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Titrate:
Viscoplastic behaviors élasto mono crystalline lenses

Date:
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:
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: 1/24

Organization (S): EDF-R & D/AMA, MMC

Handbook of Référence
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 step of time) or an implicit way
(method of Newton with local recutting of the step of time).

These behaviors can be employed for the calculation of microstructures (grid 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.

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Count

matters

1 Introduction ............................................................................................................................................ 3
2 Formulation of the behaviors mono and polycrystalline ..................................................................... 3
2.1 Relations of behavior of the monocrystal ................................................................................... 3
2.1.1 Examples of relations of flow ..................................................................................... 4
2.1.2 Examples of relations of kinematic work hardening ................................................................ 5
2.1.3 Examples of relations of isotropic work hardening ....................................................................... 5
2.2 Systems of slip and total behavior of the monocrystal .................................................. 5
2.3 Behavior of the polycrystal homogenized .................................................................................... 6
2.3.1 Recall from existing ................................................................................................................ the 6
2.3.2 Behavior of the type POLYCRISTAL ................................................................................. 6
2.3.2.1 Relation of scaling ............................................................................. 7
3 local Integration and implementation numerical .................................................................................... 8
3.1 System of equations to solve ..................................................................................................... 8
3.1.1 Behavior of the type MONOCRISTAL ............................................................................... 8
3.1.2 Behavior of the type POLYCRISTAL ................................................................................. 9
3.2 Implicit resolution ........................................................................................................................ 10
3.2.1 Operator of tangent behavior ................................................................................... 11
3.3 Resolution clarifies ........................................................................................................................ 12
4 Variables intern ................................................................................................................................ 13
4.1 Case of the monocrystal ........................................................................................................................ 13
4.2 Case of the polycrystal .......................................................................................................................... 13
5 numerical Establishment in Code_Aster ...................................................................................... 14
6 Use ............................................................................................................................................. 15
6.1 Case of the monocrystal ........................................................................................................................ 15
6.2 Case of the polycrystal .......................................................................................................................... 17
6.3 Example ......................................................................................................................................... 17
7 Bibliography ........................................................................................................................................ 20
Appendix 1
Expression of Jacobien of the integrated elastoviscoplastic equations ................ 21
Appendix 2
Evaluation of the coherent tangent operator ............................................................. 24

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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 grid 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
constraints 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:
& = G (, p), with, p & = and
S
S
S
S
S
S
& S
for an elastoplastic behavior a criterion of the type: F (, p)
and F.p&

S = 0
S
S
S
S
0
for a élasto-viscoplastic behavior, p & = F, p
S
(S S S S)

· Evolution of kinematic work hardening: & = H (, p)
S
S
S
S
S

· Evolution of the isotropic work hardening defined by a function: R (p)
S
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Titrate:
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:
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These relations become, after discretization in time:
· Relation of flow:

= G (, p), with, p
=

S
S
S
S
S
S
S
for an elastoplastic behavior a criterion
type
: F (, p)
and
S
S
S
S
0
F.p

S = 0
and for a élasto-viscoplastic behavior, p
= F,

S
(
p
S
S
S
S) T

· Evolution of kinematic work hardening:
= H (, p)
S
S
S
S
S

· Evolution of isotropic work hardening: R (p)
S

The quantities (, p) are evaluated at the moment running for an implicit discretization and with
S
S
S
S
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 command
DEFI_MATERIAU [U4.43.01].

2.1.1 Examples of relations of flow

ECOU_VISC1
- C
S
S
= G (, p) = p

S
S
S
S
S
S - C
S
S
N

- C - R (p)
S
S
S
S
p
= T
.
S
K
The parameters are: C, K, N.

ECOU_VISC2
- C - has
S
S
S

= G (, p) = p

S
S
S
S
S
S - C - has
S
S
S
N
D
- C - - R (p) has +
(C
2
)

S
S
S
S
S
S
C
p
2
=
S
K
The parameters are: C, K,
N has, D.

ECOU_VISC3

-

-


*
G


V *

0


= G, p
S
=
S

µ
& exp
exp
.
,
S
(S S S S) 0 kT


kT
S


The parameters are: K,
*
,
µ &, G
, V

0
0

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ECOU_PLAS1
F (, p)
C
R p
,
S
S
S
S
= S - S - ()
S
S
0
F.p
,
S = 0
F


= p


S
S

S
The associated parameter is: C.

2.1.2 Examples of relations of kinematic work hardening

ECRO_CINE1
= H
(, p) = - D p

S
S
S
S
S
S
S
S
The parameter is: D.

