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3.0
Titrate:
Relation of nonlinear elastic behavior
Date:
22/06/95
Author (S):
E. LORENTZ
Key:
R5.03.20-A
Page:
1/8
Organization (S): EDF/IMA/MN
Handbook of Référence
R5.03 booklet: Nonlinear mechanics
Document: R5.03.20
Elastic relation of behavior
nonlinear in great displacements
Summary:
One proposes to describe here a relation of nonlinear elastic behavior which coincides with the law
elastoplastic of Hencky-Von Mises (isotropic work hardening) in the case of a loading which induces one
radial and monotonous evolution of the constraints in any point of the structure. This model is selected in
order STAT_NON_LINE via key word RELATION: “ELAS_VMIS_LINE” or
“ELAS_VMIS_TRAC” under the key word factor COMP_ELAS.
One extends then this relation of behavior to great displacements and great rotations, in
measure where it derives from a potential (hyperelastic law); this functionality is selected via
key word DEFORMATION: “GREEN”. It is available for all the isoparametric elements 2D and 3D.
Handbook of Référence
R5.03 booklet: Nonlinear mechanics
HI-75/95/033/A
Code_Aster ®
Version
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Titrate:
Relation of nonlinear elastic behavior
Date:
22/06/95
Author (S):
E. LORENTZ
Key:
R5.03.20-A
Page:
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Contents
1 Relation of nonlinear elastic behavior: ELAS_VMIS_LINE and ELAS_VMIS_TRAC ............. 3
1.1 Objective ............................................................................................................................................ 3
1.2 Relation of behavior .............................................................................................................. 3
1.3 Resolution of the equation out of p ........................................................................................................ 4
1.4 Calculation of the relation of behavior and tangent rigidity ............................................................. 5
1.5 Taking into account of deformations of thermal origin .................................................................... 5
1.6 Particular processing of the plane constraints .................................................................................. 6
2 Elasticity in great transformations .................................................................................................... 7
2.1 Objective ............................................................................................................................................ 7
2.2 Virtual work of the external efforts: assumption of the dead loads ............................................ 7
2.3 Virtual work of the interior efforts ................................................................................................. 8
2.4 Variational formulation ............................................................................................................... 8
3 Bibliography .......................................................................................................................................... 8
Handbook of Référence
R5.03 booklet: Nonlinear mechanics
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Code_Aster ®
Version
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Titrate:
Relation of nonlinear elastic behavior
Date:
22/06/95
Author (S):
E. LORENTZ
Key:
R5.03.20-A
Page:
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1 Relation of nonlinear elastic behavior:
ELAS_VMIS_LINE and ELAS_VMIS_TRAC
1.1 Objective
Within the framework of the global solution in breaking process, one cannot give a direction to the rate of
restitution of energy that for hyperelastic relations of behavior, i.e. which derive
of a potential, free energy. In order to be able nevertheless to deal with elastoplastic problems, one
propose a relation of nonlinear elastic behavior which leads to results identical to
those obtained by the plastic relation of behavior of Hencky-Von Mises (isotropic work hardening)
in the case of an evolution of loading radial and monotonous in any point. The definition of
characteristics of the material (key word DEFI_MATERIAU) is identical to that of the behavior
isotropic plastic. For further information on the model, one will be able to refer to [bib1]. For
to illustrate the common points and the differences between the models plastic and rubber band, one presents
below a traction diagram then compression obtained for a unidimensional bar.
Final State
Final State
F max
F max
Elastoplasticity
Nonlinear elasticity
1.2
Relation of behavior
After integration in time of the relation of behavior of Hencky-Von Mises, formulated in
speeds of strains and stresses in [R5.03.02] which one adopts the notations, the expression
constraints according to the deformations is:
= K (tr) Id+
~
G (eq
)
éq 1.2 1
y
- if
eq
2µ
G = 2µ and
p = 0
y
- if
eq
>
2µ
R (p)
R (p)
2
G =
and
p such as: p +
=
eq
éq 1.2 2
eq
µ
3
3
Handbook of Référence
R5.03 booklet: Nonlinear mechanics
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Code_Aster ®
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Titrate:
Relation of nonlinear elastic behavior
Date:
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Author (S):
E. LORENTZ
Key:
R5.03.20-A
Page:
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In a way similar to plasticity, the function of work hardening R (p) is deduced from the provided data
by a simple tensile test (work hardening linear with key word ELAS_VMIS_LINE or well defined
by points with the key word ELAS_VMIS_TRAC, cf [R5.03.02]).
