Code_Aster
®
Version
5.7
Titrate:
Linear and isotropic work hardening mixed kinematic nonlinear
Date:
30/12/02
Author (S):
J.M. PROIX, E. LORENTZ
Key
:
R5.03.16-C
Page
:
1/14
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HT-66/02/004/A
Organization (S):
EDF/AMA
Manual of Reference
R5.03 booklet: Nonlinear mechanics
Document: R5.03.16
Elastoplastic relation of behavior
with linear and isotropic kinematic work hardening
nonlinear. Plane modeling 3D and stresses
Summary:
This document describes an elastoplastic law of behavior to mixed, kinematic work hardening linear and
isotropic nonlinear. Equations to solve to integrate this relation of behavior numerically
are specified, as well as the coherent tangent matrix.
This behavior is usable for modelings of continuous mediums 3D, 2D (
AXIS
,
C_PLAN
,
D_PLAN)
, and
for modelings
DKT
,
COQUE_3D
and
PIPE
.
Code_Aster
®
Version
5.7
Titrate:
Linear and isotropic work hardening mixed kinematic nonlinear
Date:
30/12/02
Author (S):
J.M. PROIX, E. LORENTZ
Key
:
R5.03.16-C
Page
:
2/14
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HT-66/02/004/A
1 Introduction
When the way of loading is not monotonous any more, work hardenings isotropic and kinematic are not
more equivalent. In particular, one can expect to have simultaneously a kinematic share and one
isotropic share. If one seeks to precisely describe the effects of a cyclic loading, it is
desirable to adopt modelings sophisticated (but easy to use) such as the model of
Taheri, for example, cf [R5.03.05]. On the other hand, for less complex ways of loading,
one can wish to include only one linear kinematic work hardening, all nonthe linearities of
work hardening being carried by the isotropic term. That makes it possible to follow a curve precisely of
traction, while representing nevertheless phenomena such as the Bauschinger effect [bib1] (see
for example it [Figure 5-a]).
The characteristics of work hardening are then given by a traction diagram and a constant,
said of Prager, for the term of kinematic work hardening linear. They are introduced into
order
DEFI_MATERIAU
:
Linear isotropic work hardening
Nonlinear isotropic work hardening
DEFI_MATERIAU (
ECRO_LINE: (
SY:
elastic limit
D_SIGM_EPSI
:
slope of the curve of
traction
)
PRAGER: (C:
constant of Prager
)
);
DEFI_MATERIAU
(
TRACTION: (SIGM:
curve of
traction
)
PRAGER:
(C:
constant of Prager
)
)
;
These characteristics can also depend on the temperature, with the proviso of employing the words then
keys factors
ECMI_LINE_FO
and
ECMI_TRAC_FO
in the place of
ECRO_LINE
and
TRACTION
. The employment of
these laws of behavior is available in the controls
STAT_NON_LINE
or
DYNA_NON_LINE
:
Linear isotropic work hardening
Nonlinear isotropic work hardening
STAT_NON_LINE
(
COMP_INCR
:
(
RELATION
:“VMIS_ECMI_LINE”
)
)
;
STAT_NON_LINE
(
COMP_INCR
:
(
RELATION:“VMIS_ECMI_TRAC”
)
)
;
In the continuation of this document, one precisely describes the model of combined work hardening. One presents
then the detail of its numerical integration in link with the construction of the tangent matrix
coherent. Lastly, a tensile test uniaxial pressing illustrates the identification of the characteristics
material.
Code_Aster
®
Version
5.7
Titrate:
Linear and isotropic work hardening mixed kinematic nonlinear
Date:
30/12/02
Author (S):
J.M. PROIX, E. LORENTZ
Key
:
R5.03.16-C
Page
:
3/14
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HT-66/02/004/A
2
Description of the model
At any moment, the state of material is described by the deformation
, the temperature
T
, deformation
plastic
p
and cumulated plastic deformation
p
. The equations of state define then in function
of these variables of state the stress
=
H
Id
+
~
(broken up into parts hydrostatic and
deviatoric), the isotropic share of work hardening
R
and the kinematic share
X
, also called forced
of recall:
()
(
)
(
)
H
ref.
tr
K tr
T-T
=
=
-
1
3
HT
HT
Id
with
=
éq 2-1
(
)
()
~
~
~
= -
=
-
= -
µ
H
tr
Id
Id
p
2
1
3
where
éq
2-2
()
R
p
=
R
éq 2-3
X
p
=
C
éq 2-4
where
K
C
, R
µ
and
are characteristics of material which can depend on the temperature.
