Code_Aster
®
Version
5.4
Titrate:
Nonlinear behaviors in plane stresses
Date:
06/03/01
Author (S):
J. Mr. PROIX, E. LORENTZ
Key:
R5.03.03-A
Page:
1/6
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HI-75/01/001/A
Organization (S):
EDF/MTI/MN
Manual of Reference
R5.03 booklet: Nonlinear mechanics
Document: R5.03.03
Taking into account of the assumption of the stresses
plane in the nonlinear behaviors
Summary:
This document describes a general method of integration of the nonlinear models of behaviors
(elastoplastic, viscoplastic, damaging,…) in plane stresses.
This is carried out by a method of static condensation due to R. of Borst.
This method makes it possible to use modeling
C_PLAN
, or modelings
COQUE_3D
,
DKT
and
PIPE
for all the models of incrémentaux behaviors of
STAT_NON_LINE
available into axisymmetric or
in plane deformations. It is not operational for the moment in
DYNA_NON_LINE
.
Code_Aster
®
Version
5.4
Titrate:
Nonlinear behaviors in plane stresses
Date:
06/03/01
Author (S):
J. Mr. PROIX, E. LORENTZ
Key:
R5.03.03-A
Page:
2/6
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HI-75/01/001/A
1 Introduction
One presents here a general method of integration of the nonlinear models of behaviors
(plasticity, viscoplasticity, damage) in plane stresses. It is activated by the key word
ALGO_C_PLAN: “DEBORST”
operand of the incrémentaux nonlinear behaviors
COMP_INCR
of
STAT_NON_LINE
, for modeling
C_PLAN
,
DKT
,
HULL 3D
and
PIPE
.
2
Difficulty of integration of the nonlinear behaviors in
plane stresses
Modeling
C_PLAN
, (as well as modelings
COQUE_3D
,
DKT
,
PIPE
) supposes that the state of
stresses room is plane, i.e. that
zz
=
0
, Z representing the direction of the normal with
surface. The tensors of stresses and deformations thus take the following form (in
C_PLAN
):
K
B
D
D
D
D
B
N
T
N
N
N
N
xx
xy
xy
yy
zz
=
-
=
!
.
!
11
12
21
22
0
0
0
0
Note:
For the hulls, it is necessary to add terms due to transverse shearing (
xz
yz
,
), but
those are treated elastically and do not intervene in the resolution of the behavior
room.
This assumption implies that the corresponding deformation is a priori unspecified (contrary
with other two-dimensional modelings where one makes an assumption directly on
zz
). It
can be given that using the relation of behavior. However the condition
zz
=
0
is not
alleviating for the integration of the behavior where one calculates an increase in stress
in
function of the increase in deformation
provided by the algorithm of Newton. In the case of
linear elasticity, the taking into account of this condition is simple and makes it possible to find:
(
)
zz
xx
yy
= - -
+
1
But if the behavior is nonlinear,
zz
cannot be calculated only from
U
and
does not result simply from the other components of the tensor of the deformations. The catch in
count EC assumption must then be made (when it is realizable) in a way specific to each
behavior, and very often brings to additional difficulties of resolution: it is the case in
private individual for the behavior of Von Mises to isotropic work hardening [R5.03.02]. So
many models of behavior are not available in plane stresses.
The method presented here has the large advantage of not requiring any particular development in
the integration of the behavior to satisfy the assumption of the plane stresses. It is usable
as soon as the model of behavior is available into axisymmetric or in plane deformations.
Code_Aster
®
Version
5.4
Titrate:
Nonlinear behaviors in plane stresses
Date:
06/03/01
Author (S):
J. Mr. PROIX, E. LORENTZ
Key:
R5.03.03-A
Page:
3/6
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HI-75/01/001/A
3
Principle of the processing of the plane stresses by
method De Borst
The idea of the method due to R. of Borst [bib1] consists in treating the condition of plane stresses not
not on the level of the law of behavior but on the level of balance. One obtains thus during
iterations of the algorithm of total resolution of
STAT_NON_LINE
stress fields which
tend towards a plane stress field progressively with the iterations:
zz
N
0
where N indicates the number of iteration of Newton.
