non-Destructive Tests and Inspections 649
Figure 16.12: Rays from two neighboring points brought to meet in the image plane.
Figure adapted from Rubber Chem. & Tech Vol. 54.
After the speckle pattern is obtained, the tire is stressed or deformed by applying a vacu-
um, vibration, heat, mechanical force or microwave energy. (It is usual to use a vacuum.)
Another photograph is taken of the tire in the deformed state. If there is a relative displace-
ment of the points P(x,y) and P(x + dx,y , y), different from that elsewhere due to the pres-
ence of a void or other anomaly, then a change will be observed in the speckle pattern.
The shearography software compares the baseline photograph to the stressed photo-
graph, producing a fringe pattern. The fringe pattern is described by the following equa-
tion17
I = 4a 2 [1+ cos(φ + ∆ / 2) ⋅ cos(∆ / 2)]
,
where I is the intensity distribution, a is the amplitude of illumination, φ is a random phase
angle and ∆ is the phase angle related to the relative displacements. Providing the camera
and the illumination lie on the z-axis18,
4π
∆=
λ
[
W (x + δ x, y ) −W (x, y) , ]
where W is the displacement component in the z-direction, λ is the wavelength of the illu-
minating light, φ x is the separation between two neighboring points that is a function of
the shear crystal angle α and the distance, Do from the lens to the object
δ x = Do ( µ −1)α ,
where µ is the refractive index of the shear crystal. Rewriting the equation for ∆ yields
4π ⎡W (x + δx, y) −W (x, y) ⎤
∆=
λ ⎢⎣ δx ⎥⎦ δx
.
shows the crown image of a section of a tire. The image has been digitally processed and
the anomalies show up as elliptical fringes. They appear to be at the belt edges, but it is
not clear whether they are between the belts, below the belts, or elsewhere. Figure 16.14
is another processed image, but only of the sidewall. It shows some anomalies around the
turn-up area of the bead. Figure 16.15 shows other anomalies on the sidewall of a tire. To
determine exactly what they are due to would require dissection of the tire. The last exam-
ple, Figure 16.16, shows an unknown anomaly in the crown area of the tire.
Shearography is a good tool for quality control in manufacturing and it can also be used
to track degradation of the tire in durability testing. However, interpretation of the shearo-
gram is often difficult: a skilled operator is needed.19 A shearogram or hologram taken
on a new or used tire without a baseline or without subsequent shearograms taken after
additional fatigue cycles provides very limited information about the condition of the tire
and its durability.
Figure 16.13: A processed shearogram of the crown area of a tire. The image is taken
from the inside of the tire. Anomalies are indicated in the belt edge area.
Figure 16.14: A processed shearogram of part of a tire sidewall. The image is taken
from outside the tire. Potential anomalies are revealed at the turn up of the sidewall.
Anomalies
Chapter 16. Non-Destructive Tests and Inspections 651
Figure 16.15: A processed shearogram of part of a tire sidewall. The image is taken
from outside the tire and shows two defects.
Anomalies
Figure 16.16: A processed shearogram of part of the crown of a tire. The image is
taken from inside the tire and indicates an anomalous feature in the crown.
Anomaly
5. Ultrasound
Another nondestructive method is ultrasound. Although it is not widely used in the tire
industry as a whole, it is regularly used by tire retreaders, and they have developed pro-
prietary equipment and techniques. Some ultrasonic equipment is available commercial-
ly. Ultrasonic inspections are effective in evaluating abnormal cord spacing, belt anom-
alies, abnormal splices, and changes in wall thickness.
Ultrasound consists of sound waves that propagate through a medium, but the frequen-
cies used, 1 to 10 MHz, are much higher than those in the audible range, 30 Hz to 20 kHz.
The velocity V is given by
V = fλ
where f is the frequency and λ is the wavelength. For longitudinal waves, the particle dis-
placement is in the direction of wave propagation, but in solids there are also shear waves,
with particle displacement normal to the direction of wave travel, and elastic surface
waves can also occur, known as Rayleigh waves.20
652 Chapter 16. Non-Destructive Tests and Inspections
where Ka is the elastic coefficient for adiabatic volume changes and ρ is the density.
Alternatively,
1/ 2
⎛ E(1−ν ) ⎞
Vc = ⎜⎜ ⎟⎟
⎝ (1+ ν )(1− 2ν ) ρ ⎠ ,
where E is the modulus of elasticity and n is Poisson’s ratio. The velocity of shear waves
within a solid are given by21
1/ 2
⎛G⎞
Vs = ⎜⎜ ⎟⎟
⎝ρ⎠ ,
where G is the modulus of rigidity of the material. The velocity of Rayleigh waves in a
solid is about 90% of that of a shear wave, and is given by22
Vr 0.87 +1.12ν
=
Vs (1+ ν ) .
In order to generate and detect sound waves, it is usual to use piezoelectric transduc-
ers. They are typically coupled acoustically to the tire surface using a gel coupling agent
or similar material. Exciting the generating transducer with either a step voltage or a con-
tinuous wave impulse launches the sound wave, which travels through the test material
and is either reflected off the far side or is received by another transducer at that point.23
Ultrasound may be used to determine the position of ply cords or the wall thickness of
a tire. Using a sensor on the outer surface of the tire, the sound wave is reflected by either
the steel or body ply, or by the inner surface.
6. Eddy currents
Eddy current technology is based on electromagnetic induction. When an AC current
passes through a coil of wire placed in proximity to a conductive material, it produces a
magnetic field and secondary, or eddy, currents within the material.24 The interaction
between the two currents is known as impedance. A change in impedance indicates a dis-
continuity in the material or a change in thickness. Although eddy currents are used exten-
sively in the metals industry for detecting discontinuities and changes in coating thick-
ness, no details of its application in the tire industry could be found in the literature.
Nevertheless, a tire causes a known change in the magnetic field. The inductive reac-
tance versus the effective resistance is seen in Figure 16.17. Placing a tire in the magnet-
ic field and rotating it about the y-axis allows an investigator, in this case a tire retreader,
to observe changes in impedance Z, given by
Z = ( X L2 + R 2 )
,
where XL = ωL, L is the inductance of the coil and ω = 2πf, where f is the frequency of
the alternating current. This information along with the phase angle Θ between the volt-
age and current
⎛X ⎞
Θ = sin −1 ⎜ L ⎟
⎝ Z ⎠
are acquired using analog electronics, and calibrated to provide distances that correlate
with the tread thickness. Figure 16.18 shows a typical setup.25
Figure 16.17: Inductive resistance from an AC current applied to a coil. Figure
adapted from Hocking: "Eddy Current NDT".
7. Summary
Nondestructive evaluation methods provide tire engineers with insight into the internal
structure of a tire without actually destroying it. X-ray methods have been mainly used,
and appear to be the most well-developed nondestructive technique. Shearography, ultra-
sound and eddy currents provide other information. All of the methods require much skill,
training and experience to interpret the data.
Application of nondestructive evaluation to tires will probably continue to increase.
Improved equipment and computer technologies have allowed some techniques to be
automated and hold significant promise for more broad-based application. However, actu-
al destructive analysis of tires will continue to be necessary in critical cases.
654 Chapter 16. Non-Destructive Tests and Inspections
References
1. Trivisonno, N.M., Rubber Chemistry and Technology, “Non-Destructive Evaluation of
Tires”, Vol. 58, 1985.
2. Bartel, Jill. P. 3.
3. J.M.Forney, “Proceedings of the Second Symposium on Nondestructive Testing of
Tires”, P. E. J. Vogle, Ed., Sponsored by the Army Materials and Mechanics Research
Center, Watertown, MA, 1974, P. 13.
4. Neuhaus, T. G., ibid.
5. YXLON International, Modular Tire Inspection System Presentation, December 2003.
6. Raj. P., Reference 5, P. 55.
7. Neuhaus, T.G., “Review of Tire X-ray Systems 1982”, Akron Rubber Group Technical
Symposium, Akron, OH, April 22, 1982.
8. YXLON International, Modular Tire Inspection System Presentation, December 2003.
9. ASTM Standard Practice F1035, “Use of Rubber – Cord Pie Disk to Demonstrate the
Discernment Capability of a Tire X-ray Imaging System.
10. (See reference 12).
11. Schematic is courtesy of YXLON Internation, Akron, OH USA
12. Y.Y. Hung, R.M. Grant, “Sherography: A New Optical Method for Nondestructive
Evaluation of Tires”, Rubber Chem. Vol. 54, P. 1042.
13. Ibid., P. 1044.
14. Ibid., P. 1045.
15. Ibid., P. 1046.
16. Y.Y. Hung, R.M. Grant, “Sherography: A New Optical Method for Nondestructive
Evaluation of Tires”, Rubber Chem. Vol. 54, P. 1046.
17. Y.Y. Hung and C.L. Liang, Applied Optics 18, (7) 1046 (1979).
18. Ibid.
19. Department of Transportation, NHTSA, FMVSS 49 CFR Part 517, Vol. 68, No. 123,
Thursday, June 26, 2003, Rules and Regulations, P. 38143.
20. Hull, B., Vernon, J., Reference 4, P. 57.
21. Hull, B., Vernon, J., Reference 4, P 58.
22. Hull, B., Vernon, J., Reference 4, P. 58.
23. Nelligan, T. J., “An Introduction to Ultrasonic Material Analysis”, Panametrics-NDT.
24. Hull, B., Vernon, J., Reference 4, P. 33.
25. Kaman Instrumentation Corporation, Measuring Systems Group, Colorado Springs,
Colorado, USA, Application Handbook, P. 72.
