Code_Aster
®
Version
6.3
Titrate:
Modeling élasto- (visco) plastic with metallurgical transformations
Date:
29/04/02
Author (S):
A. RAZAKANAIVO, A.M. DONORE, F. WAECKEL
Key
:
R4.04.02-E
Page
:
1/36
Manual of Reference
R4.04 booklet: Metallurgical behavior
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Organization (S):
EDF/MTI/MN
Manual of Reference
R4.04 booklet: Metallurgical behavior
Document: R4.04.02
Modeling élasto- (visco) plastic fascinating
in account of the metallurgical transformations
Summary:
This document presents the modeling installation in Code_Aster for the mechanical analysis
operations generating of the metallurgical transformations. The various mechanical effects are presented
resulting from structure transformations to take into account and their modelings.
Code_Aster
®
Version
6.3
Titrate:
Modeling élasto- (visco) plastic with metallurgical transformations
Date:
29/04/02
Author (S):
A. RAZAKANAIVO, A.M. DONORE, F. WAECKEL
Key
:
R4.04.02-E
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:
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R4.04 booklet: Metallurgical behavior
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Count
matters
Code_Aster
®
Version
6.3
Titrate:
Modeling élasto- (visco) plastic with metallurgical transformations
Date:
29/04/02
Author (S):
A. RAZAKANAIVO, A.M. DONORE, F. WAECKEL
Key
:
R4.04.02-E
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:
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R4.04 booklet: Metallurgical behavior
HI-75/01/001/A
1 Introduction
Certain materials undergo structure transformations when they are subjected to evolutions
thermics particular [bib1], [bib2], [bib3]. It is for example the case of the low alloy steels with
run of operations of the welding type and heat treatment or the alloys of zircaloy of
fuel sheaths for certain cases of accidental situation (APRP).
These transformations have a more or less strong influence on the thermal evolutions and
mechanics.
From a thermal point of view, structure transformations are accompanied by an amendment by
thermal characteristics (voluminal heat-storage capacity, thermal conductivity) of the material which
the sudden one, as well as production or of an energy absorption (latent heats of
transformation) [bib2]. However, the latent heats of transformation in a solid state are
relatively weak compared with the latent heats of change of state fluid-solid and one can
therefore, at first approximation, to regard the thermal and structural evolutions as
uncoupled. C `is currently the case of the established options of thermal and metallurgical calculations
in Code_Aster. [bib16]
From a mechanical point of view, the consequences of structure transformations (at the solid state) are
of four types [bib2]:
·
the mechanical characteristics of the material which undergoes them are modified. More precisely,
the elastic characteristics (YOUNG modulus and Poisson's ratio) are little
affected whereas plastic characteristics (elastic limit in particular) and it
thermal expansion factor are it strongly,
·
the expansion or the voluminal contraction which accompanies structure transformations
translated by a deformation (spherical) “of transformation” which is superimposed on the deformation
of purely thermal origin. This effect is highlighted on a test of dilatometry and,
in general, one gathers it with that due to the amendment of the one and expansion factor
speak overall about the influence of the transformations on the thermal deformation,
·
a transformation proceeding under stresses can give rise to a deformation
irreversible and this, even for levels of stresses much lower than the elastic limit
material (at the temperature and in the structural state considered). One calls “plasticity of
transformation " this phenomenon,
·
one can have at the time of the metallurgical transformation a phenomenon of restoration
of work hardening. The work hardening of the mother phase is not transmitted to the phases lately
created. Those can then be born with a virgin state of work hardening or only inherit
of a part, possibly of totality, work hardening of the mother phase.
In addition, the mechanical state also influences the metallurgical behavior. The state of
stresses can in particular accelerate or slow down the kinetics of the transformations and modify them
temperatures to which they occur. However, the experimental characterization of this
influence, in particular in the case of complex situations (three-dimensional, under temperature and
state of variable stresses) remains very delicate and it is very frequent to consider the evolution
structural like independent of the mechanical state. C `is the case of the model of transformations
structural established in Code_Aster.
If one neglects the various couplings of mechanical origin, the determination of the mechanical evolution
associated a process bringing into play structure transformations thus requires two calculations
successive and uncoupled:
·
a metallurgical thermo calculation (uncoupled) allowing the determination of the evolutions
thermics then structural,
·
a mechanical calculation (élasto-viscoplastic) taking account of the effects due to the evolutions
thermics and structural.
Code_Aster
®
Version
6.3
Titrate:
Modeling élasto- (visco) plastic with metallurgical transformations
Date:
29/04/02
Author (S):
A. RAZAKANAIVO, A.M. DONORE, F. WAECKEL
Key
:
R4.04.02-E
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:
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Manual of Reference
R4.04 booklet: Metallurgical behavior
HI-75/01/001/A
This document presents the mechanical modeling established in Code_Aster. Modeling is
available for two materials:
·
the steel which undergoes around 850° a austénito-ferritic transformation (passage of
phases
cold of cubic structure face centered (CFC) with a phase
hot of
centered cubic structure (DC)). Steel presents 4 possible ferritic phases; ferrite,
pearlite, the bainite and martensite,
·
the alloys of Zircaloy which undergo around 800°C a transformation of phase with
cold
of hexagonal structure compacts with a hot phase
of structure DC.
The models are identical for two materials, only the number of phase changes.
The model thus comprises 5 phases for steel and 3 phases for the zircaloy. The modeling of
behavior of the zircaloy indeed requires to consider 2 cold phases of behavior
mechanics different; a phase
regarded as pure and a phase
mixed with
[bib16], [bib17]. The various characteristics relating to the various phases are noted:
Zircaloy steels
Ferrite: F1_ ***
Pearlite: F2_ ***
Bainite: F3_ ***
Martensite: F4_ ***
Austenite: C_ ***
Alpha pure: F1_ ***
Mixed alpha: F2_ ***
Beta: C_ ***
Note Bucket:
Metallurgical concepts of bases necessary to the comprehension of the general problem
are gathered in [bib1].
The elastoplastic algorithm of resolution, without taking into account of the effects due to
structure transformations is clarified into [bib4].
This document to some extent is extracted from [bib5] and [bib14] where one makes a presentation more
detailed model and of some elements of validation.
The presentation of the models which one makes in this document is mainly illustrated with the case
steel.
Code_Aster
®
Version
6.3
Titrate:
Modeling élasto- (visco) plastic with metallurgical transformations
Date:
29/04/02
Author (S):
A. RAZAKANAIVO, A.M. DONORE, F. WAECKEL
Key
:
R4.04.02-E
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:
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2 Influence of structure transformations on
thermal deformation
A test of dilatometry consists in measuring the deformation (homogeneous) of a test-tube of small
dimension according to the temperature (or of time) at the time of an imposed thermal cycle (supposed
identical in all the points of the test-tube). One presents [Figure 2-a] a test of dilatometry of one
steel. The thermal cycle comprises a heating beyond the temperature of austenitization (either
850°C approximately), then a hold at this temperature and, finally, a cooling controlled until
ambient temperature. One then obtains an evolution of the deformation (variable according to the kinetics of
cooling imposed) as represented on the figure.
T
With
B
C
D
E
F
G
H
T
0
Tref
F
Appear diagrammatic 2-a: Raised of dilatometry
2.1
Areas of thermal deformation
The various areas highlighted on the figure [Figure 2-a] can be interpreted as follows:
A-B:
thermal dilation of metal in its initial metallurgical structure (of type
ferrito-perlitic
(
)
F
P
+
, bainitic
()
B
and/or martensitic
()
M
) until
initial temperature of austenitization T (B),
B-C:
austenitization and contraction of the test-tube (volume specific of the phase
austenitic
()
smaller),
CD:
thermal dilation of austenite (with an expansion factor different from that
phases known as “
“
()
F
,
()
P
,
()
B
,
()
M
),
OF:
thermal contraction of austenite,
E-F:
first transformation (partial) of austenite (for example
+
F
P
) which
be accompanied by a voluminal expansion,
F-G:
area without transformation with thermal contraction of remaining the austenite mixture -
formed phase (with a certain thermal expansion factor apparent),
G-H:
second transformation of remaining austenite (for example
M
) which
be accompanied by a voluminal expansion,
H-A:
thermal contraction of the final structure (with the same expansion factor
that with the heating).
Code_Aster
®
Version
6.3
Titrate:
Modeling élasto- (visco) plastic with metallurgical transformations
Date:
29/04/02
Author (S):
A. RAZAKANAIVO, A.M. DONORE, F. WAECKEL
Key
:
R4.04.02-E
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:
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R4.04 booklet: Metallurgical behavior
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2.2
Assumptions and notations
·
The structures ferritic, perlitic, bainitic and martensitic have a coefficient of
identical thermal dilation (noted
F
) different from that of austenite (noted
).
One defines a state of reference for which one considers that the thermal deformation is null: one
for that a metallurgical phase of reference (phase austenitic or ferritic phase) and one choose
temperature of reference
T
ref.
.
·
That is to say
HT
thermal deformation of the austenitic phase, and
fth
thermal deformation
phases ferritic, perlitic, bainitic and martensitic, we will take:
(
)
(
)
(
)
HT
=
-
- -
=
-
+
()
()
F
HT
T T
T
Z
T T
T
Z
ref.
