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
Organization (S): EDF/MTI/MN
Handbook of Référence
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.
Handbook of Référence
R4.04 booklet: Metallurgical behavior
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Code_Aster ®
Version
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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
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Count
matters
1 Introduction ............................................................................................................................................ 3
2 Influence of structure transformations on the thermal deformation ............................................. 5
2.1 Zones of thermal deformation ..................................................................................................... 5
2.2 Assumptions and notations .................................................................................................................. 6
3 Plasticity of transformation ................................................................................................................... 7
4 Restoration of work hardening ................................................................................................................... 9
4.1 Model with 2 phases with a direction of transformation ....................................................................... 9
4.2 Generalization of the model with N phases with transformations with double direction ................................. 11
5 Model of deformation (visco) plastic .............................................................................................. 12
5.1 Borderline case: Plastic model independent of time .................................................................... 14
5.2 Borderline case: Model of behavior of viscous fluid ............................................................. 15
5.3 Multiphase plasticity .................................................................................................................... 15
6 Relations of behavior ................................................................................................................. 17
6.1 Partition of the deformation:........................................................................................................... 17
6.2 Laws of behavior ................................................................................................................... 17
6.3 Various relations of elastoplastic behavior META_P_ *** .................................. 19
6.3.1 Relation META_P_IL ............................................................................................................ 20
6.3.2 Relation META_P_INL .......................................................................................................... 20
6.3.3 Relation META_P_CL ............................................................................................................ 20
6.3.4 Relation META_P_IL_PT, META_P_INL_PT, META_P_CL_PT ......................................... 21
6.3.5 Relation META_P_IL_RE, META_P_INL_RE and META_P_CL_RE .................................... 21
6.3.6 Relation META_P_IL_PT_RE, META_P_INL_PT_RE and META_P_CL_PT_RE ................ 21
6.4 Various relations of élasto-viscoplastic behavior META_V_ *** .......................... 22
6.4.1 Relation META_V_IL and META_V_INL ................................................................................. 22
6.4.2 Relation META_V_CL ............................................................................................................ 22
6.4.3 Relation META_V_IL_PT, META_V_INL_PT and META_V_CL_PT ..................................... 22
6.4.4 Relation META_V_IL_RE, META_V_INL_RE and META_V_CL_RE ..................................... 22
6.4.5 Relation META_V_IL_PT_RE, META_V_INL_PT_RE and META_V_CL_PT_RE ................ 22
7 numerical Formulation ........................................................................................................................ 23
7.1 Discretization ................................................................................................................................. 23
7.2 Algorithm of resolution of the quasi-static problem ................................................................... 24
7.2.1 Integration of relations META__ *** .................................................................................. 25
7.2.2 Stamp tangent ................................................................................................................... 26
7.2.2.1 Phase of prediction - Option RIGI_MECA_TANG .................................................... 26
7.2.2.2 Iterations of Newton - Option FULL_MECA .............................................................. 29
7.2.2.3 Tangent operator .................................................................................................... 32
8 Bibliography ........................................................................................................................................ 35
Handbook of Référence
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Code_Aster ®
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Modeling élasto- (visco) plastic with metallurgical transformations
Date:
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:
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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 a modification 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 liquid-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 dilation coefficient 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 modification of the one and dilation coefficient
speak overall about the influence of the transformations on the thermal deformation,
·
a transformation proceeding under constraints can give rise to a deformation
irreversible and this, even for levels of constraints 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
constraints 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.
Handbook of Référence
R4.04 booklet: Metallurgical behavior
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Code_Aster ®
Version
6.3
Titrate:
Modeling élasto- (visco) plastic with metallurgical transformations
Date:
29/04/02
Author (S):
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:
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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
cold phases of cubic structure face centered (CFC) with a hot phase 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 considered as pure and a phase mixed with
[bib16], [bib17]. The various characteristics relating to the various phases are noted:
Zircaloy steels
Ferrite: F1_ ***
Alpha pure: F1_ ***
Pearlite: F2_ ***
Mixed alpha: F2_ ***
Bainite: F3_ ***
Beta: C_ ***
Martensite: F4_ ***
Austenite: 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.
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Modeling élasto- (visco) plastic with metallurgical transformations
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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 maintenance 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.
