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Organization (S): EDF-R & D/AMA, CS IF
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Document: D5.04.01
To introduce a new law of behavior

Summary:

The purpose of this document is to provide to the developers the principal elements necessary to
establishment (or modification) of a law of behavior in Code_Aster. It describes the modifications with
to carry out on the catalog of commands, as well as new routine FORTRAN to create to integrate it
behavior either explicitly (method of RUNGE-KUTTA), or in an implicit way, in
environment PLASTI (method of NEWTON) or in an optimized way.

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Count

matters

1 essential Stages to introduce a new law of behavior ................................................ 3
1.1 Writing of Doc. R ......................................................................................................................... 3
1.2 Modification of the catalog of DEFI_MATERIAU .............................................................................. 3
1.3 Modification of the catalogs of STAT_NON_LINE and DYNA_NON_LINE ......................................... 4
1.4 Writing of a routine relating to a point of integration of an element .............................................. 4
1.5 Branch of this routine in routine NMCOMP (inelastic behavior) or NMCPEL
(elastic behavior) ................................................................................................................ 5
2 Modifications of the catalogs of commands ....................................................................................... 5
2.1 DEFI_MATERIAU .............................................................................................................................. 5
2.2 STAT_NON_LINE, DYNA_NON_LINE, DYNA_TRAN_EXPLI ............................................................... 6
3 Modifications of the routines ..................................................................................................................... 7
3.1 In which (S) routine (S) to intervene? .............................................................................................. 7
3.2 Programming of law while passing by the routine of shunting NMCOMP or NMCPEL ..................... 7
3.2.1 Principle ................................................................................................................................... 7
3.2.2 Example of a routine realization the integration of a law of behavior: NMCINE ......... 10
3.3 Programming of law in environment PLASTI ............................................................... 13
3.3.1 Introduction ........................................................................................................................... 13
3.3.2 Algorithm of resolution of the quasi-static problem .......................................................... 14
3.3.3 Environment Plasti ............................................................................................................ 14
3.3.4 Formalization of the equations to solve .............................................................................. 15
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1
Essential stages to introduce a new law of
behavior

1.1
Writing of Doc. R

Initially, it is necessary to write the reference material of the law of behavior connecting in one
not given the constraints to the deformations. One restricts here in the continuous mediums 2D and 3D.
method remains valid for models with local nonlinear behavior in plane constraints, such
that hulls, plates and pipes, or modelings with nonlinear behavior
monodimensional, like the multifibre beams, the bars, the grids, via the method OF BORST
[R5.03.03]. This method is applicable in a general way in small deformations, whatever it
behavior, provided that the tangent operator of the new behavior is accessible and
effective.

To solve the nonlinear problem posed on the structure, the document [R5.03.01] described
the algorithm used in Aster.

With each iteration N of the method Newton [R5.03.01 § 2.2.2.2] one must calculate the nodal forces
R (one) = QT N
N
I
I (options RAPH_MECA and FULL_MECA) constraints I being calculated from
displacements linked via the relation of behavior. One must build too
the tangent operator to calculate K nor (option FULL_MECA).

With the first iteration, one calculates K i-1.

The calculation of K i-1 (option RIGI_MECA_TANG), which is necessary to the phase of initialization
[R5.03.01 § 2.2.2.1] corresponds to the calculation of the tangent operator deduced from the problem of speed
below.

This operator is not identical to that which is used to calculate K nor by option FULL_MECA, to
run of the iterations of Newton. Indeed, this last operator is tangent with the problem discretized of
implicit way.

1.2
Modification of the catalog of DEFI_MATERIAU

The goal of DEFI_MATERIAU is to introduce parameters of behavior. These parameters
can be common to several relations of behavior.

It is possibly necessary to add in the catalog of DEFI_MATERIAU a key word corresponding factor
with the type of behavior which one wants to model and under this key word factor, key words
representing the parameters of this type of behavior.

