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Organization (S): EDF-R & D/AMA
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Document: R5.03.06
Model of Rousselier in great deformations

Summary

One presents here the model of Rousselier which makes it possible to describe the first stages of the plastic growth
cavities in a steel. The relation of behavior is elastoplastic with isotropic work hardening, allows
the changes of plastic volume and is written in great deformations. To describe the large ones
deformations, one uses the theory suggested by Simo and Miehe. The original formulation of Simo and Miehe is
modified so, on the one hand, facilitating the numerical integration of the law of behavior and, on the other hand, of
to replace the theory of Simo and Mihe within the variational framework of generalized standard materials.
This model is available in command STAT_NON_LINE via key word RELATION:
“ROUSSELIER” or “ROUSSELIER_FO” under the key word factor COMP_INCR and with the key word
DEFORMATION: “SIMO_MIEHE”.
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 model.
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Count

matters

1 Introduction ............................................................................................................................................ 3
2 Notations ................................................................................................................................................ 4
3 Theory of Simo and Miehe ...................................................................................................................... 5
3.1 Introduction ...................................................................................................................................... 5
3.2 General information on the great deformations ...................................................................................... 5
3.2.1 Kinematics ........................................................................................................................... 5
3.2.2 Constraints ............................................................................................................................. 6
3.2.3 Objectivity .............................................................................................................................. 6
3.3 Formulation of Simo and Miehe ........................................................................................................ 7
3.3.1 Original formulation ............................................................................................................. 8
3.3.2 Modified formulation ............................................................................................................. 9
3.3.3 Consequences of the approximation ......................................................................................... 9
4 Model of Rousselier .......................................................................................................................... 11
4.1 Equations of the model .................................................................................................................... 11
4.2 Processing of the singular items ................................................................................................... 12
4.3 Expression of porosity .............................................................................................................. 13
4.4 Relation `ROUSSELIER `............................................................................................................... 14
4.5 Constraints and variables intern .................................................................................................. 14
5 numerical Formulation ........................................................................................................................ 15
5.1 Expression of the model discretized .................................................................................................. 15
5.2 Resolution of the nonlinear system .............................................................................................. 17
5.2.1 Examination of the singular items .............................................................................................. 17
5.2.2 Regular solution ................................................................................................................. 18
5.3 Course of calculation .................................................................................................................. 19
5.4 Resolution ..................................................................................................................................... 19
5.4.1 Hight delimiters and lower if the function S is strictly positive with
the origin ................................................................................................................................. 19
5.4.2 Hight delimiters and lower if the function S is negative or null with the origine21
5.4.3 Hight delimiters and lower if the function S is strictly negative with
the origin and xs not solution .................................................................................................... 22
5.5 Integration of porosity ............................................................................................................... 23
5.6 Form of the tangent matrix of the behavior .................................................................. 23
6 Bibliography ........................................................................................................................................ 28
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1 Introduction

The mechanisms at the origin of the ductile rupture of steels are associated the development of
cavities within material. Three phases are generally distinguished:

· germination
: it is about the initiation of the cavities, in sites which correspond
preferentially with the defects of material,
· growth: it is the phase which corresponds to the development itself of the cavities,
controlled primarily by the plastic flow of the metal matrix which surrounds these
cavities,
· coalescence: it is the phase which corresponds to the interaction of the cavities between them to create
macroscopic fissures.

In what follows, we treat only the phases of growth and coalescence.

The model of Rousselier [bib1] presented here is based on microstructural assumptions which
introduce a microstructure made up of a cavity and a plastic rigid matrix thus isochoric.
In this case, porosity F, definite like the relationship between the volume of the cavity C
V and volume
total V of representative elementary volume, is directly connected to the macroscopic deformation
by:
1 - F
V C
J = det F
0
=
with F =

F & = (1 - F) tr D
éq
1-1
1 - F
V
where f0 indicates initial porosity, F the tensor gradient of the transformation, J the variation of volume
and D the rate of deformation.
To build the law of growth of the cavities, Rousselier takes as a starting point a phenomenologic analysis
who leads it to the following ingredients:


· great deformations figure,
· irreversible changes of volume,
· isotropic work hardening.

These considerations leads it to write the criterion of plasticity F in the following form:
(

F, R)
H
=eq + D F exp
1
- R (p) -




y
éq
1-2
1
where is the constraint of Kirchhoff, R isotropic work hardening function of the plastic deformation
cumulated p and 1, D and y of the parameters of material. The presence in the criterion of plasticity of
hydrostatic constraint H authorizes the changes of plastic volume. One also notices
that this model does not comprise a specific variable of damage because only information
microstructural reserve is porosity, directly related to the macroscopic deformation by
the equation [éq 1-1].

As for the processing of the great deformations, one adopts the theory of Simo and Miehe but in one
slightly modified formulation. The approximations brought make it possible to make easier
the numerical integration of the law of behavior but also to replace the theory of Simo and
Miehe within the variational framework of generalized standard materials.

