Code_Aster ®
Version
6.4
Titrate:
Behavior élasto-visco-plastic of J.L.Chaboche
Date:
08/12/03
Author (S):
P. of BONNIERES, J.M. PROIX, E.LORENTZ Clé
:
R5.03.04-B Page
: 1/22
Organization (S): EDF-R & D/AMA
Handbook of Référence
R5.03 booklet: Nonlinear mechanics
Document: R5.03.04
Relations of behavior élasto-visco-plastic
of Chaboche
Summary:
This document describes the integration of the model of behavior élasto-visco-plastic of J.L. Chaboche with
nonlinear and isotropic kinematic work hardening. Two versions of the model are available in
Code_Aster:
· a version with one or two variables kinematics, introduced recently, takes into account all them
variations of the coefficients with the temperature, and has an effect of work hardening on the variables
tensorial of recall. This version also makes it possible to model (in an optional way) the character
viscous of the material (viscosity of Norton). It is integrated by the solution of only one equation
nonlinear scalar.
This model is available in 3D, plane deformation, axisymetry. Modeling in plane constraint
use a method of condensation static (of Borst).
· a version with two variables kinematics which exists in Code_Aster since the version 2, which
does not take into account all the variations of the coefficients compared to the temperature, but which was
used for several studies, and for which one has sets of identified parameters. This
version is integrated in environment PLASTI. It does not make it possible to model viscosity.
It does not take into account of effect of work hardening on the tensorial variable of recall.
This model is available in 3D, plane strain, plane stress and axisymetry.
One gives also elements to identify the coefficients of the relation of behavior.
Handbook of Référence
R5.03 booklet: Nonlinear mechanics
HT-66/03/005/A
Code_Aster ®
Version
6.4
Titrate:
Behavior élasto-visco-plastic of J.L.Chaboche
Date:
08/12/03
Author (S):
P. of BONNIERES, J.M. PROIX, E.LORENTZ Clé
:
R5.03.04-B Page
: 2/22
1 Models élasto-visco-plastics of J.L. Chaboche
available in Code_Aster
For the structural analysis subjected to cyclic loadings, work hardenings isotropic (linear
or not) and linear kinematics traditional [R5.03.02] and [R5.03.16] are not sufficient any more. In
private individual, one cannot correctly describe the stabilized cycles obtained in experiments on
a tensile specimen subjected to an alternated imposed deformation or a traction and compression.
If one seeks to precisely describe the effects of a cyclic loading, it is desirable to adopt
modelings more sophisticated (but easy to use) such as the model of Said Taheri, by
example, cf [R5.03.05], or if the number of cycles is limited the model of Jean-Louis Chaboche
who is introduced here.
Actually, the model of J.L. Chaboche can be more or less sophisticated. Developed models
in Code_Aster either a kinematic variable (VISC_CIN1_CHAB comprise) or two
(VISC_CIN2_CHAB and CHABOCHE), and of isotropic work hardening.
The choice to use two variables kinematics complicates certainly the model, but makes it possible to identify
correctly uniaxial tests in a broader range of deformations [bib2], [bib7]. One
certain number of indentifications of the parameters of this model were carried out mainly for
the stainless steels A316 and A304 ([bib7], [bib8]).
The models comprise 8 parameters (a kinematic variable) or 10 (two variables
kinematics), introduced into command DEFI_MATERIAU:
CIN1_CHAB (CIN1_CHAB_FO)
= _F (
R_0 = R_0,
R_I
=
R_I, (useless
if
B=0)
B
=
B,
(defect: 0.)
C_I = C_I,
K
=
K,
(defect: 1.)
W
=
W,
(defect: 0.)
G_0
=
G_0,
A_I
=
A_I, (defect
:
0.)
)
CIN2_CHAB (CIN2_CHAB_FO)
= _F (
R_0 = R_0,
R_I
=
R_I, (useless
if
B=0)
B
=
B,
(defect: 0.)
C1_I = C1_I,
C2_I = C2_I,
K
=
K, (defect: 1.)
W
=
W, (defect: 0.)
G1_0 = G1_0,
G2_0 = G2_0,
A_I
=
A_I
,
(defect
:
0.)
)
The 8 or 10 parameters are real constants. All these parameters can depend on
temperature (key words CIN1_CHAB_FO or CIN2_CHAB_FO) and the awaited values are of type
function.
Handbook of Référence
R5.03 booklet: Nonlinear mechanics
HT-66/03/005/A
Code_Aster ®
Version
6.4
Titrate:
Behavior élasto-visco-plastic of J.L.Chaboche
Date:
08/12/03
Author (S):
P. of BONNIERES, J.M. PROIX, E.LORENTZ Clé
:
R5.03.04-B Page
: 3/22
If one wants to introduce in addition to viscosity (models VISC_CIN1_CHAB and
VISC_CIN2_CHAB), it is also necessary to provide in command DEFI_MATERIAU, under the key word
LEMAITRE (or LEMAITRE_FO) parameters NR and UN_SUR_K, which can depend on
temperature.
Parameter UN_SUR_M of key word LEMAITRE (respectively LEMAITRE_FO) must obligatorily
to be put at zero (respectively with the identically null function). In the absence of one of the key words
LEMAITRE or LEMATIRE_FO, the behavior are supposed plastic.
Model CHABOCHE is a model with two variables kinematics with isotropic work hardening, but
without the effect of work hardening on the term of recall and without taking into account of the variation of C1 and C2
with the temperature. The characteristics of work hardening are given by 9 constants, introduced
in command DEFI_MATERIAU:
CHABOCHE = _F
(
R_0
=
R_0,
R_I
=
R_I,
B = B,
K = K,
W = W,
A1
=
A1,
A2
=
A2,
C1
=
C1,
C2
=
C2,
)
In this case the characteristics do not depend any more a temperature since version 5, because these
variations were badly taken into account by this model.
The use of these laws of behavior is accessible in commands STAT_NON_LINE or
DYNA_NON_LINE by key words VISC_CIN1_CHAB, VISC_CIN2_CHAB or CHABOCHE of COMP_INCR.
In the continuation of this document, one describes models VISC_CIN1_CHAB precisely and
VISC_CIN2_CHAB. One presents then the detail of his numerical integration in link with
construction of the coherent tangent matrix. One also briefly describes the integration of the model with
two variables kinematics CHABOCHE. Lastly, one also gives some elements for
identification of the characteristics of material.
Handbook of Référence
R5.03 booklet: Nonlinear mechanics
HT-66/03/005/A
Code_Aster ®
Version
6.4
Titrate:
Behavior élasto-visco-plastic of J.L.Chaboche
Date:
08/12/03
Author (S):
P. of BONNIERES, J.M. PROIX, E.LORENTZ Clé
:
R5.03.04-B Page
: 4/22
2 Models
VISC_CIN1_CHAB and VISC_CIN2_CHAB
2.1
Description of the models
At any moment, the state of material is described by the deformation, the temperature T, the deformation
plastic p, cumulated plastic deformation p and the tensor of recall X. Equations of state
then define according to these variables of state the constraint = H Id +
~ (broken up into
parts hydrostatic and deviatoric), the isotropic share of work hardening R and the kinematic share X:
H 1
=
() =
(- HT) with HT = (ref.
tr
K tr
T-T
) Id éq 2.1-1
3
~ = - HId = µ (~
- p
2
) éq 2.1-2
R = R (p)
éq 2.1-3
X = (
X p p
) = X
éq
2.1-4
1 (p
p
) + X2 (p p
,
,
,)
where K, µ, and coefficients of (
X p) and R (p) are characteristics of the material which can
to depend on the temperature. More precisely, they are respectively the modules of compressibility
and of shearing, the thermal dilation coefficient, functions of isotropic work hardening and
kinematics. As for T ref., it is about the temperature of reference, for which one considers
thermal deformation as being null.
Note:
For model VISC_CIN1_CHAB one thus considers only the only tensorial variable X p
1 ()
X p = 0. This remains valid for all the continuation: one will describe the two models formally of
2 ()
the same way, model VISC_CIN1_CHAB resulting from VISC_CIN2_CHAB while supposing
X p = 0.
2 ()
The evolution of the plastic deformation is controlled by a normal law of flow to a criterion of
plasticity of von Mises:
(
3 ~ ~
F, R, X) = (~ - X1 - X2) - R (p)
with
With
=
A: A
éq
2.1-5
eq
eq
2
~
3
- X1 - X
p
& = & F
= &
2
éq 2.1-6
2
(~ - X1 - X2) eq
&p = &
2
=
&p &p
:
éq 2.1-7
3
As for the plastic multiplier &
, it is obtained by the condition of coherence:
if F<0 or &F < 0 & = 0
éq 2.1-8
if
F
= 0 and &F = 0 & 0
Handbook of Référence
R5.03 booklet: Nonlinear mechanics
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Code_Aster ®
Version
6.4
Titrate:
Behavior élasto-visco-plastic of J.L.Chaboche
Date:
08/12/03
Author (S):
P. of BONNIERES, J.M. PROIX, E.LORENTZ Clé
:
R5.03.04-B Page
: 5/22
Note:
The evolution of variables X and X is given by:
1
2
2
X = C p,
1
1 ()
3
1
2
X = C p,
2
2 ()
3
2
éq 2.1-9
p
&
1 =
& -
1
p p
1 ()
&
p
&
2
=
& -
2
p p
2 ()
&
The functions C (p) (p) and R (p) are defined, in accordance with [bib2] by:
R (p) = R
-
+ (R
0 - R) E LP
C
1
1
1 (p) = C
1 (+ (K -) e-wp)
C
1
1
2 (p) = C
2 (+ (K -) e-wp)
0
p =
has
-
+ 1 - has
E LP
1 ()
1 (
())
0
p =
has
-
+ 1 - has
E LP
2 ()
2 (
())
The presence of viscosity can model in a simple way (cf Lemaitre and Chaboche [bib2]) in
replacing the condition of coherence [éq 2.1-8] by:
NR
F
& =
éq 2.1-10
K
F left positive F (hooks of Macauley)
K, NR characteristic of viscosity (Norton) of material
Unchanged all the other equations of the model are left. It will be seen that such an introduction of
viscosity involves only minor modifications of the implicit algorithm of integration of the law of
behavior.
Note:
The definition of X and X in the form [éq 2.1-9]:
1
2
· allows to keep a formulation which takes into account the variations of the parameters with
temperature without introducing term in &
T as in [bib.4], in the same way that it
viscoplastic model of Chaboche. These terms necessary because their are not taken in
account would lead to inaccurate results [bib4].
· allows to have a coherent writing with the thermodynamic expression of the potential
plastic [bib2] (p.221).
Handbook of Référence
R5.03 booklet: Nonlinear mechanics
HT-66/03/005/A
Code_Aster ®
Version
6.4
Titrate:
Behavior élasto-visco-plastic of J.L.Chaboche
Date:
08/12/03
Author (S):
P. of BONNIERES, J.M. PROIX, E.LORENTZ Clé
:
R5.03.04-B Page
: 6/22
Note:
Significance of the functions C (p), (p), C (p), (p), R (p:
1
1
2
2
)
it is noted that the functions C (p), (p), C (p), (p), R (p intervening in the equations
1
1
2
2
)
the preceding ones allow all the three to model various nonlinear effects of work hardening.
The introduction of work hardening, either on the level of the kinematic part, by C (p), or on the level
term of recall, by the function (p), does not have the same effect on the classification tests
[bib2]. The use of a model with (p) makes it possible in particular to identify more easily of
strong cyclic work hardenings. Several work of identification of the coefficients of the models of
Chaboche were carried out besides on the basis of model with a work hardening represented by
(p) ([bib5], [bib6]), in particular for stainless steel A316L.
2.2
Integration of relations VMIS_CIN1_CHAB and VMIS_CIN2_CHAB
To numerically carry out the integration of the law of behavior, one carries out a discretization in
time and one adopts a diagram of implicit, famous Euler adapted for relations of behavior
elastoplastic. Henceforth, the following notations will be employed: Has, A and A
represent
respectively values of a quantity at the beginning and the end of the step of time considered thus that sound
increment during the step. The problem is then the following: knowing the state at time T - like
increments of deformation (resulting from the phase of prediction (cf Doc. R STAT_NON_LINE
[R5.03.01])) and of temperature T, to determine the state of the variables intern at time T as well as
constraints.
One takes into account the variations of the characteristics compared to the temperature while noticing
that:
H
K
=
H + K tr (-
-
HT)
éq
2.2-1
K
~
µ ~
=
-
2µ
p
E
2µ p
-
+
(~
) ~
- = -
µ
éq
2.2-2
with
~
µ
E
~-
~
= - + 2
µ
µ
Within sight of the equation [éq 2.2-1], one notes that the hydrostatic behavior is purely elastic
if K is constant. Only the processing of the deviatoric component is delicate.
In the absence of viscous term, the relation of discretized coherence is:
Elastic mode: F 0 and p = 0
Plastic mode: F = 0 and p 0
On the other hand, in the presence of viscosity, the condition of coherence is replaced by the equation
[éq 2.1-10] which, discretized, is written:
NR
1/NR
p
F
p
=
F = K
T
K
T
Handbook of Référence
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Code_Aster ®
Version
6.4
Titrate:
Behavior élasto-visco-plastic of J.L.Chaboche
Date:
08/12/03
Author (S):
P. of BONNIERES, J.M. PROIX, E.LORENTZ Clé
:
R5.03.04-B Page
: 7/22
In other words, while posing:
1/NR
~
p
F = F - K
T
the viscoplastic increment of cumulated deformation is determined by:
~
:
rubber band
Mode
F 0 and p = 0
~
éq 2.2-3
viscoplast
Mode
:
ic
F = 0 and p 0
Finally, by adopting an implicit discretization, the only difference between the laws of behavior
plastic and viscoplastic lies in the form of the function of load F: a term there is observed
complementary in the event of viscosity. In fact, incremental plasticity seems the borderline case
incremental viscoplasticity when K tends towards zero. This convergence was already described by
J.L. Chaboche and G. Cailletaud in [bib3].
In the continuation of this paragraph, one will thus detail the integration of the viscoplastic law. To find
the case of the plastic behavior, it is enough to take K = 0 in the equations below (one
recall that the user to place itself in this case must obligatorily remove key word LEMAITRE
or LEMAITRE_FO of command DEFI_MATERIAU).
~
~
2
2
2
-
-
= E - C - - C -
X
X
- 2µ p
C
C
1
2
1
1
2
2
- (1 1 + 2 2)
3
3
3
Equations of flow [éq 2.1-6] and [éq 2.1-7], once discretized, and the condition of coherence
[éq 2.2-3] are written (by noticing that p =):
2
2
2
2
~e - C -
-
p
C
µ
C
C
1
-
2
- 2
-
1
-
1
2
1
2 2
3
p =
3
3
3
3
p
éq 2.2-4
2
2
2
2
2
~e - C -
-
p
C
µ
C
C
1
-
2
- 2
-
1
-
1
2
1
2 2
3
3
3
3
eq
~
~
F 0 p 0 F p = 0
éq
2.2-5
The processing of the condition of coherence (preceding equation) is traditional. One starts with one
test elastic (p = 0) which is well the solution if the criterion of plasticity is not exceeded, it be-with
to say if:
~
2
2
E -
-
-
C
-
1 (-
p) 1 - C2 (-
p)
R
0
éq
2.2-6
3
3
2 - (p) <
eq
Handbook of Référence
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Code_Aster ®
Version
6.4
Titrate:
Behavior élasto-visco-plastic of J.L.Chaboche
Date:
08/12/03
Author (S):
P. of BONNIERES, J.M. PROIX, E.LORENTZ Clé
:
R5.03.04-B Page
: 8/22
In the contrary case, the solution is plastic (p > 0) and the condition of coherence is reduced to
~
F = 0. To solve it, it is shown that one can bring back oneself to a scalar problem by expressing p
and 1,2 according to p. By gathering the equations of the problem resulting from the discretization
implicit, the system of equations is obtained:
1/NR
2
E
-
2
-
p
2
2
~
- C 1 - C 2 - 2
µ - C 1 - C 2 = R
1
2
1
2
(p)
p
+ K
éq 2.2-7
3
3
3
3
eq
T
2
E
-
2
-
p
2
2
~
- C 1 - C 2 - 2
µ - C 1 - C
1
2
1
2
p
3
2
=
p
3
3
3
3
éq
2.2-8
1/NR
2
R (p)
p
+ K
T
=
p -
1
p
1 1
éq
2.2-9
=
p -
2
p
2 2
In this writing, it should well be noted that p = p + p
and = - + and that C, are
I
I
I
I
I
functions of p. By considering the three last equations, this linear system out of p and can
I
to be solved to express these quantities according to p. En effet, it is equivalent to:
1/NR
p
p
3~e
-
-
R (p) + 3
µ p + K
= p - 1
C 1 - C22 - 1
C 1 - C22
éq
2.2-10
T
2
1+ p = p -
-
p
1 (
1)
1 1
éq
2.2-11
+
p =
p -
-
2 (1
p
2)
2 2
By calculating C11 and C2 2 and by replacing them in the expression of p one obtains one
expression of p according to p only:
C
C
p
- p
1
C
1
1
1
p
-
1 =
-
M
1
=
1 (p)
- M1 (p)
p
1 +
p
1 +
p
1
1
1
1
C
C
p
- p
2
C
2
2
2
p
-
éq 2.2-12
2 =
-
M
2
=
2 (p)
- M2 (p)
p
1 +
p
1 +
p
2
2
2
2
C p
with M
I
I (p)
()
= 1+i (p) p
Handbook of Référence
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Code_Aster ®
Version
6.4
Titrate:
Behavior élasto-visco-plastic of J.L.Chaboche
Date:
08/12/03
Author (S):
P. of BONNIERES, J.M. PROIX, E.LORENTZ Clé
:
R5.03.04-B Page
: 9/22
By deferring this expression in the expression of p one finds:
p
1
=
3p~e - p ((C1 - M11p) -
1 + (C2 - m2 p) -
2)
1/NR
2
2
R (p) + (3µ + M1 + m2)
p
p + K
T
what is simplified in:
1
3
p
p ~e
=
-
éq
2.2-13
()
(pM -
-
1 + m2)
2
1
2
D p
with:
1/NR
D (p) = R (p) + (3µ + M1 (p) + m2 (p)
p
p
+ K
T
It now only remains to replace p in the expressions of C
1
1 and C2
2 for
to express this term according to p by:
M3
~ E
C
1
=
p
-
-
-
1
1
-
(pM + M
1 1
2 2)
M
p
D
1 1
2
1
-
M3
~ E
C
2
=
p
-
-
-
2
2
-
(pM + M
1 1
2 2)
M
p
D
2 2
2
2
-
~
then to substitute the expression obtained thus that p according to p in the equation F = 0, and
one obtains a scalar equation out of p to be solved, namely:
~
1/NR
F (p) E 2
-
2
-
p
2
2
~
p
= - C 1 - C 2 - 2
1
2
µ -
1
C 1 - C22 - R (p) - K
= 0
3
3
3
3
eq
T
what is simplified in:
1/NR
R (p)
p
+ K
~
1/NR
F (p)
T
2
-
2
~
p
=
-
E
éq 2.2-14
D (p)
- M1 1 - m2 2 - R (p) - K
= 0
3
3
eq
T
This scalar equation out of p is solved numerically, by a method of search for zero of
function (method of secants which one briefly describes in appendix 2).
Once determined p, one can calculate p using the equation [éq 2.2-13] then 1 and 2 with
assistance of the equations [éq 2.2-11]. It any more but does not remain to calculate the tensor of the constraints, by
equations [éq 2.2-1] and [éq 2.2-2], and to bring up to date the variables intern 1 and 2.
Note:
· an interesting borderline case (for the validation of this model) arises by posing = 0. One
I
finds itself then exactly in the situation of linear kinematic work hardening (if
R (p) = y, [R5.03.02]) or of mixed work hardening for R (p) unspecified (cf [R5.03.16]),
· these models are also available in plane constraints, by a total method
(static condensation due to R. of Borst) [R5.03.03].
Handbook of Référence
R5.03 booklet: Nonlinear mechanics
HT-66/03/005/A
Code_Aster ®
Version
6.4
Titrate:
Behavior élasto-visco-plastic of J.L.Chaboche
Date:
08/12/03
Author (S):
P. of BONNIERES, J.M. PROIX, E.LORENTZ Clé
:
R5.03.04-B Page
: 10/22
2.3
Calculation of tangent rigidity
In order to allow a resolution of the total problem (equilibrium equations) by a method of
Newton [R5.03.01], it is necessary to determine the coherent tangent matrix of the problem
incremental.
This matrix is composed classically of an elastic contribution and a plastic contribution:
E
p
=
- 2
µ
éq
2.3-1
µ
with E = + 2µ
p, which gives again in particular ~e
~ -
~
=
+ 2µ
-
µ
One immediately deduces from it that in elastic mode (traditional or pseudo-discharge), the matrix
tangent is reduced to the elastic matrix:
E
=
éq 2.3-2
For that, one once more adopts the convention of writing of the symmetrical tensors of command 2 pennies
form vectors with 6 components. Thus, for a tensor a:
T
= [axx ayy azz has
2axy
2axz
2ayz]
éq
2.3-3
If one introduces moreover the hydrostatic vector 1 and stamps it deviatoric projection P:
1 =t [1 1 1 0 0]
0
éq 2.3-4
1
P = Id - 1 1
éq 2.3-5
3
where is the tensorial product
Then the matrix of coherent tangent rigidity is written for an elastic behavior:
E
= K 1 1 + 2Μ P
éq 2.3-6
On the other hand, in plastic mode, the variation of the plastic deformation is not null any more.
~
One derives compared to E, knowing that one a:
p
p ~e
p
=
= 2 µ
P éq
2.3-7
~.
E
~.
E
S spaces symmetrical tensors
P projector on the diverters
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Date:
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Author (S):
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p
To calculate ~, one uses the expression of p according to ~
E
E and p:
1
3
p =
p ~
E -
()
(pM -
-
1 + m2)
2
1
2
D p
what is written in the form:
p = (
With p) ~e + B1 (p) -
-
1 + B2 (p) 2
Thus:
p
~ E
With p
B p
B p
= A p Id
1
2
~
+
+
+
E
()
() ()
-
()
-
~ E
~ E
1
~e
2
(
With p)
(
With p) (
With p) p
Quantities of the type
=
~e is calculated using: ~e
p ~
E
p
Finally, it any more but does not remain to calculate the variation of p:
~e
~
One uses for that: F (~
p, E
) = 0
1/NR
R (p)
p
+ K
~
1/NR
F (~
T
E
-
p
p, E)
2
2
~
=
-
D (p)
- M1 1 - m2 2 - R (p) - K
= 0
3
3
eq
T
~
~
~
p
p
F, p (
F
~
,
~e
p,)
~
~
~
,
p
= - F~e p,
= -
éq
2.3-8
, (
E
) E
E (
E)
~e
~
F, p (~e
p,)
The detail of calculations is given in appendix 1.
The initial tangent matrix, used by option RIGI_MECA_TANG is obtained by adopting it
behavior of the preceding step (elastic or plastic, meant by internal variable being worth 0 or
1) and while making tighten p
towards zero in the preceding equations.
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2.4
Significance of the internal variables
The internal variables of the two models at the points of Gauss (VELGA) are:
· V1 = p: cumulated plastic deformation (positive or null)
· V2 =: being worth 1 if the point of Gauss plasticized during the increment or 0 if not.
The following internal variables are, for modeling 3D:
· For model VISC_CIN1_CHAB
-
V3 = 1xx
-
V4 = 1yy
-
V5 = 1zz
-
V6 = 1xy
-
V7 = 1xz
-
V8 = 1yz
· For model VISC_CIN2_CHAB
-
V3 = 1xx
-
V4 = 1yy
-
V5 = 1zz
-
V6 = 1xy
-
V7 = 1xz
-
V8 = 1yz
-
V9 =
2 xx
-
V10 =
2 yy
-
V11 =
2zz
-
V12 =
2 xy
-
V13 =
2 xz
-
V14 =
2 yz
For modelings C_PLAN, D_PLAN, and AXIS:
-
V7 = 0
-
V8 = 0
-
V13 = 0
-
V14 = 0
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3
Model with two variables kinematics: CHABOCHE
3.1
Description of the model
This model comprises two tensorial variables which describe the kinematic share of work hardening:
X = X1 + X2. The equations of the behavior are then:
H 1
=
() =
(- HT) with HT = (ref.
tr
K tr
T-T
) Id éq 3.1-1
3
~ = - HId = µ (~
- p
2
) éq 3.1-2
R = R (p)
éq 3.1-3
X = X
p
p
1 (p,) + X2 (p,)
éq 3.1.4
where K, µ, and the coefficients of the functions R (p), X1 (p) and X2 (p) are characteristics of
material which can depend on the temperature. More precisely, they are respectively them
modules of compressibility and shearing, the thermal dilation coefficient, functions
of isotropic and kinematic work hardening. As for T ref., it is about the temperature of reference, for
which one regards the thermal deformation as being null.
The evolution of the internal variables is controlled by a normal law of flow to a criterion of
plasticity:
(
3 ~ ~
F, R, X, X) = (~ - X - X) - R (p)
with
With
=
A: A
1
2
1
2
éq
3.1-5
eq
eq
2
~
p
3
- X1 - X
& = & F
=
&
2
éq 3.1-6
2
(~ - X1 - X2) eq
&p = &
2
=
&p: &p
éq 3.1-7
3
As for the plastic multiplier &
, it is obtained by the condition of coherence:
if F<0 or &F < 0 & = 0
if F
= 0 and &F = 0 &
éq 3.1-8
0
The evolution of variables Xi is given in model CHABOCHE by:
&
2
X = C
has (p) & p
- X
I
I
I
I &p, I
,
1 2
3
=
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The functions (p) and R (p) are defined by:
(p) ((K) E wp
= +
-
-
1
1
)
R (p) = R
-
+ (R0 - R) E LP
3.2
Integration of the relation of behavior CHABOCHE
As for the relations of behavior VISC_CIN1_CHAB and VISC_CIN2_CHAB, one adopts one
diagram of implicit Euler. One takes into account the variations of the elastic characteristics of
even way that previously [éq.2.1-1], [éq.2.1-2].
One starts with an elastic test, while taking for stress field:
~ E
µ ~-
~
=
+ 2µ
-
µ
who is well the solution of the problem if:
~
2
2
E - C1 (-
p) - 1 - C2 (-
p) - 2 - R (p) < 0
3
3
In the contrary case, the solution is elastoplastic. It is then necessary to solve the system of equations not
linear according to:
µ
~
~
-
-
1 - - 2µ
p =
-
(-) 0
µ
2
X1 - 1
C 1
(p) p has
+ C X
1
1 p = 0
3
2
X2 - 2
A.c. 2 (p) p
+ C X
2
2 p = 0
3
(~ - X1 - X2) - R (p) = 0
eq
that one can write in a way more contracted in the following form:
G (y
)
H
y
X
F L (y
)
()
= 0 =
with
y
1
=
I (y
)
X
2
J
(y
)
p
One solves this system by the method of Newton proposed in environment PLASTI, (described in
detail in [R5.03.10]), is:
Fl
D (y
L
K) = - F (y
K)
y
K
y
+1 = y
+ D
K
K
(y
K)
While reiterating in K until convergence. This resolution takes place for each point of integration.
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The resolution requires the calculation of Jacobien of the local system F L. His general expression is given
hereafter; analytical calculations are not detailed in this document.
G
G
G
G
X1 X2 p
H
H
H
H
X1 X2 p
J =
I
I
I
I
X1 X2 p
J
J
J
J
X1 X2 p
3.3
Operator of tangent behavior
After resolution of the preceding discretized system, the solution obtained is such as
the equation (F L (y
) =)
0 are checked at the end of the increment. One seeks to evaluate the tangent operator in
this point, i.e.
.
T +t
For a small variation of F L, by regarding this time as variable and not
parameter, the system remains with balance and one checks dF L = 0, i.e.:
F L
F L
F L
F L
L
+
+
X1+
F
X2 +
= 0
X1
X2
p
p
This system can be still written:
H
0
F L
(y
)
, with
0
= X
X =
y
0
0
This system of equations can be put in the form:
K = H
from where the required tangent operator:
- 1
K H
=
t+t
One is led to re-use the same matrix jacobienne J as previously to evaluate the operator
tangent. The calculation of K-1 is carried out numerically by a method of decomposition of Gauss.
Its expression is detailed in appendix 1.
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3.4
Significance of the internal variables
The internal variables of the model at the points of Gauss (VARI_ELGA) are, for modeling 3D:
· V1 = X1xx
· V2 = X1yy
· V3 = X zz
1
· V4 = X1xy
· V5 = X1xz
· V6 = X1yz
· V7 = X 2xx
· V8 = X 2yy
· V9 = X 2zz
· V10 = X 2xy
· V11 = X 2xz
· V12 = X 2yz
· V13 = p: cumulated plastic deformation (positive or null)
For modelings D_PLAN, and AXIS,
· V5 = 0
· V6 = 0
· V11 = 0
· V12 = 0
4
Comparison of models VMIS_CIN2_CHAB and CHABOCHE
The principal difference between models VISC_CIN2_CHAB and CHABOCHE relates to the evolution of
variables kinematics.
In case VISC_CIN2_CHAB, one a:
&
2
2
2
X = (&C + C &) =
&C + C (p) & p
- I (p) X
I
I
I
I
I
I
I
I
I &p
3
3
3
In case CHABOCHE, one a:
&
2
X = C
has (p) & p
- X
I
I
I
I &p, I
,
1 2
3
=
These two models are not equivalent: in particular, variation of the coefficients with
temperature is not well taken into account in model CHABOCHE.
In the particular case where the coefficients C are constant, the two models are equivalent. For
I
that, it is necessary to choose:
·
0
I (p) =
I
C what implies: I = I
C, has = 1
2
·
C (p) = C = C.a.
I
I
I I and W = 0, because of the additional term:
&C
3 I which allows
to take into account the variation of Ci with the temperature and the plastic deformation
cumulated p.
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5
Principle of the identification of the parameters of the model.
In case the simplest (only one kinematic variable, 1 = cste, 1
C = cste, R (p) = y) them
coefficients of model 1, 1
C can be identified on a simple tensile test uniaxial, or
on a cyclic curve of work hardening.
Indeed in the uniaxial case, the model is reduced in 1D to [bib2]:
dX = C D p
- X D p
1
1
1 1
,
= ±1
- X1 = y
that one can integrate (in monotonous loading) in the following way:
C
C
X
1
=
+ X 0
1
1
1 -
- -
= ±
(
exp
1 (p
p
0),
1
1
1
= + X
y
1
C
whose asymptote of the traction diagram makes it possible to obtain 1
by:
1
C
C
p X
1
1
1
thus y +
1
1
and whose slope in the beginning provides C (If X 0
1
1 =)
0:
p 0 X
0
0
&1 C1 - y X
1 1
X1 = C1 - y X
1
1
For a model has two variables kinematics, without isotropic work hardening, a traction diagram
still allows to find these relations:
C
C
p
1
2
y +
and the slope in the beginning is worth C + C
+
1
2
1
2
But apart from these simple cases a numerical identification is necessary to obtain them
parameters. One will be able to make this identification for example on tensile tests compression with
imposed deformation.
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6 Bibliography
[1]
P. MIALON, Eléments of analysis and numerical resolution of the relations of elastoplasticity.
EDF - Bulletin of Direction of Etudes and Recherches - Série C - N° 3 1986, p. 57 - 89.
[2]
J.LEMAITRE, J.L.CHABOCHE, Mécanique of solid materials. Dunod 1996
[3] J.L.CHABOCHE,
G.CAILLETAUD, Integration methods for complex constitutive equations,
Methods computer in Applied Machanics Engineering, N°133 (1996), p 125-155
[4]
J.L.CHABOCHE, Cyclic viscoplastic constitutive equations, Journal off Applied Mechanics,
Vol.60, Décembre 1993, pp. 813-828
[5]
R. FORTUNIER, Loi of behavior of Chaboche
: identification of the parameters
elastoplastic and élasto-visco-plastics of steel EDF-SPH between 20°C and 600°C. Note
FRAMATOME/Novatome, NOVTUDD90011, October 1990
[6]
C.MIGNE, Recalage of the parameters of the model of kinematic plasticity nonlinear of
SYSTUS. Modeling of the phenomenon of progressive deformation with consolidation
cyclic of material. Note FRAMATOME EE/R. DC.0286. September 1992.
[7]
J.J.ENGEL, G.ROUSSELIER, Comportement in uniaxial constraint under loading
cyclic of the austenitic stainless steel 17-12 Mo with very low carbon and nitrogenizes control.
Identification of 20) C with 600°C of a model of elastoplastic behavior to work hardening
nonlinear kinematics. Note EDF/DER/EMMA N°D599 CHECHMATE/T43 (1985)
[8]
P. GEYER, C.COUTEROT, characterization of steel 304L used during the tests
“deformation, progressive” on CUMULUS and identification of the parameters of the model of
Chaboche, Note EDF/DER/HT-26/93/040/A
[9]
R of Borst “the zero normal stress condition in plane stress and Shell elastoplasticity”
Communications in applied numerical methods, Vol 7, 29-33 (1991)
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Appendix 1 Matrice of behavior tangent for the models
VMIS_CIN1_CHAB and VMIS_CIN2_CHAB
p
To obtain the tangent behavior in the elastoplastic case, ~ should be calculated [éq 2.3-7].
E
One uses for that the expression of p according to ~
and p, which is written in the form:
E
p
3 p ~
*
-
=
*
2D (p) E + B1 (p) 1 + B2 (p) -
2
with
B * I (p)
M I (p)
= - p
D (p)
C p
M
I
I (p)
()
=
1 + I (p) p
1/NR
D (p) = R (p) + (3µ + M1 (p) + m2 (p)
p
p
+ K
T
The following definitions are pointed out:
R (p) = R
-
+ (R0 - R) E LP
C
- wp
I (p) = Ci (1 + (K -)
1 E
)
0
- LP
I (p) = I (has + (1 - has) E)
thus:
3 p
p
3 p
2D (p)
*
*
~e
1
B (p)
-
B2 (p)
-
=
Id
~e
2D (p) +
+
1 +
2
~e
~e
~e
(
With p)
(
With p) (
With p) p
Quantities of the type
=
~
E are calculated using: ~e
p ~
E
These various terms are expressed by:
3 p
2 (
D p)
3
1
D (p)
·
= I (p)
with I (p) =
-
p
p
2
(
D p) D2 (p)
B *
I (p)
M I (p)
·
= -
-
.
