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Titrate:
Nonlinear behaviors in plane constraints
Date:
06/03/01
Author (S):
J. Mr. PROIX, E. LORENTZ
Key:
R5.03.03-A
Page:
1/6
Organization (S): EDF/MTI/MN
Handbook of Référence
R5.03 booklet: Nonlinear mechanics
Document: R5.03.03
Taking into account of the assumption of the constraints
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 constraints.
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 TUYAU
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.
Handbook of Référence
R5.03 booklet: Nonlinear mechanics
HI-75/01/001/A
Code_Aster ®
Version
5.4
Titrate:
Nonlinear behaviors in plane constraints
Date:
06/03/01
Author (S):
J. Mr. PROIX, E. LORENTZ
Key:
R5.03.03-A
Page:
2/6
1 Introduction
One presents here a general method of integration of the nonlinear models of behaviors
(plasticity, viscoplasticity, damage) in plane constraints. It is activated by the key word
ALGO_C_PLAN: “DEBORST” of the operand of the incrémentaux nonlinear behaviors
COMP_INCR of STAT_NON_LINE, for modeling C_PLAN, DKT, COQUE 3D and TUYAU.
2
Difficulty of integration of the nonlinear behaviors in
plane constraints
Modeling C_PLAN, (as well as modelings COQUE_3D, DKT, TUYAU) supposes that the state of
constraints room is plane, i.e. that = 0, Z representing the direction of the normal with
zz
surface. The tensors of constraints and deformations thus take the following form (in C_PLAN):
0
DNN. DNN
xx
xy
K N = BT DNN - 12 21 B
!
!
11
0
DNN
= xy
yy
22
0
0
zz
Note:
For the hulls, it is necessary to add terms due to transverse shearing (,
), but
xz yz
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). It
zz
can be given that using the relation of behavior. However the condition = 0 is not
zz
alleviating for the integration of the behavior where one calculates an increase in constraint 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, cannot be calculated only starting from U and
zz
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 constraints.
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 constraints. It is usable
as soon as the model of behavior is available into axisymmetric or in plane deformations.
Handbook of Référence
R5.03 booklet: Nonlinear mechanics
HI-75/01/001/A
Code_Aster ®
Version
5.4
Titrate:
Nonlinear behaviors in plane constraints
Date:
06/03/01
Author (S):
J. Mr. PROIX, E. LORENTZ
Key:
R5.03.03-A
Page:
3/6
3
Principle of the processing of the plane constraints by
method De Borst
The idea of the method due to R. of Borst [bib1] consists in treating the condition of plane constraints 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 of the stress fields which
tend towards a plane stress field progressively with the iterations:
N
zz
0
where N indicates the number of iteration of Newton.
One thus obtains the condition of constraint 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
! 11
D
D12
The tangent operator D =
D D D becoming D =
.d
=
.
=
D
D
zz 21
22
D
11
D
D12
where D and D indicate infinitesimal increases, and where by definition
=
is
D
D
D
21
22
the coherent tangent matrix with the behavior without the assumption of plane constraints, 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 constraints 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 which 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.A convergence, this variation must tend towards zero.
Handbook of Référence
R5.03 booklet: Nonlinear mechanics
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Code_Aster ®
Version
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Titrate:
Nonlinear behaviors in plane constraints
Date:
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Author (S):
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Key:
R5.03.03-A
Page:
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By writing n+1
N
=
+
=
zz
zz dzz 0 one obtains, for the iteration n+1:
D
N
N
n+1
! ! -
-
+
+
11
N 1
! N
! N 1! N
D
D 12
D!
+1
.
D
= N
N =
N
=
N
N
-
-
zz
zz
zz
zz
D
n+1
21
D 22
D
zz
what, by using the last equation of this system, enables us to be reduced to:
!n+1 =!N
N
+ D D
1
N 1
D
D
N 1
0
11!
+
N
-
-
.
