chapter viii elastoplastic fem part i

1Copyright by A. Erman Tekkaya 2003

Chapter 8General Elasto-Plastic Finite Element

Solution• Introduction• Part A: Review of Continuum Mechanics

– Nonlinear Kinematics of Deformation– Stress Definitions– Objectivity– Elasto-Plastic Constitutive Law

• Part B: Implicit Solution– Linearization of Principle of Virtual Work– Integration of Constitutive Law– Incremental Objectivity and Stress Update

• Part C: Explicit Solution– General Formulation– Discretization– Stability


Aim of Lecture

This lecture aims

• To review fundamentals of continuum mechanics

• To derive the elastoplastic material law for large strains and large displacements

• To overview the various discretization procedures


Introduction (1)

The principle of virtual displacements is valid for any body and reads (neglecting mass forces and inertia effects):

{ } { } { } { }t

T T

V A

T dV t u dAδε δ⋅ = ⋅∫ ∫ or:t

ij ij i iV A

T dV t u dAδε δ=∫ ∫

internal virtual work

external virtual work

internal virtual work

external virtual work

with Tij the Cauchy stress tensor, ti the traction vector and uthe displacement vector.


Introduction (2)Remarks:1. This principle must be applied at the current

(deformed) configuration, at t = t0 + ∆t2. The virtual “strain” due to the virtual displacement is

in fact the “engineering” strain as defined at the current configuration:

12

jiij

j i

uux x

ε ∂∂

= + ∂ ∂ 3. The principle is the weak form of the static equilibrium

equations, i. e. the Euler equation corresponding to the principle of virtual displacements are the differential equations of equilibrium.

4. However, since the current (deformed) configuration is not known, we end up with a non-linear problem.


Introduction (3)This nonlinearity results due to the nonlinear kinematics and due to elasto-plastic material behaviour.

The large displacement elastic behaviour as well as the continuous change of the configuration of the body necessiates the introduction of nonlinear kinematics concepts for the deformation: We can not add the Cauchy stress increment during time increment ∆t to the Cauchy stress at time t0 due to eventual rigid-body rotations.

For this purpose we will adapt the notation of Ramm & Bathe:

00 0

12

ttj

ij

ji

i

t uux x

ε ∂∂

= + ∂ ∂

configuration of quantity

reference configuration coordinate axis (components)


PART AReview of Continuum Mechanics

• Nonlinear kinematics• Stresses• Elasto-Plastic Constitutive Equations


Kinematics of Deformation (1)

x2,u2

A ,V0 0

prescribeddisplacements

prescribedforces

tti

tti

Body at timeB t0

P0

Q0

0dxi

P

Q

tF

tF

Body B

tAt

tAu

Q

P

at time t

t tA, V

0xi 0

xi

Ptxi

Qtxi

tdxi

x3 3,u

x1,u1

Qui

Pui


Kinematics of Deformation (2)

The motion can be described by:

( )0 ,t ti i ix x x t=

The neighborhood of a particle can be described by:( )0

00

,ti it

i jj

x x tdx dx

x∂

=∂

00

t ti ij jdx F dx=or:

0 0

tt i

ijj

xFx

∂=∂

is the deformation gradient tensor

where


Finite Strain Measure (1)Remarks:1. All equations are valid for finite displacements2. Only an infinitesimal neighborhood of a particle is described3. The equation of motion is invertible4. The deformation gradient tensor is not a measure of strain

A measure of strain is defined by considering material line lengths: 2 0 2 0 0

0t t

i ij jds ds dx E dx− =

where the Green-Lagrangian strain tensor is defined by:

( )0 0 0 0 0

1 12 2

t tt t t m m

ij mi mj ij iji j

x xE F Fx x

δ δ ∂ ∂

= − = − ∂ ∂


Finite Strain Measure (2)Notice that the finite strain can be written in terms of the infinitesimal strain and rotation tensor as:

( )0 0 0 0 0 0 0 0 0 012

t t t t t t t t t tij ij im mj im mj im mj im mjE ε ε ε ω ε ε ω ω ω= + − + −

where, the infinitesimal strain tensor is given by:

0 0 0

12

jt iij

j i

uux x

ε ∂∂

= + ∂ ∂ (does not measure strain for large deformations)

and the infinitesimal rotation tensor by:

0 0 0

12

jt iij

j i

uux x

ω ∂∂

= − ∂ ∂

(does not measure rigid body rotations for large deformations)


Volume and Density ChangesIt can be shown that the relation ship between infinitesimal volumes is given by:

00

t tijdV F dV=

where, the Jacobian determinant 0tJ is defined by

0 0t t

ijJ F=

Due to mass conservation note that:

00t t

ijFρ ρ=

where ρ is the density.