More precisely, they are respectively the modules of compressibility and shearing, it
thermal expansion factor average (see [R4.08.01]), the isotropic function of work hardening and
constant of Prager. As for
T
ref.
, it is about the temperature of reference, for which
thermal deformation is null.
K,
µ
are connected to the Young modulus E and the Poisson's ratio by:
K
E
E
=
+
= -
= +
3
2
1 2
2
1
µ
µ
Note:
Concerning the kinematic share of work hardening [éq 2-4], one notes that it is linear in it
model. In addition, it is necessary to take guard with the fact that in certain references, one calls
constant of Prager
2
3
C
/
and not
C
. In the same way, for the isotropic function of work hardening,
elastic limit is included there by
()
R 0
=
y
, certain references treating it separately.
Evolution of the internal variables
p
and p
is controlled by a normal law of flow associated
a criterion of plasticity
F
:
(
)
(
)
F,
~
~ ~
R
R
eq
eq
X
X
With
WITH A
=
-
-
=
with
3
2
éq
2-5
(
)
&
& F
& ~
~
p
X
X
=
=
-
-
3
2
eq
éq 2-6
Code_Aster
®
Version
5.7
Titrate:
Linear and isotropic work hardening mixed kinematic nonlinear
Date:
30/12/02
Author (S):
J.M. PROIX, E. LORENTZ
Key
:
R5.03.16-C
Page
:
4/14
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HT-66/02/004/A
&
&
& &
p
= =
2
3
p
p
éq 2-7
As for the plastic multiplier
&
, it is obtained by the condition of following coherence:
if
or
if
and
F
&F
&
F
&F
&
<0
0
0
0
0
0
<
=
=
=
éq 2-8
3
Integration of the relation of behavior
To numerically carry out the integration of the law of behavior, one carries out a discretization in
time and one adopts a diagram of implicit, famous Euler adapted for relations of behavior
elastoplastic. Henceforth, the following notations will be employed:
With
With
With
-
, and
represent
respectively values of a quantity
With
at the beginning and the end of the pitch of time considered thus that
its increment during the pitch. The problem is then the following: knowing the state at time
T
-
thus
that increments of deformation
and of temperature
T
, to determine the state at time
T
like
stresses
.
Initially, one takes into account the variations of the characteristics compared to
temperature by noticing that:
(
)
H
H
K
K
K
=
+
-
-
-
tr
HT
éq
3-1
(
)
~
~
~
-
=
+
-
-
µ
µ
µ
2
p
éq 3-2
X
X
p
=
+
-
-
C
C
C
éq 3-3
Within sight of the equation [éq 3-1], one notes that the hydrostatic behavior is purely elastic.
Only the processing of the deviatoric component is delicate. To reduce the writings to come, one
introduced
~s
E
the difference
~
-
X
in the absence of increment of plastic deformations, so that:
(
)
~
~
~
~
- =
-
+
-
+
-
-
-
-
X
X
S
p
E
µ
µ
µ
µ
C
C
C
2
2
1
2
4444
3
4444
éq 3-4
Code_Aster
®
Version
5.7
Titrate:
Linear and isotropic work hardening mixed kinematic nonlinear
Date:
30/12/02
Author (S):
J.M. PROIX, E. LORENTZ
Key
:
R5.03.16-C
Page
:
5/14
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HT-66/02/004/A
The equations of flow [éq 2-6] and [éq 2-7] and the condition of coherence [éq 2-8] are written one
times discretized and by noticing that
p
=
:
(
)
p
X
X
=
-
-
3
2 p
eq
~
~
éq 3-5
F
F
=
0
0
0
p
p
éq 3-6
The processing of the condition of coherence [éq 3-6] is conventional. One starts with a test
rubber band (
p
=
0
) which is well the solution if the criterion of plasticity is not exceeded, i.e. if:
()
F
R
=
-
-
S
p
eq
E
0
éq 3-7
In the contrary case, the solution is plastic (
p
>
0
) and the condition of coherence [éq 3-6] is reduced
with
F
=
0
. To solve it, one starts by showing that one can bring back oneself to a scalar problem
while eliminating
p
