Chapter 2 Stress and Strain
-- Axial Loading
Statics – deals with undeformable bodies (Rigid bodies)
Mechanics of Materials – deals with deformable bodies
-- Need to know the deformation of a boy under
various stress/strain state
-- Allowing us to computer forces for statically
indeterminate problems.
The following subjects will be discussed:
 Stress-Strain Diagrams
 Modulus of Elasticity
 Brittle vs Ductile Fracture
 Elastic vs Plastic Deformation
 Bulk Modulus and Modulus of Rigidity
 Isotropic vs Orthotropic Properties
 Stress Concentrations
 Residual Stresses
2.2 Normal Strain under Axial Loading
  normal strain 

L
For variable cross-sectional area A,
strain at Point Q is:
 d 
  lim

x  0  x
dx
The normal Strain is dimensionless.
2.3 Stress-Strain Diagram
Ductile Fracture
Brittle Fracture
Some Important Concepts and Terminology:
1. Elastic Modulus
2. Yield Strength – lower and upper Y.S. -- y
0.2% Yield Strength
3. Ultimate Strength, ut
4. Breaking Strength or Fracture Strength
5. Necking
6. Reduction in Area
7. Toughness – the area under the - curve
8. Percent Elongation
9. Proportional Limit
2.3 Stress-Strain Diagram
LB  Lo
Percent elongation = 100%
Lo
Percent reduction in area = 100%
A0  AB
Ao
2.4 True Stress and True Strain
Eng. Stress = P/Ao
Ao = original area
Eng. Strain =

Lo
Lo = original length
dL
L
t  
 n
Lo L
Lo
True Stress = P/A
A = instantaneous area
True Strain =
 t    ( L / L)
L = instantaneous length
L
(2.3)
2.5 Hooke's Law: Modulus of Elasticity
  E
(2.4)
Where E = modulus of elasticity or Young’s
modulus
Isotropic = material properties do not vary with
direction or orientation. E.g.: metals
Anisotropic = material properties vary with direction or
orientation. E.g.: wood, composites
2.6 Elastic Versus Plastic Behavior of a Material
2
Some Important Concepts:
1. Recoverable Strain
2. Permanent Strain – Plastic Strain
3. Creep
4. Bauschinger Effect: the early yielding behavior in the
compressive loading
2.7 Repeated Loadings: Fatigue
Fatigue failure generally occurs at a stress level that is much
lower than y
The  -N curve = stress vs life curve
The Endurance Limit = the stress for which fatigue failure
does not occur.
2.8 Deformations of Members under Axial Loading
  E

P
 

E
AE
  L
PL
 
AE
 
i
Pi Li
Ai E i
(2.4)
(2.5)
(2.6)
(For Homogeneous rods)
(For various-section rods)
Pdx
d    dx 
AE
P
(For variable cross-section rods)
 
 B/ A
L
o
Pdx
AE
PL
 B  A 
AE
(2.9)
(2.10)
2.9 Statically Indeterminate Problems
A. Statically Determinate Problems:
-- Problems that can be solved by Statics, i.e. F = 0
and M = 0 & the FBD
B. Statically Indeterminate Problems:
-- Problems that cannot be solved by Statics
-- The number of unknowns > the number of equations
-- Must involve “deformation”
Example 2.02:
Example 2.02
1   2
Superposition Method for Statically
Indeterminate Problems
1. Designate one support as redundant support
2. Remove the support from the structure & treat it as
an unknown load.
3. Superpose the displacement
Example 2.04
Example 2.04
  L  R  0
2.10 Problems Involving Temperature Changes
 T   ( T ) L
2(.21)
 = coefficient of thermal
expansion
T + P = 0
 T  T
 T   ( T ) L
P
PL