Thereafter, one briefly gives some concepts of mechanics in great deformations, then one
point out the theory of Simo and Miehe as well as the made modifications. One presents finally them
relations of behavior of the model of Rousselier and its numerical integration.
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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


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


H
With
tr A
hydrostatic part of tensor A defined by H
With =

3
:
doubly contracted product: A: B = A B = tr (
T
AB)
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

With
With =

X

gradient: X
X



, µ, E, K moduli of the isotropic elasticity


y
elastic limit



thermal dilation coefficient


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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3
Theory of Simo and Miehe

3.1 Introduction

We point out here specificities of the formulation suggested by SIMO J.C and MIEHE C. [bib2] for
to treat the great deformations. This formulation was already used for models of
thermoelastoplastic behavior with isotropic work hardening and criterion of Von Mises, [R5.03.21]
for a model without effect of the metallurgical transformations and [R4.04.03] for a model with effect
metallurgical transformations.
The kinematics choices make it possible to treat great displacements and great deformations
but also of great rotations in an exact way.
Specificities of these models 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,
· if the criterion of plasticity depends only on the deviatoric constraint, then the deformations
plastics are done with constant volume. The variation of volume is then only due to
elastic thermo deformations,
· this model led during its numerical integration to a model incrémentalement objective
(cf [§3.2.3]) what makes it possible to obtain the exact solution in the presence of great rotations.

Thereafter, one briefly points out some concepts of mechanics in great deformations.

3.2
General information on the great deformations

3.2.1 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 3.2.1-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
éq
3.2.1-1
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To define the change of metric in the vicinity of a point, one introduces the tensor gradient of
transformation F:

x$
F =
= Id + U


éq
3.2.1-2
X
X

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

D = Jdo with J
O
= det F = éq
3.2.1-3
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 éq
3.2.1-4
2
1
With =
Id - b-1
(
) with B = FFT
éq
3.2.1-5
2
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).

3.2.2 Constraints

The tensor of the constraints used in the theory of Simo and Miehe is the tensor of definite Kirchhoff
by:
J =









éq 3.2.2-1
where is the tensor eulérien of Cauchy. The tensor thus results from a “scaling” by
variation of volume of the tensor of Cauchy.

3.2.3 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 éq
3.2.3-1
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).
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3.3
Formulation of Simo and Miehe

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 3.3-a: Décomposition of the tensor gradient F in an elastic part Fe and plastic F p

By composition of the movements, one obtains the following multiplicative decomposition:
F = FeF p








éq 3.3-1
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
=

éq
3.3-2
However, one will employ alternatively another measurement of the elastic strain E, which coincides
with the opposite of the linearized deformations when the elastic strain are small:
1
E = (
E
Id - b) éq 3.3-3
2
In the case of an isotropic material, one can show that the potential free energy depends only on
left tensor of Cauchy-Green Be (where in our case of the tensor E) and in plasticity of the variable p
dependant on isotropic work hardening. Moreover, one supposes that the voluminal free energy breaks up, all
as in small deformations, in a hyperelastic part which depends only on the deformation
rubber band and another related to the mechanism of work hardening:
(,
E p)
el
= (E)
bl
+ (p) éq
3.3-4
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If instead of using the constraint of Cauchy, one uses the constraint of Kirchhoff, the inequality of
Clausius-Duhem is written (one forgets the thermal part):
: D - & 0









éq 3.3-5
expression in which D represents the rate of deformation eulérien.

Under the preceding assumptions, dissipation is still written:

E
1
p T

+
b: D +
(&G
F
F)

:
-
p & 0 éq
3.3-6

E

2nd
p
The second principle of thermodynamics then requires the following expression for the relation
stress-strain:
E
= -
B









éq 3.3-7
E

One defines finally the thermodynamic forces associated with the elastic strain and the deformation
figure cumulated in accordance with the framework of generalized standard materials:

E
S = -
that is to say

= S B
éq
3.3-8
E


= -
With










éq 3.3-9
p

where thermodynamic force A is the opposite of the isotropic variable of work hardening R.

It remains then for dissipation:
1
p T E 1
1
: (-
-
F &
G F B
) + A p & = S: (-
p T
F &
G F) + A p & 0
éq
3.3-10
2
2

3.3.1 Formulation
original

The principle of maximum dissipation applied starting from the threshold of elasticity F, function of the constraint of
Kirchhoff and of thermodynamic force A makes it possible to deduce the laws of evolution from them from
plastic deformation and of the cumulated plastic deformation, is:
1
p T E 1
-

-
F
FG & F B
= &

éq
3.3.1-1
2

F

&p = &










éq 3.3.1-2
With

&
0 F
0 F & = 0 éq
3.3.1-3
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Note:

One can show easily that the derivative compared to the time of the variation of volume
plastic p
J is written:
p

J & =
F
p
& J tr

éq
3.3.1-4

so that if the surface of load F depends only on the deviatoric part of the tensor of
forced of Kirchhoff, then the plastic deformations are done with constant volume is:
J p
p
det F
1 from where J
J E
E
=
=
=
= det F = det F éq
3.3.1-5

3.3.2 Formulation
modified

The approximation introduced here on the original formulation of Simo and Miehe relates to the expression of
law of flow, all the more reduced approximation as the elastic strain are small,
since
E
= S B. Indeed, one henceforth expresses the threshold of elasticity like a function of the forces
thermodynamic and either of the constraints F (S, A) 0, and it is compared to these variables that one
apply the principle of maximum dissipation, which leads to the following laws of flow:
1
p T

