Code_Aster
®
Version
7.4
Titrate:
Relations of behavior 1D
Date:
02/05/05
Author (S):
J.M. PROIX, B. QUINNEZ, C. CHAVANT
Key
:
R5.03.09-B
Page
:
1/36
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HT-66/05/002/A
Organization (S):
EDF-R & D/AMA, SINETICS
Manual of Reference
R5.03 booklet: Nonlinear mechanics
Document: R5.03.09
Nonlinear relations of behavior 1D
Summary:
This document describes the quantities calculated by the operator
STAT_NON_LINE
necessary to the implementation of
the quasi static nonlinear algorithm describes in [R5.03.01] in the case of the elastoplastic behaviors or
viscoplastic monodimensional. These behaviors are applicable to the elements of
BAR
, with
elements of beam and beams multifibre (direction axial only) and to the elements of concrete reinforcement
(modeling
ROAST
).
The behaviors described in this document are:
·
the behavior of Von Mises with linear isotropic work hardening:
VMIS_ISOT_LINE
, and unspecified
VMIS_ISOT_TRAC
,
·
the behavior of Von Mises with linear kinematic work hardening:
VMIS_CINE_LINE
,
·
the behavior of Von Mises with linear, nonsymmetrical work hardening in traction and compression:
with restoration of the center of the elastic range:
VMIS_ASYM_LINE
. This last was developed
to modelize the action of the ground on the Cables with Gas Insulation,
·
the behavior of
PINTO-MENEGOTTO
who allows to represent the elastoplastic behavior
uniaxial of the reinforcements of the reinforced concrete. This model translates nonthe linearity of the work hardening of the bars
under cyclic loading and takes into account the Bauschinger effect. It makes it possible of more than simulate it
buckling of the reinforcements in compression. This relation is available in Code_Aster for
elements of bar and elements of grid,
·
viscoplastic behaviors of
LMA-RC
and of J.Lemaître, usable by
ASSE_COMBU
in
axial direction of the elements of beam.
The resolution is made in all the cases by a method of integration implicit as from the moment of calculation
precedent, one calculates the stress field resulting from an increment of deformation, and the behavior
tangent which makes it possible to build the tangent matrices.
One describes finally a method, similar to the method due of Borst [R5.0303] allowing to use all them
behaviors available in 3D in the elements 1D.
Code_Aster
®
Version
7.4
Titrate:
Relations of behavior 1D
Date:
02/05/05
Author (S):
J.M. PROIX, B. QUINNEZ, C. CHAVANT
Key
:
R5.03.09-B
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:
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Manual of Reference
R5.03 booklet: Nonlinear mechanics
HT-66/05/002/A
Count
matters
2
Relation of behavior of Von Mises with linear isotropic work hardening:
VMIS_ISOT_LINE
or
Code_Aster
®
Version
7.4
Titrate:
Relations of behavior 1D
Date:
02/05/05
Author (S):
J.M. PROIX, B. QUINNEZ, C. CHAVANT
Key
:
R5.03.09-B
Page
:
3/36
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HT-66/05/002/A
1
Use of the relations of behavior 1D
1.1
Relations of behavior 1D in Code_Aster
The relations treated in this document are:
VMIS_ISOT_LINE:
Von Mises with symmetrical linear isotropic work hardening
VMIS_ISOT_TRAC:
Von Mises with unspecified isotropic work hardening
GRILL_ISOT_LINE:
Von Mises with symmetrical linear isotropic work hardening
VMIS_CINE_LINE:
Von Mises with symmetrical linear kinematic work hardening
GRILL_CINE_LINE
Von Mises with symmetrical linear kinematic work hardening
VMIS_ASYM_LINE:
Von Mises with asymmetrical linear work hardening and restoration
PINTO_MENEGOTTO:
Behavior of the reinforced concrete reinforcements
GRILL_PINTO_MEN:
Behavior of the reinforced concrete reinforcements
ASSE_COMBU
Viscoplastic behavior of the fuel assemblies:
Model of LEMAITRE or the LMA-RC
These relations of behavior (incremental) are given in the operator
STAT_NON_LINE
[U4.51.03] under the key word factor
COMP_INCR
, by the key word
RELATION
[U4.51.03]. They are not
valid that in small deformations.
One describes for each relation of behavior the calculation of the stress field from one
increment of deformation given (cf algorithm of Newton [R5.03.01]), the calculation of the nodal forces
R
and of the tangent matrix
K
in
.
1.2 Notations
general
All the quantities evaluated at the previous moment are subscripted by
-
.
Quantities evaluated at the moment
T
T
+
are not subscripted.
The increments are indicated by
. One has as follows:
(
)
()
Q
Q
Q
Q
Q
Q
=
+
=
+
=
+
-
-
T
T
T
.
tensor of the stresses (in 1D, one is interested only in the single component
nonnull uniaxial).
~
deviative operator:
~
ij
ij
kk
ij
=
-
1
3
.
()
eq
equivalent value of Von Mises, equalizes in 1D with the absolute value
increment of deformation.
With
tensor of elasticity, equal in 1D to the Young modulus E
µ
,
E K
moduli of the isotropic elasticity.
thermal expansion factor secant.
T
temperature.
()
+
positive part.
p
cumulated plastic deformation
p
plastic deformation
Code_Aster
®
Version
7.4
Titrate:
Relations of behavior 1D
Date:
02/05/05
Author (S):
J.M. PROIX, B. QUINNEZ, C. CHAVANT
Key
:
R5.03.09-B
Page
:
4/36
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R5.03 booklet: Nonlinear mechanics
HT-66/05/002/A
1.3
Change of variables
Whatever the type D `finite element referring to a law of behavior 1D, it is necessary to carry out one
change of variables to pass from the elementary quantities (efforts, displacements) to
stresses and deformations.
1.3.1 Calculation of the deformations (small deformations)
For each finite element of Code_Aster, in
STAT_NON_LINE
, the total algorithm (Newton)
provides to the elementary routine, which integrates the behavior, an increase in field in
displacement.
For the elements of bar, one calculates the deformation (only one axial component) by:
() ()
=
-
U L
U
L
0
,
and the increase in deformation by:
()
()
=
-
U L
U
L
0
,
For the elements of grid (modelings
ROAST
and
GRILL_MEMBRANE
), the deformation is calculated
membrane as for the elements of hulls DKT. Simply, only one direction corresponds
physically with the directions of reinforcements. One thus finds oneself in the presence of a behavior 1D.
In addition, in small deformations, for all the models described in this document, one writes for
any moment the partition of the deformations in the form of an elastic contribution, dilation
thermics, and of plastic deformation:
()
()
()
()
()
()
()
()
() ()
()
()
()
()
(
)
T
T
T
T
T
T T
T
E T
T
T
T T
T T
T
E
HT
p
E
HT
ref.
=
+
+
=
=
=
-
-
,
with
With
Id
1
1
1.3.2 Calculation of the generalized efforts (forced integrated)
For integration of the behavior 1D, it is necessary to integrate the component of stresses obtained, for
to provide to the total algorithm (Newton) a vector containing the generalized efforts.
