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
Organization (S): EDF-R & D/AMA, SINETICS
Handbook of Référence
R5.03 booklet: Nonlinear mechanics
Document: R5.03.09
Nonlinear relations of behavior 1D
Summary:
This document describes the quantities calculated by 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 BARRE, with
elements of beam and beams multifibre (direction axial only) and to the elements of concrete reinforcement
(modeling GRILL).
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 model the action of the ground on Câbles with Isolation Gazeuse,
· the behavior of PINTO-MENEGOTTO which makes it possible 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 the LMA-RC and 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.
Handbook of Référence
R5.03 booklet: Nonlinear mechanics
HT-66/05/002/A
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
: 2/36
Count
matters
1 Use of the relations of behavior 1D ....................................................................................... 3
1.1 Relations of behavior 1D in Code_Aster ...................................................................... 3
1.2 General notations ......................................................................................................................... 3
1.3 Change of variables ................................................................................................................ 4
1.3.1 Calculation of the deformations (small deformations) ..................................................................... 4
1.3.2 Calculation of the generalized efforts (forced integrated) ........................................................... 4
2 Relation of behavior of Von Mises with linear isotropic work hardening: VMIS_ISOT_LINE or
unspecified: VMIS_ISOT_TRAC ........................................................................................................... 5
2.1 Equations of model VMIS_ISOT_LINE ........................................................................................ 5
2.2 Integration of relation VMIS_ISOT_LINE ................................................................................... 6
2.3 Variables intern ............................................................................................................................ 8
3 Relation of behavior of Von Mises, linear kinematic work hardening 1D: VMIS_CINE_LINE9
3.1 Equations of model VMIS_CINE_LINE ........................................................................................ 9
3.2 Integration of relation VMIS_CINE_LINE ................................................................................. 10
3.3 Variables intern .......................................................................................................................... 12
4 Relation of behavior of Von Mises with asymmetrical linear work hardening: VMIS_ASYM_LINE..13
4.1 Equations of model VMIS_ASYM_LINE ....................................................................................... 13
4.1.1 Asymmetrical behavior in traction and compression ................................................ 13
4.2 Integration of behavior VMIS_ASYM_LINE .......................................................................... 15
4.3 Variables intern .......................................................................................................................... 16
5 Model of PINTO_MENEGOTTO ........................................................................................................... 17
5.1 Formulation of the model .................................................................................................................. 17
5.1.1 Monotonous loading ......................................................................................................... 17
5.1.2 Cyclic loading ............................................................................................................ 18
5.1.3 Case of inelastic buckling ................................................................................................ 21
5.2 Establishment in Code_Aster .................................................................................................. 23
5.3 Variables intern .......................................................................................................................... 25
6 Relation of behavior of LEMAITRE (ASSE_COMBU) ................................................................. 26
6.1 Formulation of the model .................................................................................................................. 26
6.2 Implicit integration ........................................................................................................................ 27
6.3 semi-implicit integration ............................................................................................................... 28
6.4 Variables intern .......................................................................................................................... 30
6.5 Identification of the parameters of the model ...................................................................................... 30
7 Relation of behavior of LMA-RC (ASSE_COMBU) ..................................................................... 31
7.1 Formulation of the model .................................................................................................................. 31
7.2 Implicit integration ........................................................................................................................ 32
7.3 Variables intern .......................................................................................................................... 33
7.4 Identification of the parameters of the model ...................................................................................... 33
8 Method to use in 1D all the behaviors 3D ................................................................... 34
9 Bibliography ........................................................................................................................................ 36
Handbook of Référence
R5.03 booklet: Nonlinear mechanics
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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
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 operator STAT_NON_LINE
[U4.51.03] under the key word factor COMP_INCR, by 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 nor.
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 par. One has as follows:
Q =
(
Q +) = Q () + Q = Q
T
T
T
+ Q
.
tensor of the constraints (in 1D, one is interested only in the single component
nonnull uniaxial).
~
1
deviative operator: ~
ij = ij - kk ij.
3
()
equivalent value of Von Mises, equalizes in 1D with the absolute value
eq
increment of deformation.
With
tensor of elasticity, equal in 1D to the Young modulus E
, µ, E, K
moduli of the isotropic elasticity.
thermal dilation coefficient secant.
T
temperature.
()
positive part.