Chapter 17. Tire Standards and Specifications 655
Chapter 17
by Joseph D. Walter
1. Tires and highway safety ...................................................................................... 656
Chapter 17
Tire Standards and Specifications
by Joseph D. Walter
bile related deaths – i.e., about 400-500 fatalities per year.2 There are a variety of reasons
why tires can be involved in loss of vehicle control leading to an accident, including fail
ure due to puncture, underinflation, excessive speed, overloading or insufficient tread
depth to stop a vehicle in a timely fashion on a slippery road surface. The precise num
ber of highway fatalities due to tire related causes is not well documented but can only be
inferred from accident reports issued by various police jurisdictions.
A variety of technologies have been developed to reduce or eliminate air loss as a cause
of tire disablement including self-sealing or sealant tires, run-flat tires, and airless tires.
Each approach involves trade-offs and compromises with vehicle weight, fuel economy,
1VehicleMiles Travelled
2TheNational Accident Sampling System (Crashworthiness Data) estimates 535 fatalities
(2001) while NHTSA estimates 414 (2002).
Chapter 17. Tire Standards and Specifications 657
Note that most state laws require a minimum tread depth of 2/32 in. Federal law only
requires that passenger car tires incorporate tread wear indicators during manufacture that
become exposed when tread depth is less than this amount. The corresponding European
(ECE) requirement for minimum tread depth is the equivalent of 2/32 in – i.e., 1.6 mm.
Most new passenger car tires are produced with tread depths in the range of 9/32-13/32
of an inch depending on tire size and application and are considered worn out when minimum
tread depth is reached. As tread depth is reduced due to tire wear, reductions in driving
and braking forces occur in wet, snow and muddy conditions compared to dry road per-
formance. The critical speed for the on-set of hydroplaning on rain covered highways is
similarly lowered with increasing tire wear due to the reduced drainage capacity of the
grooves, sipes(kerfs), and slots in the tread design (Chapter 11).
In addition to the federal standards and state laws, there are a variety of industry sanc-
tioned regulations pre-dating the governmental standards that must be met before tires can
be sold and safely used on the highway. These industry sanctioned specifications address
features such as tire load carrying capacity, tire dimensions, and interchangeability of tires
and wheels. These voluntary standards have been in existence almost from the advent of
the pneumatic tire. Some aspects of these industry standards are safety related and are dis
cussed in Section 17.3.
Finally, there are further government requirements intended to provide useful con
sumer information to the end user regarding the treadwear, traction and temperature resist
ance capabilities of passenger car tires and are known as Uniform Tire Quality Grade
(UTQG) labeling standards. The traction (wet road surface) and temperature resistance
(surrogate for high speed capability) grades are inherently safety related; the treadwear
index is not usually considered as a safety related measure of tire performance but tires
with tread designs and compounds having excellent wet grip, ceteris paribus, tend to com
promise wear and vice-versa, tread compounds formulated to deliver high mileage tend to
sacrifice wet skid resistance. Thus, even the treadwear grade, in some way, is often an
indirect measure of safety with respect to wet weather driving. The UTQG labeling sys
tem is discussed in Section 17.4.
Thus, the specifications and standards that are covered in this chapter:
a) indicate that minimum safety levels regarding, inter alia, high speed
performance and durability of the tire have been met;
b) assure interchangeability on a given vehicle of tires (and wheels) produced by
different manufacturers for tires having the same designation molded, stamped,
or otherwise imprinted on the sidewall;
c) provide somewhat useful information to the user regarding treadwear, traction
and temperature resistance capabilities of the tire.
None of the industry or government regulations specify the kind of material to be used in
either tires or wheels. That choice is determined by the manufacturer.
TREAD Act was responsible for the first major revisions to federal tire safety regulations
since originally issued when bias ply tires were the norm. An additional revision to
FMVSS 139 is anticipated as NHTSA is actively developing a test to assess the effect of
aging on tire performance. Bias ply tires are exempt from the provisions of FMVSS 139
and remain subject to the older regulations (FMVSS 109). The tire manufacturers certify
that their products meet the minimum federal safety levels with compliance sampling con
ducted by the government.
Current government safety standards applicable to passenger car tires are published in
the Code of Federal Regulations (CFR), Title 49 (Transportations), Part 571 (FMVSS).
The relevant portions of Part 571 are discussed in the remainder of this section.
Bead unseating
Underinflated tires in service may experience temporary or permanent bead dislodgement
from the rim – especially due to curb impact or hard cornering where, in the latter case,
appreciable lateral forces may be generated. Low aspect ratio tires (45 series and below)
are especially vulnerable to unseating since the circumferential restraining force generat
ed by the inflation pressure needed to maintain bead contact with the rim is considerably
reduced as tire section height is decreased.
The bead unseating test is a laboratory measure of the ability of the tire to maintain an
air tight seal at the tire-rim interface when subjected to a lateral force applied directly to
the tire sidewall. The non-rotating tire-wheel assembly is positioned horizontally in a
holding fixture and a contacting anvil applies the lateral force quasi-statically – i.e., 2
in/min. The tire is inflated to the maximum pressure allowed (normally 35 psi). Specific
details related to anvil geometry and test fixture configuration for different size tires and
wheels are described in the standard.
The bead unseating force must equal or exceed the values shown in Table 17.3 depend
ing on the section width of the tire.
Table 17.3: Bead unseating force requirements
Tire section width Bead Unseating Force
Less than 6 in 1500 lb
Equal to or greater than 6 in but less than 8 in 2000 lb
Equal to or greater than 8 in 2500 lb
It should be noted that, compared to the load and inflation conditions of this test, pas
senger car tires on a vehicle in highway service:
a) experience shock loads at the sidewall due to curb impact as well as cyclic
dynamic loads in the contact patch;
b) are normally inflated to pressures less than that required in the laboratory test.
Plunger strength
When FMVSS 109 was first developed, the structural robustness of the tread region of the
bias tires in use at the time tended to correlate directly with the number of plies in the tire
660 Chapter 17. Tire Standards and Specifications
and the cord spacing in each ply. More strength, load carrying capacity and tread impact
resistance could be achieved by adding more plies and using more densely packed cords
in each ply. The so-called plunger test was a measure of this strength and resistance to
road hazards.
To conduct this test, an inflated but otherwise unloaded tire is held fixed; a ¾ inch
diameter cylindrical steel shaft (the plunger) with a hemispherical head is forced into a
tread rib on or near the tire center line at a travel rate of 2 in/min.
The minimum value of plunger energy needed to meet the strength requirement (which
ranges from 1950-5200 in-lb – calculated as one-half the product of plunger force times
distance traveled) varies with tire section width and permissible inflation pressure.
Generally, radial passenger car tires contain a minimum of three plies in the tread region
(two belt plies and at least one radial body ply) and rarely fail to achieve the minimum
value of plunger energy necessary to meet the test requirements. This test is especially
moot for steel belted tires featuring nylon cap or overlay plies added to the belt region to
achieve high speed ratings. Also, very low aspect ratio tires tend to limit plunger travel
which can cause the tire tread region to come in contact with the rim (i.e., “bottom out”)
before the requisite level of calculated energy is achieved unless plunger force is allowed
to build up against the rigid surface of the rim without further plunger travel.
Tire Endurance
Mass market radial passenger car tires are normally designed to deliver about 40,000
miles of tread wear in the U.S. This mileage represents approximately 30 million revolu
tions of the tire under a wide spectrum of driving and environmental conditions – and
occurs on average over a three year period. The tire endurance test is meant to assess the
durability of the tire in a relatively short period of time under controlled laboratory con
ditions. Tire load, speed, inflation pressure and ambient temperature can be used in vari
ous combinations to control the test severity.
The tire to be checked is run continuously on a curved road wheel or drum at constant
speed with increasing loads per the schedule shown in Table 17.4.
6 hr @ 90% load
24 hr @100% load
Test duration is 34 hours with a total distance traveled of 4,080 km (2550 miles) or
5The percentages are applied to the maximum load rating marked on the tire sidewall.
Chapter 17. Tire Standards and Specifications 661
approximately two million tire revolutions. To successfully complete this test, the tire can
not show any evidence of component failure or experience any air loss. In contrast to
highway use, no cornering or camber angles are imposed on the tire during the test.
Again, there can be no evidence of component failure or air loss at the completion of
the test.
5ibid.
662 Chapter 17. Tire Standards and Specifications
Tire aging
Oxygen and ozone are the primary agents responsible for the aging and weathering of rub
ber as well as the degradation of the bond at the many thousand cord-rubber interfaces in
tires; contributing factors are heat, light, humidity and mechanical factors (stress and
strain) arising during highway service. Even unvulcanized rubber can be adversely affect
ed by these conditions during tire manufacturing operations. Aging of rubber is addressed
in Chapter 2.
In any case, the factors causing aging, alone or in combination with one another, are
responsible for the loss of physical properties of the constituent components of the tire and
may control its useful life.
Two long term trends have exacerbated the tire aging problem:
a) the number of years tires remain in service has increased considerably due to the
high mileage attained by many current products;
b) weight reductions achieved by employing thinner sidewalls, etc., have tended to
produce cooler running tires, albeit a small effect, allowing for reduced levels of
anti-oxidants and anti-ozidants.
