Code_Aster
®
Version
7.4
Titrate:
Modeling élasto (visco) plastic in great deformations
Date:
14/04/05
Author (S):
V. CANO, E. LORENTZ
Key
:
R5.03.21-C
Page
:
1/18
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HT-66/05/002/A
Organization (S):
EDF-R & D/AMA
Manual of reference
R5.03 booklet: Nonlinear mechanics
Document: R5.03.21
Modeling élasto (visco) plastic with
isotropic work hardening in great deformations
Summary
One describes here a thermoelastoplastic relation between behavior and isotropic work hardening written into large
deformations and proposed by Simo and Miehe. This model is available in the control
STAT_NON_LINE
via the key word
RELATION:
“VMIS_ISOT_TRAC”
or
“VMIS_ISOT_LINE”
under the key word
factor
COMP_INCR
and with the key word
DEFORMATION:
“SIMO_MIEHE”
. A viscous version of this model
is also proposed:
“VISC_ISOT_TRAC”
and
“VISC_ISOT_LINE”
.
This model is established for three-dimensional modelings (3D), axisymmetric (Axis) and in deformations
plane (D_PLAN).
One presents the writing and the digital processing of this law, as well as the associated variational formulation. It
of a variational formulation eulérienne acts, with reactualization of the geometry and which takes account of
rigidity of behavior and geometrical rigidity.
Code_Aster
®
Version
7.4
Titrate:
Modeling élasto (visco) plastic in great deformations
Date:
14/04/05
Author (S):
V. CANO, E. LORENTZ
Key
:
R5.03.21-C
Page
:
2/18
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HT-66/05/002/A
Count
matters
Code_Aster
®
Version
7.4
Titrate:
Modeling élasto (visco) plastic in great deformations
Date:
14/04/05
Author (S):
V. CANO, E. LORENTZ
Key
:
R5.03.21-C
Page
:
3/18
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HT-66/05/002/A
1 Introduction
We present here a thermoelastoplastic law of behavior written in great deformations
proposed by SIMO J.C and MIEHE C. [bib1] which tends in small deformations towards the model of
elastoplastic behavior with isotropic work hardening and criterion of Von Mises, described in [R5.03.02].
The kinematics choices allow, as with the simple reactualization available via the key word
PETIT_REAC
, to treat great displacements and great deformations but also of
great rotations in an exact way.
Specificities of this model are as follows:
·
just like in small deformations, one supposes the existence of a slackened configuration,
i.e. locally free of 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,
·
the plastic deformations are done with constant volume. The variation of volume is then
only due to the elastic thermo deformations,
·
this model led during its numerical integration to a model incrémentalement objective
(cf [§3.3]) what makes it possible to obtain the exact solution in the presence of great rotations.
A viscous version of this model is also available (law in hyperbolic sine as in
the case of the model of Rousselier ROUSS_VISC, cf [R5.03.07]).
Thereafter, one briefly points out some concepts of mechanics in great deformations, then one
present the relations of behavior of the model and its numerical integration to treat them
equilibrium equations.
One proposes a variational formulation eulérienne, with reactualization of the geometry. For this reason,
one expresses the work of the interior efforts and his variation (with an aim of a resolution by the method
of Newton) for the continuous problem, which provides respectively after discretization by elements
stop the vector of the interior forces and the tangent matrix.
Note Bucket:
One will find in [bib2] or [bib3] a presentation deepened on the great deformations.
This document is extracted from [bib4] where one makes a more detailed presentation of the model
elastoplastic, of its numerical integration and where one gives some examples of
validation.
Code_Aster
®
Version
7.4
Titrate:
Modeling élasto (visco) plastic in great deformations
Date:
14/04/05
Author (S):
V. CANO, E. LORENTZ
Key
:
R5.03.21-C
Page
:
4/18
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HT-66/05/002/A
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
X
positive part of
X
~
With
deviatoric part of the tensor
With
defined by
~
(
)
WITH A
In Id
= - 13tr
:
doubly contracted product:
WITH B
AB
:
(
)
,
=
=
WITH B
tr
ij ij
I J
T
tensorial product:
(
)
WITH B
=
ijkl
ij kl
WITH B
With
eq
equivalent value of von Mises defined by
With
eq
= 32 ~: ~
WITH A
X
With
gradient:
=
X
With
With
X
div
X
With
divergence:
(div
X
With)
I
ij
J
J
With
X
=
,
µ
, E,
, K
moduli of the isotropic elasticity
y
elastic limit
coefficent of thermal dilation
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
= -
-
Code_Aster
®
Version
7.4
Titrate:
Modeling élasto (visco) plastic in great deformations
Date:
14/04/05
Author (S):
V. CANO, E. LORENTZ
Key
:
R5.03.21-C
Page
:
5/18
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HT-66/05/002/A
3
Recalls on the great deformations
3.1 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 3.1-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
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
=
=
+
$
The transformations of the element of volume and the density are worth:
O
J
=
D
D
with
J
O
=
=
det F
where
O
and
are respectively the density in the configurations initial and current.
