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Titrate:
Élasto-viscoplastic relation of behavior polycrystalline CFC
Date:
22/06/01
Author (S):
S. TAHERI,
E. LORENTZ
Key:
R5.03.13-A
Page:
1/12
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HI-75/01/001/A
Organization (S):
EDF/MTI/MN
Manual of Reference
R5.03 booklet: Nonlinear mechanics
R5.03.13 document
Élasto-viscoplastic relation of behavior
for cubic polycrystalline materials with faces
centered
Summary:
A polycrystalline model was developed with the Center of Materials of the School of the Mines of Paris to describe it
élasto-viscoplastic cubic polycrystalline material behavior with centered faces. It is about a model
with great number of variables intern which allows, thanks to a simplified description of the microstructure (texture
crystallographic, systems of slip), to describe many phenomena observed on these materials
(examples: Bauschinger effect, anisotropy of surface of plasticity).
It is introduced into Code_Aster under the name of
POLY_CFC
on the level of the control
DEFI_MATERIAU
;
behavior is integrated by an explicit diagram of Runge-Kutta of command 2 into adaptive pitch.
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Code_Aster
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Version
5.0
Titrate:
Élasto-viscoplastic relation of behavior polycrystalline CFC
Date:
22/06/01
Author (S):
S. TAHERI,
E. LORENTZ
Key:
R5.03.13-A
Page:
2/12
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R5.03 booklet: Nonlinear mechanics
HI-75/01/001/A
Contents
1 Introduction ............................................................................................................................................ 3
2 Formulation of the model .......................................................................................................................... 4
2.1 Behavior on a grain scale .............................................................................................. 6
2.1.1 Definitions on the level of the systems of slip ................................................................ 6
2.1.2 Law of behavior on a grain scale ........................................................................... 6
2.2 Relation of scaling .................................................................................................. 7
2.3 Equations of the model ....................................................................................................................... 8
3 Implementation numerical in Code_Aster .................................................................................... 9
3.1 Calculation of the macroscopic stress ........................................................................................... 9
3.2 Evolution of the internal variables and the variables of localization .................................................. 10
4 Synthesis .............................................................................................................................................. 10
5 Bibliography ........................................................................................................................................ 11
Appendix 1 Introduction of information textures ........................................................................................ 12
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Code_Aster
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Titrate:
Élasto-viscoplastic relation of behavior polycrystalline CFC
Date:
22/06/01
Author (S):
S. TAHERI,
E. LORENTZ
Key:
R5.03.13-A
Page:
3/12
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R5.03 booklet: Nonlinear mechanics
HI-75/01/001/A
1 Introduction
Many search related during the twenty last years to a modeling
phenomenologic of the behavior of materials. The model suggested is, on the other hand, founded on
a polycrystalline approach which integrates microstructural information and offers capacities of
modeling extended compared to the phenomenologic approaches. Moreover, by comparison with
the phenomenologic models using a criterion of Von Mises, the micromechanical approach allows
to have a criterion of plasticity which can present angular parts.
The basic idea of crystallographic modelings of plasticity is as follows:
the introduction of the variables attached to the physical mechanisms is beneficial for the capacities of
modeling of the corresponding relations of behavior. Indeed, if the physical aspect of
mechanisms of deformation is well represented in the model, the predictive character of the equations
constitutive is improved, including for complex loadings located out of the field
of identification initial of the studied model, since the mechanical behavior of material is associated
its microstructure.
The model is designed to account for the mechanical polycrystalline material behavior of
cubic structure with centered faces.
The model is introduced into Code_Aster in 3D, plane deformations (
D_PLAN
), forced plane
(
C_PLAN
) and axisymetry (
AXIS
) under the name of
POLY_CFC
in
DEFI_MATERIAU
. It is about one
viscoplastic polycrystalline model with great number of internal variables, namely 1688 per point
of integration. The taking into account of the microstructure is carried out by the introduction of texture.
texture includes/understands the directions of orientation and the voluminal fraction associated each orientation
grains which make the microstructure.
One presents in this note the equations constitutive of the model and his introduction into
Code_Aster.
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Titrate:
Élasto-viscoplastic relation of behavior polycrystalline CFC
Date:
22/06/01
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E. LORENTZ
Key:
R5.03.13-A
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2
Formulation of the model
It is about a élasto-viscoplastic model which lies within the scope of the micromechanical approaches
with taking into account of microstructural information to modelize the elementary mechanisms of
inelastic deformation. The general step is to build a law of behavior
macroscopic starting from a simplified description of the microstructure. The method consists then with
to describe the microstructure (i.e. on a local scale, to identify the phases and to characterize them
behavior) and to express the total sizes like the averages of the local sizes.
The elementary ladders and mechanisms to consider are in the following way selected
[Figure 2-a]:
·
an element of elementary volume representative of material, for the scale of arrival to which
the process of modeling must succeed considered, which makes it possible to describe the behavior with
the macroscopic scale,
·
the grain, for the starting element of the scaling, for which the relation of
behavior developed is associated the mechanism of deformation retained in this
modeling, i.e. crystallographic slip, without consideration concerning
morphology and the space distribution of the grains of elementary volume (the grain is defined by
an orientation).
Orientation G
Scale of the élém
of volume
Scale of the grain
“Grain” 1
“Grain” N
“Grain” G
,
,
Appear 2-a: Scales considered for polycrystalline modeling
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Titrate:
Élasto-viscoplastic relation of behavior polycrystalline CFC
Date:
22/06/01
Author (S):
S. TAHERI,
E. LORENTZ
Key:
R5.03.13-A
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The model is based on a self-coherent approach. One replaces the detailed analysis of the interactions there
mechanics between the various phases by an approximate evaluation which consists in considering
successively interactions between each phase (grains having even orientation of network
crystalline lens are indistinguishable and constitute a phase), the whole of these phases being gathered in
an inclusion, and a fictitious homogeneous medium, which constitutes the matrix. The character of autocoherence is
ensured by the fact that the mechanical behavior of this fictitious medium is precisely that of the medium
homogeneous equivalent in the studied heterogeneous medium. The model was defined and developed in
work [bib1], [bib2], [bib3] and [bib4].
The modeling of the behavior of the phases is done via internal variables which go
to describe the work hardening of material in the grains, where plasticity is given by the law of Schmidt to
level of the systems of slip.
The sizes present on a grain scale are the inelastic deformation, the generated stress
in the grain and a certain number of variables intern associated with each grain. With the scale
macroscopic of the element of volume, one has of the total deflection, the deformation
inelastic, of the stress (the variables intern are associated the grains which constitute the element
of volume).
The presence of two scales of modeling consequently requires to define a stage of
scaling, which is the essential element of this type of modeling and by which the different ones
polycrystalline models are distinguished.
Notations:
E E E
E
ij
ije
ijth
ijvp
,
,
,
Deflections total, elastic, thermal and viscoplastic.
ij
vp
G
Viscoplastic deformation in the grain G.
F
G
Voluminal fraction of matter constituting the grain G.
!
S
,
Speed of slip of the system S.
!p
S
,
Intensity the speed of slip of the system S.
m N L
ijs
I
I
,
,
Tensor of orientation and unit vectors corresponding respectively to
normal in the plan which slips and the direction of slip.
F
S
Function threshold.
X
S
Kinematic variable of work hardening.
R
S
Isotropic variable of work hardening.
S
G
,



