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Code_Aster
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Version
7.4
Titrate:
To introduce a new law of behavior
Date
:
22/06/05
Author (S):
J.M. PROIX, G. BERTRAND
Key
:
D5.04.01-B
Page
:
1/16
Data-processing manual of Description
D5.04 booklet: -
HT-66/05/003/A
Organization (S):
EDF-R & D/AMA, CS IF















Data-processing manual of Description
D5.04 booklet: -
Document: D5.04.01



To introduce a new law of behavior




Summary:

The purpose of this document is to provide to the developers the main elements necessary to
establishment (or amendment) of a law of behavior in Code_Aster. It describes the amendments with
to carry out on the catalog of controls, 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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Code_Aster
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Version
7.4
Titrate:
To introduce a new law of behavior
Date
:
22/06/05
Author (S):
J.M. PROIX, G. BERTRAND
Key
:
D5.04.01-B
Page
:
2/16
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D5.04 booklet: -
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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
Amendment of the catalog of
DEFI_MATERIAU
.............................................................................. 3
1.3
Amendment 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 the routine
NMCOMP
(inelastic behavior) or
NMCPEL
(elastic behavior) ................................................................................................................ 5
2
Amendments of the catalogs of controls ....................................................................................... 5
2.1
DEFI_MATERIAU
.............................................................................................................................. 5
2.2
STAT_NON_LINE
,
DYNA_NON_LINE
,
DYNA_TRAN_EXPLI
............................................................... 6
3
Amendments 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 the 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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Code_Aster
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Version
7.4
Titrate:
To introduce a new law of behavior
Date
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Author (S):
J.M. PROIX, G. BERTRAND
Key
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D5.04.01-B
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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 stresses to the deformations. One restricts here in the continuous mediums 2D and 3D.
method remains valid for models with local nonlinear behavior in plane stresses, 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
method Newton [R5.03.01 § 2.2.2.2] one must calculate the nodal forces
()
R U
Q
in
T in
=
(options
RAPH_MECA
and
FULL_MECA
) stresses
in
being calculated from
displacements
U
in
via the relation of behavior. One must build too
the tangent operator to calculate
K
in
(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
in
by the option
FULL_MECA
, with
run of the iterations of Newton. Indeed, this last operator is tangent with the problem discretized of
implicit way.
1.2
Amendment 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 modelize 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 (channel 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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Code_Aster
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Version
7.4
Titrate:
To introduce a new law of behavior
Date
:
22/06/05
Author (S):
J.M. PROIX, G. BERTRAND
Key
:
D5.04.01-B
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1.3
Amendment of the catalogs of
STAT_NON_LINE
and
DYNA_NON_LINE
In the catalogs of these controls, one gives under the key words factors
COMP_INCR
or
COMP_ELAS
, the name of the relation of behavior after the 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 stresses at the moment of preceding calculation,
·
the variables intern at the moment of preceding calculation; for example
I
P
X
p,
,
,
·
the option of calculation: 3 options must be calculated:
“RIGI_MECA_TANG”
,
“RAPH_MECA”
and
“FULL_MECA”
.
The arguments of exit are according to the option of calculation:
·
the tensor of the stresses 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, stresses at the previous moment, and increment of deformation, given
in arguments of input, are such as the components except diagonal (shearing for
stresses, 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 stresses at exit must also be multiplied by the 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:
·
SMALL: in this case of the tensors deformations are calculated linearly by report/ratio
with displacements, on the initial geometry (Assumption of the Small Disturbances:
HP);
·
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
HP, but connects this time the deformations of GREEN-LAGRANGE to the stresses of
PIOLA-KICHHOFF of 2
ème
species. The transformation of stresses 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 the formulation HP like previously any more [R5.03.21].
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Code_Aster
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Version
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Titrate:
To introduce a new law of behavior
Date
:
22/06/05
Author (S):
J.M. PROIX, G. BERTRAND
Key
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D5.04.01-B
Page
:
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D5.04 booklet: -
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1.5 Branch of this routine in the routine
NMCOMP
(inelastic behavior) or
NMCPEL
(elastic behavior)
Shunting is done according to the name of the relation which was given under the key word
RELATION
of
COMP_INCR
or
COMP_ELAS
.


2
Amendments of the catalogs of controls
2.1
DEFI_MATERIAU
One introduces into the catalog of the control
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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Code_Aster
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Version
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Titrate:
To introduce a new law of behavior
Date
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22/06/05
Author (S):
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Key
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D5.04.01-B
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D5.04 booklet: -
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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 controls
STAT_NON_LINE
and
DYNA_NON_LINE
by giving the name of the relation after the 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 (
MODEL = 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 (
MODEL = 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
indicate the cumulated plastic formation,
·
X
indicate 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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Code_Aster
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Version
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Titrate:
To introduce a new law of behavior
Date
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J.M. PROIX, G. BERTRAND
Key
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D5.04.01-B
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3
Amendments 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
stresses 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 the 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 partnership School of the 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
who makes the integration of the laws of behavior
incremental (thus relative to
COMP_INCR
).
The 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 the 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 BARS, multifibre Poutres, 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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Code_Aster
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Version
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Titrate:
To introduce a new law of behavior
Date
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These calculations consist in determining at the current moment the tensor of the stresses, 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 the key word
RELATION
of
COMP_INCR
and treated in the routine.
If a relation of name were defined
“LOUSE”
, i.e one wrote:
statnl = STAT_NON_LINE (
MODEL = MOD,
CHAM_MATER
=
chmat,
COMP_INCR = _F (RELATION = “LOUSE”),
…)
One will write a routine
NMTOTO
who will be called by
NMCOMP
in the following way:
IF (COMPOR (1) (1:4) .EQ' TOTO') then CAL NMTOTO (…,…)