Thus, NHTSA has indicated that a laboratory tire aging standard will be included at a
future date in FMVSS 139. This could be as simple as an “expiration date” molded on the
tire sidewall after which use would be prohibited; or it could be as complicated as a lab
oratory roadwheel test involving the multiplicity of conditions previously mentioned
involving chemical, mechanical and thermal factors.
The development of a meaningful accelerated aging test that will accurately predict
long term tire performance subject to the vagaries of customer use in different geograph
ical regions remains a formidable challenge.
As an example of a typical sidewall inscription, consider a passenger car tire for a mid
size sport sedan; its size and service description would appear, e.g., as P215/55R16 91H.
This nomenclature is deciphered as follows:
P Passenger car tire
215 Nominal section width in millimeters
55 Aspect ratio
R Radial construction
16 Rim diameter in inches
91 Load index
H Speed symbol
The service description, comprised of the load index “91” and the speed symbol “H,”
is written in code. The load index is an assigned number varying from 0 to 279 which for
U.S. passenger car tire sizes (so-called P-metric or P-type6) ranges mostly from 80 to 109
as listed in Table 17.7.
Table 17.7: Load index (LI) range for passenger car tires
LI kg lb LI kg lb LI kg lb
80 450 992 90 600 1323 100 800 1764
81 462 1019 91 615 1356 101 825 1819
82 475 1047 92 630 1389 102 850 1874
83 487 1074 93 650 1433 103 875 1929
84 500 1103 94 670 1477 104 900 1985
85 515 1136 95 690 1521 105 925 2040
86 530 1169 96 710 1566 106 950 2095
87 545 1201 97 730 1610 107 975 2150
88 560 1235 98 750 1654 108 1000 2205
89 580 1279 99 775 1709 109 1030 2277
For the example P215/55R16 91H tire, the “91”indicates a load carrying capacity of
615 kg or 1356 lb at the maximum rated inflation pressure of 35 psi.
Tires are designed to operate over a restricted range of inflation pressures with a cor
responding range of loads. There is a minimum pressure below which tires should not be
operated due to the damaging effects of heat build-up since the internal temperature of tire
components varies inversely with pressure. One of the most important responsibilities of
TRA and similar international organizations is the establishment of tire load/inflation rela
tionships. The load carrying capacity of a passenger car tire is based on a semi-empirical
formula containing tire section width, rim diameter and the square root of inflation pres
sure (see Chapter 5).
For the P215/55R16 91H tire previously considered, the tire load limits at various cold
inflation pressures are given by TRA as follow:
6 The
European or Euro-metric designation eliminates the letter "P" and the comparable sidewall inscription
would appear as: 215/55R16 91H.
664 Chapter 17. Tire Standards and Specifications
Inflation 26 29 32 35 psi
pressure 180 200 220 240 kPa
1179 1246 1312 1356 lb
Load limits
535 565 595 615 kg
The overall tire dimensions and approved rim widths are also specified by TRA. For
the example tire on a measuring rim width of 7.00 in, the design section width is 226 mm
(8.90 in); the overall diameter is 642 mm (25.28 in); and rim widths approved for use in
highway service are 6, 6½, 7 and 7½ inches.
The speed rating of a tire indicates the maximum speed at which it can safely carry a
given load at a specified inflation pressure. The speed symbol, denoted in code by a let
ter of the alphabet, is the second part of the service description (following the load index).
For most passenger car tires, speed ratings normally vary from “Q” (applicable to many
winter tires) to “Y” (reserved for high performance sport cars) as shown in Table 17.8.
For the example P215/55R16 91H tire, the “H” indicates a speed capability of 210
km/h (130 mph) if the tire is inflated to its maximum allowable pressure (35 psi) at a load
equal to or less than the maximum allowed (1356 lb). In 2004, TRA adopted an addition
al inflation adjustment (increase) based on the maximum speed capability of the vehicle.
This makes the proper selection of both the tire size and specified inflation to ensure no
overloading a more complex task for the vehicle manufacturer.
The speed rating system evolved in Europe with the commercial development of radi
al tires in the 1950s and was originally comprised of three symbols:
S designating standard tire;
H designating high speed tire;
V designating very high speed tire.
As tire manufacturers made more speed rated tires for an increasingly complex car
population, the category was expanded to cover earthmover and agricultural tires at the
lower end of the alphabet and to high performance luxury sedans and sports cars at the
upper end. U.S. tire manufacturers voluntarily adopted the load index and speed rating
symbols in the 1990s to be in conformity with European practice. The load index and
speed rating, along with the tire size, are important safety factors when purchasing
replacement tires for automobiles.
Wheel issues
The tightness of the fit at the tire-rim interface on a given wheel is controlled by the tire
design in the bead region and is governed principally by:
a) the bead winding diameter relative to the wheel diameter;
b) the stiffness of tire components between the bead and the rim;
c) the bead seat taper angle.
Thus, the interchangeability of tires and wheels presents important design challenges to
the tire development engineer. Importantly, interchangeability assures that a tire manu
factured in one country can be fit to a wheel made in another country when placed in serv
ice on a vehicle produced in yet a third country.
Tubeless passenger car tires are designed to have an interference fit with the rim so that
an air-tight seal exists at the tire-rim interface and to prevent circumferential slip of the
inflated tire on the wheel due to braking or driving torque. Another important aspect of
specifying tire dimensions is to assure the conformity or uniformity of fitment of the tire
bead region to the rim. Rim contours are specified by TRA, ETRTO and JATMA along
with other wheel dimensions. These other dimensions, rim width, wheel diameter, and
offset8 are important parameters, in conjunction with the tire, that influence vehicle ride
and handling.
8Offset is the horizontal distance from the mid-point of the wheel rim to the wheel disk.
666 Chapter 17. Tire Standards and Specifications
Treadwear
UTQG tread wear grades for radial passenger car tires typically range from 100 to over
600 in 20 point increments. Higher numbers indicate better resistance to wear. The
grades are based on tests conducted under controlled conditions on a government pre
scribed 400 mile course comprised of public roads in the vicinity of San Angelo, Texas.
A group of not more than four test vehicles travels the course in a convoy so that all tires
experience similar conditions. The same procedure is followed for a set of control or
course monitoring tires (CMTs). Tread depths of candidate tires and CMTs are measured
every 800 miles. The vehicles travel a total distance of 7,200 miles with the first 800
miles used as a “break-in” period (i.e., no tread depths are measured). After the comple
tion of both tests, wear rates are compared and the candidate tires are assigned a UTQG
tread wear grade by the manufacturer.
The wear grades are only meant to be a relative (and not absolute) indication of tread
wear performance since the tire wear rates experienced by the driver of a given vehicle
will depend on road surface abrasiveness, weather conditions, driving habits, wheel align
ment, and other factors addressed in Chapter 13.
Traction
The UTQG traction grade is a measure of the ability of the candidate tire to stop on wet
concrete and asphalt surfaces in a straight-ahead, locked-wheel braking test from 40 mph.
Grades of AA, A, B, and C are assigned depending on the following deceleration levels
(reported as traction coefficients9) experienced by the tire:
Traction grade Traction coefficient
Asphalt Concrete
AA above 0.54 above 0.38
A above 0.47 above 0.35
B above 0.38 above 0.26
C 0.38 or less 0.26 or less
Measurements are made using an instrumented towed trailer.
The traction grade assigned may not be an accurate prediction of wet road acceleration,
friction so the coefficients as reported can not be used to predict stopping distances of
vehicles equipped with anti-lock braking systems (ABS). Further, at highway driving
speeds on rain covered roadways, tire tread designs play a dominant role in evacuating
water from the contact patch – a factor not well accounted for in this test. Rubber friction
and tire traction, including the subject of hydroplaning, are addressed in Chapter 11.
9 The so-called traction coefficient is actually a braking and not a driving coefficient of friction as the word
traction might imply.
Chapter 17. Tire Standards and Specifications 667
Temperature resistance
The UTQG temperature grade is based on the sustained speed a tire can achieve without
failure. This speed is considered to be a measure of the ability of the tire to dissipate inter
nal heat build-up; that is, speed is a surrogate for temperature. Temperature resistance rat
ings are determined by testing candidate tires on an indoor road wheel (as used in
FMVSS109 and 139) under controlled laboratory conditions. Successive 30 minute runs
are made in 5 mph increments starting at 75 mph and continuing until the tire fails or is
removed at the highest speed with no failure. Temperature grades, based on the speeds
achieved without failure, are assigned per the following schedule:
Temperature Grade Speed (mph)
A over 115
B between 100 to 115
C between 85 to100
Every tire sold in the U.S. must be capable of meeting a minimum rating of “C” in con
trast to the treadwear and traction grades where absolute minimums are not prescribed.
Note that any tire that meets the FMVSS 139 high speed test requirement will, by default,
achieve at least the minimum “C” rating for temperature.
Bibliography
- Code of Federal Regulations, Title 49, Transportation:
- U.S. Government Printing Office, Washington D.C., issued annually.
a) Federal Motor Vehicle Safety Standards
i) 49 CFR 571.109 (FMVSS 109)
ii) 49 CFR 571.139 (FMVSS 139)
b) Consumer Information Regulations
- The Tire Tech Guide, Bennett Garfield, Boca Raton, Florida, issued annually.