+
p
cumulated plastic deformation
p
plastic deformation
Handbook of Référence
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Code_Aster ®
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7.4
Titrate:
Relations of behavior 1D
Date:
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Author (S):
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:
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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
constraints 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)
0
=
,
L
and the increase in deformation by:
(
U L) - (
U)
0
=
,
L
For the elements of grid (modelings GRILL and GRILL_MEMBRANE), one calculates the deformation
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)
= E (T) + HT (T) + p (T), with
E (
1
T) =
-
To 1 (T (T) (T) =
T
E (T) ()
HT (T) = (T (T) (T (T) - Tref) Id
1.3.2 Calculation of the generalized efforts (forced integrated)
For integration of the behavior 1D, it is necessary to integrate the component of constraints 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:
- NR
F = NR
For the elements of GRILL, one calculates the efforts as for the elements of hulls DKT (efforts
membrane) by integration of the constraints in the thickness (only one layer and only one point
of integration).
Handbook of Référence
R5.03 booklet: Nonlinear mechanics
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Code_Aster ®
Version
7.4
Titrate:
Relations of behavior 1D
Date:
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Author (S):
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:
R5.03.09-B Page
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2
Relation of behavior of Von Mises with work hardening
isotropic linear
: VMIS_ISOT_LINE or unspecified
:
VMIS_ISOT_TRAC
2.1
Equations of model VMIS_ISOT_LINE
They are the restriction of the behavior 3D [R5.03.02] on the uniaxial case:
~
p
3
& =
p&
= p&
2
eq
= - p
- HT
E
eq - R (p) = - R (p)
0
p & = 0 if eq - R (p) < 0
p & 0 if eq - R (p)
= 0
with:
·
&p:
speed of plastic deformation,
·
p:
cumulated plastic deformation,
·
HT = (T -
thermal deformation,
Re
T F):
E.E
·
R (p)
T
=
. p +:
E - E
y
function of linear work hardening isotropic, or R (p) refines by
T
pieces, deduced from the traction diagram.
In 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 operator DEFI_MATERIAU [U4.43.01].
/ECRO_LINE = (D_SIGM_EPSI = AND, SY = y)
/ECRO_LINE_FO = (D_SIGM_EPSI = AND, SY = y)
In case VMIS_ISOT_TRAC, the data of the characteristics of materials are provided under
key word factor TRACTION of operator DEFI_MATERIAU [U4.43.01].
TRACTION = _F (SIGM = courbe_traction)
courbe_traction represents the traction diagram, point by point. The first point allows
to define the elastic limit y and the Young modulus E [R5.03.02].
Handbook of Référence
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Code_Aster ®
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Titrate:
Relations of behavior 1D
Date:
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Author (S):
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:
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ECRO_LINE_FO corresponds if AND 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
E
T
y
E
L
y
L = E L
if L <
E
.
y
y
L = y + LTE - if L
E
E
When the criterion is reached one a:
L
L - R (p)
= 0, therefore L - R L -
E = 0, from where:
E E
R (p)
T
=
p + = H. p +
E - E
y
y
T
In the case of a traction diagram, the step is identical to [R5.03.01].
2.2
Integration of relation VMIS_ISOT_LINE
By direct implicit discretization of the relations of behavior, a way similar to integration 3D
[R5.03.02] one obtains:
-
+ - R (-
p +p) 0
-
HT
E
+
E (-) - (-
+) +
-
= Ep
-
-
E
+
p
-
0 if + = R (-
p +p)
p =
-
0 if + < R (-
p +p)
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Titrate:
Relations of behavior 1D
Date:
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:
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Two cases arise:
HT
·
-
R (-
+
<
p + p) in this case p = 0 is = (
-)
-
+
-
thus
-
+ (
- HT
) <R (-
p
-
),
·
-
R (-
+
=
p + p) in this case p 0
-
thus
+ (
- HT
) R (-
p
-
).
One deduces the algorithm from it from resolution:
-
let us pose E
=
+
-
(
HT
-
)
if E R (p) then
p
= 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).
E
= R (p
+ p
) +E p
One thus deduces some:
E
- (+ HP
y
)
· in the case of a linear work hardening: p
=
E + H
· and in the case of an unspecified work hardening, the curve R (p) being refined per pieces, one
solves the equation out of p directly
:
p
+ R (p
E
+)
E
p = in the same way that in
3D [R5.03.02].
Handbook of Référence
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Relations of behavior 1D
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Let us notice in the passing that:
E
=
E
R (p)
then
E
E
= (- +
) =
R (p) =
E
E p
1 + R (p)
Moreover, the option
N
FULL_MECA makes it possible to calculate the tangent matrix Ki with each iteration.
The tangent operator who is used for building it is calculated directly on the preceding discretized system.
One obtains directly:
E
-
if > R (p)
= AND
if not
= E
Note:
The option
0
RIGI_MECA_TANG which makes it possible to calculate the tangent matrix Ki used in
phase of prediction of the algorithm of Newton, takes account of the indicator of plasticity with
the previous moment:
· if = 1
= T
E
· if = 0
= E
2.3 Variables
interns
The relation of behavior VMIS_ISOT_LINE produces two internal variables: p and.
