Code_Aster ®
Version
7.4
Titrate:
Viscoplastic behavior with damage of CHABOCHE
Date:
02/05/05
Author (S):
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:
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Organization (S): EDF-R & D/MMC, EDF-DPN/CAPE, ALTRAN Technologies
Handbook of Référence
R5.03 booklet: Nonlinear mechanics
R5.03.15 document
Viscoplastic behavior with
damage of CHABOCHE
Summary:
The viscoplastic model coupled with the isotropic damage of Chaboche (developed at the origin to predict
successfully the lifespan and the cracking of the paddles of the modern turbojets) is used for
calculations of prediction of the time of ruin of structures at high temperatures.
It is established in Code_Aster under the name of VENDOCHAB; the equations of speed are integrated
numerically by a diagram clarifies of Runge-Kutta of command 2 with automatic cutting in under-not
buildings according to an estimate of the error of integration (method of Runge-Kutta encased, cf [R5.03.14]),
or by a method of integration implicit [R5.03.01].
Test SSNV126 validates the explicit integration of this model. The document of validation [V6.04.126] provides
analytical solution for an isothermal uniaxial creep test.
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Code_Aster ®
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Titrate:
Viscoplastic behavior with damage of CHABOCHE
Date:
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Author (S):
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:
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Count
matters
1 Introduction ............................................................................................................................................ 3
2 Formulation of the model .......................................................................................................................... 4
2.1 Theoretical framework ............................................................................................................................... 4
2.2 Equations of the model ....................................................................................................................... 6
3 Calculation of the parameters material ............................................................................................................ 7
4 Establishment in Code_Aster ........................................................................................................... 9
4.1 Algorithm of resolution .................................................................................................................. 9
4.2 Implicit integration of the relation of behavior ....................................................................... 9
4.2.1 Implicit discretization of the equations of the model ................................................................ 10
4.2.2 Numerical resolution .......................................................................................................... 11
4.2.3 Operator of tangent behavior ................................................................................... 11
4.2.4 Particular case of the plane constraints ................................................................................. 12
4.3 Syntax of use ....................................................................................................................... 12
4.3.1 Operator: DEFI_MATERIAU .............................................................................................. 12
4.3.2 Operator: STAT_NON_LINE ............................................................................................... 13
4.3.3 Use of DEFI_NAPPE when K_D depends on the tensor of the constraints. ...................... 14
5 Significance of the variables intern ..................................................................................................... 15
6 Bibliography ........................................................................................................................................ 15
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Titrate:
Viscoplastic behavior with damage of CHABOCHE
Date:
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Author (S):
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1 Introduction
Calculations by finite elements carried out within the framework of the studies on the serious accidents of
nuclear engines highlighted the need to use models of damage in order to
to envisage the ruin of a structure such as the tank subjected to the severe thermal conditions and
complexes (high temperatures going until fusion, high thermal gradients in space or
time, etc) which the corium [bib1] would impose to him.
The major interest this choice lies in the fact that the value of the variable of damage at
rupture (or with cracking) can be regarded as an intrinsic parameter of the material which is
accessible, although that is difficult and delicate, by measurements physical (ultrasounds, X-rays diffraction,
etc). The criterion of rupture with the theory of the damage is then more “physical” that them
criteria in maximum deformation used sometimes in viscoplastic calculations without
damage or criteria of damage not coupled (rule of addition of time
actually passed under certain conditions (, T) divided by the time of ruin for these same
conditions).
The model implemented in Code Aster is a viscoplastic model of behavior to work hardening
viscosity-multiplicative coupled with the isotropic damage (model due to Chaboche [bib2]).
Note Bucket:
One will find in the reference [bib3] a detailed description of the capacities of the model, one
methodology for the identification of the parameters and the values of these parameters for
steel 22 MoNiCr 3 7.
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Titrate:
Viscoplastic behavior with damage of CHABOCHE
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2
Formulation of the model
2.1 Tally
theoretical
In this sub-chapter, one insists on the specificity of law VENDOCHAB (i.e.
the damage) compared to the usual viscoplastic models. For more details, one
will refer to [bib2].
