09 Constitutive Models 21

Constitutive models

Part 2

Elastoplastic

Elastoplastic material models

• Elastoplastic materials are assumed to behave elastically up to a certain stress limit after which combined elastic and plastic behaviour occurs.

• Plasticity is path dependent – the changes in the material structure are irreversible

Stress-strain curve of a hypothetical materialIdealized results of one-dimensional tension test

Engineering stress

Engineering strain

Yield pointYield stress

areainitialforce /

l

l0

Johnson’s limit … 50% of Young modulus value

Real life 1D tensile test, cyclic loading

Hysteresis loops moveto the right - racheting

Where is the yield point?

Conventional yield point

Lin. elast. limit

Mild carbon steel before and after heat treatment

Conventional yield point … 0.2%

The plasticity theory covers the following fundamental points

• Yield criteria to define specific stress combinations that will initiate the non-elastic response – to define initial yield surface

• Flow rule to relate the plastic strain increments to the current stress level and stress increments

• Hardening rule to define the evolution of the yield surface. This depends on stress, strain and other parameters

Yield surface, function• Yield surface, defined in stress space separates stress states

that give rise to elastic and plastic (irrecoverable) states• For initially isotropic materials yield function depends on

the yield stress limit and on invariant combinations of stress components

• As a simple example Von Mises …• Yield function, say F, is designed in such a way that

plasticity analyticalfor leinadmissib outside,0

surface theon0

surface the withinstate stress0

F

F

F

0yieldeffective F

0...),,( P KF ijij

Three kinematic conditions are to be distinguished

• Small displacements, small strains– material nonlinearity only (MNO)

• Large displacements and rotations, small strains– TL formulation, MNO analysis

– 2PK stress and GL strain substituted for engineering stress and strain

• Large displacements and rotations, large strains– TL or UL formulation

– Complicated constitutive models

Rheology models for plasticity

Ideal or perfect plasticity, no hardening

Loading, unloading, reloading and cyclic loading in 1D

stre

ss

strain

+ n

ew

yie

ld

stre

ss 1

- n

ew

yie

ld s

tre

ss 1

initi

al y

ield

str

ess

Isotropic hardening

ne

w y

ield

str

ess

2 loading

unloading

reloading

Isotropic hardening in principal stress space

321Y31 ,0)(

tensionin stress yield1D and stresses principalby expressedTresca

F

02])()()[(

tensionin stress yield1D and stresses principalby expressed Misesvon2Y

213

232

221 F

- plane

arccos (2/sqrt(3))

stre

ss

strain

+ n

ew

yie

ld

stre

ss 1

initi

al y

ield

str

ess

Kinematic hardening

loading

unloading

reloading

Loading, unloading, reloading and cyclic loading in 1D

Kinematic hardening in principal stress space

constant ..., where,0)( takewe

hardening) isotropic of case in (as0)( of insteadP ccF

F

ijijijij

ij

Von Mises yield condition, four hardening models

1. Perfect plasticity – no hardening 2. Isotropic hardening

3. Kinematic hardening 4. Isotropic-kinematic

Different types of yield functions

),,(

have could weall,at generalnot is whichGenerally,

invariant. anusually , of functionscalar a is)( e wher

hardening isotropic),(

waydifferent a in ofcomponent every on depends hardening

hardening isotropic-non),(

constant.a is and e wher

hardening kinematic)(

. strain plasticpermanent theon depends h whic

flow) (free nsdislocatio of motion theblocking on depends hardening theGenerally,

... nsdislocatio of nition Defi

ns.dislocatio of motionby caused isflow material tic Plas

region. plasticity thearound

exists whichstructure material healthy'' by the stabilized isit practiceIt

forever. so do toinclided is andflow tostarts material hardening, no means

plasticityperfect )(

P

PP

P

P

P

KFF

KK

KFF

FF

cc

FF

FF

ijij

ijij

ij

ij

ijij

ijij

ijij

ij

ij

Plasticity models – physical relevance

• Von Mises- no need to analyze the state of stress- a smooth yield sufrace- good agreement with experiments

• Tresca- simple relations for decisions (advantage for hand calculations)- yield surface is not smooth (disadvantage for programming, the normal to yield surface at corners is not uniquely defined)

• Drucker Pragera more general model

1D example, bilinear characteristics

PET ddd

d

plasticelastic

strain

stress

total

PT

ddd

EEE

EPT ddd Etan

Ttan E

Strain hardening parameter

Y

HEE

E

EE

EEE

/1 T

T

T

TP

… means total or elastoplastic… elastic modulus

… tangent modulus

Strain hardening parameter again

Elastic strains removed

Initial yield

Upon unloading and reloading the effective stress must exceed

Geometrical meaning of the strain hardening parameter is the slope of the stress vs. plastic strain plot

How to remove elastic part

T

TP EE

EEE

1D example, bar (rod) elementelastic and tangent stiffness

L

A

F F

Y

Y L

EAFk

E

L

AE

L

AFk

PE

PP

TT dd

d

d

d

d

d

P

P

P

PPT 1

/d/d

/Ed

EE

E

L

EA

EEL

AEk

Elastic stiffness

Tangent stiffness

Results of 1D experiments must be correlated to theories capable to describe full 3D behaviour of

materials

• Incremental theories relate stress increments to strain increments

• Deformation theories relate total stress to total strain

Relations for incremental theoriesisotropic hardening example 1/9

tt d

dlim:rates and increments between Relation

0

surface yield back to go0

0 that meansit - neutral0and0

ticelastoplas0and0

elastic0and0

elastic0if

and on dependsn deformatio ofincrement

0),( is surface yield Let the

Peff

eff

eff

eff

P

F

F

F

F

F

F

F

ij

ijij

Parameter only


Eq. (i) … increment of plastic deformation has a direction normal to F while its magnitude (length of vector) is not yet known

