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Code_Aster
®
Version
5.0
Titrate:
Integration of the viscoelastic relations in
STAT_NON_LINE
Date:
02/02/01
Author (S):
P. of BONNIERES
Key:
R5.03.08-A
Page:
1/14
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HI-75/01/001/A
Organization (S):
EDF/MTI/MN
Manual of Reference
R5.03 booklet: Nonlinear mechanics
Document: R5.03.08
Integration of the relations of behavior
viscoelastic in the operator
STAT_NON_LINE
Summary
This document describes the viscoelastic behaviors in the case of the products necessary to the setting in
work of the non-linear algorithm
STAT_NON_LINE
described in [R5.03.01]. Data input of all them
viscoelastic relations of behavior integrated in Aster have in a general way the same form. Only
the way of introducing the main data (the function speed of viscous deformation) varies: it is presented
according to the various key words which make it possible the user to choose the relation of behavior wished.
These quantities are calculated by a method of integration semi-implicit. From the initial state, or from
the moment of preceding calculation, one calculates the stress field resulting from an increment of deformation.
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Titrate:
Integration of the viscoelastic relations in
STAT_NON_LINE
Date:
02/02/01
Author (S):
P. of BONNIERES
Key:
R5.03.08-A
Page:
2/14
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HI-75/01/001/A
Contents
1 Introduction ............................................................................................................................................ 3
2 Relation continues ................................................................................................................................... 3
3 Nature of the function G for each relation of behavior ................................................. 4
3.1 Relation
LEMAITRE
......................................................................................................................... 4
3.2 Relations
ZIRC_CYRA2
and
ZIRC_EPRI
.......................................................................................... 4
4 Integration of the relation of behavior ........................................................................................... 7
4.1 Establishment of the scalar equation for the implicit scheme and with elastic coefficients
constant ......................................................................................................................................... 7
4.2 Resolution of the scalar equation: principle of the routine
ZEROF2
.................................................. 8
4.3 Calculation of the stress at the end of the pitch of current time ............................................................... 10
4.4 Semi-implicit diagram .................................................................................................................. 11
4.5 Taking into account of the variation of the elastic coefficients according to the temperature .......... 12
5 Calculation of the tangent operator ............................................................................................................... 13
6 Bibliography ........................................................................................................................................ 14
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Code_Aster
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Titrate:
Integration of the viscoelastic relations in
STAT_NON_LINE
Date:
02/02/01
Author (S):
P. of BONNIERES
Key:
R5.03.08-A
Page:
3/14
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HI-75/01/001/A
1 Introduction
The tubes of sheath in Zircaloy of the fuel pin of the power stations ITEM present a behavior
strongly viscous mechanics.
Within the framework of the chaining enters Code_Aster and the code of the fuel pin CYRANO3, two
non-linear viscoelastic models specific to Zircaloy were introduced into Code_Aster.
One of them is the model used in Cyrano2 and Cyrano3. Other was developed by the EPRI.
In addition, a model much more general and agent with other materials that Zircaloy has
also introduced summer. It is about the non-linear viscoelasticity of Lemaître, which can be brought back for
certain particular values of the parameters to a relation of viscoelastic behavior of
Norton.
For these three models, one supposes that the material is isotropic. They can be used in 3D, in
plane deformations (
D_PLAN
) and into axisymmetric (
AXIS
).
One presents in this note the equations constitutive of the models and their establishment in
Code_Aster.
2 Relation
continuous
One places oneself on the assumption of the small disturbances and one divides the tensor of the deformations into
an elastic part, a thermal part, a anelastic part (known) and a viscous part.
equations are then:
()
(
)

































early
E
HT
has
v
E
v
eq
eq
T
T
=
+
+
+
=
=
With
!
G
,
~
3
2
with:
: cumulated viscous deformation
!
! : !
=
2
3



v
v
~



: diverter of the stresses
()
~






= -
1
3 Tr
I



eq
: equivalent stress






eq
=
3
2 ~: ~
()
WITH T
: tensor of elasticity
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Titrate:
Integration of the viscoelastic relations in
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Date:
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Author (S):
P. of BONNIERES
Key:
R5.03.08-A
Page:
4/14
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R5.03 booklet: Nonlinear mechanics
HI-75/01/001/A
3
Nature of the function G for each relation of
behavior
3.1 Relation
LEMAITRE
In this case,
G
express yourself explicitly (
is a scalar here):
(
)
G,
,
,
/
T
K
K
m
N
m
N
=


