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Law of behavior great deformations with transformations
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Organization (S): EDF-R & D/AMA
Handbook of Référence
R4.04 booklet: Metallurgical behavior
Document: R4.04.03
Law of behavior élasto (visco) plastic
in great deformations with transformations
metallurgical

Summary

This document presents a model of behavior thermo élasto- (visco) plastic at isotropic work hardening
with effects of the metallurgical transformations writes in great deformations. This model can be used for
three-dimensional, axisymmetric modelings and in plane deformations.

One presents the writing of this model and his digital processing.

To include/understand this document, it is practically essential to read the two notes [R5.03.21] and [R4.04.02]
devoted to the written models of behavior, respectively, in great deformations without effects
metallurgical and in small deformations with metallurgical effects.

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Count

matters

1 Introduction ............................................................................................................................................ 3
2 Notations ................................................................................................................................................ 4
3 Recalls of the metallurgical model and the model great deformations ............................................... 5
3.1 Model with metallurgical transformations .................................................................................. 5
3.2 Model written in great deformations ............................................................................................ 6
3.2.1 General presentation ............................................................................................................ 6
3.2.2 Kinematics ........................................................................................................................... 6
4 Extension of the model great deformations ......................................................................................... 8
4.1 Thermodynamic aspect ................................................................................................................ 8
4.2 Extension ......................................................................................................................................... 9
4.3 Relations of behavior ............................................................................................................. 9
4.4 The various relations ................................................................................................................ 12
4.5 Constraints and variables intern ................................................................................................... 13
5 numerical Formulation ........................................................................................................................ 14
5.1 Integration of the various relations of behavior ................................................................. 14
5.2 Form of the tangent matrix ................................................................................................ 18
6 Bibliography ........................................................................................................................................ 20

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1 Introduction

This document presents a law of behavior thermo élasto- (visco) plastic at isotropic work hardening
in great deformations which takes into account the effects of the metallurgical transformations. It
model can be used for three-dimensional, axisymmetric problems and in plane deformations.

This law represents a “assembly” of two models established in Code_Aster, namely one
thermoelastoplastic model with isotropic work hardening written in great deformations (key word
factor DEFORMATION: “SIMO_MIEHE”, cf [R5.03.21]) and a model small deformations
thermo élasto- (visco) plastic with effects of the metallurgical transformations (key word factor
“META_P_ ** _ **” or “META_V_ ** _ **” of COMP_INCR of operator STAT_NON_LINE). The first
model of great deformations was thus wide to take account of the consequences of
metallurgical transformations on mechanics.

To include/understand this document, it is practically essential to read the reference documents
[R5.03.21] and [R4.04.02] which concerns, respectively, the model great deformations without effects
metallurgical and the model small deformations with metallurgical effects. Nevertheless, to facilitate
the reading of this note, we make some recalls on these two models.

To justify the extension of the model written in great deformations to the model great deformations
with metallurgical effects, we take again some theoretical aspects extracted from [bib1] related to
the writing of the model great deformations.

One presents then the relations of behavior of the complete model, his numerical integration and
forms of the tangent matrix (options FULL_MECA and RIGI_MECA_TANG).

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2 Notations

One will note by:

Id
stamp identity
tr A
trace tensor A
AT
transposed of tensor A
det A
determinant of A
X
positive part of X
~
~
1
With
deviatoric part of tensor A defined by A = A - (tr A) Id
3

T
:
doubly contracted product: A: B = A B
ij ij = tr (AB)
I, J

tensorial product: (A B) ijkl = ij
With kl
B

3
With
~ ~
eq
equivalent value of von Mises defined by Aeq =
:
WITH A
2

With
TESTSTEMXŔ
gradient: TESTSTEMXŔ =
X

ij
With
divx A
divergence: (div
)
X A.i. =
X
J
J
, µ

E
E
coefficients of Lamé: =
, µ =

(1 +) (1 -
2)
2 (1 +)

Young modulus

Poisson's ratio

E
modulate rigidity with compression: 3K = 3 + =

(1 - 2)
T
temperature
Tref
temperature of reference
Z
proportion of austenite
Zi
proportion of the four phases: ferrite, pearlite, bainite and martensite

In addition, within the framework of a discretization in time, all the quantities evaluated at the moment
precedent are subscripted by -, the quantities evaluated at the moment T + T
are not subscripted and them
increments are indicated par. One has as follows:

Q = Q - Q
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3
Recalls of the metallurgical model and the model large
deformations

3.1
Model with metallurgical transformations

We present only here the consequences of the metallurgical transformations on
mechanical behavior.

