Code_Aster ®
Version
7.4
Titrate:
Modeling élasto (visco) plastic in great deformations
Date:
14/04/05
Author (S):
V. CANO, E. Key LORENTZ
:
R5.03.21-C Page
: 1/18

Organization (S): EDF-R & D/AMA
Handbook of reference
R5.03 booklet: Nonlinear mechanics
Document: R5.03.21
Modeling élasto (visco) plastic with
isotropic work hardening in great deformations

Summary

One describes here a thermoelastoplastic relation between behavior and isotropic work hardening written into large
deformations and proposed by Simo and Miehe. This model is available in command STAT_NON_LINE
via key word RELATION: “VMIS_ISOT_TRAC” or “VMIS_ISOT_LINE” under the key word
factor COMP_INCR and with key word DEFORMATION: “SIMO_MIEHE”. A viscous version of this model
is also proposed: “VISC_ISOT_TRAC” and “VISC_ISOT_LINE”.
This model is established for three-dimensional modelings (3D), axisymmetric (Axis) and in deformations
plane (D_PLAN).

One presents the writing and the digital processing of this law, as well as the associated variational formulation. It
of a variational formulation eulérienne acts, with reactualization of the geometry and which takes account of
rigidity of behavior and geometrical rigidity.

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Titrate:
Modeling élasto (visco) plastic in great deformations
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:
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Count

matters

1 Introduction ............................................................................................................................................ 3
2 Notations ................................................................................................................................................ 4
3 Recalls on the great deformations .................................................................................................. 5
3.1 Kinematics ..................................................................................................................................... 5
3.2 Constraints ...................................................................................................................................... 6
3.3 Objectivity ........................................................................................................................................ 7
4 Presentation of the model of behavior ............................................................................................ 7
4.1 Kinematic aspect ......................................................................................................................... 7
4.2 Relations of behavior ............................................................................................................. 8
4.3 Choice of the function of work hardening ................................................................................................ 10
4.4 Constraints and variables intern ................................................................................................... 10
4.5 Field of application ...................................................................................................................... 11
4.6 Integration of the law of behavior ........................................................................................... 11
5 variational Formulation .................................................................................................................... 14
5.1 Case of the continuous medium .................................................................................................................... 14
5.2 Discretization by finite elements .................................................................................................... 14
5.3 Form of the tangent matrix of the behavior ................................................................... 16
6 Bibliography ........................................................................................................................................ 18

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Titrate:
Modeling élasto (visco) plastic in great deformations
Date:
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V. CANO, E. Key LORENTZ
:
R5.03.21-C Page
: 3/18

1 Introduction

We present here a thermoelastoplastic law of behavior written in great deformations
proposed by SIMO J.C and MIEHE C. [bib1] which tends in small deformations towards the model of
elastoplastic behavior with isotropic work hardening and criterion of Von Mises, described in [R5.03.02].
The kinematics choices allow, as with the simple reactualization available via the key word
PETIT_REAC, to treat great displacements and great deformations but also of
great rotations in an exact way.
Specificities of this model are as follows:

·
just like in small deformations, one supposes the existence of a slackened configuration,
i.e. locally free of constraint, which makes it possible to break up the total deflection into
a thermoelastic part and a plastic part,
·
the decomposition of this deformation in parts thermoelastic and plastic is not any more
additive as in small deformations (or for the models great deformations written in
rate of deformation with for example a derivative of Jaumann) but multiplicative,
·
the elastic strain are measured in the current configuration (deformed) tandis
that the plastic deformations are measured in the initial configuration,
·
as in small deformations, the constraints depend only on the deformations
thermo rubber bands,
·
the plastic deformations are done with constant volume. The variation of volume is then
only due to the elastic thermo deformations,
·
this model led during its numerical integration to a model incrémentalement objective
(cf [§3.3]) what makes it possible to obtain the exact solution in the presence of great rotations.

A viscous version of this model is also available (law in hyperbolic sine as in
the case of the model of Rousselier ROUSS_VISC, cf [R5.03.07]).