ECRO_CINE2
C

S
m
S


= H
(, p) =
- D p
- (
)

S
S
S
S
S
S
S
S
M
S
Parameters being then: D, M, m, C.

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
R (p) R
Q (
H 1
(
E
))
S
S
= 0 +
-
- LP
Sr
r=1
with:
H = 1
(-) +
Sr
Sr
Sr
The parameters are: R, Q, B, h.
0

ECRO_ISOT2
1s
2s
R (p) = R + Q (H Q) + Q Q
S
0
1
rs
2
sg
with:
dqis = B 1
(- qis) dp
I
The parameters are: R, Q, B,
H Q.
0
1
1
2

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.
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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 unit vector N).

S
1
m = (N L + L N)
ij
2
I
J
J
I

From the point of view of the behavior at the material point, this tensor intervenes for the calculation of the scission
reduced S
S
= m
:
and that the speed of total viscoplastic deformation vp
E, defined to leave
ij
ij
knowledge speeds of slip S
& for all the systems of
slip: vp
E&
S
m

ij
ij
=
S
&

S G
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 what exists

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 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 type POLYCRISTAL

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):

· total deflection macroscopic E
· viscoplastic deformation macroscopic VP
E
· macroscopic constraint:

= D (
HT
vp
E - E - E)
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· 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 fg, a relation of localization of the constraints, general form (to be expressed
in the local reference mark of each phase)

= L, E,
G
(vp vpg G)

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:
& = G (, p), with, p & = and p & = F, p
S
(S S S S)
S
S
S
S
S
S
& S
­ Evolution of kinematic work hardening: & = H (, p)
S
S
S
S
S
­ Evolution of the isotropic work hardening defined by a function: R (p)
S
­ Viscoplastic Déformations of the phase: vpg
&
S
m

ij
ij
=
S
&

S G

There remain the equations of homogenization then: vp
vp
E & = F
G &g
G

2.3.2.1 Relation of scaling

Two relations of localization of the type = L, E, are available in the version
G
(vp vpg G)
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:

vp
E
1
3
G
vp
vp
ij
G
ij = ij + µ

Eij - ij
= 1+ µ


2
J2 (ij)

· The second relation, developed more particularly for cyclic loadings
[bib4] allows to give a good description to schematize the interactions between
grains:

G
G
G
­
ij = ij + µ (Bij - ij)
Bij = F gij
G

G
vp
G
vp
vp
­
& =
G
&
- (
D -
G

) ||
G
&
||
ij
ij
ij
ij
ij

where D and are parameters characteristic of material and temperature.

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3
Local integration and implementation numerical

3.1
System of equations to be solved

3.1.1
Behavior of the type MONOCRISTAL

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 constraints at the previous moment
-
(T)
,
I 1
=
-
· variables intern at the previous moment, for each system of slip:
T
T
p T
,
S (
,
,
I 1
-)
S (I 1
-)
S (I 1
-)
· and of the tensor of increase in total deflection provided by iteration N in algorithm
total of resolution
N
E
= E
(with the notations of [R5.03.01]).
I

Integration consists in finding:

· the macroscopic tensor of constraints = (T)
I
· and the variables intern = (T), = (T), p = p (T)
S
S
I
S
S
I
S
S
I

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 noted quantities +/-
With
are evaluated at the moment T: +/-
-
With
= A = (
With T

I 1
-)
I 1
-
· implicit, if they are evaluated at the moment T: A+/- = A+ = (
With T
I)
I

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 T,
I
-
E T
(
)
: deformation at the moment T,
I 1
=
-
E
I 1
-
-
(T)
: macroscopic constraint at the moment T,
I 1
=
-
I 1
-
T
T
p T
: variables intern for each system of slip to T,
S (
,
,
I 1
-)
S (I 1
-)
S (I 1
-)
I 1
-

It is necessary to find:

= (T): macroscopic constraint at the moment T, in the reference mark corresponding to the orientation
I
I
total
= (T)
S
S
I
= (T)
S
S
I
p = p (T)
S
S
I
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checking:

1
D = (1
D)
-
- + (
HT
vp
E
- E

- E

), where D can depend on the temperature, and can correspond to one
orthotropic elasticity.
vp
E = m
S S
S
for each system of slip (of the whole of the families of systems):
N equations:
+/-
=: m
S
S
S
N relations of flow: maybe in viscoplasticity
G


p

S =
(+/-, +/-, +/-, +/-)
S
S
S
S
S
with p
F


p

S =
(+/-, +/-, +/-, +/-)
S
S
S
S
maybe in plasticity F (+/-
, +/-
, +/-
, +/-
p
)
,
F p
,
S = 0
S
S
S
S
0
with p
=


S
S
N equations of evolutions of kinematic work hardening:
H


p

S =
(+/-, +/-, +/-, +/-)
S
S
S
S
S
N equations of evolution of isotropic work hardening: R (+/-
p
)
S
S
S

This is solved either explicitly (Runge_Kutta), or implicit (Newton).