Polar Decomposition Theorem (1)The deformation gradient tensor can be decomposed into a unique product:

0 0 0t t t

ij im mjF R U=

where Rij is the rotation tensor and Uij the right stretch tensor.

The stretch tensor presents the pure deformation, whereas the rotation tensor represents the rigid body rotation of the principal axes of the strech tensor.

Alternatively, the decomposition can be done also by:

0 0 0t t t

ij im mjF V R=

where Vij the left stretch tensor.


Polar Decomposition Theorem (2)It can be shown that:

( )0 0 0t t t

ij mi mjU F F=

and

( )0 0 0t t t

ij im jmV F F=


Distortional Component of Fij

The deformation gradient can be decomposed into a distortional and volumetric portions:

( )1 3

0 0 0ˆt t t

ij ijF J F=

0ˆtijFwhere is the isochoric part of the deformation gradient.


Area ChangesThe relationship between an undeformed and deformed area element is given by:

0 00 0t t T t t

i ij jdA n J F n dA−=

where ni is the unit normal vector and dA the area of the area element considered.


Velocity Gradient, Rate of Deformation Tensor & Spin Tensor (1)

tt i

id xvdt

=Velocity Field:

tt it ij t

j

d vLd x

=The velocity gradient:

The velocity gradient can be decomposed as:t t tt ij t ij t ijL D W= +

where the rate of deformation tensor is defined by:

12

ttjt i

t ijj i

vvDx x

∂∂= + ∂ ∂

(symmetric)



and the rotation tensor is defined by:

12

ttjt i

t ijj i

vvWx x

∂∂= − ∂ ∂

(skew-symmetric)

It can be shown that the rate of deformation tensor is a measure for the strain-rate:

2 2t t t ti t ij jds dt dx D dx=

The spin tensor Wij can be interpreted as the rate of rigid-body rotation of the principal axes of the rate of deformation tensor Dij.



Recall carefully that

00 0

tij t t t

mi t mn nj

d EF D F

dt=

and

( ) ( ) ( )0 0 01 10 0 0 0 0

12

t t ttim nm mkt ij t t t t t

jm in mk nm jk

d R d U d Ud WR R U U R

dt dt dt dt− −

= + −

Also, the rate of volume change is given by:

( )0 0( )tt d Jd dV dV

dt dt=

( )00

tt t

t kk

d JJ D

dt=and


Stress Definitions (1)

x2

0 0A, V

Body B

P0

Body B

P

at time t

t tA, V

0xi

x3

x1

at time0t 0

ni

tni

0dhi

tdfi

0dA

tdA

txi

The Cauchy stress tensor Tij is defined as:

t t ti t ij jt T n=

such that:t t t t

i t ij jdf T n dA=

current area

current force

t t ti idf t dA=Note that:



0 00

t ti ij jdf H n dA=

Instead of relating the current force to the current area, we can relate the original (undeformed) area to the current force:

x2

0 0A, V

Body B

P0

Body B

P

at time t

t tA, V

0xi

x3

x1

at time0t 0

ni

tni

0dhi

tdfi

0dA

tdA

txi

This defines the first Piola-Kirchhoff stress tensor.

The first Piola-Kirchhoff stress tensor is unsymmetric. It corresponds to the “engineering” stress definition.

initial area

current force



x2

0 0A, V

Body B

P0

Body B

P

at time t

t tA, V

0xi

x3

x1

at time0t 0

ni

tni

0dhi

tdfi

0dA

tdA

txi

10 0t t t

i ij jdh F df−=

To introduce a symmetric stress tensor defined wrt the undeformed configuration, we introduce a pseudo-force vector:

Now the second Piola-Kirchhoff stress tensor is defined as:

0 00 0t t

i ij jdh S n dA=

initial area

pseudoforce


Stress Definitions (4)The various stress definitions are related to each other by:Cauchy stress ⇔ First Piola-Kirchhoff stress:

0 00

tt t tt ij im mjT F Hρ

ρ=

Cauchy stress ⇔ Second Piola-Kirchhoff stress:

0 0 0t t t

ij im jmH S F=

Second Piola-Kirchhoff stress ⇔ First Piola-Kirchhoff stress:

0 0 00

tt t t tt ij im mn jnT F S Fρ

ρ=

Note that for small displacements/strains all stress measures are approximately equal.