-
F
FG & F = &

éq
3.3.2-1
2
S

F

&p = &

éq
3.3.2-2
With

&
0 F
0 F & = 0 éq
3.3.2-3

3.3.3 Consequences of the approximation

By replacing the constraint by the thermodynamic force S associated with the elastic strain
into the expression of the criterion of plasticity, one introduces in fact a disturbance of the border of
field of reversibility of a factor 2nd. Compared to the initial formulation, it results from it
obviously an influence on the elastic limit observed but also on the direction of flow:
in particular, the derivative compared to the time of the plastic variation of volume is written then:
p
p
E 1
-

J & =
F
& J B

:
éq
3.3.3-1
S

so that if the criterion F depends only on the diverter of the tensor of the constraints S, one
do not find p
J = 1: the isochoric character of the plastic deformation is not perfectly any more
preserved.

Insofar as the elastic strain remain small, results obtained with this model
modified do not deviate significantly from those obtained with the old formulation (cf [bib3]),
while numerical integration will be simplified by it. Indeed, it will be seen thereafter that this model follows
the same diagram of integration as that of the models written in small deformations.
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Note:

This new formulation of the great deformations makes it possible to replace the theory of Simo and
Miehe within the framework of generalized standard materials. From a numerical point of view, this
results in to express the resolution of the law of behavior like a problem
of minimization compared to the internal increments of variables.
Indeed, one recalls that within the framework of generalized standard materials, the data of
two potentials free energy (, has) and potential of dissipation D (a&), function of the tensor
of deformation and a certain number of internal variables has, allows to define
completely the law of behavior (one places materials in the case of independent of
time).


=


, A = -
D (a&)
éq
3.3.3-2


has


where D (a&) is under differential of the potential of dissipation D.
The laws of generalized behavior of the standard type which do not depend on time are
characterized by a potential of dissipation positively homogeneous of degree 1, which
translated by the following property:
&
has

>
D (

0
a&) = D (has &
)

D
(a&) = D (a&) éq 3.3.3-3
Now if one writes the problem [éq 3.3.3-2] in form discretized in time and if one uses
the property of under differentials [éq 3.3.3-3], one obtains the following discretized problem:



=


, A = -
D (has
) éq
3.3.3-4


has


One can show that the equation [éq 3.3.3-4] is equivalent (cf [bib4]) to solve the problem
of minimization compared to the internal increments of variables has
according to:



-
D (has
) has
= ArgMin [(A + *)
+ D has (*)
has]
éq
3.3.3-5
has

*
has
The application of the equation [éq 3.3.3-5] to the model of Rousselier in great deformations
modified is written:

(,
E p
D (
and

)
p
D, p &
)
=>
(Tr
E +,
E p + p

D (
and

)
,
E p
) éq
3.3.3-6
tion
discretized
continuous

energy
E
discretized

energy




S = -




E
With = -
=
D (,
E p
)
has




- R = -

éq.
3.3.3-7

p

Min [(Tr
E +,
E p + p
) + D (,
E p
)]
E, p

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One will find in the paragraph [§4], the relation which binds the rate of plastic deformation
p
D
once discretized and the increment of elastic strain E
, as well as the definition of Tr
E.
One sees well here whom if one takes the initial formulation of Simo and Miehe, one cannot write any more
the problem of minimization [éq 3.3.3-7] with the constraint of Kirchhoff because of the term
in E
B in the expression:



E
= -
B
éq
3.3.3-8
E


4
Model of Rousselier

We now describe the application of the great deformations to the model of Rousselier presented
in introduction.

4.1
Equations of the model

To describe a thermoelastoplastic model with isotropic work hardening (the equivalent into small
deformations with the model with isotropic work hardening and criterion of Von Mises), Simo and Miehe propose one
elastic potential polyconvexe. By reason of simplicity, one chooses here the potential of Saint Venant which
is written:
(
1
,
E p) = [K (tre) 2
~ ~
+ 2µ E: E + 6K T
tr E] p
+ R (U) of the éq
4.1-1
2
0
In accordance with the equations [éq 3.3-8] and [éq 3.3-9], the laws of state which derive from the elastic potential
above are written then:
S = [
- K tr E Id + µ e~
2
+ 3K T
Id] éq
4.1-2
With = - R (p)







éq 4.1-3
The threshold of elasticity is given by:
HS
F (S, R) = S
+ Df exp
- R
eq
1




y
éq
4.1-4
1
According to the equations [éq 3.3.2-1] and [éq 3.3.2-2], the laws of flow are defined by:

1
p T
s~
3
Df
S
-
G
F & F = &
+
exp H Id




éq
4.1-5
2

2s
3
eq
1
&p = &










éq 4.1-6
&
0 F
0 F & = 0 éq
4.1-7
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4.2
Processing of the singular points

In fact, the equation of flow [éq 4.1-5] translated the membership of the direction of flow to the cone
normal on the surface of the field of elasticity. It is valid only at the regular points, characterized by:
seq 0









éq 4.2-1
It thus remains to characterize the normal cone at the singular points, i.e. checking:

~
S
H
S = 0

and

D F exp
1
- R =




y
éq
4.2-2
1
The normal cone with convex of elasticity in such a point is the whole of the directions of flow which
carry out the problem of maximization according to:

*
(, Sr) = sup [
p
S: D - R p & - (p
D, p&)] éq
4.2-3
p
D, p&
where *
is the indicating function of convex F and (p
D, p&) potential of dissipation obtained by
transform of Legendre-Fenchel of the indicating function of F:
p
(D, p&) = Sup [
p
S: D - R p&] éq
4.2-4
S, R
F (S, R) 0
After some calculations, one obtains:

D

p
p
tr p
2
(D, p&) = p & + tr D ln
- 1 + I + (tr p
D) + I + (
p
p & - D)
y
1


éq
4.2-5
IR
IR
eq
p
Df

&
3

with
0
if
X 0
I + (X) =
éq
4.2-6
IR
+ if not

For ~
S = 0, *
is worth:




*


p
tr D

p
p

(, Sr) = Sup
HS tr D - tr D ln
1
-

1 - R p & -
&
éq
4.2-7
p


p
y
D, p &
D F p&


p
tr D 0 1
4
4
4
4
4
4
2
4
4
4
4
4
4
3
p

p & 2D p
G (tr D)
0

-
eq
3
By noticing that for tr
p
D 0, the function G (tr p
D) is concave, the suprémum compared to
trace rate of plastic deformation
p
D is obtained for:

p
p
S
G (tr D) = 0 of
where
tr D = D F p
H
&


exp

éq
4.2-8
1

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Note:

One finds well then for the indicating function of the threshold of elasticity F.

0
if
F 0
* (S, R) = Sup [F &p] =

éq
4.2-9
p
+
&
if not
2 p
&p Deq 0
3

In a singular point, the normal cone, together of the acceptable directions of flow,
thus characterize by:

p
S
tr D = D F p
H
&


exp


éq
4.2-10
1

p & 2 peq
D 0
éq
4.2-11
3

4.3
Expression of porosity

One saw in introduction that the microscopic inspiration of the model of Rousselier is based on one
microstructure made up of a cavity and a plastic rigid matrix, therefore isochoric. In this case,
porosity is directly connected to the macroscopic deformation by:
1 - F
J = det F
0
=
F & = (1 - F) tr D
éq
4.3-1
1 - F
However, on a macroscopic scale, one supposes that the material can also become deformed
reversible elastic manner. The expression above is not thus exact any more, even if it represents
still a good approximation as long as the elastic strain are small. Unfortunately,
it prohibits even reasonable elastic compressions, because very quickly, porosity is cancelled and
impose an isochoric behavior again (J = constant bus F = 0).

Rousselier proposes as for him to express porosity while basing himself on the rate of plastic deformation
p
D. The relation is written in incremental form:
F & = (1 -)
p
F tr D éq
4.3-2
That means that the variable porosity employed to parameterize the criterion of plasticity F does not depend
that plastic deformation. In fact, the rate of plastic deformation is an evaluated quantity
in the slackened configuration. Its transport in the current configuration (as D) is expressed
still:

T
E
p E
1
p T
F D F
= -
G
F & F éq
4.3-3
2
Finally, one adopts like law of evolution of porosity:
1
F & = (1 - F)
p T
tr-
G
F & F
éq
4.3-4
2

Again, this law of evolution of porosity remains close to that employed by Rousselier when
the elastic strain are small.
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4.4 Relation
`ROUSSELIER `

This relation of behavior is available via the argument `ROUSSELIER `of key word COMP_INCR
under operator STAT_NON_LINE, with the argument `SIMO_MIEHE `of the key word factor DEFORMATION.

The whole of the parameters of the model is provided under the key words factors `ROUSSELIER `or
`ROUSSELIER_FO `and `TRACTION `(to define the traction diagram) command
DEFI_MATERIAU ([U4.43.01]).

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.

4.5
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, porosity F,
· V3 with V8, the tensor of elastic strain E,
· V9, the indicator of plasticity (0 if the last calculated increment is elastic, 1 if solution
figure regular, 2 if singular plastic solution).

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.
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5 Formulation
numerical

For the variational formulation, it is about same as that given in the note [R5.03.21] and which
refers to the law of behavior with isotropic work hardening and criterion of Von Mises into large
deformations. We recall only that it is about a eulérienne formulation, with
reactualization of the geometry to each increment and each iteration, and which one takes account of
rigidity of behavior and geometrical rigidity.
We now present the numerical integration of the law of behavior and give
the form of the tangent matrix (options FULL_MECA and RIGI_MECA_TANG).

5.1
Expression of the discretized model

Knowing the constraint -
, cumulated plastic deformation p, elastic strain -
E,
displacements U and U, one seeks to determine (, p, E).
Displacements being known, gradients of the transformation of 0 with -, noted F, and of -
with, noted F, are known.

To integrate this model of behavior, one does not choose a purely implicit algorithm because,
on the one hand, that led to the resolution of a rather complex nonlinear system, and on the other hand,
allows more to express the problem like the minimization of a functional calculus. This is why one
prefer to treat in an explicit way the variation according to porosity in the threshold of elasticity. For
other terms, one employs a diagram of implicit Euler.