Determination of the mechanical evolution associated a process bringing into play
metallurgical transformations requires a metallurgical thermo calculation as a preliminary. This calculation
thermo metallurgical is uncoupled and allows the determination of the thermal evolutions then
metallurgical. For the metallurgical models of behavior of steels, one will be able to consult
note [R4.04.01].

For the study of the metallurgical transformations of steel, there are five metallurgical phases:
ferrite, pearlite, the bainite, martensite (phases) and austenite (phase).

The effects of the metallurgical transformations (at the solid state) are of four types:

· the mechanical characteristics of the material which undergoes the transformations are modified.
More precisely, elastic characteristics (YOUNG modulus E and coefficient of
Poisson) are not very affected whereas the plastic characteristics, such as the limit
of elasticity, are it strongly,
· the expansion or the voluminal contraction which accompanies the metallurgical transformations
results in a deformation (spherical) of “transformation” which is superimposed on
purely thermal deformation of origin. In general, one gathers this effect with that due to
modification of the thermal dilation coefficient,
· 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. One calls “plasticity of transformation” this phenomenon. Total deflection
is written then:

= E + HT + p + Pt

where E, HT, p and Pt are, respectively the elastic strain, thermal,
plastics and of plasticity of transformation,
· 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 completely transmitted to the phases
lately created. Those can then be born with a virgin state of work hardening or
to inherit only one part, even totality, the work hardening of the mother phase.
cumulated plastic deformation p is not then any more characteristic of the state of work hardening and it
is necessary to define other variables of work hardening for each phase, noted rk which hold
count restoration. The laws of evolution of these work hardenings differ from the laws
usual so as to allow a “return towards zero” total, or partial, of these parameters
at the time of the transformations.

One will be able to find in the document [R4.04.02] the expressions of the various relations of
behavior.
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3.2
Model written in great deformations

3.2.1 Presentation
general

This model is a law of behavior thermo eulérienne elastoplastic written into large
deformations which was proposed by Simo and Miehe ([bib2]) which tends under the assumption of small
deformations towards the model with isotropic work hardening and criterion of von Mises describes in [R5.03.02].
It makes it possible to treat not only the great deformations, but also, in an exact way, them
great rotations.

The essential characteristics of this law are as follows:

· just like in small deformations, one supposes the existence of a slackened configuration,
i.e. locally free of constraint, which makes it possible to break up the total deflection into
a thermoelastic part and a plastic part,
· the decomposition of this deformation in parts thermoelastic and plastic is not any more
additive as in small deformations (or for the models great deformations written in
rate of deformation with for example a derivative of Jaumann) but multiplicative,
· as in small deformations, the constraints depend only on the deformations
thermoelastic,
· to write the law of behavior, one uses the tensor of the constraints of Kirchhoff, which is
connected to the tensor of Cauchy by the relation J = where J represents the variation of volume
between the configurations initial and current,
· the plastic deformations are done with constant volume. The variation of volume is then
only due to the elastic thermo deformations,
· this model led during its numerical integration to a model incrémentalement objective
what makes it possible to obtain the exact solution in the presence of great rotations.

3.2.2 Kinematics

We make here some basic recalls of mechanics in great deformations and on the model of
behavior.

Let us consider a solid subjected to great deformations. That is to say the 0 field occupied by the solid
before deformation and (T) the field occupied at the moment T by the deformed solid. In the configuration
initial 0, the position of any particle of the solid are indicated by X (Lagrangian description).
After deformation, the position at the moment T of the particle which occupied position X before deformation
is given by variable X (description eulérienne).