The theory of the damage describes the evolution of the phenomena between the virgin state and the starting of
the macroscopic fissure in a material by means of a variable continues (scalar or tensorial)
describing the progressive deterioration of this material. This idea, due to Kachanov which was the first with
to use to model the creep rupture of metals in uniaxial stress, was taken again in
France in the Seventies by Lemaitre and Chaboche. Evolution of material of its virgin state with
its damaged state is not always easy to distinguish from the phenomenon of deformation
generally accompanying and is due to several different mechanisms of which creep forms part.
The viscoplastic damage of creep corresponds to intergranular decoherences
accompanying the viscoplastic deformations for metals at average temperatures and
raised.
To define what is this variable of damage, let us consider the surface S of one of the faces of one
element of volume located by its normal directed towards outside N. On this section, them
~
microscopic cracks and the cavities which constitute the damage leave traces of various forms. That is to say S
the effective resistant surface and SD the total surface of the whole of the traces.
One a:
S
S S
D =
- ~
and one defines the variable of damage by:
S
D
D
N = S
DNN is the measurement of the local damage relative to direction N. From a point of view
physics, the variable of damage DNN is thus the relative surface of the fissures and cut cavities
by the normal plan with the direction R
N. From a mathematical point of view, while making tighten S towards 0,
variable Dr. N is the surface density of discontinuities of the matter in the normal plan with N.
DNN = 0 corresponds at the virgin state not damaged. DNN = 1 corresponds to the element of broken volume
in two parts according to a normal plan with N.
The assumption of isotropy implies that the fissures and cavities are uniformly distributed in
orientation in a point of material. In this case, the variable of damage becomes a scalar
who does not depend any more orientation and is noted D. One a:
D = DNN N
We will consider here only the isotropic variable of damage.
Total mechanical measurements (modification of the characteristics of elasticity, plasticity or of
viscoplasticity) are easier to interpret in term of variable of damage thanks to the concept
of effective constraint introduced by Rabotnov. The effective constraint represents the constraint
paid to the section which resists the efforts indeed. In the case of the isotropic damage,
it is written:
~
= (1 - D)
And one a:
·
~
= for a virgin material
·
~
+ at the instant of the failure
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Titrate:
Viscoplastic behavior with damage of CHABOCHE
Date:
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The principle of equivalence in deformation implies that any behavior with the deformation,
unidimensional or three-dimensional of a damaged material is translated by the laws of
behavior of the virgin material in which one replaces the usual constraint by the constraint
effective.
One distinguishes 2 types of variables to characterize the medium:
Observable variables (measurable):
· the temperature T
· the total deflection which breaks up as indicated below:
E
vp
HT
= + +
Internal variables:
· viscoplastic deformation vp
· the isotropic variable of work hardening R
· the isotropic variable of damage D
That is to say = (
vp
, T
, R, D), the potential of state, the laws of state describing this potential are:
=
R =
-
R
S =
-
T
Y =
-
D
According to the law of normality, one has, with, the potential of dissipation:
vp
& =
R
& =
R
D & =
Y
The modeling of the work hardening and the damage of material is done via
internal variables (or hidden). In the case of the model VENDOCHAB, variables internal
introduced into Code_Aster are:
·
vp
: tensor of the inelastic deformations
·
p: cumulated plastic deformation
·
R: variable of work hardening viscosity
·
D: scalar variable of isotropic damage
Note Bucket:
In the made programming, the variable vp
is stored in the shape of a vector with
6 components, vp
, with the result that the variables intern are stored in a table of
dimension 9 (cf [§5]).
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Viscoplastic behavior with damage of CHABOCHE
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2.2
Equations of the model
The equations of the models are written then:
= E + HT + vp
= 1
(- D) E
'
vp
& = 3 p
&
2
eq
r&
p & =
1
(- D)
NR
D
eq -
1
(
y
-)
R & =
1
1
(- D)
M
K R
R
()
- K (())
D & =
1
(- D)
With
with:
() = J () + J () + 1
(- -) J ()
0
1
2
:
where J ()
maximum
principal
constraint
is
0
J () = Tr ()
1
3
J
'
'
() = =
2
eq
ij
ij
2
X:
of
positive
part
X
where:
E
HT
vp
,
and
are respectively the deflections total, elastic, thermal and plastic,
= (
ijkl)
is the elastic tensor of rigidity,
'= 1
- Tr () Id is the deviatoric part of the tensor of the constraints,
3
p
is the cumulated plastic deformation,
R
is the variable of isotropic work hardening viscoplastic
D
is the scalar variable of isotropic damage
Note Bucket:
The whole of the parameters of the model, NR, M, K,
WITH R and K can be related to
the temperature (in °C). K can be constant, depend on the temperature or on () (in
MPa) and of the temperature.