R
F
T
F
ref.
R
F
T
ref.
ref.
1
where:
T
ref.
:
Temperature of reference,
()
T
: average expansion factor of the austenitic phase at the current temperature
T, compared to the temperature of reference.
F
T
()
: expansion factor average of the phases ferritic, perlitic, bainitic and
martensitic at the current temperature
T
, compared to the temperature of
reference.
Z
R
:
characterize the metallurgical phase of reference;
Z
R
= 1 when the phase of reference is the austenitic phase,
Z
R
= 0 when the phase of reference is the ferritic phase.
()
()
ref.
ref.
HT
F
T
F
T
T
ref.
HT
-
=
translated the difference in compactness between the structures
crystallographic cubic with centered faces (austenite) and cubic centered (ferrite) with
temperature of reference
T
ref.
.
That is to say
(
)
{
}
Z M T
Z Z Z Z
,
,
,
,
=
1
2
3
4
respective proportions of ferrite, pearlite, bainite and
martensite present in a material point
M
at the moment
T
. With the help of the assumption of a law
of mixture to define the thermal deformation of a multiphase mixture (characterized by
Z
) one a:
(
)
(
)
(
)
(
)
HT
,
.
Z
T
Z
T T
Z
Z
T
T
Z
I
I
I
ref.
R
F
T
I
I
I
F
ref.
R
F
T
ref.
ref.
=
-
-
- -
+
-
+
=
=
=
=
1
1
1
4
1
4
Code_Aster
®
Version
6.3
Titrate:
Modeling élasto- (visco) plastic with metallurgical transformations
Date:
29/04/02
Author (S):
A. RAZAKANAIVO, A.M. DONORE, F. WAECKEL
Key
:
R4.04.02-E
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:
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R4.04 booklet: Metallurgical behavior
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For the calculation of the thermal deformation it is thus necessary to be given:
·
the expansion factor of the cold phases,
·
the expansion factor of the hot phase,
·
a metallurgical phase of reference and a temperature of reference,
·
the difference in compactness between the hot and cold phase at the temperature of reference.
These data are provided by the user in the operator
DEFI_MATERIAU
[U4.23.01] under the word
key
ELAS_META_FO
except the temperature of reference which one defines in
AFFE_MATERIAU
.
depend on the temperature and are calculated for the temperature of the point of current Gauss.
3
Plasticity of transformation
In experiments, it is noted that the dilatometric statement of a test-tube in the course of
structure transformation is strongly influenced by the state of stresses and that the application of one
stress even lower than the elastic limit of material can nevertheless cause one
unrecoverable deformation (cf [Figure 3-a]).
T
Transformation
bainitic
= 0
MPa
= -
42 M
AP
= -
85 M
AP
Pt
Application of
the stress
Pt
Appear 3-a: swelling Behaviors under uniaxial stresses
of compression for a steel 16 MND5
One calls plasticity of transformation this phenomenon and one notes
Pt
unrecoverable deformation
corresponding.
The model of plasticity of transformation most frequently used is, at the origin, generalization
three-dimensional of the unidimensional phenomenologic model established by DESALOS [bib12]. If, with
to start from a dilatometric test, one traces the difference between lengthening
obtained for a stress
applied different from zero and that obtained for a null stress according to the advance of
the transformation, one notes that:
()
()
()
()
Pt
B
B
B
K
B
,
,
,
F
=
-
=
0
where:
K
is a homogeneous constant contrary to a stress,
F
is a standardized function (
()
F 0
0
=
and
()
F 1
1
=
),
and
B
is the proportion of the transformed phase.
Code_Aster
®
Version
6.3
Titrate:
Modeling élasto- (visco) plastic with metallurgical transformations
Date:
29/04/02
Author (S):
A. RAZAKANAIVO, A.M. DONORE, F. WAECKEL
Key
:
R4.04.02-E
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A three-dimensional and temporal generalization of the preceding experimental model, for only one
transformation, was proposed by LEBLOND [bib8], [bib9], [bib10], [bib11], in the form:
()
&
~ F'
&
I J
Pt
I J
K
B B
= 32
On the basis of experimental tests and for transformation of a bainitic type of a steel 16MND5
for example:
K
is taken equalizes with
10
4
1
-
-
MPa
and
()
(
)
F B
B
B
=
-
2
.
It is based on the following heuristic considerations:
·
the relation must be “incremental”, i.e. to connect the rate of plastic deformation to the rate
of transformation,
·
the speed of plastic deformation of transformation must be, as for plasticity
conventional, proportional to the deviatoric part
~
tensor forced
~
T R
=
-
1
3
I D
, (the plasticity of transformation occurs without change of
volume, from where a dependence compared to the diverter of the stresses rather than to the field of
stresses itself),
·
the rate of plastic deformation of transformation must be null apart from the ranges of
transformations,
·
the integration of this relation in the uniaxial case with constant stress
must give again
experimental relation.
The phenomenon of plasticity of transformation can exist at the time of structure transformations under
stresses of type the ferritic, perlitic, bainitic and martensitic, which possibly can
to appear simultaneously. On the other hand, it is considered that this phenomenon does not exist at the time of
austenitic transformation. The general model established in Code_Aster is thus:
(
)
(
)
&
,
&
,
~
F '
&
Pt
ipt
I
I
I
I
I
I
I
I
I
I
Z
K
Z
Z
Z
=
=
<
>
=
=
=
=
=
=
1
4
1
4
1
4
3
2
where:
<
>
X
indicate the positive part of a size.
Data
K
I
and
F
I
are provided by the user in
DEFI_MATERIAU
under the key word
META_PT
.
In Code_Aster it is possible not to take into account the phenomenon of plasticity of
transformation. If this phenomenon is taken into account, it appears as soon as there is
transformation and that even if the structure plasticizes. The model is more particularly dedicated to
steel.
Code_Aster
®
Version
6.3
Titrate:
Modeling élasto- (visco) plastic with metallurgical transformations
Date:
29/04/02
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A. RAZAKANAIVO, A.M. DONORE, F. WAECKEL
Key
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R4.04.02-E
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4 Restoration
of work hardening
In a usual way the state of work hardening of a phase I is characterized by its plastic history. Thus by
example in the case of plasticity with linear isotropic work hardening, one generally takes
like variable of work hardening noted cumulated plastic deformation
p
. The term of work hardening
is written then:
R
R p
I
I
=
0
where
R
I
0
is the linear coefficient of work hardening of the phase
I
.
At the time of the metallurgical transformations, there exists within material of displacements of atoms
more or less important. These displacements of atoms can destroy dislocations which are with
the origin of work hardening. In these cases, the work hardening of the mother phase is not transmitted to the phase
produced, it is the restoration of work hardening. The new phase can then be born with a plastic state
virgin or to inherit only one part, possibly totality, the work hardening of the mother phase.
Cumulated plastic deformation
p
is not characteristic any more of the state of work hardening and it is necessary to define
other variables of work hardening for each phase, noted
R
I
who take account of the restoration
of work hardening.
The term of work hardening of phase I is written then
R
R R
I
I I
=
0
.
4.1
Model with 2 phases with a direction of transformation
To define the variables
R
I
, one chooses the model suggested by LEBLOND [bib11].
One considers an element of two-phase volume V which undergoes a metallurgical transformation and one
plastic deformation.
Phase 1 is the mother phase characterized by
voluminal fraction V1,
proportion of phase (1 Z),
variable décrouissage 1
Phase 2 is the phase produced characterized by
voluminal fraction V2,
proportion of phase Z,
variable décrouissage 2
-
R
R
Equations of evolution of
R
I
obtained by derivation compared to time are written:
&
&
&
& &
&
R
p
R
p zz R zz R
1
2
2
1
=
= -
+
éq
4.1-1
characterize the proportion of work hardening transmitted of the mother phase to the produced phase.
&p
is the rate of equivalent plastic deformation.
Note:
p
here is not any more one internal variable of the problem as such. Only significance of
&p
is
here to be the plastic multiplier and it is equal to the rate of equivalent plastic deformation.
Code_Aster
®
Version
6.3
Titrate:
Modeling élasto- (visco) plastic with metallurgical transformations
Date:
29/04/02
Author (S):
A. RAZAKANAIVO, A.M. DONORE, F. WAECKEL
Key
:
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SJÖSTRÖM obtains the same equations by using a phenomenologic reasoning as one
defer here to clarify the model [bib13].
That is to say an increment of time
T, such as between T and t+t:
·
a fraction
V
2
mother phase is transformed into phase 2 and thus comes to be added with
volume V
2
of this phase produced,
·
the element of volume V undergoes a plastic deformation
p
.
2
1
2
1
V R T
V R T
T
p
V
V R T
p
V
R T
p
V R T
p
2 2
1 1
2
2 2
2
1
1 1
, ()
, ()
, ()
,
()
, ()
+
+
+
One supposes that at the time of the metallurgical transformation, the transformed fraction
V
2
inherit only one
part
R
1
work hardening of the mother phase
(0
1)
.