3.1.2
Behavior of the type POLYCRISTAL

The discretized relations of behavior are:

Being given (in a point of Gauss) total tensors:

· increase in total deflection E
,
· total deflection at the moment previous E (T
,
I 1)
-
=
-
E
· constraint at the previous moment: (T

I
=
-)
-,
1
· the whole of the internal variables - -
-
, p relating to the systems of slip of
S
S
S
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 = (T, = T, = T, p = p T checking:
I)
S
S (I)
S
S (I)
S
S (I)

· on the level of the point of Gauss: = D (D-1) - + D (
HT
vp
E
- E

- E

), in the total reference mark,
­ for each phase (or “grain”), defined by an orientation and a proportion fg, one
relation of localization of the constraints, the general form (to be expressed in the reference mark
room of each phase)
= L, E,
G
(vp vpg G)

­ and for each system of slip of each phase:
­
vp


= m

G
S
S
S
­ N

S equations:
=: m
S
S
­ N

=
S relations of flow:
G
, p, with p
=

S
(S S S S)
S
S
­ N

=
S evolutions of work hardening:
H
, p
S
(S S S S)
­
F (, p
F p
, (in plasticity independent of time)
S
S
S
S)
,
0. S = 0

· There remain the equations of homogenization then:
vp
vp
E

=
F

G
G
G
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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:










vp
+
+
+
1
-
1
-
-
HT
vp
S (, E

, p)
D - (D) - (E
- E

- E

)
S
S
S
-

vp
vp
+
+
+

vp
R Y
() = R (, E

, p
) = E (, E

, p)

=
E

- m
= 0
S
S
S
S
S
S
S
S

S

vp
has (,
E

, +
, +
, p+)
S
S
S

- H (+
, +
, +
, p+)
S
S
S
S
S
vp
N G (, E

, +
, +
, p+)

S
S
S
S

N - G (+
, +
, +
, p+)

S
S
S
S
S
S

vp

p (,
E

, +
, +
, p+)

S
S
S



p
- F (+
, +
, +
, p+)
S
S
S
S
S


+
+
=: m
S
S

In more contracted way, one poses:

S (y)

(ey)



vp
E


R (y) = 0 = (
y has)


avecy =


S
G (y)



S


p (y)




PS

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:

Dr.
Y
= Y - (
) 1
- R (Y)
K 1
+
K
K
dYk
Dr.
It is thus necessary to define the initial values Y, and to calculate the matrix jacobienne system:
(this one
0
dYk
is detailed in appendix for the viscoplastic behaviors described previously). It with
following form:

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S
S
S
S
S

vp

E


p
S
S
S
E
E
E
E
E
vp
E


p
S
S


S
has
has
has
has
has
J =

vp
E


p
S
S
S
G
G
G
G
G

vp

E


p
S
S
S
p
p
p
p
p
vp
E


p
S
S


S

R (Y)
The criterion of stop of the iterations relates to the nullity of the residue:
K
<. If convergence is not
R (Y)
0
reached after the maximum number of iterations, the stationnarity of the solution is also tested:

Y
Y

K +1 - K <

The method used allows a local recutting of the step 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 (R (Y) = 0) is checked in end
of increment. For a small variation of R, by regarding this time as variable and not
like parameter, the system remains with balance and one checks dF L = 0, i.e.:

R







+ R

+
R
E

R
R
R
vp
E +



p

vp
S +
S +
S = 0



E

E



p
S
S
S

This system can be still written:




E

vp


E
0
R
(Y) = X, avecY = etX

S
= 0
Y





S
0
p
0
S



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.
This one is written directly (see [§Annexe2]):







=


= (
1
-
-
Y - Y Y Y

0
1 3
) 1
2
E


T
T

+

E t+t
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By writing the matrix jacobienne in the form:

Y
1
0
0
[] []

J.Y
= []
0
[]

vp
1
Y

1


E

Y
0
2
[



] Y
3
Z


With:

Y = D-1
0
S



Z
= ×n
S
S
p
S

The submatrices have as dimensions:

dim (Y = D 1
-
0
) = [6,
6]
dimY1 = [3
,
6 * NS]

dimY2 = [3 * NS 6
,]
dimY1 = [3 * NS 3
, * NS]

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 step 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
Dy
is a function only values of their derivatives
= F (Y, T):
dt

H (-
, -
, -
, p)
S
S
S
S
S


G (-

, -
, -
, p)

S
S
S
S

S
Y

=
=

= F (-
, -
, -
, p)
p
S
S
S
S
S
S




vp
E
m



S
S
S
with =: m = - + D (
HT
vp
E
- E

- E

)
S
S

One integrates according to the following diagram:

Yt+h = Y (2) if the criterion of precision is satisfied
H
Y (2) = Y + [F (Y, T) + F (Y (1), T + H)] with Y (1) = Y + H F (Y, T)
2
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()
()
The difference between 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 step of time H which is initialized with Ti for
the first attempt. The strategy of the control of the step 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 Y is noted = (y1, y2,…, yN):

| y (2) - y (1)|
Y (T) = sup
J
J

<
J =1, NR
max [

, | yj (T)|]

The parameter is fixed at 0,001. The precision of desired integration must be coherent with
level of precision necessary for the total stage.

If the criterion is not checked, the step of time Re-is cut out according to a discovery method (a number
of under-not defined by the user via key word ITER_INTER_PAS). When the step of time becomes
too much weak (H < 1.10­20), 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 p for the system of slip S = 1
1
1
1
V10, V11, V12 correspond to the system S = 2, 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 step 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 Beta tensor
values of p for each system of slip
S
S
S
The last internal variable, Vp, (p=6+1+m (6+3n) +1), p = 7 + (6 + 3n
, m being the number of
S) + 2
G =,
1 m
phases and N being the number of systems of slip of the phase G).
S
is an indicator of plasticity (threshold exceeded in at least a system of slip to the step 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.
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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 routines LCMAFL, LCMAEI, LCMAEC. For integration, it is enough to write the definition of
increases in variables intern in routines LCMMFL (flow), LCMMEC (work hardening
kinematics) and LCMMEI (isotropic work hardening), so that explicit integration functions.

Implicit integration also uses 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 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).
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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 command
DEFI_MATERIAU. Currently, this is defined starting from 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 operator DEFI_COMPOR, for example:

MONO1=DEFI_COMPOR (MONOCRISTAL = (_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',),
),
)
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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
ECOULEMENT
ECRO_ISOT
ECRO_CINE
“CUBIQUE”
MATER1


VISC ECOU_VISC1
ECRO_ISOT1
ECRO_CIN1
“BASAL” MATER1


VISC ECOU_VISC1
ECRO_ISOT1
ECRO_CIN1
“PRISMATIQUE”
MATER1
PLAS ECOU_PLAS1
ECRO_ISOT2
ECRO_CIN2
… …





Operator DEFI_COMPOR calculates 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 sets of grains) an orientation, using key word MASSIF of
AFFE_CARA_ELEM. For example:

ORIELEM = AFFE_CARA_ELEM (MODEL = MO_MECA,
MASSIF = (
_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 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 MONOCRISTAL. 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 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 key COMPOR, obligatory word with 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 this one 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.
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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 POLYCRISTAL 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,);

Key word POLYCRISTAL makes it possible to define each phase by the data of an orientation, of one
voluminal fraction, of a monocrystal (i.e. a model of behavior and systems of
slip).
Key word LOCALIZATION makes it possible 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 key COMPOR, obligatory word with 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,),
);
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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,
MASSIF = (
_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 constraints according to Z. One notes
nonhomogeneity of the values, and one can even distinguish the contour of the grains.
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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 constraints according to
equivalent plastic deformations for the whole of the grains.