In addition, it is seen that this model considers that it can exist a viscoplastic threshold there which
depends on the temperature.
It is seen that this model is reduced to the viscoplastic model of Lemaitre if it is considered that D = 0 and
if one neglects the equation of evolution of D.M, NR, and K are coefficients characteristic of
purely viscoplastic behavior of material.
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The evolution of the damage is governed by a law with three parameters: With, R, and K. The constraint
equivalent () allows to take account of a possible effect of the spherical part of the tensor of
forced on the damage (a little as in the laws growth cavities at the base
models of Gurson and Rousselier). The fact that the maximum principal constraint can play one
role in () is difficult to imagine for materials as steel but returns the model more
General.
3
Calculation of the parameters material
The parameters of the law of behavior can be calculated starting from creep tests carried out
for various levels of constraints and temperature. For that one uses a law of behavior
unidimensional because the stress of a cylindrical test-tube in traction can be modelled in
dimension 1. The tensor of the constraints is reduced to its axial component.
2
0
0
0 0
0
3
'
1
= 0 0 0
and
= 0 0 -
0
3
0
0 0
1
0
0
-
3
One thus has: J = J = J =
0 ()
1 ()
2 ()
0
() =
(
,
0
)
The system of equations to be solved is then:
1 0
0
vp
1
&
= p&0 -
0
2
1
0
0
-
2
R
p =
&
&
1
(- D)
NR
- 1
(- D)
0
R &
y
=
1
1
(- D)
M
K R
R
0
- K ()
0
D & =
1
(- D)
With
This system of equations is integrable, which makes it possible to have only one equation for the rate of
cumulated viscoplastic deformation (which one can compare to the total deflection by neglecting them
elastic strain).
One can then correlate this expression with the experimental data to adjust the coefficients, but
the number of parameters and non-linearities make that difficult (moreover there is not unicity).
It is thus necessary to use a method of correlation calling upon “physical” assumptions on
phenomenon of creep whose curve is represented hereafter.
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Deformation (%)
Deformation of end of
primary education creep *
Time (H)
T *
tr
Time of creep
Time of
primary education
Creep
rupture
primary education
Creep
Creep
secondary
tertiary sector
Be reproduced 3-a: Les various phases of creep on a curve of creep
The curve of deformation according to time obtained after a creep test breaks up into
three parts:
· a part known as of primary education creep where the damage is negligible.
· a part known as of secondary creep where the speed of deformation is appreciably constant.
· a part known as of tertiary creep where work hardening is saturated and where phenomena
of damage are dominating.
A method of calculation of the parameters using the experimental data (, T) (one uses
also (&, T) which results some by a numerical procedure) for various levels from constraints
and various temperatures was elaborate at the ECA. It uses the expressions found higher in
the case of a homogeneous and unidimensional constraint by making assumptions according to the part of
the curve where the data are taken. For example, in the primary education phase of creep, one makes
the assumption D = 0 and in the secondary phase of creep, one uses the fact that &
is constant.
One will find in the references [bib3] and [bib4] the complete description and of the examples of calculations
carried out on the German steel of tank 22 MoNiCr 3 7.
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4
Establishment in Code_Aster
4.1
Algorithm of resolution
The algorithm used is of the total-room type.
The total iterations use the elastic matrix of rigidity calculated starting from the matrix of Hooke
damaged:
0
= 1
(- D)
On the level of the local iterations (i.e. in each point of GAUSS), the numerical integration of
equations of speed can be carried out either by an explicit diagram of Runge-Kutta of command 2 with
automatic cutting in under-not buildings according to an estimate of the error of integration
(method of Runge-Kutta encased), that is to say by an implicit scheme of Euler solved by a method of
Newton. One will refer to the references [bib3] for all the details concerning the methods
numerical, and with [R5.03.14] for the explicit algorithms employed and their programming
data processing.