Handbook of Référence
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Titrate:
Relations of behavior 1D
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:
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3
Relation of behavior of Von Mises, work hardening
linear kinematics 1D: VMIS_CINE_LINE
3.1
Equations of 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
(
3 ~ ~
F, R, X) = (~ - X) -
with
With
=
WITH A
eq
y
eq
2
F
~
p
3
- X
& = &p
= &p
2
(~ - X) eq
if
F<0
&p = 0
if
F
= 0
&p 0
In the uniaxial case, the tensors are written:
2 3
~
3
= D
X =
D
p
X
= p D
with
D =
-
1 3
2
- 1
3
As long as the loading is monotonous, the following relations immediately are obtained:
3
3
E.E
p
p
=
X = C p
= C p
+
T
y = F () = y +
p
2
2
E - AND
2 EET
EE
3
C is determined by: C =
. One poses: H
T
=
= C
3rd - AND
E - AND
2
Handbook of Référence
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Titrate:
Relations of behavior 1D
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:
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The relation of behavior 1D is written then:
p
- X
& = p&
- X
= (- HT
- p
)
E
3
X =
p
C =
p
H
2
- X - y 0
p & = 0 if - X - y< 0
p & 0 if - X - y = 0
The data of the material characteristics are those provided under the key word factor
ECRO_LINE or ECRO_LINE_FO of operator DEFI_MATERIAU [U4.43.01]:
/ECRO_LINE = (D_SIGM_EPSI = AND, SY = y)
/ECRO_LINE_FO = (D_SIGM_EPSI = AND, SY = y)
3.2
Integration of 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:
-
+ - -
X - X - y 0
E p
= E (- HT
) - (-
+) E -
+
-
E
-
+ - -
X - X
p
= p
-
+ - -
X - X
-
X
X
-
= p
-
H
H
-
-
p 0 if + - X - X = y
-
-
p = 0 if + - X - X < y
with
HT
= (T -
-
-
ref.
T
) - (T - ref.
T
)
Handbook of Référence
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Titrate:
Relations of behavior 1D
Date:
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:
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Two cases arise:
·
-
-
+ - X - X < y in this case p = 0 is
= (- HT
) E - H -
+
-
X
-
-
E
H
E
H
thus
-
-
- X
+ - HT
< R p
-
(
) (-
-
),
E
H
· if not p 0.
E
-
H
To simplify the writings one will pose: E
=
-
X - + E
.
-
-
(
HT
-
)
E
H
One deduces the algorithm from it from resolution:
E
H
HT
E
· if
y then p =,
0
-
X = X
, = E -
+
-
(
)
-.
-
H
E
· if not it is necessary to solve:
p
HT
E
-
- H
E = E (-) - = - (+) + X
-
H
-
+ - -
X - X
p
- X
= p
= p
-
+ - -
X - X
- X
H
-
p
X -
X =
H
-
H
-
+ - -
X - X - y = 0
H
Let us notice that:
-
p
X = X - H
. One deduces then from the first equation:
H -
E
p
= - X + (E + H)
One thus obtains, while eliminating - X from the second equation:
p
E
p
= (
E + H) p + y
By replacing p in the relation between E and - X, one obtains:
-
y
X = E
(E + H) p
+ y
By taking the absolute value of the two members of the preceding equation, one finds p:
(E H) p
E
+
+ y =
Handbook of Référence
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Relations of behavior 1D
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:
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Once p determined, one can calculate:
E
p
= p
E
-
E
-
HX
X = X + X
=
+ H p
-
E
H
- X
E
and while using: =
, one obtains directly:
E
y
E
= y
+ X
E
Moreover, the option
N
FULL_MECA makes it possible to calculate the tangent matrix K I with each iteration.
The tangent operator who is used for building it is calculated directly on the preceding discretized system.
One obtains directly:
E
-
if > R (p)
= AND
if not
= E
The option
0
RIGI_MECA_TANG which makes it possible to calculate the tangent matrix K I used in the phase of
prediction of the algorithm of Newton is obtained using the indicator of plasticity -
moment
precedent:
· if -
=1 then
= T
E
· if -
= 0 then
= E
3.3 Variables
interns
The relation of behavior VMIS_CINE_LINE produces two internal variables: X and.
Handbook of Référence
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Titrate:
Relations of behavior 1D
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:
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4
Relation of behavior of Von Mises with work hardening
linear asymmetrical: VMIS_ASYM_LINE
4.1
Equations of model VMIS_ASYM_LINE
4.1.1 Behavior
asymmetrical
in traction and compression
It is a behavior uncoupled in traction and compression, built starting 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 RT and RC. The two fields are independent one
other.
