Ch2-

Mechanical
Properties
 INTRODUCTION
 Many materials, when in service, are subjected to forces or
loads; examples include the aluminum alloy from which an
airplane wing is constructed and the steel in an automobile
axle.
 In such situations, it is necessary to know the
characteristics of the material and to design the member
from which it is made such that any resulting deformation
will not be excessive and fracture will not occur.
 The mechanical behavior of a material reflects the
relationship between its response or deformation to an
applied load or force. Important mechanical properties are
strength, hardness, ductility, and stiffness.
 The role of structural engineers is to determine stresses
and stress distributions within members that are
subjected to well-defined loads. This may be
accomplished by experimental testing techniques and/or
by theoretical and mathematical stress.
CONCEPT OF STRESS AND STRAIN
 If a load is static or changes relatively slowly with time and is
applied uniformly over a cross section or surface of a member,
the mechanical behavior may be ascertained by a simple stress–
strain test; these are most commonly conducted for metals at
room temperature.

There are three principal ways

in which a load may be applied:
namely, tension, compression,
and shear (Figures a, b, c).
In engineering practice many
loads are torsional rather than
pure shear; this type of loading
is illustrated in Figure d.
 Let’s go into details for tension, compression, and shear and torsional tests;

 TENSION TESTS:
One of the most common mechanical stress–strain tests
is performed in tension. The tension test can be used to
ascertain several mechanical properties of materials that
are important in design.

Standard tensile specimen with

circular cross section
An apparatus used to conduct
tensile stress-strain tests
The output of such a tensile test is recorded on a strip
chart (or by a computer) as load or force versus
elongation. These load–deformation characteristics are
dependent on the specimen size.

For example, it will require twice the load to produce the

same elongation if the cross-sectional area of the
specimen is doubled. To minimize these geometrical
factors, load and elongation are normalized to the
respective parameters of engineering stress and
engineering strain.
 Engineering stress, σ is defined by the relationship >>>

in N/mm2 or MPa

where F is the instantaneous load applied perpendicular to the specimen cross section, in units of
newtons (N) and A0 is the original cross-sectional area before any load is applied (mm2).

Engineering strain, Є is defined according to >>>

in unitless

where l0 is the original length before any load is applied, and l i is the instantaneous length. Sometimes the
quantity li-l0 is denoted as Δl, and is the deformation elongation or change in length at some instant, as
referenced to the original length.
 COMPRESSION TESTS:

Compression stress–strain tests may be conducted if in-service

forces are of this type. A compression test is conducted in a manner
similar to the tensile test, except that the force is compressive and
the specimen contracts along the direction of the stress.

Similar equations used for tension tests are utilized to compute

compressive stress and strain, respectively. By convention, a
compressive force is taken to be negative, which yields a negative
stress. Furthermore, since l0 is greater than li , compressive strains
computed from the equation are necessarily also negative.
 SHEAR TESTS:

For tests performed using a pure shear force, the shear

stress ζ is computed according to >>>

in N/mm2 or MPa

where F is the load or force imposed parallel to the upper

and lower faces, each of which has an area of A0.
 TORSION TESTS:

Torsion is a variation of pure shear, wherein a structural

member is twisted in the manner of torsional forces
which produce a rotational motion about the longitudinal
axis of one end of the member relative to the other end.

Torsional tests are normally performed on cylindrical

solid shafts or tubes. A shear stress ζ is a function of the
applied torque T, whereas shear strain ү is related to the
angle of twist, Ф.
 ELASTIC DEFORMATION
 STRESS-STRAIN BEHAVIOUR
The degree to which a structure deforms or strains depends on
the magnitude of an imposed stress. For most metals that are
stressed in tension and at relatively low levels, stress and strain
are proportional to each other through the relationship:

This is known as Hooke’s law, and the constant of

proportionality E (GPa) is the modulus of elasticity, or Young’s
modulus.
Deformation in which stress and strain are proportional is called
elastic deformation; a plot of stress (ordinate) versus strain
(abscissa) results in a linear relationship, as shown in the following
figure:
The slope of this linear segment
corresponds to the modulus of
elasticity E.
This modulus may be thought of as
stiffness, or a material’s resistance to
elastic deformation.
The greater the modulus, the stiffer
the material, or the smaller the
elastic strain that results from the
application of a given stress.
The modulus is an important design
parameter used for computing elastic
deflections.
There are some materials (e.g., gray cast iron, concrete, and many
polymers) for which this initial elastic portion of the stress–strain
curve is not linear.

Hence, it is not possible to determine a modulus of elasticity as

described previously.

For this nonlinear behavior, either tangent or secant modulus is

normally used.

Tangent modulus is taken as the slope of the stress–strain curve at

some specified level of stress, while secant modulus represents the
slope of a secant drawn from the origin to some given point of the
σ–Є curve. The determination of these modules is illustrated in the
following figure:
Stress-strain diagram showing nonlinear elastic behavior
Table shows elastic
and shear moduli as
well as poisson’s ratio
for various materials
PROPERTIES RELATED TO
STRENGTH
 Strength is the ability of a material to resist
applied forces without yielding or fracturing.
 Strength of a material may change considerably
with respect to the way it is deformed.
 Mode of stress, type of stress & rate of stress
application may affect the strength of a
material.
 Strength data are usually obtained from lab.
Tests which are performed under strictly
standardized specimens under controlled
conditions. These tests also serve for obtaining
σ-ε relationships.
 σ-ε curves can be grouped into three as:

• Ductile Materials → exhibit both elastic &

plastic behavior
• Brittle Materials → exhibit essentially elastic
behavior
• Viscoelastic Materials → exhibit large elastic
deformation
SPECIAL FEATURES OF STRESS-STRAIN
DIAGRAMS

D
σU

σF E
σY C
σE
σPL A B
Point A (Proportional Limit): The greatest stress
(σPL) that can be developed in the material
without causing a deviation from the law of
proportionality of stress to strain. In other
words it is the stress upto which the material
responds following Hooke’s Law.

Point B (Elastic Limit): Maximum stress (σE) that

can be developed in a material without
causing permanent deformation. In other
words it is the stress upto which the
deformations are recoverable upon unloading.
Point C (Yield Point): The stress at which the
material deforms appreciably without an
increase in stress. Sometimes it can be
represented by an upper and lower yield
points. σY,U represents the elastic strength of
the material and σY,L is the stress beyond which
the material behaves plastically.

Point D (Ultimate Strength): It is the maximum

stress that can be developed in a material as
determined from the original X-section of the
specimen.
Point E (Fracture Strength): The stress at which the
material breaks, fails.
 In an engineering σ-ε plot the original area
(A0) & length (l0) are used when determining
stress from the load and strain from
deformations.
 In the true σ-ε plot instantaneous area &
length are used.
 The true values of stress & strain for
instantaneous area & length of the specimen
under tension will differ markedly, particularly
close to the breaking point where reduction in
cross-section & elongation of the specimen are
observed.
 P 
E  & E  Engineering
A0 l0

P 
T  & T  True
Ai li

li li li
dl  li  dl 1 li  l 0 li
 true    ln   eng     1
l l l0 l0 l0
l0  l0  l0 0 l0

 true  ln 1   eng 
  For εeng ≤ 0.1 → ln(1+0.1) ≈ 0.1
 For small strains εtrue ≈ εeng

