chapter 1. 1d 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 1. 1D Plasticity Models

Contents 1. 1D Rate independent plasticity models

1. Motivation 2. Perfect plasticity model 3. Hardening plasticity model

2. 1D Rate dependent plasticity models 1. Motivation 2. Perfect plasticity model 3. Hardening plasticity model

1D Plasticity Models > Contents

Contents

April 1, 2015 Carlos Agelet de Saracibar 2





Contents


Physical and rheological models: Linear elastic spring

1D Plasticity Models > Rate Independent Plasticity Models

Motivation


Physical model: Rigid block sliding on a rough surface. Coulombs frictional law


Motivation


0

0

no equilibrium

F N

F N

F N

< =

=

>

Rheological model: Coulombs frictional device


Motivation


0

0

not admissible

e

e

e

< =

=

>

Rheological model: Elastic spring + frictional device


Motivation


( )( )

0, , ,

0, , 0,

not admissible

f e e e fe

e f e fe

e

E E E

E E

< = = = = =

= = = = = =

>

Rheological model: Elastic spring + frictional device


Motivation


Hypothesis Within the framework of the infinitesimal deformation theory, we introduce the following hypothesis for a 1D rate-independent linear elastic-perfect plastic model, within the incremental theory of plasticity: H1. Additive split of the infinitesimal strain H2. Set of plastic internal variables: plastic strain


Perfect plasticity model


e p = +

{ }:p p=E

H3. Free energy per unit of volume: quadratic elastic potential H4. Clausius-Planck inequality. Linear elastic constitutive equation, and reduced plastic dissipation inequality




( ) ( ) ( )21: 2e e eW E = =

( ): 0e = D( ): 0e e ee p = = + D

, : 0e e pE = = = D

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




( ){ }( ) ( ){ }

( ){ }

: 0

int : 0

: 0

Y

Y

Y

f

f

f

= =

= = Rate Independent Plasticity Models



0 if 0

0 if 0

pY

pY

= > = >

= < = = =

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

( )( )

if 0 then 0

if 0 then

Plastic loading

Elastic loading/unloadi0 ng

f

f

> =

< =

H8. Plastic consistency condition Plastic loading: plastic consistency (or persistency) condition




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

( ) ( )( ) ( )

if 0 and 0 then 0

if 0 and

Plastic loading

Elastic0 unloathe di gn n0

f f

f f

= > =

= < =

( ) ( ) ( )sgn sgn sgn 0pf f E E E E = = = =

( )1 sgn 0E E = >

( ) ( )0 and 0 0f f = > =

Trial stress rate Plastic loading: plastic multiplier (or plastic consistency) parameter




0trial p trialf f f f E f E = = = =

( )1 1 sgn 0trial trialE f E = = >

: , :trial trial pE E = =

( )1 1 sgn 0trial trialE f E = =

Plastic loading/elastic unloading from the yield surface Case 1 Case 2




if 0 then

if 0 then 0, 0 0

trial

trial

f

f f E f

= < < =

if 0 the Elastic unloadingn 0trialf < =

if 0 then Plastic loadi0 ngtrialf > >

if 0 then

if 0 then 0 0, 0

trial

trial

f

f f E f

>

= = > > =

Plastic loading/elastic unloading from the yield surface Geometric interpretation




trial trial

Y Y

trialtrial ( )f ( )f

Elastic unloading Elastic unloading

Plastic loading Plastic loading

Stress-strain curve for a rate-independent elastoplastic model with linear elastic and perfect plastic response




Stress-strain curve: hardening plasticity


Hardening plasticity model


Stress-strain curve: Bauschinger effect (kinematic hardening)




Stress-strain curve: isotropic/kinematic hardening




Hypothesis Within the framework of the infinitesimal deformation theory, we introduce the following hypothesis for a 1D rate-independent elastoplastic model with linear elastic response, and linear isotropic and kinematic hardening, within the incremental theory of plasticity: H1. Additive split of the infinitesimal strain H2. Set of plastic internal variables




e p = +

{ }: , ,p p =E

H3. Free energy per unit of volume: quadratic elastic, isotropic hardening and kinematic hardening potentials




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

0, 0, 0E K H E K E H+ + > + > + >

( ) ( )

( )

( )

2

2

2

1 Elastic potential21 Isotropic hardening potential21 Kinematic hardening potential2

e eW E

K

H

=

=

=

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




( ): , , 0e = D

( ):

0

e

e e

e

p

=

= +

D

, : , :

: 0

ee

p

E q K q H

q q

= = = = = =

= + + D

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




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

:q K = =

( )1 2: :m

Yq k k = = +

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




( ){ }( ) ( ){ }

( ){ }

: , , , , 0

int : , , , , 0

: , , , , 0

Y

Y

Y

q q f q q q q

q q f q q q q

q q f q q q q

= = +

= = + Rate Independent Plasticity Models



( )( ) ( )