D
B
C
F
E
G
With
H
T0
T
Tref
F
Appear diagrammatic 2-a: Relevé of dilatometry
2.1
Zones of thermal deformation
The various zones 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),
C-D:
thermal dilation of austenite (with a dilation coefficient different from that
phases known as ““(F), (P), (B), (M)),
D-E:
thermal contraction of austenite,
E-F:
first transformation (partial) of the austenite (for example F + P) which
be accompanied by a voluminal expansion,
F-G:
zone without transformation with thermal contraction of remaining the austenite mixture -
formed phase (with a certain thermal dilation coefficient apparent),
G-H:
second transformation of the remaining austenite (for example M) which
be accompanied by a voluminal expansion,
H-A:
thermal contraction of the final structure (with the same dilation coefficient
that with the heating).
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Modeling élasto- (visco) plastic with metallurgical transformations
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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 Tref reference.
·
That is to say HT
HT
thermal deformation of the austenitic phase, and F thermal deformation
phases ferritic, perlitic, bainitic and martensitic, we will take:
HT
R
Tref
= (T) (T - Tref) - (1 - Z)
F
HT =
T
(T)
R
ref.
F
F
(T - Tref) +Z F
where:
T ref.:
Temperature of reference,
(T): average dilation coefficient of the austenitic phase at the current temperature
T, compared to the temperature of reference.
F (T): dilation coefficient 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.
Tref
HT
= T
HT
- T
F
F (ref.)
(ref.) translated the difference in compactness between the structures
crystallographic cubic with centered faces (austenite) and cubic centered (ferrite) with
temperature of Tref reference.
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:
I = 4
I = 4
HT (
T
T
Z, T) 1
Z.
R
ref.
R
ref.
=
-
I
(T - Tref) - (1 - Z)
Z
F
I
F
(T - Tref) +Z
F
+
I = 1
I = 1
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Modeling élasto- (visco) plastic with metallurgical transformations
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For the calculation of the thermal deformation it is thus necessary to be given:
·
the dilation coefficient of the cold phases,
·
the dilation coefficient 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 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
constraint even lower than the elastic limit of material can nevertheless cause one
unrecoverable deformation (cf [Figure 3-a]).
Transformation
bainitic
= 0 MPa
T
Pt
= - 42 MPa
= - 85 MPa
Pt
Application of
the constraint
Appear 3-a: Courbes dilatometric under uniaxial constraints
of compression for a steel 16 MND5
One calls plasticity of transformation this phenomenon and one notes Pt the 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 the lengthening obtained for a constraint
applied different from zero and that obtained for a null constraint according to the advance of
the transformation, one notes that:
Pt (, b) = (, b) - (,
0 b) = K F (b)
where:
K is a homogeneous constant contrary to a constraint,
F is a standardized function (F ()
0 = 0 and F ()
1 = 1),
and
B is the proportion of the transformed phase.
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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:
3
&pt
~
=
K
F' (b) &
I J
I J
B
2
On the basis of experimental tests and for transformation of a bainitic type of a steel 16MND5
for example: K is taken equalizes at 10-4
- 1
MPa and F (b) = B (2 - b).
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
traditional, proportional to the deviatoric part ~
tensor forced
~
1
= -
T R
I D, (the plasticity of transformation occurs without change of
3
volume, from where a dependence compared to the diverter of the constraints rather than to the field of
constraints 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 constraint must give again
experimental relation.
The phenomenon of plasticity of transformation can exist at the time of structure transformations under
constraints 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:
I = 4
I = 4
I = 4
3
& Pt (, Z) = & Pt
I (, Z)
~
=
K F '
I
I
Zi
&Zi
<
>
2
I = 1
I = 1
I = 1
where: < X > indicates the positive part of a size.
The Ki data and
F
I are provided by the user in DEFI_MATERIAU under 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.
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Modeling élasto- (visco) plastic with metallurgical transformations
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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 i0 is the linear coefficient of work hardening of 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.
The 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 laughed which take account of the restoration
of work hardening.
The term of work hardening of phase I is written R then = R R
I
I
0 I.
4.1
Model with 2 phases with a direction of transformation
To define the variables laughed, 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.
F
voluminal raction V1,
Phase 1 is the mother phase characterized by Pr
oportion of phase (1 - Z),
variable décrouissage
1
R
F
voluminal raction V2,
Phase 2 is the phase produced characterized by Pr
oportion of phase Z,
variable décrouissage
2
R
The equations of evolution of laughed obtained by derivation compared to time are written:
&r1 = &p
éq
4.1-1
&z
&z
&r2 = &p - R + R
Z 2
Z
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 variable internal of the problem as such. The only significance of &p is
here to be the plastic multiplier and it is equal to the rate of equivalent plastic deformation.