- Traffic Safety Facts, A Compilation of Motor Vehicle Crash Data from the Fatality
Analysis Reporting System and the General Estimate System, National Highway
Traffic Safety Administration, DOT HS 809 620, Washington D.C., issued annually.
- Year Book of the Tire and Rim Association, TRA, Copley, Ohio, issued annually.
References
1. L. Evans, Traffic Safety, Science Serving Society, Bloomfield Hills, Michigan, 2004.
2. W. Blythe and D.E. Seguin, Legal Minimum Tread Depth for Passenger Car Tires in the
U.S.A., Paper submitted for publication, 2005.
Chapter 17. Tire Standards and Specifications 669
Test Questions
True False
____ ____ The number of estimated highway fatalities attributed to tires each year is
about 4,000-5,000 in the U.S.
____ ____ It is a violation of federal law to use worn out tires (tread depth less than
2/32 in) on rain covered highways.
____ ____ The Uniform Tire Quality Grade (UTGQ) labeling standards are the
principal federal regulations governing tire safety.
____ ____ Speed rated tires may be driven at their maximum rated speeds at inflation
pressures between 26 – 35 psi.
____ ____ The load limits specified for a given tire increase with decreasing inflation
pressure.
____ ____ Load and inflation pressure limits on a given size tire are controlled
directly by the National Highway Traffic Safety Administration (NHTSA).
____ ____ The plunger test is a laboratory measure of the ability of a tire to maintain
an air-tight seal at the tire-rim interface.
____ ____ The endurance, high-speed, and low pressure tests prescribed in FMVSS
139 are all conducted on previously untested tires.
____ ____ The UTGQ tread wear grade can be used by consumers to predict tire
mileage on a given vehicle.
____ ____ The service description on the tire sidewall, comprised of the load index and
speed rating, is mandated by federal law.
670 Chapter 18. Tire Materials: Recovery and Re-use
Chapter 18
by A. I. Isayev and J. S. Oh
Chapter 18
Tire Materials: Recovery and Re-use
by A. I. Isayev and J. S. Oh
1. Introduction
Manufacturing of pneumatic tires and other rubber products involves vulcanization, lead
ing to a three-dimensional chemical network. This makes the direct reprocessing and recy
cling of used tires and waste rubber impossible. Therefore, the environmental problems
caused by used tires and other waste rubber products have become serious in recent years.
In fact, Goodyear, who invented the sulfur vulcanization process more than 150 years ago,
was also the first to initiate efforts to recycle cured rubber wastes through a grinding
method.
A large number of tires are scrapped each year. According to a recent survey by the
Scrap Tire Management Council of the Rubber Manufacturers Association, approximate
ly 281 million scrap tires were generated in the United States alone in 2001. The market
for scrap tires consumed about 77.6% of that total while the rest was added to an existing
stockpile of an estimated 300 million scrap tires located around the USA
(www.rma.org/scrap_tires/scrap_tire_markets/facts_and_figures/). These stockpiled tires
create serious fire dangers and provide a breeding ground for rodents, snakes, mosquitoes,
and other pests causing health hazards and environmental problems.
The major use of scrap tires in the U.S. is to generate so-called tire-derived energy by
burning them. However, burning tires may contribute to air pollution. About 53% of the
consumed scrap tires were burnt in 2001, and only 19% of the total consumed amount
were turned into ground tire rubber (GRT), the initial material for tire rubber recycling
processes.
Waste tires, being made of high quality rubber, represent a large potential source of raw
material for the rubber industry. The main reasons for the low scale of current rubber recy
cling are: more stringent requirements for high quality of rubber articles; the substitution
of raw rubber by other materials, for example by plastics in some cases; rising cost of pro
ducing reclaim rubber from tires and rubber waste due to more stringent regulations for
environmental protection; comparatively high labor requirements for producing reclaim;
and, as a result of all this, the high cost of reclaimed rubber [1]. However, the increasing
trend to restrict landfills is inducing a the search for economical and environmentally
sound methods of recycling discarded tires and waste rubber. Recent aggressive policies
of the automotive industry are aimed to increase the usage of recycled plastic and rubber
materials. This is an example of the growing industrial demand for such technologies.
The main objective of this chapter is to provide an up-to-date account of recycling of
used tires and waste rubber, including existing methods and emerging technologies of
grinding, reclaiming and devulcanization, and also the possibility for utilizing recycled
rubber in products. Devulcanization is a process in which scrap rubber or vulcanized
waste product is converted using mechanical, thermal or chemical energy into a state in
which it can be mixed, processed and vulcanized again. Strictly speaking, devulcanization
of sulfur-cured rubber consists of cleaving, totally or partially, the sulfide crosslinks
formed during the initial vulcanization [2]. However, in the present context, it is defined
672 Chapter 18. Tire Materials: Recovery and Re-use
as a process that causes breakup of the chemical network and/or the macromolecular
chains.
A number of methods have been applied to tire rubber recycling and waste rubber uti
lization [1-5]. These methods include retreading, reclaiming, grinding, pulverization,
microwave and ultrasonic processes, pyrolysis and incineration. Processes for utilization
of recycled rubber are also being developed to manufacture rubber products and thermo
plastic/rubber blends and to use ground rubber to modify asphalt and cement.
2. Retreading of tires
Retreading is one way of recycling. Also it saves energy. It takes about 83 liters of oil to
manufacture one new truck tire whereas a retread tire requires only about 26 liters. The
cost of a retread tire can be from 30-50 % less than a new tire [3, 6]. Approximately 24.2
million retreaded tires were sold in North America in 2001, with sales totaling more than
$2 billion. Mostly medium and heavy duty truck tires, off-the-road vehicles, and aircraft
tires were retreaded [6]. However, high labor costs and tougher safety regulations may
hurt the retreading business [3].
and/or increase of surface area, for example by grinding. Presently, three methods are
used: ambient grinding, cryogenic grinding and wet-ambient grinding. Vulcanized scrap
rubber is first reduced to 2 inch by 2 inch or 1 inch by 1 inch chips. Then a magnetic sep
arator and a fiber separator (cyclone) remove steel and polyester fragments. The chips can
then be further reduced using ambient grinding or cryogenic grinding.
Another method for obtaining fine-mesh rubber is by cooling scrap tires in liquid nitro
gen below their glass transition temperature and then pulverizing the brittle material in a
hammer mill. But for inexpensive rubbers such as tire rubbers, the process is probably not
economical because of the large amount of liquid nitrogen or other cryogenic liquids
needed to freeze the rubber. However, little or no heat is generated in the process. This
results in less degradation of the rubber. In addition, almost all fiber or steel is liberated
from the rubber, resulting in a high yield of usable product.
Because of the high cost of cryogenic size reduction, chopping and grinding is often
used instead. The vulcanized rubber is sheared and ground into small particles using a
conventional high-powered rubber mill, set at close nip. With this relatively inexpensive
method it is common to produce 10 to 30 mesh material and relatively large crumb.
Multiple grinds can be used to further reduce the particle size. Ambient grinding produces
an irregularly-shaped particle with many small hair-like appendages that attach to the vir
gin rubber matrix producing an intimate well-bonded mixture. The lower particle limit for
the process is 40 mesh. The process, however, generates a significant amount of heat that
can degrade the rubber and if not cooled properly first, cause combustion on storage.
Other recycling processes include mechanical and thermo-mechanical methods, which
only comminute the rubber and similar to the above mentioned processes do not devul
canize it. A process using a wet grinding method to achieve a crumb fineness of approx
imately 200 mesh has been reported. When this product, which had a high surface-to
mass ratio, was then devulcanized, no chemicals and only minimal heating and mechani
cal processing were required. Wet or solution process grinding may yield the smallest par
ticle size, ranging from 400 to 500 mesh, allowing good processing and producing rela
tively smooth extrudates and calendered sheets.
Pulverization techniques for rubber are also being developed based on the process orig
inally proposed for plastics [7]. Polymer powder is produced using a twin-screw extruder
which imposes compression and shear on the polymer at temperatures that depend on the
polymer. Pulverization of rubber waste by solid-state shear extrusion is also proposed [8].
The obtained rubber particles were fluffy and exhibited a unique elongated shape.
Recently, pulverization of rubbers in a single screw extruder has been carried out, pro
ducing particles varied in size from 40 to 1700 µm [9]. A schematic diagram is shown in
Figures 18.1a and 18.1b. As indicated in figure 18.1a, the extruder consists of three zones:
feeding (Zone 1), compression (Zone 2) and pulverization (Zone 3). The screw is square-
pitched with the compression zone having a uniform taper to create a compression ratio
of 5. The water-cooling channel is located in the barrel in order to remove heat generated
by pulverization. Experiments showed that a significant amount of heat is generated by
friction, leading to partial degradation of the rubber. The rubber granules are fed into the
hopper of the extruder and conveyed into the compression zone. They emerge from the
pulverization zone as a powder with a small particle size. Surface oxidation of the parti
cles and agglomeration of a fraction of them may take place. The particles exhibit irregu
lar shapes with rough surfaces and a porous structure. Their crosslink density and gel frac
Chapter 18. Tire Materials: Recovery and Re-use 675
tion are reduced in comparison with the initial rubber granules. This indicates that partial
devulcanization has occurred. Due to this, the particles can be molded into products by
applying high temperature and high pressure for a period of at least one hour. Table 18.1
shows the dependence of the elongation at break, tensile strength and crosslink density of
compression-molded slabs of the original rubber compound and slabs prepared from pulver
ized particles of size in the range 250 to 425 , obtained from discarded by-products of natu
ral rubber (SMR-20) vulcanizates. The approximate composition of the rubber compound
was about 54 wt % of SMR-20, 27 wt % carbon black (SRF), 11 wt % aromatic oil and 8 wt
% vulcanizing ingredients. Molding temperature and pressure were 157oC and 5.11 MPa,
respectively. Slab F1, produced without adding sulfur curatives, exhibited the best strength
properties of slabs obtained from the rubber powder. In this sample, oil, vulcanization
residues and soluble materials were removed by toluene extraction. According to the authors,
this led to enhanced particle bonding and improved strength properties. On the other hand,
slabs F2 and F3, produced by adding sulfur curatives to the particles, showed inferior strength
properties due to less particle bonding. Furthermore, slabs F1, F2 and F3 showed inferior
strength to the original slab indicating that compression molding of the rubber particles did
not achieve the properties of the original vulcanizate.