Once discretized, the following system then is obtained:

-
F = FF

















éq 5.1-1
J = det F

















éq 5.1-2
J =


















éq 5.1-3
E
= S B


















éq 5.1-4
Be = Id - E
2

















éq 5.1-5

· Equations of state:
S
[-
= µe~
2
+ K tr E Id + 3K T
Id]











éq. 5.1-6
With -
= R (p)
















éq 5.1-7
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Thereafter, one expresses the laws of flow and the criterion of plasticity directly according to
tensor of the elastic strain E.

· Laws of flow


p
1
p T
1
p T
- p T
-
T
D - FG & F = -
FG F - FF G F F
2
2 T
4
1 4
23
4
1
4
2 3

E
E

-
B
B

= - 1

[Id-2nd-F {Id- -
2nd} T
F]

éq
5.1-8
2 T

=
E - 1
(
[Id-F {Id- -
2nd} T
F])/T
Tr
= (E - E)/T

2
1
4
4
4
4
2
4
4
4
4
3
Tr
E
By taking the parts traces and deviatoric of the equation [éq 4.1-5], one obtains:
Tr
-
3K T

K tr E
tr E - tr E
p

= Df exp (-
) exp (-
)








éq 5.1-9
1
1


~
Tr - 3
~
E
E
p

regular

solution

if
~
2
E
E =
eq






éq 5.1-10

2 ~ ~Tr
0


and

p (E - E)
singular

solution

if

eq
3
· Conditions of coherence

3
-
KT
K tr E

+ D F exp (-
) exp (-
) - R
regular
solution

if

eq
E
1


y
1
1
F =
3
-
KT
K tr

E
éq
5.1-11
D F
exp (-
) exp (-
) - R
singular
solution

if


1


y
1
1

with F

0 p
F

0
p = 0
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5.2
Resolution of the nonlinear system

The integration of the law of behavior is thus summarized to solve the system [éq 5.1-9], [éq 5.1-10]
and [éq 5.1-11]. We will see that this resolution is brought back to that of only one scalar equation, of which
unknown factor X is the increment of the trace of the elastic strain:
Tr
X = tr E - tr E







éq 5.2-1
Thanks to this choice, that the solution is elastic or plastic, attack in a singular point or not,
the equation [éq 5.1-9] bearing on the trace of the elastic increment is always valid and allows
to express the increment of cumulated plastic deformation:

K tr Tr


Tr
E
3K
-
T

K (tr E - tr Tr
E)
tr E - tr E = p
D F exp

-
exp (-
) exp

-






1

1

1

1
4
4
4
4
4
4
2
4
4
4
4
4
4
3
G
éq
5.2-2
K X

(X) 1
p
= X exp
G
1
This equation shows us that one can seek X 0 to guarantee a plastic deformation
cumulated positive and that the elastic solution is obtained for X = 0. It is also noticed that
the increment of cumulated plastic deformation is a continuous and strictly increasing function X.
With the help of these remarks, if one notes by S the term [éq 5.2-3] in the criterion of plasticity, it acts
then, there too, of a continuous and strictly increasing function of X:
Kx
F = 2 eq
µe - (
S X) with

S (
X) = - G exp
1
-
+



R (p (X))+ y éq
5.2-3

1
This stage, the step of resolution breaks up into two times.

5.2.1 Examination of the singular points

Such a singular point is characterized by [éq 5.1-10] (low) and [éq 5.1-11] (low), therefore in particular by
(Sx) = 0. Because of the properties of S, this equation admits with more the one positive solution, say S
X
who exists if and only if (
S 0) 0. The knowledge of S
X makes it possible to deduce the tensor from it from
elastic strain E, cumulated plastic deformation p as well as the thermodynamic forces
S and R.
Finally, this singular point will be solution if the inequality in [éq 5.1-11] (low) is checked, i.e.
if:
S
2 ~s ~Tr
p
(E - E) eq éq
5.2.1-1
3
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5.2.2 Solution
regular

The equation of flow [éq 5.1-10] (high) which determines the deviatoric part of the tensor of
elastic strain makes it possible to deduce a scalar equation from it function from the increment from
cumulated plastic deformation:

Tr
3
~
eq
E = eq
E - p
~ ~
3
2
Tr
E

E - E
-
=
p




E

éq
5.2.2-1
2
eq
E
e~= eq ~Tr
E

Tr

eq
E
One notes that because of the positivity of eq
E, the value sells by auction p
is limited:
2 Tr
p

eq
E









éq 5.2.2-2
3
The condition of coherence determines X now:
F = 2µeTr
eq - S (X) - 3
µ p 0
éq
5.2.2-3
Being given this expression, the increase of the licit value of p
is reduced to the only condition
(Sx) 0 or, in an equivalent way, with
S
X X.

