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Model of Rousselier in great deformations
Date:
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Organization (S):
EDF-R & D/AMA
Manual of Reference
R5.03 booklet: Nonlinear mechanics
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 the control
STAT_NON_LINE
via the 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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Model of Rousselier in great deformations
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Count
matters
5.4.1 Hight delimiters and lower if the function S is strictly positive with
5.4.3 Hight delimiters and lower if the function S is strictly negative with
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Model of Rousselier in great deformations
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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
representative elementary volume, is directly connected to the macroscopic deformation
by:
(
)
D
F
tr
1
1
1
det
0
F
F
V
V
F
F
F
J
C
-
=
=
-
-
=
=
&
with
éq
1-1
where
0
F
indicate initial porosity,
F
the tensor gradient of the transformation,
J
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 products:
·
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:
()
y
H
eq
p
F
D
R
-
-
+
=
)
R (
exp
,
F
1
1
éq
1-2
where
is the stress of Kirchhoff,
R
isotropic work hardening function of the plastic deformation
cumulated
p
and
1
,
D
and
y
parameters of material. The presence in the criterion of plasticity of
hydrostatic stress
H
authorize 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 amendments. One presents finally them
relations of behavior of the model of Rousselier and its numerical integration.
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Model of Rousselier in great deformations
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2 Notations
One will note by:
Id
stamp identity
With
tr
trace tensor
With
With
T
transposed of the tensor
With
det A
determinant of
With
~
With
deviatoric part of the tensor
With
defined by
Id
With
With
With
)
tr
3
1
(
~
-
=
H
With
hydrostatic part of the tensor
With
defined by
3
tr A
=
H
With
:
doubly contracted product:
)
tr (
:
,
T
J
I
ij
ij
B
With
AB
B
With
=
=
tensorial product:
(
)
WITH B
=
ijkl
ij kl
WITH B
With
eq
equivalent value of Von Mises defined by
With
eq
=
3
2
~: ~
WITH A
With
X
gradient:
X
With
With
X
=
,
µ
, E,
, K
moduli of the isotropic elasticity
y
elastic limit
thermal expansion factor
T
temperature
T
ref.
temperature of reference
In addition, within the framework of a discretization in time, all the quantities evaluated at the moment
precedent are subscripted by
-
, quantities evaluated at the moment
T
T
+
are not subscripted and them
increments are indicated by
. 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 stress, 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 stresses depend only on the deformations
thermo rubber bands,
·
if the criterion of plasticity depends only on the deviatoric stress, 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
0
the field occupied by the solid
before deformation and
()
T
the field occupied at the moment
T
by the deformed solid.
Current configuration deformation
Initial configuration
F
0
()
T
Appear 3.2.1-a: Representation 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
particle which occupied
position
X
before deformation is given by the variable
X
(description eulérienne).
The total movement of the solid is defined, with
U
displacement, by:
X X X
X U
=
= +
$ (,)
T
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
:
F
X
X
Id
U
X
=
=
+
$
éq
3.2.1-2
The transformations of the element of volume and the density are worth:
D
Jd
O
=
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:
E
C Id
=
-
1
2 (
)
with
C F F
=
T
éq
3.2.1-4
With
Id B
=
-
-
1
2
1
(
)
with
B FF
=
T
éq
3.2.1-5
E
and
With
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 is
quantities defined on
0
, and of description eulérienne by the tensors
B
or
With
(definite on
).
3.2.2 Stresses
The tensor of the stresses used in the theory of Simo and Miehe is the tensor of Kirchhoff
defined
by:
=
J
éq 3.2.2-1
where
is the tensor eulérien of Cauchy. The tensor
thus result 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
Q X
*
()
()
=
+
T
T
éq
3.2.3-1
where
Q
is an orthogonal tensor which represents 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
E
1
, for example, followed of one
rotation of 90° around
E
3
, which amounts carrying out a tensile test according to
E
2
, then the danger
with a nonobjective law of behavior is not to find a tensor of the stresses
uniaxial in the direction
E
2
(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
with
current configuration
()
T
, by
F
p
the tensor gradient which makes pass from the configuration
0
with
slackened configuration
R
, and
F
E
configuration
R
with
()
T
. The index p refers to the part
plastic, the index E with the elastic part.