Then variables of work hardening
R
I
at the moment t+
T are such as:
R T
T
R T
p
R T
T
V R T
p
V
R T
p
V
V
1
1
2
2
2
2
1
2
2
(
)
()
(
)
(()
)
(
()
)
+
=
+
+
=
+
+
+
+
Maybe, by considering that
R T
T
R T
R
I
I
I
(
)
()
+
=
+
R
p
R
p
V
V
V
R
V
V
V R
1
2
2
2
2
1
2
2
2 2
=
=
+
+
-
+
-
-
éq
4.1-2
One obtains the equations [éq 4.1-1] while passing in extreme cases.
For the discretization of the laws of evolutions of
R
I
, one chooses a diagram of integration clarifies in
using directly the equations [éq 4.1-2].
Code_Aster
®
Version
6.3
Titrate:
Modeling élasto- (visco) plastic with metallurgical transformations
Date:
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Author (S):
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Key
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4.2
Generalization of the model with N phases with transformations with double
feel
In the case of steel the existing phases are: Ferrite, Pearlite, Bainite, Martensite and Austenite
, of
respective proportions
Z Z Z Z
Z
K
K
K
1
2
3
4
1
4
1
,
,
,
and
-
=
=
.
·
In the case of a cooling, the metallurgical transformations to consider are them
transformations of
()
() () ()
()
in
or
F
P
B
M
,
.
·
In the case of a heating one considers the transformations in the other direction:
() ()
F
P
,
() ()
()
B
M
,
in
.
One can thus write in a general case (where
X
indicate the positive part of
X
.
If
if not
If
if not
and
and
Z
R
p
Z
R
Z
R
Z
R
R
Z
R
p
Z
R
Z
R
Z
R
R
K
K
K
K
K
K
K
K
K
K
K
K
K
K
K
K
K
>
=
+
< -
>
-
< -
>
-
=
=
>
=
+ <
>
- <
>
=
=
=
-
-
=
=
-
-
-
-
-
-
0
1
0
0
0
0
0
1
4
1
4
1
4
éq
4.2-1
K
: proportion of restoration of work hardening at the time of the transformation
in
K
K
: proportion of restoration of work hardening at the time of the transformation
K
in
For transformations with dissemination (ex:
out of F, P, B) implying of important displacements
atoms one will be able to take
= 0; dislocations at the origin of plastic work hardening are
completely destroyed by the transformation. For transformations without dissemination
(ex: martensitic transformation), one will be able to take
= 1, work hardening being completely transmitted.
are provided by the user in the operator
DEFI_MATERIAU
under the key word
META_RE
.
Code_Aster
®
Version
6.3
Titrate:
Modeling élasto- (visco) plastic with metallurgical transformations
Date:
29/04/02
Author (S):
A. RAZAKANAIVO, A.M. DONORE, F. WAECKEL
Key
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5
Models of deformation (visco) plastic
The main characteristic of the thermal evolutions concerned in this type of analysis is
that they sweep a broad temperature range, which has an important effect on the behavior
mechanics of the material which undergoes the thermal evolution. One is in particular in fields of
temperature where the phenomena of viscosity can not be negligible more. It can thus be
necessary to use a élasto-viscoplastic model of behavior especially when one remains in
these fields for one important length of time; for example during the processing of detensioning
associated welding.
A viscoplastic model is thus chosen whose characteristics are such as it makes it possible to describe
with the same formalism, therefore without changing model:
·
a conventional plastic behavior; to modelize the cases at low temperature when
the viscous effects are still negligible or to modelize the processes at speed
raised (welding),
·
a hammer-hardenable viscoplastic behavior at high temperature, to modelize the effects
of creep and relieving associated for example with the processing with detensioning or with
multirun weldings,
·
a behavior of the fluid type viscous for the temperatures higher than the temperature
of fusion, in order to have a reasonable description of the molten area.
The selected viscoplastic model degenerates indeed for certain borderline cases in model of plasticity
independent of time, or in model of viscous fluid.
One places oneself here within the framework of the plasticity of von Mises with additive isotropic work hardening.
The use of a kinematic work hardening being also possible (version 6.1.6).
Function threshold:
F
R R T Z
T Z
eq
C
=
-
-
(; ,)
(,)
eq
equivalent stress of von Mises,
E Q
=
3
2
1 2
~: ~
/
(
)
R R T Z
; ,
:
isotropic term of work hardening,
()
C
T Z
,
:
initial critical stress; corresponds to the minimal stress
initial to apply to have a viscoplastic flow.
Plastic rate of flow:
&
&
&
~
vp
eq
F
p
=
= 32
Cumulated plastic deformation
&p
is viscous and is written:
&
(; ,)
(; )
p
R R T Z
T Z
eq
C
N
= <
-
-
>
éq
5-1
, N
: coefficients materials of viscosity.
Code_Aster
®
Version
6.3
Titrate:
Modeling élasto- (visco) plastic with metallurgical transformations
Date:
29/04/02
Author (S):
A. RAZAKANAIVO, A.M. DONORE, F. WAECKEL
Key
:
R4.04.02-E
Page
:
13/36
Manual of Reference
R4.04 booklet: Metallurgical behavior
HI-75/01/001/A
Note:
One can rewrite the equation [éq 5-1] in the form:
0
)
,
(
)
,
;
(
1
=
-
-
-
N
p
Z
T
Z
T
R
R
C
eq
&
,
i.e. in this model, the stress can be interpreted as the sum of a stress
limit of flow (which breaks up it even into an initial ultimate stress and a term
of work hardening) and a stress “viscous” depending on the speed of deformation and null with
null speed.
Viscous restoration of work hardening
One also introduces into modeling the phenomenon of viscous restoration of work hardening
who leads to a évanescence partial of work hardening. Under the action of thermal agitation, it
product a slow restoration of the crystal structure of metal by annihilation of dislocations and
internal stress relaxation. The model used to describe this phenomenon is as follows:
-
=
=
m
R
C
p
R
R
R
R
)
(
0
&
&
The term of evolution of the variable of work hardening
R
thus comprise a term of work hardening due to
plastic deformation and a term of restoration.
The model thus makes it possible to describe the primary education phenomenon of creep (work hardening) and creep
secondary (stabilization of work hardening).
Case of linear kinematic work hardening:
In a way equivalent to the case with isotropic work hardening the equations are written;
function threshold:
C
eq
X
F
-
-
=
)
~
(
law of flow
Writing of the rate of deformation (visco) plastic
&
& (
~
~)
(
)
vp
eq
p
X
X
=
-
-
3
2
with &p
F
N
= < >
0
3
2 H
X
=
, ~
: tensor of stress and its diverter
: variable tensor of kinematic work hardening,
X
: tensor of work hardening associated with the variable tensor with work hardening
,
0
H
: kinematic coefficient of work hardening
model of evolution of the tensor of work hardening
of a material with N phases
+
-
+
=
+
>
<
-
>
<
+
=
eq
m
eq
K
K
K
K
K
K
vp
K
eq
m
eq
K
K
K
K
K
K
vp
C
Z
Z
Z
C
Z
Z
Z
)
(
2
3
)
(
2
3
)
(
)
(
&
&
&
&
&
&
&
&
Code_Aster
®
Version
6.3
Titrate:
Modeling élasto- (visco) plastic with metallurgical transformations
Date:
29/04/02
Author (S):
A. RAZAKANAIVO, A.M. DONORE, F. WAECKEL
Key
:
R4.04.02-E
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:
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Manual of Reference
R4.04 booklet: Metallurgical behavior
HI-75/01/001/A
By preoccupation with simplification and a same manner that into isotropic, one takes for the term of
viscous restoration:
=
=
=
I
I
I
I
I
I
I
I
I
m
Z
m
C
Z
C
Z
ij
: coefficients of metallurgical restoration at the time of transformation I
J
I
I
m
C,
: coefficients of viscous restoration of phase I.
5.1
Borderline case: Plastic model independent of time
One wants to describe an instantaneous elastoplastic behavior and to cancel the viscous effects. For that
viscous parameters
and
C
will be taken equal to zero. To be been free from the numerical problems
what can pose the taking into account of
and
C
null, and in a way similar to the processing carried out for
the viscoplastic model of Taheri [bib15], one rewrites the equation [éq 5-1] in the form:
F
p
N
-
&
1
0
éq 5.1-1
the strict inequality being obtained in the case
F
p
<
=
0
0
&
(elastic mode).
In the purely plastic field of behavior (
0
) the inequality [éq 5.1-1] is then reduced
with:
F
R
T
eq
C
=
- -
() 0
and
&p
can be given more only by the equation of consistency
&f = 0
.
One thus finds oneself well within the framework of instantaneous plasticity independent of time, with one
digital processing identical to that classically used for the processing of the aforementioned.
Note:
It will be noted that
C
corresponds then to the conventional definition of the yield stress
y
.
