Mse2 1
Mse2 1
Mse2 1
Learning Objectives
Define engineering stress and engineering strain.
State Hookes law, and note the conditions under which it is valid.
Given an engineering stressstrain diagram, determine (a) the modulus
of elasticity, (b) the yield strength (0.002 strain offset), and (c) the
tensile strength, and (d) estimate the percent elongation.
Name the two most common hardness-testing techniques; note two
differences between them.
Define the differences between ductile and brittle materials.
State the principles of impact, creep and fatigue testing.
State the principles of the ductile-brittle transition temperature.
Some Definitions
Tensile stress:
Where F: force, normal to
the cross-sectional area,
A0: original cross-sectional area
F
=
A0
Shear Stress
Fs: force, parallel to the cross-sectional area
A0: the cross-sectional area
unit of stress:
1Pa = 1 Nm-2;
1MPa = 106Pa; 1GPa=109Pa
Force N
= 2
area
m
Fs
=
A0
l l0 l
=
=
l0
l0
Nominal tensile strain (Axial strain)
Engineering Strain
Poissons ratio
= tan
z =
l z
l0 z
x =
l x
l0 x
lateral strain x
=
Poissons ratio: =
tensile strain
z
V
=
V
p
p
p
V-V
p
Elastic Behavior of Materials
Tensile:
Shear modulus
Shear: = G
Hydrostatic: p =
Bulk modulus
Modulus of Elasticity
Metals
Modulus of Elasticity
Ceramics
Modulus of Elasticity
- Polymers
Polymers
Elastic Modulus
(GPa)
Polyethylene (PE)
0.2-0.7
Polystyrene (PS)
3-3.4
Nylon
2-4
Polyesters
1-5
Rubbers
0.01-0.1
d 2U
S0 = 2
dx
x = x0
dF
=
dx
x = x0
F=
dU
dx
Unit area
F S0 ( r r0 )
F
= NS0 ( r r0 )
A0
Q =
Youngs modulus
Stiffness & Youngs Modulus
for different bonds
( r r0 )
r0
E=
Bonding type
S0(Nm-1)
E(GPa)
Ionic(i.e: NaCl)
8-24
32-96
Covalent
(i.e: C-C)
50-180
200-1000
Metallic
15-75
60-300
Hydrogen
2-3
8-12
0.5-1
2-4
S 0 ( r r0 ) S 0
= = E
r0
r0
r0
S0
=
r0
Material
Metals:
Ceramics:
Polymers:
E (GPa)
60 ~ 400
10 ~ 1000
0.001 ~ 10
Tensile Testing
The sample is pulled slowly
The sample deforms and then fails
The load and the deformation are measured
The load and deformation are easily transform into engineering stress () and
engineering strain ()
F
=
A0
A curve stress-strain is obtained
l l0 l
=
=
l0
l0
Modulus of Elasticity
It is a measure of material stiffness
and relates stress to strain in the
linear elastic range.
E =
=
z Ductility Parameters
Percent Elongation
Percent Reduction of
Area
Strain Hardening
Parameter
Fracture Strength
f<<UTS Due to the definition of Engineering
stress and tensile specimen necking.
f =
Pf
Ao
U = d
0
Approximated by:
G=
ys + U TS
2
* f
Elastic Recovery
After a load is released from a stress-strain test, some of the total deformation
is recovered as elastic deformation. During unloading, the curve traces a
nearly identical straight line path from the unloading point parallel to the initial
elastic portion of the curve
The recovered strain is calculated as the strain at unloading minus the strain
after the load is totally released.
Resilience
Resilience is the capacity of a material to absorb energy when it is deformed
elastically and then, upon unloading, to have this energy recovered.
Modulus of resilience Ur
U r = d
0
1
1 y y
U r = y y = y =
2
2 E 2E
Ductility
Ductility is a measure of the degree of plastic
deformation at fracture
expressed as percent elongation
also expressed as percent area reduction
lO and AO are the original gauge length and
original cross-section area respectively
lf and Af are length and area at fracture
Percentage elongation and percentage area
reduction are UNITLESS
A smaller gauge length will produce a larger overall
percentage elongation due to the contribution from
necking. Therefore, the percentage elongation
should be reported with original gauge length.
Percentage reduction is not affected by sample size,
thus it is a better measure of ductility
% EL = (
l f l0
% AR = (
l0
) * 100
A0 A f
A0
) * 100
True Stress
True stress is the stress determined by the instantaneous load acting on the
instantaneous cross-sectional area
Ao
P
P Ao
P
=
*
=
*
True stress is related to engineering stress: T =
A
A Ao
Ao
A
Assuming material volume remains constant
+lo
Ao
l
=
=
=
+ 1 = (1 + )
A lo
lo
lo
= A l
P
T =
(1 + ) = (1 + )
Ao
True Strain
The rate of instantaneous increase in the
instantaneous gauge length.
dl
l
= ln
T =
l
l
l o + l
l
l
ln o +
lo
lo lo
T = ln(1 + )
T = ln
T = F/Ai
T = ln(li/lo)
= (li-lo/lo)
Strain Hardening
Parameter (n)
T = K T
T
T
d
n =
d
T
T
Instability in Tension
Necking or localized deformation begins at maximum load, where the increase
in stress due to decrease in the cross-sectional area of the specimen
becomes greater than the increase in the load-carrying ability of the metal due
to strain hardening.
This conditions of instability leading to localized deformation is defined by the
condition P = 0.