The elastic solution is obtained for X = 0. It is the solution of the problem if and only if:
F ()
0 = 2µ eTr
eq - S (0 <
) 0 éq
5.2.2-4
In the contrary case, one must then solve:

Kx

X > xs
Tr
xs
F (X) = 2µeeq - S (X) -
X exp (
) =

if
exist
0 with



éq
5.2.2-5
G
1

X > 0
if not

This function is continuous and strictly decreasing and tends towards - with X. It thus admits with
more one solution. The demonstration of the existence of this solution is immediate. Indeed, it is enough to
to prove that F is positive on the lower limit of the interval of search.
When S
X does not exist, (
F)
0 > 0 since the solution is not elastic.
When S
X exists, the function is worth:
S
Tr
S
S
2 Tr
F (X) = 2µ eq
E - 3 p

µ
> 0 p
<
eq
E éq
5.2.2-6
3
This condition is checked since the singular solution was rejected.
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5.3
Course of calculation

The step to solve the whole of the equations of the model is as follows:

1) One seeks if the solution is elastic
· calculation of F ()
0
· if F ()
0 < 0, the solution of the problem are the elastic solution Sol
X
= 0
· if not one passes into 2)
2) If
S ()
0 > 0, the solution are plastic and regular
· one passes into 4)
3) If
S ()
0 < 0, one seek if the solution is singular
· one solves S (S
X) = 0
2
· if S
X checks the inequality
S
~s ~Tr
p
(E - E)
Ground
eq, then the solution is singular
S
X
= X
3
· if not, S
X is a lower limit to solve F (X) = 0, one passes into 4)
4) The solution is plastic and regular
· one solves F (X) = 0

5.4 Resolution

To solve the two equations S (X) = 0 and F (X) = 0, one employs a method of Newton with
controlled terminals coupled to dichothomy when Newton gives a solution apart from
the interval of the two terminals. One now presents the determination of the terminals for each case
precedents (items 2) 3) and 4) of the preceding paragraph).

5.4.1 Hight delimiters and lower if the function S is strictly
positive at the origin

One solves:

µ
Kx
2 Tr
eq
µe - S (X = 3
)
X exp (
)
F (X) = 0

4
42
1
4
43
G




1

F
1 4
4 2 4
4 3
éq
5.4.1-1
F ()
0 > 0
1

p

F ()
0
1
> 0
where the function p (X) is continuous, strictly increasing and null at the origin and the function F ()
1 X is
continue, strictly decreasing and strictly positive at the origin (see [Figure 5.4.1-a]).
One poses:
Tr
Kx
F = 2µeeq - R (X) - y + G exp (-
) then F
(X) < F (X)

X 0
1
1
2
1

éq 5.4.1-2
1 4
4
4 2
4
4
4 3
1
f2
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where the function F ()
2 X are continuous, strictly decreasing. In this case, the resolution of the equations:
inf
inf
inf
Kx
F (p
inf
2
) = 3µ p

and
inf
X
exp (
) = G p

éq
5.4.1-3
1
to deduce p successively from it
then X gives a lower limit Inf
X
who corresponds to
solution of the model with isotropic work hardening and criterion of Von Mises. If F (0)
0
2
<, the lower limit
is taken equalizes to zero: inf
X
= 0.
The upper limit Sup
X
is such as:

KxSup
Sup
G
X
exp (
) =
F (xInf)
éq
5.4.1-4

3 1
1
µ
Kx
The equation of type X exp (
) = constant is solved by a method of Newton.
1
3µ p (X)
f1
3µ p (sup
X
) = F
inf
1 (X
)
F 2
3µ p (inf
X
) = F
inf
2 (X
)
X
Ground
inf
Sup
X
X
X

Appear 5.4.1-a: chart of the hight delimiters and lower
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5.4.2 Hight delimiters and lower if the function S is negative or
null at the origin

The system to be solved is as follows:

X
Kx
-
Kx
S (X) = 0
R p + exp (
) + y = G exp (
1
-




)


G
1
1

éq
5.4.2-1
S (0) < 0

-
R (p) + y < G
1
The part of left is a continuous, strictly increasing X and strictly positive function with
the origin, the part of straight line is a continuous, strictly decreasing function X and strictly
positive at the origin. Using the properties of these two functions, a chart
(cf [Figure 5.4.2-a]) of these functions shows that the upper limit Sup
X
is such as:
Sup
G exp (
1
- Kx
G
) = R (-
p)


Sup
1

+

1
y

X
=
log

éq
5.4.2-2
K

-
1
R (p)


+ y
The lower limit Inf
X
is such as:
Inf
Kx

Sup
Sup
-
X
Kx

G exp (
1
-
) = R p +
exp (
) +

y
1

G
1

+





éq
5.4.2-3


Inf
G
X
=
1 log
1

K

Sup
Sup
-
X
Kx


R p +
exp (
) +

y

G
1


R (X)
1 G
+ y
Kx
1 G exp (-
)
R (p -)
+
1
y
X
Inf
Ground
Sup
X
X
X

Appear 5.4.2-a: chart of the hight delimiters and lower
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5.4.3 Hight delimiters and lower if the function S is strictly
negative at the origin and xs not solution

The following system is solved:

Tr

Kx
2µeeq - S (X) =
X exp (
)
4
42
1
4
43
G
1

1 4
4 2 4
4 3
F (X) = 0

f1


p

S (0) < 0



F (0)
1
> 0

éq
5.4.3-1

S

S (X) = 0
S

Tr
µ S
Kx
2µeeq = 3

X exp (
)
G

1

The Sol solution
X
is such as S (Sol
X
) > 0.
For the lower limit, one takes Inf
S
X
= X. Being given properties of the functions 1f
(strictly decreasing) and 3µ p (X) (strictly increasing), the upper limit Sup
X
is such
that (cf [Figure 5.4.3-a]):
Sup
Sup
Kx
2G Tr
X
exp (
) =
eq
E
éq
5.4.3-2
3
1
This equation is solved by a method of Newton.
S (X) < 0 S (X) > 0
2µeTr
eq - S (0)
S (X) = 0