. Indeed, by taking account of [éq 3-4] and [éq 3-5], one notes that
p
is
colinéaire with
~s
E
because:
(
)
(
)
[
]
p
E
p
X
S
=
-
-
+
3
2
2
p
C
eq
~
~
µ
éq
3-8
In addition, according to [éq 3-5], the standard of
p
is worth:
()
p
eq
p
=
3
2
éq 3-9
One thus deduces immediately the expression from it from
p
according to
p
:
p
E
S
=
3
2 p S
eq
E
~
éq 3-10
It now only remains to replace
p
by its expression [éq 3-10] in the equation [éq 3-4]
one obtains:
(
)
~
~
- =
-
+
X
S
C
p
S
E
eq
E
1
3
2 2
µ
while deferring
~
-
X
in the equation
F
=
0
, one brings back oneself to a scalar equation in
p
to solve,
with knowknowing:
(
)
(
)
S
C
p
p
p
eq
E
-
+
-
+
=
-
3
2 2
0
µ
R
éq
3-11
Code_Aster
®
Version
5.7
Titrate:
Linear and isotropic work hardening mixed kinematic nonlinear
Date:
30/12/02
Author (S):
J.M. PROIX, E. LORENTZ
Key
:
R5.03.16-C
Page
:
6/14
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HT-66/02/004/A
Insofar as the function
R
is positive, which one will admit henceforth, there is a solution
p
with this equation, characterized by:
(
)
(
)
3
2 2
0
2
3 2
µ
µ
+
+
+
=
<
<
+
-
C
p
p
p
S
p
S
C
eq
E
eq
E
R
where
éq
3-12
Let us note that in the interval specified in [éq 3-12], the solution is single. For details as for
solution of this equation, one will refer to [R5.03.02].
The particular case of the plane stresses is studied with [§6].
4
Calculation of tangent rigidity
In order to allow a resolution of the total problem (equilibrium equations) by a method of
Newton, it is necessary to determine the coherent tangent matrix of the incremental problem. For
that, one once more adopts the convention of writing of the symmetrical tensors of command 2 pennies
form vectors with 6 components. Thus, for a tensor
has
:
[
]
has
=
T
xx
yy
zz
xy
xz
yz
has
has
has
has
has
has
2
2
2
éq
4-1
If moreover the hydrostatic vector is introduced
1
and stamps it deviatoric projection
P
:
[
]
1
=
T
1 1 1 0 0 0
éq 4-2
P
Id
1 1
=
-
1
3
éq 4-3
Then the matrix of coherent tangent rigidity is written for an elastic behavior:
µ
=
+
K
1 1
P
2
éq 4-4
and for a plastic behavior:
()
(
)
µ
µ
µ
µ
=
+
-
+
- +
+
K
p
S
p
S
p
C
S
S
eq
E
eq
E
eq
E
eq
E
1 1
P
S
S
E
E
2 1 3
9
1
3
2 2
2
R
~
~
éq 4-5
The initial tangent matrix, used by the option
RIGI_MECA_TANG
is obtained by adopting it
behavior of the preceding pitch (elastic or plastic, meant by internal variable
being worth 0
or 1) and while taking
p
=
0
in the equation [éq 4-5].
Code_Aster
®
Version
5.7
Titrate:
Linear and isotropic work hardening mixed kinematic nonlinear
Date:
30/12/02
Author (S):
J.M. PROIX, E. LORENTZ
Key
:
R5.03.16-C
Page
:
7/14
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HT-66/02/004/A
Note:
RIGI_MECA_TANG
is the operator linearized compared to time (cf [R5.03.01], [R5.03.05]) and
to the problem of speed corresponds what is called; in this case, the linearization by
report/ratio with
U
, in
U
=
0
, provides the same expression.
One now proposes to show the expression [éq 4-5]. By differentiating them [éq 2-1] and [éq 2-2] with
fixed temperature, one obtains immediately:
[
]
µ
µ
=
+
-
K
1 1
P
p
2
2
éq 4-6
If the mode of behavior is plastic, the incremental law of flow [éq 3-10] provides then:
p
E
E
S
S
=
+
3
2
3
2
p S
p
S
eq
E
eq
E
~
~
éq 4-7
As for
p
, it is obtained by differentiating the implicit equation [éq 3-12]:
(
)
()
3
2 2
µ
+
+
=
C
p
p
S
eq
E
R
éq 4-8
Lastly, it any more but does not remain to provide the variations of
~s
E
:
µ
µ
µ
µ
~
~
~
~
~
~
~
~
S
S
S
S
S
E
E
E
E
E
=
=
=
-
2
3
1 2 3
S
S
S
S
S
S
eq
E
eq
E
eq
E
eq
E
eq
E
eq
E
éq
4-9
While replacing then [éq 4-7], [éq 4-8] and [éq 4-9] in [éq 4-6], one obtains well the expression [éq 4-5].