ELSE



ENDIF
Let us take
NMTOTO
like a generic routine to carry out the integration of a law of behavior.
Arguments at exit of
NMTOTO
will be:
Standard name
Significance
SIGP (6)
R
stresses 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
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 one
K16
indicating one
assumption on the deformations.
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Code_Aster
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Version
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Titrate:
To introduce a new law of behavior
Date
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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
, there is a formulation in
speed and the matrix is symmetrical,
·
if
crit (2) = 1
, there is 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 pitch of time,
·
if
crit (4) = - 1, 0 or 1,
there is not
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).
LIFO (6)
R
Increment of deformation, i.e., it acts of B.
U in
HP (see remark has).
SIGM (6)
R
Stresses 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
stresses are not calculated, (see
notice b),
·
FULL_MECA
: the tangent matrix is reactualized
at each iteration and one updates them
stresses and internal variables,
·
RAPH_MECA
: the matrix is not reactualized
tangent; one updates the stresses and them
internal variables.
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Code_Aster
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Version
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Titrate:
To introduce a new law of behavior
Date
:
22/06/05
Author (S):
J.M. PROIX, G. BERTRAND
Key
:
D5.04.01-B
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:
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Note:
a. As specified previously, the tensors deformation, stresses at the moment
precedent, and increment of deformation, given in arguments of input, are such as
components except diagonal (shearing for the stresses, 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 stresses
at exit must also be multiplied by the same coefficient
2
. This does not have
in theory of consequence on the matrix of elasticity, nor on the tangent matrix.
B. The argument OPTION is important because it makes it possible to determine calculations to carry out.
In particular, the option
RIGI_MECA_TANG
is intended to calculate only one matrix
tangent of prediction, to build from
DSIDEP
. It is necessary to take guard in
programming not to use in this case the arguments
SIGP
and
VIP
, of which the place
memory is not allocated for this option.
The 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 the routine
RCVALA
.
For example, these characteristics can be
E O C S
T
y
,
,
.
One thus will calculate
E O C
S
T
y
-
-
-
-
,
,
,
(i.e at the moment instam) and
E O C
S
T
y
+
+
+
+
,
,
,
(i.e at the moment
instap).
·
When one will handle the stresses and the deformations, one will not make loops of 1 with
6 but of the loops of 1 with
NDIMSI
.
NDIMSI
= 4 for the 2D
NDIMSI
= 6 for the 3D
·
Calculation of the threshold (for the laws with thresholds).
·
For the options
FULL_MECA
and
RAPH_MECA
: calculation of the stresses and the internal variables.
·
For the 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
carry 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].
Arguments of
NMCINE
appear among those of the generic routine
NMTOTO
described with [&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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Code_Aster
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Version
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Titrate:
To introduce a new law of behavior
Date
:
22/06/05
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J.M. PROIX, G. BERTRAND
Key
:
D5.04.01-B
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:
11/16
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D5.04 booklet: -
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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)
The coefficients of Lamé rather are used
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
,
the routine is used
RCVALA
as 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 stress of test and its standard within the meaning of von Mises:
~
~
~
S.E.
=
+
-
2
background image
Code_Aster
®
Version
7.4
Titrate:
To introduce a new law of behavior
Date
:
22/06/05
Author (S):
J.M. PROIX, G. BERTRAND
Key
:
D5.04.01-B
Page
:
12/16
Data-processing manual of Description
D5.04 booklet: -
HT-66/05/003/A
To affect the terms of
~se
, one makes a loop of 1 with
NDIMSI
like one saw it with [§3.1]:
C
C CALCULATION OF THE ELASTIC STRESSES
D0
110
K=1,3
DEPSTH (K)
= LIFO (K) ­ (ALPHA * (TP-TREF) - ALPHAM * (TM-FREF))
DEPSTH (K+3)
=
LIFO (K+3)
110 CONTINUOUS
EPSMO = (DEPSTH (1) + DEPSTH (2) + DEPSTH (3)/3/D0
C 115 K=1, NDIMSI
DEPSDV (K) = DEPSTH (K) ­ EPSMO * KRON (K)
115 CONTINUOUS
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 CONTINUOUS
SIGMO = TROISK/TROIKM * SIGMO
SIELEQ = SQRT (1.D5D0 * SIELEQ)
·
Calculation of the threshold of plasticity
Threshold =
~se sy
-
THRESHOLD = SIELEQ - SIGY
·
For the options
RAPH_MECA
and
FULL_MECA
, calculation of the stresses and the 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 STRESSES 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 stresses 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 CONTINUOUS
ENDIF
background image
Code_Aster
®
Version
7.4
Titrate:
To introduce a new law of behavior
Date
:
22/06/05
Author (S):
J.M. PROIX, G. BERTRAND
Key
:
D5.04.01-B
Page
:
13/16
Data-processing manual of Description
D5.04 booklet: -
HT-66/05/003/A
·
For the options
RIGI_MECA_TANG
and
FULL_MECA
, calculation of the matrix of behavior
tangent:
RIGI_MECA_TANG
calculate
&
FULL_MECA
calculate
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 stresses.
To write the tangent matrix, the fact is used that, when one writes:
=