ij
Cission solved on the system S and forced in the grain G.
S
S
S
Q
Q
,
,
1
2
Variables associated with the systems with slip S for the description of
work hardening in the grains.
Q
Q
S
S
1
2
and
represent the interaction between
systems of slip (dislocation).
H
rs
Components of the matrix of interaction enters dislocations (
H
rs
= 1 on
diagonal to take into account a car-work hardening when systems
are identical).
ij
Macroscopic stress.
B
ij
ijg
,
Variables of localization, sizes comparable to deformations (on the scale
macroscopic and on a grain scale).
()
J
ij
2
(
)
3
2
2
Dev.
ijd
ij
or
ijd
is the deviatoric part of
ij
.
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Titrate:
Élasto-viscoplastic relation of behavior polycrystalline CFC
Date:
22/06/01
Author (S):
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E. LORENTZ
Key:
R5.03.13-A
Page:
6/12
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R5.03 booklet: Nonlinear mechanics
HI-75/01/001/A
2.1
Behavior on a grain scale
2.1.1 Definitions on the level of the systems of slip
The mechanism of deformation considered is the crystallographic slip. It is thus advisable to define
the mechanical behavior related to this mechanism where it is supposed that a system of slip S is
credit when its solved cission



S
reached a breaking value. For a cubic material with faces
centered, there are 12 systems of slip which are defined by their tensor of orientation
m
S
to leave
crystallographic definitions of the direction of slip (defined by the unit vector
L
) and of
the normal in the plan which slips (definite by the unit vector
N
).