The elastic limit will be noted
C
in viscoplasticity and
y
in plasticity independent of time
Code_Aster
®
Version
6.3
Titrate:
Modeling élasto- (visco) plastic with metallurgical transformations
Date:
29/04/02
Author (S):
A. RAZAKANAIVO, A.M. DONORE, F. WAECKEL
Key
:
R4.04.02-E
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:
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Manual of Reference
R4.04 booklet: Metallurgical behavior
HI-75/01/001/A
5.2
Borderline case: Model of behavior of viscous fluid
At very high temperature one a:
R
C
0
0
if one takes N 1, then:
&
vp
eq
= 32
maybe into unidimensional:
&
vp
=
. A model thus is obtained
of behavior of the fluid type viscous Newtonian, viscosity
.
Note:
In Code_Aster, the relations of behavior available are is models
completely plastic independent of time, that is to say models with viscous effect. (cf [§6]).
5.3 Plasticity
multiphase
The metallurgical transformations involve amendments of the mechanical characteristics of
material.
The elastic characteristics (YOUNG modulus and Poisson's ratio) are affected little by
metallurgical changes of structures. Only their dependence compared to the temperature
is thus taken into account.
On the other hand, the plastic characteristics (elastic limit in particular) strongly depend on
metallurgical structure. It is thus necessary to take into account the differences in characteristics plastic
for each possible phase. In modeling the strain and the stress are defined in
the scale of the material point (macroscopic) which can be multiphase. One seeks to define it
plastic behavior are equivalent of material when it has a multiphase structure, with
in particular a single criterion of plasticity. The definition of the behavior of material are equivalent
fact using a law of the mixtures on the characteristics of the phases. More precisely the definition
this material equivalent would correspond in 1D to a rheological model of I bars in parallel such
that:
&
&
&
vp
ivp
I I
I
I
Ci
I
ivp
Z
R
=
=
=
+
+
with
More precisely, in the case of the plasticity of von Mises with isotropic work hardening;
·
the function threshold is expressed by:
(
)
(
)
(
)
Z
Z
Z
,
,
,
,
;
,
T
R
T
R
T
R
F
C
eq
-
-
=
where:
(
)
(
)
I
I
I
I
R
T
R
Z
R
T
R
,
,
,
=
Z
is the work hardening of multiphase material,
R
I
being that of the phase
I
.
where
(
)
=
I
Ci
I
C
Z
T
Z
,
is the elastic limit of multiphase material,
Ci
that of the phase
I
.
Code_Aster
®
Version
6.3
Titrate:
Modeling élasto- (visco) plastic with metallurgical transformations
Date:
29/04/02
Author (S):
A. RAZAKANAIVO, A.M. DONORE, F. WAECKEL
Key
:
R4.04.02-E
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:
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Manual of Reference
R4.04 booklet: Metallurgical behavior
HI-75/01/001/A
·
and the rate of plastic deformation checks the condition of consistency
~
F
= 0
data by this law
of mixture. I.e. when one is in load,
&p
is such as:
0
)
,
(
)
,
;
(
~
1
=
-
-
-
=
I
I
I
C
eq
I
N
p
Z
Z
T
Z
T
R
R
F
&
One also gives the possibility of using a nonlinear law of the mixtures [bib9] such as one has in
1D:
=
-
+
(
())
()
1 F Z
F Z
H
H
. One has then:
·
+
-
=
+
-
=
C
H
C
H
C
H
H
Z
F
Z
F
R
Z
F
R
Z
F
R
)
(
))
(
1
(
)
(
))
(
1
(
y
is the elastic limit of the austenitic phase,
Z
Z
K
K
=
=
1
4
is the total proportion of the phases “”
''
(
)
F P B M
,
Z
Z
K
C
K
K
C
=
=
4
1
is the equivalent elastic limit of the cold phases “”
''
R
Z R
Z
K
K
K
=
=
1
4
is the average work hardening of the cold phases.
·
and in load
&p
check
0
))
(
1
(
)
,
(
)
,
;
(
~
1
1
=
-
-
-
-
-
=
K
K
K
H
C
eq
K
N
N
p
Z
Z
p
Z
F
Z
T
Z
T
R
R
F
&
&
.
()
F Z
H
is a function defined by the user under the operand
SY_MELANGE
key word factor
ELAS_META_FO
.
Parameters
I
I
I
I
N C
m
,
and
are defined in DEFI_MATERIAU under the key word factor
META_VISC. The limit elastic parameters are defined under the key word factor ELAS_META_FO;
key word * _SY for the plastic models independent of time and key word * _S_VP for
viscoplastic models.
Code_Aster
®
Version
6.3
Titrate:
Modeling élasto- (visco) plastic with metallurgical transformations
Date:
29/04/02
Author (S):
A. RAZAKANAIVO, A.M. DONORE, F. WAECKEL
Key
:
R4.04.02-E
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:
17/36
Manual of Reference
R4.04 booklet: Metallurgical behavior
HI-75/01/001/A
6
Relations of behavior
6.1
Partition of the deformation:
The deformation is written as the sum of four components:
=
+
+
+
E
T H
vp
Pt
where:
E
T H
vp
,
,
and
p T
are respectively the elastic strain, thermics,
viscoplastic and of plasticity of transformation,
6.2
Laws of behavior
6.2.1 Case with isotropic work hardening
()
(
)
(
)
(
)
(
)
=
-
-
-
=
=
<
=
=
-
-
=
=
+
-
+
-
-
-
=
>
<
-
=
=
+
+
+
=
=
=
=
0
)
,
(
)
,
,
(
~
0
0
0
0
)
,
(
)
,
,
(
)
,
(
)
,
,
(
~
2
3
)
1
(
.
,
1
'
F
~
2
3
1
5
1
4
1
4
1
I
I
I
C
eq
I
I
I
I
C
eq
eq
vp
T
F
R
ref.
F
I
I
T
F
R
ref.
H
T
I
I
I
I
T
p
E
T
p
vp
H
T
E
I
N
ref.
ref.
p
Z
Z
T
R
Z
T
R
F
F
p
F
p
R
T
R
Z
R
Z
T
R
Z
T
R
Z
T
R
F
p
Z
T
T
Z
Z
T
T
Z
T
Z
K
T
&
&
&
&
&
&
&
-
check
-
and
if
if
with
Z
Z
With
(
)
(
)
=
=
=
=
=
=
=
>
-
>
<
-
>
<
+
=
=
=
>
-
-
>
-
<
-
>
-
<
+
=
5
1
4
1
4
1
4
1
0
0
,
0
0
0
,
0
1
I
I
I
moy
K
K
K
m
moy
K
K
K
K
K
K
m
moy
K
K
K
K
K
K
K
K
R
Z
R
Z
R
Z
R
C
Z
R
Z
R
Z
p
R
Z
R
Z
R
C
Z
R
Z
R
Z
p
R
with
if
if
if
if
&
&
&
&
&
&
&
&
&
&
Code_Aster
®
Version
6.3
Titrate:
Modeling élasto- (visco) plastic with metallurgical transformations
Date:
29/04/02
Author (S):
A. RAZAKANAIVO, A.M. DONORE, F. WAECKEL
Key
:
R4.04.02-E
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:
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Manual of Reference
R4.04 booklet: Metallurgical behavior
HI-75/01/001/A
F
the function threshold,
R Z R
I
I I
,
variables intern work hardening and their forces
thermodynamic associated,
()
With
= A
I J K L
the tensor of elastic rigidity, depend on the temperature,
()
T T
and
()
Z
T
the temperature and the metallurgical structure.
6.2.2 Case with kinematic work hardening
()
(
)
(
)
(
)
(
)
=
-
-
-
=
=
<
=
=
-
-
=
-
-
=
+
-
+
-
-
-
=
>
<
-
=
=
+
+
+
=
=
=
=
0
)
,
(
)
,
,
(
~
0
0
0
0
)
,
(
)
,
,
(
)
,
(
)
(
)
(
)
~
~
(
2
3
)
1
(
.
,
1
'
F
~
2
3
1
5
1
4
1
4
1
I
I
I
C
eq
I
I
I
I
C
eq
eq
vp
T
F
R
ref.
F
I
I
T
F
R
ref.
H
T
I
I
I
I
T
p
E
T
p
vp
H
T
E
I
N
ref.
ref.
p
Z
Z
T
R
Z
T
R
F
F
p
F
p
T
X
Z
Z
T
X
Z
T
X
F
X
X
p
Z
T
T
Z
Z
T
T
Z
T
Z
K
T
&
&
&
&
&
&
&
-
check
-
and
if
if
with
Z
Z
With
eq
I
I
I
eq
K
K
eq
m
eq
K
K
K
K
K
K
vp
K
eq
m
eq
K
K
K
K
K
K
vp
Z
Z
C
Z
Z
Z
Z
C
Z
Z
Z
=
=
>
+
-
+
=
=
>
+
>
<
-
>
<
+
=
with
if not
if
if not
if
0
0
)
(
2
3
0
0
)
(
2
3
)
(
)
(
&
&
&
&
&
&
&
&
&
&
Code_Aster
®
Version
6.3
Titrate:
Modeling élasto- (visco) plastic with metallurgical transformations
Date:
29/04/02
Author (S):
A. RAZAKANAIVO, A.M. DONORE, F. WAECKEL
Key
:
R4.04.02-E
Page
:
19/36
Manual of Reference
R4.04 booklet: Metallurgical behavior
HI-75/01/001/A
F
the function threshold,
R Z R
I
I I
,
variables intern work hardening and their forces
thermodynamic associated,
()
With
= A
I J K L
the tensor of elastic rigidity, depend on the temperature,
()
T T
and
()
Z T
the temperature and the metallurgical structure.