Important remark:

From a data-processing point of view, the key words factors must be of K10 (chain of
characters limited to 10 characters), and the key words under unclaimed are limited to 8 characters.
In practice, that means that if key word is longer, only the first 8 characters
will be used. It thus has a collision risk with other key words having the 8 first
joint characters.
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1.3
Modification of the catalogs of STAT_NON_LINE and DYNA_NON_LINE

In the catalogs of these commands, one gives under the key words factors COMP_INCR or
COMP_ELAS, the name of the relation of behavior after key word RELATION.

One gives also the number of variables intern associated with this relation after the name with this relation.

The name of this relation of behavior can be different from the name of the type of behavior given
in DEFI_MATERIAU.

1.4
Writing of a routine relating to a point of integration of an element

The point of integration is a point of Gauss in the case of continuous mediums, or a point of integration
in the thickness, for hulls, for example.

The arguments of input are:

· the increment of total deflection,
· the tensor of constraints at the moment of preceding calculation,
· the variables intern at the moment of preceding calculation; for example P
, p, Xi,
· the option of calculation: 3 options must be calculated: “RIGI_MECA_TANG”, “RAPH_MECA” and
“FULL_MECA”.

The arguments of output are according to the option of calculation:

· the tensor of the constraints reactualized (RAPH_MECA and FULL_MECA),
· reactualized internal variables (RAPH_MECA and FULL_MECA),
· the coherent matrix of behavior tangent or of speed (FULL_MECA and
RIGI_MECA_TANG).

Important remarks:

The tensors deformation, constraints at the previous moment, and increment of deformation, given
in arguments of input, are such as the components except diagonal (shearing for
constraints, and distortion for the deformations, are multiplied by 2 before call to the routine
of integration of the behavior. Consequently, components of shearing of
tensor of constraints at output must also be multiplied by same coefficient 2.

One describes here the integration of a new behavior under the assumption of the small deformations.
The assumptions available in Code_Aster on the deformations are:
· PETIT: in this case of the tensors deformations are calculated linearly by report/ratio
with displacements, on the initial geometry (Hypothèse of Petites Perturbations:
HPP);
· PETIT_REAC: the deformations are calculated linearly starting from displacements
on the reactualized geometry. Nothing changes in the integration of the behavior;
· GREEN
: in this case the provided deformations are the deformations of
GREEN-LAGRANGE. Under the assumption of small deformations (but the large ones
displacements), the behavior is expressed in a way similar to the behavior
HPP, but connects this time the deformations of GREEN-LAGRANGE to the constraints of
PIOLA-KICHHOFF of 2nd species. The transformation of constraints PK2 into
forced of Cauchy is managed by the appealing routines of NMCOMP. [R5.03.22].
· SIMO_MIEHE: in this case the arguments of input correspond to the gradient of
transformation F at the previous moment and of gradient of the transformation enters
configuration at the previous moment and the current configuration F
. But in this case it
is necessary to formulate the model of behavior in great transformations, and
one cannot use formulation HPP like previously any more [R5.03.21].
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1.5 Branch of this routine in routine NMCOMP
(inelastic behavior) or NMCPEL (elastic behavior)

Shunting is done according to the name of the relation which was given under key word RELATION of
COMP_INCR or COMP_ELAS.

2
Modifications of the catalogs of commands

2.1
DEFI_MATERIAU

One introduces into the catalog of command DEFI_MATERIAU a key word factor under which one goes
to be able to give the parameters necessary to the description of the behavior of material.

This key word factor is K16, whose only 10 characters are significant.

Examples:

· to describe an elastic behavior, a user will employ the key word factor ELAS
in DEFI_MATERIAU: to subdue = DEFI_MATERIAU (ELAS =…),
· an elastoplastic behavior with linear work hardening, a user will describe
to employ the key words factors ECRO_LINE and ELAS: to subdue = DEFI_MATERIAU
(ELAS =…, ECRO_LINE =…).

Under the key words factors defining the behavior of material, one gives the key words which go
to correspond to the names of the parameters of the law and after which one gives the values of these
parameters.

These key words are of K8.

The values of the parameters are either of the real numbers, or of the functions (thus of K8).