Thereafter, one briefly points out some concepts of mechanics in great deformations, then one
present the relations of behavior of the model and its numerical integration to treat them
equilibrium equations.
One proposes a variational formulation eulérienne, with reactualization of the geometry. For this reason,
one expresses the work of the interior efforts and his variation (with an aim of a resolution by the method
of Newton) for the continuous problem, which provides respectively after discretization by elements
stop the vector of the interior forces and the tangent matrix.

Note Bucket:

One will find in [bib2] or [bib3] a presentation deepened on the great deformations.
This document is extracted from [bib4] where one makes a more detailed presentation of the model
elastoplastic, of its numerical integration and where one gives some examples of
validation.

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Titrate:
Modeling élasto (visco) plastic in great deformations
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:
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2 Notations

One will note by:

Id
stamp identity


tr A
trace tensor A


AT
transposed of tensor A


det A
determinant of A


X
positive part of X


~
With
~
1
deviatoric part of tensor A defined by A = A - (
With
tr) Id
3


:
doubly contracted product: A: B = A B = tr (ABT)
ij ij

I, J



tensorial product: (A B)
=
ijkl
ij
With kl
B

3
With
~ ~
eq
equivalent value of von Mises defined by Aeq =
:
WITH A
2



With
TESTSTEMXŔ
gradient:
=
TESTSTEMXŔ

X



ij
With
div
=
X With
divergence: (divxA) I

X
J
J


, µ, E, K
moduli of the isotropic elasticity


y
elastic limit



coefficent of thermal dilation


T
temperature


Tref
temperature of reference

In addition, within the framework of a discretization in time, all the quantities evaluated at the moment
precedent are subscripted by -, the quantities evaluated at the moment T + T
are not subscripted and them
increments are indicated par. One has as follows:

Q = Q - Q
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Titrate:
Modeling élasto (visco) plastic in great deformations
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:
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3
Recalls on the great deformations

3.1 Kinematics

Let us consider a solid subjected to great deformations. That is to say the 0 field occupied by the solid
before deformation and (T) the field occupied at the moment T by the deformed solid.

Initial configuration
Current configuration deformation
F
0
(T)

Appear 3.1-a: Représentation of the initial and deformed configuration

In the initial configuration 0, the position of any particle of the solid is indicated by X
(Lagrangian description). After deformation, the position at the moment T of the particle which occupied
position X before deformation is given by variable X (description eulérienne).

The total movement of the solid is defined, with U displacement, by:

X = x$ (X, T) = X + U

To define the change of metric in the vicinity of a point, one introduces the tensor gradient of
transformation F:

x$
F =
= Id + U


X
X

The transformations of the element of volume and the density are worth:


D = J D O with J
O
= det F =

where O and are respectively the density in the configurations initial and current.

Various tensors of deformations can be obtained by eliminating rotation in
local transformation. For example, by directly calculating the variations length and angle
(variation of the scalar product), one obtains:

1
E = (C - Id) with C = FTF
2

1
With =
Id - b-1
(
) with B = FFT
2
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Modeling élasto (visco) plastic in great deformations
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:
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E and A are respectively the tensors of deformation of Green-Lagrange and Euler-Almansi and C
and B, tensors of right and left Cauchy-Green respectively.

In Lagrangian description, one will describe the deformation by the tensors C or E because it are
quantities defined on 0, and of description eulérienne by tensors B or A (definite on).

Note:

That is to say a solid undergoing two successive transformations, for example the first
transformation makes pass the solid of the initial configuration 0 to a configuration 1
(tensor F1 gradient/0 and vector u1 displacement/0), then the second transformation of
configuration 1 to 2 (tensor gradient F2 1/and vector displacement u2 1/).
F
F2/1
1/0
0
1
2
F2/0

The passage of configuration 0 to 2 is given by the tensor F2 gradient/0
(displacement U
= U
+ U
2/0
2 1
/
1/0) such as:

F
= F F
2/0
2 1
/
1/0

One obtains then, for example, for the tensor of deformation of Green-Lagrange E
E
= FT E F + E
2/0
1/0 2 1
/
1/0
1/0
where E2/0, E1/0 and E2/1 is the deformations of Green-lagrange of configurations 2 by
report/ratio with 0 associated F2/0, 1 compared to 0 associated F1/0 and 2 per report/ratio
to 1 associated F2 1/, respectively.
This constitutes one of the difficulties encountered at the time of the writing of a law of behavior in
great deformations because one cannot write any more one formula similar to that written in
small deformations, namely 2/0 = 2/1 + 1/0 where is the tensor of total deflection
linearized.
To find 2/0 = 2/1 + 1/0 in small deformations starting from the expression of E2/0,
it is necessary to neglect all the terms of command 2 of u2/0, u1/0 and u2 1/. In this case, one has
E
~
E
~
T
F E
F
~
2/0 - 2/0, 1/0 - 1/0 and 1/0 2/1 1/0 - 2/1.

