Code_Aster
®
Version
7.4
Titrate:
Viscoplastic behaviors élasto mono crystalline lenses
Date:
19/04/05
Author (S):
J.M. PROIX, T. KANIT, O. DIARD
Key
:
R5.03.11-A
Page
:
1/24
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HT-66/05/002/A
Organization (S):
EDF-R & D/AMA, MMC
Manual of Reference
R5.03 booklet: Nonlinear mechanics
R5.03.11 document
Mono behaviors elastoviscoplastic and
polycrystalline
Summary:
The goal of this document is to describe the integration of the mono and polycrystalline behaviors, while specifying
independent way the criterion, the flow, work hardening etc
One treats here integration of these laws of behavior associated with systems with slip
corresponding to the usual crystal families. This integration can be made (method explicitly
of Runge_Kutta with control of the precision and local recutting of the pitch of time) or an implicit way
(method of Newton with local recutting of the pitch of time).
These behaviors can be employed for the calculation of microstructures (mesh of an aggregate, with
geometrical representation of each physical grain) or for calculation of polycrystals, mediums
“homogenized” having in any material point (or not of integration or calculation) several phases
simultaneous, in variable proportions.
Code_Aster
®
Version
7.4
Titrate:
Viscoplastic behaviors élasto mono crystalline lenses
Date:
19/04/05
Author (S):
J.M. PROIX, T. KANIT, O. DIARD
Key
:
R5.03.11-A
Page
:
2/24
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HT-66/05/002/A
Count
matters
Code_Aster
®
Version
7.4
Titrate:
Viscoplastic behaviors élasto mono crystalline lenses
Date:
19/04/05
Author (S):
J.M. PROIX, T. KANIT, O. DIARD
Key
:
R5.03.11-A
Page
:
3/24
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HT-66/05/002/A
1 Introduction
The general objective of the development of the “microcomputer-macro” functionalities in Code_Aster is of
to be able to integrate in a modular way of the models into several scales (with a possibility of choice
laws of behaviors, rules of localization, types of microstructures, link between
not integration in the element and the “module law of behavior”). What can carry out to
types of calculation different (polycrystalline calculations, use of a law of the Berveiller-Zaoui type or one
standard law “regulates in
”, calculations of aggregates multi-crystalline lenses with mesh of a microstructure,…).
The step presented here consists in allowing decoupling, by modularity, of the various elements
who constitute a law of behavior. This flexibility is accessible directly to the user. Of
more, for the developer, it is possible to add a law of behavior (macroscopic or
microscopic) by simply defining the derivative partial of the problem, in terms of calculation of
stresses and of internal variables. This is sufficient if one is satisfied with an explicit integration;
for an implicit integration, it is necessary to define in more the tangent operator.
More precisely, for the aspect behavior of monocrystal, the modularity is total on the level of
calculation “not material”: the material, represented by some homogeneous equations in the case of them
macroscopic phenomenologic models, is now more complex: for a finite element
given, it consists of a monocrystal having an orientation given, and having some
a number of systems of slip. Each family of systems of slip has her own law
of behavior local.
In the case of a polycrystalline model, one supposes that in a material point (not of integration of one
finite element), several metallurgical phases are present simultaneously, each phase being able
to consist of grains with orientations given, each grain having a certain number
systems of slip (not inevitably the same ones for each phase). The representation of
material can also include the shape of the grains and the type of phases involved, inducing such or
such type of rule of transition from scale. Each family of systems of slip has her characteristic
local law of behavior. One finds a separation between the crystallographic structure, the law of
crystal viscoplasticity and rules of transition from scales. This mode of separation is also wide
to the law of viscoplasticity itself, with a separation enters elasticity, the criterion and the law
of flow. The representation of material can also include the shape of the grains and the type of
involved, inducing such or such type phases of rule of transition from scale.
2
Formulation of the mono and polycrystalline behaviors
2.1
Relations of behavior of the monocrystal
The behavior related to each system of slip of a monocrystal is (in the whole of
behaviors considered) of élasto-visco-plastic type. Owing to the fact that one is interested each time in
only one direction of slip, the behavior is mono dimensional. It can break up into
three types of equations:
·
Relation of flow:
)
,
,
,
(
S
S
S
S
S
p
G
=
&
, with,
S
S
p
&
&
=
and
for an elastoplastic behavior a criterion of the type:
0
)
,
,
,
(
S
S
S
S
p
F
and
0
.
=
S
p
F &
for a élasto-viscoplastic behavior,
(
)
S
S
S
S
S
p
F
p
,
,
,
=
&
·
Evolution of kinematic work hardening:
)
,
,
,
(
S
S
S
S
S
p
H
=
&
·
Evolution of the isotropic work hardening defined by a function:
)
(
S
p
R
Code_Aster
®
Version
7.4
Titrate:
Viscoplastic behaviors élasto mono crystalline lenses
Date:
19/04/05
Author (S):
J.M. PROIX, T. KANIT, O. DIARD
Key
:
R5.03.11-A
Page
:
4/24
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HT-66/05/002/A
These relations become, after discretization in time:
·
Relation of flow:
)
,
,
,
(
S
S
S
S
S
p
G
=
, with,
S
S
p
=
for an elastoplastic behavior a criterion
type
:
0
)
,
,
,
(
S
S
S
S
p
F
and
0
.
=
S
p
F
and for a élasto-viscoplastic behavior,
(
)
T
p
F
p
S
S
S
S
S
=
,
,
,
·
Evolution of kinematic work hardening:
)
,
,
,
(
S
S
S
S
S
p
H
=
·
Evolution of isotropic work hardening:
)
(
S
p
R
Quantities
)
,
,
,
(
S
S
S
S
p
are evaluated at the moment running for an implicit discretization and with
the previous moment for an explicit discretization.
To fix the ideas, here examples of relations of viscoplastic or elastoplastic flow, and
of work hardening. The names of these relations correspond to their name in the control
DEFI_MATERIAU
[U4.43.01].
2.1.1 Examples of relations of flow
ECOU_VISC1
N
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
K
p
R
C
T
p
C
C
p
p
G
)
(
.
)
,
,
,
(
-
-
=
-
-
=
=
The parameters are:
N
K
C,
,
.
