Mechanical Properties
The adaptability of a material to a particular use is determined by its
mechanical properties.
Properties are affected by
Bonding type
Crystal Structure
Imperfections
Processing
Learning Objectives
Define engineering stress and engineering strain.
State Hooke’s law, and note the conditions under which it is valid.
Given an engineering stress–strain 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.
Types of Mechanical Testing
ƒ Slow application of stress
ƒ Allows dislocations to move to equilibrium positions
ƒ Tensile testing
ƒ Rapid application of stress
ƒ Ability of a material to absorb energy as it fails. Does not allow
dislocations to move to equilibrium positions.
ƒ Impact testing
ƒ Fracture Toughness
ƒ How does a material respond to cracks and flaws
ƒ Fatigue
ƒ What happens when loads are cycled?
ƒ High Temperature Loads
ƒ Creep
Some Definitions
Tensile stress:
Where F: force, normal to
the cross-sectional area,
F
σ=
A0
A0: original cross-sectional area
Shear Stress
Fs: force, parallel to the crosssectional area
A0: the cross-sectional area
unit of stress:
Force N
= 2
area
m
Fs
τ=
A0
1Pa = 1 Nm-2;
1MPa = 106Pa; 1GPa=109Pa
Engineering Strain
Nominal tensile strain (Axial strain)
l − l0 ∆l
ε=
=
l0
l0
Engineering Shear Strain
For small strain:
γ = tan θ
γ ≅θ
Poisson’s ratio
∆l z
εz =
l0 z
∆l x
εx = −
l0 x
Nominal lateral strain
(transverse strain)
lateral strain ε x
=−
Poisson’s ratio: ν = −
tensile strain
εz
p
Dilatation (Volume strain)
Under pressure: the volume will change
∆V
∆=
V
p
p
V-∆V
p
σ
Elastic Behavior of Materials
(Hooke’s Law)
When strains are small, most of
materials are linear elastic.
Young’s modulus
E
Tensile:
σ=Εε
Shear modulus
ε
Shear:
τ=Gγ
Bulk modulus
Hydrostatic: – p = κ ∆
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
Physical Basis of Young’s Modulus
Review: Inter-atomic forces (attractive and repulsive forces)
Define: stiffness
d 2U
S0 = 2
dx
x = x0
dF
=
dx
x = x0
dU
F=
dx
Unit area
Assume the strain is small,
F ≈ S0 ( r − r0 )
F
σ=
= NS0 ( r − r0 )
A0
σ
σ
Where N: number of bonds/unit area, N=1/r02
(r − r0 )
Qε =
r0
S 0 (r − r0 ) S 0
σ=
= ε = Eε
r0
r0
r0
E=
Stiffness & Young’s Modulus for different bonds
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
Van der Waals
0.5-1
2-4
Material
Metals:
Ceramics:
Polymers:
σ So
=
ε ro
Young’s modulus
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
Standard tensile specimen
The load and deformation are easily transform into engineering stress
(σ) and engineering strain (ε)
l − l0 ∆l
F
A curve stress-strain is obtained
σ=
A0
ε=
l0
=
Parameters Obtained From Stress Strain Curve
Strength Parameters
Modulus of Elasticity
Yield Strength
Ultimate Tensile Strength
Fracture Strength
Fracture Energy
Ductility Parameters
Percent Elongation
Percent Reduction of
Area
Strain Hardening
Parameter
l0
Modulus of Elasticity
It is a measure of material stiffness and relates
stress to strain in the linear elastic range.
δσ σ 2 − σ 1
E=
=
δε ε 2 − ε 1
Yielding and Yield Strength
Proportionality Limit (P): Departure from
linearity of the stress-strain curve
Yielding Point – Elastic Limit: the turning
point which separate the elastic and plastic
regions –onset of plastic deformation
Yield strength: the stress at the yielding
point.
Offset yielding (proof stress): if it is difficult
to determine the yielding point, then draw a
parallel line starting from the 0.2% strain,
the cross point between the parallel line and
the σ−ε curve
Tensile Strength (TS)
The stress increases
after yielding until a
maximum is reached. It
is also known as the
Ultimate Tensile
Strength (UTS), or
Maximum Uniform
Strength.
Prior to TS, the stress in the specimen is uniformly distributed.
After TS, necking occurs with localization of the deformation to the necking
area, which will rapidly go to failure.
Fracture Strength
σf<<σUTS Due to the definition of engineering stress
and tensile specimen necking
σf =
Pf
Ao
Fracture Energy (Toughness)
It is a measure of the work required to
cause the material to fracture.
It is a function of strength and ductility.
Its magnitude is defined by the area
under the stress strain curve
εf
U = ∫ σdε
0
Approximated by: G =
σ ys + σ UTS
2
*ε 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.
Ur
Modulus of resilience Ur
If it is in a linear elastic region,
σ y
1
1
U r = σ y ε y = σ y 
2
2
 E
εy
= ∫ σ dε
0
 σy
 =
 2E
2
Ductility
Ductility is a measure of the degree of plastic deformation at fracture
expressed as percent elongation
%EL = (
also expressed as percent area reduction
% AR = (
lO and AO are the original gauge length and original crosssection 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 %elongation due to the contribution
from necking. Therefore %elongation
should be reported with original gauge
length.
%Reduction is not affected by sample size,
thus it is a better measure of ductility
l f − l0
l0
) *100
A0 − A f
A0
) *100