Code_Aster ®
Version
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Titrate:
Élasto-viscoplastic relation of behavior of the LMARC
Date:
11/07/96
Author (S):
P. GEYER
Key:
R5.03.10-A
Page:
1/18
Organization (S): EDF/RNE/EMMA
Handbook of Référence
R5.03 booklet: Nonlinear mechanics
Document: R5.03.10
Élasto-viscoplastic relation of behavior
LMARC for the tubes of sheath of the pencil
fuel
Summary:
A model was developed with the LMA-RC of the university of Besancon (Laboratoire de Mécanique Appliqué R.
Chaléat) to describe the élasto-viscoplastic behavior of the tubes of sheath of the fuel pin. It
the taking into account of the strong mechanical anisotropy of the tubes in Zircaloy 4 allows.
It is established in Code_Aster under the name of LMARC; the equations of speed are integrated
numerically by an implicit scheme of Euler in environment PLASTI.
Handbook of Référence
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Code_Aster ®
Version
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Titrate:
Élasto-viscoplastic relation of behavior of the LMARC
Date:
11/07/96
Author (S):
P. GEYER
Key:
R5.03.10-A
Page:
2/18
Contents
1 Introduction ............................................................................................................................................ 3
2 Formulation of the model .......................................................................................................................... 3
2.1 Theoretical framework ............................................................................................................................... 3
2.2 Description of the tensors of anisotropy ............................................................................................ 5
2.3 Equations of the model ....................................................................................................................... 6
2.4 Relation LMARC .............................................................................................................................. 7
3 Establishment of the model in Code_Aster ......................................................................................... 7
3.1 Algorithm of resolution of the quasi-static problem ..................................................................... 7
3.2 Environment PLASTI ................................................................................................................... 9
3.3 Discretization of the equations of the model ........................................................................................ 10
3.4 Operator of tangent behavior ............................................................................................ 11
4 Bibliography ........................................................................................................................................ 12
Appendix 1 Expression of Jacobien of the integrated elastoviscoplastic equations ............................... 13
Appendix 2 Evaluation of the coherent tangent operator MC ................................................................... 16
Handbook of Référence
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Code_Aster ®
Version
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Titrate:
Élasto-viscoplastic relation of behavior of the LMARC
Date:
11/07/96
Author (S):
P. GEYER
Key:
R5.03.10-A
Page:
3/18
1 Introduction
The tubes of sheath in Zircaloy of the fuel pin of power stations REP present a behavior
anisotropic and strongly viscous mechanics. A phenomenologic model was developed with
LMA-RC of Besancon [bib1] to obtain a fine description of the behavior of material in sight
to evaluate realistic states of stresses in situation of Interaction Pellet-sheath (IPG).
Being given the crystallographic texture of the tubes, one makes the assumption of orthotropism. The model was
confronted with the experimental results on tubes of sheath in Zircaloy 4 in various states
metallurgical and subjected to loadings plain and multiaxis [bib1], [bib2], [bib3]. The model is
initially intended to be used within the framework of the coupling between Code_Aster and the code of
fuel pin CYRANO3. It could however be used for other metallic materials
presenting an orthotropic viscoplastic mechanical behavior.
The model is introduced into Code_Aster in 3D, plane deformations (D_PLAN), and axisymetry
(AXIS) under the name of LMARC. It is about a unified viscoplastic model with internal variables:
cumulated viscoplastic deformation and three variables of kinematic work hardening.
The taking into account of the anisotropy is carried out by four tensors of a nature 4 affecting the sizes
equivalent mechanics but also laws of evolution of the internal variables.
One presents in this note the equations constitutive of the model and his establishment in
Code_Aster.
2
Formulation of the model
2.1 Tally
theoretical
The model of behavior developed to the LMA-RC lies within the scope of the thermodynamics of
irreversible processes and of the mechanics of the continuous mediums. It is about a model
elastoviscoplastic unified, i.e. dependant inelastic deformations or
independent of time are gathered in only one term. By considering the assumption of small
disturbances, one divides the tensor of the deformations into an elastic part, a thermal part and
a viscoplastic part:
= E + HT + vp
The elastic part is given by the law of Hooke, the anisotropy of behavior which can be neglected
in this case. The concept of surface of load used in plasticity is replaced by a family of
equipotential surfaces: they are surfaces of the space of the constraints in each point of which
the module the speed of deformation is the same one (dissipation is the same one) [bib4]. Being given
texture tubes of sheath of the fuel pin, one can make the assumption of orthotropism of
mechanical behavior and one use a formulation of the Hill type to describe surfaces
equipotential:
3
F =
ij
ij
ijkl kl
kl
R
R
2 (~ - X) M
(~ - X) - 0 = - X - 0
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Élasto-viscoplastic relation of behavior of the LMARC
Date:
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Author (S):
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Key:
R5.03.10-A
Page:
4/18
where
~
1
= - tr () I D
is the deviatoric part of the tensor of the constraints.
3
X
a kinematic variable of work hardening (tensorial).
M
tensor of command 4 for the description of the anisotropy
(with the formulation of Hill, only 6 coefficients are independent).
