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
Organization (S): EDF/IMA/MN
Handbook of Référence
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
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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
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 Lagrangienne 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 Lagrangienne 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 ®
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Titrate:
Rate of refund of energy in thermo non-linear elasticity
Date:
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Author (S):
E. SCREWS
Key:
R7.02.03-B
Page:
3/12
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 of space R2 or R3. 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 Sd part of.
S

F
G
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
commands 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 constraints,
·
° the tensor of the initial constraints,
·
(, °, °°, T) density of free energy.
is connected to the field of displacement U by:
1
U
() =
U
(
)
2 I, J + uj, I
Density of free energy (,
, °, °°, T) are 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 =
(, °, °, T)
ij
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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:
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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 Vo.
V =
v
{acceptable, v = U on Sd}
Vo =
v
{acceptable, v = O on Sd}
With the assumptions of the paragraph [§1.1] (with ° = ° = 0), relations of balance in formulation
weak are:
U
V

ij v

v
v

I, J D
=
fi

I D +
gi
S
I D, v Vo
They are obtained by minimizing the total potential energy of the system:
W v
() =

((v), T) D F
v
v

I

I D -
gi
S
I D
The demonstration is identical to that in linear elasticity [R7.02.01 §1.2].
1.3
Lagrangienne 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.

O
m
That is to say the field such as:
=
µ
{such as µn = 0 on}
by noting N the normal with.
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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
The rate of refund of energy G is solution of the variational equation:
G m dS
G (),


=

O

where G () is defined by:

G () =
ij iu, p p, J - K, K -

T, K D


K
T
1
1
+

HT
ij -
°
°


°
,
,



ij
2

ij kk - ij - ij -



ij
2

ij K kd
+
fi I
the U.K., K +

fi, K K I
U D




+
gi, K U + G the U.K., K -
nk

D
S
K
I
I
I
nk

- ij N U D
S
J
I, K
K
D
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), strains 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:
1
2

(, T) = K
3
2
2 (
- (T - T
kk
ref.) +
eq
3
with

3
2
D D
3
1

1
eq =
ij ij =
ij - kkij ij
kkij
2
2

3


-



3



3
2

1
2
eq =
ijij - kk
2

3


· in thermo non-linear elasticity (2µ eq y):
() 1
2
1
2
3
2 (
(
)

p
, T
K
T T
R p
eq
=
-
-
+
+
R S ds
kk
ref.

((eq)
()



()
0
with R ((
p eq): function of work hardening.
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Titrate:
Rate of refund of energy in thermo non-linear elasticity
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R7.02.03-B
Page:
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For a linear isotropic work hardening (RELATION: “ELAS_VMIS_LINE”) one a:
E E
R (p) = + p
T
= + p has
y
E - E
y
T
eq - y
E E
p =
with A
T
=
µ
3 + has
E - AND
(
p
1
1
p
With p) =
R (S) ds = + has p2 = p

YP
y
+
0
(+a p
y
)
2
2
2
(
p
With p) = (+ R
y
(p)
2
Postprocessing is then identical to the problem in linear elasticity except for the term
thermics:

THER = -
,
T K
T
K
If coefficients of LAME (T) and µ T
() are independent of the temperature, this term is null.

In the contrary case, it is necessary to calculate

(, T) at a given moment.
T
For a linear isotropic work hardening, one a:
(
1 dK T

D


T
, T)
()
=
kk -
3 T - T
- 3K
ref.
+
T - Tréf kk -
3 T - T
T
2 dT (
(
)
() (
) (
ref.)


dT




R (p)
Dr. (P) (T)
dA p
+

-
R
2
(p)
()

dT
dT

+ dT
Dr. (p) D y (T) D has (T)
D (
p T)
with
=
+
p + has
dT
dT
dT
dT
D has (T)
1
dET (T)
D E
2
(T) 2
=
E -
E
dT
(
2
T
E - E




T)
dT
dT
D (
p T)
1

dµ T
dA T
D T
=
(-
y
eq)
()
()
3
+

3
has)
()
dT
(

µ
2
y
3 + has)


dT
dT -
+
dT



dA (p) 1 (
dp T)


=
(
1
D T
dRp T

y


y + RP)
()
()
+ p
+
dT
2 dT
2 dT
dT
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Titrate:
Rate of refund of energy in thermo non-linear elasticity
Date:
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Author (S):
E. SCREWS
Key:
R7.02.03-B
Page:
7/12
1.5
Warning
Caution! By definition, in the general case:

(, T):
Although it is possible to not carry out an elastoplastic calculation followed by the calculation of G in elasticity
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
measurement where it derives from a potential (hyperelastic law). This functionality is started by
key word DEFORMATION: “GREEN” in commands 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 constraints 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:
1
E
(
)
ij U
() =
U
2 I, J + uj, I + the U.K., I the U.K., J
·
S is connected to the tensor of the constraints of cauchy T by:
S
- 1
- 1
ij
= det F
() Fik Tkl Fjl
F being the gradient of the transformation which makes pass from the configuration of reference O to
current configuration, connected to displacement by:
F
(
)
ij
= ij +ui, J
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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
The relation of behavior of a material hyperelastic is written in the form:


Sij =
=
= S
E
ji
ij
Ejj
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 commands 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
constraints 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 of the 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 Skj VI, J D =
R
I VI D, vV

O

They can be obtained by minimizing the total potential energy of the system:
W v
() =

(v ())D - R
v

I I D
O

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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
Indeed, if this functional calculus is minimal for the field of displacement U, then:



W =
E
v

ij D -
IH I D = 0, v V
O E

ij
=
S
(
)
ij

1 v
D - R
v

I, J + vj, I + vp, I up, J + U p, I vp, J
I
I D
O
2

=
S

(
)
ij
v
v

IP + up, I
p, I D -
IH I D
O

=
F
pi S
v

ij vp, J D -
IH I D
O

=
F
ik S
v

kj VI, J D -
IH I D = 0, v V
O

We thus find the equilibrium equations and the relation of behavior while having posed:

Sij =
E
ij
2.3 Lagrangienne 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: MR. M + M
() of the field O in a field which corresponds 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:
W U
(
())
G = -
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 (real positive and M pertaining to the field
O), we will note:

(, M) = (
, F (M)) and
! = =0
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Titrate:
Rate of refund of energy in thermo non-linear elasticity
Date:
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Author (S):
E. SCREWS
Key:
R7.02.03-B
Page:
10/12
Definite potential energy 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 m dS
G (),


=

O

with:

“#
$ %
$



- G () = ((E, T) + (E, T) K, D
K
- IH!ui + IH, K U
K
I + R U
I
I K, K -
nk


D
O


nk
However:

(“#
$ %
$



(E, T) =
!E +
!T
E
ij
T
ij


Thereafter, we will consider only the term
!

E, the thermal term being treated the same one
E
ij
ij
way that into small displacement [R7.02.01].
And according to proposal 2 of [bib4]:
1
!Eij = 2 (!ui, J +!uj, I +!the U.K., U
I
K, + U
J
K, I!
the U.K., J)
1
-




2 (ui, p p, + U
J
J, p
p, + U
I
K, p
p, U
I
K, + U
J
K, U
I
K, p
p, J)
One can eliminate!U of the 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))
D -
R U!

D
I
I
=
O

1 U! , +u! , +u! U
,
, + U
U
!
D

,
O
E 2 (I J
J I
K I
K J
K I
K J)


ij
1
- R U
I!I
D -
U,
, + U,
, + U
U
,
,
, + U
U
D



,
,
,
O
E 2 (I p p J
J p p I
K p p I
K J
K I
K p p J)
ij
= -
S U,
, + U
U
D
ij (I p p J K I, K, p p, J)
O
= -
S

(
K I,)
ij
+ U
U

D

ki
K, p
p, J
O
= -
S
ij kFi the U.K., p p, J D
O
= - ik
F Skj I
U, p p, J D
O
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Rate of refund of energy in thermo non-linear elasticity
Date:
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Author (S):
E. SCREWS
Key:
R7.02.03-B
Page:
11/12
Finally, one obtains:


G () =
F S
,
E,
,
,


(U
) - () D + R U + R the U.K.K -


D
ik
kj
I p
p J
K K
I K
K
I
I
I
O


N
K
The expression supplements for the following loadings:
· nonfollowing surface density R applied to part of the edge of O,
· nonfollowing voluminal density F applied to the field,
and by taking account of thermics:


G () = F S
ik
kj

E


(ui, p p, J) - () K, K -, T D
O
T
K
K

+ F U
I
I K, K + fi, U D
K
K
I

O



+ IH, U
K
K
I + R U
I
I K, K -
nk D


nk

2.4
Establishment in Code_Aster
The comparison of the formulas of G
() of [§1.3] and of [§1.4] shows that the terms of G
() are very
close relations. The introduction of the great transformations requires little modification in postprocessing.
The presence of key word DEFORMATION:“GREEN” under the key word factor COMP_ELAS of
commands CALC_G_THETA and CALC_G_LOCAL indicates that it is necessary to recover the tensor
constraints 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.
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Rate of refund of energy in thermo non-linear elasticity
Date:
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Page:
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3 Bibliography
[1]
BUI H.D., Mécanique of brittle fracture, Masson, 1977.
[2]
DESTUYNDER pH., DJAOUA Mr., Sur 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., “Problèmes in extreme cases in the polygons”, Mode of employment - EDF - Bulletin of
Direction from Etudes and Recherches, Série C, 1, 1986 pp. 21-59.
[4]
MIALON P., “Calcul of derived from a size compared to a bottom of fissure by
method théta ", EDF - Bulletin of Direction of Etudes and Recherches, Série C, n3, 1988,
pp. 1-28.
[5]
MIALON P., Etude 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., Cours on the great deformations, Ecole of summer, Sophia-Antipolis, 8 with
September 10, 1982.
[7]
LORENTZ E., Une nonlinear relation of behavior hyperelastic, internal Note EDF
DER HI-74/95/011/0, 1995.
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