YT
elastic limit in traction. In absolute value.
TESTSTEMYÇ
elastic limit in compression. In absolute value.
Pt
Variable interns in traction. Algebraic value.
PC
Variable interns in compression. Algebraic value.
ETT
Slope of work hardening in traction.
Etc
Slope D `work hardening in compression.
The equations of the model of behavior are:
·
678
&p = & - -
1
- &th
&p = &p p
C + &
T
p
&C = &pC
p
&T = &pT
- T
R (Pt) 0
- - RC (PC) 0
with
&pC = 0 if - - C
R (PC) <
0
&pC 0 if - = C
R (PC)
&pT = 0 if - T
R
0
(Pt) <
&pT 0 if = RT (Pt)
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& p: speed of plastic deformation in compression, p
C
&:
T speed of plastic deformation in traction,
HT:thermal deformation of origin: HT = (T - Re
T F).
It is noticed that one cannot have simultaneously plasticization in traction and compression: that is to say
&pC = 0, either &pT = 0, or both is null.
The data of the material characteristics are those provided under the key word factor
ECRO_ASYM_LINE of operator DEFI_MATERIAU [U4.43.01].
ECRO_ASYM_LINE = _F (DT_SIGM_EPSI = HT, ETT,
SY_T = yT,
DC_SIGM_EPSI = HC, ETC,
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:
E
E
R (p)
TT
=
p +
= H. p
T
+
E - E
T
yT
T
T
yT
TT
E
E
R (p)
TC
=
p +
= H. p
C
+
E - E
C
teststemyç
C
C
teststemyç
TC
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4.2
Integration of behavior VMIS_ASYM_LINE
By direct implicit discretization of the asymmetrical relation of behavior, a way similar to
the preceding one, one obtains:
p = p
p
T + C
p
HT
= - - E
- +
Pt = Pt
- +
(- +
) -
-
-
T
R (Pt + Pt) 0
p 0
if
-
-
-
T
(+) - TR (Pt + Pt) = 0
p = 0
if
-
-
-
T
(+) - TR (Pt + Pt) < 0
- +
PC = PC
- +
- (- +) -
-
-
C
R (PC + PC) 0
p
0
if - (- +
) - RC (-
PC +
-
PC)
C
= 0
PC = 0
if - (- +
) - RC (-
PC +
-
PC) < 0
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 step) for the incremental problem to solve.
Four cases arise:
· - HT > 0:
E is posed
-
HT
T =
+ E (
-
)
-
E
E
T <
-
T
R (Pt) in this case Pt = 0 thus = T and = E
- if not
:
E
-
T - (+ H p
yT
T T)
p =
, p
T
= 0
E + H
C
T
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E
E
T
=
= T R p
E p
E
T (T)
T
1 +
T
T
R (Pt)
= ETT
· - HT < 0
E is posed
-
HT
C =
+ E (
-
)
-
- E <
E
C
R (-
C
PC) in this case PC = 0 thus = C and = E
- if not
:
E
-
C - (+ H p
teststemyç
C C)
p =
, p
C
= 0
E + H
T
C
E
E
C
=
= C R p
E p
E
C (C)
C
1 +
C
C
R (PC)
= etc
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 produces 2 internal variables: PC Pt.
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 zones (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]). The 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 zone
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 zones
following:
· The linear elasticity, defined by the Young modulus elastic Y.E and limit.
=
E (zone 1, [Figure 5.1.1-a])
· The plastic bearing, ranging between the elastic strain limits 0y and the deformation
of work hardening H, higher plate in deformation limits. During the bearing
constraint remains constant.
= 0y (zone 2, [Figure 5.1.1-a])
· Work hardening, describing the traction diagram up to the ultimate point of constraint and of
deformation, (U, U). This part is represented by a polynomial of the fourth degree:
4
0
U -
= - (-)
U
U
y
(zone 2, [Figure 5.1.1-a])
U - H
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The slope of work hardening (used thereafter, for the cyclic behavior) is defined here by:
0
U - y
Eh =
0. It is the average slope of zones 2 and 3 of the following figure.
U - y
U
0y
zone 3
zone 2
zone 1
0
y
H
U
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 zone. The discharge remains in this case
rubber band,
· the starting position is in the plastic zone (0y). 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 zone 2 or of zone 3).
The relation which the deformation must satisfy so that the curve of Giuffré is activated is as follows:
0
max - > y, with
.