In term of relations of behavior of STAT_NON_LINE available, the modeling put in
place offers several possibility:
·
choice of the type of behavior for the plastic deformation; plastic independent of
time or with taking into account of the viscous effects,
·
choice of a work hardening isotropic linear, isotropic nonlinear or kinematic,
·
taking into account or not of the plasticity of transformation,
·
taking into account or not of the metallurgical restoration of work hardening.
The choice of the material (steel or zircaloy) and thus of the number of phase is done by informing the key word
KIT of STAT_NON_LINE. “STEEL” for steel with 5 phases and “ZIRC” for the zircaloy with 3
phases.
6.3 Various relations of elastoplastic behavior
META_P_ ***
There are 12 relations of elastoplastic behavior independent of time
META_P *
.
·
8 relations with isotropic work hardening according to whether a linear isotropic work hardening is considered
or not linear, that one takes into account or not the plasticity of transformations, that one
takes into account or not the metallurgical restoration of work hardening.
·
4 relations with linear kinematic work hardening according to whether one takes into account or not
plasticity of transformations and/or metallurgical restoration of work hardening.
For these 12 relations of behavior one informs under the key word
ELAS_META_FO
or
ELAS_META
elastic parameters E and Naked, expansion factors, as well as the elastic limits.
/ELAS_META_FO
: (E: E
NAKED:
F_ALPHA:
F
C_ALPHA:
C
PHASE_REFE:
“HOT”
“COLD”
EPSF_EPSC_TREF:
F C
T
ref.
F1_SY
:
yf1
F2_SY
:
yf2
F3_SY:
yf3
F4_SY:
yf4
A_SY:
teststemyç
SY_MELANGE
:
F)
with for steel:
F
:
F
C:
F
C
T
ref.
:
F
T
ref.
yfi
: elastic limit of the phase
I
Code_Aster
®
Version
6.3
Titrate:
Modeling élasto- (visco) plastic with metallurgical transformations
Date:
29/04/02
Author (S):
A. RAZAKANAIVO, A.M. DONORE, F. WAECKEL
Key
:
R4.04.02-E
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:
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Manual of Reference
R4.04 booklet: Metallurgical behavior
HI-75/01/001/A
6.3.1 Relation
META_P_IL
This relation makes it possible to treat the relation of behavior in the case of the plasticity of Von Mises
with linear isotropic work hardening, applied to a material which undergoes phase shifts
metallurgical. The phenomena of plasticity of transformation and restoration of work hardening are
neglected. The coefficients of work hardening are provided under the key word
META_ECRO_LINE
of the operator
DEFI_MATERIAU
.
/META_ECRO_LINE:
F1_D_SIGM_EPSI
:
H0f1
F2_D_SIGM_EPSI
:
H0f2
F3_D_SIGM_EPSI
:
H0f3
F4_D_SIGM_EPSI
:
H0f4
C_D_SIGM_EPSI
:
H0c
with for steel:
H0fi
: Linear coefficient of work hardening of phase I.
F
: function of
Z
defining the law of mixture for the plastic behavior.
6.3.2 Relation
META_P_INL
This relation makes it possible to treat the relation of behavior in the case of the plasticity of von Mises
with nonlinear isotropic work hardening, applied to a material which undergoes phase shifts
metallurgical. In
DEFI_MATERAU
in addition to
ELAS_META_FO
one returns under the key word
META_TRACTION
the curves R (R).
META_TRACTION:
F1_SIGM:
()
R R
1
F2_SIGM:
()
R R
2
F3_SIGM:
()
R R
3
F4_SIGM:
()
R R
4
C_SIGM:
()
R R
C
6.3.3 Relation
META_P_CL
This relation makes it possible to treat the relation of behavior in the case of the plasticity of Von Mises
with linear kinematic work hardening, applied to a material which undergoes phase shifts
metallurgical. The phenomena of plasticity of transformation and restoration of work hardening are
neglected. The coefficients of work hardening are provided under the key word
META_ECRO_LINE
of the operator
DEFI_MATERIAU
.
/META_ECRO_LINE:
F1_D_SIGM_EPSI
:
H0f1
F2_D_SIGM_EPSI
:
H0f2
F3_D_SIGM_EPSI
:
H0f3
F4_D_SIGM_EPSI
:
H0f4
C_D_SIGM_EPSI
:
H0c
with for steel:
H0fi
: Coefficient of kinematic work hardening linear of phase I.
F
: function of
Z
defining the law of mixture for the plastic behavior.
Code_Aster
®
Version
6.3
Titrate:
Modeling élasto- (visco) plastic with metallurgical transformations
Date:
29/04/02
Author (S):
A. RAZAKANAIVO, A.M. DONORE, F. WAECKEL
Key
:
R4.04.02-E
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:
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Manual of Reference
R4.04 booklet: Metallurgical behavior
HI-75/01/001/A
6.3.4 Relation
META_P_IL_PT, META_P_INL_PT, META_P_CL_PT
Compared to
META_P_IL,
META_P_INL or META_P_CL
one holds account in addition to plasticity of
transformation but one always neglects the restoration of work hardening. In addition to the data of the key word
factor
ELAS_META_FO
and of the key word relating to the data of work hardening, one must inform too
those relating to the plasticity of transformation which are provided under the key word factor
META_PT.
/META_PT
:
(
F1_D_F_META:
F'1 F1_K:
f1
F2_D_F_META:
F'2 F2_K
:
f2
F3_D_F_META:
F'3 F3_K
:
f3
F4_D_F_META:
F'4 F4_K
:
f4)
with for steel:
F' f1 =
F
F
f1 =
K
F
F' f2 =
F
p
f2 =
K
p
F' f3 =
F
B
f3 =
K
B
F' f4 =
F
m
f4 =
K
m
6.3.5 Relation
META_P_IL_RE, META_P_INL_RE and META_P_CL_RE
One takes account of the restoration of work hardening but the plasticity of transformation is neglected.
data relating to the restoration of work hardening are provided under the key word factor
META_RE
of
the operator
DEFI_MATERIAU
.
/META_RE
: (
C_F1_THETA
:
cf 1
F1_C_THETA
:
cf 2
C_F2_THETA
:
cf 2
F2_C_THETA
:
cf 2
C_F2_THETA
:
cf 2
F3_C_THETA
:
cf 3
C_F2_THETA
:
cf 2
F4_C_THETA
:
cf 4
)
with for steel:
CF1 =
F
CF2 =
P
CF3 =
B
CF4 =
M
F1C =
F
F2C =
P
F3C =
B
F4C =
M
6.3.6 Relation
META_P_IL_PT_RE, META_P_INL_PT_RE and META_P_CL_PT_RE
One holds account at the same time phenomena of plasticity of transformation and restoration
of work hardening. Data of the key words factors
ELAS_META_FO
,
META_PT
and
META_RE
must
to be well informed.
Code_Aster
®
Version
6.3
Titrate:
Modeling élasto- (visco) plastic with metallurgical transformations
Date:
29/04/02
Author (S):
A. RAZAKANAIVO, A.M. DONORE, F. WAECKEL
Key
:
R4.04.02-E
Page
:
22/36
Manual of Reference
R4.04 booklet: Metallurgical behavior
HI-75/01/001/A
6.4 Various relations of élasto-viscoplastic behavior
META_V_ ***
One has in the same way that in conventional plasticity, 12 relations of behavior which are available
according to the type of work hardening and according to whether one holds account or not phenomena of plasticity of
transformation and/or of metallurgical restoration of work hardening. One uses the same terminology as
in the case of conventional plasticity to differentiate the 12 élasto-viscoplastic relations. For
each relation one must inform in
ELAS_META
or
ELAS_META_FO
yield stresses
of flow viscous, in the place of the conventional apparent elastic limits.
F1_SC:
cf 1
F2_SC:
cf 2
F3_SC:
cf 3
F4_SC:
cf 4
C_SC:
DC
SC_MELANGE
: function for the law of the mixtures
instead of
* _SY
for the plastic case.
6.4.1 Relation
META_V_IL and META_V_INL
Élasto-viscoplastic relation of behavior applied to a material which undergoes transformations
metallurgical with or not linear linear work hardening. One does not take account of the phenomena of
plasticity of transformation and metallurgical restoration of work hardening.
6.4.2 Relation
META_V_CL
Élasto-viscoplastic relation of behavior applied to a material which undergoes transformations
metallurgical with linear kinematic work hardening. One does not take account of the phenomena of
plasticity of transformation and metallurgical restoration of work hardening.
6.4.3 Relation
META_V_IL_PT, META_V_INL_PT and META_V_CL_PT
Idem that
META_P_IL_PT
,
META_P_INL_PT
and
META_V_CL_PT
but in viscoplasticity.