ECOU_VISC2
N
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
K
C
C
D
p
R
has
C
p
has
C
has
C
p
p
G
2
)
(
2
)
(
)
,
,
,
(
+
-
-
-
=
-
-
-
-
=
=
The parameters are:
.
,
,
,
,
D
has
N
K
C
ECOU_VISC3
(
)
S
S
S
S
S
p
G
,
,
,
=
=
S
S
S
kT
V
kT
G
µ
.
exp
exp
*
0
*
0
-
-
&
,
The parameters are: K,
*
0
,
,
,
0
V
G
µ
&
Code_Aster
®
Version
7.4
Titrate:
Viscoplastic behaviors élasto mono crystalline lenses
Date:
19/04/05
Author (S):
J.M. PROIX, T. KANIT, O. DIARD
Key
:
R5.03.11-A
Page
:
5/24
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HT-66/05/002/A
ECOU_PLAS1
0
)
(
)
,
,
,
(
-
-
=
S
S
S
S
S
S
S
S
p
R
C
p
F
,
0
.
=
S
p
F
,
S
S
S
F
p
=
The associated parameter is:
C
.
2.1.2 Examples of relations of kinematic work hardening
ECRO_CINE1
S
S
S
S
S
S
S
S
p
D
p
H
-
=
=
)
,
,
,
(
The parameter is:
D
.
ECRO_CINE2
S
S
m
S
S
S
S
S
S
S
S
S
M
C
p
D
p
H
)
(
)
,
,
,
(
-
-
=
=
Parameters being then:
C
m
M
D
,
,
,
.
2.1.3 Examples of relations of isotropic work hardening
One taking the very simple shape of the matrix
H
translating the interaction enters the systems of
slip credits, isotropic work hardening can be form:
ECRO_ISOT1
=
-
-
+
=
NR
R
LP
Sr
S
S
R
E
H
Q
R
p
R
1
0
))
1
(
(
)
(
with:
Sr
Sr
Sr
H
+
-
=
)
1
(
The parameters are:
H
B
Q
R
,
,
,
0
.
ECRO_ISOT2
S
sg
S
rs
S
Q
Q
Q
H
Q
R
p
R
2
2
1
1
0
)
(
)
(
+
+
=
with:
dp
Q
B
dq
is
I
is
)
1
(
-
=
The parameters are:
2
1
1
0
,
,
,
,
Q
H
B
Q
R
.
2.2
Systems of slip and total behavior of the monocrystal
A monocrystal is composed of one or more families of systems of slip, (cubic,
octahedral, basal, prismatic,…), each family including/understanding a certain number of systems
(12 for the octahedral family for example).
To each family of system of slip are associated a law with flow, a type of work hardening
kinematics and isotropic, and of the values of the parameters for these laws. In other words, one does not envisage
not to vary the relations of behavior or the coefficients within the same family of
systems of slip. On the other hand, from one family to another, the laws of behavior can change,
as well as the values of the parameters.
Code_Aster
®
Version
7.4
Titrate:
Viscoplastic behaviors élasto mono crystalline lenses
Date:
19/04/05
Author (S):
J.M. PROIX, T. KANIT, O. DIARD
Key
:
R5.03.11-A
Page
:
6/24
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HT-66/05/002/A
A system of slip is determined by the tensor of orientation
S
m
, built from
crystallographic definitions of:
·
direction of slip (defined by the unit vector
L
)
·
and of the normal in the plan which slips (definite by the unit vector
N
).
)
(
2
1
I
J
J
I
S
ij
N
L
L
N
m
+
=
From the point of view of the behavior at the material point, this tensor intervenes for the calculation of the scission
reduced
S
ij
ij
m
:
=
S
and that the speed of total viscoplastic deformation
vp
E
, defined to leave
knowledge speeds of slip
S
&
for all the systems of
slip:
=
G
S
S
vp
ij
E
&
&
S
ij
m
Moreover, the monocrystal can be directed compared to the total axes of definition of the co-ordinates.
This orientation is defined for each mesh or groups meshs (typically for each grain)
by the data of 3 nautical angles. Components of the tensor of orientation
S
m
, defined in
reference mark related to the monocrystal, are then expressed in the total reference mark by using these nautical angles.
2.3
Behavior of the homogenized polycrystal
In the case of homogenized polycrystal, it is necessary to define each single-crystal phase by sound
orientation, its proportion (voluminal fraction) and the associated behavior. It is necessary moreover define one
regulate localization.
The single-crystal behavior is built like previously starting from the behavior
preceding elasto-visco-plastic and of the data of families of systems of slip.
2.3.1 Recall of existing system
Code_Aster has, since version 4, one only polycrystalline law of behavior
(
POLY_CFC
), specific to steels c.f. C, (thus having 12 systems obligatorily of
slip), and limited to 40 grains (40 definite phases each one by a voluminal fraction and one
orientation). The law of behavior is fixed (élasto-visco-plasticity, with kinematic work hardening
nonlinear), and the 2 methods of localization and homogenization are that of Berveiller-Zaoui, and
that of Pilvin-Cailletaud. The introduction of the orientations of the phases, the voluminal fractions and of
orientations of the systems of slip is done using the operator
DEFI_TEXTURE
. This operator
create a table, which is provided to
DEFI_MATERIAU
, in complements of the parameters of the law of
behavior [R5.03.13]. This is validated in test SSNV125.
2.3.2
Behavior of the POLYCRYSTAL type
In addition to the single-crystal behavior describes previously, one adds a scale of modeling,
who represents that of the phases.