Initial configuration
Current configuration
Slackened configuration
F
E
F
p
F
R
0
()
T
T
T
ref.
=
=
0
Appear 3.3-a: Decomposition of the tensor gradient
F
in an elastic part
F
E
and plastic
F
p
By composition of the movements, one obtains the following multiplicative decomposition:
F F F
=
E p
éq 3.3-1
The elastic strain are measured in the current configuration with the tensor eulérien of
Left Cauchy-Green
B
E
and plastic deformations in the initial configuration by the tensor
G
p
(Lagrangian description). These two tensors are defined by:
B
F F
E
E and
=
,
G
F F
p
Pt
p
=
-
(
)
1
from where
B
FG F
E
p T
=
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:
(
)
E
B
Id
E
-
=
2
1
éq 3.3-3
In the case of an isotropic material, one can show that the potential free energy depends only on
left tensor of Cauchy-Green
B
E
(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:
()
()
()
p
p
bl
el
,
+
=
E
E
éq
3.3-4
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If instead of using the stress of Cauchy
, one uses the stress of Kirchhoff
, the inequality of
Clausius-Duhem is written (one forgets the thermal part):
0
:
-
&
D
éq 3.3-5
expression in which
D
represent the rate of deformation eulérien.
Under the preceding assumptions, dissipation is still written:
(
)
0
:
2
1
:
-
+
p
p
T
p
E
&
& F
G
F
E
D
B
E
+
éq
3.3-6
The second principle of thermodynamics then requires the following expression for the relation
stress-strain:
E
B
E
-
=
éq 3.3-7
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
B
S
E
S
=
-
=
that is to say
éq
3.3-8
p
-
=
With
éq 3.3-9
where the thermodynamic force
With
is the opposite of the isotropic variable of work hardening
R
.
It remains then for dissipation:
0
With
)
2
1
(
:
With
)
2
1
(
:
+
-
=
+
-
-
p
p
T
p
E
T
p
&
&
&
&
F
G
F
S
B
F
G
F
1
éq
3.3-10
3.3.1 Formulation
original
The principle of maximum dissipation applied starting from the threshold of elasticity
F
, function of the stress of
Kirchhoff
and of the thermodynamic force
With
allows to deduce the laws of evolution from them from
plastic deformation and of the cumulated plastic deformation, is:
B
F
G
F
=
-
F
2
1
&
&
1
-
E
T
p
éq
3.3.1-1
With
F
&
&
=
p
éq 3.3.1-2
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:
=
F
tr
p
p
J
J
&
&
éq
3.3.1-4
so that if the surface of load
F
depends only on the deviatoric part of the tensor of
stress of Kirchhoff
, then the plastic deformations are done with constant volume is:
J
J
J
p
p
E
E
=
=
=
=
=
det
det
det
F
F
F
1
from where
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
B
S
=
. Indeed, one henceforth expresses the threshold of elasticity like a function of the forces
thermodynamic and either of the stresses
0
With)
,
F (
S
, and it is compared to these variables that one
apply the principle of maximum dissipation, which leads to the following laws of flow:
S
F
G
F
=
-
F
2
1
&
&
T
p
éq
3.3.2-1
With
F
&
&
=
p
éq
3.3.2-2
0
=
&
&
F
0
F
0
éq
3.3.2-3
3.3.3 Consequences of the approximation
By replacing the stress
by the thermodynamic force
S
associated 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
E
2
. 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:
S
B
=
F
:
1
-
E
p
p
J
J
&
&
éq
3.3.3-1
so that if the criterion
F
depends only on the diverter of the tensor of the stresses
S
, one
do not find
1
=
p
J
: 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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Model of Rousselier in great 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 of 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).
