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Titrate:
Model of Rousselier with gradient of internal variables
Date:
03/02/05
Author (S):
V. CANO
Key
:
R5.04.11-A
Page
:
1/22
Manual of Reference
R5.04 booklet: Nonlocal modelings
HT-66/05/002/A
Organization (S):
EDF-R & D/AMA















Manual of reference
R5.04 booklet: Nonlocal modelings
Document: R5.04.11



Model of Rousselier with gradient of variables
interns



Summary:

One presents here the model of Rousselier in great deformations in a nonlocal version i.e.
introducing gradients of variables intern in order to take into account strong space variations of
mechanical fields. One activates the nonlocal formulation of the model of Rousselier by one of modelings
“X_GRAD_VARI”
control
AFFE_MODELE
key word
MODEL
. As for the model even, it is available
in the control
STAT_NON_LINE
via the key word
RELATION = “ROUSSELIER”
under
key word factor
COMP_INCR
and with the key word
DEFORMATION =
'SIMO_MIEHE
.
This model is established for three-dimensional modelings (
3d_GRAD_VARI
), axisymmetric
(
AXIS_GRAD_VARI
) and in plane deformations (
D_PLAN_GRAD_VARI
).

One presents the writing and the digital processing of this model.
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Code_Aster
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Titrate:
Model of Rousselier with gradient of internal variables
Date:
03/02/05
Author (S):
V. CANO
Key
:
R5.04.11-A
Page
:
2/22
Manual of Reference
R5.04 booklet: Nonlocal modelings
HT-66/05/002/A
Count
matters
1
Recall on the theory of the models with gradient ....................................................................................... 3
1.1
Construction of the models with gradient .............................................................................................. 3
1.2
Discretization in time ................................................................................................................... 4
1.3
Space discretization by finite elements ........................................................................................ 4
1.4
Calculation of the variables intern at the points of gauss ......................................................................... 5
2
Application to the model of Rousselier .................................................................................................... 6
2.1
Some notations of the model of Rousselier ................................................................................. 6
2.2
Continuous model ................................................................................................................................ 7
2.3
Model discretized ............................................................................................................................. 8
2.4
Processing of the singular items .................................................................................................... 10
3
Numerical resolution .......................................................................................................................... 11
3.1
Expression of the model discretized ................................................................................................... 11
3.2
Resolution of the nonlinear system ............................................................................................... 12
3.3
Course of calculation ................................................................................................................... 13
3.4
Resolution of the functions to cancel ............................................................................................... 14
3.4.1
Hight delimiters and lower if S (0) > 0 ..................................................... 14
3.4.2
Hight delimiters and lower if S (0) = 0 ..................................................... 15
3.4.3
Hight delimiters and lower if S (0) < 0 and X
S
not solution ......................... 16
3.5
Form of the tangent matrix ................................................................................................ 17
3.5.1
Elastic case ........................................................................................................................ 18
3.5.2
Singular case ......................................................................................................................... 18
3.5.3
Regular case .......................................................................................................................... 19
4
Relation `
ROUSSELIER
`........................................................................................................................ 21
5
Bibliography ........................................................................................................................................ 21
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Titrate:
Model of Rousselier with gradient of internal variables
Date:
03/02/05
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V. CANO
Key
:
R5.04.11-A
Page
:
3/22
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R5.04 booklet: Nonlocal modelings
HT-66/05/002/A
1
Recall on the theory of the models with gradient
The models with gradient presented here were developed by E. Lorentz [bib1] in order to be able to describe it
material behavior requested by strong gradients of the mechanical fields which appear
in the damaged areas or in the vicinity of geometrical singularities. Indeed, in
case of strong gradients, the behavior of a material point is not independent any more of its entourage
but depends on the behavior of its vicinity, from where the introduction of gradients into the models.
From a numerical point of view, the calculation of a structure with a local law of damage
conventional watch which the damaged area always locates on only one layer of finite elements
and thus that the response of the structure depends on the adopted mesh: the models with gradient mitigate it
problem.
In what follows, we make a short recall of this theory. One will find in [R5.04.01] details
thorough on this theory.
1.1
Construction of the models with gradient
This formulation is restricted with generalized standard materials. The state is described there by
deformation
, of the internal variables
has
and their associated gradient
has
:
has
=
has
éq 1.1-1
According to the formalism of generalized standard materials, the data of the free energy
)
,
,
(
has
has
and of the potential of dissipation
)
,
(
a&
&a
(for the choice of these two energies, one will be able
to refer to [bib1]) allow to deduce from them the laws from state and the laws of evolution:
=
,
has
With
-
=
,
-
=
has
With
éq 1.1-2
)
,
(
)
,
(
has
With
&
&a
With
éq 1.1-3
If one calls
F
the threshold of elasticity associated with the potential
)
,
(
a&
&a
, the preceding equation is
equivalent to:
With
has
=
F
&
&
,
=
With
has
F
&
&
éq 1.1-4
The problem here is that the variables are not independent any more and are bound by the stress not
local [éq 1.1-1] so that one is not sure to check:
a&
&
&
=
=
With
has
F
éq 1.1-5
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Titrate:
Model of Rousselier with gradient of internal variables
Date:
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V. CANO
Key
:
R5.04.11-A
Page
:
4/22
Manual of Reference
R5.04 booklet: Nonlocal modelings
HT-66/05/002/A
One then proposes to forget the assumption of normal flow in each point of the structure all in
preserving the formalism of generalized standard materials but on the scale of the structure, where
variables of state are now the field of deformation
and the field of internal variables
has
. One
thus defines the total free energy and the total potential of dissipation:
=
X
X
has
X
has
X
has
D
))
(
),
(
),
(
(
)
,
(
F
éq 1.1-6
=
X
X
has
X
has
has
D
))
(
),
(
(
)
D (
&
&
&
éq 1.1-7
The relation of behavior generalized is written now:
=
F
,
has
With
-
=
F
,
)
D (has
With
&
éq 1.1-8