3.2 Constraints

For the model describes here, the tensor of the constraints used is the tensor eulérien of definite Kirchhoff
by:


J =

where is the tensor eulérien of Cauchy. The tensor thus results from a “scaling” by
variation of volume of the tensor of Cauchy; this is not the case of other tensors of constraints
used (first and second tensor of Piola-Kirchhoff).
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:
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In description eulérienne, the equilibrium equations are given by:

div + F = on

0


X

D
F
N
. = T on




where F is the voluminal force applied to the field, N the normal external with the border F and
F the part of the border of the field where are applied the surface forces td.

3.3 Objectivity

When a law of behavior in great deformations is written, one must check that this law is
objectify, i.e. invariant by any change of space reference frame of the form:

X * = C (T) + Q (T) X

where Q is an orthogonal tensor which represent the rotation of the reference frame and C a vector which translates
translation.
More concretely, if one carries out a tensile test in the direction e1, for example, followed of one
rotation of 90° around e3, which amounts carrying out a tensile test according to e2, then the danger
with a nonobjective law of behavior is not to find a tensor of the constraints
uniaxial in the direction e2 (what is in particular the case with kinematics PETIT_REAC).

4
Presentation of the model of behavior

4.1 Aspect
kinematics

This model supposes, just like in small deformations, the existence of a slackened configuration
R, i.e. locally free of constraint, which then makes it possible to break up the total deflection
in parts rubber band and plastic, this decomposition being multiplicative.
Thereafter, one will note by F the tensor gradient which makes pass from the initial configuration 0 to
current configuration (T), by F p the tensor gradient which makes pass from configuration 0 to
slackened configuration R, and Fe of the configuration R with (T). The index p refers to the part
plastic, the index E with the elastic part.
Initial configuration
Current configuration
F

(T)
0
F p
F E
T = Tref
R
= 0
Slackened configuration

Appear 4.1-a: Décomposition of the tensor gradient F in an elastic part Fe and plastic F p
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:
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By composition of the movements, one obtains the following multiplicative decomposition:

F = FeF p

The elastic strain are measured in the current configuration with the tensor eulérien of
Left Cauchy-Green Be and plastic deformations in the initial configuration by the tensor
G p (Lagrangian description). These two tensors are defined by:

Be
FeFeT
=
, G p
F pTF p
=
-
(
) 1 from where Be
FG pFT
=


The model presented is written in order to distinguish the isochoric terms from the terms of change from
volume. One introduces for that the two following tensors:

F = -
J 1/F
3 and Be = - 2/B
3rd
J
with J = det F

By definition, one a: det F = 1 and det Be = 1.

4.2
Relations of behavior

The law presented is a model thermoélasto (visco) plastic with isotropic work hardening which tends under
the assumption of the small deformations towards the model [R5.03.02] with criterion of Von Mises (it acts of
plastic model). The plastic deformations are done with constant volume so that:

J p
p
= det F = 1 from where J J E
E
=
= det F

The relations of behavior are given by:

·
thermoelastic relation stress-strain:

~
~
E
= µb
3K
2
9K
1
tr =
(J -)
1 -
(T - T) (J
)
2
2
ref.
+ J

·
threshold of plasticity (it is admitted that it is expressed with the constraints of Kirchhoff):

F = eq - R (p) - y

where R is the isotropic variable of work hardening, function of the cumulated plastic deformation p.