On the level of a point of Gauss, there are always the relations of elasticity on
total tensors (homogeneous):
·
macroscopic total deflection
E
·
macroscopic viscoplastic deformation
VP
E
·
macroscopic stress:
(
)
vp
HT
E
E
E
D
-
-
=
Code_Aster
®
Version
7.4
Titrate:
Viscoplastic behaviors élasto mono crystalline lenses
Date:
19/04/05
Author (S):
J.M. PROIX, T. KANIT, O. DIARD
Key
:
R5.03.11-A
Page
:
7/24
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HT-66/05/002/A
·
Moreover, knowing the whole of the internal variables relating to the systems of slip
of each phase, parameters of behavior of each phase, orientations and
voluminal fractions of each phase, and the type of method of localization,
for each single-crystal phase (or “grain”), defined by an orientation and one
proportion F
G
, a relation of localization of the stresses, general form (to be expressed
in the local reference mark of each phase)
(
)
G
vp
G
vp
G
E
L
,
,
,
=
and for each system of slip of each phase, of the relations of behavior
relating to each system of slip, similar to the case of the monocrystal:
Relation of flow:
)
,
,
,
(
S
S
S
S
S
p
G
=
&
, with,
S
S
p
&
&
=
and
(
)
S
S
S
S
S
p
F
p
,
,
,
=
&
Evolution of kinematic work hardening:
)
,
,
,
(
S
S
S
S
S
p
H
=
&
Evolution of the isotropic work hardening defined by a function:
)
(
S
p
R
Viscoplastic Déformations of the phase:
=
G
S
S
vp
ij
G
&
&
S
ij
m
There remain the equations of homogenization then:
vp
G
G
G
vp
F
E
&
&
=
2.3.2.1 Relation of scaling
Two relations of localization of the type
(
)
G
vp
G
vp
G
E
L
,
,
,
=
are available in the version
current:
·
The relation of Berveiller-Zaoui [bib5] established on the concept of autocoherence. This relation is
validated under certain conditions, namely: isotropy of material, elastic behavior
homogeneous and monotonous loading:
()
ijg
ij
ijvp
ij
vp
ijvp
ij
G
E
J
=
+
-
= +
µ
µ
E
1
1 32
2
·
The second relation, developed more particularly for cyclic loadings
[bib4] allows to give a good description to schematize the interactions between
grains:
µ
ijg
ij
ij
ijg
ij
G ijg
G
F
=
+
-
=
(
)
B
B
||
||
)
-
(
-
=
G
G
G
vp
ij
vp
ij
G
ij
vp
ij
G
ij
D
&
&
&
where
D
and
are parameters characteristic of material and temperature.
Code_Aster
®
Version
7.4
Titrate:
Viscoplastic behaviors élasto mono crystalline lenses
Date:
19/04/05
Author (S):
J.M. PROIX, T. KANIT, O. DIARD
Key
:
R5.03.11-A
Page
:
8/24
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HT-66/05/002/A
3
Local integration and implementation numerical
3.1
System of equations to be solved
3.1.1
Behavior of the MONOCRYSTAL type
The local behavior of the monocrystal is defined, at one moment given of the discretization in time and to
level of a point of integration of a finite element, by the data:
·
tensor of macroscopic stresses at the previous moment
-
-
=
)
(
1
I
T
,
·
variables intern at the previous moment, for each system of slip
:
() () ()
1
1
1
,
,
-
-
-
I
S
I
S
I
S
T
p
T
T
,
·
and of the tensor of increase in total deflection provided by iteration N in algorithm
total of resolution
N
I
E
E
=
(with the notations of [R5.03.01]).
Integration consists in finding:
·
the macroscopic tensor of stresses
)
(
I
T
=
·
and internal variables
)
(
I
S
S
T
=
,
)
(
I
S
S
T
=
,
)
(
I
S
S
T
p
p
=
checking the equations of behavior in each system of slip (which are relations
mono dimensional), and relations of passage between the tensors macroscopic and the unit
directions of slip. Notation: one writes the equations in the form discretized of way:
·
clarify, if the noted quantities
-
+
/
With
are evaluated at the moment
1
-
I
T
:
()
1
/
-
-
-
+
=
=
I
T
With
With
With
·
implicit, if they are evaluated at the moment
I
T
:
()
I
T
With
With
With
=
=
+
-
+
/
The equations to be integrated can be put in the following general form:
Being given, in a point of Gauss, the tensors:
E
: variation of deformation at the moment
I
T
,
-
-
=
E
T
E
I
)
(
1
: deformation at the moment
1
-
I
T
,
-
-
=
)
(
1
I
T
: macroscopic stress at the moment
1
-
I
T
,
() () ()
1
1
1
,
,
-
-
-
I
S
I
S
I
S
T
p
T
T
: variables intern for each system of slip with
1
-
I
T
,
It is necessary to find:
)
(
I
T
=
: macroscopic stress at the moment
I
T
, in the reference mark corresponding to the orientation
total
)
(
I
S
S
T
=
)
(
I
S
S
T
=
)
(
I
S
S
T
p
p
=
Code_Aster
®
Version
7.4
Titrate:
Viscoplastic behaviors élasto mono crystalline lenses
Date:
19/04/05
Author (S):
J.M. PROIX, T. KANIT, O. DIARD
Key
:
R5.03.11-A
Page
:
9/24
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HT-66/05/002/A
checking:
)
(
)
(
1
1
vp
HT
E
E
E
D
D
-
-
+
=
-
-
-
-
, where
D
can depend on the temperature, and can correspond to one
orthotropic elasticity.
=
S
S
S
vp
m
E
for each system of slip (of the whole of the families of systems):
S
N
equations:
S
S
m
:
/
-
+
=
S
N
relations of flow: maybe in viscoplasticity
)
,
,
,
(
/
/
/
/
-
+
-
+
-
+
-
+
=
S
S
S
S
S
p
G
with
)
,
,
,
(
/
/
/
/
-
+
-
+
-
+
-
+
=
S
S
S
S
S
p
F
p
maybe in plasticity
0
)
,
,
,
(
/
/
/
/
-
+
-
+
-
+
-
+
S
S
S
S
p
F
,
0
=
S
p
F
,
with
S
S
p
=
S
N
equations of evolutions of kinematic work hardening:
)
,
,
,
(
/
/
/
/
-
+
-
+
-
+
-
+
=
S
S
S
S
S
p
H
S
N
equations of evolution of isotropic work hardening:
)
(
/
-
+
S
S
p
R
This is solved either explicitly (Runge_Kutta), or implicit (Newton).
3.1.2
Behavior of the POLYCRYSTAL type
The discretized relations of behavior are:
Being given (in a point of Gauss) total tensors:
·
increase in total deflection
E
,
·
total deflection at the previous moment
()
-
-
=
E
T
E
I
1
,
·
stress at the previous moment:
()
,
1
-
-
=
I
T
·
the whole of the internal variables
-
-
-
S
S
S
p
,
,
relating to the systems of slip of
each phase,
·
parameters of behavior of each phase, orientations and fractions voluminal of
each phase, and the type of method of localization.