=
,
)
D (has
has
With
&
-
=
éq
3.3.3-2
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:
)
D (
)
(
)
D (
)
D (
0
has
has
D
has
has
has
&
&
&
&
&
=
=
>
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:
=
,
)
D (has
has
With
-
=
éq
3.3.3-4
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 (
*)
(
Min
Arg
)
D (
*
has
has
has
has
has
has
has
+
+
=
-
-
éq
3.3.3-5
The application of the equation [éq 3.3.3-5] to the model of Rousselier in great deformations
modified is written:
E
discretized
energy
tion
discretized
continuous
energy
)
,
D (
and
)
,
(
)
,
D (
and
)
,
(
p
p
p
p
p
Tr
p
+
+
=>
-
E
E
E
D
E
&
éq
3.3.3-6
[
]
)
,
D (
)
,
(
Min
)
,
D (
,
p
p
p
p
p
R
S
has
With
Tr
p
+
+
+
-
=
-
-
=
=
-
=
-
E
E
E
E
E
E
éq.
3.3.3-7
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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 stress of Kirchhoff
because of the term
in
E
B
in the expression:
E
B
E
-
=
éq
3.3.3-8
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 Coming Saint who
is written:
()
()
[
]
()
p
U
T
K
K
p
0
2
R
tr
6
~
:
~
2
tr
2
1
,
+
+
µ
+
=
E
E
E
E
E
éq
4.1-1
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:
[
]
Id
E
Id
E
S
T
K
K
+
µ
+
-
=
3
~
2
tr
éq
4.1-2
()
p
R
With
-
=
éq 4.1-3
The threshold of elasticity is given by:
y
H
S
Df
eq
S
-
-
+
=
R
1
exp
1
R)
,
F (S
éq
4.1-4
According to the equations [éq 3.3.2-1] and [éq 3.3.2-2], the laws of flow are defined by:
+
=
-
Id
S
F
G
F
1
exp
3
2
~
3
2
1
S
Df
S
H
eq
T
p
&
&
éq
4.1-5
&
&
=
p
éq 4.1-6
0
=
&
&
F
0
F
0
éq
4.1-7
Code_Aster
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Titrate:
Model of Rousselier in great deformations
Date:
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V. CANO, E. LORENTZ
Key
:
R5.03.06-A
Page
:
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R5.03 booklet: Nonlinear mechanics
HT-66/02/004/A
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:
0
eq
S
éq 4.2-1
It thus remains to characterize the normal cone at the singular points, i.e. checking:
y
H
S
F
D
=
-
=
R
exp
0
~
1
1
and
S
éq
4.2-2
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:
()
[
]
p
p
p
p
p
&
&
&
,
R
:
sup
R)
,
(
,
*
D
D
S
S
p
D
-
-
=
éq
4.2-3
where
*
is the indicating function of the convex one
F
and
)
,
(
p
p
&
D
potential of dissipation obtained by
transform of Legendre-Fenchel of the indicating function of
F
:
[
]
p
p
p
p
&
&
R
:
Sup
)
,
(
0
R)
,
F (
R
,
-
=
D
S
D
S
éq
4.2-4
After some calculations, one obtains:
)
(
I
)
(tr
I
1
tr
ln
tr
)
,
(
IR
IR
1
p
eq
p
p
p
y
p
D
p
p
Df
p
p
3
2
-
&
&
&
&
+
+
+
+
-
+
=
D
D
D
D
éq
4.2-5
with
+
=
+
if not
if
0
0
)
(
I
IR
X
X
éq
4.2-6
For
0
~
=
S
,
*
is worth:
-
-
-
-
=
-
p
p
p
F
D
S
Sup
y
p
p
p
H
D
p
p
p
p
eq
p
p
&
&
4
4
4
4
4
4
3
4
4
4
4
4
4
2
1
&
&
&
R
1
tr
ln
tr
tr
R)
,
(
)
G (tr
1
0
0
tr
,
*
3
2
D
D
D
D
D
D
S
éq
4.2-7
By noticing that for
0
tr
p
D
, the function
)
G (tr
p
D
is concave, the suprémum compared to
trace rate of plastic deformation
p
D
is obtained for:
=
=
1
exp
tr
0
)
(tr
G
S
p
F
D
H
p
p
&
D
D
where
of
éq
4.2-8
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Model of Rousselier in great deformations
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Key
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Note:
One finds well then for the indicating function of the threshold of elasticity
F
.
