background image
Code_Aster
®
Version
4.0
Titrate:
Viscoplastic relation of behavior of Taheri
Date:
08/09/97
Author (S):
E. LORENTZ
Key:
R5.03.05-A
Page:
1/16
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HI-74/97/019/A
Organization (S):
EDF/IMA/MN
Manual of Reference
R5.03 booklet: Nonlinear mechanics
R5.03.05 document
Viscoplastic relation of behavior of Taheri
Summary
One presents in this document the establishment of the relation of viscoplastic behavior of Taheri,
available for the whole of the isoparametric elements (continuous medium 2D and 3D) except for
plane stresses. After a presentation of the equations of evolution of this law, one describes the system obtained
by implicit discretization; it is shown in particular that it always admits a solution.
This model is well adapted to describe the response of the austenitic steels under cyclic stresses, and
in particular the phenomenon of progressive deformation. On the other hand, because of its complexity (two surfaces of
charge, semi-discrete internal variable), it does not appear desirable to employ it for applications
different (monotonous way of loading, for example).
background image
Code_Aster
®
Version
4.0
Titrate:
Viscoplastic relation of behavior of Taheri
Date:
08/09/97
Author (S):
E. LORENTZ
Key:
R5.03.05-A
Page:
2/16
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HI-74/97/019/A
1
Description of the model
The relation of behavior proposed by Taheri [bib5] makes it possible to describe the response of steels
austenitic under cyclic stresses: it is indeed well adapted to represent it
phenomenon of progressive deformation. Before stating the equations themselves, one can
to specify that this model differs from conventional plasticity (criterion of von Mises with work hardening
kinematics and isotropic) by two characteristics, sources of difficulties in the numerical formulation.
On the one hand, the evolution of the dissipative variables rests on two criteria of load instead of one:
first, conventional, conditions the appearance of plastic deformation, the second makes it possible to keep one
trace “maximum” work hardening reached by material to account for the phenomenon of ratchet.
In addition, satisfactorily to represent the progressive deformation, an internal variable
semi-discrete was introduced. Constant when the behavior is dissipative, it evolves/moves only in
the elastic mode of material. Of original appearance, this model does not rest any less on
physical bases, always exposed in Taheri [bib5]. It is accessible, in a wide version
viscoplastic (necessary to describe the behavior under high temperatures), by
order
STAT_NON_LINE
under the key word
RELATION: VISC_TAHERI
.
1.1 Behavior
plastic
A detailed description of the law of behavior is given in Taheri and Al [bib6].
Briefly, the state of material is described by its state of deformation, its temperature like
four internal variables:
tensor of total deflection
T
temperature
p
cumulated plastic deformation
p
tensor of plastic deformation
p
stress of peak, memory of maximum work hardening
p
N
plastic tensor deformation due to the last discharge (variable semi-discrete).
The equations of state which express the thermodynamic forces associated according to the variables
of state are written:
(
)
(
)
(
)












=
-
+
-
=
-
K Tr
T T
ref.
HT
p
HT
Id
Id
2
µ
~
éq 1.1-1
()
(
)
R
R
With
m E
has
eq
has
B p
p
S
=
+
-




= -
-
-




D
D
0
2
3
1
1






p
N
p
éq 1.1-2
[
]
X
p
N
p
=
-
=
+
-
-




C
C
S
C
C E
p
B p
p
S






1
1
éq 1.1-3
background image
Code_Aster
®
Version
4.0
Titrate:
Viscoplastic relation of behavior of Taheri
Date:
08/09/97
Author (S):
E. LORENTZ
Key:
R5.03.05-A
Page:
3/16
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HI-74/97/019/A
~
has
deviatoric part of a tensor
has
R
isotropic variable of work hardening
X
kinematic variable of work hardening
K,
µ
modules of compressibility and shearing
thermal expansion factor
T
ref.
temperature of reference
S
stress of ratchet
B R With
m C C
,
,
,
°
has
1
other characteristics of work hardening of material
Let us note that the moduli of elasticity and the thermal expansion factor are indicated by
the user by the control
DEFI_MATERIAU
, key word
ELAS
, while characteristics of
work hardening are fixed by the key word
TAHERI
. These characteristics can depend on
temperature, by employing the key words
ELAS_FO
and
TAHERI_FO
. Also let us specify that one
example of identification of the characteristics of work hardening on uniaxial situations is given
in Geyer [bib2].
The evolution of the internal variables is defined by two criteria. The first controls plasticity
traditional with work hardenings kinematics and isotropic compounds:
(
)
(
)
F
~
~
~
=
-
-
=
-
-