It is necessary to find
()
()
()
()
I
S
S
I
S
S
I
S
S
I
T
p
p
T
T
T
=
=
=
=
,
,
,
checking:
·
on the level of the point of Gauss:
()
(
)
vp
HT
E
E
E
D
D
D
-
-
+
=
-
-
1
, in the total reference mark,
for each phase (or “grain”), defined by an orientation and a proportion F
G
, one
relation of localization of the stresses, the general form (to be expressed in the reference mark
room of each phase)
(
)
G
vp
G
vp
G
E
L
,
,
,
=
and for each system of slip of each phase:
S
S
S
vp
G
m
=
N
S
equations:
S
S
m
:
=
N
S
relations of flow:
(
)
S
S
S
S
S
p
G
,
,
,
=
, with
S
S
p
=
N
S
evolutions of work hardening:
(
)
S
S
S
S
S
p
H
,
,
,
=
(
)
0
.
,
0
,
,
,
=
S
S
S
S
S
p
F
p
F
, (in plasticity independent of time)
·
There remain the equations of homogenization then:
vp
G
G
G
vp
F
E
=
Code_Aster
®
Version
7.4
Titrate:
Viscoplastic behaviors élasto mono crystalline lenses
Date:
19/04/05
Author (S):
J.M. PROIX, T. KANIT, O. DIARD
Key
:
R5.03.11-A
Page
:
10/24
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HT-66/05/002/A
The viscoplastic behaviors relating to each system of slip are identical to the case of
the microstructure.
In the current version of Code_Aster, these relations of behavior are only integrated
explicit way.
3.2 Resolution
implicit
It is thus necessary to solve a system of the following general form:
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
vp
vp
HT
S
S
S
vp
S
S
S
vp
S
S
S
vp
S
S
S
S
vp
S
S
S
vp
S
S
S
vp
m
p
F
p
p
G
p
H
N
m
E
E
E
E
D
D
p
E
p
p
E
G
p
E
has
N
p
E
E
p
E
S
p
E
R
Y
R
:
0
)
,
,
,
(
)
,
,
,
(
)
,
,
,
(
)
(
)
(
)
,
,
,
,
(
)
,
,
,
,
(
)
,
,
,
,
(
)
,
,
,
,
(
)
,
,
,
,
(
)
,
,
,
,
(
)
(
1
1
+
+
+
+
+
+
+
+
+
+
+
+
+
+
-
-
-
-
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
=
=
-
-
-
-
-
-
-
-
=
=
=
In more contracted way, one poses:
()
()
()
()
()
()
=
=
=
S
S
S
vp
p
E
y
y
p
y
G
y
has
y
E
y
S
y
R
with
0
To solve this system of 6+6+3ns nonlinear equations (in 3D), one uses a method of
Newton: one builds a vector series in the following way solution:
)
(
)
(
1
1
K
K
K
K
Y
R
Dy
Dr.
Y
Y
-
+
-
=
Thus should be defined the initial values
0
Y
, and to calculate the matrix jacobienne system:
K
Dy
Dr.
(the aforementioned
is detailed in appendix for the viscoplastic behaviors described previously). It with
following form:
Code_Aster
®
Version
7.4
Titrate:
Viscoplastic behaviors élasto mono crystalline lenses
Date:
19/04/05
Author (S):
J.M. PROIX, T. KANIT, O. DIARD
Key
:
R5.03.11-A
Page
:
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Manual of Reference
R5.03 booklet: Nonlinear mechanics
HT-66/05/002/A
=
S
S
S
vp
S
S
S
vp
S
S
S
vp
S
S
S
vp
S
S
S
vp
p
p
p
p
E
p
p
p
G
G
G
E
G
G
p
has
has
has
E
has
has
p
E
E
E
E
E
E
p
S
S
S
E
S
S
J
The criterion of stop of the iterations relates to the nullity of the residue:
<
)
(
)
(
0
Y
R
Y
R
K
. If convergence is not
reached after the maximum number of iterations, the stationnarity of the solution is also tested:
<
-
+
K
K
Y
Y
1
The method used allows a local recutting of the pitch of time, either systematic, or in the event of
not convergence.
3.2.1 Operator of tangent behavior
The formed system of the equations of the model written in discretized form
()
(
)
0
=
Y
R
is checked in end
of increment. For a small variation of
R
, by considering this time
like variable and not
like parameter, the system remains with balance and one checks
dF
L
=
0
, i.e.:
0
=
+
+
+
+
+
S
S
S
S
S
S
vp
vp
p
p
R
R
R
E
E
R
E
E
R
R
This system can be still written:
()
=
=
=
0
0
0
0
,
and
with
E
X
p
E
Y
X
Y
Y
R
S
S
S
vp
By successive substitution and elimination (cf [§Annexe2]), one deduces from it that the matrix jacobienne
calculated for implicit integration allows to calculate the tangent operator without intervention
additional in the code.
The aforementioned is written directly (see [§Annexe2]):
(
)
1
2
1
3
1
0
-
-
+
+
-
=
=
Y
Y
Y
Y
E
E
T
T
T
T
Code_Aster
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Version
7.4
Titrate:
Viscoplastic behaviors élasto mono crystalline lenses
Date:
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J.M. PROIX, T. KANIT, O. DIARD
Key
:
R5.03.11-A
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R5.03 booklet: Nonlinear mechanics
HT-66/05/002/A
By writing the matrix jacobienne in the form:
[] []
[] []
[]
=
Z
E
Y
Y
Y
Y
Y
J
vp
3
2
1
0
0
1
0
0
1
.