Figure 18.1: Schematic diagram of the single screw extruder for pulverization of rub-
bers (a) and geometry of the screw channel with variable depth (b) [from Rubber
Chem. Technol., 73, 340-555, 2000].
676 Chapter 18. Tire Materials: Recovery and Re-use
Particles obtained by other grinding processes can also be compression molded into
slabs by means of high-pressure and high-temperature sintering [10]. Rubber particles
based on several elastomers, and obtained by various grinding methods, were compres
sion molded into slabs with and without addition of various acids and chemicals. The
effects of time, pressure and temperature on the mechanical properties of sintered slabs
was studied. Figure 18.2 shows the effect of molding temperature on mechanical proper
ties of a NR/SBR slab compression molded from 80 mesh particles for 1 hour at 8.5 MPa
[10]. It clearly shows the importance of the molding temperature. Below approximately
80oC, the process was unsuccessful. The highest tensile strength, about 4 MPa, was
achieved with a reasonably high breaking elongation (about 800%). The mechanism of
consolidation of particles in this process is the creation of radicals that react with other
radicals across the particle interface and thus create chemical bonds. The authors
explained the inferior properties of sintered NR rubber particle slabs in comparison to the
original as due to the presence of voids. Less energy is required to generate voids in the
sintered slabs and this does not allow strain-induced crystallization to be achieved.
Figure 18.2: Effect of molding temperature on the mechanical properties of NR/SBR
slabs compression molded from particles of 80 mesh for 1 hour at a pressure of 8.5
MPa [10].
Chapter 18. Tire Materials: Recovery and Re-use 677
sections were inserted into the barrel through two ports. Two bronze restrictors were
placed in the barrel. These restrictors forced the rubber to flow through the gap created
between the rotating screw and the tip of the horn. In the devulcanization section, the larg
er diameter provided a converging flow of the rubber to the devulcanization zone. This
feature may enhance the devulcanization process. In the grooved barrel ultrasonic reactor,
the barrel surface has two helical channels (grooved barrel). Rubber flows through the
channels and passes through the gap between the rotating shaft and the tip of the horns
where devulcanization takes place.
ULTR ASONIC
(a) FEEDER POWER
SUPPLY
WATTME TER
CONVE RTE R
EXTRUDER
TEMPERATURE
AND P RESSURE
GAUGES DIE HORN BOOSTER
(b)
(c)
Chapter 18. Tire Materials: Recovery and Re-use 679
Figure 18.4: The entrance pressure for GRT of devulcanization zone of different reac-
tors vs ultrasonic amplitude at a flow rate of 0.63 g/s, and vs flow rate at an amplitude
of 10 µm. Gap = 2 mm.
Cavitation can also occur in bubble-free liquids when the acoustic pressure amplitude
exceeds the hydrostatic pressure acting on the liquid. In the tensile half of the pressure
cycle, voids are formed at weak points in the structure of the liquid. These voids grow in
size and collapse in the same way as gas-filled bubbles.
In the case of polymer solutions, irradiation by ultrasound waves produces cavitation,
and the formation and collapse of bubbles plays an important role in causing scission of
the macromolecules. In solid elastomers, intrinsic microvoids are thought to be responsi
ble for cavitation when the material is subjected to a sudden hydrostatic tension, or a sud
den depressurization.
Studies of ultrasonically treated rubber show that the breakup of chemical crosslinks is
accompanied by partial degradation of the rubber molecules. But the mechanism of devul
canization is not well understood, unlike the degradation of long-chain polymers in solu
tions. In particular, the way in which ultrasonic energy is converted into chemical ener
gy is not clear. The proposed models of devulcanization [13, 14] are based upon break
down of the rubber network caused by cavitation created by high-intensity ultrasonic
waves in the presence of pressure and heat.
Under some devulcanization conditions, the tensile strength of unfilled revulcanized
SBR was found to be much higher than that of the original vulcanizate with the elonga
tion at break being practically unchanged. Figure 18.6 shows the stress-strain curves of
unfilled virgin vulcanizates and revulcanized SBR obtained from rubber devulcanized at
various ultrasonic amplitudes, A. The devulcanized samples were made with the coaxial
ultrasonic reactor depicted in Figure 18.3 (a) at a barrel temperature of 120oC, a screw
speed of 20 rpm, and a flow rate of 0.63 g/s. The improvement in mechanical properties
is attributed to the presence of a bimodal network in the revulcanized rubber. Superior
properties of revulcanized rubber were also observed in unfilled EPDM and silicone rub
bers. Unfilled revulcanized NR rubber also showed good properties, with the elongation
at break remaining similar to that of the original vulcanizate, but the ultimate strength was
reduced to about 70 % of the original. Interestingly, revulcanized NR showed the charac
teristic strain-induced crystallization of NR, as indicated in Figure 18.7 by the upturn in
the stress-strain curves for both the original and the revulcanized material.
It is of interest to establish what role a filler plays in the devulcanization process.
Figure 18.8 shows stress-strain curves for vulcanizates of virgin and devulcanized NR
containing 35 phr of carbon black. The original vulcanizates were cured using 5 phr ZnO,
1 phr stearic acid, 1 phr CBS, 2 phr sulfur. The revulcanization recipe contained 2.5 phr
ZnO, 0.5 phr stearic acid, 0.5 CBS and 2 phr sulfur. The mechanical properties of revul
canized rubbers typically deteriorate depending on the devulcanization conditions. This is
evident in Figure 18.8. It is thought that ultrasonic devulcanization causes a partial deac
tivation of filler due to breakup of macromolecular chains attached to the surface of car
bon black. When devulcanized rubber was blended with virgin rubber, the vulcanizates
showed significantly improved properties. Also, vulcanizates containing fresh carbon
black exhibited better properties. However, revulcanized samples prepared from EPDM
roofing membrane material containing carbon black and a significant amount of oil
showed mechanical properties similar to or better than the original rubber. Possibly, the
oil prevents deactivation of the filler during devulcanization.
682 Chapter 18. Tire Materials: Recovery and Re-use
Figure 18.6: Stress-strain curves of unfilled virgin vulcanizates and revulcanized SBR
obtained from rubbers devulcanized in a coaxial reactor at various ultrasonic ampli-
tudes.
A, µm
2.0 virgin 1
5
7.5
10
1.5
Stress, MPa
virgin 2
1.0
Stress, MPa
0.5
0.0
0 50 100 150 200 250
Strain, %
Q,g/s A,µm
20 0.32 5 1
0.32 7.5
0.32 10
0.63 5
16 2
0.63 7.5
0.63 10
1.26 7.5
Stress, MPa
12 1.26 10
2.52 10
Stress, MPa
Strain, %
Figure 18.8: Stress-strain curves for 35 phr carbon black filled virgin NR and revul-
canized NR devulcanized in a coaxial reactor at a flow rate of 0.63 g/s and various
ultrasonic amplitudes. Gap = 2.54 mm, barrel temperature = 120°C.
35
virgin NR
30 5 µm
7.5 µm
10 µm
25
Stress, MPa
20
15
10
0
0 100 200 300 400 500 600
Strain, %
Ultrasonic devulcanization also alters the kinetics of revulcanization. For SBR the
induction period was shorter or absent. This was also the case for other unfilled and car
bon black filled rubbers, such as GRT, SBR, NR, EPDM and BR cured by sulfur-contain
ing curative systems, but not for silicone rubber cured by peroxide. The decrease or dis
appearance of the induction period for sulfur-cured rubbers is attributed to a crosslinking
reaction between rubber molecules chemically modified in the course of devulcanization
and unmodified rubber molecules. Note that approximately 85% of the accelerator
remained in the devulcanized SBR rubber.
Ultrasonically devulcanized rubber consists of sol and gel fractions, where the gel por
tion is typically soft and has a lower crosslink density than the original vulcanizate.
Crosslink density and the gel fraction were found to be correlated by a master curve that
is unique to the particular elastomer. Figure 18.9 shows the normalized gel fraction as a
function of the normalized crosslink density for devulcanized GRT, obtained from three
different reactors. The dependence of gel fraction on crosslink density was described by a
single curve, independent of processing conditions. The curve for the barrel and grooved
barrel reactors was shifted toward lower crosslink density indicating a higher efficiency
of devulcanization, possibly due to an additional shearing effect.