()
[]
*
&
, R
Sup F &
F
&
S
=
=
+
p
p
Deq
p
p
2
3
0
0
0
if
if not
éq
4.2-9
In a singular point, the normal cone, together of the acceptable directions of flow,
thus characterize by:
=
1
exp
tr
S
p
F
D
H
p
&
D
éq
4.2-10
0
p
eq
D
3
2
p&
éq
4.2-11
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:
(
)
D
F
tr
1
1
1
det
0
F
F
F
F
J
-
=
-
-
=
=
&
éq
4.3-1
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 (
0
=
=
F
constant
J
because
).
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:
(
)
p
F
F
D
tr
1
-
=
&
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 (like
D
) expresses itself
still:
T
p
T
E
p
E
F
G
F
F
D
F
&
2
1
-
=
éq
4.3-3
Finally, one adopts like law of evolution of porosity:
(
)
-
-
=
T
p
F
F
F
G
F &
&
2
1
tr
1
éq
4.3-4
Again, this law of evolution of porosity remains close to that employed by Rousselier when
the elastic strain are small.
Code_Aster
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Titrate:
Model of Rousselier in great deformations
Date:
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Key
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R5.03.06-A
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HT-66/02/004/A
4.4 Relation
`
ROUSSELIER
`
This relation of behavior is available via the argument `
ROUSSELIER
`of the key word
COMP_INCR
under the 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) control
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 stress
=
F S
/
- deformation logarithmic curve
ln (
/)
1
0
+
L L
where
L
0
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 stresses and variables
The stresses are the stresses 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. Linearized deformations
conventional measure
deformations under the assumption of the small deformations and do not have a direction into large
deformations.
Code_Aster
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Model of Rousselier in great deformations
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Key
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HT-66/02/004/A
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 stress
-
, cumulated plastic deformation
p
-
, elastic strain
-
E
,
displacements
U
-
and
U
, one seeks to determine
)
,
,
(
E
p
.
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
F
F
éq 5.1-1
F
det
=
J
éq 5.1-2
=
J
éq 5.1-3
E
B
S
=
éq 5.1-4
E
Id
B
2
-
=
E
éq 5.1-5
·
Equations of state:
[
]
Id
Id
E
E
S
T
K
K
µ
+
+
-
=
3
tr
~
2
éq. 5.1-6
()
p
R
With
-
=
éq 5.1-7
Code_Aster
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Model of Rousselier in great deformations
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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
{
}
[
]
{
}
[
]
T
T
T
T
Tr
T
T
T
T
p
T
p
T
p
p
T
E
E
-
=
-
-
-
=
-
-
-
-
=
-
-
=
-
-
-
-
/
)
(
/
)
2
2
1
(
2
2
2
1
2
1
2
1
R
-
-
E
E
F
E
Id
F
Id
E
F
E
Id
F
E
Id
F
F
G
F
F
F
FG
F
G
F
D
E
B
B
4
4
4
4
3
4
4
4
4
2
1
43
42
1
4
3
42
1
&
éq
5.1-8
By taking the parts traces and deviatoric of the equation [éq 4.1-5], one obtains:
)
tr
exp (
)
3
exp (
tr
tr
1
1
K
T
K
Df
p
Tr
E
E
E
-
-
=
-
-
éq 5.1-9
-
-
=
singular
solution
if
and
regular
solution
if
eq
Tr
eq
Tr
p
E
p
)
~
~
(
3
2
0
~
2
3
~
~
E
E
E
E
E
éq 5.1-10
·
Conditions of coherence
0
F
0
0
F
with
singular
solution
if
R
)
1
tr
exp (
)
1
3
exp (
1
regular
solution
if
R
)
1
tr
exp (
)
1
3
exp (
1
2
F
=
-
-
-
-
-
-
-
-
-
-
+
=
p
p
y
K
T
K
F
D
y
K
T
K
F
D
eq
E
µ
E
E
éq
5.1-11
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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
the unknown factor
X
is the increment of the trace of the elastic strain:
Tr
X
E
E tr
tr
-
=
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:
()
1
1
G
1
1
exp
G
1
p
)
tr
(tr
exp
)
3
exp (
tr
exp
tr
tr
X
K
X
X
K
T
K
K
F
D
p
Tr
Tr
Tr
=
-
-
-
-
=
-
-
E
E
E
E
E
4
4
4
4
4
4
3
4
4
4
4
4
4
2
1
éq
5.2-2
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
. One also notices
that
the increment of cumulated plastic deformation is a continuous and strictly increasing function of
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
:
()
(
)
y
eq
X
Kx
X
X
µe
+
+
-
-
=
-
=
)
p (
R
exp
G
)
S (
S
2
F
1
1
with
éq
5.2-3
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
()
0
S
=
X
. Because of the properties of
S
, this equation admits with more the one positive solution, say
S
X
who exists if and only if
()
0
0
S
. The knowledge of
S
X
allows 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:
eq
Tr
S
S
3
2
p
)
~
~
(
E
E
-
éq
5.2.1-1
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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
Tr
eq
eq
Tr
eq
eq
eq
Tr
E
E
p
E
E
E
p
E
E
E
E
E
~
~
2
3
~
2
3
~
~
éq
5.2.2-1
One notes that because of positivity of
eq
E
, the value sells by auction
p
is limited:
Tr
eq
E
p 32
éq 5.2.2-2
The condition of coherence determines now
X
:
0
p
3
S
2
F
-
-
=
µ
(X)
E
µ
Tr
eq
éq
5.2.2-3
Being given this expression, the increase of the value sells by auction
p
is reduced to the only condition
()
0
S
X
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:
0
0
S
2
)
0
F (
<
-
=
)
(
E
µ
Tr
eq
éq
5.2.2-4
In the contrary case, one must then solve:
>
>
=
-
-
=
if not
exist
if
with
0
0
)
exp (
G
3
S
2
)
F (
1
X
X
X
X
Kx
X
µ
(X)
E
µ
X
S
S
Tr
eq
éq
5.2.2-5
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
do not exist,
0
)
0
(
F
>
since the solution is not elastic.
When
S
X
exist, the function is worth:
Tr
eq
S
S
Tr
eq
S
E
p
p
E
X
3
2
0
3
2
)
F (
<
>
µ
-
µ
=
éq
5.2.2-6
This condition is checked since the singular solution was rejected.
Code_Aster
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Model of Rousselier in great deformations
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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
)
0
F (
·
if
0
)
0
F (
<
, the solution of the problem is the elastic solution
0
=
Ground
X
·
if not one passes into 2)
2) If
0
)
0
S (
>
, the solution is plastic and regular
·
one passes into 4)
3) If
0
)
0
S (
<
, one seeks if the solution is singular
·
one solves
0
)
S (
=
S
X
·
if
S
X
check the inequality
eq
Tr
S
S
3
2
p
)
~
~
(
E
E
-
, then the solution is singular
S
Ground
X
X
=
·
if not,
S
X
is a lower limit to solve
0
)
F (
=
X
, one passes into 4)
4) The solution is plastic and regular
·
one solves
0
)
F (
=
X
5.4 Resolution
To solve the two equations
0
)
S (
=
X
and
0
)
F (
=
X
, 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:
>
=
-
>
=
0
)
0
(
F
)
exp (
G
3
)
S (
2
0
)
0
F (
0
)
F (
1
3
1
F
1
4
4 3
4
4 2
1
4
43
4
42
1
p
µ
Tr
eq
Kx
X
X
µe
X
µ
éq
5.4.1-1
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:
0
F
)
(
F
)
exp (
G
)
R (
2
F
1
2
1
1
F
1
2
<
-
+
-
-
=
X
(X)
X
Kx
X
E
y
Tr
eq
then
4
4
4 3
4
4
4 2
1
µ
éq 5.4.1-2
Code_Aster
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Model of Rousselier in great deformations
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where the function
)
(
F
2
X
is continuous, strictly decreasing. In this case, the resolution of the equations:
inf
inf
p
µ
p
=
3
)
(
F
2
and
inf
inf
inf
p
G
Kx
X
=
)
exp (
1
éq
5.4.1-3
to deduce some successively
p
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
0
)
0
(
F
2
<
, the lower limit
is taken equalizes to zero:
0
=
inf
X
.