With:
S
S
S
S
N
p
Z
D
Y
×
=
=
-
1
0
The submatrices have as dimensions:
(
)
[]
[
]
[
]
[
]
S
S
S
S
N
N
Y
N
Y
N
Y
D
Y
*
3
,
*
3
dim
6
,
*
3
dim
*
3
,
6
dim
6
,
6
dim
1
2
1
1
0
=
=
=
=
=
-
3.3 Resolution
explicit
Another method of resolution, very simple to implement to solve the equations
differentials of the single-crystal behavior is the explicit resolution. So that it is effective
numerically, it is essential to associate an automatic control of pitch to him. As in
[R5.03.14], one uses the method of Runge and Kutta. The calculation of the variables intern at the moment
T + H
is a function only values of their derivatives
()
Dy
dt
F Y T
=
,
:
)
(
:
)
,
,
,
(
)
,
,
,
(
)
,
,
,
(
vp
HT
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
vp
S
S
S
E
E
E
D
m
with
m
p
F
p
G
p
H
E
p
Y
-
-
+
=
=
=
=
=
-
-
-
-
-
-
-
-
-
-
-
-
-
One integrates according to the following diagram:
Y
T
+
H
=
Y
(2)
if the criterion of precision is satisfied
Y
(2)
=
Y + H2 [F (Y, T) + F (Y
(1)
, T + H)]
with
Y
(1)
=
Y + H F (Y, T)
Code_Aster
®
Version
7.4
Titrate:
Viscoplastic behaviors élasto mono crystalline lenses
Date:
19/04/05
Author (S):
J.M. PROIX, T. KANIT, O. DIARD
Key
:
R5.03.11-A
Page
:
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R5.03 booklet: Nonlinear mechanics
HT-66/05/002/A
The difference enters
()
Y
2
(diagram of command 2) and
()
Y
1
(diagram of command 1, Euler) provides an estimate
error of integration and makes it possible to control the size of the pitch of time
H
who is initialized with
T
I
for
the first attempt. The strategy of the control of the pitch is defined on the standard basis of the variation enters
two methods of integration:
|| Y
(2)
-
Y
(1)
||
and of the precision required by the user
(key word:
RESI_INTE_RELA
). The criterion selected is as follows, where one notes
Y
=
(y
1
, y
2
,…, y
NR
)
:
Y (T)
=
sup
J
=
1, NR
| y
J
(2)
-
y
J
(1)
|
max [
, | y
J
(T)|]
<
The parameter
is fixed at 0,001. Precision of desired integration
must be coherent with
level of precision necessary for the total stage.
If the criterion is not checked, the pitch of time Re-is cut out according to a discovery method (a number
of under-not defined by the user via the key word
ITER_INTER_PAS
). When the pitch of time becomes
too much weak (H < 1.10
20
), calculation is stopped with an error message.
4 Variables
interns
4.1
Case of the monocrystal
The internal variables in Code_Aster are called
V1
,
V2
,
…
Vp
.
The six first are the 6 components of the viscoplastic deformation.
V7
,
V8
,
V9
are the values of
1
1
1
p
for the system of slip
1
=
S
V10
,
V11
,
V12
correspond to the system
2
=
S
, and so on.
The last internal variable, Vp, (p=6+3n+1, N being the total number of systems of slip) are
an indicator of plasticity (threshold exceeded in at least a system of slip to the pitch of time
running). If it is null, there no was increase in internal variables at the current moment. If not, it
the iteration count of Newton contains (for an implicit resolution) which was necessary for
to obtain convergence.
4.2
Case of the polycrystal
The internal variables in Code_Aster are called
V1
,
V2
,
…
Vp
.
The six first are the 6 components of the viscoplastic deformation. The seventh is
viscoplastic deformation are equivalent cumulated (macroscopic).
Then, for each phase, one finds:
Viscoplastic deformations or the tensor Beta
values of
S
S
S
p
for each system of slip
The last internal variable, Vp, (p=6+1+m (6+3n) +1),
(
)
2
3
6
7
,
1
+
+
+
=
=
m
G
S
N
p
, m being the number of
phases and
S
N
being the number of systems of slip of the phase G).
is an indicator of plasticity (threshold exceeded in at least a system of slip to the pitch of time
running). If it is null, there no was increase in internal variables at the current moment. If not, it
the iteration count of Newton contains (for an implicit resolution) which was necessary for
to obtain convergence.
Code_Aster
®
Version
7.4
Titrate:
Viscoplastic behaviors élasto mono crystalline lenses
Date:
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J.M. PROIX, T. KANIT, O. DIARD
Key
:
R5.03.11-A
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R5.03 booklet: Nonlinear mechanics
HT-66/05/002/A
5
Numerical establishment in Code_Aster
Generally, the single-crystal behaviors are integrated into the methods of Runge-
Kutta for explicit integration [R5.03.14], and with the environment “
plasti
” for implicit integration
[R5.03.10]. The tensors of orientation of the systems of slip are as for them all defined in
a routine, providing the tensor in total reference mark for the nth system of the provided family of which it
name is provided by the appealing routine.
To add a new behavior of monocrystal, or simply a new law
of collapse or work hardening, it is advisable to define its parameters in
DEFI_MATERIAU
. According to
the case (flow, isotropic work hardening or kinematics), it is necessary to add the reading of these parameters
in the routines
LCMAFL
,
LCMAEI
,
LCMAEC
. For integration, it is enough to write the definition of
increases in variables intern in the routines
LCMMFL
(flow),
LCMMEC
(work hardening
kinematics) and
LCMMEI
(isotropic work hardening), so that explicit integration functions.
Implicit integration also uses the routines
LCMMFL
,
LCMMEC
and
LCMMEI
. It asks moreover
to define the derivative of the equations compared to the various variables. The derivative are to be written
in the routines
LCMMJF
(derivative of L `equation of flow),
LCMMJI
(derivative of the relation
of isotropic work hardening) and
LCMMJC
(derivative of the kinematic relation of work hardening).
Code_Aster
®
Version
7.4
Titrate:
Viscoplastic behaviors élasto mono crystalline lenses
Date:
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J.M. PROIX, T. KANIT, O. DIARD
Key
:
R5.03.11-A
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R5.03 booklet: Nonlinear mechanics
HT-66/05/002/A
6 Use
These models are accessible in Code_Aster in
3D
, plane deformations (
D_PLAN
), forced
plane (
C_PLAN
) and axisymetry (
AXIS
).
6.1
Case of the monocrystal
In the case of microstructures with a grid, various grains of a monocrystal being represented by
groups of meshs, it is necessary to affect the parameters of materials and the behaviors of
monocrystals like their orientations with the various grains.
