Chapter 2 Stress and Strain

-- Axial LoadingStatics – 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

δε = =normal strainL

0lim

δ δε∆ →

∆= =∆x

dx dx

For variable cross-sectional area A, strain at Point Q is:

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. -- σ y

0.2% Yield Strength

3. Ultimate Strength, σut

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

2.3 Stress-Strain Diagram

100%−B o

o

L LL

0100% B

o

A AA−

Percent elongation =

Percent reduction in area =

( / )ε ε= Σ∆ = Σ ∆t L L

2.4 True Stress and True Strain

Eng. Stress = P/Ao True Stress = P/A

Ao = original area A = instantaneous area

Eng. Strain = True Strain = δoL

lε = =∫o

L

t Lo

dL Ln

L L(2.3)

Lo = original length L = instantaneous length

Where E = modulus of elasticity or Young’s

modulus

2.5 Hooke's Law: Modulus of Elasticity

Eσ ε= (2.4)

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

2.6 Elastic Versus Plastic Behavior of a Material

Some Important Concepts:

1. Recoverable Strain

2. Permanent Strain – Plastic Strain

3. Creep

4. Bauschinger Effect: the early yielding behavior in the

compressive loading

Fatigue failure generally occurs at a stress level that is much

lower than σy

The Endurance Limit = the stress for which fatigue failure does not occur.

2.7 Repeated Loadings: Fatigue

The σ -N curve = stress vs life curve

2.8 Deformations of Members under Axial Loading

Eσ ε=P

E AEσε = =

Lδ ε=PLAE

δ =

i i

i i i

P LAE

δ = ∑

Pdxd dxAE

δ ε= =

(2.4)

(2.5)

(2.6)

(For Homogeneous rods)

(For various-section rods)

(For variable cross-section rods)

P

L

o

PdxAE

δ = ∫

/δ δ δ= − =B A B A

PLAE

(2.9)

(2.10)

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:

Example 2.02

1 2δ δ=

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

0δ δ δ= + =L R

2.10 Problems Involving Temperature Changes

( )δ α= ∆T T L

ε α= ∆T T ( )δ α= ∆T T L

PPLAE

δ =

2(.21)

α = coefficient of thermal expansion

δT + δP = 0

0( )δ δ δ α= + = ∆ + =T P

PLT L

AE

Therefore:

( )σ α= = − ∆PE T

A

( )α= − ∆P AE T

2.11 Poisson 's Ratio

/ε σ=x x E

'υ = = − lateral strainPoisson s Ratioaxial strain

ε ευε ε

= − = −y z

x x

σ υσε ε ε= = = −X X

x y zE E

• Cubic → rectangular parallelepiped

• Principle of Superposition:

-- The combined effect = Σ (individual effect)

2.12 Multiaxial Loading: Generalized Hooke's Law

Binding assumptions:

1. Each effect is linear 2. The deformation is small and does not change the overall condition of the body.

Generalized Hooke’s Law

2.12 Multiaxial Loading: Generalized Hooke's Law

υσ υσσε

σ υσυσε

υσ συσε

= + − −

= − + −

= − − +

y zxx

y zxy

y zxz

E E E

E E E

E E E

Homogeneous Material -- has identical properties at all points.

Isotropic Material -- material properties do not vary with direction or orientation.

(2.28)

Original volume = 1 x 1 x 1 = 1

Under the multiaxial stress: σx, σy, σ z

The new volume =

2.13 Dilation: Bulk Modulus

1 1 1( )( )( )υ ε ε ε= + + +x y z

1 x y zυ ε ε ε= + + +

1 1 1

2 30( . )

x y z

x y z

e the hange of olume

e

υ ε ε ε

ε ε ε

= = − = + + + −

∴ = + +

Neglecting the high order terms yields:

Eq. (2.28) → Eq. (2-30)

e = dilation = volume strain = change in volume/unit volume

( )X y z X y zeE E

σ σ σ υ σ σ σ+ + + += −

2

1 2( )X y ze

Eυ σ σ σ−= + +

3 1 2( )υ−= −e pE 3 1 2( )

κυ

=−E

κ= − pe

κ = bulk modulus = modulus of compression +

(2.31)

(2.33)

(2.33)

Special case: hydrostatic pressure -- σx, σy, σz = p

Define:

3Eκ =

3e pE= −

3 1 2( )κ

υ=

−E

Since κ = positive,

Therefore, 0 < υ < ½

(1 - 2υ) > 0 1 > 2 υ υ < ½

υ = 0

υ = ½3 1 2 0( )

e pEυ−

= − = 0eκ = ∞ =

-- Perfectly incompressible materials

2.14 Shearing Strain

xy xyGτ γ=

yz yz zx zxG Gτ γ τ γ= =

(2.36)

(2.37)

If shear stresses are present

Shear Strain = xyγ (In radians)

y zXx

y zXy

y zXz

xy yz zxxy yz zx

E E E

E E E

E E E

G G G

υσ υσσε

σ υσυσε

υσ συσε

τ τ τγ γ γ

= + − −

= − + −

= − − +

= = =

The Generalized Hooke’s Law:

12EG

υ= +

2 1( )E

Gυ

=+

2.18 Further Discussion of Deformation under Axial Loading: Relation Among E, υ , and G

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

y zxy xz

x x

andε ευ υε ε

= − = −

-- orthotropic materials

xy y zx zXx

x y z

xy X y zx zy

x y z

xy X yz y zz

x y z

E E E

E E E

E E E

υ σ υ σσε

υ σ σ υ σε

υ σ υ σ σε

= + − −

= − + −

= − − +

xy yx yz zy zx xz

x y y z z xE E E E E E

υ υ υ υ υ υ= = =

xy yz zxxy yz zxG G G

τ τ τγ γ γ= = =

2.17 Stress and Strain Distribution Under Axial Loading: Saint-Venant's Principle

( )σ σ= =y y ave

PA

If the stress distribution is uniform:

In reality:

2.18 Stress Concentrations

maxσσ

=ave

K

-- Stress raiser at locations where geometric discontinuity occurs

= Stress Concentration Factor

2.19 Plastic Deformation

Elastic Deformation → Plastic Deformation

→Elastoplastic behavior

σ

σyY C

A Dε

Rupture

maxσσ= =ave

AP A

K

σ= YY

AP

K

σ=U YP A

= UY

PP

K

maxσσ

=ave

K maxσσ =ave K

For σ ave = σY

For σmax = σY

For σmax < σY

2.20 Residual Stresses

After the applied load is removed, some stresses may still remain inside the material

→ Residual Stresses