( )( ) ( )

if 0, , , 0 then

, , sgn

, ,

, , sgn

Y

p

q

q

f q q q q

f q q q

f q q

f q q q

> = + =

= =

= =

= =

H7. Kuhn-Tucker loading/unloading conditions




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

( )( )

if 0 then , , 0

i

Plastic loading

Elastic loading/unloadingf , , 0 then 0

f q q

f q q

> =

< =

H8. Plastic consistency condition Plastic loading: plastic consistency (or persistency) condition




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

( ) ( )( ) ( )

if , , 0 and 0 then , , 0

if , , 0 and

Plastic loading

Elastic unlo, , 0 the dingn 0 a

f q q f q q

f q q f q q

= > =

= < =

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

Plastic loading: plastic consistency (or persistency) condition




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

2 2

sgn sgn

sgn sgn sgn

sgn sgn sgn

sgn 0

q q

p

f f f q f q

q q q q

q E q E K q H

q E q E K q H

q E E K H

= + +

= +

= +

=

= + + =

( ) ( )1 sgn 0E K H q E = + + >

Trial stress rate Plastic loading: plastic multiplier parameter




( ) 0

q q

trial pq q

trial

f f f q f q

f f E f K f H

f E K H

= + +

=

= + + =

( ) ( ) ( )1 1 sgn 0trial trialE K H f E K H q = + + = + + >

: , :trial trial pE E = =

( ) ( ) ( )1 1 sgn 0trial trialE K H f E K H q = + + = + +

Plastic loading/elastic unloading from the yield surface Case 1 Case 2




( )if 0 and 0 then

0, 0 0

trial

trial

f

f f E K H f

< >

= + + < < =

if 0 the Elastic unloadingn 0trialf < =

if 0 then Plastic loadi0 ngtrialf > >

( )if 0 and 0 then

0 0, 0

trial

trial

f

f f E K H f

> =

= + + > > =

Plastic loading/elastic unloading from the yield surface Geometric interpretation




trial trial

( )Y q q Y q q +

trialtrial ( ), ,f q q ( ), ,f q q

Elastic unloading Elastic unloading

Plastic loading Plastic loading

Isotropic hardening Geometric interpretation: expansion (without translation) of the yield surface




Y0Y

Y q ( )Y q

Kinematic hardening Geometric interpretation: translation (without expansion) of the yield surface




Y0Y

Y q +( )Y q

Isotropic + Kinematic hardening Geometric interpretation: expansion and translation of the yield surface




Y0Y

Y q q +( )Y q q

Stress-strain curve for a rate-independent elastoplastic model with linear elastic and linear isotropic hardening plasticity




Stress-strain curve for a rate-independent elastoplastic model with linear elastic and linear kinematic hardening plasticity




Stress-strain curve for a rate-independent elastoplastic model with linear elastic, linear isotropic/linear kinematic hardening plasticity




Reduced plastic dissipation Plastic dissipation rate per unit of volume




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

( )( )

:

sgn sgn

sgn

, ,

0

p

Y

Y

q qq q q q

q q q

q q

f 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,




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

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

{ } { }: , , , : , ,q q p p = =S 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

Maximum plastic dissipation principle Proof 1. Maximum plastic dissipation implies associative plastic flow rule and Kuhn-Tucker loading/unloading conditions

Constrained minimization problem Unconstrained minimization problem: Lagrange multipliers




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

( ) ( ) ( ), , : ,p p f = + E D ET T TL( )arg min , ,p = S ET TL

The optimality conditions of the unconstrained minimization problem yield




( ) ( ) ( )( )

, , ,

0

p p

p

f

f

== =

= +

= + =

T T T T ST S T S

S

T E D T E T

E S

L

( ) ( )0, 0, 0f f =S S( )p f= SE S

Maximum plastic dissipation principle Proof 2. Maximum plastic dissipation implies convexity of the yield surface

Unconstrained minimization problem




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

( ) ( ) ( ) ( )

, , , ,

, ,

, ,

if 0 then

p p

p p

p p

f f

f f

f f f

f f f

+ +

>

S

S

S E E

D S E S D E

D E D S E S

S S S

S S S

T T

T T T

T T T

T T T

T T T

L L

Maximum plastic dissipation principle Proof 3. Associative plastic flow rule, Kuhn-Tucker loading/ unloading conditions and convexity of the yield surface implies maximum plastic dissipation

Using associative plastic flow rule Using convexity of the yield surface




( )( ) ( ) ( )

p

p

f

f

=

=

S

S

E S

S E S ST T T

( ) ( ) ( )( )p f f S E ST T T

Maximum plastic dissipation: elastic unloading (trivial case) Maximum plastic dissipation: plastic loading




( )( ) ( )

if 0 then 0 and

, ,

p

p p

=

S E

D S E D E

T

T T

( )( ) ( )

( ) ( )

if 0 then 0 and

0

, ,

p

p p

f

f

> =

S

S E

D S E D E

T T T

T T

Hypothesis H1. Additive split of the infinitesimal strain 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 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


Rheological model: dashpot

1D Plasticity Models > Rate Dependent Plasticity Models

Motivation


1 =

Rheological model: elastic spring + frictional device | dashpot


Motivation


e p = +

e p

E

No se puede mostrar la imagen en este momento.