Examples:

· for an elastic material, one must give the Young modulus E and the naked Poisson's ratio.
One has as follows:
to subdue = DEFI_MATERIAU
(ELAS= _F (E = yg, [R]











NAKED = naked, [R]










)







)

· for an elastoplastic material with linear work hardening, one must give the characteristics
rubber bands and the linear curve of work hardening which is defined by the elastic limit SY and the slope
*
traction diagram D_SIGM_EPSI, i.e.
*
One has as follows:

to subdue = DEFI_MATERIAU
(ELAS= _F (E = yg,
[R]












Naked = naked,
[R]











)
ECRO_LINE
=
_F (
SY
=
sy,
[R]
D_SIGM_EPSI
=
dsde,
[R]












),







)

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2.2
STAT_NON_LINE, DYNA_NON_LINE, DYNA_TRAN_EXPLI

The non-linear laws in the case of are placed. It is necessary to modify the catalogs of the commands
STAT_NON_LINE and DYNA_NON_LINE by giving the name of the relation after key word RELATION under
the key words factors COMP_INCR or COMP_ELAS.

Examples:

· in the case of a relation of behavior of elastoplasticity of von Mises with work hardening
isotropic linear, one a:

statnl = STAT_NON_LINE (


MODELE = MOD,
CHAM_MATER
=
chmat,


COMP_INCR = _F (RELATION = “VMIS_ISOT_LINE”),






…)

· in the case of a relation of behavior of elastoplasticity of Von Mises with work hardening
linear kinematics, one a:

statnl = STAT_NON_LINE (


MODELE = MOD,
CHAM_MATER
=
chmat,


COMP_INCR = _F (RELATION = “VMIS_CINE_LINE”),






…)

It should be noticed that these two relations use the same parameters of DEFI_MATERIAU but them
behaviors are different and the numbers of the variables intern are different:

2 for VMIS_ISOT_LINE: p,
7 for VMIS_CINE_LINE: X,

·
p indicates the cumulated plastic formation,
·
X indicates the tensor of recall (it is symmetrical, it thus has 6 components),
·
indicate an indicator of plasticity in a given point:
- if
X = 1 the point is “plastic”,
- if
= 0 it are not it.

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3
Modifications of the routines

3.1
In which (S) routine (S) to intervene?

To carry out the calculation of the behavior in a point of integration [§1.4], i.e. the calculation of
constraints and of the internal variables, and the calculation of the tangent behavior, three solutions are
possible in Code_Aster:

· that is to say to use the architecture of environment PLASTI. It is about a whole of routines
(routines of shunting and routines utility) allowing to introduce a new model “with little
expenses ", i.e. by defining some specific routines. On the other hand, PLASTI
does not allow to obtain models optimized in time calculation. This environment is described in
[§3.3]. This general framework is used in particular for the integration of the models of
behavior of the monocrystals [R5.03.11];
· maybe, which is not advised to obtain a good convergence, but can be faster
in a phase of test, to use the architecture of integration clarifies by the method of
RUNGE-KUTTA [R5.03.14]. Indeed simplicity comes owing to the fact that only the equations
differentials describing the evolution of the variables intern are to be programmed, since in it
case it does not have there a tangent operator;
· either to create a routine supplements integration of the behavior, which by the means of
the example of the other existing routines, often makes it possible to obtain powerful models
(for example, by reducing the system to be solved with only one scalar equation, not
linear). This process is described with [§3.2.2].

In version 8, it will be also possible to define a new behavior in the formalism of
Zmat (module of behavior of the Zebulon code) via the coupling Aster-Zmat. The goal of this
functionality is of prototyper new models, but not to use it on calculations of
structures of important flying bridge, because time calculation is increased to a significant degree. The interface
will be accessible only within the framework from the Ecole partnership of Mines of Paris ­ EDF R & D.

3.2 Programming of law while passing by the routine of shunting
NMCOMP or NMCPEL

3.2.1 Principle

One places oneself in the case of the routine NMCOMP which makes the integration of the laws of behavior
incremental (thus relating to COMP_INCR).

Routine NMCOMP is called on the level of the calculation of the elements, that is to say TE.

In fact NMCOMP is not called directly by TE but by called routines themselves
by TE.