AE
PL
   T   P   ( T ) L 
0
AE
Therefore:
P   AE ( T )
P
    E ( T )
A
2.11 Poisson 's Ratio
x x / E
lateral strain
  Poisson ' s Ratio  
axial strain
y
z
   
x
x
 
x

X
E
   
y
z

E
X
2.12 Multiaxial Loading: Generalized Hooke's Law
 Cubic  rectangular parallelepiped
 Principle of Superposition:
-- The combined effect =  (individual effect)
Binding assumptions:
1. Each effect is linear
2. The deformation is small and does not change
the overall condition of the body.
2.12 Multiaxial Loading: Generalized Hooke's Law
Generalized Hooke’s Law
x  
y  
z  
x
E

 x
E
 x
E
 y


E
y
E

 z

 z
 y
E
E

E
(2.28)
z
E
Homogeneous Material -- has identical properties at all points.
Isotropic Material -- material properties do not vary with direction
or orientation.
2.13 Dilation: Bulk Modulus
Original volume = 1 x 1 x 1 = 1
Under the multiaxial stress: x, y, z
The new volume =
  (1   x )(1   y )(1   z )
Neglecting the high order terms yields:
 1 x   y  z
e  the hange of olume    1  1   x   y   z  1
e   x   y  z
( 2.30)
e = dilation = volume strain = change in volume/unit volume
Eq. (2.28)  Eq. (2-30)
 X  y  z
2 ( X   y   z )
e 

E
E
(2.31)
1  2
e
( X   y   z )
E
Special case: hydrostatic pressure -- x, y, z = p
e 
3(1  2 )
p
E
e 
p

Define:  
E
3(1  2 )
(2.33)
(2.33)
 = bulk modulus = modulus of compression
+
E

3(1  2 )
Since  = positive,
1>2
(1 - 2) > 0
<½
Therefore, 0 <  < ½
= 0
3
e 
p
E
=½
3(1  2 )
e 
p
E
E

3
0
-- Perfectly incompressible materials
 
e0
2.14 Shearing Strain
If shear stresses are present
Shear Strain =
 xy
(In radians)
 xy  G  xy
 yz  G  yz  zx  G  zx
(2.36)
(2.37)
The Generalized Hooke’s Law:
x  
y  
z  
 xy 
X
E

 X
E
 X
E
 xy
G
 y


E
y
E

 z

 z
 y
E
 yz 
E

 yz
G
E
z
E
 zx 
 zx
G
2.18 Further Discussion of Deformation under Axial Loading:
Relation Among E, , and G
E
1
2G
E
G
2(1   )
Saint-Venant’s Principle:
-- the localized effects caused by any load acting on the
body will dissipate or smooth out within region
that are sufficiently removed form the location of
he load.
2.16 Stress-Strain Relationships for Fiber-Reinforced
Composite Materials
-- orthotropic materials
y
 xy  
x
x  
y  
z  
X
Ex
z
and  xz  
x

 xy X
Ex
 xy X
Ex
 xy y

 zx z

 zx z
 yz y
z
Ey


y
Ey
Ey
Ez
Ez

Ez
 xy
Ex

 xy 
 yx  yz
Ey Ey
 xy
G

 yz 
 zy  zx
Ez Ez
 yz
G

 zx 
 xz
Ex
 zx
G
2.17 Stress and Strain Distribution Under Axial Loading:
Saint-Venant's Principle
If the stress distribution is uniform:
 y  ( y )ave
In reality:
P

A
2.18 Stress Concentrations
-- Stress raiser at locations where geometric discontinuity occurs
 max
K
 ave
= Stress Concentration Factor
2.19 Plastic Deformation
Elastic Deformation  Plastic Deformation
Elastoplastic behavior

y
Y
C
Rupture

A
D
For max < Y
K
 max
 ave
 ave 
P   ave A 
 max A
For max = Y
PY 
Y A
K
For ave = Y
PU   Y A
PY 
PU
K
 max
K
K
2.20 Residual Stresses
After the applied load is removed, some
stresses may still remain inside the material
 Residual Stresses