chapter 4

Namas ChandraAdvanced Mechanics of Materials Chapter 4-1

EGM 5653

CHAPTER 4

Inelastic Material Behavior

EGM 5653Advanced Mechanics of Materials

Namas ChandraAdvanced Mechanics of Materials Chapter 4-2

EGM 5653

Objectives

Nonlinear material behavior

Yield criteria

Yielding in ductile materials

Sections

4.1 Limitations of Uniaxial Stress- Strain data

4.2 Nonlinear Material Response

4.3 Yield Criteria : General Concepts

4.4 Yielding of Ductile Materials

4.5 Alternative Yield Criteria

4.6 General Yielding

Namas ChandraAdvanced Mechanics of Materials Chapter 4-3

EGM 5653

Introduction

�When a material is elastic, it returns to the same state (at

macroscopic, microscopic and atomistic levels) upon removal of all

external load

�Any material is not elastic can be assumed to be inelastic

E.g.. Viscoelastic, Viscoplastic, and plastic

� To use the measured quantities like yield strength etc. we need

some criteria

�The criterias are mathematical concepts motivated by strong

experimental observations

E.g. Ductile materials fail by shear stress on planes of maximum

shear stress

�Brittle materials by direct tensile loading without much yielding

� Other factors affecting material behavior

- Temperature

- Rate of loading

- Loading/ Unloading cycles

Namas ChandraAdvanced Mechanics of Materials Chapter 4-4

EGM 5653

Types of Loading

Namas ChandraAdvanced Mechanics of Materials Chapter 4-5

EGM 5653

4.2.1 Models for Uniaxial stress-strain

All constitutive equations are models that are supposed to represent

the physical behavior as described by experimental stress-strain

response

Experimental Stress strain curves Idealized stress strain curves

Elastic- perfectly plastic response

Namas ChandraAdvanced Mechanics of Materials Chapter 4-6

EGM 5653

4.2.1 Models for Uniaxial stress-strain contd.

.Linear elastic response Elastic strain hardening response

Namas ChandraAdvanced Mechanics of Materials Chapter 4-7

EGM 5653

4.2.1 Models for Uniaxial stress-strain contd.

.

Rigid models

Rigid- perfectly plastic

response

Rigid- strain hardening plastic

response

Namas ChandraAdvanced Mechanics of Materials Chapter 4-8

EGM 5653

Ideal Stress Strain Curves

Namas ChandraAdvanced Mechanics of Materials Chapter 4-9

EGM 5653

4.2.1 Models for Uniaxial stress-strain contd.

.

4.4 The members AD and CF are made of

elastic- perfectly plastic structural steel, and member BE

is made of 7075 –T6 Aluminum alloy. The members

each have a cross-sectional area of 100 mm2.Determine

the load P= PY that initiates yield of the structure and the

fully plastic load PP for which all the members yield.

Soln:

Contd..

Namas ChandraAdvanced Mechanics of Materials Chapter 4-10

EGM 5653

4.2.1 Models for Uniaxial stress-strain contd.

Namas ChandraAdvanced Mechanics of Materials Chapter 4-11

EGM 5653

4.3 The Yield Criteria : General concepts

General Theory of Plasticity defines

Yield criteria : predicts material yield under multi-axial state of stress

Flow rule : relation between plastic strain increment and stress increment

Hardening rule: Evolution of yield surface with strain

Yield Criterion is a mathematical postulate and is defined by a yield

function { },

( )i j

f f Yσ=

where Y is the yield strength in uniaxial load, and is correlated with the

history of stress state.

Maximum Principal StressCriterion:- used for brittle materials

Maximum Principal Strain Criterion:- sometimes used for brittle materials

Strain energy density criterion:- ellipse in the principal stress plane

Maximum shear stress criterion (a.k.a Tresca):- popularly used for ductile materials

Von Mises or Distortional energy criterion:- most popular for ductile materials

Some Yield criteria developed over the years are:

Namas ChandraAdvanced Mechanics of Materials Chapter 4-12

EGM 5653

4.3.1 Maximum Principal Stress Criterion

Originally proposed by Rankine

2 cos

1 sinC

cY

φφ

=−

( )1 2 3max , ,f Yσ σ σ= −1

Yσ = ±

2Yσ = ±

3Yσ = ±

Yield surface is:

Namas ChandraAdvanced Mechanics of Materials Chapter 4-13

EGM 5653

4.3.2 Maximum Principal Strain

This was originally proposed by St. Venant

1 1 2 3 0f Yσ υσ υσ= − − − = 1 2 3 Yσ υσ υσ− − = ±

2 1 2 3 0f Yσ υσ υσ= − − − =2 1 3

Yσ υσ υσ− − = ±

3 3 1 2 0f Yσ υσ υσ= − − − = 3 1 2 Yσ υσ υσ− − = ±

or

or

or

Hence the effective stress may be defined as

maxe i j ki j k

σ σ υσ υσ≠ ≠

= − −

The yield function may be defined as

ef Yσ= −

Namas ChandraAdvanced Mechanics of Materials Chapter 4-14

EGM 5653

4.3.2 Strain Energy Density Criterion

.

