Chapter 3. J2 Plasticity Models v1.0

Computational Solid Mechanics Computational Plasticity

C. Agelet de Saracibar ETS Ingenieros de Caminos, Canales y Puertos, Universidad Politcnica de Catalua (UPC), Barcelona, Spain

International Center for Numerical Methods in Engineering (CIMNE), Barcelona, Spain

Chapter 3. J2 Plasticity Models

Contents 1. J2 rate independent plasticity models

1. Hardening plasticity model

2. J2 rate dependent plasticity models 1. Hardening plasticity model

J2 Plasticity Models > Contents

Contents

April 1, 2015 Carlos Agelet de Saracibar 2





Contents


Hypothesis Within the framework of the infinitesimal deformation theory, we introduce the following hypothesis for a J2 rate-independent linear elastic-hardening plasticity model, within the incremental theory of plasticity: H1. Additive split of the infinitesimal strain tensor H2. Set of plastic internal variables

J2 Plasticity Models > Rate Independent Plasticity Models

Hardening plasticity model


e p= +

{ }: , ,p p = E

H3. Free energy per unit of volume

Elastic potential for a linear elastic material model Constant isotropic elastic constitutive tensor (0, >0, >0) Elastic potential for isotropic material




( ) ( ) ( ) ( ), , :e eW = + +

( ) 1 : : Elastic potential2e e eW =

( )13: 2 2 = + = + 1 1 1 1 1 1

( ) ( ) ( )2 2 21 1: tr dev : dev tr dev2 2e e e e e eW = + = +


Isotropic hardening potential for a linear isotropic hardening material model




( ) ( ) ( ) ( ), , :e eW = + +

( ) 21 Isotropic hardening potential2

K =


Kinematic hardening potential for a linear kinematic hardening material model




( ) ( ) ( ) ( ), , :e eW = + +

( ) 21 1 : Kinematic hardening potential3 3H H = =

2 2 2 22 0, 2 0, 2 03 3 3 3

K H K H + + > + > + >

H4. Clausius-Planck inequality. Linear elastic constitutive equation, linear hardening equations and reduced dissipation




( ): : , , 0e = D

( ): : : :

: : : 0

e

e e

e

p

=

= +

D

( ) ( ): tr 2 tr 2 dev2: , :3

: : : 0

ee e e e e

p

q K H

q

= = = + = +

= = = =

= + +

1 1

q

q

D

Linear isotropic hardening Nonlinear isotropic hardening (1) Exponential saturation law + linear isotropic hardening (2) Power law isotropic hardening




( ) ( )( ): : 1 expYq K = =

:q K = =

( )1 2: :m

Yq k k = = +

H5. Space of admissible stresses, elastic domain, and yield surface. Yield function




( ) ( ){ }( ) ( ) ( ){ }

( ) ( ){ }

23

23

23

: , , , , : dev 0

int : , , , , : dev 0

: , , , , : dev 0

Y

Y

Y

q f q q

q f q q

q f q q

= =

= = Rate Independent Plasticity Models



1

2

3

1

2

1 2 3 = =1 2 3

Hydrostatic stress axis = =

1 2 3

Octahedral planecte + + =

Von Mises yield surface( )23 YR q=

( ) ( )23, , : dev 0Yf q q= = q q

Yield surface: gometrical representation in the space of principal stresses




1 21 2

3

023 Y

R =

( )23t Y t

R q= dev t

tq

0dev

0 =q 0

3

( ) ( )23, , : dev 0Yf q q= = q q

H6. Plastic flow rule Non-associative plastic flow rule where is a plastic potential defined in the (deviatoric) stress space. Taking the yield function as plastic potential, the plastic flow rule is said to be associative.




( )( )( )

, ,

, ,

, ,

p

q

g q

g q

g q

=

=

=

q

q

q

q

( ), ,g q q

H6. Plastic flow rule Associative plastic flow rule

where is the unit outward normal to the yield surface such that,




( )( )( )

, ,

, , 2 3

, ,

p

q

f q

f q

f q

= =

= =

= =

q

q n

q

q n

n

( ) dev: , ,dev

tr : 0, dev , 1

f q = =

= = = =

qn q q

n 1 n n n n

Note that, both the plastic strain tensor and the kinematic hardening tensor defined in the strain space are deviatoric,

And, therefore, the kinematic hardening tensor defined in the stress space is also deviatoric,




( ) ( ), , , , ,dev , devdev , dev

p

p p

p p

f q f q = = = =

= =

= =

q q n q n

2dev :3

H= = = q q

H7. Kuhn-Tucker loading/unloading conditions




( ) ( )0, , , 0, , , 0f q f q = q q

( )( )