S
G
=
ij
ijs
: m
m
N
L
L
N
ijs
I
J
J
I
=
+
1
2
(
)
The speed of deformation viscoplastic
!
vp
G
is defined starting from the knowledge speeds of
slip
!
S
for all the systems of slip:
!
!
ij
vp
S
S G
G
=
m
ijs
2.1.2 Law of behavior on a grain scale
The variables intern local behavior noted
S
,
S
and
p
S
allow to define the evolution
work hardening and to calculate the speed of deformation
!
vp
G
in all the grains.
S
is associated
kinematic work hardening and
p
S
is the cumulated viscoplastic slip.
The viscoplastic formulation retained in this modeling proposes a function power (by
analogy with the law of Norton in creep) to define the intensity
!p
S
speed of slip and
allows to have access to derived from the internal variables starting from the knowledge of the stress
G
and of the internal values of variables:
!
| ! |
F
p
K
S
S
S N
=
=
with <x>=0 if x<0 and <x>=x if x>0
where
K
and
N
are parameters characteristic of material and temperature.
F
S
depends on the initial threshold
0
of flow on the system of slip
S
solved cission
S
and
of two variables of work hardening
X
S
and
R
S
. The kinematic variable of work hardening
X
S
allows
to take account of local heterogeneities in the grains due to the development of dislocations.
isotropic variable of work hardening
R
S
who accounts for the interaction between dislocations (by
the intermediate of a matrix of interaction of components
H
rs
) can present a value of saturation
work hardening.
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Titrate:
Élasto-viscoplastic relation of behavior polycrystalline CFC
Date:
22/06/01
Author (S):
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E. LORENTZ
Key:
R5.03.13-A
Page:
7/12
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R5.03 booklet: Nonlinear mechanics
HI-75/01/001/A
The criterion of flow corresponding to the equipotential ones in viscoplasticity is written:
F
|
X |
R
()
S
S
S
S
C D S
X
=
=
=
=
-
-
-
-
-
-
+
0
1
2
2
where
0
,
C
and
D
are parameters characteristic of material and temperature.
The equations of state of the variables are as follows:
X
S
S
S
C
X
=
+
=
+
S
S
has
has
R
S
rs
R S
S
S
H Q
Q
=
Q
Q
1
2




+
1
2
H
rs
rs
rs
=
+
-
H (1
)
with
rs
= 0 if R
S and
rs
= 1 if R = S
where
C
,
has
,
Q
1
,
Q
2
and
H
are parameters characteristic of material and temperature.
It remains to define the laws of evolution of the variables of work hardening:
!
!
!






S
S
p
=
-
S
S
D
!
(
!
Q
B
Q
p
is
I
is
S
=
)
1
-
(I = 1,2)
where
D
,
B
1
and
B
2
are parameters characteristic of material and temperature.
2.2
Relation of scaling
It is a question of modelizing the internal stresses and heterogeneities of deformation of a phase to one
other to have access to the total sizes. It is noted that the total deflection in each grain
is never calculated. One is interested in the viscoplastic deformation of the phases, which makes it possible to define
viscoplastic deformation of elementary volume representative of material from
voluminal weighting of the fractions of the phases
F
G
:
E
ij
vp
G
ij
vp
G
F
G
=



Two relations of localization are programmed (according to the presence or not coefficients
D
and
in [éq 2.2-2]). The difficulty to justify, on macroscopic data of behavior,
relevance of the rule of scaling is avoided while having a relation [bib5] which can
to be used as reference because it validates the character of autocoherence. This validation is effective under
certain conditions, namely: isotropy of material, homogeneous elastic behavior and
monotonous loading:
()



ijg
ij
ij
vp
ij
vp
ijvp
ij
G
E
J
=
+
-




= +
µ
µ
E
1
1 32
2
éq 2.2-1
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The second relation, developed more particularly for cyclic loadings [bib6] allows
to give a good description to schematize the interactions between the grains:






µ
µ
µ
µ






ijg
ij
ij
ijg
ij
G ijg
G
F
=
+
-
=
(
)
B
B
éq 2.2-2
!
!
!
ijg
ij
vp
ijg
ij
vp
ij
vp
G
G
G
D
=
- (
-
)||
||
where
D
and
are parameters characteristic of material and temperature.
2.3
Equations of the model
Total behavior:
E
E
E
E
E
ij
ij
E
ij
HT
ij
vp
ij
vp
G ij
vp
G
F
G
=
+
+
=



Intragranular behavior:
(
)
!
!
ij
vp
S
S G
ijs
I
J
J
I
G
=
=
+
m
ijs
m
N
L
L
N
1
2
!
| ! |
F
p
K
S
S
S N
=
=
with <x>=0 if x<0 and <x>=x if x>0
Criterion:
F
|
X |
R
(X)
S
S
S
S
C D S
=
=
=
=
-
-
-
-
-
-
+
0
1
2
2
S
G
=
ij
ijs
: m
X
S
S
S
S
S
C
X
=
+
=
+
has
has
!
!
!
S
S
S S
p
=
-
D
R
S
rs
R S
S
S
H Q
Q
=
Q
Q
1
2




+
1
2
!
(
!
Q
B
Q
p
is
I
is
S
=
)
1
-
(I = 1,2)
H
rs
rs
rs
= 1
+
H (
)
with
rs
= 1 if R = S and
rs
= 0 if R
S
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Titrate:
Élasto-viscoplastic relation of behavior polycrystalline CFC
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Relations of scaling (accessible according to the presence from the parameters
D
and
):



ijg
ij
ijvp
ij
vp
G
=
+
-
µ
(
)
E
(
D
and
miss command file [bib5])
1
1 32
2
µ
=
||
+
ij
vp
ij
E ||
J (
)












ijg
ij
ij
ijg
=
+
-
µ
(
)
B
(
D
and
are defined [bib6])
B
ij
G
ijg
G
F
=



!
!
!















ijg
ij
vp
ijg
ij
vp
ij
vp
G
G
G
D
=
- (
-
)||
||
In Code_Aster, the whole of the parameters of the model
D
N K Q
H Q
C D has
,
B
,
B
,
1
2
,
,
,
1
2
can be a function of the temperature.
3
Implementation numerical in Code_Aster
The general outline adopted with the polycrystalline behavior to integrate the relation of
behavior is a method of Runge Kutta to adaptive pitch [bib7]. The programmed diagram is
of command 2.
From the state of deformation
E ()
T
, of the knowledge of the internal variables
(
)
y T
p
S
S
S
()
,
,
=
and of the variables of localization



G
T
()
at the moment T, one seeks to find, at the moment t+
T the stress
macroscopic



to represent balance. From the increment of total deflection proposed with
the total stage
E E
E
= (
)
()
T
T
T
+
-
and of an assumption of evolution of this linear deformation in
time, one explicitly integrates the differential equations of the behavior by controlling the precision
by a method with variable pitch.
One thus knows
E
vp
T
T
(
)
+
, which makes it possible to calculate



(
)
T
T
+
as specified below.
3.1
Calculation of the macroscopic stress
The stress is calculated



starting from the viscoplastic deformation
E
vp
. The aforementioned corresponds to
first size arranged in the vector which constitutes the whole of the variables of the model
y
.