684 Chapter 18. Tire Materials: Recovery and Re-use
Figure 18.9: Normalized gel fraction vs. normalized crosslink density for GRT devul-
canized under various conditions using coaxial, barrel and grooved barrel reactors.
1.0
0.9
Normalized gel fraction
0.8
0.7
0.6
0.5
Coaxial reactor
0.4 Barrel reactor
Grooved barrel reactor
0.3
0.0 0.2 0.4 0.6 0.8 1.0
ment in the impact energy for linear low density polyethylene (LLDPE) composites was
greater than that for the corresponding high density polyethylene (HDPE) composites. It
was also suggested that the low polarity and/or low crystallinity of the matrix polymer
appeared to favor compatibility with GRT.
Modification of the surface of GRT particles has been studied to improve the compat
ibility of GRT and polymer. Surface modification can be carried out by chemical treat
ments, for example, chromic acid etching, thermal oxidation or by mechanical means.
Maleic anhydride-grafted and chlorinated GRT improved the physical properties of
GRT/EPDM/acrylated high-density polyethylene and GRT/polyvinyl chloride blends. It
was found that surface treatment of ground rubber with a mixture of unsaturated curable
polymer and a curing agent could also improve the performance of the blends. The effect
of cryogenically ground rubber (CGR, approximately 250 µm) from old tires on some
mechanical properties of an unsaturated polyester resin was investigated. Composites
made from silane-treated ground rubber showed better mechanical properties than a com
posite made from untreated CGR. However, the particle size of the ground rubber was
apparently too large to produce a toughening effect.
High-energy treatments including plasma, corona discharge and electron-beam radia
tion were used to modify the surface of GRT. Oxidation of the surface, such as occurs in
plasma, and autoclaving in an oxygen atmosphere were shown to improve adhesion
between GRT and polyamide. An epoxy resin compounded with tire rubber particles mod
ified by plasma surface treatment was also studied. An improvement in mechanical prop
erties of the resulting material over those containing the untreated rubber was observed.
The effects of corona discharge treatment of GRT on the impact property of a thermoplas
tic composite containing GRT were investigated. X-ray photoelectron spectroscopy
analysis showed that the treatment increased the amount of oxygen-containing groups on
the rubber surface. For some composites it has been found that treated GRT marginally
improved the impact property. However, prolonged times of treatment and higher power
inputs for corona discharge reduced the impact strength.
A phenolic resin cure system and maleic anhydride grafted polypropylene (PP) com
patibilizer significantly improved the mechanical properties of a blend of PP and ultrason
ically devulcanized GRT, prepared by dynamic vulcanization. Also, improvement in pro
cessing efficiency and better properties of PP/GRT blends were achieved when ultrason
ic treatment was carried out during extrusion in a more efficient reactor [17]. Mechanical
properties of PP/GRT, PP/devulcanized GRT (DGRT) and PP/revulcanized GRT (RGRT),
mixed in a 40/60 proportion, are shown in Table 18.3. The tensile strength, Young’s mod
ulus and elongation at break of the blend prepared in this way are higher than those
obtained earlier. Also, properties of PP/RGRT at 10µm are higher than those of PP/GRT.
Evidently, the ultrasonic treatment of PP/GRT blends led to increased compatibilization
between the plastic and rubber phases. This was attributed to mechanochemical reactions
induced by ultrasound. In addition, ultrasonically devulcanized GRT was blended with
HDPE using a Brabender internal mixer and a twin-screw extruder. These blends were
dynamically vulcanized in the mixers. Also, HDPE and GRT blends mixed in a twin-
screw extruder were then passed through an ultrasonic devulcanization extruder and sub
sequently dynamically vulcanized by means of an internal mixer and twin-screw extrud
er. The blends mixed by using the twin-screw extruder prior to devulcanization were
found to have better tensile properties and impact strength than any other blends.
688 Chapter 18. Tire Materials: Recovery and Re-use
Table 18.3: Mechanical properties of PP/GRT, PP/DGRT and PP/RGRT blends [17]
Blend Tensile strength Young’s modulus, Elongation at
MPa MPa break, %
PP/GRT 6.7 116 16.6
PP/DGRT 2 horns, 5µm 6.6 98 21.5
2 horns, 7.5µm 6.5 100 20.6
2 horns, 10µm 5.15 102 7.6
1 horn, 5µm 6.9 110 19.3
1 horn, 7.5µm 6.6 104 17.4
1 horn, 10µm 7.0 116 21.0
earlier work 5.2 108 20.7
Novel blends of GRT and recycled HDPE from used milk containers have been stud
ied. Effects of GRT particle size and concentration on mechanical and rheological prop
erties were determined. The blend systems were optimized by using a soft rubber-plastic
binder produced from a mixture of HDPE and EPDM, wherein EPDM is dynamically vul
canized during its mixing with the HDPE. It was concluded that the softening of the
HDPE binder provides compositions of improved ultimate mechanical properties.
adhesion. Several surface modifications are also proposed including treatment of rubber
with sulfuric acid and nitric acid to chemically oxidize the rubber and introduce polar
groups. Contrary to expectation, treatment with nitric acid led to a decrease of the strength
of the composite. On the other hand, treatment with sulfuric acid improved the adhesion
of rubber to concrete. Using a combination of chemical and surface probing techniques it
was shown that the hydrophilicity of the rubber surface is greatly improved by acid or
base treatment. Typically, the rubber surface is hydrophobic because rubber usually con
tains zinc stearate that diffuses to the surface. By acid treatment the zinc stearate can be
hydrolyzed to stearic acid. Treating the rubber with base, the zinc ions are converted into
sodium ions creating the soluble sodium stearate.
It was also found that addition of rubber particles to mortar led to a decrease in com
pressive and flexural strength. However, treatment of the particles before mixing with a
bifunctional silane-coupling agent, such as gamma mercapto trimethoxy silane, improved
the interface and led to increased ductility. This research should be expanded to include
the effects of the type of coupling agent on the adhesion and the fracture behavior of rub-
ber-filled cement paste, mortar and concrete, used in highway pavement overlays, side
walks, medians, and sound barriers.
modify the asphalt binder. Thus, the blending process of GRT and asphalt before prepar
ing a mixture is most efficient in improving properties. Typical use levels range from 15
to 30 wt %.
A limited amount of work has been done on characterizing blends of GRT and asphalt.
Blends are typically mixed at temperatures of 300 to 400oC for a period of 0.5 to 2 hours.
The mix increases in viscosity and has the consistency of a slurry with discernable rubber
particles spread throughout. At room temperature the resulting composition is a tough rub
bery-elastic material. The mixing period is often referred to as reaction or digestion time.
The elastic quality of the blend is attributed to undissolved rubber particles acting as an
elastic component within asphalt, which is modified by the fraction of the rubber particles
that have dissolved. Longer mixing times, 2 hours, compared to 0.5 or 1 hour, significant
ly improve elastic recovery and reduce the amount of solid rubber in the mixture. The
addition of GRT at 10, 20 and 30 wt % levels significantly increases the softening point
and strain recovery of asphalt, with a viscosity increase of similar magnitude for all three
blends.
Rheological properties are affected by asphalt composition, particle size and amount of
dissolved rubber, and temperature. By controlling these variables binders with improved
resistance to cracking and rutting can be produced. Finally, scrap tires, used as a crumb
rubber modifier for asphalt, improve paving performance and safety and are an excellent
and cost effective material for highway pavements.
tion. However, these attempts proved to be economically unsuccessful due to the low
price of crude oil at that time. Also, pyrolysis plants are thought to produce toxic waste
as a byproduct of operation. However, significant research has been carried out recently
and various new pyrolysis processes have been developed. Despite this progress, pyrol
ysis of scrap tires is still performed on a quite limited scale, mainly due to the absence of
a large market for oil and carbon black produced in this way.
6. Concluding Remarks
Waste tires are an important problem of international significance. This chapter describes
some routes available to solve this problem. Many technologies are being developed.
Among them, in addition to the well-known grinding techniques, are continuous pulver
ization methods based on single or twin-screw extruders that may serve as a possible
means of supplying rubber powder as a feedstock for various present and future devulcan
ization and recycling technologies. These include reclaiming, surface treatment, ultrason
ic devulcanization and utilization of rubber particles for making composites with other
materials. Ultrasonic technology is considered a promising method to produce devulcan
ized rubber suitable for making products from 100% recycled rubber and for adding to vir
gin rubber, virgin and recycled plastics, asphalt, concrete and cement. High-output equip
ment is needed to make the process economically acceptable. Clearly, there is a need for
better understanding of various recycling processes. Development of the science of rub
ber recycling, novel processes for recycling, and novel materials using recycled rubber,
would significantly reduce energy consumption and reduce the problem of disposing of
scrap tires and rubber waste.
Acknowledgment
This work was supported in part by grant DMI-0084740 from the National Science
Foundation.