The total movement of the solid is defined, with U displacement, by:

X = x$ (X, T) = X + U

To define the change of metric in the vicinity of a point, one introduces the tensor gradient of
transformation F:

x$
F =
= Id + U


X
X
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The transformations of the element of volume and the density are worth:


D = Jdo with J
O
= det F =

where O and are respectively the density in the configurations initial and current.

To now write the model great deformations, the existence of a configuration is supposed
slackened R, i.e. locally free of constraint, which then makes it possible to break up
total deflection in parts thermoelastic and plastic, this decomposition being
multiplicative.

One will note by F the tensor gradient which makes pass from the initial configuration 0 to the configuration
current (T), by F p the tensor gradient which makes pass from configuration 0 to the configuration
slackened R, and Fe of the configuration R with (T). The index p refers to the plastic part, the index
E with the thermoelastic part.

Initial configuration
Current configuration
F

(T)
0
F p
F E
T = Tref
R
= 0
Slackened configuration

Appear 3.2.2-a: Décomposition of the tensor gradient F in an elastic part Fe and plastic F p

By composition of the movements, one obtains the following multiplicative decomposition:

F = FeF p

The thermoelastic deformations are measured in the current configuration with the tensor
eulérien of left Cauchy-Green Be and plastic deformations in the initial configuration by
the tensor G p (Lagrangian description). These two tensors are defined by:

Be
FeFeT
=
, G p
F pTF p
=
-
(
) 1 from where Be
FG pFT
=


The model presented is written in such manner to distinguish the isochoric terms from the terms of
change of volume. One introduces for that the two following tensors:

F = -
J 1/F
3 and Be = - 2/B
3rd
J
with J = det F
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By definition, one a: det F = 1 and det Be = 1.

In this model, the plastic deformations are done with constant volume so that:

J p
p
= det F = 1 from where J I
E
=
= det F

One will find in the reference document ([R5.03.21]) the expressions of the relations of
behavior.

4
Extension of the model great deformations

The objective of this paragraph is to justify the extension of the model written in great deformations for
to take account of the metallurgical transformations. In particular, to take account of the plasticity of
transformation, we cannot add as in small deformations a term
additional of deformation related to the plasticity of transformation. In fact, on the aspect decomposition
kinematics, the taking into account of the plasticity of transformation does not change anything. One always has
decomposition F = FeF p where F p thus contains all information on the “anelastic” deformation (
including that related to the plasticity of transformation). It is only on the level behavior that
fact, in particular, processing of the plasticity of transformation.

Initially, we point out some theoretical elements which make it possible to write it
model without metallurgical effects then we show the modifications to be made to hold account
metallurgical effects and plasticity of transformation in particular.

4.1 Aspect
thermodynamics

The writing of the law of behavior great deformations is from the thermodynamic framework with
internal variables. The thermodynamic formalism rests on two assumptions. First is that
the free energy depends only on the elastic strain Be and of the variables intern related to
the work hardening of the material (here cumulated plastic deformation associated the variable of work hardening
isotropic R). This allows, thanks to the inequality of Clausius-Duhem, to obtain the laws of state. The second
assumption is the principle of maximum dissipation, which corresponds to the data of a potential of
dissipation, which then makes it possible to determine the laws of evolution of the internal variables.

The free energy is given by:

= (Be,) = E (Be) + p
p
(p)

One obtains by the first assumption, the laws of state, that is to say:

E
p

=

2
E
0
B and R =

E
B
0 p

It remains for dissipation:

1
:(-
& p T e-1
FG F B
) - &
RP 0
2
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With the help of the introduction of a function threshold such as F (, R) 0, the principle of dissipation
maximum (or in an equivalent way the data of a pseudopotential of dissipation [bib3]) allows
to deduce some, by the property of normality, the laws of evolution, is:

F
- 1
p T e-1
F
FG
& F B
= & and &p = - &
2


R

It is here about a model of associated plasticity.

4.2 Extension

For the restoration of work hardening, there are no particular difficulties been dependant on large
deformations. It is enough that the free energy depends, either to the cumulated plastic deformation, but
variables intern work hardening rk associated with the variables with work hardenings Z .R
K
K of each one
metallurgical phases.

To take maintaining account of the deformations due to the plasticity of transformation, one proposes
to add an additional term in the law with flow of the plastic deformation G p which
derives from a potential of dissipation.