ij
ij
ijth
ijvp
C
=
-
-
(
)
E
E
E
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Élasto-viscoplastic relation of behavior polycrystalline CFC
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3.2
Evolution of the internal variables and the variables of localization
Knowing the cission
S
(calculated starting from the stress
G
and of the tensors of orientation for
all systems of slip) and variables internal
y
in the grains, one can calculate them
derived from the variables
!y
starting from the writing clarifies equations of the model and to have access to
speeds of viscoplastic deformation in the grains
!
vp
G
.
Variables characteristic of the relation of localization used to describe a behavior under
cyclic stresses are accessible for all the grains starting from information from deformation and
of speed [éq 2.2-1].
4 Synthesis
The model is accessible in Code_Aster in 3D, plane deformations (
D_PLAN
), forced
plane (
C_PLAN
) and axisymetry (
AXIS
) starting from the key word
COMP_INCR
control
STAT_NON_LINE
. The whole of the parameters of the model is provided under the key word factor
POLY_CFC
or
POLY_CFC_FO
control
DEFI_MATERIAU
[U4.23.01].
/
POCY_CFC:
(
DLL
:
D
DA
:
NR
:
N
K
:
kMPa
TAU_0
:
0
MPa
Q1
:
Q
1
MPa
B1
:
B
1
HL
:
H
Q2
:
Q
2
MPa
B2
:
B
2
C1
:
C MPa
D1
:
D (MPa)
3
C2
:
MPa has
)
Parameters
D
and
are optional. When they are present, the rule of change
of scale used [bib6] is that of the equation [éq 2.2-2]. If not, one uses the rule [bib5] of
the equation [éq 2.2-1] (valid only under monotonous loading); cf [§2.2].
The selection of the diagram of integration of the relation of behavior is done by the option
RUNGE_KUTTA_2
operand
RESO_INTE
starting from the key word
NEWTON
control
STAT_NON_LINE
.
The method does not provide a tangent matrix; one thus uses the elastic matrix for
total balance what implies that the user must force himself to define a cutting of
no relatively fine time, so that balance at the total level is facilitated.
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Élasto-viscoplastic relation of behavior polycrystalline CFC
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The significant number of internal variables used in this modeling implies to store them
that at the last moment of calculation to allow the recoveries. The option then is used
CHAM_EXCLU
key word factor
FILING
control
STAT_NON_LINE
with the only possible argument
“VARI”
. The variables intern are then backed up only with the last pitch of time.
5 Bibliography
[1]
CAILLETAUD G.: “A phenomenologic micromechanical approach of the behavior
inelastic of metals ", Thesis of Doctorate of State of the University Paris 6, 1987.
[2]
PILVIN P.: “Multiéchelles Approaches for the anelastic forecast of the behavior of
metals ", Thesis of Doctorate of State of the University Paris 6, 1990.
[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.
[6]
PILVIN P.: “Contribution of the digital simulation to the development of relations of
behavior in Mechanics of Materials ", Memory of accreditation to supervise resear
university Paris 6, 1995.
[7]
CROUZEIX Mr., MIGNOT A.L.: “Numerical Analysis of the differential equations”, Masson,
1989.
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Élasto-viscoplastic relation of behavior polycrystalline CFC
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Page:
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HI-75/01/001/A
Appendix 1 Introduction of information textures
The texture of material intervenes in this model like a data of description provided once and for all
in
DEFI_MATERIAU
, as well as the parameters material. The aforementioned makes it possible to calculate the tensors
of orientation
m
S
while carrying out a change of reference mark the reference mark of material enters (reference mark of the laboratory) and
the crystallographic reference mark (the local reference mark) and also to have access to the voluminal fractions
F
G
of
material which presents such or such crystallographic orientation.
This texture is thus defined in the following way:
1
2
,
, F
G
where
1
,
,
2
are the angles of Euler.
One lays out by defect, of a description of texture distributed in an isotropic way on 40 different orientations
for
F
G
= 1/40.
One can also introduce the results of texture obtained starting from an experimental study (diffraction of
x-rays of the crystallographic plans of the cubic structure with centered faces). One then presents them under
form of a succession of 3 angles (angles of Euler which correspond to the setting in coincidence of the reference mark related to
the sample and of that related to the crystal) associated a voluminal fraction. If one follows 40 orientations for this
modeling, the file of texture comprises 40 lines then corresponding to the 40 orientations more
represented in the matter sample analyzed.
For the modeling introduced into Code_Aster, one does not take account of the possible evolution of texture
with work hardening (assumptions of the small disturbances), which makes it possible to regard as fixed parameters
voluminal fractions
F
G
and components of the tensors
m
S
(6 components), definite for each grain
(each orientation) and for the 12 main systems of slip of structure CFC (40 * 12 * 6=2880
components).