These routines are:

· NMPL 2D and NMPL 3D for the solid elements 2D and 3D in small deformations,
· NMGP 2D and NMGP 3D for the solid elements 2D and 3D in great deformations
(SIMO_MIEHE),
· DKQNLI and DKTNLI for elements DKQ and DKT,
· VDXNLR for the thick hulls 3D,
· TE0329 for the hulls 1D,
· TUFULL for the elements pipes.
· NMCO1D for the behaviors 1D of the elements BARRE, Poutres multifibre, Grills.

The calculations carried out on the level of NMCOMP relate to a given point of integration of an element
given.
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These calculations consist in determining at the current moment the tensor of the constraints, the internal variables
and behavior tangent stamps it.

The goal of the play is to write a routine making these calculations, this routine being called by NMCOMP.
disconnection towards this routine in NMCOMP is done starting from a test consisting in comparing
variable COMPOR (1) with the name of the relation of behavior given under key word RELATION of
COMP_INCR and treated in the routine.

If one defined a relation of name “TOTO”, i.e one wrote:

statnl = STAT_NON_LINE (

MODELE = MOD,
CHAM_MATER
=
chmat,

COMP_INCR = _F (RELATION = “LOUSE”),





…)

One will write a routine NMTOTO which will be called by NMCOMP in the following way:

IF (COMPOR (1) (1:4) .EQ' TOTO') then CALL NMTOTO (…,…)

ELSE



ENDIF

Let us take NMTOTO like a generic routine to carry out the integration of a law of behavior.

The arguments at output of NMTOTO will be:

Standard name
Significance
SIGP (6)
R
constraints at the current moment
VIP (NBVARI)
R
variables intern at the current moment
DSIDEP (6,6)
R
stamp behavior tangent. It is one
square matrix dimensioned “into hard” 6 X 6 for
continuous mediums 2D and 3D

The arguments in input of NMTOTO will be:

Standard name
Significance
NDIM
I
Dimension of space (2 or 3, addresses material
coded).

· TYPMOD (1) is the type of modeling: 3D,
D_PLAN, AXIS or C_PLAN,
· TYPMOD (2) is equal to “INCO” for
incompressible elements.
COMPOR (3)
K16
Table of 3 K16 relating to relation of
behavior.

· COMPOR (1) is the name of the relation of
behavior,
· COMPOR (2) is the number of internal variables
by point of integration,
· COMPOR (3) is K16 indicating one
assumption on the deformations.
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crit (5)
R
Table of local criteria of convergence.

· crit (1): maximum iteration count with
convergence,
· crit (2): type of tangent matrix at the moment
t+dt,
· if crit (2) = 0, one has a formulation in
speed and the matrix is symmetrical,
· if crit (2) = 1, one has a formulation
incremental and the matrix can be
not-symmetrical,
· crit (3) is the value of tolerance of
convergence,
· crit (4) is the number of increments for
local recutting of the step of time,
· if crit (4) = - 1, 0 or 1, it does not have there
recutting,
· crit (5) is the type of local integration for
law of behavior,
· if crit (5) =0, integration is Euler-implicit,
· if crit (5) = 1, one makes an integration of
RUNGE_KUTTA.
instam
I
Moment of preceding calculation
instap
I
Moment of calculation
TM
R
Temperature at the moment of preceding calculation
TP
R
Temperature at the moment of calculation
TREF
R
Temperature of reference
EPSM (6)
R
Deformations at the moment of preceding calculation (see
notice has).
DEPS (6)
R
Increment of deformation, i.e., it acts of B.U in
HPP (see remark has).
SIGM (6)
R
Constraints at the moment of preceding calculation
VIM (NBVARI)
R
variables intern at the moment of preceding calculation;
NBVARI is in entirety entered “into hard” the routine,
clean with the relation of behavior and not one
variable
option
K16
Option of calculation asked.
There are the choice between:

· RIGI_MECA_TANG: this option is useful at the time of
prediction, internal variables and them
constraints are not calculated, (see
notice b),
· FULL_MECA: the tangent matrix is reactualized
at each iteration and one updates them
constraints and internal variables,
· RAPH_MECA: the matrix is not reactualized
tangent; one updates the constraints and them
internal variables.