One obtains thus for the laws of state:

E
p

=

2
E
0
B and Z .R =

E
B
K
K
0 rk

and for the laws of evolution:

Pt
- 1
p T e-1
F

FG
& F B
= & +

2


123
plasticity of transformation
R

&
R = - &
F
K


-

(Z .R)
(Z .R)
K
K
K
K
1 2
4
3
4
restoration D work hardening
metallurgical and viscous
= Pt + R
()


One chooses the potentials Pt and R, respectively related to the plasticity of transformation and on
restoration of work hardening, such manner to find, under the assumption of the small deformations, them
same laws of evolution as those of the model with metallurgical effects writes in small deformations.

4.3
Relations of behavior

A linear isotropic work hardening in the case of is placed.

The partition of the deformations implies:

Be
FG pFT
=
with F = -
J 1/F
3, J = det F and Be = - 2/B
3rd
J

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The relations of behavior are given by:

· Thermoelastic relation stress-strain:

~
~
= µbe
3K
2
9K HT
1
tr =
(J -)
1 -
(J +)
2
2
J
4
HT
R
Tref
R
Tref
=
Z [(T - Re
T F) - (1 -
Z)
F] + (iZ [) F (T - rTef) + Z
F]
i=1

where: Z R characterizes the metallurgical phase of reference
Zr = 1 when the phase of reference is the austenitic phase,
Zr = 0 when the phase of reference is the ferritic phase.
Tref
HT
HT
F
= F (Re
T F) - (Re
T F) translated the difference in compactness between the ferritic phases
and austenitic at the temperature of Tref reference,
F is the dilation coefficient of the four ferritic phases and that of the phase
austenitic.

· Threshold of plasticity:

F = - R
eq
- y

R is the variable of work hardening of the multiphase material, which is written:
F (Z) 4
4
R = (1 - F (Z))R +
Z .R, Z = Z
Z
I
I
I
i=1
i=1

where Rk is the variable of work hardening of the phase K which can be linear or not linear by
report/ratio with rk and F (Z) a function depending on Z such as F (Z) [
0,]
1.
In the linear case, there are R = R R
K
0k K where R0k is the slope of work hardening of the phase K.
(I)
(I)
(I)
(I)
In the nonlinear case, one writes: R = R
+ R (R - R
K
K
K
K
)
0
where significances of Rk,
R (I)
(I)
0k and rk
are represented on the figure below.
Rk
(3)
Rk (2)
(2)
Rk
R0 K
(1)
(1)
R
R
0 K
K
(0)
R0k
(0)
rk
Rk (0) (1)
(2)
(3)
R
R
K
rk
K
rk

Nonlinear curve of work hardening
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The elastic limit is worth there:
4
I
Z
y I
If Z 0
=
,
I 1
y = (1 - F (Z)
) y + F (Z
) y,
y
=

Z
If Z = 0, y = y

where yi is the four limit elastic of the ferritic phases,
y that of the phase
autenitic.

· Laws of evolution:

3
4
FG
& pFT = - &p
B
~ E - 3 B
~ E K F (1 - Z)
I I
&i
Z



eq
i=1
4
- &Z (R - R)
I
I I

&r
=1
= &p
I
+
- (Cr) m
Z
if
> 0
Z
moy


1 2
4
3
4
only in viscosity
&Z (R
- R)
I
I
I
&r = &p +
- (Cr) m
Z
I
if > 0
Z
moy
I
I
1 2
4
3
4
only in viscosity
5
5
5
R
= Z R
moy
K K, C = ZkCk, m = Zk K
m
K =1
k=1
k=1

where Ki, Fi, I
C and I
m are data of material associated with phase I, I it
coefficient of restoration of work hardening at the time of the transformation into I (I [
0]
1
,) and I
the coefficient of restoration of work hardening at the time of transformation I into (I [
0]
1
,).

All the data material are indicated in operator DEFI_MATERIAU ([U4.43.01]) under
various key words factors ELAS_META (_F0) and META_ **.