1.2
Discretization in time
While being based on the assumption of convexity compared to
has
potentials
F
and
D
and by adopting one
diagram of implicit Euler, the temporal discretization of the preceding problem [éq 1.1-8] is reduced to
resolution of a problem of minimization relating to the increment
has
fields of variables
interns. This problem is written for behaviors independent of time:
[
]
)
D (
)
,
(
F
Min
has
has
has
has
+
+
-
éq 1.2-1
where
-
has
is the field of internal variables at the previous moment.
1.3
Space discretization by finite elements
To solve the problem of minimization [éq 1.2-1], one carries out a space discretization by
finite elements of the fields of internal variables by means of the nodal unknown factors which one will note
With
.
=
node
K
With
X
NR
X
has
K
)
(
)
(
,
=
node
K
With
X
NR
X
has
K
)
(
)
(
éq
1.3-1
where
NR
K
and
NR
K
are related to form and their gradients associated with the node
K
, respectively.
To simplify the writing, one will pose:
)
,
(
and
)
,
(
with
)
(
)
(
NR
NR
has
has
X
X
=
=
=
B
has
With
B
has
R
R
R
R
éq
1.3-2
The equation [éq 1.2-1] is written then:
(
)
+
gauss
)
(
)
(
Min
With
B
With
B
With
G
G
G
R
R
éq 1.3-3
in which
G
indicate the weight of integration of the point of Gauss
G
.
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Code_Aster
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Titrate:
Model of Rousselier with gradient of internal variables
Date:
03/02/05
Author (S):
V. CANO
Key
:
R5.04.11-A
Page
:
5/22
Manual of Reference
R5.04 booklet: Nonlocal modelings
HT-66/05/002/A
The introduction of new unknown factors
R
who represent the values and the gradients of the variables
interns at the points of gauss allows to transfer nonthe linearities at the local level and to divide
resolution of the equation [éq 1.3-3] in a linear total part (on the structure) and a local part
nonlinear (at the points of integration). The problem of minimization is written then:
(
)
(
0
=
)




-
+
-
-
gauss
G
G
G
G
G
G
G
With
B
With
R
R
R
R
R
R
(
)
Min
(
,
éq 1.3-4
By dualisation of the stress, one builds the Lagrangian one increased problem [éq 1.3-4] for
to return to a problem without stresses:
)
,
,
(
L
Min
Max
,
µ
With
With
µ
R
R
R
R
R
éq 1.3-5
with
X
NR
X
X
With
B
µ
With
B
µ
With
NR
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
.
.
0
)
(
2
)
(
)
(
)
,
,
(
L
2
2
G
NR
G
G
G
G
G
G
G
G
G
G
R
G
G
R
R
=
>


-
+
-
+
-
+
=
-
and
éq 1.3-6
The matrix
NR
R
R
defined positive is introduced into the standard so that the coefficient of penalization
R
that is to say adimensional. This matrix is selected like a diagonal approximation of the derivative
second (cf [R5.04.01] for more detail):




