The chapter discusses stress and strain concepts including: 1. Stress-strain diagrams show the elastic and plastic deformation regions. Yield strength and ultimate strength are important properties. 2. Hooke's law defines the linear elastic region where stress is proportional to strain. The modulus of elasticity describes this relationship. 3. Materials experience both recoverable elastic strain and permanent plastic strain. Fatigue failure can occur at stresses lower than the yield strength due to repeated loading. 4. Temperature changes induce thermal stresses proportional to the coefficient of thermal expansion and temperature change. Residual stresses may remain after unloading.

1. 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.

2. 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

3. 2.2 Normal Strain under Axial Loading
δ
ε = normal strain =
L
For variable cross-sectional area A,
strain at Point Q is:
∆δ d δ
ε = lim =
∆x → 0 ∆x dx
The normal Strain is dimensionless.

4. 2.3 Stress-Strain Diagram

5. Ductile Fracture Brittle Fracture

7. Some Important Concepts and Terminology:
1. Elastic Modulus
2. Yield Strength – lower and upper Y.S. -- σ y
0.2% Yield Strength
3. Ultimate Strength, σ ut
4. Breaking Strength or Fracture Strength
5. Necking
6. Reduction in Area
7. Toughness – the area under the σ-ε curve
8. Percent Elongation
9. Proportional Limit

8. 2.3 Stress-Strain Diagram
LB − Lo
Percent elongation = 100%
Lo
A0 − AB
Percent reduction in area = 100%
Ao

9. 2.4 True Stress and True Strain
Eng. Stress = P/Ao True Stress = P/A
Ao = original area A = instantaneous area
δ
Eng. Strain = True Strain = ε t = Σ∆ε = Σ( ∆L / L)
Lo
Lo = original length L = instantaneous length
dL
L L
εt = ∫ = ln (2.3)
Lo L Lo

10. 2.5 Hooke's Law: Modulus of Elasticity
σ = Eε (2.4)
Where E = modulus of elasticity or Young’s
modulus
Isotropic = material properties do not vary with
Anisotropic = material properties vary with direction or
direction or orientation.
E.g.: metals E.g.: wood, composites
orientation.

12. 2.6 Elastic Versus Plastic Behavior of a Material
2

13. Some Important Concepts:
1. Recoverable Strain
2. Permanent Strain – Plastic Strain
3. Creep
4. Bauschinger Effect: the early yielding behavior in the
compressive loading

14. 2.7 Repeated Loadings: Fatigue
Fatigue failure generally occurs at a stress level that is much
lower than σ y
The σ -N curve = stress vs life curve
The Endurance Limit = the stress for which fatigue failure
does not occur.

16. 2.8 Deformations of Members under Axial Loading
σ = Eε (2.4)
σ P
ε = = (2.5)
E AE
δ = εL (2.6)
PL
δ = (For Homogeneous rods)
AE
Pi Li
δ = ∑
i A Ei
(For various-section rods)
i
Pdx
d δ = ε dx = (For variable cross-section rods)
AE
P

17. L Pdx
∫
(2.9)
δ =
o AE
PL
δ B/ A = δB − δA = (2.10)
AE

18. 2.9 Statically Indeterminate Problems
A. Statically Determinate Problems:
-- Problems that can be solved by Statics, i.e. ΣF = 0
and ΣM = 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:

19. Example 2.02
δ1 = δ 2

20. 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

21. Example 2.04

22. δ = δL + δR = 0

23. 2.10 Problems Involving Temperature Changes
δ T = α ( ∆T ) L 2(.21)
α = coefficient of thermal
expansion
δT + δP = 0
ε T = α∆T δ T = α ( ∆T ) L
PL
δP =
AE
PL
δ = δ T + δ P = α ( ∆T ) L + =0
AE

24. Therefore:
P = − AEα ( ∆T )
P
σ = = − Eα ( ∆T )
A

25. 2.11 Poisson 's Ratio
εx =σx / E
lateral strain
υ = Poisson ' s Ratio = −
axial strain
εy εz
υ= − = −
εx εx
σ υσ
ε = X
ε =ε = − X
x
E y z
E

26. 2.12 Multiaxial Loading: Generalized Hooke's Law
• Cubic → rectangular parallelepiped
• Principle of Superposition:
-- The combined effect = Σ (individual effect)
Binding assumptions:
1. Each effect is linear
2. The deformation is small and does not
change the overall condition of the body.