As for the variable p, it deserves a few moments of attention. In the plastic model, its
significance is clear. It is about the cumulated plastic deformation, always increasing; it is one
variable interns model. On the other hand, in the elastic case, it does not have any more the internal statute of variable,
since there is no dissipation. Moreover, it decrease during discharges. In fact, its value coincides
with that obtained in plasticity as long as the evolution of the loading is radial and monotonous.
In addition to the relation of behavior itself, it is necessary to know the value of energy
free for a state given for calculations of rate of refund of energy. Without demonstration, it
potential whose the relation derives from behavior is worth:
y
1
2
2µ
- if
2
eq
() = K (tr) +
2µ
eq
2
3
2
éq 1.2-3
y
1
R p
2
((eq)
p
-
eq
if eq >
() = K (tr)
()
+
+
R (S)
2
2
6
µ
µ
ds
0
1.3
Resolution of the equation out of p
One could note in the preceding paragraph that the expression of the constraints requires
solution of an equation relating to the variable p. Dans la mesure où the function of work hardening R is
increasing, this equation can be written by gathering the terms where appear p in the first
member (who is then increasing with p):
()
p + R p
3µ
2 eq
2
()
eq
p + R p =
X
3µ
3
3
p
More precisely, the first member is linear per pieces in p. Pour to solve the equation, it
is then enough sequentially to traverse each interval until finding that in which
locate the solution. An equation closely connected provides the value of p. then.
Handbook of Référence
R5.03 booklet: Nonlinear mechanics
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Code_Aster ®
Version
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Titrate:
Relation of nonlinear elastic behavior
Date:
22/06/95
Author (S):
E. LORENTZ
Key:
R5.03.20-A
Page:
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1.4
Calculation of the relation of behavior and tangent rigidity
The calculation of the constraints and tangent rigidity, i.e. variation of the constraints by report/ratio
with the deformations, is carried out according to the algorithm presented below. By adopting the convention of
Code_Aster, the constraints and the deformations are arranged in a vector with six components,
while tangent rigidity is a matrix 6x6.
xx
xx
1
yy
yy
1
zz
zz
{
1
} =
{}
=
{}
1 =
2 xy
2 xy
0
2
xz
2
0
xz
2 yz
2 yz
0
Relation of behavior:
{}
= K (tr) {} + G {}
~
1
Tangent rigidity:
D {}
= K = K1 + K
D {}
[]
[2]
3K - G
·
[K]
{}
1 {}
1
+ G [I]
D
1 =
3
y
[
0]
if eq
2µ
·
[K2] = 3 2µ R' (p)
y
G ~ ~
if
2
'
eq >
2
R
2
eq
(p)
-
{} {}
+ µ
3
µ
1.5
Taking into account of deformations of thermal origin
In a way identical to plasticity, one divides the total deflection into a mechanical part which checks
the preceding relation of behavior [éq 1.2-1], [éq 1.2-2] and a thermal part, function of
temperature. Let us note moreover that the various characteristics of material can also depend
temperature.
= m + HT
= K (tr m)
+ G (
~
Id
eq)
éq 1.5 1
with HT =
(T - Tréf) Id
: thermal dilation coefficient
Tréf: temperature of reference
Handbook of Référence
R5.03 booklet: Nonlinear mechanics
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Titrate:
Relation of nonlinear elastic behavior
Date:
22/06/95
Author (S):
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Key:
R5.03.20-A
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It remains to supplement the potential free energy [éq 1.2-3] to include the temperature there. Several choices
are possible, depend on the way in which one wishes to define the entropy (derivative of the free energy
compared to the temperature). In our case, the adopted potential is:
y
1
HT 2
2µ
if
2
eq
(, T) = K Tr
2
2
((-) +
µ
eq
3
2
y
1
R p
HT 2
(eq) p eq
if eq >
(, T) = K Tr
R
2
2
((-)
()
()
+
+
(S)
µ
6µ
ds
0
1.6
Particular processing of the plane constraints
Usually, one seeks to determine the constraints knowing the deformations and the temperature.
