FY Lecture1

8/11/2019 FY Lecture1

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1

Constitut ive modeling of large-strain

cyclic plasticityfor anisotropic metals

Fusahito YoshidaDepartment of Mechanical Science and Engineering

Hiroshima University, JAPAN

1: Basic framework of modeling

2: Models of orthotropic anisotropy

3: Cyclic plastici ty Kinematic hardening model

4: Applications to sheet metal forming and some

topics on material modeling

Lecture 1: Contents

Introduction:

purpose of constitutive modeling,

Stress and strain

Yielding of isotropic solids

Plastic potential and associated flow rule

Isotropic/kinematic hardening models

Isotropic hardening law


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Material behavior under uniaxial tension

(Tensile strength)

(yield strength)

yield point

stress

strain

necking

Void nucleation

Void growth/coalescence

Fracture

Necking occurs at a nominal stress peak, and it develops rapidly with increasing

strain. The specimen fractures as a consequence of void nucleation, growth and

coalescence.

Ductile fracture

Stress-strain curves of various metals

Experiment

Models

SNCM439

S35C

SUS304

BsBM1

A1100

Upper yield point

Elastic region

Lu ders bands

Lower yield point


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What material behaviors are our interests

in plasticity modeling?

Anisotropy (r-value, stress directionality)

Cyclic plasticity (the Bauschinger effect,

cyclic hardening, ratcheting, ,etc.)

Damage (evolution of voids, )

Rate-dependent behavior (viscoplasticity,

creep) Thermo-mechanical coupling

. etc.

98 TS

9 TS

Modeling ofAnisotropy andHardening ( - responses)

includingthe Bauschinger effect

Earing in cylindrical cup

deep drawing Springback

Barlat Yld2000-2d

Yoshida-Uemori


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Predictions of cracking and wrinkle

By PAM-STAMP 2G

(Yoshida-Uemori model)

Stamped panel

FE simulationBy JSTAMP

Cracking

Cracking

Sheet thinning

Photo

3D measurement

FE simulation

Deformation of solids

X

Current (t= t) configuration

Reference (t= 0) configuration

F : Deformation gradient

L : Velocity gradient

D : Rate of deformation

(stretching) tensor

W: Spin tensor

E: Lagrangian strain tensor

( ) ( )1 1

,2 2

/

iij

j

kkm

m

T T

T

e p

xd d , F X

vd d L

x

-

d dt

= =

= =

= + =

=

x F X

v L x,

D L L W L L

E F D F D

D = D + D


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Stress (1)

( ) ( )

1 2 3

[ ]

0, , ,

xx xy xz

ij yx yy yz

zx zy zz

p p

ij ij

= =

= =

Deviatoric stress and its Invariants

Cauchy stress, principal stress

( ) ( )1 2 31 1

3 3

ij ij m ij

m xx yy zz

s

=

= + + = + +

=hydrostatic stess (or mean stress)

Stress (2)

1 2 3

' ' '

1 2 3

1 13 , ,

2 3

1 10, ,

2 3

ii m ij ij ij jk ki

ii ij ij ij jk ki

J J J

J s J s s J s s s

= = = =

= = = =

Stress invariants

Jaumann rate (objective rate)

, ij ij im mj im mjW W W W = + = +& &


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Initial Yielding of Isotropic Solids

( ) ( ) ( )' '1 2 3 1 2 3 2 3, , , , ,f f f J J J f J J = = =

Since the yielding is not affected by the hydrostatic

stress component (i.e., incompressible), initial yielding

of an isotropic solid is expressed by the function (yield

function):

For example,

von Mises

Drucker

( ) ( )'3 '2 62 3 027D ijf s J J = = =

( ) ' 223

2M ij of s J = = =

Yield locus

Thin-walled tube in axial loads

& internal/outernal pressure

Yield locus is a description of yield criterion in stress space.