X
S
X
X
0
eq
eq
R
0
and
éq 1.1-4
()
eq
equivalent standard:
()
has
has has
eq
=
32
1
2
~:~
F
criterion of plasticity
S
0
normal external with the criterion
F
This criterion is matched conventional condition of load/discharge:
if
or
if
and
such as
elasticity
plasticity
=
F
! :
F
! :
!
!
!F
(
)
(
)
<
=
>
=


0
0
0
0
0
0
0






S
S
0
0
p
p
éq 1.1-5
And the law of flow associated with the criterion
F
is:
!
!
!
!



p
0
S
=
=
3
2
2
3
p
p
and thus
eq
p
éq 1.1-6
background image
Code_Aster
®
Version
4.0
Titrate:
Viscoplastic relation of behavior of Taheri
Date:
08/09/97
Author (S):
E. LORENTZ
Key:
R5.03.05-A
Page:
4/16
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HI-74/97/019/A
The second criterion controls the evolution of the stress of peak. Geometrically in the space of
diverters of the stresses, it translates the fact that the first surface of load
(
)
F
=
0
, represented by
a sphere of center
X
and of radius
R
, inside a sphere of center the origin remains and
p
. It is written simply:
G
=
+
-
X
R
eq
p
0
éq 1.1-7
G
criterion of maximum work hardening
According to the preceding geometrical considerations, the evolution of the stress of peak is:
if
or
if
and
such as
= 0
G
!
!
!
G
!
!
!
!G
<
+
=
=
+ >




0
0
0
0
0
0
X
R
X
R
eq
p
eq
p

éq 1.1-8
It should be noticed that in the natural state of material, the stress of peak is not null but is worth
initial elastic limit, namely:
(
)
(
)
p
m R
initial
=
-
1
0
Until now, we did not evoke the evolution of the semi-discrete internal variable
p
N
. In fact,
it evolves/moves only in elastic mode. More exactly, this variable takes account of the state of
plastic deformation during the last discharge; in other words, at the beginning of each discharge, this
variable should take the value of the current plastic deformation instantaneously. However, for
to preserve a continuous behavior, one regularizes the evolution of
p
N
in the following way:
In elastic mode:
(
)
!
!
!
!
!F
(





















N
p
N
p
p
N
p
p
N
p
p
=
-
=


=

if
if
tq
0
0
= 0
)
(
)
conventional elasticity
pseudo - discharge
éq 1.1-9
In plastic mode:
!



N
p
=
0
The behavior is thus completely given. Before passing to the introduction of viscosity,
the observation of two surfaces of load calls an important remark. One could think that
surface
G
=
0
is actually activated only in plastic mode. In practice, it of it is nothing. One can
for example to quote the case of a thermal loading: a cooling involves (generally) one
dilation of the surface of load
F
=
0
, so that the stress of peak is brought to evolve/move for
to preserve
G
0
, and this same in elastic mode.
background image
Code_Aster
®
Version
4.0
Titrate:
Viscoplastic relation of behavior of Taheri
Date:
08/09/97
Author (S):
E. LORENTZ
Key:
R5.03.05-A
Page:
5/16
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HI-74/97/019/A
1.2
Taking into account of viscosity
To modelize the behavior of the stainless steels under cyclic loading when
temperature is about 550°C, it is not more possible to neglect the terms of creep. To return
count these effects of viscosity while preserving the properties of the preceding model, a method
simple consists in making viscous the evolution of the plastic deformation. In other words, viscosity
intervenes only in plastic mode: no direct influence on the semi-discrete internal variable nor on
the surface of load G = 0. For that, while following Lemaitre and Chaboche [bib3], one replaces
condition of coherence [éq 1.1-5] by:
!
F
p
K p
M
NR
=


