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Code_Aster
®
Version
4.0
Titrate:
Rate of refund of energy in thermo non-linear elasticity
Date:
08/10/97
Author (S):
E. SCREWS
Key:
R7.02.03-B
Page:
1/12
Manual of Reference
R7.02 booklet: Breaking process
HI-75/97/025/A
Organization (S):
EDF/IMA/MN
Manual of Reference
R7.02 booklet: Breaking process
R7.02.03 document
Rate of refund of energy in thermo elasticity
non-linear
Summary:
One presents the calculation of the rate of refund of energy by the method théta in 2D or 3D for a problem
thermo non-linear rubber band. The relation of nonlinear elastic behavior is described in [R5.03.20].
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Code_Aster
®
Version
4.0
Titrate:
Rate of refund of energy in thermo non-linear elasticity
Date:
08/10/97
Author (S):
E. SCREWS
Key:
R7.02.03-B
Page:
2/12
Manual of Reference
R7.02 booklet: Breaking process
HI-75/97/025/A
Contents
1 Calculation of the rate of refund of energy by the method théta in thermo nonlinear elasticity ......... 3
1.1 Relation of behavior .............................................................................................................. 3
1.2 Potential energy and relations of balance ....................................................................................... 4
1.3 Lagrangian expression of the rate of refund of energy .......................................................... 4
1.4 Establishment of G in thermo nonlinear elasticity in
Code_Aster
....................................... 5
1.5 Warning .................................................................................................................................. 7
2 Calculation of the rate of refund of energy by the method théta in great transformations ................ 7
2.1 Relation of behavior .............................................................................................................. 7
2.2 Potential energy and relations of balance ....................................................................................... 8
2.3 Lagrangian expression of the rate of refund of energy in thermo non-linear elasticity and
in great transformations ............................................................................................................ 9
2.4 Establishment in
Code_Aster
.................................................................................................. 11
2.5 Restriction ...................................................................................................................................... 11
3 Bibliography ........................................................................................................................................ 12
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Code_Aster
®
Version
4.0
Titrate:
Rate of refund of energy in thermo non-linear elasticity
Date:
08/10/97
Author (S):
E. SCREWS
Key:
R7.02.03-B
Page:
3/12
Manual of Reference
R7.02 booklet: Breaking process
HI-75/97/025/A
1
Calculation of the rate of refund of energy by the method
théta in thermo nonlinear elasticity
1.1
Relation of behavior
One considers a fissured solid occupying the field
space R
2
or R
3
. That is to say:
·
U
the field of displacement,
·
T
the field of temperature,
·
F
the field of voluminal forces applied to
,
·
G
the field of surface forces applied to a part
S
of
,
·
U
the field of displacements imposed on a part
S
D
of
.
F
G
S
Sd
The behavior of the solid is supposed to be elastic non-linear such as the relation of behavior
coincide with the elastoplastic law of Hencky-Von Mises (isotropic work hardening) in the case of one
loading which induces a radial and monotonous evolution in any point. This model is selected in
controls
CALC_G_THETA
[U4.63.03] and
CALC_G_LOCAL
[U4.64.04] via the key word
RELATION:“ELAS_VMIS_LINE”
or
“ELAS_VMIS_TRAC”
under the key word factor
COMP_ELAS
[R5.03.20].
One indicates by:
·



the tensor of deformations,
·
°
°
°
°
the tensor of the initial deformations,
·



the tensor of the stresses,
·
°
°
°
°
the tensor of the initial stresses,
·
(
)
,
,
,
, °
°
°
°, °°°°,
T
density of free energy.



is connected to the field of displacement
U
by:



U
()
=
1
2 U
I, J
+
U
J, I
(
)
Density of free energy
(
)
,
,
,
, °
°
°
°, °°°°,
T
is a convex and differentiable, known function for one
state given [R5.03.20 éq 3]. The relation of behavior of material is written in the form:
(
)
ij
ij
T
=
, °°°°, °°°°,
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Code_Aster
®
Version
4.0
Titrate:
Rate of refund of energy in thermo non-linear elasticity
Date:
08/10/97
Author (S):
E. SCREWS
Key:
R7.02.03-B
Page:
4/12
Manual of Reference
R7.02 booklet: Breaking process
HI-75/97/025/A
It derives from the potential free energy. For this hyperelastic relation of behavior, one knows
to give a direction to the rate of refund of energy within the framework of the global solution in mechanics of
rupture. It is not the case for a plastic relation of behavior.
1.2
Potential energy and relations of balance
One defines spaces of the fields kinematically acceptable
V
and
V
O
.
V
=
v
acceptable,
v
=
U
on
S
D
{
}
V
O
=
v
acceptable,
v
=
O
on
S
D
{
}
With the assumptions of the paragraph [§1.1] (with



