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Titrate:
Finite elements treating the quasi-incompressibility
Date:
14/04/05
Author (S):
S. MICHEL-PONNELLE, E. Key LORENTZ
:
R3.06.08-D Page
: 1/16
Organization (S): EDF-R & D/AMA
Handbook of Référence
R3.06 booklet: Machine elements and thermal for the continuous mediums
R3.06.08 document
Finite elements treating the quasi-incompressibility
Summary:
In certain situations, the mechanical behavior of material imposes that voluminal dilation remains
null, in other words that the deformation is done with constant volume: isotropic elasticity with coefficient of
POISSON equal to 0.5, perfect plastic flows analyzes limit of it…
One proposes here to treat this condition of “incompressibility” or “quasi-incompressibility” while using
a valid formulation as well in the compressible case in the quasi-incompressible case. For that,
one uses a variational formulation with 3 fields where the unknown factors are displacement, deformation
voluminal and the multiplier of associated Lagrange (which would correspond to the pressure in the case
incompressible). Two versions of this formulation are proposed: one for the small deformations, the other
valid in the presence of great deformations.
After some recalls on the difficulties which raise the resolution of the incompressible problems, one describes
the mixed finite element established (in 3D and 2D, plan and axisymmetric), and one also present the large ones
lines of integration in Code_Aster (modeling INCO).
This modeling is necessary to practice the limiting analyzes and to model behaviors
rubber bands for Poisson's ratios close to 0.5. It can also be useful in the case of
modelings generating of strong plastic deformations and for which traditional modelings
can be insufficient and generate oscillations of constraints.
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Count
matters
1 Difficulties related to the processing of the incompressibility .............................................................................. 3
1.1 Incompressible” and “quasi-incompressible” behaviors the “............................................. 3
1.2 Some possible numerical solutions ...................................................................................... 4
1.3 Option selected and frameworks of application ............................................................................................ 5
2 mixed variational Formulation of the problem ....................................................................................... 6
2.1 Formulation within the framework of the small deformations ...................................................................... 6
2.2 Formulation in great deformations ............................................................................................. 7
3 Discretization by mixed finite elements ................................................................................................ 8
3.1 Choice of the discretization ................................................................................................................. 8
3.2 Writing of the discrete problem ............................................................................................................ 9
3.2.1 Writing in small deformations ............................................................................................. 9
3.2.2 Writing in great transformations .................................................................................... 10
4 Integration in Code_Aster of the incompressible finite elements .................................................. 11
4.1 General presentation of the incompressible element ...................................................................... 11
4.2 Use of modeling .............................................................................................................. 12
4.3 Formulation of the elementary terms of the second member ........................................................... 12
4.4 Calculation of the strains and the stresses .................................................................................. 12
5 Validation ............................................................................................................................................. 13
5.1 Incompressible elastic case ........................................................................................................ 13
5.2 Elastoplastic case ...................................................................................................................... 13
6 Bibliography ........................................................................................................................................ 15
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Finite elements treating the quasi-incompressibility
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1
Difficulties related to the processing of the incompressibility
In certain situations, the mechanical behavior of material imposes that the deformation
makes with constant volume. The materials having this property of not-dilatancy are often
qualified “incompressible” materials. We will see that these problems pose two types of
difficulties. The first difficulty is related to the writing of the condition of incompressibility, second is
dependant on the numerical problems which this constraint generates. These difficulties are found when it
material is quasi-incompressible.
One reasons here in small disturbances but the problem remains the same one within the framework of
finished transformations.
1.1
Incompressible” and “quasi-incompressible” behaviors “
Within the framework of the mechanics of the continuous mediums, deformation of an isochoric type is
characterized by the fact that the gradient of the transformation F is such as det F = 1. If one places oneself
within the framework of the small disturbances, the preceding condition is reduced to:
div U = 0 = tr ()
The tensor is thus only deviatoric:
D
=.
It results from it that in the case of isotropic materials, the invariant tr (or det F) does not intervene in
the expression of the density of free energy; thus in the case of incompressible elasticity in HPP,
one has simply:
D
D
() = µ.