Various tensors of deformations can be obtained by eliminating rotation in
local transformation. For example, by directly calculating the variations length and angle
(variation of the scalar product), one obtains:
E
C Id
=
-
1
2 (
)
with
C F F
=
T
With
Id B
=
-
-
1
2
1
(
)
with
B FF
=
T
Code_Aster
®
Version
7.4
Titrate:
Modeling élasto (visco) plastic in great deformations
Date:
14/04/05
Author (S):
V. CANO, E. LORENTZ
Key
:
R5.03.21-C
Page
:
6/18
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HT-66/05/002/A
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
).
Note:
That is to say a solid undergoing two successive transformations, for example the first
transformation makes pass the solid of the initial configuration
0
with a configuration
1
(tensor gradient
F
1 0
/
and vector displacement
U
1 0
/
), then the second transformation of
configuration
1
with
2
(tensor gradient
F
2 1
/
and vector displacement
U
2 1
/
).
0
1
2
F
1/0
F
2/1
F
2/0
The passage of the configuration
0
with
2
is given by the tensor gradient
F
2 0
/
(displacement
U
U
U
2 0
2 1
1 0
/
/
/
=
+
) such as:
F
F
F
2 0
2 1 1 0
/
/
/
=
One obtains then, for example, for the tensor of deformation of Green-Lagrange
E
E
F E F
E
2 0
1 0 2 1 1 0
1 0
/
/T
/
/
/
=
+
where
E
2 0
/
,
E
1 0
/
and
E
2 1
/
are the deformations of Green-lagrange of the configurations
2
by
report/ratio with
0
associated
F
2 0
/
,
1
compared to
0
associated
F
1 0
/
and
2
by report/ratio
with
1
associated
F
2 1
/
, respectively.
This constitutes one of the difficulties encountered at the time of the writing of a law of behavior in
great deformations because one cannot write any more one formula similar to that written in
small deformations, namely
0
1
1
2
0
2
/
/
/
+
=
where
is the tensor of total deflection
linearized.
To find
0
1
1
2
0
2
/
/
/
+
=
in small deformations starting from the expression of
E
2 0
/
,
it is necessary to neglect all the terms of command 2 of
U
2 0
/
,
U
1 0
/
and
U
2 1
/
. In this case, one has
0
2
0
2
/
/
~
E
-
,
0
1
0
1
/
/
~
E
-
and
1
2
0
1
1
2
0
1
/
/
/
T/
~
F
E
F
-
.
3.2 Stresses
For the model describes here, the tensor of the stresses used is the tensor eulérien of Kirchhoff
defined
by:
=
J
where
is the tensor eulérien of Cauchy. The tensor
thus result from a “scaling” by
variation of volume of the tensor of Cauchy
; this is not the case of other tensors of stresses
used (first and second tensor of Piola-Kirchhoff).
Code_Aster
®
Version
7.4
Titrate:
Modeling élasto (visco) plastic in great deformations
Date:
14/04/05
Author (S):
V. CANO, E. LORENTZ
Key
:
R5.03.21-C
Page
:
7/18
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HT-66/05/002/A
In description eulérienne, the equilibrium equations are given by:
F
D
=
=
+
on
on
0
T
N
F
X
.
div
where
F
is the voluminal force applied to the field
,
N
the normal external with the border
F
and
F
the part of the border of the field
where are applied the surface forces
T
D
.
3.3 Objectivity
When a law of behavior in great deformations is written, one must check that this law is
objectify, i.e. invariant by any change of space reference frame of the form:
X
C
Q X
*
()
()
=
+
T
T
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
).