Kx
Tr
X exp (
)
2µeeq
G
1

X
Inf
S
Ground Sup
X
= X X X

Appear 5.4.3-a: chart of the hight delimiters and lower
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5.5
Integration of porosity

This stage, it any more but does not remain to integrate the law of evolution of porosity. According to the equations [éq 4.3-
4] and [éq 5.1-8], it is still expressed using unknown factor X:

f&
X
=








éq 5.5-1
1 - F
T

maybe while integrating:

F df
X T



=
dt

F =1- 1
(- F) exp (-)
X
éq
5.5-2
1 - F
T
0

F
0
0
where one carried out an exact temporal integration by supposing X constant lasting the step of time. It
choice makes it possible to ensure that F is increasing and remains lower than 1, whatever the step of time.

5.6
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).
For the option FULL_MECA, this one is obtained by linearizing the system of equations which governs the law of
behavior. We give hereafter the broad outline of this linearization.
For option RIGI_MECA_TANG, they are the same expressions as those given for
FULL_MECA but with p = 0. In particular, there will be F
= Id.

The law of behavior can be put in the following general form:
= (, F
)







éq 5.6-1
= (E)









éq 5.6-2
E = E (Tr
E) éq 5.6-3
eTr = eTr (F
)







éq 5.6-4
The linearization of this system gives:



E

E
Tr

=
:
:
:
+
: F
= H: F



éq
5.6-5

E

E
Tr
F


F




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where H is the tangent matrix. Thereafter, the five terms of the equation are separately determined
the preceding one.
In the linearization of the system, one will often use the tensor C defined below and both
following equations:
1
has
ij = (ik jl + jkil) has
kl
éq
5.6-6
2
app = K has
L
kl









éq 5.5-7
1
ijkl
C
= (ik jl + jkil) éq
5.6-8
2





· Calculation of
and of



F



Linearization of the relation which binds the constraint of Cauchy and the constraint of Kirchhoff
give:
1
J
J
=


= -
: F
éq
5.6-9
J
J
F





By using the relation [éq 5.6-6], one obtains for
:


= C éq 5.6-10



and for
:
F






= - J
éq
5.6-11
F


J
F


with
J
= 22
F 33
F - 23
F 32
F
11
F
J
= 11
F 33
F - 13
F 31
F
22
F
J
= 11
F 22
F - 12
F 21
F
33
F

éq
5.6-12
J

J

= 31
F 23
F - 33
F 21
F
= 13
F 32
F - 33
F 12
F
12
F
21
F
J

J

= 21
F 32
F - 22
F 31
F
= 12
F 23
F - 22
F 13
F
13
F
31
F
J

J

= 31
F 12
F - 11
F 32
F
= 13
F 21
F - 11
F 23
F
23
F
32
F
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Model of Rousselier in great deformations


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· Calculation of

E


The relation which binds the constraint of Kirchhoff and the tensor of elastic strain E is given
by:
= S B
E = - 2µ E - Tr E Id
+ 4µ E E
+ 2 (tr E E
) éq
5.6-13
-


3K T
Id + 6K T
E
One obtains after linearization:
= (
2 tr E - µ + 3KT) E + (E
2 - Id) Tr E + 4µ (E E
+ E E)
éq
5.6-14
from where

ij = (2 tre-µ+3KT) Cijkl + (e2ij-ij) kl +2µ (ikelj+ilekj+eilkj+eikjl)
kl
E
éq 5.6-15

E
Tr
· Calculation of

F



The relation between the tensor of elastic strain Tr
E and the increment of the gradient of
transformation F
is written:
Tr
1
E = (
E
T
Id -
F B F) éq
5.6-16
2
Its linearization gives:
Tr
ij
E
= - 1 (
E
E
ik F
jpbpl + IP
F bpl jk) éq
5.6-17

kl
F
2

E

· Calculation of

Tr
E


Elastic case

E

In the elastic case, the calculation of
is immediate since
Tr
E
= E

from where
Tr
E

E
= C éq 5.6-18
E
Tr
Handbook of Référence
R5.03 booklet: Nonlinear mechanics
HT-66/02/004/A

Code_Aster ®
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Titrate:
Model of Rousselier in great deformations


Date:
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Author (S):
V. CANO, E. Key LORENTZ
:
R5.03.06-A Page
: 26/28

Plastic case ­ Solution regular

E

To determine
, one operates in two stages. By the law of flow discretized, one calculates
Tr
E

in first E
according to Tr
E

and
p

. Then the condition of coherence makes it possible to deduce some
p

according to Tr
E

. These two stages are thereafter detailed.
The deviatoric part of the law of flow discretized is written:
~
~ ~Tr
3
E
E - E = -
p

éq
5.6-19
2
eq
E
One obtains after linearization:
3 p
~
~Tr
3e~
9
e~
1
(+
)
E = E -

p + p
(e~ e~
:
) éq
5.6-20
2nd
2nd
4
3
eq
eq
eq
E
4
1
4
2 3
1/
To determine E
~ e~
:
, one contracts the equation [éq 5.6-20] with E
~ what gives:
~
~ ~
~Tr
E: E = E: E - E
p
eq