This density makes it possible to express only the deviatoric part of the tensor of the constraints:
D
D
= 2µ
In fact, the constraint is defined in a constant close p, which is opposite average pressure:
= 2µ D + pId
éq 1.1-1
Note:
· incompressible isotropic elasticity is of course a borderline case of isotropic elasticity with
a Poisson's ratio = E - 1 tending towards 0.5.
2µ
· there is not that the elastic materials whose Poisson's ratio is equal or
slightly lower than 0.5 which utilizes the condition of incompressibility. Thus, it
G
intervenes also in the case of plastic rigid material
= 0. Indeed, one
tr
G
has in this case: & =
; 0; G 0; G = 0
What leads to the condition of incompressibility tr & = 0.
In addition, in the case of elastoplasticity, when plastic deformations
become largely higher than the elastic strain, one finds itself in one
almost incompressible case with tr 0.
Lastly, materials checking a relation of behavior of the type NORTON-HOFF (law
used for calculations of analysis limits [R7.07.01]) show also the characteristic
of incompressibility:
N
(
1
v) = (
D
eq)
.
with N 1 and > 0
3
where
=
. D.D
eq
is the equivalent constraint of Von Mises.
2
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1.2
Some possible numerical solutions
If one wants to treat the condition of incompressibility exactly, we saw it, the constraint is not
completely given starting from the deformation (cf [éq 1.1-1]). It is thus necessary to use one
mixed formulation, i.e. to introduce (at least) another unknown factor of the problem which will allow
to determine the tensor of the constraints completely. Several alternatives are possible, more
simple consisting in imposing the condition of incompressibility using a multiplier of Lagrange,
who is then the pressure p.
Note:
If one chooses a procedure of penalization, one is reduced to the quasi-incompressible case
and thus with the difficulties evoked below.
One also can, in particular in the case of linear elasticity, to choose to return material
slightly compressible. In this way, the constraint is entirely defined starting from displacement
and the use of a mixed formulation is not essential any more. On the other hand, the resolution of these
problems with the traditional finite elements in displacement, raises numerical difficulties. In
effect, the kinematic constraint that a deformation with constant volume represents is very strong, even
too much strong if the degrees of freedom of the element are not important enough. Thus, the triangle with
3 nodes can present phenomena of blocking, i.e. the “grid” cannot
to deform. In a less extreme way, majority of the elements traditional, in particular linear,
comprise in an abnormally rigid way. New elements must thus be used in order to
“to slacken” the system. These elements can be based on various types of formulation:
·
only in displacement
·
mixed: displacements/forced, displacements/pressures, deformations/forced,
voluminal displacements/pressures/dilations…
In all the cases, if guard there is not taken, one can have numerical difficulties. Several tracks
are used to facilitate the deformation of the elements:
·
to use under-integration makes it possible to improve the results but it presents one
disadvantage: it can lead to the appearance of parasitic modes or hourglass. For
to solve this problem, one can is to enrich the matrix by rigidity thanks to matrices of
stabilization which comes to neutralize the hourglass modes, is to use methods of
projection which consists in projecting in a smaller space the condition of incompressibility
in order to eliminate the phenomena of blocking. Most known is the method B-Bar [bib1],
·
to enrich the element using additional degrees of freedom: one speaks then about methods with
increased deformations, incompatible modes,] [bib2]…
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1.3
Option selected and frameworks of application
We chose here to choose a formulation which covers theincompressible one as well
(until the incompressible one) that the compressible one. For that, the tr term is treated like a variable
independent. With the multiplier of Lagrange associated, that led to a formulation with 3 fields.
A version great deformations was also developed on the same principle. In this case,
variable independent related to the condition of incompressibility is not any more tr but det F.
The advantage of this formulation is qu `it makes it possible to use in a transparent way all the laws of
behavior elastoplastic available in Aster (not need to separate the deviatoric part
and the spherical part of the tensor of the constraints). It is thus not limited to elasticity or to
the elastoplasticity of Von Mises. On the other hand, one will not be able to treat the case where the coefficient of
Poisson is strictly equal to 0.5, because one uses for the calculation of the elastic constraint the term
E
tr, whose denominator is null when = 0.5.
1
(+ 1
) (- 2)
Consequently, this formulation INCO must be used:
·
to deal with the limiting analysis problems for which one supposes that the flow is done
with constant volume [R7.07.01],
·
to deal with elastic problems whose Poisson's ratio is higher than 0.45.