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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 V2 fraction of the mother phase is transformed into phase 2 and thus comes to be added with
V2 volume of this phase produced,
·
the element of volume V undergoes a plastic deformation p.
1
1
2
2
T
V
, R (T) + p
V
, R (T
2 2
2 2
)
p
V
, R (T) + p
V, R (T
2
1
1 1)
V
V
, R (T) + p
2
1 1
It is supposed that at the time of the metallurgical transformation, the transformed fraction V2 inherits only one
r1 part of the work hardening of the mother phase (0 1).
Then the variables of work hardening laughed at the moment t+t are such as:
R (T + T) = R (T) + p
1
1
V (R (T) + p
) + V (R (T) + p)
R
(T + T
2
2
2
1
2
) =
V + V
2
2
Maybe, by considering that R (T + T
) = R (T) + R
I
I
I
R = p
1
éq
4.1-2
V
V
R = p
2
+
R
2
-
R
2
V + V
1
V + V 2
2
2
2
2
One obtains the equations [éq 4.1-1] while passing in extreme cases.
For the discretization of the laws of evolutions of laughed, one chooses a diagram of integration clarifies in
using directly the equations [éq 4.1-2].
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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, Perlite, Bainite, Martensite and Austénite, of
K =4
respective proportions Z, Z, Z, Z
and 1 - Z
1
2
3
4
K.
K 1
=
·
In the case of a cooling, the metallurgical transformations to consider are them
transformations of () in (F), (P)
, (B)
or (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 indicates the positive part of X.
4
4
< - Z
> R - - < - Z
> R -
K
K K
K
=1
=
If Z > 0
R
= p K
K 1
+
4
1 - Zk
K =1
if not
R - = 0
and
R -
= 0
éq
4.2-1
< Z
> R - - < Z
> R -
K
K
K
K
If Z > 0
R
= p
K
K
+
Zk
if not
R - = 0
and
R -
K
K = 0
K: proportion of restoration of work hardening at the time of the transformation into K
K: proportion of restoration of work hardening at the time of the transformation K into
For transformations with diffusion (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 diffusion
(ex: martensitic transformation), one will be able to take = 1, work hardening being completely transmitted.
Are provided by the user in operator DEFI_MATERIAU under key word META_RE.
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5
Models of deformation (visco) plastic
The principal 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 traditional plastic behavior; to model the cases at low temperature when
the viscous effects are still negligible or to model the processes at speed
raised (welding),
·
a hammer-hardenable viscoplastic behavior at high temperature, to model 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 zone.
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
)
3
1/2
~ ~
eq
equivalent constraint of von Mises,
:
E Q =
2
R (R; T, Z):
isotropic term of work hardening,
(T, Z)
initial critical stress; corresponds to the minimal constraint
:
C
initial to apply to have a viscoplastic flow.
Plastic rate of flow:
F
~
3
&vp = &
=
&
p
2 eq
The cumulated plastic deformation &p is viscous and is written:
< - R (R;T, Z) - (T; Z)
N
eq
C
>
&p =
éq
5-1
, N: coefficients materials of viscosity.
Handbook of Référence
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Modeling élasto- (visco) plastic with metallurgical transformations
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:
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Note:
1
One can rewrite the equation [éq 5-1] in the form:
- R (R;T, Z) - (T, Z) -
N
p & = 0
eq
C
,
i.e. in this model, the constraint can be interpreted as the sum of a constraint
limit of flow (which breaks up it even into an initial ultimate stress and a term
of work hardening) and a constraint “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:
R = R R
0
R & = p & -
m
C
(R)
The term of evolution of the variable of work hardening R thus comprises 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:
F = ~
(- X) -
eq
C
law of flow
Writing of the rate of deformation (visco) plastic
3
(~
~
- X)
< F N
>
&vp = &p
with =
2
(- X)
&p
eq
2
X = H
0
3
, ~: tensor of constraint and its diverter
: variable tensor of kinematic work hardening,
X: tensor of work hardening associated with the variable tensor with work hardening,
H: kinematic coefficient of work hardening
0
model of evolution of the tensor of work hardening of a material with N phases
(< Z & >) - (Z &)
K
K
K
< >
K
3
& = vp
& + K
K
+
m
C
()
eq
Z
2
eq
Z & - Z &
3
& = vp
& + K K K
K
K +
m
C
()
K
eq
Z
2
K
eq
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By preoccupation with simplification and a same manner that into isotropic, one takes for the term of
viscous restoration:
= Z
I
I
I
C = Z C
I
I
I
m = Z m
I
I
I
ij: coefficients of metallurgical restoration at the time of transformation I
J
C m
,
: coefficients of viscous restoration of phase I.