1
éq 1.2-1
F
positive part of F (hooks of Macauley)
K NR M
,
characteristics of viscosity of material
The characteristics of viscosity of material are indicated in the control
DEFI_MATERIAU
,
maybe by the key word
LEMAITRE
if they do not depend on the temperature, that is to say by the key word
LEMAITRE_FO
in the contrary case. In the absence of one of these key words, the behavior is
supposed plastic.
Unchanged all the other equations of the model are left. It will be seen that such an introduction of
viscosity involves only minor amendments of the implicit algorithm of integration of the law of
behavior.
1.3
Description of the internal variables calculated by Code_Aster
The internal variables calculated by Code_Aster are 9. They are arranged in
the following command:
1
p
cumulated plastic deformation
2
p
stress of peak
3 to 8
p
N
plastic tensor of deformation due to the last discharge
(arranged in order xx, yy, zz, xy, xz, yz)
9
loadmeter/discharge (cf [§2.3])
0 elastic discharge
1 conventional plastic load
2 plastic load on two surfaces
3 pseudo-discharge
As for the tensor of the viscoplastic deformations, it is not arranged among the internal variables but
can be calculated in postprocessing via the control
CALC_ELEM
, options
`
EPSP_ELGA
'or `
EPSP
_
ELNO
', (cf [U4.61.02]).
background image
Code_Aster
®
Version
4.0
Titrate:
Viscoplastic relation of behavior of Taheri
Date:
08/09/97
Author (S):
E. LORENTZ
Key:
R5.03.05-A
Page:
6/16
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HI-74/97/019/A
2
Numerical formulation of the relation of behavior
In order to be able to treat within the same framework plasticity and viscoplasticity, one chooses to proceed to one
implicit discretization of the relations of behavior, (cf [R5-03-02]). Let us note moreover that one
explicit procedure of integration is delicate for two reasons: on the one hand, processing of the variable
semi-discrete is necessarily implicit and can lead to numerical oscillations (one
pseudo-discharge, therefore one solves
F
=
0
, and of the blow,
F
can be (very weak but) higher than zero,
from where load with the pitch following instead of continuing the discharge), and in addition, the equation [éq 1.1-2] is not
not derivable when
p
p
N
=
.
2.1
Implicit discretization of the equations of behavior
Henceforth, one adopts the convention of following notation. If
U
indicate a quantity, then:
U
-
quantity
U
at the beginning of the pitch of time
U
increment of the quantity U during the pitch of time
U
quantity U at the end of the pitch of time (not of exponent +)
Let us start by introducing the elastic stress, i.e. the stress in the absence of increment
of plastic deformation. One can notice besides that only the term deviatoric cheek a role in
the nonlinear part of the behavior:
(
)
(
)


















E
HT
p
E
p
Id
E
=
-
+
-
=
-
-
K tr
2
2
µ
µ
~
~
~
~
“#
$
$
%
$$
and
éq 2.1-1
By taking account of the equations of state [éq 1.1-1] and [éq 1.1-3] and of the law of flow [éq 1.1-6], one a:
(
)
(
)
S
X
S
E
p
N
p
0
=
-
=
-
-
-
+
-
déf
p
S
S
p
~
~
C
C












µ
3
2 2
éq 2.1-2
By noting that
S
°
is not other than
S
normalized, one deduces some immediately:
(
)
(
)
S
C S
p
C S
eq
p
+
+




=
-
-
-
3
2 2
µ
S
0
E
p
N
p
S
E
~









#
$
%
$
éq 2.1-3
Consequently, S is entirely determined by:
(
)
S
S
S
S
0
0
E
=
=
=
-
+
S
S
S
S
C S
p
eq
eq
E
eq
eq
E
with
and
3
2 2
µ
éq 2.1-4
Finally, the functions of load are:
()
F
S
D R
With
p
eq
has
eq
has
=
-
+
-
+