This formulation can also be used:
·
to deal with the problems where the plastic deformations are important, which generates
oscillations on the level of the constraints (example: in the case of calculations on test-tubes
notched). Of course, this formulation being more expensive than the formulation in displacement
traditional, it is to be held for the case posing problem and where one is interested in the values of
constraints.
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2
Mixed variational formulation of the problem
2.1
Formulation within the framework of the small deformations
That is to say a solid subjected to:
·
a field of displacement D
U on D
·
a field of effort G on NR
·
a voluminal field of effort F on
In the traditional case of the finite elements in displacement (modeling 3D or D_PLAN or AXIS in
Code_Aster), when the problem derives from an energy, the solved problem is as follows:
to find
D
U V with checking the relation of behavior, which minimizes the potential energy:
1
U
() =
.
D - fu
D - gud
2
NR
As we explained to [§1], this formulation is not appropriate when one seeks with
to bring closer the incompressible solution, i.e. of the condition div U = 0 or tr = 0. For
to circumvent this difficulty, a solution is to separately treat the spherical part of the tensor of
deformations (the part which poses numerical problems) and its deviatoric part. One will have
thus:
G
1
(U, G) = D (U) + I where D (U) = (U)
D
- (tr (U))I and G = tr (U)
D
éq 2.1-1
3
3
The preceding problem is thus reduced to the resolution of a problem to 2 variables, U and G, under
constraint G = tr. It can be brought back to the resolution of an unconstrained problem by introducing one
multiplier of Lagrange p; it is written:
to find
D
U V, p and G (problem of point-saddle), such as:
(
G
L U, p, G)
D
=
.
(U) + I
D
(pdivu G) D F ud
gu D
éq 2.1-2
3
+
-
-
-
NR
This problem can be solved, by writing the conditions of optimality:
L
= (D
+ pi). D - F U
D - G U
D =
D
0
U
NR
L
= (divu - G) p D =
0
éq
2.1-3
p
L
=
1 tr - p
G D =
0
G
3
Note:
·
the first equation corresponds to the equilibrium equation,
·
the second equation translates the kinematic relation binding G to U,
·
the third equation gives the expression of the multiplier of Lagrange p,
·
when the problem does not derive from an energy, one can directly use the system
equations [éq 2.1-3].
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2.2
Formulation in great deformations
It is possible to extend the preceding variational formulation [éq 2.1-2] to the great deformations.
The principle is identical, but one is based in this case on the decomposition of the tensor gradient of
transformation F proposed by Flory [bib3]:
1
1
F = S
S
-
F F with F
3
= J I
and F
3
= J
F and J = det F
D
The problem is brought back there still to a problem of point saddle:
to find
D
U V, G and p not saddle of the Lagrangian one:
L (U, p, G) = [W (13
G F)] -
D
+ p (J - G)
D
- FUD
U
G D
éq 2.2-1
0
-
0
-
0
0
NR
where W is the deformation energy expressed according to the variation of the gradient of
transformation F.
The choice which was made here is to write the deformation energy on the configuration - i.e. with
beginning of the step of time. One notes classically:
· F the gradient of the transformation of 0 with -
· F the gradient of the transformation of - with.
-
-
One has then: F = F F and J = J J
This problem can be solved as in small deformations by writing the conditions of optimality.
Derivation does not raise difficulties particular to condition of only remembering:
W
1
T
= and = F, being the first tensor of the constraints of Piola-Kirchhoff.
F
J
The system to be solved is thus the following:
13
L
= G
D
+ pi. U D - F ud - G U D =
D
X
0
U
J
NR
L
= (J - G) p
D
=
0
0
p
0
-
23
L
J J
1
=
Tr - p G D =
0
G
-
0
3 G G
0
With regard to obtaining the tangent matrix, it of course asks a little more calculation
that in small deformations.
K
K
K
uu
up
ug
K = Kup K p K pg
K
K
K
ug
pg
gg
For the Kuu term, the method used is the same one as that used in [bib4] or [R5.03.21].
principle consists in deriving with fixed configuration, then to choose as configuration that which coincides
with the current configuration at the moment of calculation, i.e. This matrix is not a priori
symmetrical. But in practice, a symmetrical tangent matrix is used. Other derivations
do not pose particular problems.