This expression is formally identical to that defined in R5.03.02: [éq 4-3] and is written:
(
)
µ
µ
µ
µ
=
+
-
-
+
- +
+
K
p
S
p
S
C
S
S
eq
E
eq
E
eq
E
eq
E
1 1
Id
1 1
S
S
E
E
2 1 3
1
3
9
1
3
2 2
2
R
~
~
with
=
1
if
conduit with a plasticization, and
=
0
if not.
Code_Aster
®
Version
5.7
Titrate:
Linear and isotropic work hardening mixed kinematic nonlinear
Date:
30/12/02
Author (S):
J.M. PROIX, E. LORENTZ
Key
:
R5.03.16-C
Page
:
8/14
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HT-66/02/004/A
While using [éq 3-12], one finds:
()
()
(
) ()
()
()
()
()
()
()
(
)
()
()
µ
µ
µ
µ
µ
µ
µ
=
=
-
=
+
-
-
+
+
= -
=
-
-
= +
+
= +
-
*
*
R.
R
R
R
R
*
*
~
~
R R
1 1
Id
X
X
Dev.
Dev.
Dev.
Dev.
2
9
1
1
3
2 2
2
3
2
2
1
3
2 2
1
2
H p
p
p
C
p
p
K
G p
H p
G p
H p
H p
C
p
R p
G p
with
for the option
for the option
with
and
:
:
FULL_ MECA
RIGI_ MECA_ TANG
()
3
2 C
p
R p
5
Identification of the characteristics of material
Let us consider a tensile test uniaxial pressing, [Figure 5-a]. One proposes to show
how it makes it possible to identify the constant of Prager and the isotropic function of work hardening. In such
test, the various tensors are with fixed directions, i.e.:
~
=
=
=
=
-
-
D
X
D
D
D
p
X
p
3
2
2 3
1 3
1 3
with
5-1
As long as the loading is monotonous, therefore in phase of traction, one obtains them immediately
following relations:
()
p
X
C
C
p
p
T
p
p
=
=
=
+
3
2
3
2
R
5-2
The constant of Prager is determined by the first plasticization in compression, since one a:
()
()
With
T
Ap
Ap
With
C
Ap
Ap
With
T
With
C
Ap
C
C
C
=
+
=
-
=
+
3
23
2
3
R
R
éq
5-3
Code_Aster
®
Version
5.7
Titrate:
Linear and isotropic work hardening mixed kinematic nonlinear
Date:
30/12/02
Author (S):
J.M. PROIX, E. LORENTZ
Key
:
R5.03.16-C
Page
:
9/14
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HT-66/02/004/A
3 2
C
Ap
()
T
p
p
E
= -
()
C
p
With
T
With
C
Appear 5-a: Test tensile uniaxial pressing
The curve of work hardening
()
T
p
F
=
is deduced from the traction diagram
()
F
T
=
provided by
the user under the key words
ECRO_LINE
((
SY
and
D_SIGM_EPSI
(linear work hardening)) or
TRACTION
(unspecified work hardening). It finally makes it possible to obtain the isotropic function of work hardening
by [éq 5-2]:
() ()
R
p
T
p
p
C
=
-
3
2
.
For the effective calculation of R (p), according to the R5.03.02 document, one titrates party of the linearity (
ECMI_LINE
)
or of the linearity per pieces of the traction diagram (
ECMI_TRAC
):
ECMI_LINE
:
()
()
T
p
y
T
T
y
T
T
y
F
E E
E
E p
R p
E E
E
E
C p
R p
=
=
+ -
=
+
-
-
=
+
.
.
.
3
2
éq
5-4
The equation [éq 3-12] becomes then:
(
)
(
)
3
2 2
µ
+
+
+
+
=
C
p
R p
p
S
y
eq
E
.