0
R
initial elastic limit.
The direction of evolution of the viscoplastic tensor of deformation is given by the rule of normality
on equipotential surfaces:
vp
F
!ij =!v ij
where v represent the cumulated viscoplastic deformation, obtained starting from the equation of state of which
formulation was established in experiments with the LMA-RC [bib1]:
N
F
!v =! sinh
0
K
with S
X = 0 I X 0 and X = X if X 0
To entirely define the model, it remains to give the equations of evolution of the variables
of work hardening representing the state of internal stress of the material which is opposed to the deformation
(constraints induced by the interactions on various scales between mobile dislocations and
substructure). The kinematic work hardening of nature is described in the model via
three nonlinear variables.
m
2
vp
1
X
X
!X
mn
ij = p
Y (v)
()
Nijkl! - Q
kl
ijkl
3
(X - X
kl
kl)!v
R sinh
NR R
m
ijkl
klmn
-
X0
X
()
2
! 1
1
2
X
vp
ij = p
Y
1
(v)
()
()
Nijkl! - Q
kl
ijkl
3
(X - X
kl
kl)!v
(2)
2
!
2
X
vp
ij
= p
Y
2
(v)
()
Nijkl! - Q
X
kl
ijkl
kl
!v
3
3
with
X =
X R
X
2
ij
ijkl
kl
Xij ()
()
0 = X 1
ij ()
(2)
0 = Xij ()
0 = 0
(
Y) = Y
- v
v
+ (Y0 - Y) E B
(
Y v) makes it possible to describe hardening or softening under cyclic loading.
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Code_Aster ®
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Titrate:
Élasto-viscoplastic relation of behavior of the LMARC
Date:
11/07/96
Author (S):
P. GEYER
Key:
R5.03.10-A
Page:
5/18
These equations comprise for each variable of kinematic work hardening a first term
of linear kinematic work hardening compared to the deformation and a second term of restoration
dynamics. Lastly, the first equation comprises a third static term of restoration for
to take into account the effects dependant on time.
For the description of the anisotropy, one distinguishes an intrinsic anisotropy with the structure of material
(form of equipotential) and an additional anisotropy induced by the viscoplastic flow.
The anisotropy is introduced into the model of the LMA-RC by the means of tensors of command 4 in
relations between the various tensorial variables. Anisotropy induced by the flow
viscoplastic is translated in the laws of evolution of the variables of work hardening. Terms of these
equations are connected to different mechanisms of deformation in material: work hardening
linear kinematics, dynamic and static restorations. The taking into account of the induced anisotropy
by the viscoplastic deformation is thus made by the introduction of three distinct tensors NR, Q and R.
Note Bucket:
The model suggested by the LMA-RC [bib2] is without threshold. The initial elastic limit R0 was
added during integration in Code_Aster to widen the possibilities of the model. It is enough
to consider a zero value to work with a model without threshold.
2.2
Description of the tensors of anisotropy
To simplify the writings, thereafter a matric notation, image of the notation are used
tensorial intrinsic in a orthonormé reference mark. Moreover, one uses notations of a lower command
(tensor 2 = vector, tensor 4 = matrix), in order to be identified with what is in the code. Because of
symmetry of the handled tensors of command 2, one reduces them to vectors (1 X 6) by multiplying them
components of shearing by root of 2.
T = [1 = 11, 2 = 22, 3 = 33, 4 = 2 12, 5 = 2 13, 6 = 2 23]
T = [1 = 11, 2 = 22, 3 = 33, 4 = 2 12, 5 = 2 13, 6 = 2 23]
According to cylindrical co-ordinates' related to the tube, one considers thereafter 11 = rr, 22 =, 33 = zz.
As we mentioned in the first chapter, being given the crystallographic texture of
tubes, one can make the assumption of orthotropism and confuse the axes of anisotropy with the axes
hardware. The conditions of symmetry which result from it lead to nine independent components
for each tensor of anisotropy. The incompressibility of the viscoplastic flow is translated
by three additional relations and the component count independent is tiny room to six [bib1].
With the matric notation, each of the four tensors takes a form identical to that of M is:
M
M
M
11
12
13
0
0
0
M
M
M
12
22
23
0
0
0
M11+ M12 + M
13 = 0
M
M
M
13
23
33
0
0
0
M =
with M12 + M22 + M23 = 0
0
0
0
M
44
0
0
M13+ M23+ M
33 = 0
0
0
0
0
M
0
55
0
0
0
0
0
M
6
6
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Code_Aster ®
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Titrate:
Élasto-viscoplastic relation of behavior of the LMARC
Date:
11/07/96
Author (S):
P. GEYER
Key:
R5.03.10-A
Page:
6/18
To find the isotropic version of the model, it is necessary to take the following values for the four
tensors:
2
1
M
= M = M =, M = M = M = - and M = M = M =.
11
22
33
12
13
23
44
55
66
1
3
3
In experiments, one cannot reach in our study the components 44, 55 and 66 which
correspond to shear tests. Within the framework of the work of thesis realized to the LMA-RC,
only component 66 was determined by tests of traction-torsion, the two last (R and
rz) not being able to be reached because the low thickness of the tubes.
