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

normal strainL

0lim

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. -- 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

100%B o

o

L LL

0100% B

o

A AA

Percent elongation =

Percent reduction in area =

( / ) 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 = oL

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

E (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 y

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

2.7 Repeated Loadings: Fatigue

The -N curve = stress vs life curve

2.8 Deformations of Members under Axial Loading

E P

E AE

L PLAE

i i

i i i

PLAE

Pdxd dx

AE

(2.4)

(2.5)

(2.6)

(For Homogeneous rods)

(For various-section rods)

(For variable cross-section rods)

P

L

o

PdxAE

/ 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. 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

0 L R

2.10 Problems Involving Temperature Changes

( ) T T L

T T ( ) T T L

P

PLAE

2(.21)

= coefficient of thermal expansion

T + P = 0

0( ) T P

PLT L

AE

Therefore:

( ) P

E TA

( ) P AE T

2.11 Poisson 's Ratio

/ x x E

' lateral strain

Poisson s Ratioaxial strain

y z

x x

X X

x y zE E

Cubic rectangular parallelepiped

Principle of Superposition:

-- The combined effect = (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

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: x, y, z

The new volume =

2.13 Dilation: Bulk Modulus

1 1 1( )( )( ) x y z

1 x y z

1 1 1

2 30( . )

x y z

x y z

e the hange of olume

e

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

2

1 2( )X y ze

E

3 1 2( ) e p

E 3 1 2( )

E

pe

= bulk modulus = modulus of compression +

(2.31)

(2.33)

(2.33)

Special case: hydrostatic pressure -- x, y, z = p

Define:

3E

3e p

E

3 1 2( )

E

Since = positive,

Therefore, 0 < < ½

(1 - 2) > 0 1 > 2 < ½

= 0

= ½3 1 2 0( )

e pE

0e

-- Perfectly incompressible materials

2.14 Shearing Strain

xy xyG

yz yz zx zxG G

(2.36)

(2.37)

If shear stresses are present

Shear Strain = xy (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

The Generalized Hooke’s Law:

12EG

2 1( )E

G

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

and

-- 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

xy yx yz zy zx xz

x y y z z xE E E E E E

xy yz zxxy yz zxG G G

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

( ) y y ave

PA

If the stress distribution is uniform:

In reality:

2.18 Stress Concentrations

max

ave

K

-- Stress raiser at locations where geometric discontinuity occurs

= Stress Concentration Factor

2.19 Plastic Deformation

Elastic Deformation Plastic Deformation

Elastoplastic behavior

yY C

A D

Rupture

max ave

AP A

K

Y

Y

AP

K

U YP A

UY

PP

K

max

ave

K max ave K

For ave = Y

For max = Y

For max < Y

2.20 Residual Stresses

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

Residual Stresses