This was originally proposed by Beltrami

Strain energy density is found as

( )2 2 2

0 1 1 1 1 2 1 3 2 3

12 0

2U

Eσ σ σ υ σ σ σ σ σ σ = + + − + + >

Strain energy density at yield in uniaxial tension test

2

02

Y

YU

E=

Yield surface is given by

( )2 2 2 2

1 1 1 1 2 1 3 2 32 0Yσ σ σ υ σ σ σ σ σ σ+ + − + + − =

2 2

ef Yσ= −

( )2 2 21 1 1 1 2 1 3 2 32eσ σ σ σ υ σ σ σ σ σ σ= + + − + +

Namas ChandraAdvanced Mechanics of Materials Chapter 4-15

EGM 5653

4.4.1 Maximum Shear stress (Tresca) Criterion

.

This was originally proposed by Tresca

2e

Yf σ= −

Yield function is defined as

where the effective stress is

maxeσ τ=

2 3

1

3 1

2

1 2

3

2

2

2

σ στ

σ στ

σ στ

−=

−=

−=

Magnitude of the extreme values of the stresses

are

Conditions in which yielding

can occur in a

multi-axial stress state

2 3

3 1

1 2

Y

Y

Y

σ σ

σ σ

σ σ

− = ±

− = ±

− = ±

Namas ChandraAdvanced Mechanics of Materials Chapter 4-16

EGM 5653

4.4.2 Distortional Energy Density (von Mises) Criterion

Originally proposed by von Mises & is the most popular for ductile materials

Total strain energy density = SED due to volumetric change +SED due to distortion

( ) ( ) ( ) ( )2 2 2 2

1 2 3 1 2 2 3 3 1

018 12

UG

σ σ σ σ σ σ σ σ σ− − − + − + −= +

( ) ( ) ( )2 2 2

1 2 2 3 3 1

1 2D

UG

σ σ σ σ σ σ− + − + −=

The yield surface is given by

2

2

1

3J Y=

( ) ( ) ( )2 2 2 2

1 2 2 3 3 1

1 1

6 3f Yσ σ σ σ σ σ = − + − + − −

Namas ChandraAdvanced Mechanics of Materials Chapter 4-17

EGM 5653

4.4.2 Distortional Energy Density (von Mises) Criterion contd.

Alternate form of the yield function

2 2

ef Yσ= −

where the effective stress is

( ) ( ) ( )2 2 2

1 2 2 3 3 1 2

13

2e Jσ σ σ σ σ σ σ = − + − + − =

( ) ( ) ( ) ( )2 2 2 2 2 213

2e xx yy yy zz zz xx xy yz xzσ σ σ σ σ σ σ σ σ σ = − + − + − + + +

J2 and the octahedral shear stress are related by

2

2

3

2oct

J τ= −

Hence the von Mises yield criterion can be written as

2

3octf Yτ= −

Namas ChandraAdvanced Mechanics of Materials Chapter 4-18

EGM 5653

4.4.3 Effect of Hydrostatic stress and the π- plane

Hydrostatic stress has no influence on yielding

Definition of a π- plane

Namas ChandraAdvanced Mechanics of Materials Chapter 4-19

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4.5 Alternate Yield Criteria

Generally used for non ductile materials like rock, soil, concrete and other anisotropic materials

4.5.1 Mohr-Coloumb Yield Criterion

� Very useful for rock and concretes

� Yielding depends on the hydrostatic stress

( )1 3 1 3 sin 2 cosf cσ σ σ σ φ φ= − + + −

( )max sin 2 cosi j i j

i jf cσ σ σ σ φ φ

≠ = − + + −

2 cos

1 sinT

cY

φφ

=+

2 cos

1 sinC

cY

φφ

=−

Namas ChandraAdvanced Mechanics of Materials Chapter 4-20

EGM 5653

4.5.2 Drucker-Prager Yield Criterion

1 2f I J Kα= + −

2sin 6 cos,

3(3 sin ) 3(3 sin )

cK

φ φα

φ φ= =

− −

2sin 6 cos,

3(3 sin ) 3(3 sin )

cK

φ φα

φ φ= =

+ +

This is the generalization of von Mises

criteria with the hydrostatic stress effect

includedYield function can be written as

Namas ChandraAdvanced Mechanics of Materials Chapter 4-21

EGM 5653

4.5.3 Hill’s Yield Criterion for Orthotropic Materials

This is the criterion is used for non-linear materials

The yield function is given by

( ) ( )2 2 2

22 33 33 11 11 22

2 2 2 2 2 2

23 32 13 31 12 21

( )