3 0
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
0y
3
max
nonlinear discharge:
activation of the curve of
Giuffré
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]:
ny: Elastic limit of the nth semi-cycle. (Calculation clarified with [§5.1.2.2])
n-1
R
: Constraint at the last point of inversion (forced maximum attack with the n-1ième semi-cycle).
n-1
R
: Deformation at the last point of inversion (maximum deformation attack with the n-1ième semi-cycle).
N - n-1
N
N
N
N
y
R
- 1
y: Deformation corresponding to y: y = R
+
E
F (T): Plastic excursion of the nth cycle
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n-1
p
n-1
n-1
(N, 1 n-1
(,
R
)
R
n-1
y)
y
n-1
Eh
n+1
N
Eh
(N, N
y
)
y
(N, N
R)
R
Np
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 Eh (asymptotic slope of work hardening).
N is thus determined
N
n-1
n-1
n-1
y in the following way: y = y
.
(
sign - p) +
where the function
(
sign X) = 1 if x<0 and 1 if x>0 and where n-1 is the plastic increment of constraint of the semi-cycle
precedent [Figure 5.1.2.1-a] which is defined by: n-1
N
E
-
=
1
H p
.
For each semi-cycle one thus determines N
n-1
n-1
N
y according to y
and p, one deduces y from them, 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
N
N
N
p = R - Y.
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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:
1 -
B
* = B * +
1 R
(
*
1 + ()
/
* R)
E
With B
H
=
report/ratio of the slope of work hardening on the slope of elasticity.
E
- n-1
* =
R
N
N 1
y - -
R
- n-1
* =
R
N
N 1
y - -
R
n-1
N
p
1
p
= N N 1
y - -
R
The size R makes it possible 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:
A1.
R () = R0 - G () where G () =
A2 +
Parameters R, A and A
0
1
2 are constants without unit depending on the mechanical properties
steel. Their values are obtained in experiments and Menegotto [bib3] proposes:
R = 20 0
.
To = 18 5
.
With
0
1
2 = 0 1
. 5
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 zone
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]:
E0: Initial elastic Young modulus (correspondent with E without buckling).
bc: Report/ratio of the slope of work hardening on the elastic slope in compression.
LT: Report/ratio of the slope of work hardening on the elastic slope in traction (refill after compression
with buckling).
Er: Modulus Young reduced in traction (slope of the curve of refill after compression with
buckling).
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S
E
B × E
C
5
ny
B × E
E
R
S
Appear 5.1.3-a
5.1.3.1 Compression
One introduces a negative slope B
E
C ×
, where bc is defined by:
E
B
'0
B = has (5 0
. -
-
L/D) E
y
C
0y
With
N
= 4 0
.
and '= max
the greatest plastic way carried out during
L/D
(p)
loading.
It is necessary then, as in the model without buckling, to determine ny. The method is identical, but
one adds a complementary constraint * S in order to position the curve correctly compared to
the asymptote [Figure 5.1.3-a].
-
B
*
C
B
.
110 - L/D
S = S B E
where S is given by: S =
Cl/D
1 - C
B
(
10th
- .1) 0
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And one thus has: N
N
*
y = (y)
+
without buckling
S
N - n-1
R
This modifies also the value of N
N
y
- 1
y = R
+
E
5.1.3.2 Traction
At the time of the semi-cycle in traction according to one adopts a reduced Young modulus defines by:
- has 2
E = E has
6
0
5 + (10
. - a5) (
p)
E
with a5 10
.
(50. L/D
R
)
=
+
-
/7 5
.
Note:
The parameters has, C and a6 is constants (without unit) depend on the properties
mechanics of steel and is in experiments given. Values adopted by
Monti and Nuti [bib4] are:
has = 0 006
.
C = 0 500
.
= 620 0 have
6
.
5.2
Establishment in Code_Aster
This model is accessible in Code_Aster starting from key word COMP_INCR
(RELATION = “PINTO_MENEGOTTO”) or (RELATION = “GRILL_PINTO_MEN”) of the command
STAT_NON_LINE [U4.51.03]. The whole of the parameters of the model are given via the command
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
0
First loading
_
y
First loading
_
U
First loading
_
U
First loading
_
H
E
Cycles
If no value entered one takes
B
H
=
computed value with the first loading
E
R
Cycles 20
0
has
Cycles 18.5
1
has
Cycles 0.15
2
L/D
Cycles with buckling
4 (to be by defect except buckling)
(if L/D >5)
has
Buckling 620
6
C Flambage
0.5
Flambage has
0.006
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The parameters R, has, has, has, C and has
0
1
2
6
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].
E
One gives in [Figure 5.2-a] a comparison of the model following the value of B
H
=
for two
E
values: B = 0 0
. 1 and B = 0 001
.
