Code_Aster ®
Version
4.0
Titrate:
Rate of refund of energy in thermo elastoplasticity
Date:
25/04/97
Author (S):
G. DEBRUYNE, E. SCREWS
Key:
R7.02.07-A
Page:
1/8
Organization (S): EDF/IMA/MN
Handbook of Référence
R7.02 booklet: Breaking process
R7.02.07 document
Rate of refund of energy in thermo elastoplasticity
Summary:
One presents the calculation of the rate of refund of total mechanical energy by the method théta in 2D or 3D
for an elastoplastic thermo problem. The relations of elastoplastic thermo behavior are described
in detail in the document [R5.03.02].
This rate of refund of total mechanical energy makes it possible to analyze the situations of loadings not
monotonous of the defect, for irreversible material behaviors.
Let us note that the problem of the elastoplastic thermo rupture is a delicate problem. It is advised of
to consult the references before a first use.
Caution:
The defect must be modelled by a notch and not by a fissure [§5].
Handbook of Référence
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Code_Aster ®
Version
4.0
Titrate:
Rate of refund of energy in thermo elastoplasticity
Date:
25/04/97
Author (S):
G. DEBRUYNE, E. SCREWS
Key:
R7.02.07-A
Page:
2/8
Contents
1 Choice of the formulation of the rate of refund of energy in thermo elastoplasticity .............................. 3
2 Relation of behavior ..................................................................................................................... 4
3 Lagrangienne Expression of the rate of refund of energy in thermo elastoplasticity ......................... 6
4 Establishment in Code_Aster ........................................................................................................... 8
5 Restrictions ............................................................................................................................................ 8
6 Bibliography .......................................................................................................................................... 8
Handbook of Référence
R7.02 booklet: Breaking process
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Code_Aster ®
Version
4.0
Titrate:
Rate of refund of energy in thermo elastoplasticity
Date:
25/04/97
Author (S):
G. DEBRUYNE, E. SCREWS
Key:
R7.02.07-A
Page:
3/8
1
Choice of the formulation of the rate of refund of energy in
thermo elastoplasticity
One considers a notched elastic solid occupying the field of space R2 or R3. That is to say:
·
U the field of displacement,
·
T the field of temperature,
·
the tensor of the constraints,
·
F the field of the voluminal forces applied to,
·
G the field of the surface forces applied to a part S of,
·
U the field of displacements imposed on a Sd part of.

F
S
G
Sd
In thermo or not-linear linear elasticity, the rate of refund of energy G is defined by the opposite
derivative of the potential energy compared to the field [bib1]:
W
G = -
Total potential energy with the balance of the system is:
W (U) =
D - F U D - G U



D
I
I
I
I



S
where is the density of free energy. In elasticity, is equal to the density of free energy elastic
[R7.02.01]
One extends this definition for the elastoplastic thermo problem, while choosing to replace by
~
total mechanical energy. This choice is justified in the document [bib2].
~
is a function of the following variables of state:
·
the tensor of the total deflections,
·
p the tensor of the plastic deformations,
·
T the field of temperature,
·
p the variable interns scalar isotropic work hardening (cumulated plastic deformation),
·
one or more tensorial or scalar variables of kinematic work hardening.
Handbook of Référence
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Code_Aster ®
Version
4.0
Titrate:
Rate of refund of energy in thermo elastoplasticity
Date:
25/04/97
Author (S):
G. DEBRUYNE, E. SCREWS
Key:
R7.02.07-A
Page:
4/8
~
(
D
T
T
D
, p
T
, T,
p) = () D =

(, p, p T,) + Dp () D + S T () D



0
D
O
O

D

where
is the density of free energy

Dp is the density power density of plastic dissipation
S
(T) be
T density of entropy