6.4.4 Relation
META_V_IL_RE, META_V_INL_RE and
META_V_CL_RE
Idem that
META_P_IL_RE
,
META_P_INL_RE
and
META_V_CL_RE
but in viscoplasticity
6.4.5 Relation
META_V_IL_PT_RE, META_V_INL_PT_RE and
META_V_CL_PT_RE
Idem that
META_P_IL_PT_RE
,
META_P_INL_PT_RE
and
META_V_CL_PT_RE
but in viscoplasticity
Note:
·
For the whole of the relations
META_ **
, internal variables produced in
Code_Aster are:
R
I
: variables of effective work hardening for I phases,
D
: indicator of plasticity (0 if the last calculated increment is elastic; 1 if not),
R
: the term of work hardening of the function threshold
·
In addition, these modelings can be carried out with the functionality of
geometrical reactualization
PETIT_REAC
.
Code_Aster
®
Version
6.3
Titrate:
Modeling élasto- (visco) plastic with metallurgical transformations
Date:
29/04/02
Author (S):
A. RAZAKANAIVO, A.M. DONORE, F. WAECKEL
Key
:
R4.04.02-E
Page
:
23/36
Manual of Reference
R4.04 booklet: Metallurgical behavior
HI-75/01/001/A
7 Formulation
numerical
One will treat the viscoplastic law of behavior with isotropic work hardening.
7.1 Discretization
Knowing the fields
, U
and
p
at the moment
T
, one chooses an implicit scheme to discretize in
time equations of the continuous problem, except for the parameters of work hardening where they are used
equations [éq 4.2-1].
It is noticed that with an implicit discretization, only two points differentiate the two types from
viscoplastic behavior and plastic independent of time:
·
the form of the function of load, for which one has a complementary term in the case of
viscosity,
·
the presence of the term of restoration of work hardening in the evolution of the variable
of work hardening for the viscoplastic case.
Moreover, incremental conventional plasticity seems the borderline case (without numerical difficulty
associated) of incremental viscoplasticity when
0
0
C
C
y
.
This type of processing was already carried out by LORENTZ [bib15].
If one poses
~
F
F
p
T
N
= -
1
()
(
)
(
)
(
)
[
]
(
)
=
+
+
+
=
=
-
- -
+
-
+
=
-
<
>
=
<
=
=
>
=
=
E
T H
p
Pt
E
T H
R
F
T
I
I
F
R
F
Pt
I
I
I
I
p
eq
T
T
Z
T
T
Z
Z
T
T
Z
K
Z
p
F
p
F
p
ref.
With
Z
Z
,
.
(
)
~
F'
~
~
~
1
3
2
1
3
2
0
0
0
1
4
1
4
elastic mode:
and
mode (visco) plastic
and
0
Code_Aster
®
Version
6.3
Titrate:
Modeling élasto- (visco) plastic with metallurgical transformations
Date:
29/04/02
Author (S):
A. RAZAKANAIVO, A.M. DONORE, F. WAECKEL
Key
:
R4.04.02-E
Page
:
24/36
Manual of Reference
R4.04 booklet: Metallurgical behavior
HI-75/01/001/A
(
)
(
)
(
)
()
R
p
Z
R
Z
R
Z
C R
Z
R
Z
R
p
Z
R
Z
R
Z
C R
Z
R
Z
X
X T
T
X
X T
X
X T
K
K
K
K
K
K
K
K
moy
m
K
K
K
K
K
K
moy
m
K
K
=
+
< -
>
-
< -
>
-
-
>
=
=
=
+ <
>
- <
>
-
>
=
=
=
+
=
=
+
-
-
=
=
=
-
-
-
-
-
1
4
1
4
1
4
1
0
0
0
0
0
0
if
if
if
if
with:
,
,
(
)
()
T
X T
-
7.2
Algorithm of resolution of the quasi-static problem
The incremental problem posed on the structure is a non-linear problem. Its formulation
variational, in the case of the small deformations, is form:
To find
U
such as:
(
)
(
)
()
()
()
U
U
U
U
-
+
=
=
, T
v D
L T
v
T
T
D
kinematically acceptable and
B
where:
U
indicate the field of displacement
B U
= U
D
T
()
corresponds to the boundary conditions in displacement (connections kinematics)
and
()
L T
F v D
G v D
=
+
.
.
is the virtual work of the mechanical loadings at the moment
T.
In Code_Aster, this non-linear problem is solved by a method of NEWTON [bib6],
[bib7]. The algorithm of resolution comprises:
·
a phase of prediction at the beginning of each pitch of time,
·
iterations of Newton inside a pitch of time.
We do not detail here the algorithm implemented (one will refer for that to the documents of
reference [R5.03.01] and [R5.03.02]), but we endeavor to highlight them
amendments made to the diagram of integration by the taking into account of the metallurgical evolution
()
Z T
and of the plasticity of transformation.
Code_Aster
®
Version
6.3
Titrate:
Modeling élasto- (visco) plastic with metallurgical transformations
Date:
29/04/02
Author (S):
A. RAZAKANAIVO, A.M. DONORE, F. WAECKEL
Key
:
R4.04.02-E
Page
:
25/36
Manual of Reference
R4.04 booklet: Metallurgical behavior
HI-75/01/001/A
7.2.1 Integration of the relations
META__ ***
One gives the expression of
according to;
·
(
)
or
U
unknown of the problem,
·
known terms such variables calculated with the preceding pitch (
-
, variables intern…),
the characteristics materials,
HT
…
()
(
)
()
(
)
µ
µ
µ
µ
µ
µ
µ
=
=
-
-
-
=
-
=
=
+
-
-
=
=
+
=
+
-
-
-
-
-
-
-
-
WITH T
tr
K tr
K
K tr
K tr
F Z Z
p
E
E
early
HT
vp
Pt
early
HT
E
vp
Pt
E
eq
One poses:
~
~
~
~
~
~
~
~
,
~
2
2
3
3
3
3
2
3
2
3
2
from where
~
(,
)
~
~
~
µ
µ
µ
µ
µ
= +
+
-
-
-
1
1 3
2
3
F Z Z
p
eq
with:
·
expression of
~
eq
~
~
~
~
=
+
+
E
Pt
vp
~
~
~
~ (,)
~
µ
µ
=
-
+
+
-
-
2
2
3
2
3
2
F Z Z
p
eq
(
)
2
1 3
3
µ
µ
µ
µ
µ
~
~
~
(
(,
))
+
=
+
+
-
-
eq
eq
F Z Z
p
one poses:
2
µ
µ
µ
~
~
~
+
=
-
-
E
Code_Aster
®
Version
6.3
Titrate:
Modeling élasto- (visco) plastic with metallurgical transformations
Date:
29/04/02
Author (S):
A. RAZAKANAIVO, A.M. DONORE, F. WAECKEL
Key
:
R4.04.02-E
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:
26/36
Manual of Reference
R4.04 booklet: Metallurgical behavior
HI-75/01/001/A
one a:
µ
µ
eq
E
eq
F Z Z
p
= +
+
(
(,
))
1 3
3
and
~
~
E
eq
E
eq
=
·
expression of
p
That is to say the function of load:
F
F Z Z
R R
T Z
T Z
eq
E
C
= +
-
-
-
µ
1 3
(,
)
(
; ,)
(,)
R R
T Z
(
; ,)
-
is the term of work hardening
(
)
R R T Z
;
calculated for
p
= 0.
-
If F < 0 then one is in elastic mode and
p
= 0
-
If not one is in load and
p
check;
)
,
(
)
,
;
(
)
,
(
3
1
3
0
1
Z
T
p
R
Z
T
R
R
Z
Z
F
p
T
p
C
E
eq
N
µ
µ
-
-
-
+
-
=
-
That is to say the function
N
T
p
Z
T
p
R
Z
T
R
R
Z
Z
F
p
F
C
E
eq
1
)
,
(
)
,
;
(
)
,
(
3
1
3
~
0
-
-
-
-
+
-
=
-
µ
µ
,
p
is thus
solution of the nonlinear scalar equation
~
F
= 0
.
The resolution is made in Code_Aster by a method of the secants with interval of search
[bib15].
Note:
Whenever the plasticity of transformation is not taken into account, expressions
obtained are the same ones while taking
F Z,
Z
(
)
= 0
.
Whenever it is the restoration of work hardening which is neglected then one also has them
same expressions but by taking all them
equal to 1.
0
H
is the slope of work hardening of the traction diagram. In the case of isotropic work hardening not
linear where the traction diagram is linear per piece,
0
H
is defined for the segment to which
p belongs. Stamp tangent
7.2.2.1 Phase of prediction - Option
RIGI_MECA_TANG
One linearizes the continuous problem compared to time, and one determines
U
0
as solution of
problem of speed:
()
(
)
()
()
&
,
&
U
0
T
v D
L T
v
=
kinematically acceptable
where
()
&
&.
&.
L T
F v D
G v D
=
+
.