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To simplify the writing of the various terms, one is brought to define the following tensors:
=
H
the algorithmic tangent matrix given by the law of behavior
1
G
3
F
J
1
P = H - I
(I: H) which corresponds to some extent to the deviatoric part of the matrix
3D
D
algorithmic tangent
1
J 3
D
T = P: F
+
G
Finally, the tangent matrix is made up of the following terms:
K
=
uu
[div U
-
U
eq
eq
X]:
v
D
X
(
géométriqu
rigidity
E)
G 2/3
1
+
P: - U
- div U
T:
v
D
(
comporteme
of
rigidity
NT)
J
3
X
X
K
= tr
up
(U
X) p
D
1 G 2/3
K
=
ug
T: + U
G
D
3 J
X
K
=
p
0
K
= - p
G
pg
D O
O
1 G 2/3 -
G
1/3
K
=
G
gg
- 2tr + Id: H F ×
D
-
9g J
J
J
3
Discretization by mixed finite elements
3.1
Choice of the discretization
When a mixed formulation is used, it is necessary to discretize at the same time the space of
displacements, of the multiplier of Lagrange p and “swelling” G. The experience gained on
mixed elements, in particular 2 fields for the incompressible elements, makes it possible to know that
discretization of these fields cannot be unspecified, under penalty of obtaining phenomena
oscillations (in particular on the level of the pressures) or phenomena of blocking (elements
not being able to become deformed or too rigid). Thus it is necessary to have a number of points of Gauss
of pressure sufficiently important to check the condition of incompressibility almost everywhere and one
a number of points of sufficiently weak Gauss of pressure to have more degrees of freedom with
to calculate that constraints to be checked. One of the conditions necessary to obtain results
satisfactory is the checking by the finite element considered of condition LBB (LADYJENSKAIA,
BREZZI, BABUSKA). One can find in [bib5] and [bib6] of the examples of elements satisfying
condition LBB.
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Here the problem is a little different since the formulation contains 3 fields. We are
inspired by the uses of this kind of formulation (e.g. [bib7]), by using an element of the type P2/P1/P1.
In other words, displacement is quadratic, the pressure and swelling is both
linear.
The finite elements used are thus the following:
in 2D:
U
triangle with 6 nodes
\
quadrilateral with 8 nodes
p, G
triangle with 3 nodes
\
quadrilateral with 4 nodes
in 3D:
U
tetrahedron with 10 nodes
\
cubic with 20 nodes \ pentahedron with 15 nodes
p, G
tetrahedron with 4 nodes
\
cubic with 8 nodes \ pentahedron with 6 nodes
For each type of element, one uses only one family of points of Gauss:
·
3 points for the triangles
·
9 points for the quadrilaterals
·
4 points for the tetrahedrons
·
27 points for the cubes
·
6 points for the pentahedrons
3.2
Writing of the discrete problem
That is to say U.E. , EP and Ge, vectors of the elementary nodal unknown factors (resp. displacement, pressure and
swelling). If Nq and Nl are related to forms (respectively quadratic and linear)
associated the finite element considered:
E
U = NR U
Q
E
p = NR p
L
E
G = NR G
L
3.2.1 Writing in small deformations
B is the traditional matrix of derivation making it possible to pass from
E
U with:
E
= DRUNK
In the formulation, one distinguishes Dev. and dil, which leads us to define the Bdev operators and
tr
B
D
dil such as:
E
= B
U
Dev.
and
E
= B U
dil
3
The discretized form of the equations of the problem [éq 2-3] is written:
T
D
U
F = B
(
+ pi
D) D =
ext.
F
F =
T
NR (dil
B U - L
NR G) D = 0
p
L
F =
T
NR (1 tr - p) D = 0
G
L
3
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The tangent matrix of the problem is symmetrical and is based on the following terms:
F
K
=
U =
T
uu
B
DB
E
Dev.
Dev.
D
U
F
K
=
U =
up
B
NR
E
T
dil
D
L
p
F
1
K
=
U =
T
ug
(B
D) NR
E
Tr Dev.