4
Presentation of the model of behavior
4.1 Aspect
kinematics
This model supposes, just like in small deformations, the existence of a slackened configuration
R
, i.e. locally free of stress, which then makes it possible to break up the total deflection
in parts rubber band and plastic, this decomposition being multiplicative.
Thereafter, one will note by
F
the tensor gradient which makes pass from the initial configuration
0
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 4.1-a: Decomposition of the tensor gradient
F
in an elastic part
F
E
and plastic
F
p
Code_Aster
®
Version
7.4
Titrate:
Modeling élasto (visco) plastic in great deformations
Date:
14/04/05
Author (S):
V. CANO, E. LORENTZ
Key
:
R5.03.21-C
Page
:
8/18
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HT-66/05/002/A
By composition of the movements, one obtains the following multiplicative decomposition:
F F F
=
E p
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
=
The model presented is written in order to distinguish the isochoric terms from the terms of change from
volume. One introduces for that the two following tensors:
F
F
=
-
J
1 3
/
and
B
B
E
E
J
=
- 2 3
/
with
J
= det F
By definition, one a:
det F
= 1
and
det
B
E
= 1
.
4.2
Relations of behavior
The law presented is a model thermoélasto (visco) plastic with isotropic work hardening which tends under
the assumption of the small deformations towards the model [R5.03.02] with criterion of Von Mises (it acts of
plastic model). The plastic deformations are done with constant volume so that:
J
p
p
=
=
det
F
1
from where
J
J
E
E
=
= det F
The relations of behavior are given by:
·
thermoelastic relation stress-strain:
E
B
~
~ µ
=
)
1
) (
(
2
9
)
1
(
2
3
tr
2
J
J
T
T
K
J
K
ref.
+
-
-
-
=
·
threshold of plasticity (it is admitted that it is expressed with the stresses of Kirchhoff):
y
eq
p
-
-
=
)
R (
F
where
R
is the isotropic variable of work hardening, function of the cumulated plastic deformation
p
.
·
laws of flow:
eq
E
eq
T
p
µ
B
B
F
G
F
~
~
tr
3
1
3
~
3
E
+
-
=
-
=
&
&
&
&
&
p
=
For the model of plasticity, the plastic multiplier is obtained by writing the condition of
coherence
0
F
=
&
and one a:
0
F
0
F
,
0
=
p
p
&
&
and
Code_Aster
®
Version
7.4
Titrate:
Modeling élasto (visco) plastic in great deformations
Date:
14/04/05
Author (S):
V. CANO, E. LORENTZ
Key
:
R5.03.21-C
Page
:
9/18
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HT-66/05/002/A
In the viscous case, one takes
&p
equalize with:
m
p
=
0
0
F
HS
&
&
where
0
&
,
0
and
m
are the viscosity coefficients. Let us announce that this law is reduced to a law of the type
Norton when the 2 parameters materials
0
&
and
0
are very large.
It is reminded the meeting that:
B
B
E
E
J
=
- 2 3
/
F
F
=
-
J
1 3
/
and that the partition of the deformations is written:
B
FG F
E
p T
=
For metallic materials where the report/ratio
µ
eq
/
is small in front of 1, the expression of the law
flow can be approached by:
µ
+
-
=
eq
eq
E
T
p
O
~
tr B
F
G
F
&
&
éq.4.2-1
where
O
eq
µ
is negligible in front of the first term.
It is this last expression which is established in Code_Aster.
Note:
If the deformations are small, one a:
p
p
E
E
J
Id
G
Id
B
2
2
1
-
+
+
tr
where
is the total deflection,
E
elastic strain and
p
plastic deformation in
small deformations.
By replacing these three expressions in the equations of the law of behavior presented
here, one finds well the elastoplastic thermo conventional model with isotropic work hardening and
criterion of Von Mises.
Code_Aster
®
Version
7.4
Titrate:
Modeling élasto (visco) plastic in great deformations
Date:
14/04/05
Author (S):
V. CANO, E. LORENTZ
Key
:
R5.03.21-C
Page
:
10/18
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HT-66/05/002/A
4.3
Choice of the function of work hardening
This relation of behavior is available in the operator
STAT_NON_LINE
, under the key word
factor
COMP_INCR
and the argument `
SIMO_MIEHE'
key word factor
DEFORMATION
. One can choose
for the function of work hardening, a linear work hardening or to provide a diagram traction. Five
relations can be used.