I
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
the 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 in and C null, and 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:
1
F
p N
- & 0 éq 5.1-1
F < 0
the strict inequality being obtained in the case
(elastic mode).
p & =
0
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 this one.
Note:
It will be noted that C corresponds then to the traditional definition of the yield stress.
y
The elastic limit will be noted C in viscoplasticity and plasticity independent of time
y
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5.2
Borderline case: Model of behavior of viscous fluid
R 0
At very high temperature one a:
C 0
3
if one takes N 1, then: &vp
eq
=
maybe into unidimensional: &vp =
. A model thus is obtained
2
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 modifications 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 = &vp
I
=
= +
R + vp
zi I
I
Ci
I
&i
with
I
More precisely, in the case of the plasticity of von Mises with isotropic work hardening;
·
the function threshold is expressed by:
F (, R;T, Z) =
- R
eq
(T, Z, R) - C (T, Z)
where:
R (T, Z, R) = Zi I
R (T, ir)
I
is the work hardening of multiphase material, IH being that of phase I.
where
C (T, Z) = Zici
I
is the elastic limit of multiphase material, that of phase I.
Ci
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~
·
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:
~f = - R (;rT, Z) - (T, Z)
1
-
I
N
Z p
eq
C
& = 0
I I
I
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:
R = 1
(- F (Z))R + F (Z) R
H
H
·
= 1
(- F (Z
))
+ F (Z
)
C
H
C
H
C
y is the elastic limit of the austenitic phase,
4
Z = Zk is the total proportion of the phases “” '' (F, P
, B
, M
)
K =1
4Z kck
K
=
= 1
is the equivalent elastic limit of the cold phases “” ''
C
Z
4
Z R
K K
R
K 1
= =
is the average work hardening of the cold phases.
Z
~
1
1
Z
·
and &p of it checks F = - R charges (;
R T, Z) - (T, Z) - 1
(- F (Z))
N
p
& -
K
N
p
.
eq
C
H
K & = 0
K
K
Z
F (Z)
H
is a function defined by the user under operand SY_MELANGE of the key word factor
ELAS_META_FO.
Parameters I, I
N, I
C and I
m 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.
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Modeling élasto- (visco) plastic with metallurgical transformations
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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
= E
+ HT
+ vp
+ Pt
=A (T) E
4
Pt
& = 3 ~
K 'F
I
I (1 - Z)
< &Z >
I
2 i=1
4
HT
(Z, T)
=Z.
(T -
ref.
ref.
ref.
T
)
T
R
- 1
(- Z)
+
Z
T T
Z
F
I
F (- ref.)
T
R
+
F
i=1
~
vp
3
& = p&
2 eq
5
F = - R (T, Z, R) - (T, Z) with R (T, Z, R) =
eq
C
Z R (T, R)
I I
I
i=1
p & = 0si F < 0
-
- ~
1
p & 0 if F = 0 and checks F = - R (T, Z, R) - (T, Z) -
I
N
eq
C
Z p&
= 0
I I
I
4
4
< - Z & > R -
K
K K
< - Z & >
K
R
k=1
k=
R & = p
& +
1
- (C moy
R
) m
if Z >,
0 R
& = 0si Z = 0
4
1 - Z
K
k=
1
&
&
< Z >
R - < Z > R
R & = p & +
K
K
K
K -
K
(C moy
R
) m if Z >, 0r & = 0siZ = 0
K
K
K
Z
K
5
with R
=
moy
Ir Zi
i=1
Handbook of Référence
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F
the function threshold,
R, Z R
variables intern work hardening and their forces
I
I I
thermodynamic associated,
WITH = (A
the tensor of elastic rigidity, depend on the temperature,
I J K L)
T (T) and Z (T)
the temperature and the metallurgical structure.
6.2.2 Case with kinematic work hardening
= E
+ HT
+ vp
+ Pt
=A (T) E
4
Pt
& = 3 ~
K 'F
I
I (1 - Z)
< &Z >
I
2 i=1
4
HT
(Z, T)
=Z.
(T -
ref.
ref.
ref.
T
)
T
R
- 1
(- Z)
+
Z
T T
Z
F
I
F (- ref.)