For the elements of bar, one calculates the effort (uniform in the element, by supposing that the section
is constant) by:
NR
S
=
. ,
and the vector forces nodal equivalent (as for the elements of beam, [R3.08.01]) by:
F
NR
NR
= -
For the elements of
ROAST
, one calculates the efforts as for the elements of hulls DKT (efforts
membrane) by integration of the stresses in the thickness (only one layer and only one point
of integration).
Code_Aster
®
Version
7.4
Titrate:
Relations of behavior 1D
Date:
02/05/05
Author (S):
J.M. PROIX, B. QUINNEZ, C. CHAVANT
Key
:
R5.03.09-B
Page
:
5/36
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HT-66/05/002/A
2
Relation of behavior of Von Mises with work hardening
isotropic linear
:
VMIS_ISOT_LINE
or unspecified
:
VMIS_ISOT_TRAC
2.1
Equations of the model
VMIS_ISOT_LINE
They are the restriction of the behavior 3D [R5.03.02] on the uniaxial case:
()
()
()
()
=
-
<
-
=
-
=
-
-
-
=
=
=
0
0
0
0
0
~
2
3
p
R
p
p
R
p
p
R
p
R
E
p
p
eq
eq
eq
HT
p
eq
p
if
if
&
&
&
&
&
with:
·
&
p
:
speed of plastic deformation,
·
p
:
cumulated plastic deformation,
·
(
)
HT
ref.
T T
=
-
:
thermal deformation,
·
()
R p
E E
E E
p
T
T
y
= -
+
.
.
: function of linear work hardening isotropic, or
()
p
R
refine by
pieces, deduced from the traction diagram.
In the case
VMIS_ISOT_LINE
, the data of the material characteristics are those provided
under the key word factor
ECRO_LINE
or
ECRO_LINE_FO
of the operator
DEFI_MATERIAU
[U4.43.01].
/ECRO_LINE = (D_SIGM_EPSI =
E
T
, SY =
y
)
/ECRO_LINE_FO = (D_SIGM_EPSI =
E
T
, SY =
y
)
In the case
VMIS_ISOT_TRAC
, the data of the characteristics of materials are provided under
key word factor
TRACTION
of the operator
DEFI_MATERIAU
[U4.43.01].
TRACTION = _F (SIGM = courbe_traction)
courbe_traction
represent the traction diagram, point by point. The first point allows
to define the elastic limit
y
and it Young modulus
E
[R5.03.02].
Code_Aster
®
Version
7.4
Titrate:
Relations of behavior 1D
Date:
02/05/05
Author (S):
J.M. PROIX, B. QUINNEZ, C. CHAVANT
Key
:
R5.03.09-B
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:
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R5.03 booklet: Nonlinear mechanics
HT-66/05/002/A
ECRO_LINE_FO
corresponds if
E
T
and
y
depend on the temperature and are then calculated
for the temperature of the point of current Gauss. The Young modulus
E
and the Poisson's ratio
are those provided under the key words factors
ELAS
or
ELAS_FO
. In this case the traction diagram is
the following one:
L
L
L
y
L
y
T
L
y
L
y
E
E
E
E
E
=
<
=
+
-
if
if
.
When the criterion is reached one a:
()
L
R p
-
=
0
, therefore
L
L
L
R
E
-
-
=
0,
from where:
()
R p
E E
E
E p
H p
T
T
y
y
=
-
+
=
+
.
In the case of a traction diagram, the step is identical to [R5.03.01].
2.2
Integration of the relation
VMIS_ISOT_LINE
By direct implicit discretization of the relations of behavior, a way similar to integration 3D
[R5.03.02] one obtains:
(
)
(
) (
)
(
)
(
)
+
<
+
=
+
=
+
+
+
=
+
+
-
-
+
-
+
-
-
-
-
-
-
-
-
-
-
-
p
p
R
p
p
p
R
p
p
E
E
E
E
p
p
R
HT
if
if
0
0
0
E
L
y
E
T
L
Code_Aster
®
Version
7.4
Titrate:
Relations of behavior 1D
Date:
02/05/05
Author (S):
J.M. PROIX, B. QUINNEZ, C. CHAVANT
Key
:
R5.03.09-B
Page
:
7/36
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HT-66/05/002/A
Two cases arise:
·
(
)
-
-
+
<
+
R p
p
in this case
(
)
-
-
+
-
=
=
HT
p
that is to say
0
thus
(
) ()
,
-
-
-
<
-
+
p
R
HT
·
(
)
-
-
+
=
+
R p
p
in this case
p
0
thus
(
) ()
.
-
-
-
-
+
p
R
HT
One deduces the algorithm from it from resolution:
let us pose
(
)
HT
E
-
+
=
-
-
if
()
E
R p
p
=
-
then
0
and
(
)
=
-
HT
if
()
E
R p
>
-
then it is necessary to solve:
E
E p
=
+
+
+
+
-
-
-
(
)
E
E p
=
+
+
+
-
-
1
thus by taking the absolute value:
E
E p
=
+
+
+
-
-
1
maybe, while using
(
)
-
-
+
=
+
R p
p.
(
)
p
E
p
p
R
E
+
+
=
-
One thus deduces some:
·
in the case of a linear work hardening:
(
)
H
E
HP
p
y
E
+
+
-
=
-
·
and in the case of an unspecified work hardening, the curve
()
p
R
being refined per pieces, one
solves the equation directly in
p
:
(
)
E
p
p
R
p
=
+
+
-
E
in the same way that in
3D [R5.03.02].
Code_Aster
®
Version
7.4
Titrate:
Relations of behavior 1D
Date:
02/05/05
Author (S):
J.M. PROIX, B. QUINNEZ, C. CHAVANT
Key
:
R5.03.09-B
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:
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R5.03 booklet: Nonlinear mechanics
HT-66/05/002/A
Let us notice in the passing that:
()
E
E
R p
=
then
(
)
()
()
=
+
=
= +
-
E
E
E
R p
E p
R p
1
Moreover, the option
FULL_MECA
allows to calculate the tangent matrix
K
in
with each iteration.
The tangent operator who is used for building it is calculated directly on the preceding discretized system.
One obtains directly:
()
if
if not
E
T
R p
E
E
>
=
=
-
Note:
The option
RIGI_MECA_TANG
who allows to calculate the tangent matrix
K
i0
used in
phase of prediction of the algorithm of Newton, takes account of the indicator of plasticity with
the previous moment:
·
if
T
E
=
=
1
·
if
E
=
=
0
2.3 Variables
interns
The relation of behavior
VMIS_ISOT_LINE
product two internal variables:
and
p
.