=
NR
R
R
R
R
Diag
2
éq 1.3-7
This problem is then solved by a method of Newton to solve the primal problem (calculation
of
With
and
R
) and a method BFGS with linear search Wolfe to solve the dual problem
(calculation of the multipliers of Lagrange).
1.4
Calculation of the variables intern at the points of gauss
At the time of the resolution of the local problem, one seeks to minimize the equation [éq 1.3-6] compared to
R
,
with
With
and
µr
fixed what is equivalent to:
)
(
)
(
)
,
,
(
L
Min
G
G
fixed
G
G
fixed
G
G
fixed
G
G
fixed
R
R
G
G
With
B
NR
µ
µ
With
R
R
R
R
R
R
R
R
R
R
-
+
+
-
éq
1.4-1
With convergence, the third term in the member of left of the expression above becomes null
(the stress is carried out) and multipliers of Lagrange
G
µr
seem a force then
complementary thermodynamics resulting from the nonlocal condition
0
=
-
G
G
With
B
R
R
.
From a practical point of view, to write in an incremental way the model of behavior not
room, one will write classically the equivalent of the equation [éq 1.4-1]:
)
,
(
with
F
R
With
With
With
=
=
R
R
R
With
R
R
&
R
éq
1.4-2
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Titrate:
Model of Rousselier with gradient of internal variables
Date:
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V. CANO
Key
:
R5.04.11-A
Page
:
6/22
Manual of Reference
R5.04 booklet: Nonlocal modelings
HT-66/05/002/A
with
With
With
With
B
NR
µ
With
With
R
R
R
R
R
R
R
R
R
-
=
=
-
+
+
=
)
,
(
with
)
(
With
R
R
éq
1.4-3
where
F
is the threshold of elasticity associated with the potential with dissipation
, calculated by transform of
Legendre-Fenchel, and
R
With
and
R
With
thermodynamic forces associated the internal variables
and
, respectively. It is noted here that these two thermodynamic forces must be corrected,
on the one hand, by the multipliers of Lagrange associated with the nonlocal stress
=
, and
in addition, by a measurement (balanced) of the variation enters the fields at the points of gauss
)
,
(
=
R
and the nodal field
With
.

2
Application to the model of Rousselier
We now describe the application of this theory to gradient to the model of Rousselier
(cf [R5.03.06] for more detail on this model).
2.1
Some notations of the model of Rousselier
One points out below some definitions and notations used in the model of Rousselier.
F
: tensor gradient which makes pass from the initial configuration
0
with the current configuration
()
T
F
p
: “plastic” tensor gradient which makes pass from the configuration
0
with the slackened configuration
R
F
E
: “elastic” tensor gradient which makes pass from the configuration
R
with
()
T
p
E
F
F
F
=
éq 2.1-1
J
: variation of volume
F
det
=
J
éq 2.1-2
B
E
: left tensor eulérien of Cauchy-Green of elastic strain
B
F F
E
E and
=
éq 2.1-3
G
p
: Lagrangian tensor of plastic deformation
G
F F
p
Pt
p
=
-
(
)
1
éq 2.1-4
B
FG F
E
p T
=
éq 2.1-5
E
: tensor of the deformations used in the model of Rousselier
(
)
E
B
Id
E
-
=
2
1
éq 2.1-6
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Model of Rousselier with gradient of internal variables
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:
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D
: rate of deformation
p
D
: rate of plastic deformation
T
p
p
F
G
F
D
&
2
1
-
éq 2.1-7
: stress of Cauchy
: stress of Kirchhoff
J
=
éq 2.1-8
S
: tensor of the stresses used for the model of Rousselier
E
B
S
=
éq 2.1-9
With
R
-
=
: isotropic work hardening
p
: cumulated plastic deformation
F
: porosity
(
)


-
-
=
T
p
F
F
F
G
F &
&
2
1
tr
1
éq 2.1-10
F
: criterion of plasticity of the model of Rousselier
y
eq
p
With
Df
S
With
-
+




+
=
)
(
3
tr
exp
)
,
F (
1
1
S
S
éq
2.1-11
where
y
is limited of elasticity and
1
,
D
two parameters materials specific to this law.
2.2 Model
continuous
To preserve a simple model, one will be satisfied to introduce, to control the modes of
localization poro-plastic, a quadratic term in gradient of cumulated plastic deformation
p
in the free energy of the model of local Rousselier. As for the potential of dissipation, there remains unchanged
compared to the local version.
(
)
()
[
]
()
+
+
µ
+
=
p
p
E
E
E
p
E
.
D
D
13
2
~
:
~
2
tr
2
1
,
,
2
0
2
p
R
L
U
R
K
p
B
p
éq
2.2-1
)
3
2
-
(
I
)
(tr
I
1
tr
ln
tr
)
,
,
(
IR
IR
1
p
eq
p
p
p
y
p
D
p
p
Df
p
p
&
&
&
&
&
+
+
+
+




-
+
=
D
D
D
p
D
éq
2.2-2
This potential of dissipation corresponds to the criterion of plasticity [éq 2.1-11].
B
L
is the length characteristic of the material which corresponds to the average distance between two
inclusions, privileged sites of germination and nucleation of cavities.
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Model of Rousselier with gradient of internal variables
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:
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R5.04 booklet: Nonlocal modelings
HT-66/05/002/A
2.3 Model
discretized
To adopt a purely implicit algorithm to integrate this law of led behavior, on the one hand,
the resolution of a rather complex nonlinear system any more, and in addition, does not allow to express it
problem like the minimization of a functional calculus. This is why one prefers to deal with manner
clarify the variation according to porosity in the potential of dissipation [éq 2.2-2] as well as the variation
according to
p
quadratic term in
p
free energy [éq 2.2-1]. For the other terms, one
employ a diagram of implicit Euler. It will be noted that the discretization of the rate of plastic deformation
p
D
express yourself directly according to the elastic strain
E
:
[
]
[
]
T
T
T
Tr
T
p
T
p
E
T
p
p
T
-
=