·
laws of flow:

~
~
p T
3
E
1
~
E

FG & F = - &
B
= -

3 &
tr B +






eq
3
µ eq
&p = &

For the model of plasticity, the plastic multiplier is obtained by writing the condition of
coherence F & = 0 and one a:
p &,
0 F 0 and p & F = 0
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Modeling élasto (visco) plastic in great deformations
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:
R5.03.21-C Page
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In the viscous case, one takes &p equalizes with:

m
F
p & = &0 HS




0

where &, and m are the viscosity coefficients. Let us announce that this law is reduced to a law of the type
0
0
Norton when the 2 parameters materials & and are very large.
0
0

It is pointed out that:

Be = - 2/B
3rd
J


F = -
J 1/F
3

and that the partition of the deformations is written:

Be
FG pFT
=


For metallic materials where the report/ratio
µ
eq/
is small in front of 1, the expression of the law
flow can be approached by:

~
p T
E
eq
FG & F = - & tr B
+O








éq.4.2-1
eq
µ
eq
where O
is negligible in front of the first term.
µ
It is this last expression which is established in Code_Aster.

Note:

If the deformations are small, one a:
J 1+ tr
E
E
B Id +
2

p
p
G Id -
2

where is the total deflection, E
elastic strain and p
plastic deformation in
small deformations.
By replacing these three expressions in the equations of the law of behavior presented
here, one finds well the elastoplastic thermo traditional model with isotropic work hardening and
criterion of Von Mises.

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:
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4.3
Choice of the function of work hardening

This relation of behavior is available in operator STAT_NON_LINE, under the key word
factor COMP_INCR and argument “SIMO_MIEHE” of the key word factor DEFORMATION. One can choose
for the function of work hardening, a linear work hardening or to provide a diagram traction. Five
relations can be used.

RELATION =
/“ELAS”
/
“VMIS_ISOT_TRAC”
[DEFAUT]
/
“VMIS_ISOT_LINE”
/
“VISC_ISOT_TRAC”
/
“VISC_ISOT_LINE”

For a purely thermoelastic behavior, the user chooses argument “ELAS” (it
behavior is then hyperelastic); for an isotropic work hardening given by a curve of
traction, the user chooses argument “VMIS_ISOT_TRAC” in the plastic case or
“VISC_ISOT_TRAC” in the viscous case and for a linear isotropic work hardening, the argument
“VMIS_ISOT_LINE” in the plastic case or “VISC_ISOT_LINE” in the viscous case.
The various characteristics of material are indicated in operator DEFI_MATERIAU
([U4.23.01]) under the key words:
· ELAS some is the law (one gives the Young modulus, the Poisson's ratio and
possibly the thermal dilation coefficient),
· TRACTION for “VMIS_ISOT_TRAC” and “VISC_ISOT_TRAC” (one gives the curve of
traction),
· ECRO_LINE for “VMIS_ISOT_LINE” and “VISC_ISOT_LINE” (one gives the limit
of elasticity and the slope of work hardening),
· VISC_SINH for “VISC_ISOT_TRAC” and “VISC_ISOT_LINE” (one gives the three
viscosity coefficients).

Note:

The user must make sure well that the “experimental” traction diagram used, is
directly, that is to say to deduce the slope from it from work hardening is well given in the plan
rational constraint = F/S - deformation logarithmic curve ln (1+ L/L)
0 where l0 is
initial length of the useful part of the test-tube, L variation length afterwards
deformation, F the force applied and S current surface. It will be noticed that

F L 1
F L
F L
= F/S =
0
from where =
J =
0. In general, it is well the quantity
0 which is
S L J
S L
S L
0
0
0
measured by the experimenters and this the constraint of Kirchhoff gives directly
used in the model of Simo and Miehe.

4.4
Internal constraints and variables

The constraints are the constraints of Cauchy, thus calculated on the current configuration (six
components in 3D, four in 2D).
The internal variables produced in Code_Aster are:

·
V1, cumulated plastic deformation p,
·
V2, the indicator of plasticity (0 if the last calculated increment is elastic, 1 if not),
1
·
V3, the trace divided by three of the tensor of elastic strain E
B is
E
trb.
3
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Modeling élasto (visco) plastic in great deformations
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Note:

If the user wants to possibly recover deformations in postprocessing of sound
calculation, it is necessary to trace the deformations of Green-Lagrange E, which represents a measurement of
deformations in great deformations. The traditional linearized deformations measure
deformations under the assumption of the small deformations and do not have a direction into large
deformations.