Code_Aster
®
Version
7.4
Titrate:
Relations of behavior 1D
Date:
02/05/05
Author (S):
J.M. PROIX, B. QUINNEZ, C. CHAVANT
Key
:
R5.03.09-B
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:
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R5.03 booklet: Nonlinear mechanics
HT-66/05/002/A
3
Relation of behavior of Von Mises, work hardening
linear kinematics 1D:
VMIS_CINE_LINE
3.1
Equations of the model
VMIS_CINE_LINE
They are the restriction of the behavior 3D ([R5.03.02] and [R5.03.16]) on the uniaxial case.
behavior 3D is written:
(
)
=
-
-
K
p
HT
with
K
operator of elasticity
X
p
=
C
(
)
(
)
F,
~
~ ~
R
eq
y
eq
X
X
With
WITH A
=
-
-
=
with
3
2
(
)
&
& F
&
~
~
p
X
X
=
=
-
-
p
p
eq
3
2
if
if
F
&
F
&
<
p
p
0
0
0
0
=
=
In the uniaxial case, the tensors are written:
~
=
=
=
=
-
-
D
X
D
D
D
p
X
p
3
2
2 3
1 3
1 3
with
As long as the loading is monotonous, the following relations immediately are obtained:
p
X
C
C
p
p
p
y
=
=
=
+
3
2
3
2
()
=
=
+ -
F
E E
E
E p
y
T
T
.
C is determined by:
C
EE
E
E
T
T
=
-
2
3
. One poses:
H
EE
E E
C
T
T
=
-
=
3
2
Code_Aster
®
Version
7.4
Titrate:
Relations of behavior 1D
Date:
02/05/05
Author (S):
J.M. PROIX, B. QUINNEZ, C. CHAVANT
Key
:
R5.03.09-B
Page
:
10/36
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HT-66/05/002/A
The relation of behavior 1D is written then:
(
)
=
-
-
<
-
-
=
-
-
=
=
-
-
=
-
-
=
0
0
0
0
0
2
3
y
y
y
p
p
p
HT
p
X
p
X
p
X
H
C
X
E
X
X
p
if
if
&
&
&
&
The data of the material characteristics are those provided under the key word factor
ECRO_LINE
or
ECRO_LINE_FO
of the operator
DEFI_MATERIAU
[U4.43.01]:
/ECRO_LINE = (D_SIGM_EPSI =
E
T
, SY =
y
)
/ECRO_LINE_FO = (D_SIGM_EPSI =
E
T
, SY =
y
)
3.2
Integration of the relation
VMIS_CINE_LINE
By direct implicit discretization of the relations of behavior, a way similar to integration 3D
([R5.03.02] and [R5.03.16]) one obtains:
(
) (
)
<
-
-
+
=
=
-
-
+
=
-
-
-
+
-
-
+
=
+
+
-
-
=
-
-
-
+
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
y
y
p
p
HT
p
y
X
X
p
X
X
p
H
X
H
X
X
X
X
X
p
E
E
E
E
X
X
if
if
0
0
0
with
(
)
(
)
ref.
ref.
HT
T
T
T
T
-
-
-
=
-
-
Code_Aster
®
Version
7.4
Titrate:
Relations of behavior 1D
Date:
02/05/05
Author (S):
J.M. PROIX, B. QUINNEZ, C. CHAVANT
Key
:
R5.03.09-B
Page
:
11/36
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HT-66/05/002/A
Two cases arise:
·
-
-
+
-
-
<
X
X
y
in this case
0
=
p
that is to say
(
)
-
-
-
-
-
+
-
=
X
H
H
E
E
HT
thus
(
) ()
,
-
-
-
-
-
<
-
+
-
p
R
H
H
X
E
E
HT
·
if not
p
0
.
To simplify the writings one will pose:
(
)
HT
E
E
X
H
H
E
E
-
+
-
=
-
-
-
-
.
One deduces the algorithm from it from resolution:
·
if
(
)
.
,
,
0
-
-
-
-
+
-
=
=
=
E
E
E
H
H
X
X
p
HT
y
E
then
·
if not it is necessary to solve:
(
)
=
-
-
-
+
=
-
-
-
=
-
-
+
-
-
+
=
+
+
-
=
-
-
=
-
-
-
-
-
-
-
-
-
-
-
0
)
(
y
p
p
E
HT
p
X
X
H
X
H
H
X
X
X
p
X
X
X
X
p
H
H
X
E
E
Let us notice that:
p
H
X
X
H
H
-
=
-
-
. One deduces then from the first equation:
p
E
H
E
X
+
+
-
=
)
(
One thus obtains, while eliminating
-
X
second equation:
(
)
p
E
y
p
E H p
=
+
+
While replacing
p
the relation enters
E
and
-
X
, one obtains:
(
)
-
=
+
+
X
E
H p
E
y
y
By taking the absolute value of the two members of the preceding equation, one finds
p
:
(
)
E H p
y
E
+
+
=
Code_Aster
®
Version
7.4
Titrate:
Relations of behavior 1D
Date:
02/05/05
Author (S):
J.M. PROIX, B. QUINNEZ, C. CHAVANT
Key
:
R5.03.09-B
Page
:
12/36
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HT-66/05/002/A
Once
p
determined, one can calculate:
p
E
E
p
=
E
E
p
H
H
HX
X
X
X
+
=
+
=
-
-
-
and while using:
-
=
X
y
E
E
, one obtains directly:
=
+
y
E
E
X
Moreover, the option
FULL_MECA
allows to calculate the tangent matrix
K
in
with each iteration.
The tangent operator who is used for building it is calculated directly on the preceding discretized system.
One obtains directly:
()
if
if not
E
T
R p
E
E
>
=
=
-
The option
RIGI_MECA_TANG
who allows to calculate the tangent matrix
K
I
0
used in the phase of
prediction of the algorithm of Newton is obtained using the indicator of plasticity
-
moment
precedent:
·
if
T
E
=
=
-
then
1
·
if
E
=
=
-
then
0
3.3 Variables
interns
The relation of behavior
VMIS_CINE_LINE
product two internal variables:
and
X
.
Code_Aster
®
Version
7.4
Titrate:
Relations of behavior 1D
Date:
02/05/05
Author (S):
J.M. PROIX, B. QUINNEZ, C. CHAVANT
Key
:
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4
Relation of behavior of Von Mises with work hardening
linear asymmetrical:
VMIS_ASYM_LINE
4.1
Equations of the model
VMIS_ASYM_LINE
4.1.1 Behavior
asymmetrical
in traction and compression
It is a behavior uncoupled in traction and compression, built from
VMIS_ASYM_LINE
,
but with elastic limits and different modules of work hardening in traction and in
compression. We adopt an index T for traction and C for compression. The behavior
rubber band in traction and compression identical and is characterized by the same Young modulus. There is
two fields of isotropic work hardening defined by R
T
and R
C.
The two fields are independent one
other.
YT
elastic limit in traction. In absolute value.
TESTSTEMYÇ
elastic limit in compression. In absolute value.
p
T
Variable interns in traction. Algebraic value.
p
C
Variable interns in compression. Algebraic value.
E
TT
Slope of work hardening in traction.
E
TC
Slope D `work hardening in compression.
The equations of the model of behavior are:
()
()
()
()
()
()
&
&
&
&
&
&
&
&
&
&
&
&
&
&
p
HT
p
CP
Tp
CP
C
Tp
T
T
T
C
C
C
C
C
C
C
C
T
T
T
T
T
T
p
p
R p
R
p
p
R
p
p
R
p
p
R p
p
R p
= -
-
=
+
=
=
-
- -
=
- -
<
- =
=
-
<
=
-
·
1
0
0
0
0
0
0
0
0
678
with
if
if
if
if
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Titrate:
Relations of behavior 1D
Date:
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(
)
&:
, &:
:
:
.