-
0
2
3
3
2






p
N
p
0
S
éq 2.1-5
()
G C S
S p
D R
With
p
p
eq
has
eq
has
p
=
-
+




+
+
-
+










-
-
-












p
p
0
p
N
p
0
S
S
N
3
2
3
2
0
2
3
éq 2.1-6
background image
Code_Aster
®
Version
4.0
Titrate:
Viscoplastic relation of behavior of Taheri
Date:
08/09/97
Author (S):
E. LORENTZ
Key:
R5.03.05-A
Page:
7/16
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HI-74/97/019/A
2.2
Taking into account of the viscous terms
In the absence of viscous terms, the relation of discretized coherence is:
Elastic mode:
and
Plastic mode:
and
F
F
0
p = 0
= 0
p 0
éq 2.2-1
On the other hand, in the presence of viscosity, the condition of coherence is replaced by the equation
[éq 1.2-1] which, discretized, is written:
p
T
K p
K p
p
T
M
NR
M
NR
=


=




F
F
1
1
1
éq 2.2-2
In other words, while posing:
~
F
F
=
-




K p
p
T
M
NR
1
1
éq 2.2-3
the viscoplastic increment of cumulated deformation is determined by:
Elastic mode:
and
Viscoplastic mode:
and
~
F
~
F
0
p = 0
= 0
p 0
éq 2.2-4
Finally, by adopting an implicit discretization, the only difference between the laws of behavior
plastic and viscoplastic lies in the form of the function of load
F
: a term there is observed
complementary in the event of viscosity. In fact, incremental plasticity seems the borderline case
(without associated numerical difficulty) of incremental viscoplasticity when viscosity
K
tends towards
zero. Let us note that this remark was already mentioned by Chaboche and Al [bib1].
2.3
Discretization of the conditions of coherence
Before discretizing the conditions of coherence and describing the various modes of behavior
possible, a remark is essential as for the processing of the semi-discrete variable. Like
only intervenes “to control”
p
N
, one can always bring back oneself during a pitch of time to:
(
)