The values of the parameters of the relations of behavior are provided using the control
DEFI_MATERIAU
. Currently, this is defined starting from the key words
ECOU_VISC1
,
ECOU_VISC2
,
ECOU_VISC3
for the flow,
ECRO_ISOT1
,
ECRO_ISOT2
for isotropic work hardening and
ECRO_CINE1
,
ECRO_CINE2
for kinematic work hardening [U4.43.01]. For example [V6.04.172]:
MATER1=DEFI_MATERIAU (
ELAS_ORTH=_F (E_L=192500,
E_T=128900,
NU_LT=0.23,
G_LT=74520,),
# RELATIONS Of FLOW
ECOU_VISC1=_F (N=10, K=40, C=6333),
ECOU_VISC2=_F (N=10, K=40, C=6333, D=37, A=121),
ECOU_VISC3=_F (K=40, V=, GAMMA0=),
# WORK HARDENING ISOTROPIC
ECRO_ISOT1=_F (R_0=75.5, Q=9.77, B=19.34, H=2.54),
ECRO_ISOT2=_F (R_0=75.5, Q1=9.77, B1=19.34, H=2.54, Q2=-33.27, B2=5.345,),
# WORK HARDENING KINEMATIC
ECRO_CINE1=_F (D=36.68),
ECRO_CINE2=_F (D=36.68, GM=, PM=,),
);
One can thus dissociate, on the level of the data, the flow of the isotropic work hardening and of
kinematic work hardening.
It is now necessary to define it (or them) standard of studied monocrystal. For that, one defines the behavior
in an external way with
STAT_NON_LINE
, via the operator
DEFI_COMPOR
, for example:
MONO1=DEFI_COMPOR (MONOCRYSTAL = (_F (MATER=MATER1,
ECOULEMENT=ECOU_VISC1,
ECRO_ISOT=ECRO_ISOT1,
ECRO_CINE=ECRO_CINE1,
FAMI_SYST_GLIS= (“CUBIQUE1”,),
_F (MATER=MATER1,
ECOULEMENT=ECOU_PLAS1,
ECRO_ISOT=ECRO_ISOT2,
ECRO_CINE=ECRO_CINE2,
FAMI_SYST_GLIS=' CUBIQUE2',),
),
_F (MATER=MATER2,
ECOULEMENT=ECOU_PLAS1,
ECRO_ISOT=ECRO_ISOT2,
ECRO_CINE=ECRO_CINE2,
FAMI_SYST_GLIS=' PRISMATIQUE',),
),
)
Code_Aster
®
Version
7.4
Titrate:
Viscoplastic behaviors élasto mono crystalline lenses
Date:
19/04/05
Author (S):
J.M. PROIX, T. KANIT, O. DIARD
Key
:
R5.03.11-A
Page
:
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Manual of Reference
R5.03 booklet: Nonlinear mechanics
HT-66/05/002/A
The structure of produced data contains names of systems of slip, associated names
material parameters, for each behavior of monocrystal.
FAMI_SYST_GLIS MATE_SYST
TYPE_LOI
FLOW
ECRO_ISOT
ECRO_CINE
“CUBIC”
MATER1
VISC ECOU_VISC1
ECRO_ISOT1
ECRO_CIN1
“BASAL” MATER1
VISC ECOU_VISC1
ECRO_ISOT1
ECRO_CIN1
“PRISMATIC”
MATER1
PLAS ECOU_PLAS1
ECRO_ISOT2
ECRO_CIN2
… …
…
The operator
DEFI_COMPOR
calculate the total number of variables intern associated with the monocrystal.
Lastly, to carry out a calculation of microstructure, it is necessary to give, grain by grain, or group of
meshs (representing assemblies of grains) an orientation, using the key word
SOLID MASS
of
AFFE_CARA_ELEM
. For example:
ORIELEM = AFFE_CARA_ELEM (MODEL = MO_MECA,
SOLID MASS = (
_F (GROUP_MA=' GRAIN1',
ANGL_REP= (348.0, 24.0, 172.0),
),
_F (GROUP_MA=' GRAIN2',
ANGL_REP= (327.0, 126.0, 335.0),
),
_F (GROUP_MA=' GRAIN3',
ANGL_REP= (235.0, 7.0, 184.0),
),
_F (GROUP_MA=' GRAIN4',
ANGL_REP= (72.0, 338.0, 73.0),
),
…)
Note:
·
Contrary to the current operator
DEFI_TEXTURE
, one gives only the name of
crystallographic structure, knowing that directions of slip of each family
systems of slip will be defined once and for all in the source.
·
For the same monocrystal, the values of the parameters can be different from one
family of systems of slip to the other. This is why one can define a material
different by occurrence from the key word factor
MONOCRYSTAL
. But in this case, how
to provide to transmit to
STAT_NON_LINE
information stipulating that in a point of gauss
(all those of the group of meshs concerned), are there several materials present? This is
possible thanks to an evolution of
AFFE_MATERIAU
[U4.43.03] and of the structure of
data material [D4.06.18]):
MAT=AFFE_MATERIAU (MAILLAGE=MAIL,
AFFE =_F (GROUP_MA=' GRAIN1',
MATER= (MATER1, MATER2),),
);
The other data of calculation are identical to a usual structural analysis.
Lastly, in
STAT_NON_LINE
, the behavior resulting from
DEFI_COMPOR
is provided, under the key word
COMP_INCR
via the key word
COMPOR
, obligatory with the key word
RELATION=' MONOCRISTAL'
.
COMP_INCR = _F (RELATION = ' MONOCRISTAL',
COMPOR
=
COMP1
Specified that for explicit integration, (
RESO_INTE=' RUNGE_KUTTA'
), it is useless to ask
the reactualization of the tangent matrix since the aforementioned is not calculated. To begin from
iterations of Newton of the total algorithm, it can be useful to specify
PREDICTION=' EXTRAPOL'
[U4.51.03].
One will be able to find an example of use in the tests: SSNV171 and SSNV172.
Code_Aster
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Version
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Titrate:
Viscoplastic behaviors élasto mono crystalline lenses
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Key
:
R5.03.11-A
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HT-66/05/002/A
6.2
Case of the polycrystal
In the case of multiphase polycrystals, each phase corresponds to a monocrystal. One will use
thus parameters preset previously in
DEFI_MATERIAU
for the monocrystal. Here, it is about
to lay down, for each phase, the orientation, the voluminal fraction, the monocrystal used, and the type of law of
localization. This is carried out under the key word factor
POLYCRYSTAL
of
DEFI_COMPOR
.