However, it is not completely any more the case under the assumption of the plane constraints insofar as
three of the components of the tensor of the deformations are henceforth unknown, the dual sizes
being fixed:
xz, yz and zz unknown
xz = yz = zz = 0
It is thus necessary to start by determining these unknown components. The adopted method is exposed
in [bib1] and [R5.03.02]. One can however recall here that the components xz and yz do not pose
of problem, being given the form of the relation of behavior [éq 1.2-1]:
xz = yz = 0
On the other hand the determination of the component zz requires the solution (numerical) of an equation
nonlinear scalar.
Lastly, a last warning is essential. With the difference in the plane deformations, solutions that
one obtains under the assumption of the plane constraints are generally not exact in measurement
where they do not check the conditions of geometrical compatibility (integrability of the field of
deformations). They are only approximate solutions.
Handbook of Référence
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Titrate:
Relation of nonlinear elastic behavior
Date:
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Author (S):
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Key:
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Page:
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2
Elasticity in great transformations
2.1 Objective
Henceforth, one proposes to take into account great displacements and great rotations,
functionality accessible by key word DEFORMATION: “GREEN” in the command
STAT_NON_LINE. Let us specify as of now that one restricts oneself with finite elements
isoparametric (D_PLAN, C_PLAN, AXIS and 3D) for which discretization of the continuous problem
do not raise particular difficulties, cf [R3.01.00].
To this end, it is admitted that the second tensor of the constraints of Piola-Kirchhof, S, drift of the potential
of Hencky-Von Mises expressed using the deformation of Green-Lagrange E:
S =
(E)
E
Also let us point out the definitions of E and S. One can also find information
complementary in [bib1].
1
F = Id + Gra (
D U) E = (TFF - I)
D
2
S = Det (F) F-1
T
F-1
Such a relation of behavior, known as hyperelastic, makes it possible in any rigor to take into account
great deformations and great rotations. However, we limit ourselves to the small ones
deformations, and this for two reasons. First of all, the relation of behavior adopted does not present
not the good properties (polyconvexity) to ensure the existence of solutions and does not control not
more important compressions. Then, the plastic behavior differs notably from one
behavior hyperelastic as soon as the deformations become appreciable. It is for these
reasons which we chose to preserve the assumption of small deformations, thus escaping
polemic of the great deformations.
2.2 Virtual work of the external efforts: assumption of the loads
died
To deal with the problem of hyperelastic structural analysis, one seeks to write balance under
variational form on the initial configuration. In particular, it is necessary to express the virtual work of
external efforts on this same initial configuration what requires the additional assumption of
dead loads: it is supposed that the loading does not depend on the geometrical transformation.
Typically, an imposed force is a dead load while the pressure is a loading
follower since it depends on the orientation of the face of application, therefore of the transformation. Under this
assumption, the virtual work of the external efforts is written like a linear form:
W. v = F v D +
Td v dS
ext.
O I I O I I O
O
F O
F: voluminal loading
Td: surface loading being exerted on the edge F O
Handbook of Référence
R5.03 booklet: Nonlinear mechanics
HI-75/95/033/A
Code_Aster ®
Version
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Titrate:
Relation of nonlinear elastic behavior
Date:
22/06/95
Author (S):
E. LORENTZ
Key:
R5.03.20-A
Page:
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2.3
Virtual work of the interior efforts
We will not give here a demonstration of the expressions presented. For that, one will be able
to defer to [bib1] and [R7.02.03]. There still, we choose the initial configuration like
configuration of reference, to express the work of the interior efforts:
SW. v = - F S
v
D
int
ik kl I L, O
O
v
with: v
I
I L, = Xl
In the optics of a resolution by a method of Newton, it is important to also express
variation second of the virtual work of the interior efforts, namely:
2Wint. U. v = - U.S.
I, K kl v D
I L,
O
geometrical rigidity
O
2
- U F
F
I Q, IP
jk
v
D
elastic rigidity
!
E
J L,
O
pq Ekl
O
2.4 Formulation
variational
We now have at our disposal all the ingredients to write the variational formulation
problem:
W
v + SW
int.
ext. v = 0
v acceptable Cinématiquement
3 Bibliography
[1]
LORENTZ E.: A nonlinear relation of behavior hyperelastic. Note intern EDF
DER, HI-74/95/011/0, 1995.
Handbook of Référence
R5.03 booklet: Nonlinear mechanics
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