Stress Definitions (5)An important issue is that the the stresses are “work-conjugate” with certain deformation measures. The stress-power per unit mass is given by:

( )000

1 1stress power t

ijt t tt ij t ij ijt

d ET D S

dtρ ρ≡ =

From above considerations we can also derive the Kirchhoff stress tensor:

0

1 1stress power t t t tt ij t ij t ij t ijt T D G D

ρ ρ≡ =

0t tt ij t ijtG Tρ

ρ=or:


Stress Definitions (6)To interpret physically the second Piola Kirchhoff stress tensor, visualize a rigid-body deformation:

0 0 0 0t t t t

ij im mj imF R U R= =Then:0

0 0 1t tijtJ Fρ

ρ= = =

0 0 00

tt t t tt ij im mn jnT F S Fρ

ρ=

Hence, from:0

0 0 0t t t t

mn mi t ij njtS F T Fρρ

=

0 0 0t t t t

mn mi t ij njS R T R=or for rigid-body rotations:


Objectivity (1)

x1

x2

x3

A ,V0 0

Body B

P0

Q00dxi

Body B

Q

P

at time t1

at time t0

P

Q

Body Bat time t2

0 ij

1F Qij

1dxi

2dxi

0 ij

2F

rigid body rotatio

ndefo

rmation

1 2 0 2 0 1 00i ij jds ds dx E dx− =

2 2 0 2 0 2 0 0 1 00 0i ij j i ij jds ds dx E dx dx E dx− = =

110 0

iij

j

xFx

∂=∂

220 0

iij

j

xFx

∂=∂

2 10 0ij im mjF Q F=

2 10 0ij im mn njE Q E Q=

2 10 0ij mnE E≠

1 Tij ijQ Q− =

Although

they have the same physical meaning!


Objectivity (2)By definition any vector vi and any tensor Tij is objective, if it transforms under a rigid-body rotation Qij according to:

i im mv Q v=

ij im mn jnT Q T Q=

where Qij is an orthogonal transformation:

1 Tij ijQ Q− = 1 T

im mj ijQ Q δ− =or


Objectivity (3)The Cauchy stress tensor, for instance, is objective since:

t tt ij im t mn jnT Q T Q=

However taking the material time derivative of it yields:

( )t t t tt ij im t mn jn im t mn jn im t mn jnT Q T Q Q T Q Q T Q= + +

Hence, the rate of Cauchy stress tensor is not objective unless the rigid-body rotation is not time-dependent.


Objectivity (4)To illustrate this fact, consider a rod under axial loads and rotated by 90o:

F

F

x

y

F

F

time t

time t+ t�

rigidbody rotation

The stress state at time t is:

0 00 0 00 0 0

xx

ij

TT

=

The stress state at time t+Dt is:

0 0 00 00 0 0

ij yyT T = 0ijT ≠ 0ijD =Obviously: but:


Objectivity (5)So, a constitutive equation of type

ij ijkl klT C D=

can not be used.

An objective stress rate can be defined by the so-called Jaumann rate:

ij ij im mj im mjT T W T T W= − +

where Wij is the already defined rotation tensor.

Note that for the rotated by example the Jauman rate of stress vanishes!


x1

x2

x3

A ,V0 0

Body BP0

Q00dxi

Body B

Q

P

at time t1

at time t0

Body B

at ntermedi iateconfiguration

0 ij

1F

1dxi purely elastic

tota

l defo

rmation

0 ij

1F

el

0 ij

1F

pl

purely plastic

Q

P1dxi

pl

Hyperelastic-Plastic Model:

0 0 0t t el t pl

ij im mjF F F=

Elastic work is independent of the path and the elastic strains are derived from a potential.

Elastic-Plastic Constitutive Law (1)



Hypoelastic-Plastic Model:t t el t plt ij t ij t ijD D D≈ +

This equation is only valid if

1t elt ijD 1t el

t ijWand

This is however the case for metals during large plastic deformation. Therefore, we will cover here only the hypo-elastic model.