References
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16. Naskar, A. K., Pramanik, P. K., Mukhopadhyay, R., De, S. K., and Bhowmick, A. K.,
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Index 693
Index
Abradability, 68, 69, 72, 73, 254, 277-279, Bead breaks, 100, 612, 638
537, 538, 541, 546, Bead hang-up, 614
Abrasion, 8, 13, 28, 38, 41, 68-73, 241, Bead seat pressure, 224
278, 282, 422, 425, 440, 446, 448, 455, Bead unseating, 658
463, 473, 504, 533-593 Bead wire, 6, 22, 80, 100, 638, 646, 684
Abrasion tester, Akron, 564 Belted bias tire, i, 2, 80, 86, 228, 314
Abrasion tester, laboratory (LAT 100), 440, Bias angle, 208, 210
446, 463, 467, 562, 563, 577, 583 Bias-ply tire, i, 80, 86, 90, 206, 224, 408,
Abrasion tester, Lambourn , 564 416, 560, 580
Abrasion, chemical effects in, 72, 551 Blow-out, 630
Ackermann steering geometry, 596, 597 Body ply, 2, 4, 6, 8, 10, 20, 22, 24, 218,
Aerodynamic drag, 475, 477, 515-519, 523, 372, 488, 500, 636, 642, 646, 652, 660
525 Boom, 364, 396, 398, 404, 406
Aerodynamic noise, 364, 378 Boundary Element Methods (BEM), 364,
Air loss, 603, 605, 612, 614, 630, 656, 661, 392
662, Braking and driving forces, 304, 459, 609,
Air pumping, 364, 378, 379, 389, 406 657
Airborne noise, 366, 398, Braking coefficient, 338, 420, 434, 436,
Akasaka-Hirano equations, 131 452, 454, 456, 458, 460, 470
Akron abrasion tester, 440, 463, 562, 563, Braking test procedures, 420, 452
583 Braking torque, 226, 270, 284, 290, 296,
Aligning moment (torque), 316 322, 378, 496
Aligning torque, 215, 291, 315-318, 329, Brush model, 346, 352, 358, 444, 450, 452,
336, 337, 348, 355, 479, 559-562 458, 460, 556, 558, 560, 562, 564,
Anisotropy, 107, 149, 413 584
Antidegradents, 6 Burst pressure, 212, 220
Anti-lock braking system (ABS), 324, Butyl rubber, 30, 32, 44, 52, 74, 426, 552,
332, 666, 667 632
Aquaplaning, 435-437, 458
Aramid, 7, 21, 80-82, 85-89, 91, 93-95, Camber, 152, 214, 216, 224, 226, 256, 290,
103, 126, 174, 175 316, 328, 356, 478, 480, 498, 514, 598,
Aspect ratio, 4, 20, 190-194, 196, 197, 200, 602, 604, 660
201, 229, 307-311, 313, 321, 384, 414, Camber angle, 226, 256, 290, 480, 602,
485, 502, 531, 604, 659, 660, 662 604, 660
Axis system, footprint (ISO), 231, 234, 235, Cap ply, 10, 22, 24, 164, 492, 500, 502, 624
243, 283, 286, 289, 290, 292, 302, Carbon black, 6, 20, 28, 30, 38, 54, 106,
304, 307, 313, 324, 359 426, 428, 440, 468, 504, 506, 540, 548,
Axis system, tire (SAE), 231, 234, 235, 564, 574, 670, 674, 678, 680, 682, 684,
243, 283, 286, 289, 290, 292, 302, 304, 690
307, 313, 324, 359 Clark equations, 130
Coastdown, 494, 518, 520, 522, 524, 530,
Bead, 376, 377, 411-420, 475, 483, 486- 532
489, 509, 604, 612, 614, 632, 635, 638, Compliance matrix, 110, 112, 114, 116
639, 642, 548, 650, 658, 659, 665, 684 Conicity, 310, 316, 318, 348, 360, 598
694 Index
Contact patch, ii, 18, 64, 72, 214-217, 222- Diagonal waves, 411, 413
232, 281, 282, 293, 308, 309, 374, 410, Drift, 16, 316, 355
411, 415, 416, 422, 435, 483-488, 493, Driving torque, 270, 285, 291, 296, 302,
495, 497, 498, 502, 513, 514, 557, 591, 322, 497, 655
619, 624, 625, 626, 659, 666 Durability, 15, 36, 41, 84, 86, 93, 100, 101,
Cord angle, 108, 118, 119, 137, 146, 150, 105, 123, 151, 161, 168, 169, 175, 177,
154-156, 169, 172, 207, 209, 228, 230 180, 196, 248, 391, 556, 575, 604, 608,
Cord dipping, 93 613-620, 624, 625, 640, 650, 658, 660
Cord load, 157, 207, 213, 214, 224, 229,
263, 265, 275, 637 Eddy currents, for NDE, 652, 653
Cord material, aramid, 87, 91, 94, 175 Endurance, 2, 5, 16, 17, 19, 233, 298, 658,
Cord material, nylon, 83, 150, 164 660, 661, 669, 679
Cord material, polyester, 81, 85, 87, 92, Energy dissipation, 40-42, 45, 52, 55, 62,
623, 626, 627, 635 66, 67, 77, 180, 254, 480-508, 512, 513,
Cord material, polyethylene naphthalate 538, 551, 561, 567, 570, 571, 583-586
(PEN), 88 Energy loss, 55, 73, 161, 162, 168, 185,
Cord material, rayon, 83, 91, 94, 95 214, 410-411, 429, 472, 477, 478, 482-
Cord material, steel, 95-100, 104, 175, 484, 490, 496, 502, 503, 510, 512, 513,
491, 626, 628, 632, 633, 636, 637, 644 515, 530, 543, 561, 619
Cord pull-out, 174
Cord tension, 200, 208, 209, 217-220, 222, Failure analysis, 613
224, 228-230, 266, 408, 413, 415, 419 Failure rates, 613
Cord twist, 85, 89, 104 Fatigue, 8, 9, 47, 57-61, 70-72, 75, 84-99,
Cord-rubber laminates, 63, 154, 183 149, 151, 169-181, 316, 538, 550, 566,
Cornering coefficient, 599 614, 618, 626, 628, 650
Cornering force, 152, 153, 215, 357, 379, Fatigue failure, 60, 93, 175, 566
465, 556, 581, 595, 598-608 Federal Motor Vehicle Safety Standards
Cornering force machines, 152, 153, 215, (FMVSS), 191, 657-662, 667, 668
379, 465, 556, 581, 595, 598-608 Fiber, denier, 10, 21, 83, 85, 89, 90, 91, 96
Cornering stiffness, 305-307, 311, Fiber, tenacity, 82-90, 96
335, 336, 339-342, 347, 348, 363, 443- Fillers, reinforcement by, 6, 20, 30, 32, 38,
445, 498, 557-562, 586, 588, 595, 598- 40, 45, 55, 56, 57, 473, 503, 504, 592,
604, 608, 610 673, 685, 690
Corporate average fuel economy, 17, 476 Finite element methods (FEA), 106, 108,
Crack growth, 28, 48, 57-60, 70-72, 169, 109, 124, 159, 167, 168, 177, 181, 207,
171-178, 543, 550, 566, 569, 626 210, 211, 229, 239, 287, 344, 345, 391,
Creep, 91, 158, 161, 164-166, 170, 171, 393, 398, 415, 475, 480, 483, 509-512,
178, 184, 210 531, 620, 621
Critical velocity, 410, 411, 413-415, 418 Flex fatigue, 8, 628
Cross-ply tire, 408, 413, 414 Footprint, 2, 3, 9, 10, 152-155, 169, 192,
Curatives, 6, 20, 25, 29, 675 193, 226, 227, 229, 233-283, 293-296,
Debonding, 121, 170 309, 313, 317, 319, 323, 334-336, 345,
Delamination, 170, 171, 173, 176-178, 181, 346, 368, 372, 379, 383, 389, 392, 407,
182 595, 619, 620
Denier, 10, 21, 83, 85, 89-91, 96 Footprint displacements, 237
Devulcanization, 671-673, 675, 677, 678, Footprint physics, 233, 261
679, 680, 681, 683, 684, 687, 691 Footprint pressure, 226, 227, 250
Index 695
Footprint stress, 235, 239, 250, 256, 261, Hydroplaning, 3, 10, 16, 239, 283, 319,
265, 268, 272, 274, 283, 323, 335 345, 608, 657, 666
Footprint temperatures, 254 Hysteresis, 9, 84, 161, 162, 169, 254, 409,
Footprint width, 192, 193 425, 477, 480-484, 490, 495, 503, 504,
Friction on ice, 437, 439, 440, 472 507, 509, 513, 626
Friction, coefficient of, 29, 63, 64, 66, 215,
216, 254, 283, 307, 323, 343, 347, 423, Impact damage, 97, 635
428-433, 438, 439, 444, 450, 606, 607, Inclination angle, 252, 257, 258, 262-264,
666 270, 272, 274, 291, 292, 295, 296, 298,
Friction, rolling, 64, 65, 78, 477 301, 304, 311-315, 321, 324, 328, 329,
Friction, sliding, 65, 66, 68, 555 334, 351, 359
Fuel consumption, 214, 476, 524-529 Innerliner, 7, 19, 23, 632, 633, 637
Fuel economy, 6, 9, 17, 26, 476, 498, 520, Interlaminar shear, 154-158, 169, 176, 179
524, 527-529, 607, 656 Intra-carcass pressurization, 618
Irregular wear, 15, 617
Glass plate photography, 238, 239
Glass transition temperature, 43, 44, 87, 88, Laboratory abrasion tester, 562
93, 423, 426, 438-440, 442, 443, 446, Lambourn abrasion tester, 564