defines outer normal to Fin six dimensional stress space

0d so ns,deformatio plastic during zero bemust which

ddd

aldifferenti totala as expressed becan

}{

andscalar unknown far so is where

1947) (Drucker, form in the assumed is rule Flow

PP

PP

T

3111

P

F

FFFFF

F

FF

F

ijij

ijij

ijij

ijij

ijij

q

q


elastic total plastic deformations

matrix of elastic moduli

(iii) eq. )(

are increments stress

(ii) eq.0dd

form thein expressed be can 0d condition the

}{ Denoting

PE

PTTPTT

TP31

P11

εεEεEσ

εpqεpq

p

F

FF


qEqqp

EqTT

T

get we(iii) increments stressfor and (ii) 0d

(i), ruleflow for relations theCombining

F

Dot product and quadratic form … scalar

Row vector

Column vector

Lambda is the scalar quantity determining the magnitude of plastic strain increment in the flow rule

Still to be determined


PPE with)(

writecan weincrement stress for the Now,

εεεEεEσ q

determined be tohas still where

)(

with

form theinincrement strain totalof functiona as

increment stress get the wefor ngSubstituti

TT

TEP

EP

p

Eqqqp

EqEqEE

εEσ

equal to zero for perfect plasticity

diadic product


ijij

tt

ijij

tt

t

t

ijt

ijij

ijij

t

WF

AW

W

W

F

WWf

fF

ssJ

JF

FF

P

P

Y32

Y

PY

Y32

Pij

P

PY

YPij

Pij

PPY

PYP

21

D2

2Y3

1D2

TP31

P11

and using

F

rule Chain

increments plasticby donework d),(

at suggest th sExperiment

)( need we evaluate to

invariant deviatoric second theis where

0 condition yield Mises vonAssume

}{ of ionDeterminat

p

A new constant defined

At time t


PE

PETE

Y0

Yt

P0 Pt

PW

T312211

T

TP

Y

P

YPY

Y

P

2Y

02YP

P

PPY

0Y

t

PY

0Y2

1P

}{finaly so

3

2

3

2

3

2

)(2

1

)(stics characteribilinear 1D

)( done work elastic the1D in

A

EE

EEE

EA

E

W

EW

E

W

tt

t

t

t

t

tt

p

W


εEσ

bbEE

bqEqqEqbqp

p

q

s

σε

EP

TEP

TTT

T312312332211

T

T

T312312332211

T312312m33m22m11

T312312332211

33221131

m

Y

,,

}{

3

2

}222{

}{}{

)(

follows as compute can we and and given For Summary.

ca

ca

A

EE

EEA

ssssss

ssssss

ijij

J2 theory, perfect plasticity 1/6 alternative notation … example of numerical treatment

)2,2,2,1,1,1(diag][ with},]{[}{or

)222(

deviator stress ofinvariant second

}{}{

deviator stress

stress mean)(

}{}{

}{}{

law sHooke'}...]{[}{

T21

2

22222221

2D2

Tmmm

31

m

T

T

MsMsJ

ssssssJJ

s

E

zxyzxyzzyyxx

zxyzxyzzyyxx

zzyyxx

zxyzxyzzyyxx

zxyzxyzzyyxx

J2 theory, numerical treatment …2/6

Yeff

T2eff

TT

behaviour plasticperfectly for criterion yield

2/}]{[}{33 stress effective Misesvon

)1/(with},{2}]{][[ also and

0since},]{[}{}]{[}{

thatprove can one

sMsJ

EGsGsME

ssssMsMs zzyyxx


endif

0else

,0then if

by expressed be can region elastic in ndeformatio plastic no

increment ,derivative timeits...}{2}]{[}]{[}]{[}{

law sHooke'...}]{[}]{[}{

parameter unknownfar so is...}]{[}{

}{

hypothesis Reuss- Prandtl toaccording ruleFlow

Yeff

P

PE

T

sGEEE

EE

sMF

Six nonlinear differential equations + one algebraic constraint (inequality)There is exact analytical solution to this. In practice we proceed numerically


T2Y

EPEP

2Y

T

2Y

T

2Y

2eff2

T

TT

TT

eff

T

Teff

eff

Yeff

3with

finally

2

3

4

3

get we

3/43/442

that realizing and

2

for ngSubstituti

0 also and 0

02

3

condition plasticity atingDifferenti

ssEEεEσ

εsεMEs

Mss

MssεMEs

σ

σMsMss

Msss

s

G

G

GGJGG

G

System of six nonlineardifferential equationsto be integrated

J2 theory, numerical treatment …5/6predictor-corrector method, first part: predictor

1. known stress

2. test stress (elastic shot)

3a. elastic part of increment

T)1( s r

Tsr

3b. plastic part of increment

Tc.4 sss rt

tσ

)2/()1(3.5 2Y

Tc εs r

εEσσσσ tt TT

cT' 2.6 sσσ Gtt

J2 theory, numerical treatment …6/6predictor-corrector method, second part: corrector

''

''

'eff

Y

Y'

eff

Y'

eff

Yeff

'

)1(

have weionsconsiderat plasticity intoenter not does

tensor stress theofpart spherical thesince and

)1(

)(

)(

)(

)(

a way that such in findFor

Correction

tttttt

tttttt

tt

tt

tt

tt

tttt

sσσ

sss

s

s

s

s

ss

Secant stiffness method and the method of radial return

09 Constitutive Models 21

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