One thus obtains the condition of stress planes not exactly, but in an approached way, with
convergence of the iterations of Newton, for each calculated increment. One checks, as specified by
the continuation, that the component above is lower than a given tolerance.
The method consists in breaking up the fields (strains or stresses) into a part
purely planes (specified by a “cap”) and a component according to Z. One then reveals
explicitly the component
zz
in the expression of the tangent operator in plasticity:
= =
! ,
!
zz
zz
The tangent operator
D
D
D D
= =
.
becoming
D
D
D
zz
11
11
11
11
= =
!
.
D
D
D
12
21
22
where
D
and
D
indicate infinitesimal increases, and where by definition
D
D
D
11
11
11
11
=
D
D
D
12
21
22
is
the coherent tangent matrix with the behavior without the assumption of plane stresses, is in
axisymetry, is in plane deformation to see for example [R5.03.02] for the models of Von Mises).
4
Implementation of the method
The method consists of each point of integration of each element with:
1) to use the axysimetric relation of plane behavior or deformation (they are
identical) to calculate the stresses starting from the deformations,
2) to carry out a static condensation on the relation stress-strain
3) to write the infinitesimal increases
D
and
D
who are connected above by the operator
tangent in the form of increase between two iterations in Newton N and n+1:
(
)
D
====
+
+
+
+
+
+
+
+
+
+
+
+
N 1
N
N 1
N
N 1
N
-
=
+
-
+
=
-
-
-
and the same for
D
. With convergence, this variation must tend towards zero.
Code_Aster
®
Version
5.4
Titrate:
Nonlinear behaviors in plane stresses
Date:
06/03/01
Author (S):
J. Mr. PROIX, E. LORENTZ
Key:
R5.03.03-A
Page:
4/6
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HI-75/01/001/A
While writing
zz
N
zz
N
zz
D
+
=
+
=
1
0
one obtains, for the iteration n+1:
D
D
D
D
D
N 1
N
N 1
N
12
21
N 1
N 1
!
!
!
!
!
.
!
+
+
+
+
+
+
+
+
11
11
11
11
+
+
+
+
+
+
+
+
D
D
D
zz
zz
N
zz
N
zz
N
N
N
N
N
zz
=
-
-
=
-
-
=
+
1
22
what, by using the last equation of this system, enables us to be reduced to:
() (
)
()
(
)
!
!
!
.
!
!
+
+
+
+
+
+
+
+
+
+
+
+
N
N
N
N
N
zz
N
N
zz
N
N
zz
N
N
D
D
D
D
D
D
+
-
-
=
+
-
+
=
= -
+
1
11
12
22
1
21
22
1
21
0
D
D
D
D
N 1
N 1
N 1
with the stress field which is written:
!
.
!
!
!
!
+
+
+
+
+
+
+
+
+
+
+
+
N
N
N
N
N
N
N
N
zz
N
zz
N
zz
N
zz
N
N
N
N
N
N
D
D
D
D
D
D
+
+
=
-
-
=
-
+
-
1
11
12
21
22
12
22
1
22
21
22
21
22
D
D D
D
D
D
N 1
N
N 1
By using the preceding expression of the stress field, one finds then:
B
B
L
T
T
+
+
=
= =
.
! . !
N
N
FD
FD
1
1
!
.
!
!
!
.
! .
!
!
!
!
B
D
D
D
D
B
D
D
D
B U
B
D
K
U
B
D
T
N 1
T
N 1
T
N
N 1
T
11
12
21
22
12
22
11
12
21
22
12
22
12
22
N
N
N
N
N
zz
N
N
N
N
N
N
zz
N
N
N
zz
N
D
D
D
FD
D
D
FD
D
FD
D
D
FD
-
-
=
-
-
=
+
-
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
It is thus noted that the taking into account of the plane stresses intervenes on two levels:
·
in the matrix of tangent rigidity, by a corrective term (second term of the expression
below) compared to the expression 2D of the tangent matrix:
K
B D
D D
B
N
T
=
-
!
.
!
11
12
21
22
N
N
N
N
D
FD
·
in the writing of the second member by a corrective term (second term of expression Ci
below) compared to the expression 2D of the tensor of the stresses:
(
)
R U
B
D
N 1
T
+
+
+
+
=
-
!