T
R
+
F
i=1
~
vp
3
(- ~
X)
& = p&
2 (- X) eq
5
F = (- X) - (T, Z) with X (T, Z,) =
eq
C
Z X (T,)
I
I
I
i=1
p & = 0 if F < 0
-
- ~
1
p & 0 if F = 0 and checks F = - R (T, Z, R) - (T, Z) -
I
N
eq
C
Z p&
= 0
I I
I
(< z&k >kk) - (< z&k >)
3
vp
K
K
m
& =
& +
+
C
(eq)
if
Z >
0 & = 0 if not
Z
2
eq
Z & - Z &
3
vp
K
K K
K K
m
&k = & +
+
C
(eq)
if
Zk > 0 &k = 0 if not
Z
K
2
eq
with
=
eq
Zi I
I
eq
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F
the function threshold,
R, Z R
variables intern work hardening and their forces
I
I I
thermodynamic associated,
WITH = (A
the tensor of elastic rigidity, depend on the temperature,
I J K L)
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. “ACIER” 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 key word ELAS_META_FO or ELAS_META
elastic parameters E and Naked, dilation coefficients, as well as the elastic limits.
/ELAS_META_FO
: (E: E
NAKED:
F_ALPHA:
F
C_ALPHA:
C
PHASE_REFE:
“CHAUD”
“FROID”
Tref
EPSF_EPSC_TREF:
F C
F1_SY
:
yf1
F2_SY
:
yf2
F3_SY: yf3
F4_SY: yf4
A_SY:
teststemyç
SY_MELANGE
:
F)
with for steel:
F: F
C:
Tref
Tref
F C: F
yfi: elastic limit of phase I
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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 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 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.
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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 = KB
F' f4 = F
m
f4 = km
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
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
F1C = F
CF2 = P
F2C = P
CF3 = B
F3C = B
CF4 = M
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. The data of the key words factors ELAS_META_FO, META_PT and META_RE must
to be well informed.
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6.4 Various relations of élasto-viscoplastic behavior
META_V_ ***
One has in the same way that in traditional 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 traditional plasticity to differentiate the 12 élasto-viscoplastic relations. For
each relation one must inform in ELAS_META or ELAS_META_FO the yield stresses
of flow viscous, in the place of the traditional 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 * the _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 relations META_ **, the internal variables produced in
Code_Aster are:
laughed: 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.
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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 traditional plasticity seems the borderline case (without numerical difficulty
0
associated) of incremental viscoplasticity when C 0
.
C
y
This type of processing was already carried out by LORENTZ [bib15].
1
~
p
N
If one poses F = F -
T
= E + HT + p + Pt
= A
(T) E
4
HT (Z, T) =
Z
.
(T - T) - (1 - R
Tref
R
Z)
F
Z
T - T
+
+
I [F (
) Z F]
I = 1
3
4
Pt
=
~
iK F' I (1 - Z) < Z
>
I
2
I = 1
~
3
p
=
p
2
eq
~
elastic mode:
F < 0 and
p
= 0
~
D
gime (visco) plastic F = 0 and
p
> 0
Handbook of Référence
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Code_Aster ®
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Titrate:
Modeling élasto- (visco) plastic with metallurgical transformations
Date:
29/04/02
Author (S):
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:
R4.04.02-E Page
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4
4
< - Z > R - < - Z > R
K
K K
K
m
R = p k=1
K =1
-
+
-
if
0,
0 if
0
4
(Crmoy)
Z >
R =
Z
=
1 - Zk
K =
1
< Z > R - - < Z > R -
m
if Z > 0, R = 0 if Z = 0
K
K
K
K
K
K
R = p
-
K
+
- (Crmoy)
Z
K
with:
X = X (T + T)
X - = X (T)
X = X (T + 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), T) (v) D
= L (T) v kinematically acceptable
and T
B U = ud (T)
where:
U indicates the field of displacement
B U = ud 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 step of time,
·
iterations of Newton inside a step 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
modifications made to the diagram of integration by the taking into account of the metallurgical evolution
Z (T) and of the plasticity of transformation.