CP
Tp
HT
HT
ref.
T T
speed of plastic deformation in compression
speed of plastic deformation in traction,
thermal deformation of origin
=
-
It is noticed that one cannot have simultaneously plasticization in traction and compression: that is to say
&p
C
=
0
, that is to say
&p
T
=
0
, that is to say both are null.
The data of the material characteristics are those provided under the key word factor
ECRO_ASYM_LINE
of the operator
DEFI_MATERIAU
[U4.43.01].
ECRO_ASYM_LINE = _F (DT_SIGM_EPSI = HT,
E
TT
,
SY_T
=
yT
,
DC_SIGM_EPSI = HC,
E
TC
,
SY_C
=
teststemyç
,)
The Young modulus E is provided under the key words factors
ELAS
or
ELAS_FO
.
One calculates the functions of work hardening by:
()
R p
E
E
E
E
p
H p
T
TT
TT
T
yT
T
T
yT
=
-
+
=
+
.
()
R p
E
E
E
E
p
H
p
C
TC
TC
C
teststemyç
C
C
teststemyç
=
-
+
=
+
.
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Relations of behavior 1D
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4.2
Integration of the behavior
VMIS_ASYM_LINE
By direct implicit discretization of the asymmetrical relation of behavior, a way similar to
the preceding one, one obtains:
(
) (
)
(
) (
)
(
) (
)
(
) (
)
if
if
p
Tp
CP
p
HT
Tp
T
T
T
T
T
T
T
T
T
T
T
T
CP
C
C
C
C
C
E
p
R p
p
p
R p
p
p
R p
p
p
R p
p
p
=
+
=
-
-
=
+
+
+
-
+
+
-
+
=
=
+
-
+
<
=
+
+
-
+
-
+
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
0
0
0
0
0
0
(
) (
)
(
) (
)
0
0
0
0
if
if
-
+
-
+
=
=
-
+
-
+
<
-
-
-
-
-
-
R p
p
p
R p
p
C
C
C
C
C
C
C
Integration is similar to that of
VMIS_ISOT_LINE
for each direction of traction and of
compression. It should well be seen that the centers of the fields of elasticity are data (calculated
explicitly with the preceding pitch) for the incremental problem to solve.
Four cases arise:
·
-
>
HT
0
:
one poses
(
)
Te
HT
E
=
+
-
-
-
()
Te
T
T
R p
<
-
in this case
p
T
=
0
thus
=
Te
and
=
E
- if not
:
(
)
p
H p
E
H
p
T
Te
yT
T T
T
C
=
-
+
+
=
-
,
0
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()
()
=
+
=
Te
T
T
T
Te
Te
T
T
E p
R p
R p
1
=
E
TT
·
-
<
HT
0
one poses
(
)
It
HT
E
=
+
-
-
-
()
-
<
-
It
C
C
R p
in this case
p
C
=
0
thus
=
It
and
=
E
- if not
:
(
)
p
H p
E
H
p
C
It
teststemyç
C C
C
T
=
-
+
+
=
-
,
0
()
()
=
+
=
It
C
C
C
It
It
C
C
E p
R p
R p
1
=
E
TC
Note:
The initial tangent matrix (option
RIGI_MECA_TANG)
is taken equal to the elastic matrix.
4.3 Variables
interns
The relation of behavior
VMIS_ASYM_LINE
product 2 internal variables:
p
C
p
T
.
It is not usable for the elements of grid
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5 Model
of
PINTO_MENEGOTTO
The model presented in this chapter describes the behavior 1D reinforcing steels of the concrete
armed [bib1]. The law constitutive of these steels is made up of two distinct parts: the loading
monotonous composed of three successive areas (linear elasticity, plastic bearing and work hardening) and
the cyclic loading whose analytical formulation was proposed by A. Giuffré and P. Pinto in 1973
[bib2] and was then developed by Mr. Menegotto [bib3].
During cycles, the way of loading between two points of inversion (semi-cycle) is described by
an analytical curve of expression of the type
()
=
F
. The interest of this formulation is that the same one
equation controls the discharge and load diagrams (see for example the figures [Figure 5.1.1-a] and
[Figure 5.1.1-b]). Parameters attached to the function
F
are reactualized after each inversion
of loading. The reactualization of these parameters depends on the way carried out in the plastic area
during the preceding semi-cycle.
In addition, this model can treat the inelastic buckling of the bars (G. Monti and C. Nuti [bib4]).
The introduction of new parameters into the equation of the curves then makes it possible to simulate
the softening of the answer stress-strain in compression.
5.1
Formulation of the model
5.1.1 Loading
monotonous
This chapter describes the first loading which the bar undergoes, i.e. the part preceding activation
curve of Giuffré [Figure 5.1.1-a].
The monotonous traction diagram of steel is typically described by the three successive areas
following:
·
The linear elasticity, defined by the Young modulus
E
and elastic limit
y
.
=
E
(area 1, [Figure 5.1.1-a])
·
The plastic bearing, ranging between the limiting elastic strain
y0
and deformation
of work hardening
H
, higher limit of the plate in deformation. During the bearing
stress remains constant.
=
y0
(area 2, [Figure 5.1.1-a])
·
Work hardening, describing the traction diagram up to the ultimate point of stress and of
deformation,
(
)
U
U
,
. This part is represented by a polynomial of the fourth degree:
(
)
=
-
-
-
-
U
U
y
U
U
H
0
4
(area 2, [Figure 5.1.1-a])
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The slope of work hardening (used thereafter, for the cyclic behavior) is defined here by:
E
H
U
y
U
y
=
-
-
0
0
. It is the average slope of areas 2 and 3 of the following figure.
area 1
area 3
area 2
H
U
U
y
0
y
0
Appear 5.1.1-a
5.1.2 Loading
cyclic
One places oneself now if the bar undergoes a consecutive discharge with the first
loading. Two cases arise then:
·
the starting position is in the elastic area. The discharge remains in this case
rubber band,
·
the starting position is in the plastic area (
y0
). The answer is first of all
rubber band, then, for a certain value of the deformation, the discharge becomes nonlinear
[Figure 5.1.2-a] (this is true for a discharge starting from area 2 or of area 3).
The relation which the deformation must satisfy so that the curve of Giuffré is activated is as follows:
max
.
- >
y0
3 0
, with
max
maximum deformation reached in load.
As soon as one crossed this limit with the first discharge, it is the cyclic behavior (curve of
Giuffré [Figure 5.1.2-a]) which is activated.
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discharge elastic
nonlinear discharge:
activation of the curve of
Giuffré
max
y
0
3
Appear 5.1.2-a
5.1.2.1 Presentation of the nth semi-cycle
The shape of the curve of the nth semi-cycle depends on the plastic excursion carried out during
preceding semi-cycle. The following quantities are defined [Figure 5.1.2.1-a]:
yn
: Elastic limit of the nth semi-cycle. (Calculation clarified with [§5.1.2.2])
RN
-
1
: Stress at the last point of inversion (forced maximum attack with the n-1
ième
semi-cycle).