RELATION =
/“ELAS”
/
“VMIS_ISOT_TRAC”
[DEFECT]
/
“VMIS_ISOT_LINE”
/
“VISC_ISOT_TRAC”
/
“VISC_ISOT_LINE”
For a purely thermoelastic behavior, the user chooses the argument
“ELAS”
(it
behavior is then hyperelastic); for an isotropic work hardening given by a curve of
traction, the user chooses the argument
“VMIS_ISOT_TRAC”
in the plastic case or
“VISC_ISOT_TRAC”
in the viscous case and for a linear isotropic work hardening, the argument
“VMIS_ISOT_LINE”
in the plastic case or
“VISC_ISOT_LINE”
in the viscous case.
The various characteristics of material are indicated in the operator
DEFI_MATERIAU
([U4.23.01]) under the key words:
·
ELAS
some (one is the law gives the Young modulus, the Poisson's ratio and
possibly the thermal expansion factor),
·
TRACTION
for
“VMIS_ISOT_TRAC”
and
“VISC_ISOT_TRAC”
(one gives the curve of
traction),
·
ECRO_LINE
for
“VMIS_ISOT_LINE”
and
“VISC_ISOT_LINE”
(the limit is given
of elasticity and the slope of work hardening),
·
VISC_SINH
for
“VISC_ISOT_TRAC”
and
“VISC_ISOT_LINE”
(the three are given
viscosity coefficients).
Note:
The user must make sure well that the “experimental” traction diagram used, is
directly, that is to say to deduce the slope from it from work hardening is well given in the plan
rational 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. It will be noticed that
=
=
F S
F
S
L
L J
/
0
0
1
from where
=
=
J
F
S
L
L
0
0
. In general, it is well the quantity
F
S
L
L
0
0
who is
measured by the research workers and this the stress of Kirchhoff gives directly
used in the model of Simo and Miehe.
4.4
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, the indicator of plasticity (0 if the last calculated increment is elastic, 1 if not),
·
V3, the trace divided by three of the tensor of elastic strain
E
B
that is to say
E
B
tr
3
1
.
Code_Aster
®
Version
7.4
Titrate:
Modeling élasto (visco) plastic in great deformations
Date:
14/04/05
Author (S):
V. CANO, E. LORENTZ
Key
:
R5.03.21-C
Page
:
11/18
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HT-66/05/002/A
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.
4.5 Field
of use
The choice of a kinematics
DEFORMATION
: `
PETIT_REAC'
also allows to treat a law of
elastoplastic thermo behavior with isotropic work hardening and criterion of von Mises into large
deformations. The law is written in small deformations and the taking into account of the great deformations
is done by reactualizing the geometry.
Between the law presented here (
SIMO_MIEHE
) and
PETIT_REAC
,
·
there is no difference if the deformations are small
·
there is no difference if the deformations are large but small rotations
·
there are differences if rotations are important.
In particular, the solution obtained with kinematics
PETIT_REAC
can deviate notably from
exact solution in the presence of great rotations and this whatever the size of the pitches of time
chosen by the user, contrary to kinematics
SIMO_MIEHE
.
4.6
Integration of the law of behavior
In the case of an incremental behavior, key word factor
COMP_INCR
, knowing the stress
-
, cumulated plastic deformation
p
-
, the trace divided by three of the tensor of deformations
rubber bands
-
E
B
tr
3
1
, displacements
U
-
and
U
and temperatures
T
-
and
T
, one seeks with
to determine
)
tr
3
1
,
,
(
E
p
B
.
Displacements being known, gradients of the transformation of
0
with
-
, noted
F
-
, and of
-
with
, noted
F
, are known.
The implicit discretization of the law gives:
F
FF
=
-
J
= det F
F
F
=
-
J
1 3
/
B
FG F
E
p T
=
=
J
E
B
~
~ µ
=
)
1
) (
(
2
3
)
1
(
2
1
tr
3
1
2
J
J
T
T
K
J
K
ref.