° = ° =
0
), relations of balance in formulation
weak are:
U
V
ij
v
I, J
D
=
F
I
v
I
D
+
G
I
S
v
I
D
,
v
V
O
They are obtained by minimizing the total potential energy of the system:
W v
()
=
v
()
, T
(
)
D
-
F
I
v
I
D
-
G
I
S
v
I
D
The demonstration is identical to that in linear elasticity [R7.02.01 §1.2].
1.3
Lagrangian expression of the rate of refund of energy
That is to say
m
the unit normal with
O
located in the tangent plan at
in
.



m
O
That is to say the field



such as:



====
µ
µ
µ
µ
such as
µ
µ
µ
µ
N
=
0
on
{
}
while noting
N
the normal with
.
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Code_Aster
®
Version
4.0
Titrate:
Rate of refund of energy in thermo non-linear elasticity
Date:
08/10/97
Author (S):
E. SCREWS
Key:
R7.02.03-B
Page:
5/12
Manual of Reference
R7.02 booklet: Breaking process
HI-75/97/025/A
The rate of refund of energy
G
is solution of the variational equation:
()
G
dS
O
=









m
G
,
where
()
G



is defined by:
()
G



=
-
-
+
-




-
-
-




+
+
+
+
-




-



°
°
°
ij
I p
p J
K K
K
K
ij
ij
ij K K
ij
ijth
ij
ij K K
I
I
K K
I K
K
I
I K
S
K
I
I
I
K K
K
K
ij
S
J
U
T T
D
D
F U
F
U D
G
U
G U
N N
D
N
D
,
,
,
,
,
,
,
,
,
,
1
2
1
2
U
D
I K
K
,
The demonstration is identical to that of the calculation of
G
in linear elasticity [R7.02.01]. The expression is
the same one, postprocessing is thus identical.
1.4 Establishment of G in thermo nonlinear elasticity in
Code_Aster
The types of elements and loadings, the environment necessary are the same ones as for
establishment of
G
in thermo linear elasticity [R7.02.01 §2.4].
For the calculation of the various terms of
G



()
, in a given state, one recovers the density of free energy
,
T
()
, deformations



and stresses



, calculated for the relation of behavior
non-linear (routine
NMELNL
).
It is supposed that



° = ° =
0
(identical term in thermo or not-linear linear elasticity). Density
of free energy is written then [R5.03.20 §1.5]:
·
in thermo linear elasticity:
()
(
)
(
)
µ
, T
K
T T
kk
ref.
eq
eq
ijD ijD
ij
kk ij
ij
kk ij
eq
ij ij
kk
=
-
-
+
=
=
-




-




=
-




1
2
3
2
3
3
2
3
2
1
3
1
3
3
2
1
3
2
2
2
2
2
with
·
in thermo non-linear elasticity
(
)
2
µ
eq
y
:
()
(
)
(
)
()
()
()
()
µ
, T
K
T T
R p
R S ds
kk
ref.
eq
p
eq
=
-
-
+
+
1
2
3
1
6
2
2
0
with
()
()
R p
eq
: function of work hardening.
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Code_Aster
®
Version
4.0
Titrate:
Rate of refund of energy in thermo non-linear elasticity
Date:
08/10/97
Author (S):
E. SCREWS
Key:
R7.02.03-B
Page:
6/12
Manual of Reference
R7.02 booklet: Breaking process
HI-75/97/025/A
For a linear isotropic work hardening (
RELATION: “ELAS_VMIS_LINE”
) one a:
()
()
()
(
)
()
()
(
)
R p
p E E
E
E
p has
p
has
has
E E
E E
With p
R S ds
p has
p
p
p has
With p
p
R p
y
T
T
y
eq
y
T
T
YP
p
y
y
y
=
+
-
=
+
=
-
+
=
-
=
=
+
=
+
+
=
+
µ
3
1
2
1
2
2
2
2
0
with
Postprocessing is then identical to the problem in linear elasticity except for the term
thermics:
THER
,
= -
T T
K
K
If coefficients of BLADE
T
()
and
µ
T
()
are independent of the temperature, this term is null.
In the contrary case, it is necessary to calculate
T