RN
-
1
: Deformation at the last point of inversion (maximum deformation attack with the n-1
ième
semi-cycle).
yn
: Deformation corresponding to
yn
yn
RN
yn
RN
E
:
=
+
-
-
-
1
1
()
F T
: Plastic excursion of the nth cycle
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n+1
n-1
N
p
N
p
N
-
1
(,
)
R
N
R
N
(,
)
y
N
y
N
(,
)
y
N
y
N
-
-
1
1
(
,
)
R
N
R
N
-
-
1
1
E
H
E
H
N
-
1
Appear 5.1.2.1-a
5.1.2.2 Law
of work hardening
The model is based on a kinematic law of work hardening. The branches of the semi-cycles are
included/understood between two asymptotes of slope
E
H
(asymptotic slope of work hardening).
One thus determines
yn
in the following way:
(
)
yn
yn
pn
N
sign
=
-
+
-
-
-
1
1
1
.
where the function
()
sign X
= 1 if x<0 and 1 if x>0 and where
N
-
1
is the plastic increment of stress of the semi-cycle
precedent [Figure 5.1.2.1-a] which is defined by:
N
H pn
E
-
-
=
1
1
.
For each semi-cycle one thus determines
yn
according to
yn
pn
-
-
1
1
and
, one deduces some
yn
, then
the following semi-cycle is calculated (by the law of behavior below). Maximum deformation (in
absolute value) attack before changing direction will make it possible to calculate the plastic excursion
pn
RN
yn
=
-
.
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5.1.2.3 Analytical description of the curves
()
=
F
The expression chosen in the model to follow the curves of loading is as follows:
()
(
)
*
*
*
*
/
=
+
-
+
B
B
R
R
1
1
1
With
B
E
E
H
=
report/ratio of the slope of work hardening on the slope of elasticity.
*
*
=
-
-
=
-
-
=
-
-
-
-
-
-
-
-
RN
yn
RN
RN
yn
RN
pn
pn
yn
RN
1
1
1
1
1
1
1
Size
R
allows to describe the pace of the curvature of the branches. It is a function of the way
plastic carried out during the preceding semi-cycle:
()
()
()
R
R
G
G
With
With
=
-
=
+
0
1
2
where
.
Parameters
R WITH
With
0
1
2
,
and
are constants without unit depending on the mechanical properties
steel. Their values are obtained in experiments and Menegotto [bib3] proposes:
R
With
With
0
1
2
20 0
18 5
015
=
=
=
.
.
.
5.1.3 Case of inelastic buckling
Monti and Nuti [bib4] show that for a relationship between the length L and the diameter D of the bar
lower than 5, the curve of compression is identical to that of traction. On the other hand, when
L D
/
>
5
a buckling of the bar is observed. In this case the curve of compression in the plastic area
has a lenitive behavior. The model available in Code_Aster makes it possible to describe
also this phenomenon.
The following variables are defined [Figure 5.1.3-a]:
E
0
: Initial elastic Young modulus (agent with E without buckling).
B
C
: Report/ratio of the slope of work hardening on the elastic slope in compression.
B
T
: Report/ratio of the slope of work hardening on the elastic slope in traction (refill after compression
with buckling).
E
R
: Modulus Young reduced in traction (slope of the curve of refill after compression with
buckling).
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E
R
y
N
E
5
B
E
C
×
B E
×
S
S
Appear 5.1.3-a
5.1.3.1 Compression
A negative slope is introduced
B
E
C
×
, where
B
C
is defined by:
(
)
B
has
L D E
C
B
E
y
=
-
-
5 0
0
.
/
'
With
()
=
=
4 0
0
.
/
'max
y
pn
L D
and
the greatest plastic way carried out during
loading.
It is necessary then, as in the model without buckling, to determine
yn
. The method is identical, but
a complementary stress is added
S *
in order to position the curve correctly compared to
the asymptote [Figure 5.1.3-a].
(
)
S
S
C
C
S
S
Cl D
B E B B
B
L D
E
*
/
.
/
.
=
-
-
=
-
-
1
110
10
10
where
is given by:
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And one thus has:
()
yn
yn without buckling
S
=
+
*
This modifies also the value of
yn
RN
yn
RN
E
=
+
-
-
-
1
1
5.1.3.2 Traction
At the time of the semi-cycle in traction according to one adopts a reduced Young modulus defines by:
(
)
(
)
(
)
E
E has
has
E
has
L D
R
has
p
=
+
-
=
+
-
-
0
5
5
5
10
10
5 0
7 5
6 2
.
.
.
/
/.
with
Note:
Parameters
C has
has
,
and
6
are constants (without unit) depend on the properties
mechanics of steel and is in experiments given. Values adopted by
Monti and Nuti [bib4] are:
has
C
has
=
=
=
0 006
0 500
620 0
6
.
.
.
5.2
Establishment in Code_Aster
This model is accessible in Code_Aster starting from the key word
COMP_INCR
(
RELATION = “PINTO_MENEGOTTO”
) or (
RELATION = “GRILL_PINTO_MEN”
) of the control
STAT_NON_LINE
[U4.51.03]. The whole of the parameters of the model are given via the control
DEFI_MATERIAU
(key word factor
PINTO_MENEGOTTO
) [U4.43.01]. The parameters here are indexed
intervening in the model:
Parameter of the model
Intervenes in
value adopted by defect in Aster
y
0
First loading
_
U
First loading
_
U
First loading
_
H
First loading
_
B
E
E
H
=
Cycles
If no value entered one takes
computed value with the first loading
R
0
Cycles 20
has
1
Cycles 18.5
has
2
Cycles 0.15
L D
/
Cycles with buckling
(if
L D
/
>5)
4 (to be by defect except buckling)
has
6
Buckling 620
C Buckling
0.5
flambage has
0.006
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Parameters
R has has has C
has
0
1
2
6
,
,
,
and
depend on the mechanical properties of steel and are
determined in experiments. The values adopted by defect in Code_Aster are those
proposed in the literature [bib1].
One gives in [Figure 5.2-a] a comparison of the model following the value of
B
E
E
H
=
for two
values:
B
B
=
=
0 01
0 001
.
.
and
.
THERMAL DEFORMATION
E
FFOR
T
NOR
BADLY
(
NR)
EDF
Electricity
from France
Mechanical department and Digital Models
TEST LOADING THERMAL AND CYCLIC ON A BAR (ELEMENT MECA_BARRE)
agraf 29/06/98 (c) EDF/DER 1992-1998
B=0.001
B=0.01
ELASTOPLASTIC BEHAVIOR
MODEL OF PINTO-MENEGOTTO
WITHOUT BUCKLING.
COMPARISON BETWEEN TWO VALUES OF B
- 5
0
5
10
X
4
10
0
5
10
15
20
25
30
35
40x10-4
Appear 5.2-a
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One gives in [Figure 5.2-b] a comparison of the model without buckling and the model and buckling.