éq
5.6-21
from where

~
9 p


~
~ ~
~Tr
E =
E E + C
E
: E - 3
p


éq
5.6-22
E
4 3

2nd
eq
eq

1
4
4
4 2
4
4
4 3
3
2
1A
With
2
1
For the part law of flow traces discretized, one a:

Tr
3KT
K

Tr E - Tr E = Df p


exp




exp -
Tr E éq
5.6-23

1

1

what gives:
1
Tr E =
Tr
Tr
E
DfK p
3KT
K

1+
exp
exp -
Tr






E
1

1

1

1
4
4
4
4
4
4
4
2
4
4
4
4
4
4
4
3
1
K

3KT

éq
5.6-24
Df exp -
Tr E exp











1


1

+
p


DfK p
3KT
K

1+
exp
exp -
Tr






E
1

1

1

1
4
4
4
4
4
4
4
2
4
4
4
4
4
4
4
3
2
Handbook of Référence
R5.03 booklet: Nonlinear mechanics
HT-66/02/004/A

Code_Aster ®
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Titrate:
Model of Rousselier in great deformations


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:
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: 27/28

In the plastic case, the condition of coherence is worth:
3KT
K

2 eq
µe + Df exp -
exp -
Tr E - R -
0
1
y =









éq
5.6-25

1

1

from where
3µ (~ ~
E:E)
3KT
K

- DfK exp -
exp -
Tr E Tr E - HP = 0









éq
5.6-26
eeq

1

1

By injecting the relation [éq 5.6-21] in the equation above, one obtains then:

1
~ ~Tr
p
=
E:E
eq
E

3K T

K

3µ + H + DfK exp -
exp (-
Tr)
E
2












1

1
1
4
4
4
4
4
4
4
4
4
2
4
4
4
4
4
4
4
4
4
3
3
3K T

K

éq 5.6-27
DfK exp -
exp (-
Tr)
E
1







1

1
Tr
-
Tr E

3K T

K

3µ + H + DfK exp -
exp (-
Tr)
E
2












1

1
1
4
4
4
4
4
4
4
4
4
2
4
4
4
4
4
4
4
4
4
3
4
While replacing
p

by its value obtained above in the equations [éq 5.6­22] and [éq 5.6-
24], one obtains:




1

µ
3
3 ~
Tr
1
~

1

Tr
E
= A1 + A2 + Id

3

E: E +
Id
1
+ 4A2 + Id
2
Tr E




3



eq
E

3

3



1
4
4
4
4
4
2
4
4
4
4
4
3
1
4
4
4
4
4
4
2
4
4
4
4
4
4
3
D etr
ddvetr
tr
éq 5.6-28
from where
E


1

= ddvetr + dtretr - ddvetr: Id Id éq
5.6-29
E
Tr

3


Plastic case ­ Solution singular

The step is identical to that used previously.
One obtains for the law of flow discretized:
~
~
E = 0


E = 0
éq
5.6-30
for the deviatoric part and the part traces, the relation is identical to that found for
regular solution.
Tr
Tre = 1Tre +
p
2


éq
5.6-31
where 1
and 2 have the same definitions as in the preceding paragraph.
Handbook of Référence
R5.03 booklet: Nonlinear mechanics
HT-66/02/004/A

Code_Aster ®
Version
6.0
Titrate:
Model of Rousselier in great deformations


Date:
13/11/02
Author (S):
V. CANO, E. Key LORENTZ
:
R5.03.06-A Page
: 28/28

The condition of coherence then makes it possible to find
p

according to Tr
E

.
3KT
K

Df exp
exp -
Tr E - R -
0
1
y =









éq
5.6-32

1

1

maybe after linearization:
3K T

K
DfK exp
exp (-
Tr)
E
1







1

1
Tr
p
= -
Tr E

éq
5.6-33

3K T

K

H + DfK exp
exp (-
Tr)
E


2










1

1

1
4
4
4
4
4
4
4
4
2
4
4
4
4
4
4
4
4
3
4
that is to say finally:
1
E = [1 +
4 2]
Tr
Id Tr E éq
5.6-34
31 4
4 2 4
4 3
dtretr
from where
E
= dtretrId
éq
5.6-35
E
Tr

6 Bibliography

[1]
ROUSSELIER G.: “Finite constitutive deformation relations including ductile fracture ramming,
In three dimensional constitutive relations and ductile fracture ", ED. Nemat-Nasser, North
Holland publishing company, pp. 331-355.
[2]
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.SIDOROFF F., “Cours on the great deformations”, Rapport Greco
n51, 1982.
[3]
LORENTZ E., CANO V.: “A minimization principle for finite strain plasticity: incremental
objectivity and immediate implementation ", article subjected in the Communications review in
numerical methods in engineering,
[4]
MIALON P.
: “Elements of analysis and numerical resolution of the relations of
elastoplasticity ", EDF, bulletin of DER, data-processing series C mathematical, 3, pp. 57-89,
1986

Handbook of Référence
R5.03 booklet: Nonlinear mechanics
HT-66/02/004/A

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