+
-
-
-
=
y
eq
p
p
-
+
-
=
-
)
R (
F
p
eq
E
T
p
p
-
=
-
-
~
tr
)
(
B
F
G
G
F
from where
p
eq
T
p
E
-
=
-
~
tr
E
B
F
G
F
B
In the plastic case:
0
F
0
F
,
0
=
p
p
and
In the viscous case:
0
1
0
1
0
=
-
-
+
-
-
-
m
y
eq
T
p
p
p
&
HS
)
R (
Code_Aster
®
Version
7.4
Titrate:
Modeling élasto (visco) plastic in great deformations
Date:
14/04/05
Author (S):
V. CANO, E. LORENTZ
Key
:
R5.03.21-C
Page
:
12/18
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HT-66/05/002/A
Note:
This formulation is incrémentalement objective because the only tensorial quantity
incremental which intervenes in the discretization is
&G
p
. Like
G
p
and
G
p
-
are
measured on the same configuration, i.e. initial configuration, discretization of
&G
p
, that is to say
G
G
G
p
p
p
=
-
-
, is incrémentalement objective.
One introduces
Tr
, the tensor of Kirchhoff which results from an elastic prediction (Tr: trial, in English
test):
eTr
Tr
B
~
~
µ
=
where
B
FG F
Bfr
F
eTr
p
T
E
T
=
=
-
-
,
F
F
= (J)
/
- 1 3
and
J
(
)
= det F
One obtains
B
E
-
starting from the stresses
-
by the thermoelastic relation stress-strain and
trace of the tensor of the elastic strain.
-
-
-
-
+
µ
=
E
E
B
B
tr
~
3
1
Note:
The interest of this formulation is that it is not necessary to calculate the deformation
plastic
G
p
-
who would oblige us to reverse the gradient of the transformation
F
. One needs
to only know
FG F
p
T
-
.
If
y
Tr
eq
p
+
<
-
)
R (
, one remains elastic. In this case, one a:
p
p
=
-
,
Id
=
Tr
R
T
~
3
1
+
and
eTr
E
B
B
tr
tr
3
1
3
1
=
if not, one obtains:
eTr
E
B
B
tr
tr
=
This last relation is not possible that if one makes simplification on the law of flow.
)
p
(
eq
eTr
Tr
µ
+
=
B
tr
~
~
1
While calculating the equivalent stress, one brings back oneself to a nonlinear scalar equation in
p
:
0
=
µ
-
-
p
eTr
eq
Tr
eq
B
tr
In the plastic case:
)
R (
p
p
y
eq
+
+
=
-
, which leads to
p
solution of the equation:
0
=
µ
-
+
-
-
-
p
p
p
eTr
y
Tr
eq
B
tr
)
R (
Code_Aster
®
Version
7.4
Titrate:
Modeling élasto (visco) plastic in great deformations
Date:
14/04/05
Author (S):
V. CANO, E. LORENTZ
Key
:
R5.03.21-C
Page
:
13/18
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HT-66/05/002/A
In the viscoplastic case:
+
+
+
=
-
-
m
y
eq
T
p
p
p
1
0
1
0
&
HS
)
R (
, which leads to
p
solution of the equation:
0
1
0
1
0
=
µ
-
-
+
-
-
-
-
p
T
p
p
p
eTr
m
y
Tr
eq
B
tr
HS
)
R (
&
The solution of this last equation is made in Code_Aster by a method of the secants
with interval of search (cf [R5.03.05]). Integration can be controlled by the parameters
RESI_INTE_RELA
and
ITER_INTE_MAXI
under
STAT_NON_LINE
key word
CONVERGENCE
.
Once
p
known, one can then deduce the tensor from it from Kirchhoff, that is to say:
Id
+
-
-
-
+
=
)
) (
(
)
(
~
J
J
T
T
K
J
K
ref.
Tr
Tr
eq
eq
1
2
3
1
2
2
Once calculated cumulated plastic deformation, the tensor of the stresses and the tangent matrix,
one carries out a correction on the trace of the tensor of the elastic strain
E
B
to hold account
plastic incompressibility, which is not preserved with the simplification made on the law
of flow [éq 4.2.1]. This correction is carried out by using a relation between the invariants of
E
B
and
E
B
~
and by exploiting the plastic condition of incompressibility
1
=
p
J
(or in an equivalent way
1
det
=
E
B
). This relation is written:
X
J X (1 J) 0
E
E
3
2
3
-
- -
=
with
2
2
2
2
2
2
1
µ
=
=
eq
E
eq
E
B
J
,
µ
=
=
B
~
det
~
det
E
E
J
3
and
E
X
B
tr
3
1
=
The solution of this cubic equation makes it possible to obtain
E
B
tr
and consequently
thermoelastic deformation
B
E
-
with the pitch of next time. If this equation admits
several solutions, one takes the solution nearest to the solution of the pitch of previous time. It is
moreover why one stores in an internal variable
E
B
tr
3
1
.