For a model of plasticity, the plastic multiplier is obtained by writing the condition of
coherence &f = 0 and one a:

&p,
0 F 0 and &pf = 0
In the viscous case, &p is written:
N
F
&p =







or in an equivalent way:
1/
4
F =
F
1
(- F (Z))
N
1/I
N
p&
+ Zii p&

Z i=1
where in and I
are the viscosity coefficients of material associated with phase I which depend
possibly of the temperature.
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The calculation of FG
& pFT gives:
~
~~
1


FG
& pFT = - (
E
eq
3 Aeq + &p) (tr B
+
)
3
2
eq
µ eq
4
where one posed A = K F &
Z
I I
I.
i=1
Since ~
/
2
eq 1 and ~~
/eq 1, the second term of the expression above can be neglected
(in front of 1) for metallic materials insofar as:

eq R +y
- 3
E
µ =
µ
10 << 1 tr B
tr Be 1 bus the tensor Be is symmetrical, definite positive and det Be = 1.
It is this simplification of the law of evolution of Gp which makes it possible to integrate the law easily of
behavior i.e. to bring back it to the solution of a nonlinear scalar equation. One
will thus take thereafter:
tr Be
FG
& pFT - (&p +eq A)
~


éq
4.3-1
eq

4.4
Various relations

In operator STAT_NON_LINE, one reaches these various models by using the key words factors
following:

| COMP_INCR: (
RELATION
:
/
“META_P_IL”
/
“META_P_INL”
/
“META_P_IL_PT”
/
“META_P_INL_PT”
/
“META_P_IL_RE”
/
“META_P_INL_RE”
/
“META_P_IL_PT_RE”
/
“META_P_INL_PT_RE”
/
“META_V_IL”
/
“META_V_INL”
/
“META_V_IL_PT”
/
“META_V_INL_PT”
/
“META_V_IL_RE”
/
“META_V_INL_RE”
/
“META_V_IL_PT_RE”
/
“META_V_INL_PT_RE”

DEFORMATION
:
/“SIMO_MIEHE”




)

We point out only here the significance of the letters for behaviors META:

· P_IL: plasticity with linear isotropic work hardening,
· P_INL: plasticity with nonlinear isotropic work hardening,
· V_IL: viscoplasticity with linear isotropic work hardening,
· V_INL: viscoplasticity with nonlinear isotropic work hardening,
· Pt: plasticity of transformation,
· RE: restoration of metallurgical work hardening of origin.
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Example: “META_V_INL_RE” = elastoviscoplastic law with nonlinear isotropic work hardening with
restoration of work hardening but without taking into account of the plasticity of transformation

The various characteristics of material are given in operator DEFI_MATERIAU. One
return the reader to the note [R5.04.02] for the significance of the key words factors of this operator.

Caution:

If isotropic work hardening is linear, one informs under key word META_ECR0_LINE of
DEFI_MATERIAU, the module of work hardening i.e. the slope in the plan forced ­
deformation.
On the other hand, if isotropic work hardening is nonlinear, one gives directly under the key word
META_TRACTION of DEFI_MATERIAU, the isotropic curve work hardening R (R = - y) in

function of the cumulated plastic deformation p (p = -
).
E

Note:

The user must make sure well that the “experimental” traction diagram used for in
to deduce the slope from work hardening is well given in the plan forced rational = F/S
- deformation logarithmic curve ln (1+ L/L)
0 where l0 is the initial length of the useful part of
the test-tube, L variation length after deformation, F the force applied and S
F L 1
F L
current surface. It will be noticed that = F/S =
from where = J =
. In
S L J
0 0
S L
0 0
F L
General, it is well the quantity
who is measured by the experimenters and this gives
S L
0 0
directly the constraint of Kirchhoff used in the model of Simo and Miehe.

4.5
Internal constraints and variables

The constraints of output are the stresses of Cauchy, therefore measured on the configuration
current.
For the whole of relations META_ **, the internal variables produced in Code_Aster are:

· V1: r1 variable of work hardening for ferrite,
· V2: r2 variable of work hardening for the pearlite,
· V3: r3 variable of work hardening for bainite,
· V4: r4 variable of work hardening for martensite,
· V5: r5 variable of work hardening for austenite,
· V6: indicator of plasticity (0 if the last calculated increment is elastic; 1 if not),
· V7: R the isotropic term of work hardening of the function threshold,
1
· V8: the trace divided by three of the tensor of elastic strain E
B is
E
trb.
3

Handbook of Référence
R4.04 booklet: Metallurgical behavior
HT-66/02/004/A

Code_Aster ®
Version
6.3
Titrate:
Law of behavior great deformations with transformations
Date:
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Author (S):
V. CANO
Key:
R4.04.03-B Page
: 14/20


5 Formulation
numerical

For the variational formulation, it is about same as that given in the note [R5.03.21] and which
refers to the law of behavior great deformations. We point out only that it acts
of a eulérienne formulation, with reactualization of the geometry to each increment and with each
iteration, and which one takes account of the rigidity of behavior and geometrical rigidity.
We now present the numerical integration of the law of behavior and give
the form of the tangent matrix (options FULL_MECA and RIGI_MECA_TANG).

5.1
Integration of the various relations of behavior

In the case of an incremental behavior, key word factor COMP_INCR, knowing the tensor of
constraints -, the variables intern R -
K, the trace divided by three of the tensor of deformations
1
rubber bands
E
trb, displacements U and U, the temperatures T - and T, and proportions of
3
1
various metallurgical phases Z
E
K, Zk, one seeks to determine (,
R,
b)
K
tr
.
3
Displacements being known, gradients of the transformation of 0 with -, noted F, and of -
with (T), noted F, are known.
One will pose thereafter:
4
- Z (R - R)
4
I
I I

Z R
- R
(
)
WITH = K F Z
i=1
I
I
I
I I
I, G =
and Gi =
(I = 1, 4)
i=1
Z
Zi

The implicit discretization of the law gives:

F = FF


J = det F
F = -
J 1/F
3
Be = - 2/B
3rd
J

J =
~
~
= µbe
3K
9K
1
tr =
(2
J -)
1 -
HT
(J +)
2
2
J
4
HT
R
Tref
R
Tref
=
Z [(T - Re
T F) - (1 -
Z)
F] + (iZ [) F (T - rTef) + Z
F]
i=1
F 4
F = - (1 - F) R -
Z R
eq
-
Z
I I
y
i=1
tr Be
Be = FG pFT = FG p-FT -
~ - tr Be
p
With
~



eq
If Z
-
m
-
> 0 then R = p + G -
T (Crmoy)
, if not R = 0 and R
1 2
4
3
4

= 0
only in viscosity
If Z
-
m
-
I > 0, R = p + G -
T (Cr
I
I
moy)
, if not R = 0 and R
1 2
4
3
4
I
I = 0
only in viscosity
Handbook of Référence
R4.04 booklet: Metallurgical behavior
HT-66/02/004/A

Code_Aster ®
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6.3
Titrate:
Law of behavior great deformations with transformations
Date:
20/01/03
Author (S):
V. CANO
Key:
R4.04.03-B Page
: 15/20


In the resolution of this system, only the deviatoric constraint ~
is unknown because the trace of is
function only of J (known).
One introduces Tr, the tensor of Kirchhoff which results from an elastic prediction (Tr: trial, in English
test):

~
~
Tr
eTr
= µb


where

beTr
FG p-FT
Fbe-

FT
=
=
, F
= (J) - 1/3F and J = det
(
)
F

One obtains Be starting from the constraints - by the thermoelastic relation stress-strain and
starting from the trace of the tensor of the elastic strain.

~-
E
1
E
B
=
+ trb
-

µ
3

One obtains for the tensor of Kirchhoff:

eTr
~
~
tr B
eTr
~
= µb
- µ p

- µ A
tr eTr~


B



eq

If F < 0, one has p then = 0 and:
~Tr
~

=

1+ µ tr eTr
With B

if not one obtains:

tr Be
tr beTr
=



eTr

~
tr B
1
+ µp
+ A tr eTr
~

B
= Tr



µ
eq



By calculating the equivalent constraint, one obtains the scalar equation out of p following:


eTr
eTr
Tr
eq + µ p
tr B
+ µ
With eq tr B
= eq
Handbook of Référence
R4.04 booklet: Metallurgical behavior
HT-66/02/004/A