Handbook of Référence
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Modeling élasto- (visco) plastic with metallurgical transformations
Date:
29/04/02
Author (S):
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:
R4.04.02-E Page
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7.2.1 Integration of relations META__ ***
One gives the expression of according to;
·
(or U) unknown of the problem,
·
known terms such variables calculated with the preceding step (-, variables intern…),
the characteristics materials, HT…
= (
WITH T) E
E = early - HT - vp - Pt
One poses: = early - HT
µ
~ = 2µ~e =
- + 2µ ~
~vp
~ Pt
-
(- -)
µ
3K
tr = 3K tr (E) =
-
tr
+ 3K tr
-
3K
µ
~
~
-
3
3
=
+ 2µ ~
- ~
-
F (Z, Z) -
p
µ
2
2
eq
from where
~
~
1
µ
~ -
~
=
+ 2µ
- µ
3
p
1 + µ
3 F (Z, Z
) µ-
eq
with:
~
·
expression of
eq
~ = ~e + ~ Pt + ~
vp
~
~ -
~
3
3
~
~
=
-
+ F (Z, Z) + p
µ
µ-
2
2
2
2
eq
µ
~
2µ
~ +
~ - =
-
(1 (+3ΜF (Z, Z)) +3µ
eq
p)
µ
eq
µ
one poses: 2µ
~
~ - ~
+
= E
µ-
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Modeling élasto- (visco) plastic with metallurgical transformations
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Author (S):
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:
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one a: E = (1 + µ
3 F (Z, Z
)) + µ
eq
eq
3
p
and
~e
~
=
E
eq
eq
·
expression of p
E
eq
That is to say the function of load: F =
- R (R -;T, Z) - (T, Z)
1 + µ
3 F (Z, Z
C
)
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 checks;
1
p
E
N
- 3µ p
eq
=
- R (R -;T, Z) - R p
- (T, Z)
T
1+ 3Μ F (Z, Z
)
0
C
E
1n
~
- 3µ p
eq
-
p
That is to say the function F =
- R (R T
; , Z) - R p
- T
(, Z) -
, p is thus
1+ 3Μ F (Z, Z
0
C
)
T
~
solution of the scalar equation nonlinear 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 by 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 the equal ones to 1.
H is the slope of work hardening of the traction diagram. In the case of isotropic work hardening not
0
linear where the traction diagram is linear per piece, H is defined for the segment to which
0
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 u0 as solution of
problem of speed:
&
u0 T
v D
= L T
v kinematically acceptable
((),) () & ()
where & (
L T) =
&f.v D +
&g.v D
.
Handbook of Référence
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Modeling élasto- (visco) plastic with metallurgical transformations
Date:
29/04/02
Author (S):
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:
R4.04.02-E Page
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The problem from of speed is obtained by deriving compared to time the equations from the problem
continuous:
~&
= 2µ (~ & - &vp - &pt)
3
&PT = F (Z, Z
) ~&
2
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 &vp term. As for the plastic case one a:
~
3
&p
if eq - R (T, Z, R) - y (T, Z) = 0
&vp = 2
eq
0
if eq - R (T, Z, R) - y (T, Z)
0
Derivation compared to the time of the equation
- R (T, Z, R) -
, Z =
E Q
y (T
) 0 give
the expression of &p (relation of consistency).
D
~
~vp
~ Pt
:
~
With & - &
- &
eq
D R
D y
3
(
) D R Dy
-
-
=
-
-
D T
D T
D T
2
D T
D T
eq
Note:
The derivation of A was neglected in this phase of prediction
i=5
i=5
i=5
R & T
(, Z
EFF
,) = Z & R R +
0
Z R & R +
0
Z R R
I
I
I
I I
I
I I
I
0i I
&
i=1
i=1
i=1
i=5
i=5
i=5
= Z & R R +
0
Z R & R +
0
Z R p
I
I I
I
I I
I
0i &
i=1
i=1
i=1
K =4
K =4
+ < Z & > R R +
0
< - Z & > R R
K
K
K
K
0
K K
K =1
K =1
i=5
- < Z & > R R
I
0i I
i=1
= R p
0 & + B
y
&
=
T
y
& +
Z & = C
y
T
Z
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Modeling élasto- (visco) plastic with metallurgical transformations
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:
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from where:
~ ~
I = 4
D
&
eq
D R D
y
:
-
-
= 3µ
- 3Μ K 'F -
<Z & >
- µ +
& - - =
I
I (1
Z)
I
eq
(3 R p B C 0
0)
dt
dt
dt
eq
I =1
~ ~
I = 4
:&
<3µ
- 3Μ K 'F -
<Z & > - - >
I
I (1
Z)
B C
I
eq
eq
I =1
p&=
3µ + R0
From where, finally the expression of & vp:
~ ~
&
vp
3
:
I = 4
& =
3
< µ
3
- µK 'F
&
I
I (1 - Z)
~
<Z >
- B C>
2 (3µ+H
I
eq
0)
eq
I =1
eq
if - R
eq
(T, Z, effi) - y (T, Z) =0
vp
& =0
if - R
eq
(T, Z, effi) - y (T, Z) 0
Taking into account the variations of H and
0i
y according to the temperature and of the structure
metallurgical, one chooses by convenience to neglect the term (B+C) and one thus leads to one
expression of ~&
form:
~
I = 4
~
~
&
& =
3
:
~
µ~
2 & -
<3µ
- 3Μ K 'F
Z
Z&
I
I (1 -
) <
>
>
2 (3µ +
I
eq
R
0)
eq
I =1
eq
I = 4
- 3 K 'F
Z
Z &
I
I (-
)
< > ~
1
I
2 i=1
The expression of ~&
depends on the sign of the term (criterion of load-discharge)
~
I = 4
: ~&
3µ
- 3µ K F'
I
I (Zi) < &
Z >
I
eq.