, T
()
at a given moment.
For a linear isotropic work hardening, one a:
()
()
(
)
(
)
()
(
) (
)
()
()
()
()
()
()
()
()
()
()
(
)
()
()
()
(
)
µ
µ
µ
µ
T
T
dK T
dT
T T
K
D T
dT
T T
T T
R p
Dr. P
dT
D T
dT R p
dA p
dT
Dr. p
dT
D
T
dT
D has T
dT
p has D p T
dT
D has T
dT
E E
T
dT
E
D E T
dT
E
D p T
dT
has
kk
ref.
ref.
kk
ref.
y
T
T
T
y
,
=
-
-
-
+
-








-
-
+
-




+
=
+
+
=
-
-




=
+
-
1
2
3
3
3
6
2
1
1
3
2
2
2
2
2
with
(
)
()
()
(
)
()
()
()
(
)
()
()
µ
µ
eq
y
y
p
y
D T
dT
dA T
dT
D has
T
dT
dA p
dT
dp T
dT
R
p D
T
dT
dRp T
dT
3
3
1
2
1
2
+




-
+




=
+
+
+






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Code_Aster
®
Version
4.0
Titrate:
Rate of refund of energy in thermo non-linear elasticity
Date:
08/10/97
Author (S):
E. SCREWS
Key:
R7.02.03-B
Page:
7/12
Manual of Reference
R7.02 booklet: Breaking process
HI-75/97/025/A
1.5
Warning
Caution!
By definition, in the general case:
, T
()
:



Although it is possible to carry out an elastoplastic calculation followed by the calculation of
G
in elasticity not
linear, it should well be known that that does not have any thermodynamic direction and that it is normal that the result
depends on the field
.
2
Calculation of the rate of refund of energy by the method
théta in great transformations
One extends the relation of behavior of [§1] to great displacements and great rotations, in
the measurement where it derives from a potential (hyperelastic law). This functionality is started by
key word
DEFORMATION: “GREEN”
in the controls
CALC_G_THETA
[U4.63.03] and
CALC_G_LOCAL
[U4.64.04].
2.1
Relation of behavior
One indicates by:
·
E
the tensor of deformations of Green-Lagrange,
·
S
the tensor of the stresses of Piola-Lagrange called still second tensor of
Piola-Kirchoff,
·
E
()
density of energy internal.
The behavior of the solid is supposed to be hyperelastic, namely that:
·
E
is connected to the field of displacement
U
measured compared to the configuration of reference
O
by:
E
ij
U
()
=
1
2 U
I, J
+
U
J, I
+
U
K, I
U
K, J
(
)
·
S
is connected to the tensor of the stresses of cauchy
T
by:
S
ij
=
det F
()
F
ik
-
1
T
kl
F
jl
-
1
F
being the gradient of the transformation which makes pass from the configuration of reference
O
with
current configuration
, connected to displacement by:
F
ij
=
ij
+
U
I, J
(
)
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Code_Aster
®
Version
4.0
Titrate:
Rate of refund of energy in thermo non-linear elasticity
Date:
08/10/97
Author (S):
E. SCREWS
Key:
R7.02.03-B
Page:
8/12
Manual of Reference
R7.02 booklet: Breaking process
HI-75/97/025/A
The relation of behavior of a material hyperelastic is written in the form:
S
ij
=
E
ij
=
E
jj
=
S
ji
This relation describes a non-linear elastic behavior, similar to that of [§1.1]. It offers
possibility of dealing with the problems of breaking process without integrating plasticity into it. And in
case of a monotonous radial loading, it makes it possible to obtain strains and stresses of
structure similar to those which one would obtain if the material presented an isotropic work hardening.
material hyperelastic A a reversible mechanical behavior, i.e. any cycle of
loading does not generate any dissipation.
This model and selected in the controls
CALC_G_THETA
[U4.63.03] and
CALC_G_LOCAL
[U4.64.04]
via the key word:
RELATION: “ELAS”
for an elastic relation “linear”, i.e. the relation between the deformations and them
stresses considered is linear,
RELATION: “ELAS_VMIS_LINE”
or
“ELAS_VMIS_TRAC”
for a “nonlinear” relation of elastic behavior (law of HENCKY-VON MISES with
linear isotropic work hardening).
Such a relation of behavior makes it possible in any rigor to take into account the large ones
deformations and of great rotations. However, one confines oneself with small deformations to ensure
the existence of a solution and to be identical to an elastoplastic behavior under one
monotonous radial loading [R5.03.20 §2.1].
2.2
Potential energy and relations of balance
The loading considered is reduced to a nonfollowing surface density
R
applied to a part
O
edge of
O
(assumption of the dead loads [R5.03.20 §2.2]).
One defines a space of the fields kinematically acceptable
V
:
V
=
v
acceptable,
v
=
0
on
O
{
}
The relations of balance in weak formulation are:
F
ik
O
S
kj
v
I, J
D
=
R
I
v
I
D
,
v
V
They can be obtained by minimizing the total potential energy of the system:
W v
()
=
v
()
()
O
D
-
R
I
v
I
D
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Code_Aster
®
Version
4.0
Titrate:
Rate of refund of energy in thermo non-linear elasticity
Date:
08/10/97
Author (S):
E. SCREWS
Key:
R7.02.03-B
Page:
9/12
Manual of Reference
R7.02 booklet: Breaking process
HI-75/97/025/A
Indeed, if this functional calculus is minimal for the field of displacement
U
, then:
W
=
E
ij
O
E
ij
D
-
R
I
v
I
D
=
0,
v
V
=
S
ij
O
1
2
v
I, J
+
v
J, I
+
v
p, I
U
p, J
+
U
p, I
v
p, J
(
)
D
-
R
I
v
I
D
=
S
ij
O
IP
+
U
p, I
(
)
v
p, I
D
-
R
I
v
I
D
=
F
pi
O
S
ij
v
p, J
D
-
R
I
v
I
D
=
F
ik
O
S
kj
v
I, J
D
-
R
I
v
I
D
=
0,
v
V
We thus find the equilibrium equations and the relation of behavior while having posed:
S
E
ij
ij
=
2.3 Lagrangian expression of the rate of refund of energy in
thermo non-linear elasticity and in great transformations
By definition, the rate of refund of energy
G
is defined by the opposite of derived from energy
potential with balance compared to the field
[bib1]. It is calculated by the method théta, which is one
Lagrangian method of derivation of the potential energy [bib4] and [bib2]. One considers
transformations
F
: M
M
+
M
()
field
O
in a field
who correspond to
propagations of the fissure. With these families of configuration of reference thus defined
correspond of the families of deformed configurations where the fissure was propagated. The rate of
restitution of energy
G
is then the opposite of derived from the potential energy
W U
()
()
with balance
compared to the initial evolution of the bottom of fissure
:
G
= -
W U
()
()
D
=
0
One notes as in [bib 4] par. Lagrangian derivation in a virtual propagation of fissure
of speed
. That is to say
, M
(
)
an unspecified field (
positive reality and
M
belonging to the field
O
), we will note:
()
()
(
)
,
,
!
M
M
=
=
=
F
and
0
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Code_Aster
®
Version
4.0
Titrate:
Rate of refund of energy in thermo non-linear elasticity
Date:
08/10/97
Author (S):
E. SCREWS
Key:
R7.02.03-B
Page:
10/12
Manual of Reference
R7.02 booklet: Breaking process
HI-75/97/025/A
Potential energy definite on
is brought back on
O
,
R
is supposed to be independent of
,
derivation compared to the parameter of propagation
is then easy and the rate of refund of
energy in this propagation is solution of the variational equation:
()
G
dS
O
=









m
G
,
with:
()
()
(
)
()
-
=
+
-
+
+
-




G
E
E
,
,
!
,
,
,
T
T
D
R U
R
U
R U
N N
D
O
K K
I
I
I K
K
I
I
I
K K
K
K
“#
$ %
$
However:
()
(
)
E,
!
!
T
E E
T T
ij
ij
=
+
“#
$ %
$
Thereafter, we will consider only the term
E E
ij
ij
!
, the thermal term being treated the same one
way that into small displacement [R7.02.01].
And according to proposal 2 of [bib4]:
(
)
(
)
!
!
!
!
!
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
E
U
U
U
U
U
U
U
U
U
U
U
U
ij
I J
J I
K I
K J
K I
K J
I p
p J
J p
p I
K p
p I
K J
K I
K p
p J
=
+
+
+
-
+
+
+
1
2
1
2
One can eliminate
!U
expression of
G
as in small deformations by noticing that
!U
is
kinematically acceptable (cf [bib3] for the problems of regularity) and by using the equation
of balance:
()
(
)
E
-
=
“# %
O
D
R U D
I
I
!
(
)
(
)
(
)
E
U
U
U
U
U
U
D
R U D
E
U
U
U
U
U
U
D
S U
U
U
D
S
U
ij
I J
J I
K I
K J
K I
K J
I
I
ij
I p
p J
J p
p I
K p
p I
K J
K I
K p
p J
ij
I p
p J
K I
K p
p J
ij
ki
O
O
O
O