Code_Aster
®
Version
7.4
Titrate:
Modeling élasto (visco) plastic in great deformations
Date:
14/04/05
Author (S):
V. CANO, E. LORENTZ
Key
:
R5.03.21-C
Page
:
14/18
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HT-66/05/002/A
5 Formulation
variational
Insofar as the stresses provided by the law of behavior are eulériennes, one is chosen
variational formulation written on the current configuration (eulérienne) and not on the configuration
initial, that is to say:
=
+
X
F
v
F
v
v F v
T v
D
D
dS
int
F
ext.
.
D
.
1 2
44
3
44
1
2
4444
3
4444
v
Kinematically acceptable
We are interested only here in work of the interior forces and its variation in optics
of a resolution by the method of Newton. One will find in [bib 4] the demonstration of the expressions
presented.
5.1
Case of the continuous medium
One rewrites here the work of the interior efforts in indicielle form, that is to say:
F
v
int
ij
I
J
.
v
X D
=
We need also to express the variation of the interior efforts in the configuration
current
that is to say:
F
U v
int
.
=
-
-
ij
p
p
ik
J
K
I
J
ij
pq
p
Q
I
J
U
X
U
X
v
X D
F
U
X
v
X D
+
geometrical rigidity
rigidity of behavior
where
X
-
are the punctual coordinates on the configuration
-
.
5.2
Discretization by finite elements
Displacements are discretized
U
and virtual displacements
v
by finite elements. The notations are
the following ones, by adopting the convention of summation of the repeated indices:
()
()
U X
NR
X U
U
X
D X U
U
X
D
X U
I
N
in
I
J
jn
in
I
J
J N
in
()
()
=
=
=
-
-
Code_Aster
®
Version
7.4
Titrate:
Modeling élasto (visco) plastic in great deformations
Date:
14/04/05
Author (S):
V. CANO, E. LORENTZ
Key
:
R5.03.21-C
Page
:
15/18
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HT-66/05/002/A
where:
NR X
N
()
is related to form associated with the node
N
U
in
, the component
I
nodal displacement of the node
N
D X
jn
()
, components of the gradient of the functions of form on the configuration
D
X
J N
-
()
, components of the gradient of the functions of form on the configuration
-
One obtains for the vector of the interior forces:
(F
D D
I
N
ij
jn
int
)
=
and for the tangent matrix, which is not a priori symmetrical:
[
]
K
D
D
D
D D
D
F
D D
I p
N m
p
m ij jn
km ik
p
N
Q N
ij
pq
jn
+
=
-
-
In the case of a two-dimensional modeling (deformation planes), expressions of the vector of
interior forces and of the tangent matrix are identical to this ready that the corresponding indices
with the components only vary from 1 to 2.
In the case of an axisymmetric modeling, by numbering the axes in the order (
R
,
Z
,
), the vector
interior forces is written:
(
+
33
F
D
NR
R
D
axi N
N
N
int
)
=
1
,
{}
1 2
,
,
{}
1 2
,
and the tangent matrix:
[]
[]
[
]
K
K
K
axi
corr
=
+
with:
[]
K
(1)
corr
1
33
N m
N
m
N
m
NR
R
D D
NR
R
F D D
=
+
-
[]
K
(2)
corr
1
33
33
N m
N
m
N
m
D
NR
R D
D
F
NR
R D
=
+
-
[]
K
(3)
corr
11
33
33
N m
N
m
NR
R
F
NR
R
=
-
From an algorithmic point of view, we symmetrized the tangent elementary matrix
K
who is not it
not a priori.
Code_Aster
®
Version
7.4
Titrate:
Modeling élasto (visco) plastic in great deformations
Date:
14/04/05
Author (S):
V. CANO, E. LORENTZ
Key
:
R5.03.21-C
Page
:
16/18
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HT-66/05/002/A
5.3
Form of the tangent matrix of the behavior
One gives the form of the tangent matrix here (option FULL_MECA during iterations of
Newton, option RIGI_MECA_TANG for the first iteration). The aforementioned is obtained by linearizing it
system of equations which governs the law of behavior. We give here the final result of this
linearization. One will find in [bib4] the detail of this calculation.