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Note:

a. Comme specified previously, the tensors deformation, constraints at the moment
precedent, and increment of deformation, given in arguments of input, are such as
components except diagonal (shearing for the constraints, and distortion for
the deformations, are multiplied by 2 before call to routine NMCOMP. This was
bench to facilitate calculations of standards intervening in a certain number of
nonlinear behaviors, depend on the second invariant of the tensors. It is thus
necessary to take into account this characteristic in the routine of integration with
to write.
Consequently, components of shearing of the tensor of constraints
at output must also be multiplied by same coefficient 2. This does not have
in theory of consequence on the matrix of elasticity, nor on the tangent matrix.
B. Argument OPTION is important because it makes it possible to determine calculations to carry out.
In particular, option RIGI_MECA_TANG is intended to calculate only one matrix
tangent of prediction, to build starting from DSIDEP. It is necessary to take guard in
programming not to use in this case arguments SIGP and VIP, of which the place
memory is not allocated for this option.

Routine NMTOTO will be organized in the following way:

subroutine NMTOTO (NDIM, IMATE, TYPMOD, COMPOR, crit, instam, instap, TM, TP,
TREF, EPSM, LIFO, SIGM, VIM, OPTION, SIGP, VIP, DSIDEP)

· Reading of the characteristics of material (elastic and different) and calculation of these characteristics
at the moments instam and instap by using routine RCVALA.
For example, these characteristics can be E, v, E, S
T
y.
One thus will calculate E - v - E -, S
,
,
+
+
+
+
T
y (i.e at the moment instam) and E, v, E, S
T
y (i.e at the moment
instap).

· When one will handle the constraints and the deformations, one will not make loops of 1 with
6 but of the loops of 1 to NDIMSI.

NDIMSI = 4 for the 2D
NDIMSI = 6 for the 3D

· Calculation of the threshold (for the laws with thresholds).

· For options FULL_MECA and RAPH_MECA: calculation of the constraints and the internal variables.

· For options FULL_MECA and RIGI_MECA_TANG: calculation of the matrix of behavior


tangent
or

&

.



3.2.2 Example of a routine realization the integration of a law of behavior:
NMCINE

NMCINE carries out the integration of a relation of behavior of elastoplasticity of von Mises with
linear kinematic work hardening.

For the integration of this relation, one will refer to Doc. [R5.03.02].

The arguments of NMCINE appear among those of generic routine NMTOTO described in [&3.1].

One a:

subroutine NMCINE (NDIM, IMATE, COMPOR, CRIT, INSTAM, INSTAP, TM, TP, TREF,
EPSM, LIFO, SIGM, VIMP, OPTION, SIG, VIP, DSIDEP).
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This routine is organized in the following way:

· Calculation of the elastic characteristics of material with temperatures TM and TP.
For that, one uses routine RCVALA in the following way:

C
C READING OF THE ELASTIC CHARACTERISTICS OF THE MATERIAL (TIME ­ AND +)
NOMRES (1) = ' E'
NOMRES (2) = ' NU'
NOMRES (3) = ' ALPHA4


CAL RCVALA (IMATE, “ELAS”, 1, “TEMP”, TM, 2, NOMRES, VALRES, CODRET, FB2)


CAL RCVALA (IMATE, “ELAS”, 1, “TEMP”, TM, 1,
+
NOMRES (3), VALRES (3)
=
0.D0

IF (CODRET (3) .NE. “OK”) VALRES (3) = 0.D0
EM
=
VALRES (1)
NUM
=
VALRES (2)
DEUMUM
=
EM/(1.D0+NUM)
TROIKM
=
EM/(1.D0-2.D0 * NUM0
ALPHAM
=
VALRES (3)

CAL RCVALA (IMATE, “ELAS”, 1, “TEMP”, TP, 2, NOMRES, VALRES, CODRET, FB2)

CAL RCVALA (IMATE, “ELAS”, 1, “TEMP”, TP, 1?
+
NOMRES (3), VALRES (3), CODRET (3), BL2)

IF (CODRET (3) .NE. “OK”) VALRES (3) = 0.D0
E
=
VALRES (1)
NAKED
=
VALRES (2)
LAMBDA
=
E * NAKED/551.D0-2 * NAKED) * (1.D0+NU))
DEUXMU
=
E/(1.D0+NU)
ALPHAP
=
VALRES (3)

One rather uses the coefficients of Lamé and µ and the model of compressibility K.