>
1
1
0 1
0
0
1
with
The data of the material characteristics are those provided under the key words factors
LEMAITRE
or
LEMAITRE_FO
of the operator
DEFI_MATERIAU
.
/LEMAITRE
NR:
UN_ SUR_ K: UN_ SUR_ M:
:
N
K
m
1
1




The Young modulus
E
and the Poisson's ratio
are those provided under the key words factors
ELAS
or
ELAS_FO
.
3.2 Relations
ZIRC_CYRA2
and
ZIRC_EPRI
For these relations,
G
do not express yourself explicitly. The behavior is represented by one
unidimensional creep test, with constant stress, which utilizes time passed since
the moment when the stress is applied. The relation of behavior is defined here by the data of
four functions
F, G, F, G
1
1 2
2
describing the evolution of the viscous deformation in the course of time:
()
()
()
()
v
T
T
T
T
= =
+
F
G
,
F
G
,
1
1
2
2
éq 3.2-1
The function
G
is calculated then numerically by eliminating time
T
in the following way:
1) for a given triplet
(
)
, T
, one solves in
T
the equation [éq 3.2-1] by the method of
Newton (see [bib2]). An approximation of the solution is found
(
)
T
T
,
,
2) the value of the function is obtained
G
in
(
)
, T
by deriving compared to time the equation
[éq 3.2-1] (see [bib1]):
(
)
()
()
()
()
!
! G,
F
G
,
F
G
,
'
'
v
T
T
T
T
T
= =
=
+
1
1
2
2
and in substituent in this new equation the value of
(
)
T
T
,
found previously.
One finds the formulation uniaxial following:
(
)
(
)
(
)
()
(
)
(
)
()
!
! G,
F
,
G
,
F
,
G
,
'
'
v
T
T
T
T
T
T
T
= =
=
+
1
1
2
2
For each of the two relations
ZIRC_CYRA2
and
ZIRC_EPRI
, the form of the four functions
F, G, F, G
1
1 2
2
is preset and the user introduces only some parameters into the file of
order.
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Code_Aster
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Version
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Titrate:
Integration of the viscoelastic relations in
STAT_NON_LINE
Date:
02/02/01
Author (S):
P. of BONNIERES
Key:
R5.03.08-A
Page:
5/14
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HI-75/01/001/A
Thus, for
ZIRC_CYRA2
, one a:
()
(
)
[
]
()
(
)
(
)
(
)
()
(
)
[
]
()
(
)
(
)
F
G
,
exp
/
,
F
G
,
,
,
,
exp
/
,
,
,
1
1
2
2
3
1
17
4
3
1
1
2
3
273 15
2
3
1
4
3
4450
4 5 10
9.529 10
39000
1.816 10
6400
273 15
4000
3 10
0 00266
0 413
T
C
E
T F
T
With
K T
T
C
E
T F
T
With
C
H
With
K
K
F
T
C
K
H
With
HT fab
T
rec
HT
E
irr fab
K T
rec
irr
HT
HT
rec
rec
irr
irr
irr
=
-
+
=
-
+