THERMAL DEFORMATION (- ALPHA * DT)
E
FFOR
T
NOR
BADLY
(
NR
)
EDF
Electricity
from France
Mechanical department and Digital Models
CYCLIC TEST TRACTION AND COMPRESSION ON A BAR (MECA_BARRE)
agraf 29/06/98 (c) EDF/DER 1992-1998
TOKEN ENTRY WITH BUCKLING
P.M. WITHOUT BUCKLING
MODEL PINTO-MENEGOTTO
COMPARISON OF THE MODEL WITHOUT BUCKLING
AND OF THE MODEL WITH BUCKLING.
- 5
0
5
10
X
4
10
- 2
- 1
0
1
2
3
4x10-3
Appear 5.2-b
5.3 Variables
interns
They 8, and are defined by:
(
)
(
)
buckling
of
indicator
)
plasticity
of
R
(indicateu
opposite
case
in
linear
evolution
one
with
corresponds
time
of
not
if
opposite
case
in
activated
not
is
cyclic
NT
comporteme
if
cycl
=
=
=
=
=
=
=
-
-
=
-
-
+
=
=
=
=
-
-
-
-
8
1
0
7
1
0
6
5
4
3
2
1
1
V
V
V
T
T
V
T
T
V
V
V
V
N
R
N
R
N
R
Code_Aster
®
Version
7.4
Titrate:
Relations of behavior 1D
Date:
02/05/05
Author (S):
J.M. PROIX, B. QUINNEZ, C. CHAVANT
Key
:
R5.03.09-B
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:
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HT-66/05/002/A
6
Relation of behavior of LEMAITRE (
ASSE_COMBU
)
The model presented in this chapter describes the nonlinear viscoelastic behavior 1D of
J. Lemaître developed for the modeling of the fuel assemblies, and applicable to
elements of beams, in the axial direction, with the behavior
ASSE_COMBU
[bib6].
6.1
Formulation of the model
The equations are as follows:
}
0
,
0
1
,
1
,
0
,
.
.
0
1
>
-
-
-
=
+
=
=
·
-
R
Q
m
K
N
E
L
K
E
p
p
p
HT
G
vp
RT
Q
N
m
vp
&
&
&
&
&
&
&
&
The coefficients are provided under the key word
LEMAITRE
of
CHALLENGE
_
MATERIAL
and
&
is flow
neutronics (derivative of the fluence compared to time).
() (
) (
)
S
G
Z
y
X
B
At
T
).
,
,
(
.
+
=
Note:
·
Neutron flux
)
,
,
(
Z
y
X
&
express yourself obligatorily of them 10
20
N/cm
2
/S. This implies
that the units of the other sizes are fixed:
-
E K
,
are in MPa,
-
times are in seconds,
-
co-ordinates in mm
-
T Q
R
,
in Kelvin
Two types of integrations are available according to the value of the key word
PARM_THETA
:
·
purely implicit integration, if
PARM_THETA =1.0
(default value)
·
implicit semi integration, if
PARM_THETA =0.5
Only these two values are authorized.
Code_Aster
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Titrate:
Relations of behavior 1D
Date:
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Key
:
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HT-66/05/002/A
6.2 Integration
implicit
By direct implicit discretization of the relations of behavior, one obtains:
(
)
(
)
(
)
(
)
(
)
()
() (
)
+
-
+
=
-
-
-
=
-
-
-
=
-
+
-
+
+
=
+
+
=
-
-
+
-
-
-
-
-
-
-
-
-
-
B
At
B
At
T
T
T
T
T
T
E
E
T
L
T
K
E
p
p
p
p
S
S
G
ref.
ref.
HT
HT
G
vp
RT
Q
N
m
vp
.
.
)
(
)
(
.
.
0
1
with
One can still bring back oneself there to only one nonlinear scalar equation in
p
, while posing:
(
)
E
G
HT
E
E
E
=
+
-
-
-
-
then the system is reduced to:
(
)
+
=
+
=
+
=
+
+
=
-
-
p
E
p
E
E
T
L
T
K
E
p
p
K
p
vp
E
RT
Q
N
m
1
.
.
1
0
1
and by taking the absolute value of the two members of the last equation, one obtains:
E
E p
=
+
Code_Aster
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Titrate:
Relations of behavior 1D
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Key
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what results in solving the equation:
(
)
T
L
T
K
E
p
p
p
E
K
p
RT
Q
N
m
E
+
+
-
=
-
-
.
.
1
0
1
Once this solved equation (by a method of search for zero of function scalar), one
obtains the stresses by:
= + - = -
E
E
E
E
E p
E p
E p
1
1
6.3 integration
semi-implicit
In fact, in elastoplasticity, one uses the implicit integration of the models of behavior, because
convergence towards the solution of the problem continuous in time, excellent, and is led moreover to
unconditionally stable diagrams.
For viscoelastic or viscoplastic behaviors, utilizing explicitly time
physics, the implicit discretization always leads to unconditionally stable diagrams, but
convergence towards the solution is not also any more fast. It is preferable to use an integration then
semi-implicit. It is the choice which we made here, following in that the integration of the model of
Lemaître in Aster and Cyrano3 [bib5]. The method implemented here is not one
general theta-method: it functions only for theta=0.5. It makes it possible however to obtain
correct results. For more general information, it would be necessary to use more sophisticated method, by
example method of RUNGE KUTTA of command 2 or 4.
Here, one writes simply:
(
)
(
)
(
) () ()
(
)
+
-
+
=
-
-
-
=
-
-
-
=
-
+
+
+
=
+
+
=
-
-
-
-
-
+
-
-
-
-
-
B
At
B
At
T
T
T
T
T
T
E
E
T
L
T
K
E
p
p
K
p
p
S
S
G
ref.
ref.
HT
HT
G
vp
T
T
R
Q
N
m
vp
.
.
.
)
(
)
(
2
2
2
2
2
1
.
.
2
2
1
2
2
0
2
1
Code_Aster
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Version
7.4
Titrate:
Relations of behavior 1D
Date:
02/05/05
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Key
:
R5.03.09-B
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HT-66/05/002/A
One seeks to calculate
-
+
2
. One can write:
2
2
2
2
2
2
2
2
=
+
=
+
-
-
-
-
-
-
E
E
E
E
E
E
vp
G
HT
thus
-
-
-
-
+
=
+
+
-
-
-
2
2
2
2
2
2
2
E
E
E
E
G
HT
v
As previously, one solves while posing:
E
G
HT
E
E
E
=
+
+
-
-
-
-
-
2
2
2
2
2
then the system is reduced to:
2
.
.
2
2
1
2
0
2
1
T
L
T
K
E
p
p
K
p
T
T
R
Q
N
m
+
+
+
=
+
-
-
-
E
E p
E p
=
+
+
+
+
=
+
+
+
-
-
-
-
-
2
2
2
2
2
1
2
2
from where:
E
E p
=
+
+
-
2
2
The equation to be solved is exactly same form as the implicit equation:
2
.
.