Code_Aster ®
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Titrate:
Law of behavior great deformations with transformations
Date:
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Author (S):
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Key:
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Expression of eq:

In plasticity:
-
eq = y + R p
+ D (R;T, Z)
with
F 4
R = (1 - F) R0 +
Z R
Z
I 0i
i=1
4
F
and D (R -;T, Z) = [1 - F] R (R -
+ G)
+
Z R (R + G)
Z
I I I
I
i=1

4
1/N
F
In viscosity:
-

eq = y + R
p + D (R;T, Z) + 1
(- F (Z))
1/I
N

(p/T)
+ Zii
(p/T)

Z i=1
with
D (R;T, Z) = [1 - F] R (R
+ G - T (Cr) m)
moy
4
F

+
Z R (R + G - T (Cr) m)
moy
Z
I I I
I
i=1

p checks:

Tr
eTr
4
1/N
F
- µ p
tr B
1
(- F (Z
))
(p
/T


1/

)
+ Z (p
/T N
eq
I
)
=
- D (R;T, Z) - - R p


Z
I I
y
i=1
1+ µ A
eTr
'tr B


The resolution is made in Code_Aster by a method of the secants with interval of search
[bib4].

Note:

In the case of a nonlinear isotropic work hardening, slopes of R0k work hardening and them
work hardenings R
-
K in the expressions of R.&.d (R;T, Z) correspond to the variables
R
-
-
m
K taken at the moment T, i.e. R = R + G + p - T (Cr
K
K
K
moy). However, like one
does not know a priori the value of these variables rk, one solves the equation out of p by taking them
slopes R
-
-
m
0k and Rk work hardenings for the quantities R + G

- T (Cr
K
K
moy). Once
solved the equation out of p, one checks, for each phase, which one is well in the good
interval during the calculation of work hardening and the slope. In the contrary case, for
phases concerned, the following interval is taken and the equation in p. again is solved.
One continues this process until finding the good interval for all the phases.

One finds then for the diverter of the constraints:


eTr
~
1
p tr B
~
=
1 - µ
Tr
1+ µA tr eTr
Tr
B


eq


Handbook of Référence
R4.04 booklet: Metallurgical behavior
HT-66/02/004/A

Code_Aster ®
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Titrate:
Law of behavior great deformations with transformations
Date:
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Author (S):
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Once calculated cumulated plastic deformation, the tensor of the constraints and the tangent matrix,
one carries out a correction on the trace of the tensor of the elastic strain E
B to hold account
plastic incompressibility, which is not preserved with the simplification made on the law
of flow [éq 4.3.1]. This correction is carried out by using a relation between the invariants of E
B
~
and E
B and by exploiting the plastic condition of incompressibility p
J = 1 (or in an equivalent way
det E
B = 1). This relation is written:

x3- J ex- (1 - J E) = 0
2
3

2
1 ~
()
~
~
E
E 2
eq
E
E

1
with J2 = (b) eq =
, J = det
3
B = det and
E
X = trb
2
2
(
2 µ)
µ
3
The solution of this cubic equation makes it possible to obtain
E
trb and consequently
thermoelastic deformation Be with the step of next time. If this equation admits
several solutions, one takes the solution nearest to the solution of the step of previous time. It is
1
moreover why one stores in an internal variable
E
trb.
3


Note:

If the plasticity of transformation is not taken into account, expressions
obtained while taking A = 0 are the same ones.
If 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.
Handbook of Référence
R4.04 booklet: Metallurgical behavior
HT-66/02/004/A

Code_Aster ®
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Titrate:
Law of behavior great deformations with transformations
Date:
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Author (S):
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5.2
Form of the tangent matrix

We give only here the forms of the tangent matrix (option FULL_MECA to the course
iterations of Newton, option RIGI_MECA_TANG for the first iteration). For the assumptions
concerning the metallurgical part, they are the same ones as those of the document [R4.04.02]. For
part great deformations, one will find in appendix of [bib1], the detail of the linearization of the law of
behavior.