N
p
N
p
p
=
+ -
-
-
1
0
1
éq 2.3-1
The value of
is then fixed by the conditions of coherence, which translates the equation of evolution
[éq 1.1-9] on the continuous level. Such a parameter setting with each pitch of time makes it possible to be freed from
storage of
, in condition well-sure of preserving the values of
p
N
.
After this opening remark, one can be interested in the conditions of coherence. For the criterion
G
who controls the evolution of the stress of peak, the discretized form of the condition of coherence is:
(
)
(
)
G
,
,
G
,
,
p
p
p
p
p
p
=
0
0
0
éq 2.3-2
background image
Code_Aster
®
Version
4.0
Titrate:
Viscoplastic relation of behavior of Taheri
Date:
08/09/97
Author (S):
E. LORENTZ
Key:
R5.03.05-A
Page:
8/16
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HI-74/97/019/A
The condition of coherence relating to
F
is more delicate insofar as it controls the evolution
plastic deformation in plastic mode of load and evolution of
in mode of discharge.
Once discretized, she is written:
In plastic mode of load
(
)
=
1
:
(
)
(
)
F
,
,
F
,
,
p
p
p
p
p
p
= =
= =
1
0
0
1
0
éq 2.3-3
In mode of discharge
(
)
p
=
0
:
(
)
(
)
F
,
,
F
,
,
p
p
p
p
=
=
=
=
0
0
0
1
0
0
éq 2.3-4
To be able to select the mode of behavior of material, and thus equations to be solved,
first question is:
Are we in plastic situation or rubber band?
In fact, there is a solution in elastic mode (pseudo-discharge
>
0
or conventional elasticity
=
0
) if one can find an increment of stress of peak such as:
Incremental condition of discharge (equation scalar in
p
):
(
)
(
)
(
)
F
,
,
G
,
,
G
,
,
p
p
p
p
p
p
p
p
=
=
=
=
=
= =
0
1
0
0
1
0
0
0
1
0
éq 2.3-5
In the event of plastic load, i.e. when there does not exist
p
satisfying [éq 2.3-5], one has then with
to solve the nonlinear system in
p
and
p
according to:
Plastic mode (nonlinear system in
p
and
p
):
(
)
(
)
(
)
F
,
,
G
,
,
G
,
,
p
p
p
p
p
p
p
p
p
= =
=
= =
1
0
0
1
0
0
1
0
éq 2.3-6
On the other hand, in elastic situation, two choices are still possible: pseudo-discharge
(
)
>
0
or
conventional elasticity
(
)
=
0
. The second case being more favorable, one starts by examining whether it is not
not realizable, i.e. if there is an increment of stress of peak such as:
Incremental condition of conventional elastic mode (scalar equation in
p
):
(
)
(
)
(
)
F
,
,
G
,
,
G
,
,
p
p
p
p
p
p
p
p
=
=
=
=
=
= =
0
0
0
0
0
0
0
0
0
0
éq 2.3-7
Lastly, if it were to be a question of a discharge pseudo-rubber band, it remains to solve the nonlinear system in
and
p
according to:
background image
Code_Aster
®
Version
4.0
Titrate:
Viscoplastic relation of behavior of Taheri
Date:
08/09/97
Author (S):
E. LORENTZ
Key:
R5.03.05-A
Page:
9/16
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HI-74/97/019/A
Discharge pseudo-rubber band (system nonlinear in
and
p
):
(
)
(
)
(
)
F
,
,
G
,
,
G
,
,
p
p
p
p
p
p
p
p
=
=
=
=
=
0
0
0
1
0
0
0
0
0
éq 2.3-8
Let us note as of now that nonlinear systems [éq 2.3-6] and [éq 2.3-8] can be reduced to
solution of a simple scalar equation if
p
=
0
allows to obtain a solution.
One can summarize the algorithm of choice of the equations to be solved by the decision tree below.
Charge/discharge
(2.3-5)
Plastic resolution
(2.3-6)
Elasticity
conventional/pseudo
(2.3-7)
Resolution
conventional elasticity
Resolution
pseudo-discharge
(2.3-8)
Appear 2.3-Error! Argument of unknown switch. : Decision tree to choose the mode of
behavior
background image
Code_Aster
®
Version
4.0
Titrate:
Viscoplastic relation of behavior of Taheri
Date:
08/09/97
Author (S):
E. LORENTZ
Key:
R5.03.05-A
Page:
10/16
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HI-74/97/019/A
2.4
Framing of the solutions
With the reading of the preceding paragraph, one could note the need for solving (numerically) one
certain number of scalar equations or nonlinear systems. For that, it is always
interesting to have an interval on which to seek the solution. On the one hand, a framing of
solution shows its existence (what strongly reinforces the chances of success of an algorithm of
resolution!), and in addition, it allows a suitable digital processing, therefore surer.
Concerning
p
, one undervaluing is of course 0. In addition, the stress of ratchet S represents
a limit beyond which the model does not have any more a smell. In fact, with the examination of constant materials
obtained by identification, cf Taheri and Al [bib6],
G
becomes indeed negative when
p
S
=
if
the difference between the plastic deformation and the plastic deformation due to the last discharge too is not
important (a few %):
(
)
(
)
(
)
(
)
&
()
(
)
G
,
,
%
,
%
,
,
p
S
S
C
C
m
R
S
With
S
p
eq
has
eq
has
=
=
+
-
+ -
+
-




-
1
13
3
0 75
0
20
2
3
0 5
0 7
1
1
0
“#
$ %
$ “#
$$
%
$$
“# %
“# %” #
$
%
$












p
N
p
p
N
p
éq 2.4-1
One can also seek one raising for
p
. By examining the expression of
F
:
(
)
(
)
(
)
()
F
,
,
D
C
D
D
max
p
S
R
S
S
p
R
S
C S
p
p
R
p
eq
eq
E
E
µ
µ
-
-
+
-
-
+
-
-
0
0
0
3
2 2
3
2 2
éq 2.4-2
One deduces one from them raising for
p
, such as
(
)
F
,
,
p
p
0
:
() (
)
(
)
p S
p
R
C S
E
p
p
-
+
-
max
D
,
µ
0
3
2 2
éq 2.4-3
()
(
)
(
)
(
)
S
p
S
C
S
E
p
p
p
eq
p
eq
max
max
~
C
,
~
=
-
-