MONO1=DEFI_COMPOR (MONOCRISTAL=_F (MATER=MATPOLY,
ECOULEMENT=' ECOU_VISC2',
ECRO_ISOT=' ECRO_ISOT2',
ECRO_CINE=' ECRO_CINE1',
ELAS=' ELAS',
FAMI_SYST_GLIS=' OCTAEDRIQUE',),);
POLY1=DEFI_COMPOR (POLYCRISTAL= (_F (MONOCRISTAL=MONO1,
FRAC_VOL=0.025,
ANGL_REP= (- 149.676, 15.61819, 154.676,),),
_F (MONOCRISTAL=MONO1,
FRAC_VOL=0.025,
ANGL_REP= (- 150.646, 33.864, 55.646,),),
_F (MONOCRISTAL=MONO1,
FRAC_VOL=0.025,
ANGL_REP= (- 137.138, 41.5917, 142.138,),),
......
_F (MONOCRISTAL=MONO1,
FRAC_VOL=0.025,
ANGL_REP= (- 481.729, 35.46958, 188.729,),),),
LOCALIZATION=' BETA',
DL=321.5,
DA=0.216,);
The key word
POLYCRYSTAL
allows to define each phase by the data of an orientation, one
voluminal fraction, of a monocrystal (i.e. a model of behavior and systems of
slip).
The key word
LOCALIZATION
allows to choose the method of localization for the whole of the phases
polycrystal.
Lastly, in
STAT_NON_LINE,
the behavior resulting from
DEFI_COMPOR
is provided, under the key word
COMP_INCR
via the key word
COMPOR
, obligatory with the key word
RELATION=' POLYCRISTAL'
.
COMP_INCR = _F (RELATION = ' POLYCRISTAL',
COMPOR
=
COMP1
)
This behavior is tested for example in
SSNV125A
(where one can check that the results are
identical to those obtained with
POLY_CFC
).
6.3 Example
As example of implemented, one presents here briefly a calculation of aggregate, of form
cubic (elementary volume) including/understanding 100 single-crystal grains, definite each one by a group of
meshs. The total number of elements is 86751. With meshs of command 1 (TETRA4) it comprises
15940 nodes. With meshs of command 2 (TETRA10), it comprises 121534 of them.
The loading consists of a homogeneous deformation, applied via one
normal displacement imposed on a face of the cube (direction Z). One reaches a deformation of 4% in
1s and 50 increments.
Calculation (tetra4) lasts 140000 seconds is 39 hours of CPU Alphaserveur.
ACIER=DEFI_MATERIAU (ELAS=_F (E =145200.0, NU=0.3,),
ECOU_VISC1=_F (N=10., K=40., C=10.,),
ECRO_ISOT2=_F (R_0=75.5,
B1 =19.34,
B2 =5.345,
Q1 =9.77,
Q2 =33.27,
H=0.5),
ECRO_CINE1=_F (D=36.68,),
);
Code_Aster
®
Version
7.4
Titrate:
Viscoplastic behaviors élasto mono crystalline lenses
Date:
19/04/05
Author (S):
J.M. PROIX, T. KANIT, O. DIARD
Key
:
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:
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Manual of Reference
R5.03 booklet: Nonlinear mechanics
HT-66/05/002/A
COEF=DEFI_FONCTION (NOM_PARA = ' INST', VALE = (0.0, 0.0, 1.0, 1.0,),);
MAT=AFFE_MATERIAU (MAILLAGE=MAIL, AFFE=_F (ALL = " YES ", MATER= (STEEL),),);
COMPORT=DEFI_COMPOR (MONOCRISTAL= (
_F (MATER =ACIER,
ECOULEMENT= " ECOU_VISC1 ",
ECRO_ISOT = " ECRO_ISOT2 ",
ECRO_CINE = " ECRO_CINE1 ",
ELAS= " ELAS ",
FAMI_SYST_GLIS=' OCTAEDRIQUE',),
),);
ORIELEM = AFFE_CARA_ELEM (MODEL = MO_MECA,
SOLID MASS = (
_F (GROUP_MA=' GRAIN1', ANGL_REP= (348.0, 24.0, 172.0),),
_F (GROUP_MA=' GRAIN2', ANGL_REP= (327.0, 126.0, 335.0),),
_F (GROUP_MA=' GRAIN3', ANGL_REP= (235.0, 7.0, 184.0),),
.................
_F (GROUP_MA=' GRAIN99', ANGL_REP= (201.0, 198.0, 247.0),),
_F (GROUP_MA=' GRAIN100', ANGL_REP= (84.0, 349.0, 233.0),),
))
FO_UZ = DEFI_FONCTION (NOM_PARA = “INST”,
VALE = (0.0, 0.0, 1.0, 0.04,),)
CHME4=AFFE_CHAR_MECA_F (MODELE=MO_MECA,
DDL_IMPO=_F (GROUP_NO=' HAUT', DZ=FO_UZ,),)
LINST = DEFI_LIST_REEL (DEBUT= 0.,
INTERVAL = (_F (JUSQU_A = 1., NOMBRE= 50),))
SIG=STAT_NON_LINE (MODEL =MO_MECA,
CARA_ELEM=ORIELEM,
CHAM_MATER =MAT,
EXCIT= (_F (CHARGE=CHME1),
_F (CHARGE=CHME2),
_F (CHARGE=CHME3),
_F (CHARGE=CHME4),),
COMP_INCR= (_F (RELATION
= ' MONOCRISTAL',
COMPOR =COMPORT,
ALL = ' OUI',),),
INCREMENT= (_F (LIST_INST=LINST,
SUBD_PAS =4,
SUBD_PAS_MINI=0.000001,
),),
NEWTON =_F (REAC_ITER =5,),),
);
The following figures represent isovaleurs of the deformations the stresses according to Z. One notes
nonhomogeneity of the values, and one can even distinguish the contour of the grains.
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Viscoplastic behaviors élasto mono crystalline lenses
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To be able to exploit this type of results, one can for example calculate average fields by
grains. On the following figure, one represented the equivalent stresses according to
equivalent plastic deformations for the whole of the grains.
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Viscoplastic behaviors élasto mono crystalline lenses
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R5.03 booklet: Nonlinear mechanics
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7 Bibliography
[1]
MERIC L., CAILLETAUD G.: “individual extremely structural hook modeling calculations” in Newspaper
off Engineering Material and Technology, January 1991, flight 113, pp171-182.