4.5 Field
of use

The choice of a kinematics DEFORMATION: “PETIT_REAC” also makes it possible to treat a law of
elastoplastic thermo behavior with isotropic work hardening and criterion of von Mises into large
deformations. The law is written in small deformations and the taking into account of the great deformations
is done by reactualizing the geometry.
Between the law presented here (SIMO_MIEHE) and PETIT_REAC,

·
there is no difference if the deformations are small
·
there is no difference if the deformations are large but small rotations
·
there are differences if rotations are important.

In particular, the solution obtained with kinematics PETIT_REAC can deviate notably from
exact solution in the presence of great rotations and this whatever the size of the steps of time
chosen by the user, contrary to kinematics SIMO_MIEHE.

4.6
Integration of the law of behavior

In the case of an incremental behavior, key word factor COMP_INCR, knowing the constraint
-
, cumulated plastic deformation p, the trace divided by three of the tensor of deformations
1
rubber bands
E
tr B, displacements U and U and the temperatures T - and T, one seeks with
3
1
to determine (, tr E
p
b).
3
Displacements being known, gradients of the transformation of 0 with -, noted F, and of -
with, noted F, are known.
The implicit discretization of the law gives:
F = FF


J = det F
F = -
J 1/F
3
Be
FG pFT
=


J =
~
~
E
= µb
1
1
2
3
1
tr = K (J -)
1 - K (T - T
) (J
)
3
2
2
ref.
+ J
F =
-
eq - R (p
+ p
) - y ~
~
p
p
T
E
E
p
T
F (G -
-
G
) F = - tr B
p
from where B =
-
FG
F - tr E
B
p

eq
eq
In the plastic case: p,
0 F
0 and F
p = 0
1


-
- p m
In the viscous case:
1
- R (p + p) - - HS
0
eq
y
0
=



&0

T


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Modeling élasto (visco) plastic in great deformations
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:
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Note:

This formulation is incrémentalement objective because the only tensorial quantity
incremental which intervenes in the discretization is &
G p. Comme G p and G p are
measured on the same configuration, i.e. initial configuration, discretization of
&G p, is G p G p G p
=
-
-, is incrémentalement objective.

One introduces Tr
, the tensor of Kirchhoff which results from an elastic prediction (Tr: trial, in English
test):

~
~Tr
eTr
= µb


where

beTr
FG p-FT
Fbe-

FT
=
=
, F

= (J) - 1/3F and J = det
(
)
F

One obtains Be starting from the constraints -
by the thermoelastic relation stress-strain and
trace of the tensor of the elastic strain.

~-
E
1
E
B
=
+ tr B
-
µ
3

Note:

The interest of this formulation is that it is not necessary to calculate the deformation
plastic G p which would oblige us to reverse the gradient of the transformation F. One needs
only to know FG p-FT.

If Tr

-
eq < R (p) + y, one remains elastic. In this case, one a:
1
1
1
p = p, =~Tr + T R Id
and
E
eTr
tr B = tr B

3
3
3
if not, one obtains:

E
eTr
tr B = tr B


This last relation is not possible that if one makes simplification on the law of flow.

eTr
~Tr
µ tr
p


=
B
~ (1+
)
eq

While calculating the equivalent constraint, one brings back oneself to a nonlinear scalar equation out of p:

Tr - - µ
eTr
tr B
p = 0
eq
eq


In the plastic case:
= + R (p + p
)
eq
y
, which leads to p solution of the equation:

Tr - -
-
R (p + p) - µ
eTr
tr B
p = 0
eq
y

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1
-
- 1 p m
In the viscoplastic case:
= + R (p + p) +
eq
y
0 HS


, which leads to
& T
0



p solution of the equation:
1


Tr
-
1
- p m
- - R (p + p) - HS
- µ
eTr
tr B
p 0
eq
y
0
=



&0

T



The solution of this last equation is made in Code_Aster by a method of the secants
with interval of search (cf [R5.03.05]). Integration can be controlled by the parameters
RESI_INTE_RELA and ITER_INTE_MAXI under STAT_NON_LINE key word CONVERGENCE.