One poses:
J
= det F
,
J
-
-
= det F
and
J
= det F
·
For option FULL_MECA, one a:
B
Id
B
B
F
H
H
F
With
-
-
-
-
=
=
-
-
)
) (
(
(
)
(
/
2
-
2
-
3
1
1
2
3
+
-
)
3
1
J
T
T
K
KJ
J
J
J
J
J
J
J
J
ref.
where
B
is worth:
B
F
F
F
F
B
F
F
F
F
B
F
F
F
F
B
F
F
F
F
B
F
F
F
F
B
F
F
F
F
B
F
F
F
F
B
F
F
F
F
B
F
F
F
F
11
22
33
23
32
22
11
33
13
31
33
11
22
12
21
12
31
23
33
21
21
13
32
33
12
13
21
32
22
31
31
12
23
22
13
23
31
12
11
32
32
13
21
11
23
=
-
=
-
=
-
=
-
=
-
=
-
=
-
=
-
=
-
H
and
H F
are given by:
In the elastic case (F < 0):
H
(
B
F
F B
)
F B
ijkl
ik lpe
jp
IP pl
E
jk
ij
kp lpe
=
+
-
-
-
-
µ
µ
2
3
and
H F
B
= 2µ~
eTr
Code_Aster
®
Version
7.4
Titrate:
Modeling élasto (visco) plastic in great deformations
Date:
14/04/05
Author (S):
V. CANO, E. LORENTZ
Key
:
R5.03.21-C
Page
:
17/18
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HT-66/05/002/A
In the plastic case (
F
= 0
) or viscoplastic:
-
-
-
-
+
-
+
+
+
-
+
=
E
LP
qp
kq
ij
eTr
3
eq
eq
eTr
2
E
LP
kp
eTr
eq
ij
ij
jk
E
pl
IP
jp
E
LP
ik
ijkl
B
F
~
~
)
tr
R
(
has
)
p
R
(
tr
3
B
F
)
tr
R
(
~
p
R
has
3
2
)
B
F
F
B
(
has
H
µ
µ
µ
µ
µ
B
B
B
and
B
B
B
B
Id
B
B
F
H
~
)
:
~
(
)
tr
R
(
)
R
(
tr
)
tr
R
(
~
R
tr
eq
eTr
eTr
eq
eq
eTr
eTr
eTr
eTr
has
p
p
has
has
µ
+
-
µ
+
µ
+
+
µ
-
µ
3
2
3
3
2
2
=
where
has
eq
Tr
eq
=
and
(
)
()
4
4
4
4
4
4
4
4
4
3
4
4
4
4
4
4
4
4
4
2
1
&
&
viscous
case
only
1
1
0
2
1
2
0
0
1
1
-
-
×
×
+
×
+
=
m
m
1
p
T
m
T
p
p
)
(
R'
R
,
)
(
R p
being the derivative of
isotropic work hardening compared to the cumulated plastic deformation
p
.
·
For option RIGI_MECA_TANG, they are the same expressions as those given for
FULL_MECA but with
p = 0
and with all the variables and coefficients of material taken with
the moment
T
-
(in theory, it would be necessary in the viscous case, to take the expressions of FULL_MECA
in the elastic case, all the variables being taken at the moment
T
-
). In particular, one will have
F Id
=
.
Code_Aster
®
Version
7.4
Titrate:
Modeling élasto (visco) plastic in great deformations
Date:
14/04/05
Author (S):
V. CANO, E. LORENTZ
Key
:
R5.03.21-C
Page
:
18/18
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HT-66/05/002/A
6 Bibliography
[1]
SIMO J.C., MIEHE C.: “Associative coupled thermoplasticity At finite strains: Formulation,
numerical analysis and implementation ", comp. Meth. Appl. Mech. Eng., 98, pp. 41-104,
North Holland, 1992.
[2]
SIDOROFF F.: “Elastoplastic Formulations in great deformations”, Greco Report/ratio
n29, 1981.
[3]
SIDOROFF F.: “Course on the great deformations”, Report/ratio Greco n51, 1982.
[4]
CANO V., LORENTZ E.: “Introduction into Code_Aster of a model of behavior in
great elastoplastic deformations with isotropic work hardening ", internal Note EDF DER,
HI-74/98/006/0, 1998