éq
5-5
ECMI_TRAC
:
()
(
)
()
(
)
T
p
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
F
p
p p p
p
p
p
R p
p
p p p
CP
p
p p
R p
=
=
+
-
-
-
=
+
-
-
-
-
=
-
-
-
+
+
-
+
+
-
+
-
1
1
1
1
1
1
1
3
2
,
.
for
éq 5-6
Code_Aster
®
Version
5.7
Titrate:
Linear and isotropic work hardening mixed kinematic nonlinear
Date:
30/12/02
Author (S):
J.M. PROIX, E. LORENTZ
Key
:
R5.03.16-C
Page
:
10/14
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HT-66/02/004/A
Note:
For the use: the correspondence enters the model of behavior
VMIS_CINE_LINE
and it
behavior
VMIS_ECMI_LINE
is as follows:
With
VMIS_CINE_LINE
, it is necessary to introduce into
DEFI_MATERIAU
a linear work hardening of slope
And by:
D_SIGM_EPSI
: And
With
VMIS_ECMI_LINE
, to reproduce same behavior with kinematic work hardening
linear, it is necessary to give in
DEFI_MATERIAU
.
·
a linear work hardening of slope And:
D_SIGM_EPSI
: And
·
The constant of Prager C:
PRAGER
: C
C
is determined by:
C
EE
E E
T
T
=
-
2
3
It should well be noticed that the identification of
C
and of
()
R
p
have directions only in one field
deformations limited (small deformations). In particular, if
()
T
p
present an asymptote
max
T
for
p
sufficient large, then the kinematic contribution of work hardening does not have any more
significance. It is thus advised to restrict itself with the field where work hardening is strictly
positive.
6
Particular case of the plane stresses: calculation of
p
It is necessary to add to the equations [éq 3-1] with [éq 3-4] the condition of plane stresses
33
0
=
, which
add an unknown factor (corresponding deformation):
(
)
H
H
K
K
K
=
+
-
-
-
tr
HT
éq 6-1
(
)
~
~
~
-
=
+
-
-
µ
µ
µ
2
p
éq 6-2
X
X
p
=
+
-
-
C
C
C
éq 6-3
33
0
=
éq 6-4
Then, the equation [éq 3-4] becomes:
~
~
~
(
)
~
~
(
)
~
µ
µ
µ
µ
µ
µ
µ
-
=
-
+
-
+
+
=
-
+
+
-
-
-
-
X
C
C X
C
S
C
C
p
y
E
p
y
2
2
2
2
2
éq 6-5
where
~
C
is entirely determined by the elastic behavior:
(
)
~
~
~
, ~
~, ~
~
C
C
C
C
C
33
11
22
11
11
22
22
1
= --
+
=
=
Code_Aster
®
Version
5.7
Titrate:
Linear and isotropic work hardening mixed kinematic nonlinear
Date:
30/12/02
Author (S):
J.M. PROIX, E. LORENTZ
Key
:
R5.03.16-C
Page
:
11/14
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HT-66/02/004/A
and
~
y
Y
=
0 0
0
0 0
0
0 0
is unknown. It is also supposed that
13
23
13
23
0
=
=
=
=
.
One always has:
(
)
p
X
X
=
-
-
3
2 p
eq
~
~
éq 6-6
()
F
F
=
-
=
eq
R p
p
p
0
0
0
éq
6-7
Elastic test:
·
If
()
F
R
=
-
-
S
p
eq
E
0
éq 6-8
then
~
~
=
S
E
,
p
=
0
,
Y
=
0
éq 6-9
(
)
H
H
C
K
K
K
=
+
-
-
-
tr
HT
éq
6-10
·
If not, the solution is plastic:
p
>
0
,
Y
0
. One can still bring back oneself to a problem
scalar in
p
.
By taking account of [éq 6-5] and [éq 6-6], one notes that
~
-
X
is colinéaire with
~
~
S
E
+
2
µ
y
because:
(
)
(
)
(
)
()
[
]
~
()
~
~
~
-
+
+
=
-
=
+
X
X
S
E
1
3
2 2
2
µ
µ
C p
R p
H p
y
éq
6-11
Thus:
(
)
()
~
~s
µ
33
33
X
H p
Y
-
=
+
33
4
3
E
éq
6-12
We will express the equation [éq 6-12] according to
p
only. According to [éq 6-4]:
33
33
33
0
~
~
.