27. 2.12 Multiaxial Loading: Generalized Hooke's Law
Generalized Hooke’s Law
σ x υσ y υσ z
εx = + − −
E E E
υσ x σ y υσ z
εy = − + − (2.28)
E E E
υσ x υσ y σ z
εz = − − +
E E E
Homogeneous Material -- has identical properties at all points.
Isotropic Material -- material properties do not vary with
direction or orientation.

28. 2.13 Dilation: Bulk Modulus
Original volume = 1 x 1 x 1 = 1
Under the multiaxial stress: σ x, σ y, σ z
The new volume = υ = (1 + ε x )(1 + ε y )(1 + ε z )
Neglecting the high order terms yields:
υ =1+ εx + ε y + εz
e = the hange of olume = υ − 1 = 1 + ε x + ε y + ε z − 1
∴e = ε x + ε y + εz ( 2.30)

29. e = dilation = volume strain = change in volume/unit volume
Eq. (2.28) → Eq. (2-30)
σ X + σy + σz 2υ (σ X + σ y + σ z )
e = − (2.31)
E E
1 − 2υ
e= (σ X + σ y + σ z )
E
Special case: hydrostatic pressure -- σx, σy, σz = p
3(1 − 2υ ) E
e= − p Define: κ = (2.33)
E 3(1 − 2υ )
p
e= − (2.33)
κ
κ = bulk modulus = modulus of compression +

30. E
Since κ = positive,
κ=
3(1 − 2υ )
(1 - 2υ) > 0 1>2υ υ <½
Therefore, 0 < υ < ½
3 E
υ= 0 e= −
E
p κ=
3
3(1 − 2υ ) κ =∞ e=0
υ =½ e= −
E
p =0
-- Perfectly incompressible materials

31. 2.14 Shearing Strain
If shear stresses are present
Shear Strain = γ xy (In radians)
τ xy = G γ xy (2.36)
τ yz = G γ yz τ zx = G γ zx (2.37)

32. The Generalized Hooke’s Law:
σ X υσ y υσ z
εx = + − −
E E E
υσ X σ y υσ z
εy = − + −
E E E
υσ X υσ y σ z
εz = − − +
E E E
τ xy τ yz τ zx
γ xy = γ yz = γ zx =
G G G

33. 2.18 Further Discussion of Deformation under Axial Loading:
Relation Among E, υ, and G
E
=1+υ
2G
E
G=
2(1 + υ )

34. 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.

35. 2.16 Stress-Strain Relationships for Fiber-Reinforced
Composite Materials
-- orthotropic materials
εy εz
υ xy = − and υ xz = −
εx εx
σ X υ xyσ y υ zxσ z
εx = + − −
Ex Ey Ez
υ xyσ X σ y υ zxσ z
εy = − + −
Ex Ey Ez
υ xyσ X υ yzσ y σ z
εz = − − +
Ex Ey Ez

36. υ xy υ yx υ yz υ zy υ zx υ xz
= = =
E x E y E y Ez Ez E x
τ xy τ yz τ zx
γ xy = γ yz = γ zx =
G G G

37. 2.17 Stress and Strain Distribution Under Axial Loading:
Saint-Venant's Principle
If the stress distribution is uniform:
P
σ y = (σ y )ave =
A
In reality:

38. 2.18 Stress Concentrations
-- Stress raiser at locations where geometric discontinuity occurs
σ max
K= = Stress Concentration Factor
σ ave

40. 2.19 Plastic Deformation
Elastic Deformation → Plastic Deformation
→Elastoplastic behavior
σ
σy Y C
Rupture
ε
A D

41. For σ max < σ Y
σ max σ max
K= σ ave =
σ ave K
σ max A
P = σ ave A =
K
For σ max = σ Y
σY A
PY =
K
For σ ave = σ Y
PU = σ Y A
PU
PY =
K

42. 2.20 Residual Stresses
After the applied load is removed, some
stresses may still remain inside the material
→ Residual Stresses