Once p known, one can then deduce the tensor from it from Kirchhoff, that is to say:

eq ~Tr K 2
3K
1
=
+
(J -)
1 -
(T - T
) (J +) Id
Tr

ref.


2
2
J
eq

Once calculated cumulated plastic deformation, the tensor of the constraints and the tangent matrix,
one carries out a correction on the trace of the tensor of the elastic strain E
B to hold account
plastic incompressibility, which is not preserved with the simplification made on the law
of flow [éq 4.2.1]. This correction is carried out by using a relation between the invariants of E
B
~
and E
B and by exploiting the plastic condition of incompressibility p
J = 1 (or in an equivalent way
det E
B = 1). This relation is written:

x3- J ex- (1 - J E) = 0
2
3

2
~
~
E
1
2

E
eq
E
E
1
with J 2 = B
=
eq
, J = det B = det
= tr

2
2
3
and
E
X
B

µ
3

The solution of this cubic equation makes it possible to obtain
E
tr B and consequently
thermoelastic deformation Be with the step of next time. If this equation admits
several solutions, one takes the solution nearest to the solution of the step of previous time. It is
1
moreover why one stores in an internal variable
E
tr B.
3
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Modeling élasto (visco) plastic in great deformations
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:
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5 Formulation
variational

Insofar as the constraints provided by the law of behavior are eulériennes, one is chosen
variational formulation written on the current configuration (eulérienne) and not on the configuration
initial, that is to say:

D =
v fv D +
D
X
T v dS v Cinématiquement acceptable


F

14 2
4
3
44
1444 2
4
3
4444
F
. v
int
F
.v
ext.

We are interested only here in work of the interior forces and its variation in optics
of a resolution by the method of Newton. One will find in [bib 4] the demonstration of the expressions
presented.

5.1
Case of the continuous medium

One rewrites here the work of the interior efforts in indicielle form, that is to say:


v
F .v
I
int
= ij
D




X J


We need also to express the variation of the interior efforts in the configuration
current is:


U
U
p
J

v
F U v


I
int.
.
=
-

ij
ik
D rigidity géométriq
ue

X

p
xk

X J


U
ij
statement
+

I

D rigidity of behavior
pq
F
-
X
X
Q


J


where X are the punctual coordinates on the configuration -.

5.2
Discretization by finite elements

One discretizes virtual displacements U and displacements v by finite elements. The notations are
the following ones, by adopting the convention of summation of the repeated indices:

U
U
U (X) = NR N (X) U N
I = DNN (X) U N
I = DNN (X) U N
I
I


X
J
I
-
J
I
J
X J
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Modeling élasto (visco) plastic in great deformations
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:
R5.03.21-C Page
: 15/18

where:

NR N (X) is related to form associated with node N
U nor, component I of the nodal displacement of node N
DNN (X
J
), components of the gradient of the functions of form on the configuration
DNN (X
J
), components of the gradient of the functions of form on the configuration -

One obtains for the vector of the interior forces:

(F
N
int) = DND
I
ij
J


and for the tangent matrix, which is not a priori symmetrical:

KN m
m
N
m
N
I p = [D D - D D
p ij
J
K
ik
p] D





+ DNN
ij
DNN D
Q


F
J

pq




In the case of a two-dimensional modeling (deformation planes), expressions of the vector of
interior forces and of the tangent matrix are identical to this ready that the corresponding indices
with the components only vary from 1 to 2.

In the case of an axisymmetric modeling, by numbering the axes in order (R, Z,), the vector
interior forces is written:


NR N

(F axi N = DNN +
int)
33
1d, {1}
2
, {1}
2
,

R






and the tangent matrix:

[Kaxi] [K] [Kcorr
=
+
]
with:
[
N m
N
N
NR
NR
Kcorr
=

m
m
(1)
]
D D
-

D D
1



+
R
R 33
F


[
N m
m
m

NR

N
NR
Kcorr
N
= D
-
33
(2)
]
D
D
D

33

1


+
R



F
R


[
N m
N
m
NR
NR
Kcorr
33
(3)
] =

11
R 33
F
R


From an algorithmic point of view, we symmetrized the tangent elementary matrix K which is not it
not a priori.
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Modeling élasto (visco) plastic in great deformations
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:
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5.3
Form of the tangent matrix of the behavior

One gives the form of the tangent matrix here (option FULL_MECA during iterations of
Newton, option RIGI_MECA_TANG for the first iteration). This one is obtained by linearizing it
system of equations which governs the law of behavior. We give here the final result of this
linearization. One will find in [bib4] the detail of this calculation.