-
-
=
-
-
=
-
/
)
(
2
1
1
2
1
2
1
R
E
E
F
FG
Id
E
F
FG
B
F
G
F
D
E
4
4 3
4
4 2
1
&
éq 2.3-1
So that the free energy and the discretized potential of dissipation are given by the expressions
following:
(
)
()
[
]
()
-
+
+
µ
+
=
p
p
E
E
E
p
E
.
)
(
D
D
13
2
~
:
~
2
tr
2
1
,
,
2
0
2
p
p
R
L
U
R
K
p
B
p
éq 2.3-2
)
3
2
-
(
I
)
(tr
I
1
tr
ln
tr
)
,
(
IR
IR
1
eq
y
p
p
Df
p
p
E
E
E
E
E
+
+




-
+
=
+
+
-
éq
2.3-3
with
-
-
=
T
T
T
,
tr
E
E
E
-
=
,
-
-
=
p
p
p
and
-
-
=
p
p
p
éq
2.3-4
-
Q
is the quantity known at the previous moment
-
T
.
In accordance with the paragraph [§1.4] [éq 1.4-1], the integration of the model of nonlocal Rousselier
express yourself like the minimization of the following functional calculus:
(
)
(
)
()


-
+
-
+
+
+
+
-
p
P
µ
p
P
E
p
p
E
E
NR
p
E
R
R
R
R
R
R
R
R
R
R
2
,
2
,
,
min
R
p
Tr
éq.
2.3-5
What is equivalent to [éq 1.4-1] and [éq 1.4-2]:




=
=
=




-
+
+
=
-
=
0
F
F
)
,
(
)
(
R
R
R
With
p
p
R
With
S
E
E
p
P
NR
µ
With
With
E
S
R
R
R
R
R
R
R
éq 2.3-6
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Titrate:
Model of Rousselier with gradient of internal variables
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03/02/05
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V. CANO
Key
:
R5.04.11-A
Page
:
9/22
Manual of Reference
R5.04 booklet: Nonlocal modelings
HT-66/05/002/A
According to the equation [éq 1.3-7], the matrix of weighting
NR
R
R
is worth:
2
2
2
2
2
2
13
4
D
D
0
0
0
0
0
0
0
0
0
0
0
0
1
Diag
B
C
C
C
C
L
L
p
p
R
H
L
L
L
H
=
=






=




=
-
-
and
with
)
(
-
p
p
NR
R
R
R
R
éq
2.3-7
In all these equations, one adopted the following notations:
)
,
(
)
,
(
-
-
=
=
R
With
With
R
With
R
éq 2.3-8
)
,
(
R
R
R
With
=
With
With
R
éq 2.3-9
)
,
(
=
p
p
p
R
éq 2.3-10
)
,
(
=
=
P
P
B
P
P
R
R
éq 2.3-11
The vector
Pr
represent the cumulated plastic deformation and its gradient, calculated at the points of
Gauss while
P
represent the plastic deformation calculated with the nodes.
)
,
(
µ
=
µ
µr
éq 2.3-12
The whole of the equations to be solved is thus the following:
Equations of state:
[
]
E
D
E
S
~
2
tr
µ
+
-
=
K
éq 2.3-13
()
(
)
()
2
1
C
p
C
p
R
p
P
H
R
p
R
With
R
+
-
-
=
-
+
µ
+
-
=
-
éq
2.3-14
(
)
0
4
3
2
2
=
+
-
=
-
+
+
-
=
-
-
C
p
p
P
µ
p
With
C
L
H
R
L
H
C
C
R
éq
2.3-15
Laws of flow:




=
-
1
3
tr
exp
tr
S
E
F
D
p
éq 2.3-16
eq
S
p S
E
~
2
3
~
=
éq 2.3-17
3
4
C
C
p
=
éq 2.3-18
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Model of Rousselier with gradient of internal variables
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V. CANO
Key
:
R5.04.11-A
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:
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R5.04 booklet: Nonlocal modelings
HT-66/05/002/A
Condition of coherence:
()
0
F
0
0
F
3
tr
exp
,
F
1
1
=
-
+




+
=
-
p
p
With
F
D
S
With
y
R
eq
R
S
S
éq 2.3-19
Definition of the various coefficients:
0
1
=
-
Rh
C
,
P
C
C
1
2
+
µ
=
,
0
)
1
(
2
3
+
=
-
C
L
H
R
C
and
+
=
P
µ
C
2
1
4
C
L
C
éq 2.3-20
2.4
Processing of the singular points
The expression of the normal cone to which the direction of flow belongs is licit only at the points where
the criteria are derivable, i.e. if
0
eq
S
. While proceeding classically by a prediction
rubber band followed by a plastic correction only if necessary, one can be satisfied to examine
singular points confined on the border of convex of elasticity, i.e. points such as:
y
R
With
F
D
=
+