~
It is noted that is the density of increased free energy of dissipated voluminal energy
T
D
plastically during all the evolution, and to which energy is added
S
T () D
O D
(contribution of the temperature to the variation of free energy).
Caution:
One limits oneself to a notched solid (cf [§5]).
2
Relation of behavior
The behavior of the solid is supposed to be thermo elastoplastic associated a criterion of Von Mises
with isotropic or kinematic work hardening. This type of behavior is currently treated in
operator STAT_NON_LINE [U4.32.01] under the key word factor COMP_INCR. Treated relations
in this document are:
VMIS_ISOT_LINE: Von Mises with linear isotropic work hardening,
VMIS_ISOT_TRAC: Von Mises with isotropic work hardening given by a traction diagram,
VMIS_CINE_LINE: Von Mises with linear kinematic work hardening.
For more details, to consult the documents [R5.03.02] and [U4.32.01].
is connected to the field of displacement U by:
(
1
U) = (U, + U,)
2
I J
J I
The density of free energy is written:
(, p,) = E (, p
T p
, T) + H (, p, T) + Z (T)
where
H is the density of energy of work hardening
Z an arbitrary function of the temperature
E density of thermo energy elastic defined by:
E (
1
1
, p, T) = E
p
p
ij ij =
Aijkl


,
2
2
(ij - ij - (T - ref.
T) ij) (kl - -
-
kl
(T ref.
T) kl)
with the thermal dilation coefficient, and (Aijkl) the tensor of elasticity.
In the particular case where there no was plastic evolution, one finds the expression of the density
of elastic energy for a thermo problem elastic linear [R7.02.01 §1.1].
Handbook of Référence
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Code_Aster ®
Version
4.0
Titrate:
Rate of refund of energy in thermo elastoplasticity
Date:
25/04/97
Author (S):
G. DEBRUYNE, E. SCREWS
Key:
R7.02.07-A
Page:
5/8
The free energy of work hardening H is deduced from:
H (, p T) = R (, p T), for isotropic work hardening where R (, p T) is the radius of the surface of load
p
H
and
(, T) = X (, T), for kinematic work hardening where X (, T) is the translation of surface

of load in the space of the constraints (in the case of linear kinematic work hardening = p)
For the relation of behavior of Von Mises with linear isotropic work hardening:
E E
R (,
p T)
T
=
p
E - AND

AND
y
E

Traction diagram
Characteristics of the material (Module of Young E and D_SIGM_EPSI: AND) can depend on
temperature [R5.03.02 §3.2.1].
Plastic dissipation for a law of behavior of Von Mises checks:
T Dp () D p

=
O
y
where is the initial linear elastic limit there.
~
Finally the density of total mechanical energy is written:
~ (
T
T
, p, T,
p) = (, p, p, T) + Dp


() D + S!T () D



O
O
= 1
p


p
ij (ij - ij - (T - ref.
T) ij) + R
2
(, p T) dp + X
(, T) D + p
O
y
O
For a linear isotropic work hardening:
~
~
= (
1
1 E E
, p, T, p) =
p
T
2
ij


2
(ij - ij - (T - ref.
T) ij) +
p +
p
2nd -
y
T
E
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Code_Aster ®
Version
4.0
Titrate:
Rate of refund of energy in thermo elastoplasticity
Date:
25/04/97
Author (S):
G. DEBRUYNE, E. SCREWS
Key:
R7.02.07-A
Page:
6/8
3
Lagrangienne expression of the rate of refund of energy
in thermo elastoplasticity
The rate of refund is calculated in Code_Aster by the method théta [R7.02.01 §1.3]. One notes by
!Q the Lagrangian derivative of the quantity Q in a virtual propagation of the notch of,
being a small real parameter and a field of vector representing the direction of propagation of
Q
the notch (one thus has!
Q (X (),

) =
+ Q.).