eq
I = 1
~& is approximated
by:
~
I = 4
~
9 µ
3
3µ
&
~
<
>
= 2 µ &
: ~&
~
-
K
1 Z
Z
1
D
2
2 (
-
-
<
>
I
I
-
3µ + R
I
µ
R
0)
F' (
) & ~
2
3 +
eq
I
0
1
=
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.
Handbook of Référence
R4.04 booklet: Metallurgical behavior
HI-75/01/001/A
Code_Aster ®
Version
6.3
Titrate:
Modeling élasto- (visco) plastic with metallurgical transformations
Date:
29/04/02
Author (S):
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:
R4.04.02-E Page
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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:
i=4
3
µ
3
Lpt = 2µ
K '
F
Z
Z & ~
1
1 D
v D
I I (-)
< >
I
-
()
2 1
µ
I =
3 + R0
To determine U 0, it is necessary to solve after discretization spaces the following linear system of it:
K
BT u0
L
0
Lth Lpt
0
+
+
+
B
0
0
=
0
ud
0
0
On simple cases tests for which there is an analytical solution, one noted that the fact of
to neglect the second member LP 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 one, one determines one as well as possible +1 checking:
F ((un+1) =
((un+1), T) (v) D - L (T) 0
F
F ((un+1) F ((one) +
((un+1) - (one) = 0
((one), T)
From where:
F
K
N
N =
= - F ((U)
((one), T)
With each iteration one solves the linear system:
K
BT one
L
Fn
N
-
with
Fn
N
=
, T v D
N =
((U)) ()
B
0
0
0
During iterations of a step 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.
Handbook of Référence
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Code_Aster ®
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Titrate:
Modeling élasto- (visco) plastic with metallurgical transformations
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29/04/02
Author (S):
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:
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One gives the expression of for the constitution of the consistent tangent matrix of the method
iterative of Newton.
~ 1 (tr)
~ ~ ~
=
+
Id
with
=
3
~
~
=
~
Pt
vp
~
~ (2µ (
-
-
)
One a:
(~
) = Id
with
Id
=
~
(ijkl)
ik
jl
~
~ Pt
3
~ (
) = F (Z, Z)
2
~
~
~
~
3
vp
p
~ (
)
()
=
+
2 ~
p
~
eq
eq
with
~e = 2µ
~
Id
~eeq
~
= µ
~
3 eq
~
1
~
~
2µId µ
~
3
E
eq =
-
eq
eq
eq
(p)
~
The expression of ~ is obtained by deriving F = 0 compared to, which gives:
(p
)
3µ
~
~ =
1-n
p
N
3µ + R +
+
0
(1 3µF (Z,)) eq
Z
N T
T
Handbook of Référence
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Code_Aster ®
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Titrate:
Modeling élasto- (visco) plastic with metallurgical transformations
Date:
29/04/02
Author (S):
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:
R4.04.02-E Page
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from where:
3µp
2µ Id1-
- (3µ)
2
E
eq
~
1
=
~ 1+ 3ΜF (Z, Z)
1
p ~
~
-
1-n
E
N
p
3µ + R +
eq
eq
eq
0
(1+ 3ΜF (Z, Z))
NT T
éq 7.2.2.2-1
Handbook of Référence
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Modeling élasto- (visco) plastic with metallurgical transformations
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:
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7.2.2.3 Operator
tangent
That is to say = (,
,
2
,
2
,
2
)
11
22
33
12
23
the 13 virtual increase in constraint and
that is to say = (,
, 2, 2, 2)
11
22
33
12
23
the 13 virtual increase in deformation,
the operator who binds to is given by the following expression:
2µ
µ
3 p
C
~
p
~e ~
=
1 - C
-
E ~
ij
3rd ik jl
ij
kl
kl
has
has
eq
tr
() = 3K tr ()
with:
1
die
with
drank
[E
(cf.