=
zz
0
One deduces some
zz
according to
xy
yy
xx
and
,
, and one injects this expression of
zz
in the other relations.
Therefore in the processing of the tangent matrices in the case of plane stresses,
one finds the instructions following:
C ­ - 8.3 CORRECTION FOR THE PLANE STRESSES:
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 CONTINUOUS
136 CONTINUOUS
ENDIF

3.3
Programming of law in the environment
PLASTI
3.3.1 Introduction
By means of computer, one passes by the routine
NMCOMP
who calls the routine
REDECE
.
PLASTI
is called
by
REDECE
.
Environment
PLASTI
is described in documentation [R5.03.10]: `Relation of behavior
élasto-viscoplastic of
LMARC
'.
background image
Code_Aster
®
Version
7.4
Titrate:
To introduce a new law of behavior
Date
:
22/06/05
Author (S):
J.M. PROIX, G. BERTRAND
Key
:
D5.04.01-B
Page
:
14/16
Data-processing manual of Description
D5.04 booklet: -
HT-66/05/003/A
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
U U T
T
D
+
=
=



,
B
kinematically
acceptable and
where
U
indicate the field of displacement,
()
Drunk U T
D
=
corresponds to the boundary conditions in
displacement and
)
(T
L
is the loading at the moment
T
.
One is thus led to solve, for each increment of time
T
:
(
)
F
U
U
F
U
U
T U
T
T
T
T
+
+
=
=
0
0
0
on the basis of a state with balance
being the increment of the solution on
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
U D U
F U
U
U
D U
K
K
K
K
K
K
= -
=
+



+
1
This diagram requires, starting from the estimate of displacements to the interation
K
, to calculate in
each point of Gauss:
T
T
+
who checks the law of behavior
M
T
T
C
T
T
+
+
=




the operator of tangent behavior
F
U
K
K
K
B
Data base
E
E
E
T
E
=
=
=








with
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.
background image
Code_Aster
®
Version
7.4
Titrate:
To introduce a new law of behavior
Date
:
22/06/05
Author (S):
J.M. PROIX, G. BERTRAND
Key
:
D5.04.01-B
Page
:
15/16
Data-processing manual of Description
D5.04 booklet: -
HT-66/05/003/A
The step to establish a new model can be schematized in the following way:

Writing of the equations of the model of speed
Choice of a diagram of integration
Writing of the discretized system
()
y
F y T
=
,
()
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
(the algorithm proposes a method of Newton for one
implicit nonlinear system)
()
R y
=
0
+ Amendment 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 stresses 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 vari
p
,
,
,
,
,…
=
0
éq 3.3.4-1
·
laws of evolution of the various internal variables:
That is to say
(
)
L
p vari
p
,
,
,
,
,…
=
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
indicate the variables intern others that
p
p
and
,
·
X
I
center 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].
background image
Code_Aster
®
Version
7.4
Titrate:
To introduce a new law of behavior
Date
:
22/06/05
Author (S):
J.M. PROIX, G. BERTRAND
Key
:
D5.04.01-B
Page
:
16/16
Data-processing manual of Description
D5.04 booklet: -
HT-66/05/003/A
The equation [éq 3.3.4-1] to 6 unknown factors (6 components of the symmetrical tensor of the stresses).
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
(
)
y
vari
p
T
=
One solves this system by a method of Newton, that is to say:
[]
()
()
F
y
D yk
F yk
y
yk D yk
K
K
+
+
= -
=
+
1
1
In addition, one has to calculate the tangent matrix.
It is considered that the system
()
F y
=
0
is checked at the end of the increment. One disturbs
F
according to one
small variation. One considers
like a variable and not like a parameter.
The system remains with balance and one thus checks that
dF
=
0
.
That is to say
F D
F D
F
vari D vari
F
p D p
+
+
+
=
0
One is thus led to use the same matrix jacobienne which was used for to calculate
()
F y
=
0
because
one can write:
()
F
y D y
X
=
with
[
]
y
vari
p
T
=
and
[
]
X
Hd
=
0 0
By successive substitutions and eliminations, one obtains
Kd
Hd
=
from where the required tangent operator
[
]




=
+
-
T
T
K H
1

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