Recall the Levy-Mises flow rule for isotropic hardening:32

pli i

dYE TH Y

′=⋅

Dividing both sides of the equation by an infinitesimal time dt and switching to tensorial notation yields:

32

plij ij

YD TH Y

′=⋅

where the rate of change of the yield (flow) stress is given as:

dYYdt

=


Elastic-Plastic Constitutive Law (4)The elastic strain rates are given by the generalized Hooke’s law as:

1 1 22

elij ij H ijD T T

G Eυ δ−′= +

el plij ij ijD D D= +Applying the hypoelastic-plastic assumption:

yields the well-known Prandtl-Reuß equations:

1 1 2 32 2ij ij H ij ij

YD T T TG E H Y

υ δ β−′ ′= + +⋅

. . . .

. . . . . .

1 for and > 0

0 for or and > 0

v M v M

v M v M v M

Y

Y Y

σ σβ

σ σ σ

= == = < =

with


Elastic-Plastic Constitutive Law (5)The Prandtl Reuß equations can be transformed in a more convinient way following the following procedure:

2 3ij ijT T

YY

′ ′=Taking the time derivative

of the flow condition 22

3 ij ijY T T′ ′= yields:

2 3ij ijT T

YY

′=

Or, (*)

Furthermore:

1 1 2 12 ij H ij ij mm ijT T T TG E E E

υ υ υδ δ− +′ + = − (**)


Elastic-Plastic Constitutive Law (6)Inserting the equations (*) and (**) into the Prandtl-Reuß equation yields:

2

1 94

ij nmij in jm ij nm nm ijnm nm

T TD T T

E E H Yυ υδ δ δ δ β

′ ′ += − + =

where the elastic-plastic consititutive tensor is given by:

2

1 94

ij nmijnm in jm ij nm

T TE E H Yυ υδ δ δ δ β

′ ′+= − +

For the use in the finite element discretization, this equation has to be converted however into:

1ij ijnm nmT D−=


Elastic-Plastic Constitutive Law (7)For inverting the Prandtl Reuß equations, we start from:

ij ij H ijT T T δ′= +It can be shown easily that:

( )3 1 2H ij ij nm nm

K

ET Dδ δ δυ

=−

On the other hand

( )2ij ij ijT G D Tβ λ′ ′ ′= −where

249

22

nm nm

nm nm

GT DG T T Y H

λβ

′ ′=

′ ′ +



After some computation, the inverted equation is found as:

( )22 13 3

21 2 1

ij nmij in jm ij nm nm

G

T TT G D

Y Hυδ δ δ δ βυ

′ ′ = + − − +

Hence, in1

ij ijnm nmT D−=

the inverse constitutive fourth order tensor reads:

( )1

22 13 3

21 2 1

ij nmijnm in jm ij nm

G

T TG


−′ ′

= + − − +



The Prandtl-Reuß constitutive equation is only valid for small total strains and small rotations, since in the given form the relation does not fulfill the axiom of objectivity. A generalization of the Prandtl-Reuß equation reads:

( )22 13 3

21 2 1

ij nmij in jm ij nm nm

G

T TT G D


′ ′ = + − − +

where the material time rate of the Cauchy stress tensor is replaced the Jaumann rate of it.


Elastic-Plastic Constitutive Law (10)The deviatoric part of the generalized Prandtl-Reuß relationship can be in the nine-dimensional stress-space as:

{ } { } { }1

3

12 21 2 3 2 3

i ii i i

G

T TT G D G DH Y Y

β ′ ′ ′ ′= − ⋅ +

It can be shown easily that

{ } { }1

3

11 2 3 2 3

pl i ii i

G

T TD DH Y Y

′ ′ ′= ⋅ + So that:

{ } { } { }2 2 pli i iT G D G Dβ′ ′= −



T2'

T1'

T3'

2 {G Dpl}

{Ti}'

2 {G D}'

{Ti}'

total strainloaded elastically

plastic strain partloaded elastically

Radius = 2 3Y

In the three-dimensional principal stress space this can be visualized as:

{ } { } { }2 2 pli i iT G D G Dβ′ ′= −

chapter viii elastoplastic fem part i

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