Code_Aster ®
Version
6.2
Titrate:
Law of behavior in great rotations and small deformations
Date:
05/08/03
Author (S):
V. CANO Key
:
R5.03.22-A Page
: 1/4
Organization (S): EDF-R & D/AMA
Handbook of reference
R5.03 booklet: Nonlinear mechanics
Document: R5.03.22
Law of behavior in great rotations and
small deformations
Summary:
One describes here the formulation adopted to treat great rotations and small deformations. This
formulation is valid for all the laws of behavior defined under COMP_INCR of the command
STAT_NON_LINE and provided with modelings three-dimensional (3D), axisymmetric (AXIS), in deformations
plane (D_PLAN) and in plane constraints (C_PLAN).
This functionality is selected via the key word DEFORMATION = “GREEN” under COMP_INCR.
Handbook of Référence
R5.03 booklet: Nonlinear mechanics
HT-66/03/005/A
Code_Aster ®
Version
6.2
Titrate:
Law of behavior in great rotations and small deformations
Date:
05/08/03
Author (S):
V. CANO Key
:
R5.03.22-A Page
: 2/4
Count
matters
1 Some definitions .............................................................................................................................. 3
2 Assumption of the small deformations and great rotations .................................................................... 3
3 Bibliography .......................................................................................................................................... 4
Handbook of Référence
R5.03 booklet: Nonlinear mechanics
HT-66/03/005/A
Code_Aster ®
Version
6.2
Titrate:
Law of behavior in great rotations and small deformations
Date:
05/08/03
Author (S):
V. CANO Key
:
R5.03.22-A Page
: 3/4
1 Some
definitions
One points out here some definitions of tensors related to the great deformations.
One calls tensor gradient of the transformation F, the tensor which makes pass from the initial configuration
0 with the deformed current configuration (T).
x$
F =
= Id + U
with X = x$ (X, T) = X + U
éq
1-1
X
X
where X is the position of a point in 0, X the position of this same point after deformation in
(T) and U displacement.
Various tensors of deformations can be obtained by eliminating rotation in
local transformation. This can be done in two manners, that is to say by using the theorem of
polar decomposition, is by directly calculating the variations length and angle (variation of
scalar product).
Of Lagrangian description is obtained (i.e. on the initial configuration):
· By the polar decomposition:
F = RU
éq 1-2
where R is the tensor of rotation (orthogonal) and U the tensor of pure deformations right
(symmetrical and definite positive).
· By a direct calculation of the deformations:
1
E = (C - Id) with C = FTF
éq 1-3
2
where E is the tensor of deformation of Green-Lagrange and C the tensor of right Cauchy-Green.
The tensors U and C are connected by the following relation:
2
C = U
éq 1-4
2
Assumption of the small deformations and great rotations
When the deformations are small, there are no fundamental difficulties to write the laws of
behavior: the various models “great deformations” lead to the same model
“small deformations”, and this as well for isotropic behaviors anisotropic. Only
difficulty of a geometrical nature related to finished rotation remains.
To write the model in great rotations and small deformations, one leaves the polar decomposition
of F = RU is F. As the tensor U is a tensor of deformation pure and in addition small, one
can calculate, by a law of behavior small deformations, the tensor of the constraints *
associated the this history in deformation U. It is then enough to subject to this tensor *
, rotation R
to obtain the tensor of the constraints associated with the history in deformation F, as follows:
* T
= R R
éq 2-1
Handbook of Référence
R5.03 booklet: Nonlinear mechanics
HT-66/03/005/A
Code_Aster ®
Version
6.2
Titrate:
Law of behavior in great rotations and small deformations
Date:
05/08/03
Author (S):
V. CANO Key
:
R5.03.22-A Page
: 4/4
One can summarize this diagram as follows:
ldc HPP *
* T
F U
= Id +
= R
R
éq 2-2
The disadvantage of this computation channel is that it requires the polar decomposition of F. Deux
assumptions are made then to avoid it.
On the one hand, to avoid the calculation of U, one can approach deformation HPP, by the deformation of
Green E, by benefitting owing to the fact that the deformations are small:
1
1
1
E =
FTF - Id =
U - Id U + Id = +
2
(
)
(
) (
)
éq
2-3
2
2
2
One then deduces from them by the law from behavior “small deformations”.
In addition, in the same manner to avoid the calculation of R, one can approach the tensor of
constraints HPP by the second tensor of Piola-Kirchhoff S:
S = F-1 F = Det (U) U-1 - 1
=
+
(
O)
J
U
éq
2-4
One deduces some then by:
= 1
T
FSF
éq 2-5
J
Finally, in the presence of great rotations and to small deformations, it is enough to write the law of
behavior “small deformations” with, in input, history of the deformations of Green E, and in
output, the history of the constraints of Piola-Kirchhoff S. Cette approach is valid as well for
isotropic laws of behavior that anisotropic.
As for the adapted variational formulation, it is about that adopted in hyper-elasticity
(behavior ELAS, ELAS_VMIS_XXX under COMP_ELAS with the deformations of the type GREEN). For
more details, one will refer to the associated reference document [R5.03.20]. It is however necessary to be
sure that the problem studied induced many small deformations bus if not one cannot do them any more
simplifications [éq 2-3] and [éq 2-4]. Without this assumption, the variation with a plastic behavior
increases quickly with the intensity of the deformations.
3 Bibliography
[1]
CANO V., LORENTZ E., “Introduction in Code_Aster of a model of behavior in
great elastoplastic deformations with isotropic work hardening ", internal Note EDF DER,
HI-74/98/006/0, 1998
Handbook of Référence
R5.03 booklet: Nonlinear mechanics
HT-66/03/005/A
Outline document