Plastic loading

Elastic loading/unl

if 0 then , , 0

if , , 0 then 0 oading

f q

f q

> =

< =

q

q

H8. Plastic consistency condition Plastic loading




( ) ( ) ( )if , , 0 then 0, , , 0, , , 0f q f q f q = = q q q

( ) ( )( ) ( )

if , , 0 and 0 then , , 0

if , , 0 and

Plastic loading

Elastic unloading, , 0 then 0

f q f q

f q f q

= > =

= < =

q q

q q

( ) ( ), , 0 and 0 , , 0f q f q= > = q q

Plastic loading: plastic consistency condition




0

: : : 2 3 :

2: : 2 3 :3

2: : : : 2 3 :3

2 2: : : : :3 32 2: : : : 03 3

q

e

p

f f f q f q

K H

K H

K H

K H

>

= + + = +

= +

= +

= + + = + + =

q q n n q

n n

n n n

n n n n n

n n n

Plastic loading: plastic multiplier or plastic consistency parameter

Taking into account that The plastic multiplier takes the form




12 2: : : :3 3

K H

= + + n n n

( )( )

: 0, : : : 1: : : 2 : 2

: : : 2 : 2 : 2 : dev

= = =

= + =

= + = =

n 1 n n n nn n n 1 1 n

n n 1 1 n n

12 22 2 : dev3 3

K H

= + +

n

Trial deviatoric stress state The plastic multiplier takes the form




dev 2 devtrial =

12 22 : dev 03 3

trialK H

= + +

n

12 22 : dev 03 3

trialK H

= + +

n

Plastic loading Elastic unloading Neutral loading




: dev 0 0, 0, 0trial f f > > = =n

: dev 0 0, 0, 0trial f f < = < =n

: dev 0 0, 0, 0trial f f = = = =n

12 22 : dev 03 3

trialK H

= + +

n

Plastic loading, elastic unloading, and neutral loading




( ): , ,f q= n q

trial

trial

Neutral loading

Plastic loading

Elastic unloading

( ): , ,f q= n q

dev trial

dev trialdev trial

Reduced plastic dissipation Plastic dissipation rate per unit of volume




( )( )( )( ) ( )( )( )

23

2 23 3

23

23

: : :

dev : dev

,

0

p

Y

Y

q q

q q

f q

= + + = +

= + = +

= +

=

q q n

q n q

q

D

,

Maximum plastic dissipation principle Given a strain rate and a rate of the plastic internal variables the maximum plastic dissipation principle states that the stress state satisfies where,




{ } { }: ,0, , : , ,p p = = 0 E E

( ) ( ), ,p p D S E D ET T

{ } { }: , , , : , ,q p= = q pS T

( ) ( ), : , :p p p p= = D S E SE , D E ET T

Maximum plastic dissipation principle Maximum plastic dissipation Constrained minimization problem




( ) ( ), ,p p D S E D ET T( ) ( ), : 0p p = D S E S ET T T

( )arg max , p = S D ET T( )( )arg min , p = S D ET T

Maximum plastic dissipation principle Maximum plastic dissipation, given by, is equivalent to associative plastic flow rule, Kuhn-Tucker loading/unloading conditions, and convexity of the yield surface, given by,




( )( )arg min , p = S D ET T

( )( ) ( )

( ) ( ) ( ) ( )0, 0, 0

p f

f f

f f f

=

=

S

S

E S

S S

S S ST T T

Hypothesis H1. Additive split of the infinitesimal strain tensor H2. Set of plastic internal variables H3. Free energy per unit of volume H4. Clausius-Planck inequality. Linear elastic constitutive equation, linear hardening equations and reduced dissipation H5. Space of admissible stresses, elastic domain, and yield surface. Yield function H6. Associative plastic flow rule H7. Kuhn-Tucker loading/unloading conditions




Hypothesis H1. Additive split of the infinitesimal strain tensor H2. Set of plastic internal variables H3. Free energy per unit of volume H4. Clausius-Planck inequality. Linear elastic constitutive equation, linear hardening equations and reduced dissipation H5. Space of admissible stresses, elastic domain, and yield surface. Yield function H6. Maximum plastic dissipation








Contents


Hypothesis Within the framework of the infinitesimal deformation theory, we introduce the following hypothesis for a J2 rate-dependent linear elastic-hardening plasticity model, within the incremental theory of plasticity: H1. Additive split of the infinitesimal strain tensor H2. Set of plastic internal variables

J2 Plasticity Models > Rate Dependent Plasticity Models



e p= +

{ }: , ,p p = E


Elastic potential for a linear elastic material model Constant isotropic elastic constitutive tensor (0, >0, >0) Elastic potential for isotropic material