!
N
N
N
zz
N
D
12
22
FD
Code_Aster
®
Version
5.4
Titrate:
Nonlinear behaviors in plane stresses
Date:
06/03/01
Author (S):
J. Mr. PROIX, E. LORENTZ
Key:
R5.03.03-A
Page:
5/6
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HI-75/01/001/A
To implement the method De Borst for the unit of the incrémentaux behaviors, it is enough
thus to calculate these corrective terms and to add them to the stresses and tangent matrices obtained
by integration 2D (in fact axisymmetric or deformation planes) these behaviors. With this intention, it
is necessary to store some additional information during iterations of Newton. One thus adds
(in a transparent way for the user) 4 variables intern with the behavior used.
The data-processing realization is as follows:
1) during the iteration n+1 of the algorithm of Newton, one has in input of the routine calculating it
behavior:
U
N
+
1
,
,
,
,
-
-
-
-
-
-
-
-
,
and 4 following additional variables internal resulting
preceding iteration: 1 scalar variable
-
+
zz
N
N
N
N
D
D
22
21
22
D
N
!
and 3 variables
-
D
21
22
N
N
D
,
2) before carrying out the integration of the nonlinear behaviors (which will be made in
axisymmetric), one calculates
zz
N
zz
N
zz
N
N
N
N
N
N
D
D
D
+
=
-
+
-
1
22
21
22
21
22
D
D
N
N 1
!
!
+
+
+
+
,
3) One lets the routines of integration of the behavior calculate the stresses as well as
tangent behavior D from
!
+
+
+
+
N 1
zz
N
+
1
as if modeling were axisymmetric or
of plane deformation,
4) one modifies at exit the second member and the tangent matrix (if the reactualization of
stamp tangent was asked) so that:
K
B
D
D
D
B
N
T
=
-
!
.
!
11
12
21
22
N
N
N
N
D
FD
and
(
)
R U
B
D
N 1
T
+
+
+
+
=
-
!
!
N
N
N
zz
N
D
FD
12
22
,
5) the new internal variables are stored
-
+
zz
N
N
N
N
D
D
22
21
22
D
N
!
and
-
D
21
22
N
N
D
.
To check convergence, one checks, always on the level of each point of integration of each
finite element if
zz
N 1
+
+
+
+
<
, where
1
N
+
+
+
+
=
with
provided by the user under the key word
RESI_INTE_RELA.
The default value is 10
- 6
.
At the time of testing the convergence of the total iterations of Newton (defined by
RESI_GLOB_RELA
and
RESI_GLOB_MAXI)
one examines whether all the points of integration check the condition
zz
N 1
+
+
+
+
<
. If
it is not the case, one carries out additional iterations of Newton until complete checking
of this condition.
Code_Aster
®
Version
5.4
Titrate:
Nonlinear behaviors in plane stresses
Date:
06/03/01
Author (S):
J. Mr. PROIX, E. LORENTZ
Key:
R5.03.03-A
Page:
6/6
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HI-75/01/001/A
5
Aspects practice use
To use this method, it is necessary to specify under the key word factor
COMP_INCR
the key word
ALGO_C_PLAN:“DEBORST”
. It is necessary also that the modeling (specified in
AFFE_MODELE
)
elements concerned with this behavior is “
C_PLAN
“or a model of the hull type to plasticity
local:
COQUE_3D, DKT, PIPE
.
In practice, that increases (automatically) by 4 the number of internal variables of the behavior.
For converging well, it is advised to reactualize the tangent matrix if possible (, with all them
iterations:
REAC_ITER: 1,
or all N iterations, with N small).
This method thus allows a great flexibility in use compared to the behaviors: it is enough
that a behavior is available in axisymetry or plane deformation so that it is too
usable in plane stresses.
As for all integrations of models of behaviors nonlinear, it is highly
advised to give a criterion of convergence
small (to leave the default value to 10
6
.).
6 Bibliography
[1]
R of Borst “the zero normal stress condition in plane stress and Shell elastoplasticity”
Communications in applied numerical methods, Flight 7, 29-33 (1991)