-
-








-
-
-


















E
p
N
p
E
p
N
p
éq 2.4-4
background image
Code_Aster
®
Version
4.0
Titrate:
Viscoplastic relation of behavior of Taheri
Date:
08/09/97
Author (S):
E. LORENTZ
Key:
R5.03.05-A
Page:
11/16
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HI-74/97/019/A
In particular, one can give one raising (coarse)
p
independent of
p
:
(
)
(
)
p
S
m R
C S
E
max
max
max
=
- -
+
1
3
2 2
0
µ
éq 2.4-5
(
)
[
]
[
]
S
C
C
S
S
S
E
eq
E
p
eq
eq
max
max
max
=
+
+
-
-



-
-
-
1












p
N
p
p
N
p
éq 2.4-6
One can then notice that the systems [éq 2.3-6] and [éq 2.3-8] always admit a solution. In
effect, if for each system, one writes respectively
()
()
p
p
p
and
solutions of
G
=
0
, then one a:
·
Nonlinear system of plastic load:
(
)
(
)
(
)
(
)
F
,
,
F
,
,
p
max
p
max
p
p
p
p
=
=
=
=
0
0
1
0
1
0
E
T
éq 2.4-7
·
Nonlinear system of pseudo-discharge:
(
)
(
)
(
)
(
)
F
,
,
F
,
,
p
p
p
p
=
=
=
=
=
=
0
1
1
0
0
0
0
0
and
éq 2.4-8
3
Methods of numerical resolution
The resolution of the incremental equations confronts us either with a nonlinear scalar equation, or
with a nonlinear system with two unknown factors. The numerical methods below are exposed
employees. One also examines the calculation of the tangent matrix, possibly used by
the total algorithm of
STAT_NON_LINE
, (cf [R5.03.01]).
3.1
Scalar equation: method of secants
It is a question of solving a nonlinear scalar equation by seeking the solution in an interval of
confidence. For that, one proposes to couple a method of secant with a control of
the interval of search. That is to say the following equation to solve:
()
[]
()
()
F
,
F
F
X
X
B has
has
B
=
<
>
0
0
0
éq 3.1-1
The method of the secant consists in building a succession of points
X
N
who converges towards the solution.
It is defined by recursion (linear approximation of the function by its cord):
() () ()
X
X
X
X
X
X
X
N
N
N
N
N
N
N
+
-
-
-
-
=
-
-
-
1
1
1
1
1
F
F
F
éq 3.1-2
background image
Code_Aster
®
Version
4.0
Titrate:
Viscoplastic relation of behavior of Taheri
Date:
08/09/97
Author (S):
E. LORENTZ
Key:
R5.03.05-A
Page:
12/16
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HI-74/97/019/A
In addition, if
X
N
+
1
was to leave the interval, then one replaces it by the terminal of the interval in
question:
if
then
if
then
X
has
X
has
X
B
X
B
N
N
N
N
+
+
+
+
<
=
>
=


1
1
1
1
:
:
éq 3.1-3
On the other hand, if
X
N
+
1
is in the interval running, then one reactualizes the interval:
[]
()
[]
()
if
and
then
if
and
then
X
B has
X
has
X
X
B has
X
B
X
N
N
N
N
N
N
+
+
+
+
+
+
<
=
>
=



1
1
1
1
1
1
0
0
,
F
:
,
F
:
éq 3.1-4
One considers to have converged when
F
is sufficiently close to 0 (tolerance to be informed). As
at the first two leader characters, one can choose the terminals of the interval, or, if one lays out
of an estimate of the solution, one can adopt this estimate and one of the terminals of the interval.
3.2
Nonlinear systems: method of Newton and linear search
One presents here a method of Newton which one associated a linear technique of search and
a control of the direction of descent not to leave the field of search (terminals on
unknown factors).
That is to say the following nonlinear system:
()
()
F,
G,
min
max
min
max
X y
X y
X
X X
y
y y
=
=