12 (
N
D22) (N
N
+
D
21
=
zz
! +)
N
N
- 1
N
N
D = - D
D
N 1
22
+
D
zz
()
(
21!
+
zz
)
with the stress field which is written:
N
D D
D
1
. N
N
12
21
!n+
N
N 1
N
12
N
= D
11 -
D
-
N
! + +!
N
zz
D22
D
22
N
N
N
D
D
n+1
N
zz
=
-
+ 21
N
- 21
N 1
zz
zz
N
N
!
N
! +
D
D
D
22
22
22
By using the preceding expression of the stress field, one finds then:
LT n+
. FD
1
LT n+
=! . ! FD
1
= L
=
N
N
N
!
D.D
D
+
BT
DNN
12
21
N
12
N
11 -
D! N 1
+!
zz FD
-
22
D
22
D
N
N
N
=!
D.D
D
BT DNN
12
21
-
!B.dun+1dv +!BT N
12
N
11
!
zz FD
-
22
D
22
D
DNN
= K ndun+1 +!BT
! N
12
N
-
zz FD
22
D
It is thus noted that the taking into account of the plane constraints 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:
DNN DNN
N
12
21
K N
BT
=
D
FD
11 -
N
B
!
.
!
D22
· 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 constraints:
Dn12 N
R (one 1) = BT
+
-
FD
N
zz
! !N
D22
Handbook of Référence
R5.03 booklet: Nonlinear mechanics
HI-75/01/001/A
Code_Aster ®
Version
5.4
Titrate:
Nonlinear behaviors in plane constraints
Date:
06/03/01
Author (S):
J. Mr. PROIX, E. LORENTZ
Key:
R5.03.03-A
Page:
5/6
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 constraints 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: un+1, -
-
, and 4 following additional variables internal resulting
N
N
D
DNN
preceding iteration: 1 scalar variable -
zz
21
N
! and 3 variables - 21,
N
+ N
D
D
N
D
22
22
22
2) before carrying out the integration of the nonlinear behaviors (which will be made in
N
N
N
D
D
n+1
N
axisymmetric), one calculates
zz
=
-
+ 21
N
- 21
N 1
!
! +,
zz
zz
N
N
N
D
D
D
22
22
22
3) One lets the routines of integration of the behavior calculate the constraints as well as
!n+1
tangent behavior D from
as if modeling were axisymmetric or
n+1
zz
of plane deformation,
4) one modifies at output the second member and the tangent matrix (if the reactualization of
DNN DNN
N
12
21
stamp tangent was asked) so that: K N
BT
=
D11 -
B
!
.
! FD and
N
22
D
DNN
R (one 1) = BT
+
- 12 N
! ! N
FD,
N
zz
D22
N
N
D
DNN
5) one stores the new internal variables -
zz + 21
N
! and - 21.
N
N
D
D
N
D
22
22
22
To check convergence, one checks, always on the level of each point of integration of each
finite element if n+1 <, where
N 1
+
=
with provided by the user under the key word
zz
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
N +1
RESI_GLOB_MAXI) one examines whether all the points of integration check the condition
<. If
zz
it is not the case, one carries out additional iterations of Newton until complete checking
of this condition.
Handbook of Référence
R5.03 booklet: Nonlinear mechanics
HI-75/01/001/A
Code_Aster ®
Version
5.4
Titrate:
Nonlinear behaviors in plane constraints
Date:
06/03/01
Author (S):
J. Mr. PROIX, E. LORENTZ
Key:
R5.03.03-A
Page:
6/6
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 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 constraints.
As for all integrations of models of behaviors nonlinear, it is highly
advised to give a small criterion of convergence (to leave the default value with 106.).
6 Bibliography
[1]
R of Borst “the zero normal stress condition in plane stress and Shell elastoplasticity”
Communications in applied numerical methods, Vol 7, 29-33 (1991)
Handbook of Référence
R5.03 booklet: Nonlinear mechanics
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