=
-
1
1
3
tr
exp
0
~
S
S
and
éq 2.4-1
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:
(
)
[
]








-
-
-
+
=
-
+
+
-
-
-
1
tr
ln
tr
With
tr
tr
3
1
sup
,
,
With
:
sup
1
0
0
tr
,
,
,
,
3
2
p
F
D
p
p
p
p
p
p
y
R
p
D
L
p
p
p
R
R
p
p
p
eq
B
p
p
p
&
&
&
&
&
&
&
&
&
&
&
&
&
D
D
D
S
p
D
p
With
D
S
p
D
p
D
p
D
éq 2.4-2
They are the directions of flow
(
)
p
D
&
&,
, p
p
characterized by:








=
p
D
p
F
D
p
eq
p
&
&
3
2
3
tr
exp
tr
1
S
D
éq 2.4-3
Thus, in a singular point of the border of the field of elasticity, the increments of the variables
interns
(
)
p
E
,
, p
check simply:








=
-
p
p
F
D
eq
)
(
3
2
3
tr
exp
tr
1
E
S
E
éq 2.4-4
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Code_Aster
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7.2
Titrate:
Model of Rousselier with gradient of internal variables
Date:
03/02/05
Author (S):
V. CANO
Key
:
R5.04.11-A
Page
:
11/22
Manual of Reference
R5.04 booklet: Nonlocal modelings
HT-66/05/002/A
3 Resolution
numerical
3.1
Expression of the discretized model
Knowing
µr
and
Pr
, elastic strain
-
E
, cumulated plastic deformation
-
p
, it
gradient of the cumulated plastic deformation
-
p
and displacements
U
-
and
U
, one seeks with
to determine
)
,
,
,
,
(
F
p
p
E
.
Displacements being known, gradients of the transformation of
0
with
-
, noted
F
-
, and of
-
with
, noted
F
, are known.
The system of equations to be solved is as follows:
-
=
F
F
F
éq 3.1-1
F
det
=
J
éq 3.1-2
=
J
éq 3.1-3
E
B
S
=
éq 3.1-4
E
Id
B
2
-
=
E
éq 3.1-5
{
}
[
]
T
T
F
E
Id
F
Id
E
-
-
=
-
2
2
1
R
éq 3.1-6
Equations of state:
[
]
E
Id
E
S
~
2
tr
µ
+
-
=
K
éq 3.1-7
()
2
1
C
p
C
p
R
With
R
+
-
-
=
éq 3.1-8
0
C
p
With
=
+
-
=
4
3
C
R
éq 3.1-9
Definition:
0
1
=
-
Rh
C
,
P
C
C
1
2
+
µ
=
,
0
)
1
(
2
3
+
=
-
C
L
H
R
C
,
+
=
P
µ
C
2
1
4
C
L
C
,
2
2
13
4
B
C
L
L
=
éq 3.1-10
Thereafter, one expresses the laws of flow and the criteria of plasticity directly according to
tensor of the elastic strain
E
.
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Titrate:
Model of Rousselier with gradient of internal variables
Date:
03/02/05
Author (S):
V. CANO
Key
:
R5.04.11-A
Page
:
12/22
Manual of Reference
R5.04 booklet: Nonlocal modelings
HT-66/05/002/A
Laws of flow:
)
tr
exp (
tr
tr
1
K
Df
p
Tr
E
E
E
-
=
-
-
éq
3.1-11




-
=
singular
solution
if
and
regular
solution
if
eq
eq
Tr
p
E
p
)
(
3
2
0
~
2
3
~
~
E
E
E
E
éq
3.1-12
3
4
C
C
p
=
éq 3.1-13
Condition of coherence:
()
0
F
0
0
F
tr
exp
tr
exp
2
,
F
1
1
1
1
=




-
+




-
-
+




-
+
µ
=
-
-
p
p
With
K
F
D
With
K
F
D
E
With
y
R
y
R
eq
R
with
singular
solution
if
regular
solution
if
E
E
S
éq
3.1-14
Porosity:
The law of evolution of porosity is treated same manner as in the model of Rousselier in
local version. One obtains (cf [R5.03.06] for more detail):
(
)
E
E
tr
tr
exp
1
1
-
-
-
=
Tr
0
)
F
(
F
éq
3.1-15
where
0
F
is initial porosity.
3.2
Resolution of the nonlinear system
The integration of the law of behavior is thus summarized to solve only the equations
[éq 3.1-11], [éq 3.1-12] and [éq 3.1-14] (the equation [éq 3.1-13] gives directly
p
since
4
C
and
3
C
are known). Once determined
p
and
E
by the whole of these three equations, one deduces some
stress
S
by the equation [éq 3.1-7], the stress of Cauchy
by the equations [éq 3.1-3] and
[éq 3.1-4] and porosity
F
by the equation [éq 3.1-15].
It is noticed that the three equations to be solved are identical to those of the model of Rousselier
room where one changed only the thermodynamic force
With
by
R
With
. The resolution is thus
identical to this model. For this reason, we give only the broad outline. For more
detail, the reader will refer to the document [R5.03.06].
If one poses:
0
tr
tr
-
=
Tr
X
E
E
éq 3.2-1
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Key
:
R5.04.11-A
Page
:
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Manual of Reference
R5.04 booklet: Nonlocal modelings
HT-66/05/002/A
then, the equation [éq 3.1-11] is written:
()
1
exp
1
X
K
X
G
X
p
=
with