.
EDF
Department Mécanique and Modèles Numériques
Electricity
TEST LOADING THERMAL AND CYCLIC ON A BAR (ELEMENT MECA_BARRE)
from France
ELASTOPLASTIC BEHAVIOR
X
4
10
10
MODEL OF PINTO-MENEGOTTO
WITHOUT BUCKLING.
COMPARISON BETWEEN TWO VALUES OF B
5
(
NR)
MAL
NOR
T
0
B=0.001
FFOR
B=0.01
E
- 5
0
5
10
15
20
25
30
35
40x10-4
THERMAL DEFORMATION
agraf 29/06/98 (c) EDF/DER 1992-1998
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.
EDF
Department Mécanique and Modèles Numériques
Electricity
CYCLIC TEST TRACTION AND COMPRESSION ON A BAR (MECA_BARRE)
from France
X
4
10
10
MODEL PINTO-MENEGOTTO
COMPARISON OF THE MODEL WITHOUT BUCKLING
AND OF THE MODEL WITH BUCKLING.
5
)
NR
(
MAL
NOR
T
0
TOKEN ENTRY WITH BUCKLING
FFOR
TOKEN ENTRY WITHOUT BUCKLING
E
- 5
- 2
- 1
0
1
2
3
4x10-3
THERMAL DEFORMATION (- ALPHA * DT)
agraf 29/06/98 (c) EDF/DER 1992-1998
Appear 5.2-b
5.3 Variables
interns
They 8, and are defined by:
V1 =
n-1
R
V 2 =
N
R
V 3 =
N
R
V 4 =
-
+ - (T - -
T)
V 5 = - (T - -
T)
V 6 = cycl
= 0
comporteme
if
cyclic
NT
activ
not
is
E
= 1
opposite
case
in
V 7 =
= 0
flax
evolution
one
with
corresponds
time
of
not
if
éaire
= 1
(indicateu
opposite
case
in
R
plasticity
of
)
V 8 = indicating
buckling
of
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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 behavior ASSE_COMBU [bib6].
6.1
Formulation of the model
The equations are as follows:
vp
& = p &
N
Q
-
&
RT
p & =
. E
.
+ L
1 1
Q
1
K
, N >,
0
,
,
0
0
0
m
p
K m
R
}·
vp
G
HT
= & - & - & -
&
E
The coefficients are provided under key word LEMAITRE of DEFI_MATERIAU and
& is flow
neutronics (derivative of the fluence compared to time).
G
(T) = (At + b).((X, y, Z) S
).
Note:
20
2
· The neutron flux & (X, y, Z) is expressed obligatorily in 10 N/cm/S. Ceci implies
that the units of the other sizes are fixed:
-
E, K, are in MPa,
-
times are in seconds,
-
co-ordinates in mm
-
Q
T,
in Kelvin
R
Two types of integrations are available according to the value of 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.
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6.2 Integration
implicit
By direct implicit discretization of the relations of behavior, one obtains:
-
vp
+
=p
-
+
N
-
Q
+
RT
-
-
p=
. E
(-)
+ L
.
1
T
K0t
(-
p + p) m
-
-
= - vp
- G
- HT
-
E
E
with
HT
-
-
= T
() (T - Tref) - T
(
) (T - Tref)
S
G
= (+
At + b) () S
. - (-). (-
At + b)
One can still bring back oneself there to only one nonlinear scalar equation out of p, while posing:
E
E
=
- + E
-
(- G
- HT
)
E
then the system is reduced to:
N
Q
1
-
RT
p=
. E
+ L.
1
T
K
-
K0t
m
(p + p)
E
vp
E p
= +
E = +
E p
= 1+
and by taking the absolute value of the two members of the last equation, one obtains:
E = + E p
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what results in solving the equation:
N
E
- E p
Q
1
-
p
=
. E RT
+ L. T
K
1
0
(
-
p + p
)
K
T
m
Once this solved equation (by a method of search for zero of function scalar), one
E
E p
obtains the constraints by: =
= E 1
1
E
E p
+
E - E p
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:
-
+
vp
=p
2
-
+
2
N
Q
-
-
T
+
1
2
RT +
2
p=
E
.
+ L.
1
T
K
p
0
m
K T
-
p +
2
1
-
vp
G
HT
-
=
-
-
-
-
2nd E
2
2
2
2
HT = T
() (T - Tref) -
-
T
(
) (-
T - Tref)
= (At +b) () S
G
. - () S
. (-
At + b)
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-
One seeks to calculate +
. One can write:
2
-
E -
E
E
vp
G
HT
=
+
=
+
-
- E
- E
2
2
2
-
E
2
2
2
2
2
thus
-
-
E
G HT
v
-
E
+
=
+
E
-
+
-
-
-
2
2
E
2
2
2
2
2
As previously, one solves while posing:
-
-
G
HT
E
E
=
+
+ E
-
-
E - 2
2
2
2
2
then the system is reduced to:
N
Q
-
-
T
+
p
1
2
RT
+
T
=
.