Thin-walled tube in axial loads

& torsion

von Misesvon Mises Tresca

Tresca

Crystal plasticity theory (FCC)

Carbon steel (S25C)

Stainless steel (SUS304)

Brass (BsBM1)

Aluminum alloy (A2017)


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Physical background:Plastic deformations in a crystal

{1 1 1}

[1 0 1]

atom( )

slip plane

grain

grain

boundary

1m100m

A few

Slip occurs most readily in specific directions (slip directions) on

certain crystallographic planes (slip planes) .

Why is the yielding not affected by hydrostatic stress?

Schmids law: Slip of a crystal occurs when the

resolved shear stress reaches its critical value, CRSS.

( ) cos cosR =

Resolved shear stress

Schmid factor

Yield cri terion for a crystal

( )R

crk = Critical resolved shear stress

(CRSS)Slip direction Slip plane

Keywords: Slip system = Slip plane and slip direction


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Resolved shear stress is not changed by the

hydrostatic stress (pressure):

At the atmosphere Under hydrostatic

pressure

a b =

Plastic potential & associated flow rule

Unloading

Neutral loading

LoadingInitial yield locus F

Subsequent yield locus f

Loading

Unloading


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Druckers postulate on stable stress-strain response

( ) ( )2

e p e pd d d d d E d d d = + = +

0, 0p pij ij

d d d d

(a) Stable (b) Unstable (c) Multiaxial stress state

( ) ( ) ( )

( )

( ) ( )

* * *

*

* *

0

0 0

e p

p

p p

ij ij ij ij ij ij

d d d

d

d d

= +

=

Stress cycle


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Principle of maximum plastic work

Convexity of yield locus

Normality rule for plastic

strain rate vector

( ) ( )* *0 or 0p pij ij ij ij ij ijd s s d

or

p

ij

ij

p

ij

ij

fd d

fd d

s

=

=

Associated flow rule

Yield locus of mild steel

Kuwabara


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Kinematic hardening model

Subsequent yield function

( ) ( )

( ) ( )

2

' 2

0

or 0

ij ij ij

ij ij ij

f Y

f s s Y

= =

= =

( )'3

2

p

ij ij ij

dd s

Y

=

backstress)Associated flow rule

Combined hardening model

Subsequent yield function

( ) ( )

( ) ( )

2

' 2

0

or 0

ij ij ij o

ij ij ij o

f

f s s

= =

= =

Associated flow rule

( )'3

2

p

ij ij ij

o

dd s

=

Appropriate evolution equations for isotropic hardening

and kinematic hardening is of vital importance.

ijY

oij

O

p

ijd


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Stress-Strain Response of a High Strength

Steel Sheet of 590 MPa Grade

Isotropic hardening (IH) model

Permanent

stress offsetTransient Bauschingereffect

Early

re-yielding

Hardening law

= description of

expansion of yield locus

(isotropic hardening)

movement of the center

of yield locus (kinematic

hardening: evolution of the

back stress)

ijY

oij

O

p

ijd


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Isotropic Hardening Law by means of

Effective Stress and Effective Plastic Strain

( ) ( ) 0ijF Y s Y = = =s

( ) ( )ij

s = =s

( ) ( ) ( ) ( ) 0ijf Y R s Y R = + = + =s

For initial yielding:

For the subsequent yielding:

Isotropic hardening stress

Effective stress:

For von Mises material:

'

2

33

2 i j i j

J s s= =

Effective Plastic Strain Increment d

p p p

ij ij ij ijdw d s d d = = =

2

3

p p

ij ijd d d =

Work conjugate formulation:

When using von Mises effective stress:

Effective plastic strain:

d dt = = &


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Linear hardening

Ludwik

Swift

Voce

Isotropic Hardening Laws

nY C = +

( )0n

C = +

For example,

( ) ( )Y R = = +

'Y H = +

[ ]1 exp( )SatY R = +

Uniaxial tension stress-strain curves

( )

( )

( )

p n

p n

p n

o

C

Y C

C

=

= +

= +

Ludwik

Swift

Perfectly plastic solid Linearly hardening

plastic solid

Non-linearly hardening

plastic solids

Power-law hardening

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FY Lecture1