D
L
G
3
F
K
=
p = 0
p
E
p
F
K
=
p = -
pg
NR NR
E
T
D
L
L
G
F
1
K
=
G =
gg
NR
(D) NR
E
Ttr
D
L
L
G
9
3.2.2 Writing in great transformations
One notes D the matrix of derived from the functions of form (quadratic) on the current configuration
and D on the configuration -, that is to say:
E
E
U = U
-
D
U D U
X
and
X
=
In addition, one defines the constraint of balance eq and the size Q by the following relations:
1
2
G
3
-
3
D
1 J J
eq =
+ pId and Q =
tr - p
J
-
3 G G
is here the tensor of the constraints resulting from the law of behavior.
The vector of the interior forces is written in form the following discretized form:
F =
U
D D
eq
F =
p
T
NR (J - L
NR G)
D
L
0
0
F =
G
T
NR Q
D
L
0
0
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4 Integration in Code_Aster of the finite elements
incompressible
4.1
General presentation of the incompressible element
The finite elements are integrated in Code_Aster in 2D plane deformations, in axisymmetric 2D
and in 3D. 3 modelings are accessible by using the following options for AFFE_MODELE:
·
“3d_INCO” for the 3D,
·
“D_PLAN_INCO” for the 2D in plane deformations,
·
“AXIS_INCO” for the axisymmetric 2D.
In the catalog of the elements, the incompressible elements can apply to the meshs:
Meshs
A number of nodes in displacements
A number of nodes in pressure
TRIA 6.6
3
QUAD8 8
4
HEXA20 20
8
TETRA10 10
4
PENTA15 15
6
In the routines of initializations of the incompressible elements, one defines:
·
1 only family of points of GAUSS (the first family of points of GAUSS) [R3.01.01],
·
2 families of functions of forms respectively associated with displacements (functions with
forms of degree 2) and under the terms of pressure (functions of forms of degree 1).
Let us take as example the tetrahedral element with 10 nodes: degrees of freedom in displacement
are carried by all the nodes, on the other hand, only the 4 nodes nodes have the degrees of
freedom p and G.
The accessible components for field DEPL are thus
·
displacements: DX, DY and DZ in 3D with all the nodes,
·
pressure: PRES for the nodes node,
·
swelling: GONF for the nodes node.
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4.2
Use of modeling
By choice, modeling INCO is accessible only with STAT_NON_LINE and option COMP_INCR.
Under this key word, the version small deformations is accessible by using DEFORMATION=' PETIT',
the version great deformations by using DEFORMATION=' SIMO_MIEHE'.
It is thus not possible to use modeling INCO with the commands:
·
MECA_STATIQUE
·
CALC_MATR_ELEM/CALC_VECT_ELEM/ASSE_MATRICE/ASSE_VECTEUR/RESO_LDLT
·
STAT_NON_LINE (COMP_ELAS =…)
Note:
For the moment, only the tangent matrix can be used for the phase of prediction.
However, of new developments in Code_Aster, should return the matrix
soon accessible rubber band.
4.3
Formulation of the elementary terms of the second member
The loads can be gravity, of the surface forces distributed, the pressures. Terms
elementary are calculated in a traditional way for the degrees of freedom of displacement and one affects
the zero value for the degrees of freedom of pressure and swelling.
4.4
Calculation of the strains and the stresses
In this formulation, it is advisable to distinguish the stress field resulting from the law from behavior
D
=
ldc, of the stress field which checks balance and which is defined by the relation
Id
p
ldc +
it is the latter field which is stored in SIEF_ELGA as well as the relation binding the multiplier p and
ldc.
In short, the components of SIEF_ELGA are:
· SIXX, SIYY, SIZZ, SIXY in 2D like SIXZ and SIYZ in 3D: components of the tensor
D
=
Id
p
ldc +
,
1
· SIP which is equal to tr
- p
ldc
in small deformations,
3
2
1 J - J
3
or
tr
p in great transformations.
-
ldc -
3 G G
It is also possible to recompute EPSI_ELGA_DEPL, which is the field of deformation to the direction
traditional.
One can also carry out a calculation of limiting load with POST_ELEM.