447, 449, 468, 472, 548, 674 Laminate, 62, 63, 108, 109, 137-158, 161,
Government regulations, 5, 658 164, 169, 174, 179, 181, 182, 248
Groove wander, 16, 315, 318-321 Lateral force, 18, 152, 217, 225, 226, 258,
Grooves, tread, 389 274, 275, 288, 290, 291, 293, 295, 299,
Ground scrap rubber (GRT), 671, 679, 680, 304-316, 320, 321, 324-329, 332, 335-
683-690 337, 340, 344, 346, 348, 349, 352, 355,
Group velocity, wave, 410, 412 359, 360, 363, 374, 479, 498, 556, 598-
600, 606, 626, 659
Halpin-Tsai equations, 130, 133 Load formula, 187-192, 196, 197, 200, 201
Handling, 2, 4, 6-16, 26, 27, 62, 83, 84, 87, Load index, 4, 663-665, 667, 669
95, 161, 169, 220, 227, 254, 288, 306, Load limits, 187, 663, 664, 669
330, 334, 335, 363, 381, 382, 595-610, Load rating, 4, 187, 188, 264, 658, 660
614, 618, 668 Loss modulus E″, 542
Harshness, ride, 16, 152, 168, 397, 405 Loss tangent, tan δ, 181, 482, 492
Heat conductivity, 123, 448, 544, 569, 570, Low pressure, 17, 188, 383, 658, 661, 669
688 Lubricated sliding, 66, 434
Heat dissipation, 477, 513
Heat generation, 41, 84, 86, 87, 480, 492, Magic formula, 348, 349, 351, 353
499, 510, 511, 513, 621, 625, 631 Master curve, friction, 424-431, 438, 440,
Heat transfer, 123, 124, 133, 136, 180, 442, 443, 447, 450, 452, 456, 472
181, 254, 259, 261, 470, 471, 477, 480, Match mounting of tires, 377
499, 505, 507-509, 513 Minimum tread depths, 657
Modal analysis, 215, 364, 391, 392
High speed, 4, 5, 10, 15, 17, 19, 68, 80, 84, Modal analysis, mode shape, 215, 370, 391,
86-88, 187, 207, 210, 298, 299, 319, 392
373, 374, 409, 419, 420, 449, 489, 494, Moire fringes, 239, 250
502, 509, 587, 608, 618, 625, 656, 658, Mold contour, 10, 13
660, 661, 664, 667, 669 Mooney-Rivlin strain energy function, 180,
511
696 Index
Mullins’ effect (in rubber), 38, 158, 170 Pressure monitoring systems, 191, 657, 661
Pull, 16, 25, 62, 63, 174, 175, 177, 219,
Natural rubber, 30-32, 55, 58-60, 71, 72, 94, 222, 262, 304, 311, 315, 316, 318, 324,
535, 539, 550, 675, 686 331, 455, 490, 605, 636, 688
Nibbling , 315 Pulverization of rubber, 674, 675
Noise, 3, 9, 10, 12, 16, 168, 233, 256, 319, Puncturing, 618, 632
366, 367, 373, 378-385, 388, 389, 393, Purdy equation, 208
394, 397-407, 457, 458, 689 Pyrolysis of tires, 672, 690, 691
Noise, air pumping, 378, 379, 389, 406
Noise, airborne, 366, 398 Quality control, 23, 26, 94, 638, 650
Noise, boom, 405, 406
Noise, cavity resonances, 373, 383 Radial force variation, 374-377, 396
Noise, due to stick/slip, 379 Rayon, 7, 21, 80-95, 104, 126, 133, 646
Noise, harmonics, 374-376, 380, 382, 398, Reclaimed rubber , 671
406 Reclaiming, 671, 672, 691
Noise, organ pipe, 378, 388 Reinforcement by fillers, 75
Noise, road roar, 397, 405 Relaxation length, 330-332, 335, 356, 361
Non-destructive testing (NDE), 642 Relaxation, of rubber, 91-94, 161-164, 167,
Non-uniformities, 373 265, 266, 286, 323, 330, 331, 332, 335,
Nylon, 6, 7, 10, 21, 22, 24, 80-83, 85-95, 356, 361, 424, 426, 472, 500, 543
126, 127, 150, 158, 170, 211, 414, 493, Repair of tires, 26, 614, 616, 617, 632-634
502, 646, 660 Residual aligning torque, 317, 318
Resonant frequency, 39, 370, 371, 375, 395,
Obstacle envelopment, 152 410
Operation, conditions of, 258, 392, 499
Oversteer, 598, 601-606, 609 Resorcinol-formaldehyde-latex cord
Overturning moment, 290, 293, 294, 304, adhesive (RFL), 94
307-309, 312-315, 321, 329, 334, 350, Retreading, 672
373, 479, 498 Reversion of rubber, 75, 178, 621
Oxidative degradation, 72, 544, 554, 568 Ride, 2, 8-10, 13, 14, 16, 18, 41, 81, 84, 87,
Ozone, cracking due to, 74 215, 233, 245, 255, 366, 372, 376, 377,
383, 386, 390, 391, 395-397, 399-407,
Parallel steering geometry, 596 595, 604, 613, 616, 618, 647, 657, 665,
Payne effect, 39, 40, 504 673, 687, 688
Phase velocity, wave, 410, 412, 420 Rim contour, 665
Pitch length, of tread elements, 10, 16, 379- Rim diameter, 4, 19, 187-191, 195-197,
382, 410 220, 228, 663
Pitch ratio, 379, 380,382 Road hazard, 5, 19, 612, 632, 635, 660
Pitch sequence, of tread elements, 10, 380- Road roar, 397, 405
382, 403, 407 Road test ratings, 462-473, 574-577, 587,
Plunger strength, 659 588, 589
Plysteer, 152, 265, 311, 317, 318, 348, 355, Rolling loss, 477, 510
360 Rolling radius, 265, 267, 269, 273, 295,
Pneumatic trail, 272, 293, 336 316, 324, 519
Poisson’s ratio, 32, 34, 109, 111, 113, 114, Rolling resistance, 4, 9, 13-18, 25, 68, 77,
120, 127, 128, 130, 131, 180, 652 161, 162, 168, 169, 214, 262, 288, 290,
Post-cure inflation, 92, 93 291, 296, 455, 472, 476-529, 581, 583,
Index 697
Thermal conductivity, 106, 123, 133, 134, Tire slip, 583, 599, 600
136, 181, 513, 568 Tire specifications, 85, 304, 318, 344, 518,
Thermal degradation of rubber, 448, 455, 662
554 Tire stress analysis, 206,
Thermal expansion, 33, 44, 124-126, 490 Tire surface temperature, 577-579
Thermo-oxidative degradation of rubber, Tire textiles, 80, 84, 85, 88
568 Tire transients, 330, 334, 335, 346, 347, 519
Time domain model, 415-419 Tire uniformity, 18, 91, 92, 373, 396, 398,
Tire aging, 338, 339, 662 647
Tire and Rim Association, Inc. (TRA), Tire velocity, 413, 418
187, 201, 662-665, 667 Tire vibrations, 367, 369
Tire aspect ratio, 307, 309, 384 Tire, radial, components, 6, 7, 38, 39, 75,
Tire beads, 100, 638 85, 624, 663, 665, 685
Tire break-in, 340 Tire, types, 3
Tire cavity resonance, 373, 383 Tire/wheel mismatch, 614
Tire components, 6, 7, 38, 39, 75, 85, 624, Tire-derived fuel, 691
663, 665, 685 Tire-pavement noise, 233
Tire defects, 637 Tire-rim interface, 659, 665, 669
Tire deformation, 292, 334, 489 Tires, incineration of, 690
Tire design process, 10 Toe angle, 498
Tire diameter, 25, 196, 297 Torque steer, 32, 324
Tire durability, 123, 169, 196, 391, 608, Traction, 2, 3, 5, 6, 9, 10, 12, 14, 19, 36,
613, 617, 618, 620, 625 215, 233, 272, 318, 319, 321, 324, 345,
Tire failure, 17, 27, 613, 615-617, 628, 631, 356, 381, 382, 422, 423, 434, 440, 444,
632, 635, 656, 657 446-448, 453, 458-461, 463, 467, 468,
Tire forces and moments, 235, 270, 272, 470, 472, 504, 555, 556, 560, 604, 618,
289, 290, 292, 301, 330, 335, 339, 341, 658, 665-667
344, 345 Traction control systems, 324
Tire harshness, 16, 152, 168, 397, 405 Transient effects, 330, 334, 346, 499, 508
Tire imbalance, 373 Transverse isotropy, 257
Tire inflation pressure, 15, 27, 188, 192, Traveling waves, 657
485 TREAD Act, 5, 659
Tire load, 4, 27, 187-189, 191, 196, 201, Tread depth, minimum, 657, 668
205, 265, 266, 291, 301, 371, 384, 455, Tread pattern, 10, 12, 16, 25, 215-217, 226,
480, 488, 516, 585, 588, 589, 604, 610, 234, 242, 243, 259-262, 272, 274, 278,
617, 657, 658, 660, 663 284, 304, 319-321, 341, 373, 378-380,
Tire manufacturing, 4, 20, 22-26, 662, 685 388-391, 406, 435-437, 441, 457, 458,
Tire mounting, 19 461, 472, 485, 492, 500, 509, 577, 580,
Tire retreading, 672 585, 586, 589
Tire-road friction, 595, 608 Tread rubber, grinding of, 26, 377
Tire shape, 84, 190, 207 Ultrasound, for NDE, 642, 651-653, 677,
Index 699
February 2006