( ) ( ) ( ) ( ), , :e eW = + +

( ) 1 : : Elastic potential2e e eW =

1: 2 23

= + = +

1 1 1 1 1 1

( ) ( ) ( )2 2 21 1: tr dev : dev tr dev2 2e e e e e eW = + = +


Isotropic hardening potential for a linear isotropic hardening material model




( ) ( ) ( ) ( ), , :e eW = + +

( ) 21 Isotropic hardening potential2

K =


Kinematic hardening potential for a linear kinematic hardening material model




( ) ( ) ( ) ( ), , :e eW = + +

( ) 21 1 : Kinematic hardening potential3 3H H = =

2 2 2 22 0, 2 0, 2 03 3 3 3

K H K H + + > + > + >

H4. Clausius-Planck inequality. Linear elastic constitutive equation, linear hardening equations and reduced dissipation




( ): : , , 0e = D

( ): : : :

: : : 0

e

e e

e

p

=

= +

D

( ) ( ): tr 2 tr 2 dev2: , :3

: : : 0

ee e e e e

p

q K H

q

= = = + = +

= = = =

= + +

1 1

q

q

D

H5. Elastic domain, plastic domain and yield surface. Yield function




( ) ( ){ }( ) ( ) ( ){ }

( ) ( ){ }

23

23

23

: , , , , : dev 0

ext : , , , , : dev 0

: , , , , : dev 0

Y

Y

Y

q f q q

q f q q

q f q q

= =

= = >

= = =

q q q

q q q

q q q

H6. Associative plastic flow rule Associative plastic flow rule

where is the unit outward normal to the yield surface such that,




( )( )( )

, ,

, , 2 3

, ,

p

q

f q

f q

f q

= =

= =

= =

q

q n

q

q n

n

( ) dev: , ,dev

tr : 0, dev , 1

f q = =

= = = =

qn q q

n 1 n n n n

H7. Plastic multiplier Note that, for the non-trivial case of plastic loading the following expression holds,




( )1 , , 0f q

= q

( ), , 0f q = > q

Reduced plastic dissipation Plastic dissipation rate per unit of volume




( )( )( )( ) ( )( )( )

( )

23

2 23 3

23

23

: : :

dev : dev

, ,

0

p

Y

Y

q q

q q

f q

= + + = +

= + = +

= +

= +

q q n

q n q

q

D

Perzyna model The associative plastic flow rule can be obtained as the solution of an unconstrained minimization problem, arising from the maximization of a regularized plastic dissipation, given by,




( )( )arg min , p= T TS D E( ) ( ) ( ) 21, : , 2

p p f = D S E D S E S

The solution of the unconstrained minimization problem, arising from the maximization of a regularized plastic dissipation, yields the associative plastic flow rule,




( ) ( ) ( )

( ) ( )

21, : ,2

1:

p p

p

f

f f

=

= = 0

S S

S

D S E D S E S

E S S

( )( )arg min , p= T TS D E

( ) ( )1p f f

= SE S S

The associative plastic flow rule takes the form,




( ) ( )1p f f

= SE S S

( )( )( )

, ,

, , 2 3

, ,

p

q

f q

f q

f q

= =

= =

= =

q

q n

q

q n

Duvaut-Lions model The associative plastic flow rule can be obtained as the solution of an unconstrained minimization problem, arising from the maximization of a regularized plastic dissipation, given by,




( )( )arg min , p= T TS D E( ) ( ) ( )

( ) 12

1, : , *

1: , *2

p p

p

=

=

D S E D S E S S

D S E S SC

The solution of the unconstrained minimization problem, arising from the maximization of a regularized plastic dissipation, yields the associative plastic flow rule,




( )( )arg min , p= T TS D E( ) ( ) ( )

( )

1

1

1, : , *

1: *

p p

p

=

= = 0

S SD S E D S E S S

E S S

C

C

( )11 *p

= E S SC

Closest-point-projection (cpp) The closest-point-projection (cpp) is obtained as the solution of the following constrained minimization problem, written in terms of the complementary energy norm as,

Using the Lagrange multipliers method, it can be transformed into the following unconstrained minimization problem




( )* arg min * * = T T S S

( ) ( ) ( )121 1 12 2* * * * = = S S S S S S S SC C

( ) ( ) ( ), *; * : * * *f = +SS S S SL( )* arg min , *; * *= T TS SL

Closest-point-projection (cpp) The solution of the unconstrained minimization problem yields the closest-point-projection,




( ) ( ) ( )( ) ( )

* * *

1*

, *; * : * * *

: * * * 0

f

f

= +

= + =S S S

S

SS S S S

S S SC

L

( )* arg min , *; * *= T TS SL

( )** * *f= SS S SC

( ) ( )* 0, * 0, * * 0f f =S S

Closest-point-projection (cpp) Geometric interpretation




S

S*

1 CS S*

Closest-point-projection (cpp) The solution of the unconstrained minimization problem yields the closest-point-projection,