( ) ( ) ( ): sgn sgnex Y f = =

( ) ( ) ( ) ( )1 1 1sgnp ex f f f = = =

Hypothesis Within the framework of the infinitesimal deformation theory, we introduce the following hypothesis for a1D rate-dependent linear elastic-perfect plastic model, within the incremental theory of plasticity: H1. Additive split of the infinitesimal strain H2. Set of plastic internal variables: plastic strain




e p = +

{ }:p p=E

H3. Free energy per unit of volume: quadratic elastic potential H4. Clausius-Planck inequality. Linear elastic constitutive equation, and reduced plastic dissipation inequality




( ) ( ) ( )21: 2e e eW E = =

( ): 0e = D( ): 0e e ee p = = + D

, : 0e e pE = = = D

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




( ){ }( ) ( ){ }

( ){ }

: 0

ext : 0

: 0

Y

Y

Y

f

f

f

= =

= = >

= = =

YY

H6. Associative plastic flow rule

The associative plastic flow rule can be cast as

Geometric interpretation




( ) ( ) ( )1sgn , 0p f f = = =

( ) ( ) ( ) ( )1 1 sgnp f f f = =

( ) ( ) ( ) ( )( )1 1 1sgn sgnp Y Yf f = = =

Closest-point-projection (cpp)

Natural relaxation time

Associative plastic flow rule




( )sgn if extif

Y

= =

* P

1: E =

( )11 *p E

=

Associative plastic flow rule: geometric interpretation




( )11 *p E

=

Y Y

( )EP=*

Y Y

( )EP=*

Stress-strain curve for a rate-dependent linear elastic, perfect plastic model




Perzyna model The associative plastic flow rule given by

is equivalent to the maximization of a regularized plastic dissipation, and can be obtained as the solution of the following unconstrained minimization problem




( ) ( ) ( ) ( )1 1 sgnp f f f = =

( )( )arg min , p = D( ) ( ) ( ) 21, : , 2

p p f = D D

Perzyna model Proof. The solution of the unconstrained minimization problem reads,




( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( )

2

:

1, : ,2

1, : , 0

1 0

p p

p p

p

f

f f

f f

=

=

==

=

D D

D D

( )( )arg min , p = D

Duvaut-Lions model The associative plastic flow rule given by

is equivalent to the maximization of a regularized plastic dissipation, and can be obtained as the solution of the following unconstrained minimization problem written in terms of the cpp




( )( )arg min , p = D

( )11 *p E

=

( ) ( ) 121, : , 2p p

E

= D D *

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

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




( )arg min = * * *

( ) ( ) ( )121 1 12 2E E = = * * * *

( ) ( ) ( ), ; : f = +* * * * *L( )arg min , ; = * * * *L

Proof 1. The solution of the unconstrained minimization problem reads,




( ) ( )( ) ( ) ( )

( )

1

212

1 1

1 1:

, : ,

, : , 0

0

p pE

p p

p

E

E

=

= =

= =

D D *

D D *

*

( )( )arg min , p = D

( )11p E

= *

Proof 2. The solution of the closest-point-projection, obtained as the solution of an unconstrained minimization probem written in terms of the complementary energy, reads,




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

( ) ( )1:

, ; :

, ; : 0

0

f

f

E f

= +

= + =

+ ==* * *

*

* * * * *

* * * * *

* * *

L

L

( )arg min , ; = * * * *L

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

The solution of the unconstrained minimization problem yields the following results for the closest-point-projection,




( ) ( )( )

( )

1 0

sgn sgn sgn

sgn sgn

, sgn sgn

E f

E f

E

E

E

+ =

+ =

+ =

+ =

= =

*

*

* * *

* * *

* * * *

* * *

* * *

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

and the closest-point-projection takes the form,




( )( ) ( )

( )1

if 0 then 0

0

0

Y

Y

f

f E f E

E f

> = =

= = =

= >

* * *

* * *

*

( ) , sgn sgnsgn sgn

Y

Y

f

= = =

= =

* *

* * *

Hypothesis Within the framework of the infinitesimal deformation theory, we introduce the following hypothesis for a 1D rate-dependent linear elastic-hardening plasticity model, within the incremental theory of plasticity: H1. Additive split of the infinitesimal strain H2. Set of plastic internal variables




e p = +

{ }: , ,p p =E

H3. Free energy per unit of volume: quadratic elastic, isotropic hardening and kinematic hardening potentials