=
-
+
=
=
=
=
=
°
=
+
=
=
-
-
-
-
-
-
-
-
with:
irr
=
-
1 2 10
22
,
Positive parameters
fab
rec
T
,
and
are those provided under the key word factor
ZIRC_CYRA2
of
the operator
DEFI_MATERIAU
:
(
)
/ZIRC_ CYRA2
EPSI_ FAB:
TEMP_ REHEATED:
FLUX_ PHI:
:
fab
rec
T
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Code_Aster
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Titrate:
Integration of the viscoelastic relations in
STAT_NON_LINE
Date:
02/02/01
Author (S):
P. of BONNIERES
Key:
R5.03.08-A
Page:
6/14
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HI-75/01/001/A
In the same way, for
ZIRC_EPRI
, one a:
()
()
(
)
(
)
(
)
()
()
(
)
(
)
F
G
,
F
G
,
cos
,
,
,
,
,
,
,
,
,
,
/
,
/
,
max
1
5
1
3
4
6
7
273 15
2
2
2
3
4
5
273 15
6
7
8
5
4
21
1
2
1
1 1.603 10
2 4.567 10
3 2 28
4 0 997
5 0 77
6 0 956
7 2 3 10
1 3.296 10
2 0 811
3 0 595
4
T
T
T
WITH HS A
R E
T
T
T
B
E
R
With
With
With
With
With
With
With
B
B
B
B
With
With
With
AP
With
T
B
B
B
B
T
Pb
B
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
-
+
-
+
-
-
with:
1 352
5 22 91
6 1 58
7 2 228
,
,
,
,
B
B
B
=
=
=
Positive parameters
,
max
max
R
p
and
0
2




are those provided under the key word factor
ZIRC_EPRI
of the operator
DEFI_MATERIAU
:
(
)
/ZIRC_ EPRI
FLUX_ PHI:
R P:
THETA_ MAX:
:
_
max
R
p
It will be noted that, for the two relations of behavior and all the functions:
T
express yourself in hours
T
express yourself in °C
express yourself in MPa
The document of Young
E
and the Poisson's ratio
are those provided under the key words factors
ELAS
or
ELAS_FO
.
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Code_Aster
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Titrate:
Integration of the viscoelastic relations in
STAT_NON_LINE
Date:
02/02/01
Author (S):
P. of BONNIERES
Key:
R5.03.08-A
Page:
7/14
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HI-75/01/001/A
4
Integration of the relation of behavior
4.1 Establishment of the scalar equation for the implicit scheme and
with constant elastic coefficients
One indicates by



early
total deflection at the moment
T
T
+
and by
early
variation of deformation
total during the pitch of current time. One calls



O
deformation imposed on the moment
T
T
+
and
O
variation of deformation imposed during the pitch of current time.
This imposed deformation results from thermal dilation and the anelastic deformations. One has
thus:
(
) (
)
(
)
() ()
(
)
[
]
(
)
()
O
ref.
ref.
has
has
T
T
T
T
T
T
T
T
T
T
T
=
+
+
-
-
-
+
+
-
T
T
I
3






where
I
3
is the tensor identity of command 2 in dimension 3.
One poses









=
-
early
O
As it is supposed here that
µ
is constant, one with the following relation between the diverters of






and
:
(
)
~
~









=
-
2
µ
v
éq 4.1-1
However, the law of flow is written, an implicit way:
()















v
eq
v eq
eq
T
T
=
+




-
3
2 G
,
,
~
éq 4.1-2
One thus has, while eliminating
v
between [éq 4.1-1] and [éq 4.1-2]:
()
(
)
()
2
3
2
1 3
µ
µ
µ
µ
~
~
G
,
,
~
~
~
G
,
,
~




































=
+
+




+
= +
+








-
-
-
T
T
T
T
eq
v eq
eq
eq
v eq
eq
éq 4.1-3
While posing
~
~
~









E
=
+
-
2
µ
, one thus has:
()












eq
E
eq
v eq
eq
T
T
=
+




+
-
3
µ
G
,
,
éq 4.1-4
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Integration of the viscoelastic relations in
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Date:
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Author (S):
P. of BONNIERES
Key:
R5.03.08-A
Page:
8/14
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R5.03 booklet: Nonlinear mechanics
HI-75/01/001/A
However, one has according to [éq 4.1-2]:
()
()









v eq
eq
v eq
T
T
=
+




-
G
,
,
From where:
()
()
(
)


