2
E
+ L.
2
1
K
K T
2
0
-
p
p
m
+
2
-
p
p
+
E
2
-
-
E
= +
+
E
1
2
2
2
2
-
= +
+
-
+
+
2
2
from where:
p
E
-
= +
+ E
2
2
The equation to be solved is exactly same form as the implicit equation:
N
Q
-
E
p
T
p
1 - E
RT +
2
T
=
.
2
E
+ L.
2
1
K
K T
2
-
0
p
p
m
+
2
Handbook of Référence
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:
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Once this solved equation, one obtains the internal variables while multiplying by 2 the value obtained
and constraints by:
E p
-
E
+
= 1
2
E
-
-
+
= 2 +
-
2 -
One can thus use the same routines of resolution as in the implicit case, while calculating
G HT
simply E in
,
,
.
2
2
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 steps of time (instead of 100 steps 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 step
current time.
6.5
Identification of the parameters of the model
It is done starting from creep tests (uniaxial test with constant constraint imposed under flow
neutronics constant). By integration of the equations of the model, one obtains then:
m
Q
n+m
+ &
-
vp
(T) N m
N
=
RT
+ L E
m K
0
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:
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7
Relation of behavior of LMA-RC (ASSE_COMBU)
The model presented in this chapter describes the viscoplastic behavior 1D LMA-RC
(Laboratory of Mécanique Appliquée 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:
}
= & - v p
&
- G
& - HT
&
E
v p
&
= p &.
(- X)
=
- X
N
- X
p & = &0 sin H
K
m
X
X & = Q (Y (p) v p
()
&
- (X - X 1) p&)
-
X
R sinh
m
X 0
X
()
X & 1 = q1 (Y (p) v p
() 1
(2)
&
- (X - X) p&)
(2)
X&
= Q Y
2
(p) v p
(2)
(& - X p&)
with: Y () Y (Y
Y
0
) ebp
v =
+
-
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 of the key word
LMARC).
The law of growth is identical to that used for the model of Lemaitre:
G
(T) = (At + b) ((X, y, Z) S
The 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 (X, y, Z).
· Only one diagram of integration is available: a purely implicit diagram.
20
2
· The neutron flux (X, y, Z) is expressed obligatorily in 10 N/cm/S. Ceci implies that
the units of the other sizes are fixed:
-
E, K, are in MPa,
-
times are in seconds,
-
co-ordinates in Misters.
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:
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7.2 Integration
implicit
To integrate these relations of behavior, while bringing back itself if possible to only one equation to
(- X)
to solve, it is necessary to make an assumption on =
. Indeed, it can take only two values:
- X
(- - -
X)
+1 or - 1. One thus supposes this known sign (initialized by =
). If one cannot solve
- - -
X
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
+
G HT - p
E
p
E
-
- -
= -
E
-
N
X
p =
&0 tsi
= F
nh T
,
v (X)
K
1
m
X X
X =
Q p (Y (p)
()
- (X - X)
()
- R sinh
,
1
m
= F (p X X)
X0
X
()
X 1 = Q
1
2
() 1
(2)
(, X, X)
1 (
p Y (p)
()
()
- (X - X) = F p
1
(2)
X
= Q p Y
2
2
-
=
2
(p)
()
(
X)
()
f2 (,
p X)
There is thus a system of 5 equations to 5 unknown factors:
() 1
(2)
, p, X, X, X
The second equation is also written:
1
2
- X
p N
p N
= log
+ 1+
K
& T
&0 T
0
By using the first equation, one can express X according to p:
- X =
p
E -
E
X = - X
1
2
p N
p N
X = F (
log
1
1 p) =
p
E -
-
E
X - K
+
+
0 & T
0 & T
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:
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In addition, by integration successive of the functions f2 and f1 one can also bring back oneself to one
equation utilizing only X and p:
Q Y (p)
(2)
-
-
p
2
2
(
X
)
()
X =
1+ Q p
2
-
Q
1
2 -
2
p
-
-
-
1
1
(Y (p) () ()
()
(X X X)
()
X =
1+ Q p
1
()
()
like X = F (p, X, X 1), and X 1 = G (p) according to the preceding expressions one can
to write: X = F2 (p) = F1 (p). The equation to solve to find p is thus:
F (p) = F
0
2 (p) - F1 (p) =
Once calculated p, one obtains the constraints by: =
E p
E -
7.3 Variables
interns
They are 5:
V1 = p
V 2 = X (1)
V 3 = X (2)
V 4 = X
In Code_Aster, one adds a last internal variable: V 5 = 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.