-
=
-
1
tr
exp
K
F
D
G
Tr
E
éq 3.2-2
After some calculations, the equation [éq 3.1-12] becomes:








-
=
singular
solution
if
and
regular
solution
if
Tr
eq
Tr
Tr
eq
E
p
E
X
p
X
3
2
0
~
)
(
2
3
1
)
(
~
E
E
éq
3.2-3
Lastly, if one poses:
(
)




-
-
-
+
+
=
1
1
2
1
exp
)
(
)
(
)
S (
Kx
G
C
X
p
C
X
p
R
X
y
éq
3.2-4
where S (X)
is a continuous and strictly increasing function of
X
, then the condition of coherence
[éq 3.1-14] is written (while using [éq 3.2-3]):
()
singular
solution
if
regular
solution
if



=
-
µ
-
µ
=
0
)
S (
0
)
S (
3
2
F
X
X
p
E
X
Tr
eq
éq 3.2-5
One thus brings back oneself to solve this scalar equation in
X
. The variable
X
is positive or null X
0
to guarantee a positive cumulated plastic deformation and the elastic solution is obtained for
x=0
.
3.3
Course of calculation
The general step to determine
X
is as follows:
1) One seeks if the solution is elastic
·
calculation of
)
0
F (
·
if
0
)
0
F (
<
, then 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
Tr
eq
S
E
p
~
3
2
, then the solution is singular
S
Ground
X
X
=
·
if not,
S
X
is a lower limit to solve
0
)
F (
=
Ground
X
, one passes into 4)
4) The solution is plastic and regular
·
one solves
0
)
F (
=
Ground
X
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V. CANO
Key
:
R5.04.11-A
Page
:
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R5.04 booklet: Nonlocal modelings
HT-66/05/002/A
3.4
Resolution of the functions to be cancelled
To solve the equations
0
)
S (
=
X
,
0
)
F (
=
X
, one employs a method of Newton with terminals
controlled 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.
3.4.1 Hight delimiters and lower if S (0) > 0
One solves:




>
µ
=
-


>
=
0
)
0
(
F
)
exp (
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
G
X
µe
X
éq
3.4.1-1
where the function
)
(X
p
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 3.4.1-a]).
One poses:
0
F
)
(
F
)
exp (
)
(
2
F
1
2
1
1
F
2
1
1
2
<
-
+
+
-
-
-
µ
=
X
(X)
X
Kx
G
C
p
C
X
R
E
y
Tr
eq
then
4
4
4
4
4
3
4
4
4
4
4
2
1
éq 3.4.1-2
where the function
)
(
F
2
X
is continuous, strictly decreasing. Two cases will be considered.
Case where
0
)
0
(
F
2
>
In this case, the successive resolution of the equations:
Inf
Inf
p
µ
p
=
3
)
(
F
2
éq 3.4.1-3
to deduce some
p
, then
Inf
Inf
Inf
p
G
Kx
X
=
)
exp (
1
éq 3.4.1-4
to deduce some
X
give a lower limit
Inf
X
.
Note:
In Code_Aster, the routine rcfonc solves the equation corresponding to the solution of the model to
isotropic work hardening and criterion of Von Mises, i.e.
p
p
R
y
el
eq
µ
=
-
-
3
)
(
. One provides
in input of this routine
el
eq
, the Young modulus
E
and the Poisson's ratio
. If one poses
2
1
2
C
p
C
E
Tr
eq
el
eq
+
-
µ
=
-
and
[
]
1
3
3
)
1
(
2
C
E
+
µ
+
=
, the function
Inf
Inf
p
µ
p
=
3
)
(
F
2
bring back to solve an equation of the type
p
p
R
y
el
eq
µ
=
-
-
3
)
(
.
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Titrate:
Model of Rousselier with gradient of internal variables
Date:
03/02/05
Author (S):
V. CANO
Key
:
R5.04.11-A
Page
:
15/22
Manual of Reference
R5.04 booklet: Nonlocal modelings
HT-66/05/002/A
Case where
0
)
0
(
F
2
In this case, the lower limit is taken equalizes to zero:
0
=
Inf
X
.
The upper limit
Sup
X
is such as:
X
µ
G
Kx
X
X
p
Inf
Sup
Sup
Inf
Sup
)
(
F
3
)
exp (
)
(
F
3
1
1
1
=
=
µ
éq
3.4.1-5
The equation of the type
constant
Kx
X
=
)
exp (
1
is solved by a method of Newton.
X
()
X
p
µ
3
()
X
F
1
() ()
Inf
Sup
X
F
X
p
1
3
=
µ
() ()
Inf
Inf
X
F
X
p
2
3
=
µ
()
X
F
2
Inf
X
Ground
X
Sup
X
Appear 3.4.1-a: Chart of the hight delimiters and lower