eq
E
v eq
eq
v eq
eq
E
eq
=
+
=
-
3
1
3
µ
µ
In substituent this last expression in [éq 4.1-4], one a:
(
)















eq
E
eq
eq
E
eq
eq
T
T
=
+
-




+
-
3
1
3
µ
µ
G
,
,
If one poses,



eq
E
T
T
,
,
-
and
being known:
()
(
)
F
G,
,
X
T
X
X T
X
eq
E
eq
E
=
+
-




+ -
-
3
1
3
µ
µ






one can then calculate the quantity
(
)









eq
eq
=
+
-
as being the solution of the scalar equation:
()
F X
=
0
where
X
eq
=
, convention adopted for the following paragraphs.
4.2
Resolution of the scalar equation: principle of the routine
ZEROF2
One easily shows that, if the requirements in the paragraph [§3] on the characteristics of
materials are checked, the function
F
is strictly increasing and the equation
()
F X
=
0
one admits
single solution.
If



eq
E
=
0
, then the solution is
X
=
0
. If not, one a:
()
F 0
0
= -
<



eq
E
The problem thus consists in finding for a function
F
unspecified the solution of the equation
()
F X
=
0
knowing that this solution exists, that
()
F 0
0
<
and that
F
is strictly increasing.
The algorithm adopted in
ZEROF2
is as follows:
·
one leaves
has
B
X
X
ap
ap
0
0
0
=
=
and
where
is an approximation of the solution. If it is
necessary (i.e. if
()
F B
0
0
<
), one brings back oneself by the method of the secants
(
()
()
() ()
Z
has
B
B
has
B
has
N
N
N
N
N
N
N
=
-
-
F
F
F
F
then
has
B
B
Z
N
N
N
N
+
+
=
=
1
1
and
) in one or more iterations with the case
where
()
()
F
F
has
B
<
>
0
0
and
:
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Code_Aster
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Titrate:
Integration of the viscoelastic relations in
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Date:
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Author (S):
P. of BONNIERES
Key:
R5.03.08-A
Page:
9/14
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HI-75/01/001/A
0
B
1
has
B
1
0
=
()
F B
1
()
F B
0
()
F 0
(In the case of the figure above, this first sentence was done in an iteration:
()
has
B
B
1
0
1
0
=
>
and
F
).
·
one
calculate
NR
D
= whole part
(
)
NR
max
or
NR
max
is the maximum number of iterations that
one was given. One then solves the equation by the method of the secants while using however
method of dichotomy with each time
N
is multiple of
NR
D
:
1)
If
NR
D
divided
N
Z
has
B
N
N
N
=
+
2
if not
()
()
() ()
Z
has
B
B
has
B
has
N
N
N
N
N
N
N
=
-
-
F
F
F
F
finsi
N N
= +
1
if
()
F Z
>
if
()
F Z
<
0
has
Z
B
B
N
N
N
N
+
+
=
=
1
1
if not
has
has
B
Z
N
N
N
N
+
+
=
=
1
1
finsi
to go into 1)
if not
The solution is:
X Z
END
N
=
finsi
This second part of the algorithm makes it possible to treat in a reasonable iteration count the cases
where
F
is very strongly non-linear, whereas the method of the secants would have converged too much
slowly. These cases of strong non-linearity meet in particular with the law of LEMAITRE, for
values of
N
m
large.
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Titrate:
Integration of the viscoelastic relations in
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Date:
02/02/01
Author (S):
P. of BONNIERES
Key:
R5.03.08-A
Page:
10/14
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R5.03 booklet: Nonlinear mechanics
HI-75/01/001/A
4.3
Calculation of the stress at the end of the pitch of current time
According to [éq 4.1-3], if
X
is the solution of the scalar equation, while posing:
()
(
)
B,
G,
,
X
T
X
X T
X
X
eq
E
eq
E
eq
E