Handbook of Référence
R3.06 booklet: Machine elements and thermal for the continuous mediums
HT-66/05/002/A
Code_Aster ®
Version
7.4
Titrate:
Finite elements treating the quasi-incompressibility
Date:
14/04/05
Author (S):
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:
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5 Validation
5.1
Incompressible elastic case
Test SSLV130 (cf [V3.04.130]) makes it possible to check the validity of modeling in the case of one
roll elastic incompressible subjected to an internal pressure. Its equivalent into large
deformations also exist: test SSNV112 (cf [V6.04.112]).
5.2 Case
elastoplastic
The goal of this example is to illustrate the contribution of modeling INCO if the deformations
plastics are important compared to the elastic strain. One studies for that one
notched sample into axisymmetric, subjected to an imposed displacement. Geometry and it
loading are represented on the figure below. The grid consists of 548 TRI6.
U0
E
D
F
C
With
B
Appear 5.2-a: Géométrie and boundary conditions
The behavior of material is of elastoplastic type with isotropic work hardening
linear (VMIS_ISOT_LINE). The parameters are as follows:
·
E = 200.000 MPa
·
= 0.3
·
y = 200 MPa
·
AND = 1000 MPa
Handbook of Référence
R3.06 booklet: Machine elements and thermal for the continuous mediums
HT-66/05/002/A
Code_Aster ®
Version
7.4
Titrate:
Finite elements treating the quasi-incompressibility
Date:
14/04/05
Author (S):
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:
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On the figure [Figure 5.2-b], one compares the constraint yy obtained on path FC (cf [Figure 5.2-a])
with traditional modeling AXIS and modeling AXIS_INCO.
EDF
Department Mecanique and Modeles Numeriques
Electricity
SIGMAyy
from France
300
250
200
150
AXIS
AXIS_INCO
100
50
0
0
1
2
3
4
5
agraf 26/02/2002 (c) EDF/DER 1992-1999
Appear 5.2-b: yy along line FC
It is seen very clearly that the solution obtained with formulation INCO makes it possible to be freed from
parasitic oscillations.
Handbook of Référence
R3.06 booklet: Machine elements and thermal for the continuous mediums
HT-66/05/002/A
Code_Aster ®
Version
7.4
Titrate:
Finite elements treating the quasi-incompressibility
Date:
14/04/05
Author (S):
S. MICHEL-PONNELLE, E. Key LORENTZ
:
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6 Bibliography
[1]
J.R. HUGUES: The finite element method, Prentice-Hall, Inc. Englewood Cliffs, N-J. 07632,
1987.
[2]
J.C. SIMO, Mr. S. RIFAI: With class off mixed assumed strain methods and the method off
incompatible modes, Int. Jnal Num. Meth. Engg, vol. 29, pp1595-1638, 1990.
[3]
R.J. FLORY: Thermodynamic relations off high elastic materials. Trans. Faraday Plowshare, vol.57,
1961, pp. 829-838.
[4]
V. CANO, E. LORENTZ: Introduction into Code_Aster of a module of behavior in
great deformations elastoplastic with isotropic work hardening. Note EDF/DER
HI-74/98/006/0 of the 26/08/1998
[5]
Mr. GIRAULT, P. RAVIART: Finite element methods for Navier-Stokes equations. Theory and
algorithms. Springer Verlag, 1986.
[6]
P. MIALON, B. THOMAS: Incompressibility in plasticity: under-integration and others
digital techniques. Note EDF/DER HI-72/6404 of the 19/01/1990.
[7]
A.G.K. JINKA, Mr. BELLET, L. FOURMENT: With new three-dimensional finite element model for
the simulation off powder forging processes: application to hot forming off P/M connecting
road. Int. Jnal Num. Meth. Engg, vol. 40, pp3955-3978, 1997.
Handbook of Référence
R3.06 booklet: Machine elements and thermal for the continuous mediums
HT-66/05/002/A
Code_Aster ®
Version
7.4
Titrate:
Finite elements treating the quasi-incompressibility
Date:
14/04/05
Author (S):
S. MICHEL-PONNELLE, E. Key LORENTZ
:
R3.06.08-D Page
: 16/16
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Handbook of Référence
R3.06 booklet: Machine elements and thermal for the continuous mediums
HT-66/05/002/A
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