The upper limit
Sup
X
is such as:
X
G
Kx
X
Inf
Sup
Sup
)
(
F
3
)
exp (
1
1
µ
=
éq
5.4.1-4
The equation of the type
constant
Kx
X
=
)
exp (
1
is solved by a method of Newton.
Ground
X
inf
X
X
Sup
X
)
p (
3
X
µ
1
F
2
F
)
(
F
)
p (
3
2
inf
inf
X
X
=
µ
)
(
F
)
p (
3
1
inf
sup
X
X
=
µ
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:
()
<
+
-
=
+
+
<
=
-
-
G
R
)
exp (
G
)
exp (
G
R
0
0
S
0
S
1
1
1
1
p
Kx
Kx
X
p
)
(
(X)
y
y
éq
5.4.2-1
The part of left is a continuous function, strictly increasing of
X
and strictly positive with
the origin, the part of straight line is a continuous function, strictly decreasing of
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:
()
()
+
=
+
=
-
-
-
y
Sup
y
Sup
p
K
X
p
Kx
R
G
log
R
)
exp (
G
1
1
1
1
éq
5.4.2-2
The lower limit
Inf
X
is such as:
+
-
-
+
+
=
+
+
=
-
y
Sup
Sup
Inf
y
Sup
Sup
Inf
Kx
X
p
K
X
Kx
X
p
Kx
)
exp (
G
R
G
log
)
exp (
G
R
)
exp (
G
1
1
1
1
1
1
éq
5.4.2-3
X
Inf
X
Sup
X
Ground
X
y
X
+
)
R (
y
p
+
-
)
R (
G
1
)
exp (
G
1
1
Kx
-
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 X
S
not solution
The following system is solved:
=
>
=
-
=
<
=
)
exp (
G
3
2
0
)
0
(
F
)
exp (
G
3
)
S (
2
0
)
S (
0
)
0
S (
0
)
F (
1
1
3
1
F
1
Kx
X
µe
Kx
X
X
µe
X
X
S
S
Tr
eq
p
µ
Tr
eq
S
µ
µ
4
4 3
4
4 2
1
4
43
4
42
1
éq
5.4.3-1
The solution
Ground
X
is such as
0
)
S (
>
Ground
X
.
For the lower limit, one takes
S
Inf
X
X
=
. Being given properties of the functions
1
F
(strictly decreasing) and
)
p (
3
X
µ
(strictly increasing), the upper limit
Sup
X
is such
that (cf [Figure 5.4.3-a]):
Tr
eq
Sup
Sup
E
Kx
X
3
G
2
)
exp (
1
=
éq
5.4.3-2
This equation is solved by a method of Newton.
X
Sup
X
Ground
X
)
exp (
3
1
µ
Kx
X
G
)
0
(
S
E
2
Tr
eq
-
µ
0
)
(
<
X
S
0
)
(
=
X
S
0
)
(
>
X
S
Tr
eq
E
2
µ
S
Inf
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 the unknown factor
X
:
T
X
F
F
=
-
1
&
éq 5.5-1
maybe while integrating:
)
(
exp
)
1
(
1
1
F
X
F
F
dt
T
X
F
df
0
T
0
F
0
-
-
-
=
=
-
éq
5.5-2
where one carried out an exact temporal integration while supposing
X
constant during the pitch of time. It
choice makes it possible to ensure that
F
is increasing and remains lower than 1, whatever the pitch 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
, the aforementioned is obtained by linearizing the system of equations which governs the law of
behavior. We give hereafter the broad outline of this linearization.
For the option
RIGI_MECA_TANG
, they are the same expressions as those given for
FULL_MECA
but with
p
=
0
. In particular, one will have
Id
F
=
.