=
+
+
-




=
-
1
1 3
1
3
µ
µ
one a:
()
~
,
~






=
B X
eq
E
E
éq 4.3-1
If



eq
E
=
0
, which is equivalent according to the scalar equation to
X
=
0
, one prolongs
B
by
continuity. For that, one poses
()
(
)
y X
X
eq
E
=
+
-
-
µ
1
3



and
()
()
(
)
G
, y
,
X
G X
X T
=
. The derivative of
G
express yourself according to the derivative partial of
G
at the point
()
(
)
X
X T
, y
,
:
()
()
(
)
()
(
)
G'
G, y,
G, y,
X
X X
X T
y X
X T
=
-
µ
1
3
Prolongation of
B
by continuity gives then:
()
()
B,
G'
0 0
1
1 3
0
= + µ
T
and one has, always if






eq
E
=
=
0
0
, ~
.
Once one calculated
~



, one obtains



by the relation (
K
is supposed to be constant here):
()
()












=
+
= +
+




-
-
~
1
3
3
Tr
K Tr
I
éq 4.3-2
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Code_Aster
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Titrate:
Integration of the viscoelastic relations in
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Date:
02/02/01
Author (S):
P. of BONNIERES
Key:
R5.03.08-A
Page:
11/14
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HI-75/01/001/A
4.4 Diagram
semi-implicit
With an implicit numerical diagram [éq 4.1-2], in the case, for example, where
G
does not depend on
,
only intervenes by the calculation of
v
the value of the stress at the end of the pitch of time. It can in
to result from the important numerical errors if the stress strongly varies in the course of time (see
[bib2]).
To cure that and to improve the resolution, one discretizes the law of flow in way
semi-implicit:
()
























v
eq
v eq
eq
T
T
T
=
+




+
+


+




+




-
-
-
-
-
3
2
2
2
2
2
2
G
,
,
~
~
éq 4.4-1
To transform in the most economic way what was programmed previously (while following
implicit formulation [éq 4.1-2]), it is enough to divide each member of the equation [éq 4.4-1] by 2:
(
)
()
























v
eq
v eq
eq
T
T
T
/
G
,
,
~
~
2
3
2
2
2
2
2
2
2
=
+




+
+


+




+




-
-
-
-
-
and to make the same thing with the relation [éq 4.1-1]:
~
~









2
2
2
2
=
-




µ
v
It is noted that this system is same form as that consisted the equations [éq 4.1-1] and
[éq 4.1-2], the data being
2
instead of
, unknown factors being respectively






2
2
and
v
with
place of






and
v
and the function
G
2
replacing the function
G
.
One can thus use the resolution of the paragraphs [§4.1] with [§4.3] as well as the corresponding algorithm
while introducing
2
and by dividing the function
G
by 2. It then remains to multiply the results






2
2








and
v
by 2 to obtain the increments of calculated stress and viscous deformation
by the semi-implicit diagram (it






and it
v
equation [éq 4.4-1]).
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Titrate:
Integration of the viscoelastic relations in
STAT_NON_LINE
Date:
02/02/01
Author (S):
P. of BONNIERES
Key:
R5.03.08-A
Page:
12/14
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HI-75/01/001/A
It will be noticed that the calculation of the tangent operator is not affected by this amendment of the diagram
numerical. Indeed, one has obviously:
()












=
2
2
4.5 Taking into account of the variation of the elastic coefficients in
function of the temperature
One has, if
With
is the tensor of elasticity:
()









=
+
-
v
With
1
with:
()
(
) (
)
()
With
With
With
-
-
-
-
-
-
-
=
+
+
-
1
1
1












T
T
T
This is translated in the equations of [§4.4] by:
()
2
2
2
3
2
2
2
2
2
2
2
2
4
µ
µ
µ
µ
µ
~
~
~
G
,
,
~
~





