= =
+
=
+
+
E
K Y
, with
(
)
E
H
H
C
K
K
K
=
+
-
-
-
tr
HT
éq
6-13
Code_Aster
®
Version
5.7
Titrate:
Linear and isotropic work hardening mixed kinematic nonlinear
Date:
30/12/02
Author (S):
J.M. PROIX, E. LORENTZ
Key
:
R5.03.16-C
Page
:
12/14
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HT-66/02/004/A
While using [éq 6-5], [éq 6-6] and the incompressibility of the plastic deformations, one can show that:
~s
C
C X
E
E
33
33
+
= -
-
-
éq 6-14
Then:
~
~
.
33
=
-
+
-
-
S
K Y
C
C X
E
33
33
éq 6-15
As according to [éq 6-3]:
()
X
C
C X
C
C
C X
C
p
X
R p
p
33
33
33
33
33
33
3
2
=
+
=
+
-
-
-
-
-
.
.
~
éq
6-16
()
()
X G p
C
C X
C p R p
33
33
33
3
2
.
~
=
+
-
-
, with
()
()
G p
C
p
R p
= +
1 32
éq 6-17
From [éq 6-12], [éq 6-15], [éq 6-17], one obtains an equation flexible
p
and
Y
:
()
()
()
()
Y
K H p
G p
H p
G p
.
~
4
3
1
33
µ
+
=
-
S
E
éq
6-18
The equation [éq 6-11] makes it possible to obtain the scalar equation in
p
to solve, namely:
(
)
()
()
[
]
~
(
)
~
~
-
=
+
=
+
-
X
S
E
eq
y
eq
H p
R p
p H p
2
µ
éq
6-19
Equation in which
Y
is related to
p
by the equation [éq 6-18].
As in the case of isotropic work hardening, one obtains a scalar equation in
p
, always not
linear, which is solved by a method of secant.
Once
p
known, the calculation of
~
and
X
be carried out starting from the expression of
Y
, therefore of
entirely known, by a step similar to that of the equation [éq 3-10].
(
)
()
(
)
p
E
E
E
S
S
X
S
=
+
+
=
-
+
3
2
2
2
3
2
2
p
p
H p
y
y
eq
y
eq
~
~
~
~
~
~
~
µ
µ
µ
éq
6-20
(
)
~
~
~
-
=
+
-
-
µ
µ
µ
2
p
éq 6-21
Code_Aster
®
Version
5.7
Titrate:
Linear and isotropic work hardening mixed kinematic nonlinear
Date:
30/12/02
Author (S):
J.M. PROIX, E. LORENTZ
Key
:
R5.03.16-C
Page
:
13/14
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HT-66/02/004/A
One obtains while eliminating
p
from [éq 6-6], [éq 6-3] and [éq 6-2]:
()
()
()
~
~
~
()
=
+
+
-
-
-
-
µ
µ
µ
µ
2
3
2 2
G p
H p
p
R p H p
C
C X
éq
6-22
()
()
X
X
=
+
+ -
-
-
-
-
3
2
2
1 32
C
p
R p H p
C
p
R p H p
C
C
()
~
~
()
µ
µ
µ
éq 6-23
7
Significance of the internal variables
The variables intern model at the points of Gauss (
VARI_ELGA
) are for all them
modelings:
·
VARI_1
= p: cumulated plastic deformation (positive or null)
·
VARI_2
=
: being worth 1 if the point of Gauss plasticized during the increment or 0 if not.
X: tensor of recall:
For modeling
3D
:
·
VARI_3 =
X
11
·
VARI_4 =
X
22
·
VARI_5 =
X
33
·
VARI_6 =
X
12
·
VARI_7 =
X
13
·
VARI_8 =
X
23
For modelings
D_PLAN, C_PLAN, AXIS
·
VARI_3 =
X
11
·
VARI_4 =
X
22
·
VARI_5 =
X
33
·
VARI_6 =
X
12
For modelings of hulls (DKT, COQUE_3D), in local reference mark and each point of integration
of each layer:
·
VARI_3 =
X
11
·
VARI_4 =
X
22
·
VARI_5 =
X
33
·
VARI_6 =
X
12
Code_Aster
®
Version
5.7
Titrate:
Linear and isotropic work hardening mixed kinematic nonlinear
Date:
30/12/02
Author (S):
J.M. PROIX, E. LORENTZ
Key
:
R5.03.16-C
Page
:
14/14
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HT-66/02/004/A
8 Bibliography
[1]
J. LEMAITRE, J.L. CHABOCHE: “Mechanical of solid materials”. Dunod 1992