0
0
with
éq 3.2-1
If
()
X y
,
is a point of the field of search, then one builds a succession of points
(
)
X y
N
N
,
who
converge towards a solution (or at least, it is hoped for) by the following process.
·
Determination of the direction of descent
A direction of descent
(
)
X y
,
is given by the resolution of the linear system 2 X 2:
F
F
G
G
X
y
F
G
X
N
y
N
X
N
y
N
N
N
,
,
,
,






= -

éq 3.2-2
background image
Code_Aster
®
Version
4.0
Titrate:
Viscoplastic relation of behavior of Taheri
Date:
08/09/97
Author (S):
E. LORENTZ
Key:
R5.03.05-A
Page:
13/16
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HI-74/97/019/A
·
Correction of the direction of descent
The direction of descent is corrected
(
)
X y
,
so that the points considered are
in the field of search (with
max
the maximum length which one is authorized to describe it
length of the direction of descent):
if
if
if
if
X
X X
X
X
X
X
X X
X
X
X
y
y y
y
y
y
y
y y
y
y
y
+
<
=
-
+
>
=
-
+
<
=
-
+
>
=
-




max
min
min
max
max
max
max
max
max
min
min
max
max
max
max
max
:
:
:
:
éq 3.2-3
·
Seek
linear
It any more but does not remain to minimize the quantity
(
)
E
F
G
/
=
+
2
2
2
in the direction of descent.
Let us note that the standard
E
that one minimizes thus is a measurement of the error made in
resolution of the system: it is null when
()
X y
,
is solution of the system [éq 3.2-1]. For
to minimize
E
, one simply will seek to cancel his derivative, i.e. to solve
the scalar equation:
(
)
[
]
(
)
(
)
E
,
F F
G G
F F
G G
max
,
,
,
,
X
X y
y
X
X
y
y
X
y
+
+
=
+
+
+
#
$
%
$
0
0
and
éq 3.2-4
·
Criterion of convergence
One considers to have converged when the error
E
is lower than a prescribed size. By
elsewhere, if the standard of the direction of descent becomes too weak (another size with
to inform), one can think that the algorithm does not manage to converge.
3.3 Criteria
of stop
Until now, values of stop and iteration counts maximum of the methods of resolution
the preceding ones were not specified. Two cases should be distinguished.
·
When one seeks to check the conditions of coherence (scalar equation or system not
linear according to the situation), one awaits precise results, of which the relative tolerance
is
fixed by the user in the control
STAT_NON_LINE
under the key word
RESI_INTE_RELA
,
(cf [U4.32.01]). According to whether one seeks to solve
F
, G
=
=
0
0
or simultaneously
F G
= =
0
, the criterion of stop is expressed respectively:
F
G
F
G
R
R
R
0
0
0
2
2
1
2
+
or
or
R
0
initial elastic limit, provided by the user, cf [§ 1.1].
background image
Code_Aster
®
Version
4.0
Titrate:
Viscoplastic relation of behavior of Taheri
Date:
08/09/97
Author (S):
E. LORENTZ
Key:
R5.03.05-A
Page:
14/16
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HI-74/97/019/A
In addition, the user always specifies a maximum iteration count in the control
STAT_NON_LINE
under the key word
ITER_INTE_MAXI
, (cf [U4.32.01]).
·
When one carries out iterations of linear search, one seeks to obtain a convergence
faster (or at least sourer). One should not therefore devoting an excessive time to it.
This is why one fixed once for a a whole iteration count maximum equal to 3, one
limit maximum
max
equalize to 2 and one criterion of relative stop of 1%:
(
)
[
]
(
)
[
]
E
,
E
,
X
X y
y
X
X y
y
+
+
+
+
-
=
10
2
0
3.4 Stamp
tangent
In the optics of a resolution of the equilibrium equations (total) by a method of Newton, it is
essential to determine the consistent matrix of the tangent behavior, (cf Simo and Al [bib4]).
This matrix is composed classically of an elastic contribution and a plastic contribution:
µ















=
-
E
p
2
éq 3.4-1
One immediately deduces from it that in elastic mode (conventional or pseudo-discharge), the matrix
tangent is reduced to the elastic matrix:
Elastic mode:












=
E
éq 3.4-2
On the other hand, in plastic mode, the variation of the plastic deformation is not null any more. Rules
of made up derivation allow to obtain:












p
E
0
E
0
E
0
E
0
0
S
S
S
Id
S
S
~
~
~
~
=
+




=
+
-








3
2
3
2
3
2
p
p
p
p
S
eq
E
éq 3.4-3
tensorial product
One can note that one preferred to derive compared to
~



E
, knowing that one a:
µ





















p
p
E
E
p
E
P
P
=
=
~.
~
~.
:
~
2
with
S
S
'
éq 3.4-4
S
space symmetrical tensors
P
projector on the diverters
Finally, it any more but does not remain to calculate the variation of
p
. For that, it is necessary to distinguish if it is about one
conventional mode of plasticity
(
)
p
=
0
or of plasticity on two surfaces. As follows:
background image
Code_Aster
®
Version
4.0
Titrate:
Viscoplastic relation of behavior of Taheri
Date:
08/09/97
Author (S):
E. LORENTZ
Key:
R5.03.05-A
Page:
15/16
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HI-74/97/019/A
Conventional plasticity:
()
F, ~
p



E
=
0
()
()
()
()
F
, ~
F
, ~
~
~
F
, ~
F
, ~
,
, ~
, ~
,
p
p
p
p
p
p
p
p
























E
E
E
E
E
E
E
E
= -
= -
éq 3.4-5
Plasticity on two surfaces:
(
)
(
)
F,
, ~
G,
, ~
p
p
p
p






E
E
=
=
0
0
and
F
F
G
G
F
G
. ~
~
F
F
G
G
F
F
G
G
,
,
,
,
, ~
, ~
,
, ~
,
, ~
,
,
,
,
p
p
p
p
p
p
p
p
p
p
p
p
p










= -
=


















E
E
E
E
E
E
éq 3.4-6
An attentive examination of the expressions [éq 2.1-5] and [éq 2.1-6] makes it possible to note that the variations of
F
and
G
compared to
~



E
are not necessarily colinéaires with
S
°. By taking account of
[éq 3.4-3], one from of deduces whereas the tangent matrix is in general not symmetrical in mode
plastic. Rather than to impose the use of a nonsymmetrical solvor, much more expensive in time
calculation, one prefers to symmetrize this matrix.
3.5 Stresses
plane
The processing of the plane stresses adds a nonlinear equation to solve, coupled with
systems [éq 2.3-6] and [éq 2.3-8], (cf [R5-03-02]). In front of this considerable difficulty and the absence
of expressed need, one preferred not to make it possible to force a state of plane stresses with
level of the law of behavior. In other words, modeling
C_PLAN
is not available for
law of behavior
VISC_TAHERI
.
background image
Code_Aster
®
Version
4.0
Titrate:
Viscoplastic relation of behavior of Taheri
Date:
08/09/97
Author (S):
E. LORENTZ
Key:
R5.03.05-A
Page:
16/16
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HI-74/97/019/A
4 Bibliography
[1]
J.L. Chaboche, G. Cailletaud, 1996. Constitutive integration methods for complex plastic
equations. Comp. Meth. Appl. Mech. Eng., n° 133, pp. 125-155
[2]
PH. Geyer, 1992. Study of the elastoplastic model of behavior for the loading
cyclic develops with
EDF
/
DER
/
MN
. Note intern EDF/DER, HT-26/92/39/A.
[3]
J. Lemaitre, J.L. Chaboche, 1988. Mechanics of solid materials. Dunod.
[4]
J.C. Simo, R.L. Taylor, 1985. Consist tangent operators for misses-independent
elastoplasticity. Comp. Meth. Appl. Mech. Eng., n° 48, pp. 101-118.
[5]
S. Taheri, 1990. A uniaxial law of behavior in elastoplasticity for the loading
cyclic. Note intern EDF/DER, HI-71/6812.
[6]
S. Taheri, pH. Geyer, J.M. Proix, 1995. Constitutive Three dimensional elastic-plastic law for
the description off ratchetting off 316 stainless steel. Note intern EDF/DER, HI-74/95/012/0.