-
+




=
+




+
+


+




+




-
+




-
-
-
-
-
-
-
-
-
T
T
T
eq
v eq
eq
While posing:
~
~
~









E
=
+




+




-
-
-
2
2
4
2
2
µ
µ
µ
µ
and
()
()
Tr
K
K
K
Tr
K Tr
E









=
+




+




-
-
-
3
3
6
3
2
one is reduced exactly to the preceding case [§4.4].
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Code_Aster
®
Version
5.0
Titrate:
Integration of the viscoelastic relations in
STAT_NON_LINE
Date:
02/02/01
Author (S):
P. of BONNIERES
Key:
R5.03.08-A
Page:
13/14
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HI-75/01/001/A
5
Calculation of the tangent operator
If



eq
E
X
=
=
0
0
and
, one takes the tensor of elasticity as tangent operator.
If not, one obtains this operator by deriving the equation [éq 4.3-1] compared to
:
()
()

~
~
B,
~
,
~































=
=
+
X
B X
eq
E
E
eq
E
E
then while also deriving [éq 4.3-2] compared to
:
()
























=
+
=
+
~
~
K
Tr
K
T
I
I I
3
3 3
It will be noted that, in these equations, the tensors of command 2 and command 4 are respectively compared to
vectors and with matrices.
I
3
is here a tensor of a nature 2, compared to a vector:
(
)
T
I
3
111 0 0 0
=
,
One has moreover:
() ()
()
B,
B,
B
,
X
X X
X
X
eq
E
eq
E
eq
E
eq
E
eq
E






















=
+
It is thus necessary to calculate
X
. For that, one derives the scalar equation implicitly compared to
.
To simplify, one will omit thereafter in the writing of
G
and of its derivative the parameter
T
.
One has then:
()
[
]
()
3
1
µ
T
X
X
T there X y
eq
E
eq
E
+
+
=
G
G,















From where:
()
()
()
()
µ
µ
µ
X
T there X y
T
X
X
T there X y
T
X
eq
E
eq
E T
E


















=
-
+
=
-
+
1
1 3
1
1 3
3
G,
G
G,
G
~
with the expression of
()
G' X
obtained with [§4.3].
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Code_Aster
®
Version
5.0
Titrate:
Integration of the viscoelastic relations in
STAT_NON_LINE
Date:
02/02/01
Author (S):
P. of BONNIERES
Key:
R5.03.08-A
Page:
14/14
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HI-75/01/001/A
One obtains finally the following expression of the tangent operator:
()
[
]
()
()
()
µ
µ

























=
+
+
=
=
-
=
-
+
-






K
X
T there X y
T
X
X
T
E T E
eq
E
T
eq
E
eq
E
I I
With
With
J
I I
J
3
3
6
3
3
6
3
2
1
3
6
3
2
1
1 3
~
~
B,
~
.
G,
G
with:
where
is the matrix identity of row
Note:
In the case of laws
ZIRC_CYRA2
and
ZIRC_EPRI
, it is checked easily that:
()
(
)
(
) (
)
(
)
=
+
-




+
-




+
-


+
-
-
+


=
+
+
G
F G
F G G G F
F F
G G F
F F
G G F F
F F
G G F F
F F
F G
F G
G,
F G
F G
F G
F
'
'
'
'
''
'
'
''
'
''
''
'
''
''
''
''
''
''
'
'
X
X y T
1
1
3
1 1
2 2
1 1 1
2
1 1
2 2
2
2
2 2
1 2 1 2
1 2
2 1 1 2
1 2
1 1
2 2
1 1
2 2
1 1
2
µ
G
2
where
F, F F F, F F
, '
, ''
,
'
''
1 1 1 2 2 2
the values indicate of
F
F
1
2
and
and their derivative at the point
(
)
T X y T
,
and
where
G, G, G, G
'
'
1
1
2
2
the values indicate of
G
G
1
2
and
and of their derivative compared to
at the point
()
X T
,
(see [bib1]).
6 Bibliography
[1]
BONNIERES P.: Writing in generalized standard form of the laws of behavior
viscoplastic of Zircaloy, notes EDF-DER HI-71/7940-Index A, 1992
[2]
BONNIERES P., ZIDI Mr.: Introduction of viscoplasticity into the module of
thermomechanics of Cyrano3: principle, description and validation, note EDF-DER HI-71/8334,
1993