The rate of refund of total mechanical energy in this propagation is:



-
~
~
G () = -

F U
I
I + (- F U
I
I) K, K D








-
G U
I
I +

G U
I
I K, K -
nk D


N
K

S
~
~
~
~
~
~
However!
(






p


p



,
ij
, T,
p
ij
I) =
. !ij +
.
T
p
T
p
!ij +
! +
! +
. !I + S!
ij
ij
T
p
I


~
=

ij
ij
~

= -

p
ij
ij
with ~
= - S

~T


= R (p, T) + p
p
Y
~
= Xi (, T
I
)

I
Handbook of Référence
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Code_Aster ®
Version
4.0
Titrate:
Rate of refund of energy in thermo elastoplasticity
Date:
25/04/97
Author (S):
G. DEBRUYNE, E. SCREWS
Key:
R7.02.07-A
Page:
7/8

~
~


that is to say - G () =

p
!
~
ij!ij - ij!
!
ij +
T + R p!
I
D


+
+

T


K, K
I
+ traditional terms (F, G)

1 ui, J
U J, I 1
!ij =
+
- (ui, kk, J + uj, kk, I)
2


2
with
p

p
ij
p
!ij =
+
ij, K

K



T
T! =
+ T

, K K


p
p! =
+

p, K K





ij
! ij =
+ ij, K


K

One can eliminate!U of the expression of G () by noticing that!U is kinematically acceptable and
by using the equilibrium equation [R7.02.01 §1.3]. In the same way, terms
p
p

p
~ T
T
(-

.
+ R
+ X.
) + are eliminated as well as the terms
.
- S
.



Y
T

The following expression then is obtained:
~
-
~
p

G () =

K, K - ij iu, K K, J - ij K +
T K


ij, K
,


K
T
~
+ (

R (,
p T) + y), km No K +
ij, K D

K
ij
+ traditional terms (F, G)
and finally:
~
~


~
G () = -



p
K, K + ij
I
U, K

K, J -
,
T K + (R+ y) p K
ij K ij D


,
, -


+
ij,
T

K
K

ij


+ traditional terms (F, G)
~
p

For a radial and monotonous loading: ij ij K = (R+ y) p
,
, K +
and one finds

ij, K
ij
the expression of G () in nonlinear thermoelasticity [R7.02.03].
Handbook of Référence
R7.02 booklet: Breaking process
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Code_Aster ®
Version
4.0
Titrate:
Rate of refund of energy in thermo elastoplasticity
Date:
25/04/97
Author (S):
G. DEBRUYNE, E. SCREWS
Key:
R7.02.07-A
Page:
8/8
4
Establishment in Code_Aster
The comparison of the formulations of G () in thermo linear elasticity and thermo elastoplasticity
show that the terms of the two formulations differ only by terms from transport from
internal variables.
The presence of the key word factor COMP_INCR, and the key word factor RELATION: “VMIS_ISOT_LINE”
(or “VMIS_ISOT_TRAC” or “VMIS_CINE_LINE”) indicates that it is necessary to recover the field
displacements U, constraints, and characteristics of elastoplastic material. It is
also necessary to recover the fields of the tensors of plastic deformation by the operator
CALC_ELEM [U4.61.02].
The types of finite elements which support these options are the same ones as in elasticity [R7.02.01 §2.4].
They are the isoparametric elements 2D and 3D.
The supported loadings are the same ones as in the elastic case.
5 Restrictions
Caution:
This formulation of G for an elastoplastic thermo relation is valid only for one
solid notched and not for a fissured solid. One will choose for example (but the user will be able
to choose its own regular notch):
OK
OK
NON
Indeed, the principal difficulty in the establishment of this formulation is impossibility of showing
the existence of derived from total mechanical energy for a field comprising a fissure, and this
mainly by the absence of knowledge of the singularities of the fields in plasticity. For
to circumvent the problem, one regularizes the field by representing the defect in the form of notch. For
more details, it is advised to consult [bib2].
The validation of this formulation is carried out in test SSNP102 [V6.03.102] - Calcul of the rate of
restitution of energy for an elastoplastic problem.
6 Bibliography
[1]
BUI H.D.: Fragile breaking process, Masson, 1977.
[2]
DEBRUYNE G.: Proposal for an energy parameter of ductile rupture in
thermo plasticity, HI-74/95/027/0, 23/02/96
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