Code_Aster
®
Version
6.3
Titrate:
Modeling élasto- (visco) plastic with metallurgical transformations
Date:
29/04/02
Author (S):
A. RAZAKANAIVO, A.M. DONORE, F. WAECKEL
Key
:
R4.04.02-E
Page
:
27/36
Manual of Reference
R4.04 booklet: Metallurgical behavior
HI-75/01/001/A
The problem from of speed is obtained by deriving compared to time the equations from the problem
continuous:
(
)
~&
~ & &
&
µ
=
-
-
2
vp
Pt
&
(,
) ~&
Pt
F Z Z
= 32
In the case of the élasto-viscoplastic models, one uses, for the phase of prediction, the matrix
“elastic” in the direction where one will not take account of the term
&
vp
. As for the plastic case one a:
(
)
(
)
(
)
(
)
&
&
~
,
,
,
,
,
,
vp
eq
eq
y
eq
y
p
R T
R
T
R T
R
T
=
-
-
=
-
-
3
2
0
0
0
if
if
Z
Z
Z
Z
Derivation compared to the time of the equation
(
)
()
E Q
y
R T
R
T
-
-
=
,
,
Z
Z
0
give
the expression of
&p
(relation of consistency).
(
)
D
D T
D R
D T
D
D T
D R
D T
D
D T
eq
y
vp
Pt
eq
y
-
-
=
-
-
-
-
3
2
~:
~ & ~&
~&
With
Note:
The derivation of A was neglected in this phase of prediction
C
T
T
B
p
R
R
R
Z
R
R
Z
R
R
Z
p
R
Z
R
R
Z
R
R
Z
R
R
Z
R
R
Z
R
R
Z
Z
T
R
y
y
y
I
I
I
I
I
K
K
K
K
K
K
K
K
K
K
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
EFF
I
I
=
+
=
+
=
>
<
-
>
-
<
+
>
<
+
+
+
=
+
+
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
Z
Z &
&
&
&
&
&
&
&
&
&
&
&
&
&
0
5
1
0
4
1
0
4
1
0
5
1
0
5
1
0
5
1
0
5
1
0
5
1
0
5
1
0
)
,
,
(
Code_Aster
®
Version
6.3
Titrate:
Modeling élasto- (visco) plastic with metallurgical transformations
Date:
29/04/02
Author (S):
A. RAZAKANAIVO, A.M. DONORE, F. WAECKEL
Key
:
R4.04.02-E
Page
:
28/36
Manual of Reference
R4.04 booklet: Metallurgical behavior
HI-75/01/001/A
from where:
(
)
(
)
(
)
0
4
1
4
1
0
3
1
'
F
3
~
:
~
3
0
3
1
'
F
3
~
:
~
3
R
C
B
Z
K
p
C
B
p
R
Z
K
T
D
D
T
D
R
D
T
D
D
I
I
Q
E
I
I
I
Q
E
I
I
Q
E
I
I
I
Q
E
y
Q
E
+
>
-
-
>
<
-
-
<
=
=
-
-
+
-
>
<
-
-
=
-
-
=
=
=
=
µ
µ
µ
µ
µ
µ
Z
Z
&
&
&
&
&
&
From where, finally the expression of
&
vp
:
(
)
(
)
(
)
()
(
)
()
0
,
,
,
0
0
,
,
,
~
1
'
F
3
~
:
~
3
3
2
3
if
if
4
1
0
-
-
=
=
-
-
>
-
-
>
<
-
-
<
+
=
=
=
Z
Z
Z
Z
Z
T
T
R
T
T
R
C
B
Z
K
H
y
EFF
I
Q
E
vp
y
EFF
I
Q
E
Q
E
Q
E
I
I
I
I
I
Q
E
vp
µ
µ
µ
&
&
&
&
Taking into account the variations of
I
H
0
and
y
according to the temperature and structure
metallurgical, one chooses by convenience to neglect the term (B+C)
and one thus leads to one
expression of
~&
form:
(
)
(
)
(
)
>
<
-
-
>
>
<
-
-
<
+
-
=
=
=
=
=
4
1
4
1
0
~
1
'
F
2
3
~
1
'
F
3
~
:
~
3
3
2
3
~
2
~
I
I
I
I
I
Q
E
I
I
Q
E
I
I
I
Q
E
Z
K
Z
K
R
µ
µ
µ
µ
Z
Z
&
&
&
&
&
The expression of
~&
depends on the sign of the term (criterion of load-discharge)
()
3
3
1
4
µ
µ
~: ~&
F'
&
eq
I
I
I
I
eq
I
I
K
-
<
>
=
=
Z
Z
.
One approximates
~&
by:
(
)
(
)
~&
~&
~: ~&
~
F'
&
~
=
-
<
>
+
-
-
<
>
-
+
=
=
2
9
2
3
3
2
1
1
3
3
0
2
1
4
0
µ
µ
µ
µ
µ
R
K
Z
D
R
eq
I
I
I
I
I
Z
éq 7.2.2.1-1
with
D
= 1
if one plasticizes and if one is in load at the moment
T
and
D
= 0
in the contrary case.
Code_Aster
®
Version
6.3
Titrate:
Modeling élasto- (visco) plastic with metallurgical transformations
Date:
29/04/02
Author (S):
A. RAZAKANAIVO, A.M. DONORE, F. WAECKEL
Key
:
R4.04.02-E
Page
:
29/36
Manual of Reference
R4.04 booklet: Metallurgical behavior
HI-75/01/001/A
It is noticed that
~&
is a function closely connected of
~&
. The plasticity of transformation, like
thermal deformation, generate in the problem of speed a second member.
That introduced by the plasticity of transformation is form:
(
)
()
+
-
>
<
-
=
=
=
D
v
R
D
Z
K
L
I
I
I
I
I
Pt
µ
µ
µ
4
1
0
3
3
1
~
1
'
F
2
3
2
Z
&
To determine
U
0
, it is necessary to solve after discretization spaces the following linear system of it:
K
B
B
U
L
U
L
L
0
0
0
0
0
0
0
0
T
D
T H
p T
= + +
+
On simple cases tests for which there is an analytical solution, one noted that the fact of
to neglect the second member
L
p T
()
could lead, to converge, with a significant number
iterations. This is why this term is taken into account for the phase of prediction.
7.2.2.2 Iterations of Newton - Option
FULL_MECA
In the method of NEWTON, knowing
U
N
, one determines as well as possible
U
N
+1
checking:
(
)
(
)
(
)
(
)
()
()
(
)
(
)
()
(
)
()
(
)
(
) ()
(
)
F
,
F
,
U
U
U
U
U
U
U
N
N
N
N
T
N
N
T
v D
L T
F
F
N
+
+
+
+
=
-
+
-
=
1
1
1
1
0
0
From where:
()
(
)
()
(
)
K
F
U
U
N
T
N
N
=
= -
,
F
With each iteration one solves the linear system:
()
(
)
()
K
B
B
U
L
F
F
U
N
T
N
N
N
N
N
T
v D
0
0
0
= -
=
with
,
During iterations of a pitch of time given, the method of NEWTON thus uses the calculation of
the tangent operator
K
N
, which is given by the derivation of the implicit problem according to
the increment of deformation
. The tangent operator
K
N
can be recomputed or not with each
iteration.
Code_Aster
®
Version
6.3
Titrate:
Modeling élasto- (visco) plastic with metallurgical transformations
Date:
29/04/02
Author (S):
A. RAZAKANAIVO, A.M. DONORE, F. WAECKEL
Key
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R4.04 booklet: Metallurgical behavior
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One gives the expression of
for the constitution of the consistent tangent matrix of the method
iterative of Newton.
(
)
(
)
(
)
µ
=
+
=
=
-
-
~
~
~
~
~
~
~
~
~
1
3
2
tr
Id
Pt
vp
with
(
)
()
(
)
(
)
(
)
()
One a:
with
with
µ
µ
µ
µ
~
~
~
~
,
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
=
=
=
=
+
=
=
=
-
Id
Id
F Z Z
p
p
Id
Id
ijkl
ik
jl
Pt
vp
eq
eq
E
eq
E
eq
eq
eq
E
eq
3
2
3
2
2
3
1
2
3
eq
The expression of
(
)
~
p
is obtained while deriving
~
F
= 0
compared to
, which gives:
()
(
)
eq
Z
Z
F
T
p
T
N
R
p
N
N
µ
µ
µ
~
)
,
(
3
1
3
3
~
1
0
+
+
+
=
-
Code_Aster
®
Version
6.3
Titrate:
Modeling élasto- (visco) plastic with metallurgical transformations
Date:
29/04/02
Author (S):
A. RAZAKANAIVO, A.M. DONORE, F. WAECKEL
Key
:
R4.04.02-E
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:
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Manual of Reference
R4.04 booklet: Metallurgical behavior
HI-75/01/001/A
from where:
()
(
)
-
+
+
+
-
-
+
=
-
eq
eq
E
eq
E
eq
p
Z
Z
F
T
p
T
N
R
p
Id
Z
Z
F
N
N
µ
µ
µ
µ
µ
µ
~
~
)
,
(
3
1
3
1
3
3
1
2
)
,
(
3
1
1
~
~
1
0
2
éq 7.2.2.2-1
Code_Aster
®
Version
6.3
Titrate:
Modeling élasto- (visco) plastic with metallurgical transformations
Date:
29/04/02
Author (S):
A. RAZAKANAIVO, A.M. DONORE, F. WAECKEL
Key
:
R4.04.02-E
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:
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R4.04 booklet: Metallurgical behavior
HI-75/01/001/A
7.2.2.3 Operator
tangent
That is to say
= (
,
,
,
,
,
)
11
22
33
12
23
13
2
2
2
the virtual increase in stress and
that is to say
= (
,
,
,
,
,
)
11
22
33
12
23
13
2
2
2
the virtual increase in deformation,
the operator who binds
with
is given by the following expression:
~
~ ~
~
(
)
(
)
µ
µ
ij
eq
E
ik
jl
p
ije
kle
kl
has
C
p
C
has
tr
K tr
=
-
-
=
2
1
3
3
3
with:
()
()
()
()
(
)
(
)
(
)
(
)
=
>
<
-
-
=
-
+
+
+
+
=
=
+
=
=
=
-
-
if not
plasticize
one
if
and
charge
in
is
one
if
if not
and
plastic
elasto
if
:
where
iterations
at the time
time
of
not
each
of
beginning
with
and
current
iterations
at the time
time
of
not
each
of
beginning
with
)
(option
1])
-
7.2.2.2
[éq
(cf.