· Calculation of the characteristics of work hardening AND, SY and C at the temperatures TM and TP; for AND and SY,
one uses routine RCVALA like previously:

C
C READING OF the CHARACTERISTICS Of WORK HARDENING
NOMRES (1) = ' D_SIGM_EPSI'
NOMRES (2) = ' SY'


CAL RCVALA (IMATE, “ECRO_LINE”, 1, “TEMP”, TM, 2,
+
NOMRES, VALRES, CODRET, FB2)
DSDEM=VALRES (1)
SIGYM=VALRES (2)
CM=2.D0/3.D0 * DSDEM/(1.DO-DSDEM/EM)
NOMRES (1) = ' D_SIGM_EPSI'
NOMRES (2) = ' SY'
CAL RCVALA (IMATE, “ECRO_LINE”, 1, “TEMP”, TP, 2,
+
NOMRES, VALRES, CODRET, FB2)
DSDE=VALRES (1)
SIGY=VALRES (2)
C
=
2.D0/3.D0 * DSDE/(1.D0-DSDE/E)

· Calculation of the constraint of test and its standard within the meaning of von Mises:

~
~-
~
S.E. = +
2

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To affect the terms of ~
S.E. , one makes a loop of 1 to NDIMSI like one saw it with [§3.1]:

C
C CALCULATION OF THE ELASTIC CONSTRAINTS
D0
110
K=1,3
DEPSTH (K)

= LIFO (K) ­ (ALPHA * (TP-TREF) - ALPHAM * (TM-FREF))
DEPSTH (K+3)

=
DEPS (K+3)
110 CONTINUE
EPSMO = (DEPSTH (1) + DEPSTH (2) + DEPSTH (3)/3/D0
C 115 K=1, NDIMSI

DEPSDV (K) = DEPSTH (K) ­ EPSMO * KRON (K)
115 CONTINUE
C CALCULATION OF THE CRITERION OF VON MISES OF SIGEL
C ONE SEES HERE the INTEREST OF the COEFF RACINE (2) ON
C SHEARINGS.
SIGMO = (SIGM (1) + SIGM (2) + SIGM (3)/3/D0
SIELEQ = 0.D0
C 114 K=1, NDIMSI

SIGDV (K) = SIGM (K) ­ SIGMO * KRON (K)

SIGDV (K) = DEUXMU/DEUMUM * SIGDV (K)

SIGEL (K) = SIGDV (K) + DEUXMU * DEPSDV (K)

SIELEQ = SIELEQ + (SIGEL (K) - C/CM * VIM (K))** 2
114 CONTINUE
SIGMO = TROISK/TROIKM * SIGMO
SIELEQ = SQRT (1.D5D0 * SIELEQ)

· Calculation of the threshold of plasticity

Threshold = ~
S.E. - sy
THRESHOLD = SIELEQ - SIGY

· For options RAPH_MECA and FULL_MECA, calculation of the constraints and variables intern with
the current moment

-
if threshold < 0
One is in the elastic range and the increments of the variables intern are null:

C
C CALCULATION OF THE ELASTOPLASTIC CONSTRAINTS AND THE INTERNAL VARIABLES


IF (OPTION (1:9) .EQ.“RAPH_MECA”. GOLD.
+
OPTION (1:9) .EQ.“FULL_MECA” THEN
IF
(SEUIL.LT.0.D0)
THEN
VIP (7)
=
0.D0
DP
=
0.D0
SIELEQ
=
1.D0
A1
+
0.D0
A2
+
0.D0

- if
threshold

One is in the elastoplastic field and one calculates the increments of the constraints and of
internal variables.