P P
 true  &  eng 
Ai A0

If you assume no volume change:

l0
V  A0  l 0  Ai  l i Ai  A0 
li
  eng   eng 1   eng 
P li
 true 
l0 l0
A0
li
 true   eng 1   eng 
 DUCTILITY & BRITTLENESS
 Ductility can be defined as strain at fracture.
 Ductility is commonly expressed as:
a) Elongation
b) % reduction in cross-sectional area
 A ductile material is the one which deforms
appreciably before it breaks, whereas a brittle
material is the one which does not.
 Ductility in metals is described by:
A0  A f  If %RA > 50 % →
%RA   100 Ductile metal
A0
 TOUGHNESS & RESILIENCE
 Toughness is the energy absorption capacity
during plastic deformation.
 In a static strength test, the area under the
σ-ε curve gives the amount of work done to
fracture the specimen.
 This amount is specifically called as Modulus
of Toughness.
 It is the amount of energy that can be
absorbed by the unit volume of material
without fracturing it.
 σ T (Joule/m3)
σu
σf
σPL 2
T   u f
3
εPL εu εf ε

The area under the σ-ε diagram can be determined

by integration.
If the σ-ε relationship is described by a parabole.
 Resilienceis the energy absorption capacity
during elastic deformation.
1
σ R   PL 
2
σPL  PL
R
Since 
E
εPL ε 1   PL 
2

R
2 E
2
If you assume σPL = σy
y
R
2E
 YIELD STRENGTH
 It is defined as the maximum stress that can
be developed without causing more than a
specified permissible strain.
 It is commonly used in the design of any
structure.
 If a material does not have a definite yield
point to measure the allowable strains, “Proof
Strength” is used.
 Proof strength is determined by approximate
methods such as the 0.2% OFF-SET METHOD.
 At 0.2% strain, the initial tangent of the σ-ε
diagram is drawn & the intersection is located.
 DETERMINATION OF E FROM σ-ε
DIAGRAMS
 For materials like concrete, cast iron & most non-
ferrous metals, which do not have a linear
portion in their σ-ε diagrams, E is determined by
approximate methods.
1. Initial Tangent Method: Tangent is drawn to the
curve at the origin
2. Tangent Method: Tangent is drawn to the curve
at a point corresponding to a given stress
3. Secant Method: A line is drawn between the
origin & a point corresponding to a given stress
σ 1 3
2

ε
 HARDNESS
 Hardness can be defined as the resistance of a
material to indentation.
 It is a quick & practical way of estimating the
quality of a material.
 Early hardness tests were based on natural
minerals with a scale constructed solely on the
ability of one material to scratch another that
was softer.
 A qualitative & somewhat arbitrary hardness
indexing scheme was devised, temed as Mohs
Scale, which ranged from 1 on the soft end for
talc to 10 for diamond.
1. Talc An unknown material will
2. Gypsum scratch a softer one & will
be scratched by harder
3. Calcite
one.
4. Fluorite
EX:
5. Apatite HARDER •Fingernail-(2.5)
6. Orthoclase
•Gold, Silver-(2.5-3)
7. Quartz
8. Topaz •Iron-(4-5)

9. Corundum •Glass-(6-7)

10. Diamond •Steel-(6-7)

 The hardness of a metal is determined by
pressing an indenter onto the surface of the
material and measuring the size of an
indentation.

 The bigger the indentation the softer is the

material.

 Common hardness test methods are:

 Brinell Hardness
 Vicker’s Hardness
 Rockwell Hardness
1. Brinell Hardness
P
• Load P is pressed for 30
D sec. and the indentation
diameter is measured as
d.

2P
d Brinell Hardness =

D D  D 2  d 2 
(kgf/mm2)
 2. Rockwell Hardness
Initial P1 • Instead of the indentation
load diameter, indentation depth
is measured.
H1 • However, the surface
roughness may affect the
results.
• So, an initial penetration is
Final P2 measured upto some load,
load
and the penetration depth is
measured with respect to
H2
this depth.
ΔH = H 2 – H 1
 3. Vickers Hardness
P
• Instead of a sphere a
conical shaped indenter is
used.

d1  d 2
d
Top 2
View Indentation

d1
P
Vicker’s Hardness = 1.854 2
d
d2
(kgf/mm2)