The law of behavior can be put in the following general form:
(
)
F
=
,
éq 5.6-1
()
E
=
éq 5.6-2
()
Tr
E
E
E
=
éq 5.6-3
)
(F
E
E
=
Tr
Tr
éq 5.6-4
The linearization of this system gives:
F
H
F
F
F
E
E
E
E
=
+
=
:
:
:
:
:
Tr
Tr
éq
5.6-5
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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:
(
)
kl
it
jk
jl
ik
ij
has
2
1
has
+
=
éq
5.6-6
kl
K
p
has
has
L
=
éq 5.5-7
(
)
it
jk
jl
ik
ijkl
C
+
=
2
1
éq
5.6-8
·
Calculation of
and of
F
Linearization of the relation which binds the stress of Cauchy
and the stress of Kirchhoff
give:
F
F
-
=
=
:
1
J
J
J
J
éq
5.6-9
By using the relation [éq 5.6-6], one obtains for
:
C
=
éq 5.6-10
and for
F
:
F
F
-
=
J
J
éq
5.6-11
with
23
11
21
13
32
32
11
12
31
23
13
22
23
12
31
31
22
32
21
13
12
33
32
21
21
33
23
31
12
21
12
22
11
33
31
13
33
11
22
32
23
33
22
11
F
F
F
F
F
J
F
F
F
F
F
J
F
F
F
F
F
J
F
F
F
F
F
J
F
F
F
F
J
F
F
F
F
F
J
F
F
F
F
F
J
F
F
F
F
F
J
F
F
F
F
F
J
-
=
-
=
-
=
-
=
-
=
-
=
-
=
-
=
-
=
13
F
éq
5.6-12
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·
Calculation of
E
The relation which binds the stress of Kirchhoff
and the tensor of elastic strain
E
is given
by:
E
Id
E
E
E
E
Id
E
E
B
S
6
3
-
)
(tr
2
4
Tr
2
E
T
K
T
K
µ
µ
+
+
+
-
-
=
=
éq
5.6-13
One obtains after linearization:
(
)
(
)
(
)
E
E
E
E
E
Id
E
E
E
+
+
-
+
+
-
=
4
Tr
2
3
tr
2
µ
T
K
µ
éq
5.6-14
from where
(
)
(
)
(
)
jl
ik
kj
it
kj
it
lj
ik
kl
ij
ij
ijkl
kl
ij
E
E
E
E
2
E
2
C
3
2
+
+
+
µ
+
-
+
+
µ
-
=
T
K
tr
E
E
éq 5.6-15
·
Calculation of
F
E
Tr
The relation between the tensor of elastic strain
Tr
E
and the increment of the gradient of
transformation
F
is written:
(
)
T
E
Tr
F
B
F
Id
E
-
-
=
2
1
éq
5.6-16
Its linearization gives:
(
)
jk
-
E
pl
IP
-
E
pl
jp
ik
kl
Tr
ij
B
F
B
F
F
E
+
-
=
2
1
éq
5.6-17
·
Calculation of
Tr
E
E
Elastic case
In the elastic case, the calculation of
Tr
E
E
is immediate since
Tr
E
E
=
from where
C
E
E
=
Tr
éq 5.6-18
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Plastic case regular Solution
To determine
Tr
E
E
, one operates in two stages. By the law of flow discretized, one calculates
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:
eq
Tr
E
p E
E
E
~
2
3
~
~
-
=
-
éq
5.6-19
One obtains after linearization:
(
)
E
E
E
E
E
E
~
:
~
~
4
9
~
2
3
~
~
)
2
3
1
(
3
/
1
+
-
=
+
eq
eq
Tr
eq
E
p
p
E
E
p
43
42
1
éq
5.6-20
To determine
E
E
~
:
~
, one contracts the equation [éq 5.6-20] with
e~
what gives:
p
E
eq
Tr
-
=
E
E
E
E
~
:
~
~
:
~
éq
5.6-21
from where
p
E
E
p
eq
Tr
eq
-
+
=
3
2
1
4
4
4 3
4
4
4 2
1
2
1
~
2
3
~
:
~
~