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

0, 0, 0, 0E E K H E K E H> + + > + > + >

( ) ( )

( )

( )

2

2

2

1 Elastic potential21 Isotropic hardening potential21 Kinematic hardening potential2

e eW E

K

H

=

=

=

H4. Clausius-Planck inequality. Linear elastic constitutive equation, linear isotropic/linear kinematic hardening, and reduced plastic dissipation inequality




( ): , , 0e = D

( ):

0

e

e e

e

p

=

= +

D

, : , :

: 0

ee

p

E q K q H

q q

= = = = = =

= + + D

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




( ){ }( ) ( ){ }

( ){ }

: , , , , 0

ext : , , , , 0

: , , , , 0

Y

Y

Y

q q f q q q q

q q f q q q q

q q f q q q q

= = +

= = + >

= = + =

Y q q +( )Y q q

H6. Associative plastic flow rule H7. Plastic multiplier




( ) ( )( )( ) ( )

, , sgn

, ,

, , sgn

p

q

q

f q q q

f q q

f q q q

= =

= =

= =

( )1 , , 0f q q

=

Reduced plastic dissipation Plastic dissipation rate per unit of volume




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

( )( )( )

:

sgn sgn

sgn

, ,

0

p

Y

Y

q qq q q q

q q q

q q

f q q

= + +

= +

= +

= +

= +

= +

D

Perzyna model The 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 following 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

Duvaut-Lions model The 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 following 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 The closest-point-projection (cpp) is obtained as the solution of the following constrained minimization problem, written in terms of the complementary energy 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 The solution of the unconstrained minimization problem, defining the closest-point-projection, yields,




( ) ( ) ( )( ) ( )

* * *

1*

, *; * : * * *

: * * *

f

f

= +

= + = 0S S S

S

SS S S S

S S S

L

C

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

( )** * *f= SS S SC

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





( ) ( )( )( ) ( )

*

*

*

* * *, *, * * sgn * *

* * *, *, * *

* * *, *, * * sgn * *q

q

E f q q E q

q q K f q q q K

q q H f q q q H q

= =

= =

= = +

( )** * *f= SS S SC

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





( ) ( )( ) ( )

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

* * * sgn * *

* * sgn * * sgn

* sgn * *

* * * sgn * * sgn

q q E H q

q q q q

E H q

q E H q q q

= +

=

+

+ + =

( ) ( ) ( )* * * , sgn * * sgnq q E H q q = + =

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




( )if * 0 then *, *, * * * * 0Yf q q q q > = + =( ) ( )

( )( )

( ) ( )

*, *, * * *

* *

*

, , * 0

Y

Y

Y

f q q q E H q

q E H q K

q E K H q

f q q E K H

= + +

= + +

= + + +

= + + =

( ) ( )1* , , 0E K H f q q = + + >

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





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

1

1

1

* , , sgn

* , ,

* , , sgn

E E K H f q q q

q q K E K H f q q

q q H E K H f q q q

= + +

= + +

= + + +

( )** * *f= SS S SC

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




( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( ) ( )

11

11

11

1 1* , , sgn

1 1* , ,

1 1* , , sgn

p E E K H f q q q

K q q E K H f q q

H q q E K H f q q q

= = + +

= = + +

= = + +

( )11 *p

= E S SC

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

where the relaxation time takes the form,




( )

( )

sgn

sgn

p q

q

=

=

=

( ) ( ) ( )11 1, , , ,E K H f q q f q q

= + + =

( ) 1: E 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 27Nmero de diapositiva 28Nmero de diapositiva 29Nmero de diapositiva 30Nmero de diapositiva 31Nmero 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 58Nmero de diapositiva 59Nmero de diapositiva 60Nmero de diapositiva 61Nmero de diapositiva 62Nmero de diapositiva 63Nmero de diapositiva 64Nmero de diapositiva 65Nmero de diapositiva 66Nmero de diapositiva 67Nmero de diapositiva 68Nmero de diapositiva 69Nmero de diapositiva 70Nmero de diapositiva 71Nmero de diapositiva 72Nmero de diapositiva 73Nmero de diapositiva 74Nmero de diapositiva 75Nmero de diapositiva 76Nmero de diapositiva 77Nmero de diapositiva 78Nmero de diapositiva 79Nmero de diapositiva 80Nmero de diapositiva 81Nmero de diapositiva 82Nmero de diapositiva 83Nmero de diapositiva 84Nmero de diapositiva 85

chapter 1. 1d plasticity models v1.0

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