current
iterations
at the time
)
'
'
(option
1])
-
7.2.2.1
[éq
(cf.
time
of
not
each
of
beginning
with
0
0
~
~
1
0
0
1
'
F
3
~
:
~
3
1
,
F
3
1
3
1
3
3
1
3
1
0
)
,
(
3
1
1
2
4
1
1
1
0
2
2
2
0
2
2
1
3
'
'
µ
µ
µ
µ
µ
µ
µ
µ
C
Z
K
C
p
T
p
T
N
R
C
R
C
C
C
Z
Z
F
has
I
I
Q
E
I
I
I
Q
E
eeq
N
N
eqe
eqe
p
Z
Z
Z
&
&
FULL_MECA
TANG
RIGI_MECA_
Let us note then
K
the operator such as
= K
and the vector of the diverter of the stresses is S:
S
= (~, ~, ~,
~, ~, ~)
11
22
33
12
23
13
2
2
2
.
Code_Aster
®
Version
6.3
Titrate:
Modeling élasto- (visco) plastic with metallurgical transformations
Date:
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Author (S):
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Key
:
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:
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Manual of Reference
R4.04 booklet: Metallurgical behavior
HI-75/01/001/A
(
)
D
D
D
D
K
D
D
D
D
Id
D
D
has
C
p Id Ca S S
D
D
has
C
p
Id
C
S S has
K
eq
E
p
eq
E
p
µ
µ
µ
µ
*
*
*
*
*
*
*
*
*
*
*
*
~
~
~
~
~
~
:
=
+
=
-
=
-
-
=
-
-
-
+
1 1
1 1
1 1
1
3
2
1
3
2
1
3
1
3
3
3
from where
(
)
1 1
The operator
K
is written:
K
K
has
C
p
C
S S has
eq
E
p
11
3
1 1
2
3
2
1
3
=
+
-
-
µ
µ
K
K
has
C
p
C
S S has
K
has
C
p
C
S S has
eq
E
p
eq
E
p
22
3
1 2
3
2 2
1
3
2
1
3
2
3
2
1
3
=
-
-
-
+
-
-
µ
µ
µ
µ
K
K
has
C
p
C
S S has
K
has
C
p
C
S S has
K
has
C
p
C
S S has
eq
E
p
eq
E
p
eq
E
p
33
3
1 3
3
2 3
3
3 3
1
3
2
1
3
1
3
2
1
3
2
3
2
1
3
=
-
-
-
-
-
-
+
-
-
µ
µ
µ
µ
µ
µ
Code_Aster
®
Version
6.3
Titrate:
Modeling élasto- (visco) plastic with metallurgical transformations
Date:
29/04/02
Author (S):
A. RAZAKANAIVO, A.M. DONORE, F. WAECKEL
Key
:
R4.04.02-E
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Manual of Reference
R4.04 booklet: Metallurgical behavior
HI-75/01/001/A
K
C
S S has
C
S S has
C
S S has
has
C
p
C
S S has
p
p
p
eq
E
p
44
1 4
2 4
3 4
3
4 4
2
1
3
=
-
-
-
-
-
µ
µ
K
C
S S has
C
S S has
C
S S has
C
S S has
has
C
p
C
S S has
p
p
p
p
eq
E
p
55
1 5
2 5
3 5
4 5
3
5 5
2
1
3
=
-
-
-
-
-
-
µ
µ
K
C
S S has
C
S S has
C
S S has
C
S S has
C
S S has
has
C
p
C S S
has
p
p
p
p
p
eq
E
p
66
1 6
2 6
3 6
4 6
5 6
3
6.6
2
1
3
=
-
-
-
-
-
-
-
µ
µ
where them “
I
“component of the vector
K
II
correspond to “
I
“terms of the higher part of
ičme column of the symmetrical matrix
K
It is noticed that the operator
K
0
and the operator
K
N
are different. The plasticity of transformation
does not intervene in the same way in the calculation of the two operators.
Note:
These various terms were obtained by developing the case with isotropic work hardening but
one obtains the same thing in the case of kinematic work hardening,
0
R
is then replaced
by the kinematic coefficient of work hardening
0
H
.
Code_Aster
®
Version
6.3
Titrate:
Modeling élasto- (visco) plastic with metallurgical transformations
Date:
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A. RAZAKANAIVO, A.M. DONORE, F. WAECKEL
Key
:
R4.04.02-E
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Manual of Reference
R4.04 booklet: Metallurgical behavior
HI-75/01/001/A
8 Bibliography
[1]
F. WAECKEL: Metallurgy elements for the study of structure transformations in
steels - Note EDF-DER HI-71/8075.
[2]
S. ANDRIEUX, F. WAECKEL: Metallurgical thermal modeling and Mechanics of one
operation of welding - Study Bibliographical. Note intern EDF-DER HI-71/7595.
[3]
S. ANDRIEUX, F. WAECKEL: A law of thermo behavior metallurgical with
cooling for steels slightly allied. Note intern EDF-DER HI-71/7459.
[4]
P. MIALON: Elements of analysis and numerical resolution of the relations of the élasto-
plasticity, E.D.F. Bulletin of the D.E.R., series C, n°3 (ISSN 013-4511).
[5]
A. Mr. DONORE, F. WAECKEL: Influence structure transformations in the laws of
elastoplastic behavior, Note E.D.F-D.E.R HI 74/93/024.
[6]
P. MIALON, J.P. LEFEBVRE: Quasi-static nonlinear algorithm, Note E.D.F-D.E.R.
HI-75/7832 [R5.03.01].
[7]
P. MIALON: Integration of the relations of elastoplastic behavior in the Code Aster,
Note intern E.D.F-D.E.R. HI-75/7833 [R5.03.02].
[8]
J.B. LEBLOND, G. MOTTET, J.C. DEVAUX (1986): With theorical (and numerical) approach to
the plastic behavior off steels during phase transformations. Derivation off general equations.
Newspaper off the Mechanics and Physics Solids, vol. 34, n°4 pp. 395-406.
[9]
J.B. LEBLOND, G. MOTTET, J.C. DEVAUX (1986): With theorical (and numerical) approach to
the plastic behavior off steels during phase transformations. Study off classical plasticity for
ideal plastic phases. Newspaper off the Mechanics and Physics Solids, vol. 34, n°4 pp. 411-432.
[10]
J.B. LEBLOND, J. DEVAUX, J.C. DEVAUX (1989): Mathematical modelling off
transformation plasticity in steels. Put ideal plastic phases off. International Newspaper off
Plasticity, vol. 5, pp. 551-572.
[11]
J.B. LEBLOND (1989): Mathematical modelling off transfromation plasticity in steels. Coupling
with strain hardening phenomena. International Newspaper off Plasticity, vol. 5, pp. 573-591.
[12]
Y. DESALOS (1981): Report/ratio IRSID n° 9534901 MET44.
[13]
S. SJÖSTRÖM (1987): The problem off calculating quench stress. Swedish symposium one
residual stress, Sunne 1987, pp. 131-159.
[14]
A. RAZAKANAIVO (1997): Introduction into Code_Aster of a model of behavior
élasto-viscoplastic fascinating of account metallurgy. Note HI-74/97/020/0.
[15]
E. LORENTZ (1997): Numerical formulation of the viscoplastic law of behavior of
Taheri. Note EDF-DER HI-74/97/019/A [R5.03.05].
[16]
A. RAZAKANAIVO (2000): Operator CALC_META, Document of Code_Aster use
[U4.85.01].
[17]
A. RAZAKANAIVO (2000): Introduction of model EDGAR into Code_Aster. Note
HI-74/00/013/A.
Code_Aster
®
Version
6.3
Titrate:
Modeling élasto- (visco) plastic with metallurgical transformations
Date:
29/04/02
Author (S):
A. RAZAKANAIVO, A.M. DONORE, F. WAECKEL
Key
:
R4.04.02-E
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:
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Manual of Reference
R4.04 booklet: Metallurgical behavior
HI-75/01/001/A
Intentionally white left page.