3.4.2 Hight delimiters and lower if S (0) = 0
The system to be solved is as follows:
()




<
-
+
+
-
=
-
+
+




+


<
=
-
-
-
G
C
p
C
p
R
Kx
G
C
p
C
Kx
G
X
p
R
X
y
y
1
2
1
1
1
2
1
1
)
exp (
)
exp (
0
)
0
S (
0
)
S (
éq
3.4.2-1
The part of left is a continuous function, strictly increasing of
X
, the part of straight line is one
continuous function, strictly decreasing of
X
and strictly positive at the origin.
Two cases must be considered.
Case where
()
0
2
1
>
-
+
+
-
-
C
p
C
p
R
y
:
Using the properties of the two functions, a chart (cf [Figure 3.4.2-a]) of these
functions shows that the upper limit
Sup
X
is such as:
()
()




-
+
+
=
-
+
+
=
-
-
-
-
-
2
1
1
1
2
1
1
1
log
)
exp (
C
p
C
p
R
G
K
X
C
p
C
p
R
Kx
G
y
Sup
y
Sup
éq 3.4.2-2
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V. CANO
Key
:
R5.04.11-A
Page
:
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R5.04 booklet: Nonlocal modelings
HT-66/05/002/A
The lower limit
Inf
X
is such as:
()
()
+




-
+
+
=
-
+
+
=
-
2
1
1
1
2
1
1
1
log
)
exp (
C
p
C
p
R
G
K
X
C
p
C
p
R
Kx
G
Sup
y
Sup
Inf
Sup
y
Sup
Inf
éq
3.4.2-3
Case where
()
0
2
1
-
+
+
-
-
C
p
C
p
R
y
:
In this case, one will take for the lower limit:
0
=
Inf
X
éq 3.4.2-4
The successive resolution of the two equations:
Sup
y
Sup
p
C
p
R
p
C
C
G
=
-
-
-
+
-
1
1
2
1
)
(
and
Sup
Sup
Sup
p
G
Kx
X
=
)
exp (
1
éq 3.4.2-5
allows to deduce some
p
then
X
to give an upper limit
Sup
X
. To solve the equation of
left above, it is enough (see remark of the preceding paragraph) to use the routine rcfonc in
posing
2
1
1
C
p
C
G
el
eq
+
-
=
-
,
1
3
)
1
(
2
C
E
+
=
and
1
=
.
X
()
2
1
R
C
p
X
y
-
+
+
()
2
1
R
C
p
C
p
y
-
+
+
-
-
G
1
Ground
Inf
X
X
Sup
X




-
1
1
exp
G
Kx
Appear 3.4.2-a: Chart of the hight delimiters and lower

3.4.3 Hight delimiters and lower if S (0) < 0 and X
S
not solution
The following system is solved:






µ
=
>
µ
=
-


=
<
=
)
exp (
3
2
0
)
0
(
F
)
exp (
3
)
S (
2
0
)
S (
0
)
0
S (
0
)
F (
1
1
3
1
F
1
Kx
X
G
µe
Kx
X
G
X
µe
X
X
S
S
Tr
eq
p
µ
Tr
eq
S
4
4 3
4
4 2
1
4
43
4
42
1
éq
3.4.3-1
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:
R5.04.11-A
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:
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HT-66/05/002/A
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 3.4.3-a]):
Tr
eq
Sup
Sup
E
G
Kx
X
3
2
)
exp (
1
=
éq
3.4.3-2
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 3.4.3-a: chart of the hight delimiters and lower

3.5
Form of the tangent matrix
The resolution of the primal problem (calculation of
P
and
Pr
) by a method of Newton, calculation requires
following tangent matrix:






=
P
p
p
P
H
P
p
P
p
P
p
R
R
éq 3.5-1
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V. CANO
Key
:
R5.04.11-A
Page
:
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HT-66/05/002/A
3.5.1 Case
rubber band
It is reminded the meeting that the elastic solution is given by:
-
=
p
p
éq 3.5.1-1
3
4
C
C
p
=
éq 3.5.1-2
what gives for the tangent matrix:
ij
J
I
I
I
R
R
P
p
P
p
P
p
P
p
+
=
=
=
=
1
0
0
0
éq
3.5.1-3
3.5.2 Case
singular
The singular solution is given by the equations:
1
/
-
+
=
Kx
E
G
X
p
p
éq 3.5.2-1
3
4
C
C
p
=
éq 3.5.2-2
with
X
who checks:
0
)
(
)
S (
1
/
1
2
1
=
-
+
-
+
=
-
Kx
y
Ge
C
p
C
p
R
X
éq
3.5.2-3
and
P
C
C
1
2
+
µ
=
,
-
+
=
P
µ
C
2
4
C
L
Rh
éq
3.5.2-4
The linearization of this system gives:
X
E
Kx
G
p
Kx
coeff
+
=
1
1
/
1
)
1
(
1
43
42
1
éq 3.5.2-5
+
=
P
p
R
R
1
éq 3.5.2-6
0
)
(
)
S (
1
/
1
1
=
+
-
+
=
-
X
KGe
P
C
p
C
H
X
Kx
éq
3.5.2-7
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Code_Aster
®
Version
7.2
Titrate:
Model of Rousselier with gradient of internal variables
Date:
03/02/05
Author (S):
V. CANO
Key
:
R5.04.11-A
Page
:
19/22
Manual of Reference
R5.04 booklet: Nonlocal modelings
HT-66/05/002/A
While replacing
p
in the equation above, one obtains:
P
KGe
C
H
coeff
E
C
X
Kx
Kx
+
+
=
-
-
1
1
/
2
1
1
/
1
)
(
éq
3.5.2-8
what gives for the expression of
p
:
P
coeff
C
KGe
coeff
C
H
p
coeff
Kx
+
+
=
-
1
1
/
2
1
1
2
1
)
(
1
4
4
4
4
4
3
4
4
4
4
4
2
1
éq 3.5.2-9
maybe for the tangent matrix:
ij
J
I
I
I
R
R
P
p
P
p
P
p
coeff
coeff
C
P
p
+
=
=
=
=
1
0
0
2
1
1
éq
3.5.2-10
3.5.3 Case
regular
The regular solution is given by the equations:
1
/
-
+
=
Kx
E
G
X
p
p
éq 3.5.3-1
3
4
C
C
p
=
éq 3.5.3-2
X
check:
() ()
0
S
3
2
F
=
-
µ
-
=
X
X
p
µe
Tr
eq
éq
3.5.3-3
The linearization of the whole of these equations gives:
X
E
coeff
p
Kx
=
1
/
1
éq 3.5.3-4
+
=
P
p
R
R
1
éq 3.5.3-5
()
()
0
S
3
F
=
-
µ
-
=
X
X
p
éq
3.5.3-6
X
KGe
P
C
p
C
H
X
Kx
+
-
+
=
-
1
/
1
1
)
(
)
S (
éq 3.5.3-7
background image
Code_Aster
®
Version
7.2
Titrate:
Model of Rousselier with gradient of internal variables
Date:
03/02/05
Author (S):
V. CANO
Key
:
R5.04.11-A
Page
:
20/22
Manual of Reference
R5.04 booklet: Nonlocal modelings
HT-66/05/002/A
While replacing in the equation [éq 3.5.3-6], the expression of
p
[éq 3.5.3-4], one obtains for
X
:
(
)
P
C
X
E
KGe
C
H
coeff
Kx
coeff
Kx
=
+
µ
+
+
-
1
/
/
2
1
1
1
2
1
)
3
(
4
4
4
4
4
4
3
4
4
4
4
4
4
2
1
éq. 3.5.3-8
One finds then:
P
coeff
coeff
C
p
=
2
1
1
éq 3.5.3-9
+
=
P
p
R
R
1
éq
3.5.3-10
what makes for the tangent matrix:
ij
J
I
I
I
R
R
P
p
P
p
P
p
coeff
coeff
C
P
p
+
=
=
=
=
1
0
0
2
1
1
éq
3.5.3-11
background image
Code_Aster
®
Version
7.2
Titrate:
Model of Rousselier with gradient of internal variables
Date:
03/02/05
Author (S):
V. CANO
Key
:
R5.04.11-A
Page
:
21/22
Manual of Reference
R5.04 booklet: Nonlocal modelings
HT-66/05/002/A
4 Relation
`
ROUSSELIER
`
One activates the nonlocal formulation of the model of Rousselier by one of modelings
“X_GRAD_VARI”
control
AFFE_MODELE
key word
MODEL
. As for the model even, it is
available in the control
STAT_NON_LINE
via the key word
RELATION:
“ROUSSELIER”
under the key word factor
COMP_INCR
and with the key word
DEFORMATION:
“SIMO_MIEHE”
.
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
([U4.23.01]). The characteristic length
B
L
is given under the key word
LONG_CARA
of
DEFI_MATERIAU
.
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 with V4, the gradient following axes X, y and Z of
p
,
·
V5, porosity
F
,
·
V6 with V11, the tensor of elastic strain
E
,
·
V12, 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.


5 Bibliography
[1]
LORENTZ E.: “Laws of behavior to gradients of internal variables: construction,
variational formulation and implementation numerical ", Thesis of doctorate of the university
Paris 6, April 27, 1999.
background image
Code_Aster
®
Version
7.2
Titrate:
Model of Rousselier with gradient of internal variables
Date:
03/02/05
Author (S):
V. CANO
Key
:
R5.04.11-A
Page
:
22/22
Manual of Reference
R5.04 booklet: Nonlocal modelings
HT-66/05/002/A

























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