ELSE
VIP (7)
=
1.D0
DP
=
THRESHOLD/(1.5D0 * (DEUXMU+C))
A1
=
(DEUXMU/(DEUXMU+c)) * (THRESHOLD/SIELEQ)



A2 = (C/(DEUXMU+c)) * (THRESHOLD/SIELEQ)
ENDIF
PLASTI=VIP (7)


C 160 K = 1, NDIMSI



SIGDV (K) = SIGEL (K) ­ A1 * SIGEL (K) - VIM (K) * C/CM)
SIGP (K) =
SIGDV (K)
+
(SIGMO+TROISK * EPSMO) * KRON (K)
VIP (K)
=
VIM (K) * C/CM
+
A2 * (SIGEL (K) - VIM (K) * C/CM)
160 CONTINUE
ENDIF
Handbook of Descriptif Informatique
D5.04 booklet: -
HT-66/05/003/A

Code_Aster ®
Version
7.4
Titrate:
To introduce a new law of behavior
Date
: 22/06/05
Author (S):
J.M. PROIX, G. BERTRAND Clé
:
D5.04.01-B Page
: 13/16

· For options RIGI_MECA_TANG and FULL_MECA, calculation of the matrix of behavior
tangent:


RIGI_MECA_TANG calculates
&

FULL_MECA calculates


But one shows in our case that & =
if p = 0, which corresponds well to
the use of RIGI_MECA_TANG at the time of the phase of prediction.


The matrix is thus calculated
.
This calculation is not the object of this document and is too long to be exposed clearly here
[R5.03.02].


It is thus admitted that one has affected the square matrix dsidep with the values of
.
Note:

Particular case of the plane constraints.

To write the tangent matrix, the fact is used that, when one writes:


=



zz = 0

One deduces zz from it according to
xx,
yy and
xy, and this expression of zz is injected
in the other relations.

Therefore in the processing of the tangent matrices in the case of plane constraints,
one finds the instructions following:

C ­ - 8.3 CORRECTION FOR THE PLANE CONSTRAINTS:

IF (CPLAN) THEN


C 136 K=1, NDIMSI
IF
(K.EQ.3)
GO
TO
136
C
137
L=1,
NDIMSI
IF
(L.EQ.3)
GO
TO
137
DSIDEP
(K, L) =DSIDEP (K, L)
+
-
1.D0/DSIDEP (3,3) * DSIDEP (K, 3) * DSIDEP (3, L)
137 CONTINUE
136 CONTINUE
ENDIF

3.3
Programming of law in environment PLASTI

3.3.1 Introduction

By means of computer, one passes by the routine NMCOMP which calls routine REDECE. PLASTI is called
by REDECE.
Environment PLASTI is described in documentation [R5.03.10]: `Relation of behavior
élasto-viscoplastic of the LMARC'.
Handbook of Descriptif Informatique
D5.04 booklet: -
HT-66/05/003/A

Code_Aster ®
Version
7.4
Titrate:
To introduce a new law of behavior
Date
: 22/06/05
Author (S):
J.M. PROIX, G. BERTRAND Clé
:
D5.04.01-B Page
: 14/16

3.3.2 Algorithm of resolution of the quasi-static problem

One seeks to check the balance of the structure at every moment. In incremental form, it is about one
nonlinear problem whose variational formulation in the case of the small deformations can
to put in the form:

To find U
such as:


((U + U
), T) () D = L (T)

kinematically


Drunk


= ud (T)
acceptable and T

where U indicates the field of displacement, Bu
ud
=
(T) corresponds to the boundary conditions in
displacement and L (T) are the loading at the moment T.

One is thus led to solve, for each increment of time T
:

Ft+t (U + U
T
) = 0 on the basis of a state with balance F = 0
0

U
being the increment of
U
solution on
T
, C being known

The general outline adopted by Aster to solve this discretized total system is a method of
Newton which is written, K being an indication of iteration:

F

D (U
K) = - F (U
K)
U
K

U
+1 = U
+ D

K
K
(U
K)

This diagram requires, starting from the estimate of displacements to the interation K, to calculate in
each point of Gauss:

T T
+ which checks the law of behavior


MR. C
t+ T =

the operator of tangent behavior
T + T

F

T


with



= K =
K
K =
B
Data base
U
E
E
E




E


3.3.3 Environment
Plasti

It is thus necessary, with each total iteration and in each point of Gauss, to integrate them
equations of the model for calculation T T
+ and to calculate the operator of tangent behavior.

