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
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HT-66/03/005/A
Organization (S):
EDF-R & D/AMA
Manual 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
control
STAT_NON_LINE
and fitted with modelings three-dimensional (3D), axisymmetric (AXIS), in deformations
plane (D_PLAN) and in plane stresses (C_PLAN).
This functionality is selected via the key word
DEFORMATION
=
“GREEN”
under
COMP_INCR
.
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
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HT-66/03/005/A
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
.
F
X
X
Id
U
X
=
=
+
$
with
X X X
X U
=
= +
$ (,)
T
éq
1-1
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:
RU
F
=
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:
E
C Id
=
-
1
2 (
)
with
C F F
=
T
éq 1-3
where
E
is the tensor of deformation of Green-Lagrange and
C
the tensor of right Cauchy-Green.
Tensors
U
and
C
are connected by the following relation:
2
U
C
=
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
that is to say
RU
F
=
. Like 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 stresses
*
associated the this history in deformation
U
. It is then enough to subject to this tensor
*
, rotation
R
to obtain the tensor of the stresses
associated the history in deformation
F
, as follows:
T
R
R
*
=
éq 2-1
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
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HT-66/03/005/A
One can summarize this diagram as follows:
T
R
R
Id
U
F
*
*
=
+
=
HP
ldc
éq 2-2
The disadvantage of this computation channel is that it requires the polar decomposition of
F
. Two
assumptions are made then to avoid it.
On the one hand, to avoid the calculation of
U
, one can approach the deformation HP
, by the deformation of
Green
E
, by benefitting owing to the fact that the deformations are small:
E
F F Id
U Id U Id
=
-
=
-
+
= +
1
2
1
2
1
2
2
(
)
(
) (
)
T
éq
2-3
One deduces some then
by the law of behavior “small deformations”.
In addition, in the same manner to avoid the calculation of
R
, one can approach the tensor of
stresses HP
by the second tensor of Piola-Kirchhoff
S
:
()
()
S
F
F
U U
=
=
=
+
-
-
-
-
J
U
1
1
1
Det
O
éq
2-4
One deduces some then
by:
=
1
J
T
FSF
éq 2-5
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, the history of the deformations of Green
E
, and in
exit, history of the stresses of Piola-Kirchhoff
S
. This 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 into 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