time
of
not
each
of
Q 7.2.2.1
-
'
(option
1])
'
TANG
RIGI_MECA_
)
= 1+3µF has (Z, Z)
[E
(cf.
current
iterations
at the time
Q 7.2.2.2
-
(option
1])
'
'
FULL_MECA)
0
beginning
with
time
of
not
each
of
C =
3
1
current
iterations
at the time
(3µ) 2
1
C
D
with the ébut
time
of
not
each
of
1
(eeq) 2 3µ + 0
R
and C =
p
2
(3µ)
1
p
C
-
2
2
at the time
iterations
-
1 N
E
E
(eq)
N
p
eq
3µ + R +
0
(+
1 3µ (
F Z, Z)
NT T
:
where
~ ~
I =
:
4
&
1
if elastoplastic and 3µ
- 3Μ K 'F
I
I (1 - Z) <Z
& >
C =
0
I
eq
1
eq
I =1
0 if not
1
(
charge
in
is
one
if
~ ~
0)
plasticize
one
if
and
C =
2
0 if not
Let us note K then the operator such as
= K
and the vector of the diverter of the constraints is S: S = (~
, ~
, ~
,
~
2
,
~
2
,
~
2
)
11
22
33
12
23
13.
Handbook of Référence
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Modeling élasto- (visco) plastic with metallurgical transformations
Date:
29/04/02
Author (S):
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:
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D *
d~ *
=
+ K (1)
1
D *
D *
d~ *
d~ *
1
=
Id - 1 1
D *
d~ *
3
d~ *
2µ
µ
3
p
C
=
1 - C
Id
p S S
d~ *
has
3
E
-
has
eq
from where:
D *
2µ
µ
3
p
1
C
=
1 - C
Id
p
1 1
S S
D *
has
3
E
-
3
-
has
eq
+ K (1)
1
The operator K is written:
2 2
µ
3µ p
C
K = K +
1 - C
p
- S S
11
3 A
3
E
1 1 has
eq
1 2
µ
3µ p
C
K - 1 - C
p
- S S
3 A
3
E
1 has 2
eq
K
22 =
2 2
µ
3µ p
C
K +
1 - C
p
- S S
3 A
3
E
has 2 2
eq
1 2
µ
3µ p
C
K - 1 - C
p
- S S
3 A
3
E
1 has 3
eq
1 2
µ
3µ p
C
K
33 = K -
1 - C
p
-
S S
3 A
3
E
has 2 3
eq
2 2
µ
3µ p
C
K + 1 - C
p
- S S
3 A
3
E
has 3 3
eq
Handbook of Référence
R4.04 booklet: Metallurgical behavior
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Code_Aster ®
Version
6.3
Titrate:
Modeling élasto- (visco) plastic with metallurgical transformations
Date:
29/04/02
Author (S):
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:
R4.04.02-E Page
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CP
-
S S
1 has 4
C
p
-
S S
has 2 4
K = C
44
p
-
S S
has 3 4
2
µ
3µ p
C
1 - C
p
S S
has
3
E -
has 4 4
eq
CP
-
S S
1 has 5
C
p
-
S S
has 2 5
C
p
K
-
S S
55 =
has 3 5
CP
-
S S
has 4 5
2
µ
3µ p
C
1 - C
p
S S
has
3
E -
has 5 5
eq
CP
-
S S
1 has 6
C
p
-
S S
has 2 6
C
p
-
S S
has 3 6
K = C
66
p
-
S S
has 4 6
C
p
- S S
has 5 6
2
µ
3µ p
C S S
1 - C
p 6.6
has
3
E -
has
eq
where “I” components of the Kii vector correspond to “I” terms of the higher part of
ičme column of the symmetrical matrix K
It is noticed that the K0 operator and operator kN 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, R is then replaced
0
by the kinematic coefficient of work hardening H.
0
Handbook of Référence
R4.04 booklet: Metallurgical behavior
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Code_Aster ®
Version
6.3
Titrate:
Modeling élasto- (visco) plastic with metallurgical transformations
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29/04/02
Author (S):
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:
R4.04.02-E Page
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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: Thermal modeling Métallurgique and Mécanique of one
operation of welding - Etude Bibliographique. 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]
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Handbook of Référence
R4.04 booklet: Metallurgical behavior
HI-75/01/001/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 Page
: 36/36
Intentionally white left page.
Handbook of Référence
R4.04 booklet: Metallurgical behavior
HI-75/01/001/A
Outline document