4.2
Implicit integration of the relation of behavior
With each total iteration of resolution of the variational problem of balance and for each point
of elementary integration, it is necessary to integrate the equations of the model described into [§3] to obtain it
tensor of the constraints and if required to calculate the operator of tangent behavior.
The problem written in a generic form at the moment T is consisted of the four systems of equations
nonlinear following:
T-1
Rb (, p, state) = 0
,
éq
4.2-1
T-1
RP (, p, state) = 0
with
T-1
(, p, state) = 0
éq
4.2-2
T-1
(
, p,
state
state) = 0
Rb is a system of six equations (six unknown factors) describing the unknown factors associated with
constraints. One notes the vector 6 components of these unknown factors. The connection enters and is
realized by means of the system of equations and the vector p contains the variables R and D.
RP is a system of equations describing the internal unknown factors. One chooses a system of
2 equations with D & and R & like internal unknown factors. The evolution of the variables of state is described by
the system of equations.
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The implicit scheme of Euler is used and the algorithm is presented in the following form:
Initialization of the unknown factors of the discretized problem and recovery of the values of the variables of state
obtained with the preceding step
iterations of the method of Newton (a maximum number of pre iterations defined by the user):
· Recovery of the values of the parameters intervening in the material law (the operator
of elasticity)
· Calculation of the criteria of constraint and their derivative compared to the constraints
· Recovery of the values of the parameter K intervening in the evolution of
the damage and its derivative
· Calculation of the current price of the variables of state, the equations describing the unknown factors
interns and of the equations describing the constraints
· Calculation of derived from the equations compared to the unknown factors
· Resolution of the linear system
N
Rb
N
Rb
N
p
D
Rb
N
N
= -
éq
4.2-3
RP
RP
N
dp
RP
p
· Test of convergence
Evaluation of the tangent operator
4.2.1 Discretization
implicit of the equations of the model
Considering that an increment of time characterizes a new state of the system [éq 4.2-1] and [éq 4.2-2]
solved by an algorithm of Newton, one chooses to identify the state of one quantity at the previous moment
by exhibitor T-1 whereas its current state is noted without exhibitor. Thus variation of a quantity
for the increment of time considered is presented by U = U T-1 + U
= U T-1 + T
U & (T
)
For =0, one obtains an explicit diagram and for =1, one obtains a purely implicit diagram.
With these notations, the discretized form of the vectorial system is written:
T
T
Rb - (
1 - (1
D
+ T D&)
1
-
3
()
-
T
R
HT - vp +
&
=
2
T -
(1 (1
D
+ T D&)
0
(
eq)
éq 4.2.1-1
or more simply Rb - (1 - (T1
D
+ T D&)
el = 0
=
where is the vector 6 components from the tensor of the constraints.
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Viscoplastic behavior with damage of CHABOCHE
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NR
(
D
T D&
eq) - y (1 - (T-1 +
)
R & -
= 0
(1 (T-1
D
+ T D&) K (-
R
+ T r&) 1
1
M
T
RP
éq
4.2.1-2
() R
D & -
(1 (T1
D
+ T D&) -
-
K (()) = 0
With
The evolution of the variables of state is described by the system of equations:
T
D = D - 1 + T
D&
- 1
3
T
r&
- 1
R
= + T
= + T
&
éq 4.2.1-3
vp
vp
2 (1 - (- 1
- 1
&
T
D
+ T
D&)
T
vp
1 - D + T
D&
eq
((T
) eq
T
R = R - 1 + T
r&
where D, and R are the variables of state whose history is preserved.
vp
The deformations and the variables of states are not unknown factors of the problem. These sizes
will be filed with each increment of time converged to be re-used with the following increment.
4.2.2 Resolution
numerical
Rb = 0
The resolution of the nonlinear system
use the method of Newton-Raphson associated with
RP = 0
a technique of tangential approximation in order to seek the solutions in a field where
functions are correctly conditioned.
According to the algorithm of Newton-Raphson, one solves this system in an iterative way on the sequence
following:
· Initialization of the unknown factors
· Seek of a direction of descent by the resolution of the system [éq 4.2-3]
X
· Test of convergence err =
X
4.2.3 Operator of tangent behavior
The tangent operator is obtained by deriving the constraints compared to the total deflections according to
made up rules of derivation:
D
p
=
+
+
éq
4.2.3-1
D
p
where the function of constraints (, p, T-1
, v
. The derivative of the unknown factors compared to
state) =
total deflections are obtained by deriving the system [éq 4.2-1] that is to say:
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Rb Rb
Rb
p
= -
éq
4.2.3-2
RP RP p
RP
p
4.2.4 Particular case of the plane constraints
The elements 2D in plane constraints having to be usable for this model of behavior, one
carry out an additional processing in on-layer of the general processing carried out in 3D.