( ) ( ) ( ) 1

f F G H

L M N

σ σ σ σ σ σ

σ σ σ σ σ σ

= − + − + −

+ + + + + + −

2 2 2

2 2 2

2 2 2

2 2 2

23 13 12

1 1 12

1 1 12

1 1 12

1 1 12 , 2 , 2

FZ Y X

GZ X Y

HX Y Z

L M NS S S

= + −

= + −

= + −

= = =

For an isotropic material

6 6 6F G H L M N= = = = =

Namas ChandraAdvanced Mechanics of Materials Chapter 4-22

EGM 5653

General Yielding

The failure of a material is when the structure cannot support the

intended function

For some special cases, the loading will continue to increase even

beyond the initial load

At this point, part of the member will still be in elastic range. When

the entire member reaches the inelastic range, then the general

yielding occurs

2

,6

Y Y

bhP Ybh M Y= =

P YP Ybh P= =2

1.54

P Y

bhM Y M= =

Namas ChandraAdvanced Mechanics of Materials Chapter 4-23

EGM 5653

4.6.1 Elastic Plastic Bending

Consider a beam made up of elastic-perfectly plastic material

subjected to bending. We want to find the maximum bending

moment the beam can sustain

1 ( )

,

( )

zz Y

Y

k a

where

Yb

E

ε ε ε

ε

= =

=

( )2

Y

hy c

k=

0 ( )Z zzF dA dσ∑ = =∫/2

0

/ 2

0

2 2 0

2 2 ( )

Y

Y

Y

Y

y h

x ZZ y

y

y h

EP zz

y

M M ydA Y dA

or

M M ydA Y ydA e

σ

σ

∑ = − − =

= = +

∫ ∫

∫ ∫

Namas ChandraAdvanced Mechanics of Materials Chapter 4-24

EGM 5653

4.6.1 Elastic Plastic Bending contd.

2

2 2

3 1 3 1(4.43)

6 2 2 2 2EP Y

YbhM M

k k

= − = −

3

2EP Y PM M M→ =

2, /6Y

where M Ybh=

as k becomes large

Namas ChandraAdvanced Mechanics of Materials Chapter 4-25

EGM 5653

4.6.2 Fully Plastic Bending

Definition: Bending required to cause yielding either in tension or compression over the entire cross section

0z zzF dAσ= =∑ ∫Equilibrium condition

Fully plastic moment is

2P

t bM Ybt

+ =

Namas ChandraAdvanced Mechanics of Materials Chapter 4-26

EGM 5653

Comparison of failure yield criteria

For a tensile specimen

of ductile steel the

following six quantities

attain their critical

values at the same load PY

1. Maximum principal stress reaches the yield strength Y

2. Maximum principal strain reaches the value

3. Strain energy Uo absorbed by the material per unit volume reaches

the value

4. The maximum shear stress reaches the

tresca shear strength

5. The distortional energy density UD reaches

6. The octahedral shear stress

max( / )YP Aσ =

max max( / )Eε σ= /Y

Y Eε =

2

0 / 2YU Y E=

max( / 2 )

YP Aτ =

( / 2)Y Yτ =2 / 6

DYU Y G=

2 / 3 0.471oct Y Yτ = =

Namas ChandraAdvanced Mechanics of Materials Chapter 4-27

EGM 5653

Failure criteria for general yielding

Namas ChandraAdvanced Mechanics of Materials Chapter 4-28

EGM 5653

Interpretation of failure criteria for general yielding

Namas ChandraAdvanced Mechanics of Materials Chapter 4-29

EGM 5653

Combined Bending and Loading

2 2 2

2

According to Maximum shear stress criteria, yielding starts when

or 4 12 2 2

Y

Y

σ σ ττ + = + =

2 22 2

According to the octahedral shear-stress criterion, yielding starts when

2 6 2 or 3 1

3 3

Y

Y Y

σ τ σ τ+ = + =

Namas ChandraAdvanced Mechanics of Materials Chapter 4-30

EGM 5653

Interpretation of failure criteria for general yielding

Comparison of von Mises and Tresca criteria

Namas ChandraAdvanced Mechanics of Materials Chapter 4-31

EGM 5653

Problem 4.24

4.24 A rectangular beam of width b and depth h is subjected to pure

bending with a moment M=1.25My. Subsequently, the moment is released.

Assume the plane sections normal to the neutral axis of the beam remain

plane during deformation.

a. Determine the radius of curvature of the beam under the applied bending

moment M=1.25My

b. Determine the distribution of residual bending stress after the applied

bending moment is releasedSolution:

Namas ChandraAdvanced Mechanics of Materials Chapter 4-32

EGM 5653

Problem 4.24 contd.

Namas ChandraAdvanced Mechanics of Materials Chapter 4-33

EGM 5653

Problem 4.24 contd.

Namas ChandraAdvanced Mechanics of Materials Chapter 4-34

EGM 5653

Problem 4.40

4.40 A solid aluminum alloy (Y= 320 Mpa)

shaft extends 200mm from a bearing support

to the center of a 400 mm diameter pulley.

The belt tensions T1and T2 vary in magnitude

with time. Their maximum values of the belt

tensions are applied only a few times during

the life of the shaft, determine the required

diameter of the shaft if the factor of safety is

SF= 2.20Solution: