Metal Plasticity

Outline

• Introduction（引言） • 1D plasticity theory（一维塑性） • Rough values of yield stress（常用工程材料屈服应力值） • 3D plasticity theory（三维塑性理论） • Decomposition of strain（应变分解） • Yield criterion（屈服判据） • Yield surface（屈服面） • Isotropic strain hardening（各向同性强化） • Kinematic strain hardening（运动强化） • Principal of maximum plastic resistance（最大塑阻原理） • Law of plastic flow（塑性流动定律） • Elastic unloading conditions（弹性卸载的条件）

2

• Linear elastic first; Plastic (permanent deformation); Unloading

follows linear curve; Relaxation; Creep; Bauschinger effect; Cyclic

hardening/softening; Rate, loading history and temperature

dependent

Introduction

3

• In 1930’s, Taylor and scientists experimentally measured the response of thin-

walled tubes under combined torsion, axial loading, and hydrostatic pressure.

• Hydrostatic stress has no effects on plastic deformation.

• Plastic behavior doesn’t induce volume change of a material.

• Plastic deformation is caused by shearing of atomic planes via propagation of a

type of lattice defects called dislocations.

• During plastic loading, the principal components of the plastic strain rate tensor

are parallel to the components of stress acting on the solid.

• Levy–Mises flow rule relates the principal plastic strain increment to the

principal stresses.

Introduction

4

p p p p p p

I II II III III I

I II II III III I

d d d d d d

• Decomposition of strain, yield criteria, strain hardening

rules, plastic flow rule, elastic unloading criterion

• We restrict attention to small deformations (≤10%).

Introduction

5

• Decomposition of strain:

1D Plasticity

6

, e p ed d d d Ed

• Yield criterion: p

Y

• Strain hardening rules govern the

functional dependence of yield stress

on plastic strain.

Rigid-perfectly plastic model

const.Y

1D Plasticity

7

Elastic-perfectly plastic model

const.Y

Linear hardening model

0 0

10 0

1

,

,

Y Y

pY

Y Y Y

EE

E E

1D Plasticity

8

Power law hardening model

0 0

0 0

0

,

1 ,

Y Y

Np

Y

Y Y

Y

E

• Here σY0, E, and N can be treated as fitting parameters to

experimental data. 0≤N<1 is called the hardening index.

• Tangent modulus of the stress-plastic strain curve

1

0

1

N

P

P Y

Edh EN

d

1D Plasticity

9

• Law of plastic flow:

, , 0e p

Y

d dd d d d

E h

0h consth 0

0

1

N

PY Y

Y

E

• Elastic unloading condition:

0d

Rough Values of Yield Stress

10

3D Plasticity Theory

11

• von Mises yield criterion: take the distortional part of

elastic strain energy as a criterion for the onset of plastic

deformation

2 2 21 3, + 0.

2 2 ij ij

p p p

ij I II II III III I Y Yf

• Decomposition of strain: , e p e

ij ij ij ij ijkl kld d d d C d

2

2 2 2

2

2 2 2

2

2 2 2

8

3 3 3 13

2 2 2 3

1 +

2

3 31 , 0 :

2 2

1 2 1+

3 3 3

ij ij

ij ij ij ij

ij

I II III I I II III

I II II III III I

I Y II III Y e Y

I II II III III I e

J

D

2

2

3ijJ

• von Mises yield function:

• Tresca yield criterion:

Yield Criterion

12

3

2 3 3

1

1 1: 2 2

YY

• Tresca yield function:

, max , , 0.p p

ij I II II III I III Yf

• Given the current stress σij applied to the material, we need to

determine current yield stress σY based on an effective plastic strain:

11 11 22 22 33 33

11 11 11 11 11 11 11

2 2,

3 3

21 , 0 :

3

2 1 12

3 2 2

p p p p p p p

ij ij ij ij

p p p p p p p

I Y II III

p p p p p p p

d d d d d d

D d d d d d d d

d d d d d d d

Yield Surface

13

• Geometric representation of von Mises

yield condition in stress space.

• If the state of stress falls within the

cylinder, the material is below yield and

responds elastically.

• If the state of stress lies on the surface of

the cylinder, the material yields and

deforms plastically.

• The stress state cannot lie outside the

cylinder; this would lead to an infinite

plastic strain.

2 2 2

i.e. : , , 02 2

1+

2

1 1 1 3 22 +

2 2 2 2 3

I II III

e I II II III III I

Y Y

R R

R R R

Isotropic Strain Hardening

14

• Plastic deformation typically causes the metal to strain

harden, as obviously seen in 1D.

• Strain hardening can be modeled by relating the size and

shape of the yield surface to plastic strain in some

appropriate way.

• The easiest way to model strain hardening is to make the

yield surface increase in size but remain the same shape.

2 2, ,

3 3

p p p p p p p p

ij ij ij ij Yd d d d d d

• Determine the updated radius via the

effective plastic strain, 1D hardening

functions, and new yield stress.

• An isotropic hardening law does not

account for the Bauschinger effect.

• Kinematic hardening drag the yield

surface in the direction of increasing

stress as you deform the material in

tension.

• This softens the material in compression,

however.

• So, this law can model cyclic plastic

deformation.

Kinematic Strain Hardening

0

3, 0.

2 ij ij

p p p

ij ij ij ij ij Yf

2

3

p

ij ijd cd • Linear kinematic hardening law:

15

• The von Mises yield surface is convex:

• The plastic strain increment is normal to the yield surface

for plastically stable solids.

Principal of Maximum Plastic Resistance

16

* 0p

ij ij ijd p p

ij ijd d f

• Experimental results (Levy-Mises theory) suggest that

plastic strains can be derived from the yield criterion.

Law of Plastic Flow for Isotropic Hardening

• Law of plastic flow for isotropic hardening:

3 3 1 3

2 2 2

ij ijp p p klij kl

ij Y Y Y

fd d d d

h

3, 0 0 , ,

2 ij ij

p p p p p

ij Y ij ij ijf f d d f

1 1 3

2

1 1 3 3 1 32

1

32 3

3

2

1

2

2 2

pq i

k

pq

p

j

pq p

l

k

q

q k

l k

p

ij p

ij

ijp p pYij ij ij ijp

ij ij ij Y

ip jq ij pq

ij ij

ij ij YY

f fd d

f f fd d d hd d d d

h h

f

1 1

3 3pq ip jq ik jk pq ip jq ij pq

• Some intermediate results

17

• Law of plastic flow for kinematic hardening:

Law of Plastic Flow for Kinematic Hardening

18

0

3, 0 0 , ,

2 ij ijij ij ij ij Y ij ij ij ij ij ijf f d d f

2

0 0

1

0

000

3 3

2 2

3,

3 3

2

2

2

ij ij ij ij

ij

jij i

ij

ij ij ij p

ij ij

Y Y

ijp

ij

Y

ij

ij

ij

ij

ijp

Y Y

ij

Y

d

d d cd

f

dc

fd

cd

d

0 0

0 0

1 1 3 32 .

2 2 23

2 2

2

3;

3 3

3

;2

2

ij ij

pq

kl

ij ijij ij

ij

ip jq pq

ij Y

kl

ijp

p p p

ij ij

Y

i

l

j

Y

Y

kl k

d cd c d c

f f

d

d

0 0 0

3 3 3

2 2 2

ij ijklij ijklp p p

ij kl

ij Y Y Y

fd d d d

c

• Some intermediate results

Elastic Unloading Conditions

19

p p p p p p

I II II III III I

I II II III III I

d d d d d d

• Both plastic flow laws are consistent with the Levy-Mises

theory, which is based on experimental observations.

• Elastic unloading conditions

0, 0.ij ij ij ij ijd d

• In both cases, the solid deforms

elastically (no plastic strain) if the

condition is satisfied.

• Summary of isotropically hardening elastic-plastic model

• Given:

Summary of 3D Plasticity Theory

20

, , ,p p

Y YE h d d

1 1 1

2 9

0, 0

3 1 3, 0

2 2

e p

ij

kl kl

ij ij

e

ij ij kk ij ij ij kk

p

e Y

p

ij ij p

e Y

Y Y

d d d

d d d d d

d

E E G K

d

h

• It will correctly predict the conditions necessary to initiate

yield under multiaxial loading.

• It will correctly predict the plastic strain rate under an

arbitrary multiaxial stress state.

• It can model accurately any uniaxial stress-strain curve.

• Summary of linear kinematically hardening model

• Given:

Summary of 3D Plasticity Theory

21

0, , ,YE c

0

0

0 0

1 1 1

2 9

30, 0

2

3 3 3, 0

2 2 2

ij

k

ij

ij

ij ij

l kl kl

e p

ij ij ij

e

ij ij kk ij ij ij kk

ij ij Y

p

ij

ij

ij ij Y

Y Y

d d

d

d

d d d d dE E G K

d

c

• This constitutive equation is used primarily to model

cyclic plastic deformation or plastic flow under

nonproportional loading (in which principal axes of stress

rotate significantly during plastic flow).