One poses:

J = det F, J -
-
= det F and J = det F

·
For option FULL_MECA, one a:



(-
J)/
1 3
-
1
J
With =
=
H -
(HF) B -
B
F


J
3
2
J J
J

-
J
3
- 2
+

KJ - K (T - T
) (1 - J
) Id B

ref.

J
2


where B is worth:

B = F

F

- F

F
11
22
33
23 32
B = F

F

- F F
22
11
33
13
31
B = F
F - F F
33
11
22
12
21
B = F

F

- F F
12
31
23
33
21
B = F

F - F F
21
13
32
33
12
B = F

F

- F F
13
21
32
22
31
B = F

F

- F F
31
12
23
22
13
B = F

F

- F F
23
31
12
11
32
B = F

F

- F F
32
13
21
11
23

H and HF are given by:

In the elastic case (F < 0):


H
= µ (B E-F + F B E) -
F B E
ijkl
ik LP
jp
IP pl
jk
ij
kp LP
3
and
~
HF = 2µbeTr
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R5.03 booklet: Nonlinear mechanics
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Code_Aster ®
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Titrate:
Modeling élasto (visco) plastic in great deformations
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:
R5.03.21-C Page
: 17/18

In the plastic case (F = 0) or viscoplastic:

µ
E
E
H
=
(B F + F b)
ijkl
ik
LP
jp
IP
pl
jk
has

R ~
p

ij
ij
E
-

+
F B

eTr
kp
LP
3a R
(+ µ tr B
)
eq


2
eTr
3µ tr B
R
(p)
eq
~ ~
E
+

F B
3
has
R
(+
eTr
ij kq
qp
LP
µ tr b)
eq
and


µ
2
2
p


µ
3
p
-
eTr
eTr Id
~
R
tr B eTr R
(
)
HF =
B
- µ
2 tr B
eq

+
+
~
(: B eTr ~
)
has
3a
R
(+ µ tr B eTr)
a3 R
(+ µ tr B eTr)

eq

eq

Tr

where has
eq
=
eq
1
-
2

2

p
1
1 -
and R = R' (p) +

0 × 1+
×
× (p)
1



m
, R (p) being the derivative of
&

1
0

T
m (&0t)
1
4
4
4
4
4
4
4
4
2
m
4
4
4
4
4
4
4
4
4
3
only
viscous

case

isotropic work hardening compared to the cumulated plastic deformation p.


·
For option RIGI_MECA_TANG, they are the same expressions as those given for
FULL_MECA but with p = 0 and all the variables and coefficients of material taken with
the moment T - (in theory, it would be necessary in the viscous case, to take the expressions of FULL_MECA
in the elastic case, all the variables being taken at the moment T -). In particular, one will have
F = Id.

Handbook of Référence
R5.03 booklet: Nonlinear mechanics
HT-66/05/002/A

Code_Aster ®
Version
7.4
Titrate:
Modeling élasto (visco) plastic in great deformations
Date:
14/04/05
Author (S):
V. CANO, E. Key LORENTZ
:
R5.03.21-C Page
: 18/18

6 Bibliography

[1]
SIMO J.C., MIEHE C.: “Associative coupled thermoplasticity At finite strains: Formulation,
numerical analysis and implementation ", Comp. Meth. Appl. Mech. Eng., 98, pp. 41-104,
North Holland, 1992.
[2]
SIDOROFF F.: “Elastoplastic Formulations in great deformations”, Rapport Greco
n29, 1981.
[3]
SIDOROFF F.: “Course on the great deformations”, Rapport Greco n51, 1982.
[4]
CANO V., LORENTZ E.: “Introduction into Code_Aster of a model of behavior in
great elastoplastic deformations with isotropic work hardening ", internal Note EDF DER,
HI-74/98/006/0, 1998

Handbook of Référence
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