A positive test on the case of the plane constraints means:
· To the resolution of the system [éq 4.2-1], one adds the additional equations
0
33 =
13 =
23 =
· One modifies the tangent operator to assure energy balance
4.3 Syntax
of use
The key word retained for this model is:
“Viscoplasticity with ENDOmmagement de CHABoche” - > “VENDOCHAB”
This model is accessible in Code_Aster starting from key word COMP_INCR
(RELATION: “VENDOCHAB”) of command STAT_NON_LINE [U4.32.01]. The whole of
parameters of the model is given via command DEFI_MATERIAU (key words factors VENDOCHAB
or VENDOCHAB_FO if the coefficients depend on the temperature and/or the equivalent constraint
()) [U4.23.01].
4.3.1 Operator
:
DEFI_MATERIAU
The following table gives the exact correspondences between the symbols of the equations and the words
keys of Code_Aster:
Parameter material
Symbol in
Key word in Aster
equations
Threshold of viscoplasticity
y
“S_VP”
Coefficient 1 of the equivalent constraint of creep
“SEDVP1”
Coefficient 2 of the equivalent constraint of creep
“SEDVP2”
First exhibitor of the viscoplastic law
NR
“N_VP”
Second exhibitor of the viscoplastic law
M
“M_VP”
Coefficient of the viscoplastic law
K
“K_VP”
Coefficient of the law of damage
With
“A_D'
First exhibitor of the law of damage
R
“R_D'
K [[] T
,]
Second exhibitor of the law of damage
“K_D'
Note:
“_VP” => coefficient intervening in an equation of the viscoplastic behavior.
“_D' => coefficient intervening in an equation of the behavior of damage.
“SEDVP” => (Sigma) Equivalent in Dommage ViscoPlastique.
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Viscoplastic behavior with damage of CHABOCHE
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In short, that thus gives following syntax in Code_Aster.
my [to subdue] = DEFI_MATERIAU (
# Behavior
Rubber band
/ELAS = _F (
)
/
ELAS_FO
=
_F
(
)
# Mechanical Behavior Nonlinear
/
VENDOCHAB
=
_F (
S_VP
=
[R]
SEDVP1
= [R]
SEDVP2
= [R]
N_VP
=
[R]
M_VP
=
[R]
K_VP
=
[R]
A_D
=
[R]
R_D
=
[R]
K_D
=
[R]
)
/
VENDOCHAB_FO
=_F
(
S_VP
=
[function]
SEDVP1
= [function]
SEDVP2
= [function]
N_VP
=
[function]
M_VP
=
[function]
K_VP
=
[function]
A_D
=
[function]
R_D
=
[function]
K_D
=
[function],
[tablecloth]
)
)
4.3.2 Operator
:
STAT_NON_LINE
One does not detail here all the options of operator STAT_NON_LINE, but only those which it is
possible to use for the law of behavior “VENDOCHAB”. One will pay attention to the fact that it is necessary
to use:
· that is to say
NEWTON = _F (MATRIX = “ELASTIC”)
with
CONVERGENCE
=
_F
(
RESO_INTE
=
“RUNGE_KUTTA_2”)
(An error message is transmitted if not)
· that is to say
CONVERGENCE
=
_F
(
RESO_INTE
=
“IMPLICITE”)
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7.4
Titrate:
Viscoplastic behavior with damage of CHABOCHE
Date:
02/05/05
Author (S):
O. DIARD, P. DECEIVED, J.R. SCHNEITER Clé
:
R5.03.15-B Page
: 14/16
4.3.3 Use
of
DEFI_NAPPE when K_D depends on the tensor of the constraints.