( )( )

( )

*

*

*

* * : *, *, * * : * *2 *

* * *, *, * * 2 3

2 2* * *, *, * * *3 3

q

f q

q q K f q q K

H f q H

= = =

= =

= = +

q

q n n

q

q q q q n

( )** * *f= SS S SC

( ) ( )* 0, *, *, * 0, * *, *, * 0f q f q = q q

Closest-point-projection (cpp) Taking into account that the projection takes place in the deviatoric space (octahedral plane), the solution of the unconstrained minimization problem, defining the closest-point-projection, yields,




dev * dev *2 *

* * 2 32* * *3

q q K

H

=

=

= +

n

q q n

( ) ( )* 0, *, *, * 0, * *, *, * 0f q f q = q q

Closest-point-projection (cpp) The solution of the unconstrained minimization problem, defining the closest-point-projection, yields,




( )( )( )( )

23

23

23

dev * * dev * 2 *

dev * * * dev * 2 *

dev dev * * * 2 *

H

H

H

= +

= +

= + +

q q n

q n q n n

q n q n

2dev * * dev * 2 , *3

H = + =

q q n n

For the non-trivial case, using the Kuhn-Tucker complementarity conditions for the cpp, the Lagrange multiplier reads,




( ) ( )23if * 0 then *, *, * dev * * * 0Yf q q > = = q q

( ) ( ) ( )( ) ( )

( ) ( )

2 23 3

2 2 23 3 3

2 23 3

*, *, * dev * 2 *

dev * 2

, , * 2 0

Y

Y

f q H q

K H q

f q K H

= +

= + +

= + + =

q q

q

q

( ) ( )12 23 3* 2 , , 0K H f q

= + + > q

( ) ( )* 0, *, *, * 0, * *, *, * 0f q f q = q q

Closest-point-projection The solution of the unconstrained minimization problem yields the closest-point-projection,




( ) ( )( ) ( )( ) ( )

12 23 3

12 23 3

12 23 3

* 2 , , 2

* 2 , , 2 3

2* 2 , ,3

K H f q

q q K H f q K

K H f q H

= + +

= + +

= + + +

q n

q

q q q n

( )** * *f= SS S SC

Associative plastic flow rule The associative plastic flow rule takes the form,




( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

11 2 23 3

12 213 3

112 2

3 3

1 12 * 2 , ,

1 1* 2 , , 2 3

1 2 1* 2 , ,3

p K H f q

K q q K H f q

H K H f q

= = + +

= = + +

= = + +

q n

q

q q q n

( )11 *p

= E S SC

Associative plastic flow rule The associative plastic flow rule can be recast in the form,

where the relaxation time takes the form,




2 3

p

=

=

=

n

n

( ) ( ) ( )12 23 31 12 , , , ,K H f q f q

= + + = q q

( ) 12 23 3: 2 K H

= + +

Computational Solid MechanicsComputational PlasticityNmero de diapositiva 2Nmero de diapositiva 3Nmero de diapositiva 4Nmero de diapositiva 5Nmero de diapositiva 6Nmero de diapositiva 7Nmero de diapositiva 8Nmero de diapositiva 9Nmero de diapositiva 10Nmero de diapositiva 11Nmero de diapositiva 12Nmero de diapositiva 13Nmero de diapositiva 14Nmero de diapositiva 15Nmero de diapositiva 16Nmero de diapositiva 17Nmero de diapositiva 18Nmero de diapositiva 19Nmero de diapositiva 20Nmero de diapositiva 21Nmero de diapositiva 22Nmero de diapositiva 23Nmero de diapositiva 24Nmero de diapositiva 25Nmero de diapositiva 26Nmero de diapositiva 32Nmero de diapositiva 33Nmero de diapositiva 34Nmero de diapositiva 35Nmero de diapositiva 36Nmero de diapositiva 37Nmero de diapositiva 38Nmero de diapositiva 39Nmero de diapositiva 40Nmero de diapositiva 41Nmero de diapositiva 42Nmero de diapositiva 43Nmero de diapositiva 44Nmero de diapositiva 45Nmero de diapositiva 46Nmero de diapositiva 47Nmero de diapositiva 48Nmero de diapositiva 49Nmero de diapositiva 50Nmero de diapositiva 51Nmero de diapositiva 52Nmero de diapositiva 53Nmero de diapositiva 54Nmero de diapositiva 55Nmero de diapositiva 56Nmero de diapositiva 57Nmero de diapositiva 58

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Chapter 3. J2 Plasticity Models v1.0

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