P =
=
A
T
A
P = T A + A T = 0
L
L
A
A
= d T
V = Ao Lo = AL
T
=T
T
T = K T
T
T
n 1
= Kn T = n
T
T
T L / Lo L
=
=
=
T T L / L LO
1+
T
1
1+
=
(1 + ) = T
Fracture Behavior
Ductile material Significant plastic
deformation and energy absorption
(toughness) before fracture.
Characteristic feature of ductile material necking
Brittle material Little plastic
deformation or energy absorption before
fracture.
Characteristic feature of brittle materials
fracture surface perpendicular to the
stress.
Steel
Before and after fracture
fs =
fs =
3F f L
2bd 2
Ff L
R 3
Torsion Test
Ductile material twist
Brittle material fractures
max =
Tr
I Polar
max = G
max =
Gr
L
TL
I PG
Impact Test
(testing fracture characteristics under high strain rates)
Notched-bar impact tests are used to measure the impact energy (energy
required to fracture a test piece under impact load), also called notch
toughness. It determines the tendency of the material to behave in a brittle
manner.Due to the non-equilibrium impact conditions this test will detect
differences between materials which are not observable in tensile test.
We can compare the absorption energy capacity before fracture of different
materials.Two classes of specimens have been standardized for notchedimpact testing, Charpy (mainly in the US) and Izod (mainly in the UK)
Steel
20
Titanium
20
Aluminum
14
Magnesium
Low-Grade Plastic
Charpy
The impact toughness is
determined from finding the
difference in potential energy
before and after the hammer has
fractured the material.
Units are J (Joules) when testing
Metals, J/cm2 when testing
polymers (Polymers will stretch,
metals will snap).
h
h
Energy ~ h - h
Ductile-to-brittle transition
As temperature decreases a
ductile material can become brittle
- ductile-to-brittle transition.
FCC metals show high impact
energy values that do not change
appreciably with changes in
temperature.
Hardness
Hardness: a measure of a materials resistance to localized plastic
deformation (eg. Small dent or scratch).
1. Scratch hardness
Hardness: Different Techniques 2. Indentation hardness
3. Rebound hardness
Mohs Hardness
Mineral
Absolute Hardness
Talc (Mg3Si4O10(OH)2)
Gypsum (CaSO42H2O)
Calcite (CaCO3)
Fluorite (CaF2)
21
Apatite? (Ca5(PO4)3(OH-,Cl-,F-))
48
Orthoclase (KAlSi3O8)
72
Quartz (SiO2)
100
Topaz (Al2SiO4(OH-,F-)2)
200
Corundum (Al2O3)
400
10
Diamond (C)
1500
Indentation Hardness
Rebound Hardness
Energy absorbed under impact loads
Examples
Shore Scleroscope (ASTM E 448) - Measures the rebound of a small
pointed device dropped from a 254mm height.
Schmidt Hammer - Measures rebound of a spring loaded hammer. The test
has been correlated with concrete compressive strength.
1,500 kg load
300 kg load
79
238
476
40
119
238
26
79
159
20
60
119
10
16
48
95
Proximity to edge or other test locations: The distance of the center of the
indentation to the edge or from the center of adjacent indentations 2.5
times the diameter of the indentation.
Applied load:
1500 kg can be used for 48<BHN<300
1000 kg can be used for 32<BHN<200
750 kg can be used for 24<BHN<150
500 kg can be used for 16<BHN<100
HV =
1.854 F
D2
Vickers Test
Opposing indenter faces are set at
a 136 degree angle to each other
HK =
14.2 F
D2
Knoop Test
Long side faces are set at a 172
degree, 30 minute angle to each
other. Short side faces are set at a
130 degree angle to each other
Note:
No method of measuring
hardness uniquely indicates any
other single mechanical property.
Some hardness tests seem to be
more closely associated with tensile
strength, others with ductility, etc.
Fracture Mechanics
It studies the relationships between:
material properties
stress level
crack producing flaws
crack propagation mechanisms
Basic Concepts
The measured or experimental fracture strengths for most brittle
materials are significantly lower than those predicted by theoretical
calculations based on atomic bond energies.
This discrepancy is explained by the presence of very small,
microscopic flaws or cracks that are inherent to the material.
The flaws act as stress concentrators or stress raisers, amplifying the
stress at a given point.
This localized stress diminishes with distance away from the crack tip.
E/10
materials
E/100
typical ceramic
0.1
Fracture Toughness
Fracture toughness measures the resistance of a material to brittle
fracture when a crack or flaw is present.
It is a measure of the amount of stress required to propagate a
preexisting flaw.
Flaws may appear as cracks, voids, metallurgical inclusions, weld
defects, design discontinuities, or some combination thereof. The
occurrence of flaws is not completely avoidable in the processing,
fabrication, or service of a material/component.
It is common practice to assume that flaws are present and use the
linear elastic fracture mechanics (LEFM) approach to design critical
components.
This approach uses the flaw size and features, component geometry,
loading conditions and the fracture toughness to evaluate the ability
of a component containing a flaw to resist fracture.
K I = Y a
KI is the fracture toughness in
is the applied stress in MPa or psi
a is the crack length in meters or inches
Y is the component geometry factor that is different for each specimen, dimensionless.
49
The value of KIc (Critical SIF) represents the fracture toughness of the
material independent of crack length, geometry or loading system.
Kc
1
a max <
Ydesign
design <
Kc
Y a max
fracture
no
fracture
amax
fracture
amax
no
fracture
Design B
Design A
-- largest flaw is 9 mm
-- failure stress = 112 MPa
Kc
c =
Y amax
Use...
-- Result:
K I = Y a
112 MPa
(c
9 mm
4 mm
amax A = c amax B