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:
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8
Method to use in 1D all the behaviors 3D
As for the processing of the plane constraints [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 of the stress fields which tend towards a field
one-way. It is checked, with convergence of the total iterations of Newton, that the fields of
constraints 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 constraints 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 connects the increase in constraints to
the increase in deformation by:
D =
D =
Dd
that one rewrites:
D
X
D
D
D
11
12
13 D
X
D y =
D
D
D
21
22
23 D Y. By writing these increases like the difference between
D
D
D
D
Z
31
32
33 D Z
iterations N and N + 1 of Newton, one obtains:
n+1
N
n+1
N
n+1
N
D =
- =
-
, D =
-
With convergence, this variation must tend towards zero.
By introducing the conditions
N 1
+
N 1
: y = 0
+
and 2 (one-way behavior), one obtains,
for the iteration N + 1:
1
1
D
n+
X - N
n+
X
X - N
X
X D
D
D
11
12
13 D
X
n+1
N
N
D y = y - y =
- y
=
D
D
D
21
22
23 D y
D
n+1
N
N
D
D
D
Z
31
32
33
D
Z
- Z
-
Z
Z
The two last equations make it possible to express D y and D Z according to D X:
1
D y =
(N
- - D
y
21d
- D
X
23d Z)
D22
1
D Z
=
(N
- Z - 31
D D X - 32
D D y)
D33
1
that is to say
D y = (
N
N
- 33
D + D
y
23
+ D D
Z
y X)
1
D Z = (
N
N
- 32
D - D
y
22
+ D D
Z
Z X)
with
= 33
D D22 - D23 32
D, D = D
y
23 31
D - D21 33
D, Dz = 32
D D21 - 31
D D22
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:
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by deferring these expressions in the first equation, one obtains:
D D
1
12
+ D D
13
n+
N
y
Z
D D
12
23 - D
D
22 13
N
D D
12 32 - D D
12 33
N
X = X + D11 +
D X +
Z +
y
Balance with the iteration N + 1 is written:
D D
T
N 1
T
N 1
T
12
y + D D
+
B
FD =
+
B
FD
13
X
=
Z
B D11 +
D X
+ T N D D
12
23 - D
D
22 13
N
D D
12 32 - D D
12 33
N
B X +
Z +
y FD
= N n+
K of 1 + T N D D
12
23 - D
D
22 13
N
D D
12 32 - D D
12 33
N
B X +
Z +
y FD
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:
D D
12
+ D D
BT
y
Z B FD
13
· in the writing of the second member, by the corrective term:
BT
((D D - D D
12
23
22 13) N
Z + (D D - D D
12 32
12 33) N
y) FD
To implement this method, it is enough to calculate these corrective terms and to add them to
constraints 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 N + 1, the data is:
n+1
-
-
U
, and the 4 variables intern (calculated with
1
D
V1 = N
y + (
N
D23 Z -
N
D33 y - Dy N
X), V 2 = y,
iteration N):
,
1
V 3 = N
D
Z + (
N
D32 y -
N
D22 Z - Dz N
X), V 4 = Z
2) before carrying out the integration of the behavior (carried out into axisymmetric) one calculates
n+1
N
1
y
=
y + (
N
N
- D
33 y + D
23 Z + Dy D X)
,
n+1
N
1
Z
=
Z + (
N
N
- D
32 y - D
22 Z + Dz D X)
3) the integration of the behavior provides constraints N 1
+
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
n+
1
Z
<
n+
and 1
y
<, with =
n+
1,
X
=
RELA
RESI_INTE_
.
Handbook of Référence
R5.03 booklet: Nonlinear mechanics
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Code_Aster ®
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Titrate:
Relations of behavior 1D
Date:
02/05/05
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:
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9 Bibliography
[1]
J. GUEDES, P. PEGON, EP PINTO: '' A Fiber/Timoshenko Beam Element in Castem 2000 '',
Joint Research Center, 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 Civile, Maggio 1970.
[3]
Mr. MENEGOTTO, EP PINTO: '' Method off Analysis for Cyclically Loaded Reinforced
Concrete Plane Frames Including Changes in Geometry and Nonelastic Behavior off
Elements under Combined Normal Force and Bending '', IABSE Symposium one Resistance
and Ultimate Deformability off Structures Acted 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 fuel pins “Note HT-B2/95/018/A
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