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7 Bibliography

[1]
MERIC L., CAILLETAUD G.: “individual extremely structural hook modeling calculations” in Journal
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 “Note 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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Appendix
1
Expression of Jacobien of the equations
elastoviscoplastic integrated

The system to be solved is form:










vp
+
+
+
1
-
1
-
-
HT
vp
S (,
E

, p)
D - (D) - (E
- E

- E

)
S
S
S
-

vp
vp
+
+
+

vp
R Y
() = R (, E

, p
) = E (,
E

, p)

=
E

- m
= 0
S
S
S
S
S
S
S
S


S

vp
has (, E

, +
, +
, p+)
S
S
S

- H (+
, +
, +
, p+)
S
S
S
S
S
vp
N G (,
E

, +
, +
, p+)

S
S
S
S

N - G (+
, +
, +
, p+)

S
S
S
S
S
S

vp

p (, E

, +
, +
, p+)

S
S
S



p
- F (+
, +
, +
, p+)
S
S
S
S
S


+
+
=: m
S
S

That is to say thus to evaluate the terms of the hypermatrice jacobienne J at the moment T + T


S
S
S
S
S

vp

E


p
S
S
S
E
E
E
E
E
vp
E


p
S
S


S
has
has
has
has
has
J =

vp
E


p
S
S
S
G
G
G
G
G

vp

E


p
S
S
S
p
p
p
p
p
vp
E


p
S
S


S

With regard to the first line of the matrix, independently of the equations of work hardening and
of flow, one a:

S




1
= -
S
D
=
S
Id
=
S = S = 0



vp
E


p
S
S
S

The second line can be written also independently of the flow and work hardenings:
E = E
E
E
E
0
= Id
= 0
= - m
vp
S
= 0

E


p
S
S
S

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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
: 22/24

The first column of the lines corresponding to the equations (A), (G) and (p) is written:

has

has

=
S



S
G

G

=
S



S
p

p

=
S



S
with

S = (m) Ts


The second column is identically null (because of equation (E): relations of flow and
of work hardening can express itself only according to
and not of
vp
E

.
S



The last block of equations, depends as for him on the selected behaviors:

has
has
has


p

S
S
S
G
G
G


p

S
S
S
p
p
p


p

S
S
S


Example

Let us choose the viscoplastic relation of flow ECOU_VISC1

C
S -
(G)
p
S -
S
S
= 0

C
S -
S


C
R p
S -
S -
() N
(p) p
T
S -.
S
S
= 0
K

NR
with isotropic work hardening
- LP
ECRO_ISOT1:
R
R (p) R
Q (
H 1
(
E
)) , H = 1
(-) +
S
S
= 0 + Sr -
Sr
Sr
Sr
r=1
and a kinematic work hardening defined by ECRO_CINE1
(A)

D p

S - S +
S S = 0

then:
has
= 0

S
G
= 0

S
p

- N T

- 1 - C
=
- C - R p
N
S
S
S (S) N
S


K
- C
S
S
S
Handbook of Référence
R5.03 booklet: Nonlinear mechanics
HT-66/05/002/A


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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has =
1+ D p
S

S
G
=
0

S
p
nc T

- 1 - C
=
- C - R p
N
S
S
S (S) N
S
S


K
- C
S
S
S
has


= - 1

S
G

= 1

S
p

= 0

S
has


=
ds
p
S
G

- C
S
S
=
p

- C
S
S
S
p

N T


- 1 Dr. p
= 1+
- C - R p
N
S
S
S (S) N
S (S)
p

K
D p

S
S
Dr. p
S (S)
- bps
= Qbh E
S
D p
S

and, concerning the interaction between systems of slip, it there only one term not no one:

p

N T

- 1 Dr. p
=1+
- C - R p
N
S
S
S (S) N
S (S)
p

K
D p

R
R

Dr. p
S (S)
- bpr
= Qbh E
Sr
D p
R
Handbook of Référence
R5.03 booklet: Nonlinear mechanics
HT-66/05/002/A

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
: 24/24

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 (R (Y) = 0) in end
of increment. For a small variation of R, by regarding this time as variable and not
parameter, one obtains:

R







+ R

+
R
E

R
R
R
vp
E +



p

vp
S +
S +
S = 0



E

E



p
S
S
S

This system can be written:




E

vp


E
0
R
(Y) = X, avecY = etX

S
= 0
Y





S
0
p
0
S



By writing the matrix jacobienne in the form:

Y
1
0
0
[] []

J.Y
= []
0
[]

vp

1
Y

1 E

Y
0
2
[] Y
3
Z



With:
Y = D-1
0
S



Z
= ×n
S
S
p
S

While operating by successive eliminations and substitutions, the third block of the system of equation gives:

Z
= - (Y3) - 1Y2
Evp

= Y
-
Z
1
= Y1 (Y3) - 1Y2
(Y0 +Y1 (Y3) - 1Y2) = E


the required tangent operator can thus be written directly:








=


= (
1
-
-
Y - Y Y Y

0
1 3
) 1
2
E


T
T

+

E t+t

Handbook of Référence
R5.03 booklet: Nonlinear mechanics
HT-66/05/002/A

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