=
p
D (p) p Semi (p) I (p) Hi (p)
p
It remains to calculate:
~e
~
~
F ~e p,
p
, (
E
)
One thus uses, following [éq 2.3-8]:
= -
~
E
~
F, p (~e
p,)
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~
E
E
E
F (
1/
1/
p ~
,)
p
p
= S
~
~
eq (p,)
NR
NR
- R (p) - K
= G (p,) - R (p) - K
T
T
1/NR
1/NR
R (p)
p
M
+ K
I (p) R (p)
p
+ K
T
E
-
-
T
2
~
with S = A + B11 + B
With
2 2
=
= -
D ()
B
p
I
3
D (p)
1/NR
p
Then, by posing v
R (p) = R (p) + K
:
T
3 v
R p S
v
R
~ E
-
-
~e
WITH + B + B
G ~e p,
p
, (
)
()
(
1 1
2 2)
2D (p) S
DS
= -
= -
eq
= -
eq
~
E
G
E
S
S
p (~
p,)
3
'
3
- R p
2
3
,
v ()
'
: S, p - v
R (p)
'
: S, p - v
R (p)
2 S
2
eq
Seq
~
3
E
-
-
1
L (p). + L21 (p) 1 + L22 (p) 2
= - 2
3
L (p)
2
v
R p
With p
with
1
L (p)
()
2 ()
=
=
2
D (p) * S
S
eq
eq
v
R p
1
v
R p
1
21
L (p)
()
= D (p) 1B (p)
L22 (p)
()
=
B p
Seq
D (p) 2 () Seq
1 1
3
-
S
~ E
-
-
K p NR
3
L (p) =
: (A (p)
'
+ 1
B (p)
'
1 + B2 (p) 2) - R (p) -
2 Seq
NT T
p
Finally, ~ is put in the form:
E
p
3 p
3
=
Id +
I
p
S
~e + I
p
has
-
-
~ E
~
+ I
p
has
E
1
2
2 (
D p)
(()
()
())
2
1
2
+ (1
Hs~e + 1
Ha -
1
1 1 + Ha2 -
2) - 1
+ (2
Hs ~e + 2
Ha -
2
1 1 + Ha2 -
2) - 2
with:
3
L p
1
3 I p L
p
I
21
S (p) = -
I (p)
()
.
I
p
1
= -
2
L3 (p)
has ()
() ()
2
L3 (p)
3 I p L
p
I
22
a2 (p)
() ()
= - 2 L3 (p)
3 H p. L p
3 H p L
p
3 H p L
p
H1
1
1
1
1
21
1
1
22
S (p)
() ()
= -
H
p
1
= -
H
p
2
= -
2
L3 (p)
has ()
() ()
2
L3 (p)
has ()
() ()
2
L3 (p)
3 H p. L p
3 H p L
p
3 H p L
p
H2
2
1
2
2
21
2
2
22
S (p)
() ()
= -
H
p
1
= -
H
p
2
= -
2
L
L (p
3
)
3 (p)
has ()
() ()
2
L3 (p)
has ()
() ()
2
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Appendix 2 Résolution of the equation F (p) = 0
It is a question of solving a nonlinear scalar equation by seeking the solution in a confidence interval.
For that, one proposes to couple a method of secant with a control of the interval of search. That is to say
the following equation to solve:
F (X) = 0
X [has, B] F (A) < 0 F (b) > 0
éq A2-1
The method of the secant consists in building a succession of points xn which converges towards the solution. It is
defined by recurrence (linear approximation of the function by its cord):
xn
1
1
1
1
- xn-
xn+
= xn-
- F (xn-)
éq A2-2
F (xn) - F (xn-1)
In addition, if xn+1 were to leave the interval, then one replaces it by the terminal of the interval in question:
if
xn+1 < has
then
xn+1:= has
éq A2-3
if
xn+
1 > B
then
xn+1:= B
On the other hand, if X n+1 is in the interval running, then the interval is reactualized:
if
xn+1 [has, B] and F (xn+1) < 0
then
= xn+1 has
éq A2-4
if
xn+1 [has, B] and
1
1
F (xn+) > 0
then B = xn+
One considers to have converged when F is sufficiently close to 0 (tolerance to be informed). As for both
first leader characters, one can choose the terminals of the interval, or, if one has an estimate of
the solution, one can adopt this estimate and one of the terminals of the interval.
Note:
This method functions well if there is only one solution in the interval [has, B]. Without that being
formally shown, one can note that F ()
0 > 0.
One seeks then B such as F (b) < 0.
~e 2
-
2
-
C
1 1 -
C2
R -
3
3
2
- (p)
eq
One leaves for that B =
µ
3
If F (b) is > 0, one multiplies B by 10 and one tests if F (b) > 0, and so on, until finding a value
B such as F (b) < 0.
One is sure that there is then at least a solution on [has, B].
Handbook of Référence
R5.03 booklet: Nonlinear mechanics
HT-66/03/005/A
Code_Aster ®
Version
6.4
Titrate:
Behavior élasto-visco-plastic of J.L.Chaboche
Date:
08/12/03
Author (S):
P. of BONNIERES, J.M. PROIX, E.LORENTZ Clé
:
R5.03.04-B Page
: 22/22
Intentionally white left page.
Handbook of Référence
R5.03 booklet: Nonlinear mechanics
HT-66/03/005/A
Outline document