2
2
1
2
0
2
1
T
L
T
K
E
p
p
p
E
K
p
T
T
R
Q
N
m
E
+
+
-
=
+
-
-
Code_Aster
®
Version
7.4
Titrate:
Relations of behavior 1D
Date:
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Key
:
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HT-66/05/002/A
Once this solved equation, one obtains the internal variables while multiplying by 2 the value obtained
and stresses by:
-
-
-
-
+
=
-
+
=
+
-
2
1
2
2
E
E
E p
One can thus use the same routines of resolution as in the implicit case, while calculating
simply
E
in
2
,
G
2
,
HT
2
.
On an elementary test of creep (test SSNL109A), one obtains by the semi-implicit method one
correct result (to 0.02% of the analytical solution) if one uses 2 pitches of time (instead of 100 pitches of
times required to have a correct solution with implicit integration).
6.4 Variables
interns
Two variables intern are calculated in this model: p and neutron fluence calculated with the pitch
current time.
6.5
Identification of the parameters of the model
It is done starting from creep tests (uniaxial test with constant stress imposed under flow
neutronics constant). By integration of the equations of the model, one obtains then:
()
m
N
m
N
RT
Q
vp
E
L
K
m
m
N
T
+
-
+
+
=
0
&
Code_Aster
®
Version
7.4
Titrate:
Relations of behavior 1D
Date:
02/05/05
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J.M. PROIX, B. QUINNEZ, C. CHAVANT
Key
:
R5.03.09-B
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:
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R5.03 booklet: Nonlinear mechanics
HT-66/05/002/A
7
Relation of behavior of the LMA-RC (
ASSE_COMBU
)
The model presented in this chapter describes the viscoplastic behavior 1D LMA-RC
(Laboratory of Mechanics Applied R. Chaléat of Besancon) developed for the modeling of
fuel assemblies, and applicable to the elements of beams, in the axial direction, with
behavior
ASSE_COMBU
[bib6].
7.1
Formulation of the model
The élasto-viscoplastic model developed with the LMA-RC to describe the orthotropic behavior of
tubes of sheaths of the fuel pin [R5.03.10] is written in 1D isotropic:
}
(
)
()
()
(
)
(
)
()
()
()
()
(
)
(
)
()
()
()
(
)
-
=
-
-
=
-
-
-
=
-
=
-
-
=
=
-
-
-
=
p
X
p
Y
Q
X
p
X
X
p
Y
Q
X
X
X
X
X
R
p
X
X
p
Y
Q
X
K
X
p
X
X
p
E
p
p
m
m
p
N
p
HT
G
p
&
&
&
&
&
&
&
&
&
&
&
&
&
&
&
&
&
2
v
2
2
2
1
v
1
1
0
1
v
0
v
v
sinh
H
sin
.
with:
()
(
)
Y
Y
Y
Y E
LP
v
=
+
-
0
The coefficients, as in 3D, are provided by key word LMARC (one does not use the coefficients here
dependant on the anisotropy) (
Q Q Q
,
1
2
correspond respectively to the parameters
p p p
,
,
1
2
key word
LMARC).
The law of growth is identical to that used for the model of Lemaitre:
() (
) (
)
(
)
S
G
Z
y
X
B
At
T
,
,
+
=
Neutron flux
is the product of a function of X (clevis pin, having to be confused
with one of the axes of the total reference mark) and a function of y and Z.
Note:
·
The fluence is worth
)
,
,
(
Z
y
X
.
·
Only one diagram of integration is available: a purely implicit diagram.
·
Neutron flux
(,)
X y Z
express yourself obligatorily of them 10
20
N/cm
2
/S. This implies that
the units of the other sizes are fixed:
-
E K
,
are in MPa,
-
times are in seconds,
-
co-ordinates in Misters.
Code_Aster
®
Version
7.4
Titrate:
Relations of behavior 1D
Date:
02/05/05
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J.M. PROIX, B. QUINNEZ, C. CHAVANT
Key
:
R5.03.09-B
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:
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R5.03 booklet: Nonlinear mechanics
HT-66/05/002/A
7.2 Integration
implicit
To integrate these relations of behavior, while bringing back itself if possible to only one equation to
to solve, it is necessary to make an assumption on
(
)
=
-
-
X
X
. Indeed, it can take only two values:
+1 or - 1. One thus supposes this known sign (initialized by
(
)
=
-
-
-
-
-
-
X
X
). If one cannot solve
the equation obtained with this assumption, one takes the opposite sign. The remainder of the equations can
to be integrated in a purely implicit way. The system is written:
(
)
()
()
(
)
(
)
()
(
)
()
()
()
()
(
)
(
)
=
+
=
+
-
-
=
-
=
-
=
=
-
-
-
=
=
-
-
=
-
-
-
E E
E
X
K
F
X
X
Q
Y p
X
X
R
X
X
X
X
F p X X
X
Q
Y p
X
X
F p
G
HT
E
N
v
m
m
- p
p
p
T sinh T
p
p
&
,
sinh
,
0
1
0
1
1
1
1
2
1
()
()
(
)
()
()
()
(
)
()
(
)
,
,
,
X
X
X
Q
Y p
X
F p X
1
2
2
2
2
2
2
=
-
=
p
There is thus a system of 5 equations to 5 unknown factors:
()
()
,
,
,
p, X
X
X
1
2
The second equation is also written:
-
=
+
+
X
K
N
N
log
&
&
p
T
p
T
0
1
0
2
1
By using the first equation, one can express
X
according to
p
:
- =
-
=
-
=
-
-
+
+
-
X
E
X
X
F
E
X
K
E
E
N
N
p -
X
p) =
p -
p
T
p
T
1
0
1
0
2
1
(
log
&
&
Code_Aster
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Titrate:
Relations of behavior 1D
Date:
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Key
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HT-66/05/002/A
In addition, by integration successive of the functions
F
2
and
F
1
one can also bring back oneself to one
equation utilizing only
X
and
p
:
()
()
()
(
)
X
Q
Y p
X
Q
2
2
2
2
1
=
-
+
-
p
p
()
()
()
()
()
(
)
(
)
X
Q
Y p
X
X
X
Q
1
1
1
2
2
1
1
=
-
-
-
+
-
-
p
p
like
()
(
)
X
F p X X
=
,
1
, and
()
()
X
G p
1
=
according to the preceding expressions one can
to write:
()
()
X
F
p
F
p
=
=
2
1
. The equation to solve to find
p
is thus:
()
() ()
F
F
F
p
p
p
=
-
=
2
1
0
Once calculated
p
, one obtains the stresses by:
=
-
E
E
p
7.3 Variables
interns
They are 5:
()
()
2
1
4
3
2
p
1
X
V
X
V
X
V
V
=
=
=
=
In Code_Aster, one adds a last internal variable:
=
5
V
neutron fluence calculated with
no current time.
7.4
Identification of the parameters of the model
The identification of the parameters is carried out in the reference [bib7]. It relates to the ZIRCALOY 4 with
350°C.