4
9
~
3
With
With
E
E
C
E
E
E
éq
5.6-22
For the part law of flow traces discretized, one a:
=
-
E
E
E
Tr
exp
3
exp
Tr
Tr
1
1
K
-
T
K
p
Df
Tr
éq
5.6-23
what gives:
p
Tr
exp
3
exp
1
3
exp
Tr
exp
Tr
Tr
exp
3
exp
1
1
Tr
2
1
1
1
1
1
1
Tr
1
1
1
+
+
+
=
4
4
4
4
4
4
4
3
4
4
4
4
4
4
4
2
1
4
4
4
4
4
4
4
3
4
4
4
4
4
4
4
2
1
E
E
E
E
E
K
-
T
K
p
DfK
T
K
K
-
Df
K
-
T
K
p
DfK
éq
5.6-24
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In the plastic case, the condition of coherence is worth:
0
Tr
exp
3
exp
2
1
1
1
=
-
-
+
y
eq
R
K
-
T
K
-
Df
µe
E
éq
5.6-25
from where
(
)
0
Tr
Tr
exp
3
exp
~
~
3
1
1
=
-
-
p
H
K
-
T
K
-
DfK
:
E
µ
eq
E
E
E
E
éq
5.6-26
By injecting the relation [éq 5.6-21] in the equation above, one obtains then:
Tr
Tr
eq
)
K
(-
T
K
-
DfK
H
µ
)
K
(-
T
K
-
DfK
:
)
K
(-
T
K
-
DfK
H
µ
E
µ
p
E
E
E
E
E
E
Tr
Tr
exp
3
exp
3
Tr
exp
3
exp
~
~
Tr
exp
3
exp
3
1
3
4
3
1
1
2
1
1
1
1
1
2
4
4
4
4
4
4
4
4
4
3
4
4
4
4
4
4
4
4
4
2
1
4
4
4
4
4
4
4
4
4
3
4
4
4
4
4
4
4
4
4
2
1
+
+
-
+
+
=
éq 5.6-27
While replacing
p
by its value obtained above in the equations [éq 5.622] and [éq 5.6-
24], one obtains:
Tr
Tr
eq
E
E
Id
With
Id
E
E
Id
With
With
E
etr
D
ddvetr
+
+
+
µ
+
+
=
Tr
3
1
3
1
~
:
~
3
3
1
tr
2
2
4
1
3
3
2
1
4
4
4
4
4
3
4
4
4
4
4
2
1
4
4
4
4
4
4
3
4
4
4
4
4
4
2
1
éq 5.6-28
from where
Id
Id
ddvetr
dtretr
ddvetr
E
E
-
+
=
:
3
1
Tr
éq
5.6-29
Plastic case singular Solution
The step is identical to that used previously.
One obtains for the law of flow discretized:
0
~
0
~
=
=
E
E
éq
5.6-30
for the deviatoric part and the part traces, the relation is identical to that found for
regular solution.
p
Tr
+
=
2
1
Tr
Tr
E
E
éq
5.6-31
where
1
and
2
the same definitions as in the preceding paragraph have.
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The condition of coherence then makes it possible to find
p
according to
Tr
E
.
0
Tr
exp
3
exp
1
1
1
=
-
-
y
R
K
-
T
K
Df
E
éq
5.6-32
maybe after linearization:
Tr
)
K
(-
T
K
DfK
H
)
K
(-
T
K
DfK
p
E
E
E
+
-
=
Tr
Tr
exp
3
exp
Tr
exp
3
exp
4
1
1
2
1
1
1
4
4
4
4
4
4
4
4
3
4
4
4
4
4
4
4
4
2
1
éq
5.6-33
that is to say finally:
[
]
Tr
E
Id
E
dtretr
Tr
3
1
2
4
1
4
4 3
4
4 2
1
+
=
éq
5.6-34
from where
Id
dtretr
E
E
=
Tr
éq
5.6-35
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., “Course on the great deformations”, Greco Report/ratio
n51, 1982.
[3]
LORENTZ E., CANO V.: “A minimization principle for finite strain plasticity: incremental
objectivity and immediate implementation ", article subjected in the review Communications 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