+
+
+
-
-
+
+
+
= -
+
= -
+
1
2
1
2
!
!
!
!
!
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
(
)
K I
K p
p J
ij
ki
K p
p J
ik
kj
I p
p J
U
D
S F U
D
F S U
D
O
O
,
,
,
,
,
,
,
= -
= -

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Code_Aster
®
Version
4.0
Titrate:
Rate of refund of energy in thermo non-linear elasticity
Date:
08/10/97
Author (S):
E. SCREWS
Key:
R7.02.03-B
Page:
11/12
Manual of Reference
R7.02 booklet: Breaking process
HI-75/97/025/A
Finally, one obtains:
()
(
)
()
G






=
-
+
+
-




F S U
D
R
U
R U
N
D
ik
kj
I p
p J
K K
I K
K
I
I
I
K K
K
O
,
,
,
,
,
E
The expression supplements for the following loadings:
·
nonfollowing surface density
R
applied to a part
edge of
O
,
·
nonfollowing voluminal density
F
applied to the field
,
and by taking account of thermics:
()
(
)
()
,
,
,
,
,
,
,
,
G






=
-
-
+
+
+
+
-




F S
U
T T
D
F U
F
U D
R
U
R U
N N
D
ik
kj
I p
p J
K K
K
K
I
I
K K
I K
K
I
I K
K
I
I
I
K K
K
K
O
O
E
2.4
Establishment in Code_Aster
The comparison of the formulas of
G



()
[§1.3] and [§1.4] shows that the terms of
G



()
are very
close relations. The introduction of the great transformations requires little amendment in postprocessing.
The presence of the key word
DEFORMATION:“GREEN”
under the key word factor
COMP_ELAS
controls
CALC_G_THETA
and
CALC_G_LOCAL
indicate that it is necessary to recover the tensor
stresses of Piola-Lagrange
S
and the gradient of the transformation
F
(routines
NMGEOM
and
NMELNL
).
The types of finite elements are the same ones as in linear elasticity [R7.02.01 §2.4]. They are them
isoparametric elements 2D and 3D.
The supported loadings are those supported in linear elasticity provided that it is
dead loads: typically an imposed force is a dead load while the pressure is one
following loading since it depends on the orientation of surface, therefore of the transformation.
2.5 Restriction
With the relation of behavior specified to the 2, there is a formulation of
G
valid for the large ones
deformations for materials hyper-rubber bands, but… if one wishes a coherence with
actual material which, let us recall it, is elastoplastic, it is imperative to be confined with deformations
small, displacements and rotations being able to be large.
Conditions of loadings proportional and monotonous, essential to ensure
coherence of the model with actual material, lead to important restrictions of the field of
capable problems being dealt with by this method (thermics in particular can lead to
local discharges). It can thus be a question only of one palliative solution before being in measurement of
to give a direction to the rate of refund of energy within the framework of plastic behaviors.
background image
Code_Aster
®
Version
4.0
Titrate:
Rate of refund of energy in thermo non-linear elasticity
Date:
08/10/97
Author (S):
E. SCREWS
Key:
R7.02.03-B
Page:
12/12
Manual of Reference
R7.02 booklet: Breaking process
HI-75/97/025/A
3 Bibliography
[1]
BUI H.D., fragile Breaking process, Masson, 1977.
[2]
DESTUYNDER pH., DJAOUA Mr., On an interpretation of the integral of Rice in theory of
brittle fracture, Mathematics Methods in the Applied Sciences, vol. 3, pp. 70-87, 1981.
[3]
GRISVARD P., “Problems in extreme cases in the polygons”, Instructions - EDF - Bulletin of
the Management of the Studies and Search, Series C, 1, 1986 pp. 21-59.
[4]
MIALON P., “Calculation of derived from a size compared to a bottom of fissure by
method théta ", EDF - Bulletin of the Management of the Studies and Search, Series C, n3, 1988,
pp. 1-28.
[5]
MIALON P., Study of the rate of refund of energy in a direction marking an angle
with a fissure, notes intern EDF, HI/4740-07, 1984.
[6]
SIDOROFF F., Course on the great deformations, School of summer, Sophia-Antipolis, 8 with
September 10, 1982.
[7]
LORENTZ E., a nonlinear relation of behavior hyperelastic, internal Note EDF
DER HI-74/95/011/0, 1995.