Code_Aster
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Titrate:
Relations of behavior 1D
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Key
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HT-66/05/002/A
8
Method to use in 1D all the behaviors 3D
As for the processing of the plane stresses [R5.03.03], it is possible to profit for
modelings 1D of the behaviors available in 3D. One extends for that the method due to R. of
Borst with the case 1D, by treating this condition (unidimensional stress field) not with the level
law of behavior but on the level of balance. One obtains thus during iterations of
the algorithm of
STAT_NON_LINE
stress fields which tend towards a field
one-way. It is checked, with convergence of the total iterations of Newton, that the fields of
stresses are indeed one-way, except for a precision, if not the iterations are continued.
The method consists in breaking up the fields of strains and stresses into a part
purely one-way (direction X) and a part relating to the other directions, and to carry out one
static condensation by writing that components of the stresses relating to the other directions
are null. One does not consider in the tensors (command 2) only the diagonal terms, written under
form vectors with 3 components. Direction X corresponds to the direction of the element (bars,
multifibre beam) or with the direction of the reinforcements of grid. At one unspecified moment of the resolution of
incremental behavior, the tangent operator
D
connect the increase in stresses to
the increase in deformation by:
Dd
D
D
=
=
that one rewrites:
=
Z
y
X
Z
y
X
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
33
32
31
23
22
21
13
12
11
. By writing these increases like the difference between
iterations
N
and
1
+
N
from Newton, one obtains:
N
N
N
N
N
N
D
D
-
=
-
=
-
=
+
+
+
1
1
1
,
With convergence, this variation must tend towards zero.
By introducing the conditions
1
2
1
0
:
+
+
=
N
N
y
and
(one-way behavior), one obtains,
for the iteration
1
+
N
:
=
-
-
-
=
-
-
-
=
+
+
+
+
Z
y
X
nz
N
y
nx
N
X
nz
N
Z
N
y
N
y
nx
N
X
Z
y
X
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
33
32
31
23
22
21
13
12
11
1
1
1
1
The two last equations make it possible to express
Z
y
D
D
and
according to
X
D
:
(
)
(
)
(
)
(
)
22
31
21
32
33
21
31
23
32
23
22
33
22
32
23
33
32
31
23
21
,
,
1
1 33
1
22
1
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
Z
y
X
Z
N
Z
N
y
Z
X
y
N
Z
N
y
y
y
X
N
Z
Z
Z
X
N
y
y
-
=
-
=
-
=
+
-
-
=
+
+
-
=
-
-
-
=
-
-
-
=
with
that is to say
Code_Aster
®
Version
7.4
Titrate:
Relations of behavior 1D
Date:
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Key
:
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HT-66/05/002/A
by deferring these expressions in the first equation, one obtains:
N
y
nz
X
Z
y
N
X
nx
D
D
D
D
D
D
D
D
D
D
D
D
D
D
-
+
-
+
+
+
+
=
+
33
12
32
12
13
22
23
12
13
12
11
1
Balance with the iteration
1
+
N
is written:
-
+
-
+
+
=
-
+
-
+
+
+
+
=
=
+
+
+
FD
D
D
D
D
D
D
D
D
B
K
FD
D
D
D
D
D
D
D
D
B
D
D
D
D
D
D
B
FD
B
FD
B
N
y
nz
nx
T
N
N
N
y
nz
nx
T
X
Z
y
T
N
X
T
N
T
33
12
32
12
13
22
23
12
1
33
12
32
12
13
22
23
12
13
12
11
1
1
It is thus noted that the taking into account of the unidimensional behavior intervenes to two
levels:
·
in the tangent matrix, by the corrective term:
FD
B
D
D
D
D
B
Z
y
T
+
13
12
·
in the writing of the second member, by the corrective term:
(
)
(
)
(
)
FD
D
D
D
D
D
D
D
D
B
N
y
nz
T
33
12
32
12
13
22
23
12
-
+
-
To implement this method, it is enough to calculate these corrective terms and to add them to
stresses and tangent matrix obtained of the resolution 3D of the behavior. For that it is
necessary to store information of an iteration of Newton to the other, by the means of 4 variables
additional interns. The stages of the resolution are:
1) with
the iteration
1
+
N
, the data are:
-
-
+
,
,
1
N
U
and the 4 variables intern (calculated with
the iteration
N
):
(
)
(
)
=
-
-
+
=
=
-
-
+
=
Z
N
X
Z
nz
N
y
N
Z
y
N
X
y
N
y
nz
N
y
D
V
D
D
D
V
D
V
D
D
D
V
4
,
1
3
,
2
,
1
1
22
32
33
23
,
2) before carrying out the integration of the behavior (carried out into axisymmetric) one calculates
(
)
(
)
X
Z
nz
N
y
N
Z
N
Z
X
y
N
Z
N
y
N
y
N
y
D
D
D
D
D
D
D
D
+
-
-
+
=
+
+
-
+
=
+
+
22
32
1
23
33
1
1
1
,
3) the integration of the behavior provides the stresses
1
+
N
and the tangent operator
D
,
4) one modifies the second member and the tangent matrix as indicated above,
5) one stores the new variables intern and one checks if
RELA
RESI_INTE_
=
=
<
<
+
+
+
,
,
1
1
1
nx
N
y
nz
with
and
.
Code_Aster
®
Version
7.4
Titrate:
Relations of behavior 1D
Date:
02/05/05
Author (S):
J.M. PROIX, B. QUINNEZ, C. CHAVANT
Key
:
R5.03.09-B
Page
:
36/36
Manual of Reference
R5.03 booklet: Nonlinear mechanics
HT-66/05/002/A
9 Bibliography
[1]
J. GUEDES, P. PEGON, EP PINTO: '' A Fiber/Timoshenko Beam Element in Castem 2000 '',
Joint Research Centers, European Commission, Institute for Safety Technology, 1994.
[2]
A. GIUFFRE, EP PINTO: '' It Comportemento del Cemento Armatoper Sollecitazioni Cicliche
di Forte Intensita ''', Giornale del Genio Civil, Maggio 1970.
[3]
Mr. MENEGOTTO, EP PINTO: '' Method off Analysis for Cyclically Loaded Reinforced
Concrete Frames Including Changes in Geometry and Nonelastic Behavior Planes off
Elements under Combined Normal Forces and Bending '', IABSE Symposium one Resistance
and Ultimate Deformability off Acted Structures One by Well-Defied Repeated Loads, Final
Carryforward, Lisbon, 1973.
[4]
G. MONTI, C. NUTI: '' Nonlinear Cyclic Behavior off Reinforcing Bars Including Buckling '',
Newspaper off Structural Engineering, vol. 118, No 12, December 1992.
[5]
P. BONNIERES, Mr. ZIDI: “Introduction of viscoplasticity into the modules of
thermomechanics of CYRANO3: principle, description and validation “Note HI-71/8334.
[6]
J.M. PROIX, B.QUINNEZ, P. MASSIN, P. LACLERGUE: “Fuel assemblies under
irradiation. Feasibility study “. Note HI-75/97/017/0
[7]
I. the PICHON, P. GEYER: “Modeling of the anisotropic viscoplastic behavior of
tubes of sleeving of the fuel pins “Notes HT-B2/95/018/A