[2]
LECLERCQ S., DIARD, O., PROIX J.M.: “Which microcomputer-macro in Aster? Impact study of
establishment of a library of laws of behavior and rules of transition
scales “Notes EDF R & D HT-26/03/053/A.
[3]
CAILLETAUD G.: “A micromechanical approach to inelastic behavior off metals”, Int. J. off
Plasticity, 8, pp. 55-73, 1992.
[4]
PILVIN P.: “The contribution off micromechanical approaches to the modelling off inelastic
behavior off polycrystals ", Int. Conf. one Biaxial/Multiaxial tires, France, ESIS/SF2M,
pp. 31-46, 1994.
[5]
BERVEILLER Mr., ZAOUI A.: “Year extension off the coil-consistent design to plasticity flowing
polycrystal " J. Mech. Phys. Solids, 6, pp. 325-344, 1979.
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R5.03 booklet: Nonlinear mechanics
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Appendix
1
Expression of Jacobien of the equations
elastoviscoplastic integrated
The system to be solved is form:
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
vp
vp
HT
S
S
S
vp
S
S
S
vp
S
S
S
vp
S
S
S
S
vp
S
S
S
vp
S
S
S
vp
m
p
F
p
p
G
p
H
N
m
E
E
E
E
D
D
p
E
p
p
E
G
p
E
has
N
p
E
E
p
E
S
p
E
R
Y
R
:
0
)
,
,
,
(
)
,
,
,
(
)
,
,
,
(
)
(
)
(
)
,
,
,
,
(
)
,
,
,
,
(
)
,
,
,
,
(
)
,
,
,
,
(
)
,
,
,
,
(
)
,
,
,
,
(
)
(
1
1
+
+
+
+
+
+
+
+
+
+
+
+
+
+
-
-
-
-
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
=
=
-
-
-
-
-
-
-
-
=
=
=
That is to say thus to evaluate the terms of the hypermatrice jacobienne
J
at the moment
T
T
+
=
S
S
S
vp
S
S
S
vp
S
S
S
vp
S
S
S
vp
S
S
S
vp
p
p
p
p
E
p
p
p
G
G
G
E
G
G
p
has
has
has
E
has
has
p
E
E
E
E
E
E
p
S
S
S
E
S
S
J
With regard to the first line of the matrix, independently of the equations of work hardening and
of flow, one a:
0
1
=
=
=
=
=
-
S
S
S
vp
p
S
S
S
Id
E
S
D
S
The second line can be written also independently of the flow and work hardenings:
0
0
0
=
-
=
=
=
=
S
S
S
S
vp
p
E
E
E
Id
E
E
E
m
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The first column of the lines corresponding to the equations (A), (G) and (p) is written:
=
=
=
S
S
S
S
S
S
p
p
G
G
has
has
with
()
T
S
S
m
=
The second column is identically null (because of equation (E): relations of flow and
of work hardening can express itself only according to
S
and not of
vp
E
.
The last block of equations, depends as for him on the selected behaviors:
S
S
S
S
S
S
S
S
S
p
p
p
p
p
G
G
G
p
has
has
has
Example
Let us choose the viscoplastic relation of flow
ECOU_VISC1
0
)
(
.
)
(
0
)
(
=
-
-
-
=
-
-
-
N
S
S
S
S
S
S
S
S
S
S
S
K
p
R
C
T
p
p
C
C
p
G
with isotropic work hardening
ECRO_ISOT1:
=
-
-
+
=
NR
R
LP
Sr
S
S
R
E
H
Q
R
p
R
1
0
))
1
(
(
)
(
,
Sr
Sr
Sr
H
+
-
=
)
1
(
and a kinematic work hardening defined by
ECRO_CINE1
0
)
(
=
+
-
S
S
S
S
p
D
has
then:
()
S
S
S
N
S
S
S
S
N
S
S
S
C
C
p
R
C
K
T
N
p
G
has
-
-
-
-
-
=
=
=
-
1
0
0
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()
S
S
S
S
N
S
S
S
S
N
S
S
S
S
C
C
p
R
C
K
T
nc
p
G
p
D
has
-
-
-
-
=
=
+
=
-
1
0
1
0
1
1
=
=
-
=
S
S
S
p
G
has
()
()
()
S
LP
S
S
S
S
S
S
S
N
S
S
S
S
N
S
S
S
S
S
S
S
S
E
Qbh
p
D
p
Dr.
p
D
p
Dr.
p
R
C
K
T
N
p
p
C
C
p
G
D
p
has
-
-
=
-
-
+
=
-
-
=
=
1
1
and, concerning the interaction between systems of slip, it there only one term not no one:
()
()
()
R
LP
Sr
R
S
S
R
S
S
N
S
S
S
S
N
R
E
Qbh
p
D
p
Dr.
p
D
p
Dr.
p
R
C
K
T
N
p
p
-
-
=
-
-
+
=
1
1
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Viscoplastic behaviors élasto mono crystalline lenses
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R5.03 booklet: Nonlinear mechanics
HT-66/05/002/A
Appendix 2 Evaluation of the coherent tangent operator
It is a question of finding the operator tangent coherent, i.e. calculated starting from the solution of
()
(
)
0
=
Y
R
in end
of increment. For a small variation of
R
, by considering this time
like variable and not like
parameter, one obtains:
0
=
+
+
+
+
+
S
S
S
S
S
S
vp
vp
p
p
R
R
R
E
E
R
E
E
R
R
This system can be written:
()
=
=
=
0
0
0
0
,
and
with
E
X
p
E
Y
X
Y
Y
R
S
S
S
vp
By writing the matrix jacobienne in the form:
[] []
[] []
[]
=
Z
E
Y
Y
Y
Y
Y
J
vp
3
2
1
0
0
1
0
0
1
.
With:
S
S
S
S
N
p
Z
D
Y
×
=
=
-
1
0
While operating by successive eliminations and substitutions, the third block of the system of equation gives:
()
()
()
(
)
E
Y
Y
Y
Y
Y
Y
Y
Z
Y
E
Y
Y
Z
vp
=
+
=
-
=
-
=
-
-
-
2
1
3
1
0
2
1
3
1
1
2
1
3
the required tangent operator can thus be written directly:
(
)
1
2
1
3
1
0
-
-
+
+
-
=
=
Y
Y
Y
Y
E
E
T
T
T
T