In the command files where one wishes to make depend parameter K_D on the constraint, it
is necessary to use operator DEFI_NAPPE. It is necessary for that to use DEFI_NAPPE in the following way (it
key word opposite NOM_PARA must be “TEMP” imperatively, and NOM_PARA_FONC is then “X”):
An example below is shown:
kd_t = DEFI_NAPPE (NOM_PARA = ' TEMP',
PROL_DROITE = ' LINEAIRE',
PROL_GAUCHE = ' LINEAIRE',
PARA = (900, 1000., 1025., 1050),
NOM_PARA_FONC = ' X',
DEFI_FONCTION = _F (PROL_DROITE = ' LINEAIRE',
PROL_GAUCHE = ' LINEAIRE',
VALE= (0.1, 14.355,
100. , 14.855,
200. , 14.355),
),
_F (PROL_DROITE = ' LINEAIRE',
PROL_GAUCHE = ' LINEAIRE',
VALE= (0.1. , 14.5,
100. , 15.,
200. , 15.5),
),
_F (PROL_DROITE = ' LINEAIRE',
PROL_GAUCHE = ' LINEAIRE',
VALE= (900. , 14.5363,
1000. , 15.0363,
1050. , 15.5363),
_F (PROL_DROITE = ' LINEAIRE',
PROL_GAUCHE = ' LINEAIRE',
VALE= (0.1, 14.5725,
100. , 15.0725,
200. , 15.5725),
))
If there is only one dependence in constraint, it is imperatively necessary to use a DEFI_NAPPE with
thus a “virtual” dependence in temperature.
For example:
kd_t = DEFI_NAPPE (NOM_PARA = ' TEMP',
PROL_DROITE = ' LINEAIRE',
PROL_GAUCHE = ' LINEAIRE',
PARA = (0., 10000.),
NOM_PARA_FONC = ' X',
DEFI_FONCTION = (_F (PROL_DROITE = ' LINEAIRE',
PROL_GAUCHE = ' LINEAIRE',
VALE= (0.1. , 14.5,
100. , 15.,
200. , 15.5),
),
_F (PROL_DROITE = ' LINEAIRE',
PROL_GAUCHE = ' LINEAIRE',
VALE= (0.1. , 14.5,
100. , 15.,
200. , 15.5),
)),
)
It should be noted that the use of a tablecloth for K_D slows down calculations because the value of K_D in function
tensor of the constraints is reactualized with each local iteration.
Handbook of Référence
R5.03 booklet: Nonlinear mechanics
HT-66/05/002/A
Code_Aster ®
Version
7.4
Titrate:
Viscoplastic behavior with damage of CHABOCHE
Date:
02/05/05
Author (S):
O. DIARD, P. DECEIVED, J.R. SCHNEITER Clé
:
R5.03.15-B Page
: 15/16
5
Significance of the internal variables
The internal variables of the model at the points of Gauss (key word VARI_ELGA) are accessible by:
· V1 = 11
vp
· V2 = 22
vp
· V3 = 33
vp
· V4 = 12
vp
· V5 = 13
vp
· V6 = 23
vp
· V7 = p, cumulated plastic deformation
· V8 = R, the variable of isotropic work hardening viscoplastic
· V9 = D, the variable of damage
6 Bibliography
[1]
P. MONGABURE, A. SIRVENT, Mr. DESMET: “RUPTHER: test 1”, ratio DMT/95/367
[2]
LEMAITRE J., CHABOCHE J.L. : “Mechanical of solid materials”, ED. Dunod (1985)
[3]
P. DECEIVED, J. - R. SCHNEITER
: “
Introduction of a coupled model of damage
viscoplastic in Aster “- Rapport of advance of phase 4 of the CERD RNE 533, note
EDF/MTC to be appeared
[4]
H. JAMET: “Determination of the parameters of the viscoplastic law of coupled behavior
with the damage for steel 20MnMoNI55. Application to calculation by finite elements of one
pipe of the primary education circuit “, notes ECA/DRN/DMT/SEMT/LAMS/DMT/95/406
Handbook of Référence
R5.03 booklet: Nonlinear mechanics
HT-66/05/002/A
Code_Aster ®
Version
7.4
Titrate:
Viscoplastic behavior with damage of CHABOCHE
Date:
02/05/05
Author (S):
O. DIARD, P. DECEIVED, J.R. SCHNEITER Clé
:
R5.03.15-B Page
: 16/16
Intentionally white left page.
Handbook of Référence
R5.03 booklet: Nonlinear mechanics
HT-66/05/002/A
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