An environment was created in Code_Aster with an aim of parameterizing the establishment of models
elastoviscoplastic presenting a function threshold (field of elasticity).

This algorithm:

· manage the choices of integration elastic or (visco) plastic,
· propose various routines to contribute to the resolution of the nonlinear system (local) formed
by the equations of the model,
· updates the variables at the end of the increment,
· call the routines user for the calculation of the operator of tangent behavior.
Handbook of Descriptif Informatique
D5.04 booklet: -
HT-66/05/003/A

Code_Aster ®
Version
7.4
Titrate:
To introduce a new law of behavior
Date
: 22/06/05
Author (S):
J.M. PROIX, G. BERTRAND Clé
:
D5.04.01-B Page
: 15/16

The step to establish a new model can be schematized in the following way:

Writing of the equations of the model of speed
y = F (y, T)
Choice of a diagram of integration
Writing of the system discretized R (y) = 0
Writing routines specific to the models:
· recovery of the data materials,
· evaluation of the function threshold,
· evaluation of the operator of tangent behavior
· routine for the resolution of the system R (y) = 0
(the algorithm proposes a method of Newton for one
implicit nonlinear system)

+ Modification of the routines of shunting of the algorithm

3.3.4 Formalization of the equations to be solved

One has to solve the following equations:

· The law of behavior connecting the increment of the constraints to the increment of the total deflections
with internal variables (cumulated total deflection, center of the surface of load,…).

That is to say G (
p
, p
, v
ari. ). = 0 éq 3.3.4-1

· laws of evolution of the various internal variables:

That is to say L (
p
, p
, v
ari,)
… = 0









éq 3.3.4-2
· the criterion of plasticity
F (, X, p
I
) = 0
That is to say - X - R (p
I
) éq 3.3.4-3
·
vari indicates the variables intern others that p and p,
·
Xi center of the surface of load is an example of component of vari,
· The increment of cumulated plastic deformation p is calculated with [éq 3.3.4-3].
Handbook of Descriptif Informatique
D5.04 booklet: -
HT-66/05/003/A

Code_Aster ®
Version
7.4
Titrate:
To introduce a new law of behavior
Date
: 22/06/05
Author (S):
J.M. PROIX, G. BERTRAND Clé
:
D5.04.01-B Page
: 16/16

The equation [éq 3.3.4-1] to 6 unknown factors (6 components of the symmetrical tensor of the constraints).

The equation [éq 3.3.4-3] to 1 unknown factor: p.
The number of unknown factors relating to the equation [éq 3.3.4-2] is equal to the component count of
variables intern others that the deformation (visco-) plastic and the deformation (visco-) plastic
cumulated.

One has to solve, as indicated in [§3.2.2]:

F (y
) = 0

with yT = (vari p)

One solves this system by a method of Newton, that is to say:

F D [yk] = - F (yk)
yk+1

y

1
= y
K + D
K +
(y
K)

In addition, one has to calculate the tangent matrix.

It is considered that the system F (y
) = 0 are checked at the end of the increment. One disturbs F according to one
small variation. One regards a variable and not as a parameter.

The system remains with balance and one thus checks that dF = 0.

F
F
F
F
That is to say D
+ D
+

+
= 0


D vari
D p


vari

p


One is thus led to use the same matrix jacobienne which was used for to calculate F (y
) = 0 bus
one can write:

F D (y)

= X
y


with yT = [vari p]

and X = [Hd 0
] 0

By successive substitutions and eliminations, one obtains Kd = Hd



from where the required tangent operator

[- K1H]





=
T +t

The expression of K-1 is difficult to determine, also uses one a solvor LU to evaluate it.

Handbook of Descriptif Informatique
D5.04 booklet: -
HT-66/05/003/A

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