One poses:
J = det F, J -
-
= det F and J = det F

· For option FULL_MECA, one a:

-

()/
J 1 3
1
J -
With =
=
H -
(HF) B -
B
F

J
3
J J
J 2

J -
3
HT
- 2
+
KJ -

K
(1 - J) Id B
J
2


where B is worth:
B = F

F

- F

F
11
22
33
23 32
B = F

F

- F

F
22
11
33
13 31
B = F

F

- F

F
33
11
22
12 21
B = F

F

- F

F
12
31
23
33 21
B = F

F

- F

F
21
13
32
33 12
B = F

F

- F

F
13
21
32
22 31
B = F

F

- F

F
31
12
23
22 13
B = F

F

- F

F
23
31
12
11 32
B = F

F

- F

F
32
13
21
11 23

and where H and HF are given by:

In the elastic case (F < 0):

µ
2
H
(B E-F

+ F
B E - F
B E - 2 A
~ F
B E
=
-)
ijkl

(1+ µ A
eTr
ik LP
jp
IP pl
jk
tr b)
3 ij
kp LP
ij
kp pl

and

~
HF =
(beTr - A tr beTr~)
(1+
µ A tr beTr)
Handbook of Référence
R4.04 booklet: Metallurgical behavior
HT-66/02/004/A

Code_Aster ®
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Titrate:
Law of behavior great deformations with transformations
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Author (S):
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if not in plastic or viscoplastic load, one a:

µ
H
= (B E-F

+ F
B E)
ijkl
ik LP has
jp
IP pl
jk

R (

With
+ p
) ~


ij
eq
ij
- 2µ
+
F
B E
has
3
(R
eTr
+ µ tr B (1+ R A
))
kp LP

eq


2
eTr
µ
3
tr B
(R p
-)

eq
+
~ ~ F
B E
3 (R have
eTr
+ µ tr B (1+ R A
)) ij kq qp LP
eq

and


Id
R
(
With + p
) ~

HF =
beTr - 2µ tr beTr
eq

+

has
3a (R
eTr

eq
+ µ tr B
1
(+ R
With))
3 2
µ tr beTr (R
p - eq)
+
(~:beTr) ~


3
(R has
eTr
eq
+ µ tr B
1
(+ R
With))

with

4
F
1
(- N)/N
F


1
(-)/
R = 1
(- F)
I
N
I
N
0
R +
Zi 0
R + 1
(- F (Z)) (p
/T
)
/N T
+
Z (p
/T
)
/N T

Z
I
I I
I
i=,
1 4
Z
1
4
4
4
4
4
4
4
4
4
4
4
4
4
2
I 1
=
4
4
4
4
4
4
4
4
4
4
4
4
4
3
you
in viscosi

only

Tr

has
eq
=
eq

· For option RIGI_MECA_TANG

for the plastic model: they are the same expressions as those given for
FULL_MECA but with p = 0 and A = 0, all variables and coefficients of material
being taken at the moment T -. In particular, there will be F = Id.

for the viscous model: one takes only the expressions of FULL_MECA in the case
rubber band with A = 0, all variables being taken at the moment T -.

Handbook of Référence
R4.04 booklet: Metallurgical behavior
HT-66/02/004/A

Code_Aster ®
Version
6.3
Titrate:
Law of behavior great deformations with transformations
Date:
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Author (S):
V. CANO
Key:
R4.04.03-B Page
: 20/20


6 Bibliography

[1]
CANO V., LORENTZ E., “Introduction in Code_Aster of a model of behavior in
great elastoplastic deformations with isotropic work hardening ", internal Note E.D.F
D.E.R., HI-74/98/006/0, 1998
[2]
SIMO J.C., MIEHE C., “Associative coupled thermoplasticity At finite strains: Formulation,
numerical analysis and implementation ", Comp. Meth. Appl. Mech. Eng., 98, pp. 41-104,
North Holland, 1992.
[3]
LEMAITRE J., CHABOCHE J.L., Mécanique of the continuous mediums, Editions Dunod 1985
[4]
E. LORENTZ, numerical Formulation of the viscoplastic law of behavior of Taheri,
Note intern E.D.F-D.E.R. HI-74/97/019/A [R5.03.05].

Handbook of Référence
R4.04 booklet: Metallurgical behavior
HT-66/02/004/A

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