Being given the preceding considerations, only components 11, 22, 33 and 66 are put in
reading of the command file, the other components being is given starting from the equations
had with the plastic incompressibility, is taken equal to the isotropic values for the components of
shearing.
Note Bucket:
The matric notations used in the references [bib 1, 2 and 3] are those of Voight.
Only the terms of shearing are influenced; conversion to work with
notations of Code_Aster is obtained using the following formulas (I = 4,5,6):
1
1
M =
MVoight, Q = QVoight, NR = 2 NVoight and R =
RVoight
II
II
II
II
II
II
II
II
2
2
2.3
Equations of the model
= E + HT + vp
= A (T) E
3
F =
(~ - X) T M (~ - X) - R0 = ~ - X - R
2
0
~
vp
F
3
M (- X)
! = v!
=
v
!
2
~ - X
if F,
0
()
1
(2)
v! =,
0!X =!X
=!X
= 0
if F >,
0
N
2
vp T
- 1 vp
F
v! =!0
(! ) M! =
!0 sinh
3
K
X m
!
2
X
X =
p
Y (v)
vp
()
NR! - Q
1
rm sinh
NR R
3
(X-X) v!
-
X0
X
()
!
2
2
X 1 =
p
Y
1
2
2
2
1
(v)
vp
()
()
NR! - Q
3
(X - X) ()
v
p
Y
!
!X
2
(v)
vp
()
NR!
Q X
v!
=
-
3
with
(
3
Y v) =
- bv
T
Y + (Y0 -
Y) E
X =
X R X
2
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Code_Aster ®
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Titrate:
Élasto-viscoplastic relation of behavior of the LMARC
Date:
11/07/96
Author (S):
P. GEYER
Key:
R5.03.10-A
Page:
7/18
Note Bucket:
In Code_Aster, the whole of the parameters of the model
R! , K, N, Y, B X, R, m, p, p, p
0
0
0
1
2, M, NR, Q, R (I
m
II
II
II
II
= 1, 2, 3,)
6 can
to be a function of the temperature.
2.4 Relation
LMARC
The model is accessible in Code_Aster in 3D, plane deformations (D_PLAN), and axisymetry
(AXIS) starting from key word COMP_INCR of command STAT_NON_LINE. The whole of
parameters of the model is provided under the key word factor LMARC or LMARC_FO of the command
DEFI_MATERIAU [U4.23.01].
/
LMARC: (
R_0
:
R0
DE_O
:
0
NR
:
N
K
:
K
y_0
:
yo
y_I
:
y
B
:
B
A_0
:
X0
RM
:
rm
M
:
m
P
:
p
P1
:
p1
P2
:
p2
M11
:
M11
N11
:
N11
M22
:
M22
N22
:
N22
M33
:
M33
N33
:
N33
M66
:
M66
N66
:
N66
Q11
:
Q11
R11
:
R11
Q22
:
Q22
R22
:
R22
Q33
:
Q33
R33
:
R33
Q66
:
Q66
R66
:
R66
)
3
Establishment of the model in Code_Aster
3.1
Algorithm of resolution of the quasi-static problem
One seeks to check the balance of the structure at every moment. In incremental form, it is about one
nonlinear problem whose variational formulation in the case of the small deformations can
to put in the form:
To find U such as:
kinematically
((U + U
), T) () D = L (T)
acceptable and T
Drunk = ud
(T)
where U indicates the field of displacement, Bu
ud
=
(T) corresponds to the boundary conditions in
displacement and L (T) are the loading at the moment T.
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One is thus led to solve, for each increment of time T:
Ft+t (U + U
T
) = 0 on the basis of a state with balance F
= 0
0
U
being the increment of the solution U on T
, C being known
The general outline adopted by Aster to solve this discretized total system is a method of
Newton [bib5] which is written, K being an indication of iteration:
F D
(U
K) = - F (U
K)
U
K
U
+1 = U
+ D
K
K
(U
K)
This diagram requires, starting from the estimate of displacements to the iteration K, to calculate in each
not Gauss:
T T
+ which checks the law of behavior
MCt+ T =
the operator of tangent behavior
t+t
F
T
= K = K with K =
B
Data base
U
E
E
E
E
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Élasto-viscoplastic relation of behavior of the LMARC
Date:
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Author (S):
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Key:
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Page:
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3.2 Environment
PLASTI
It is thus necessary, with each total iteration and in each point of Gauss, to integrate the equations
model described in [§2.3] for the calculation of t+t and to calculate the operator of behavior
tangent.
An environment was created in Code_Aster with an aim of parameterizing the establishment of models
elastoviscoplastic presenting a function threshold (field of elasticity).
This algorithm:
· manage the choices of integration elastic or (visco) plastic,
· propose various routines to contribute to the resolution of the nonlinear system (local) formed
by the equations of the model,
· updates the variables at the end of the increment,
· call the routines user for the calculation of the operator of tangent behavior.
The step to establish a new model can be schematized in the following way:
Writing of the equations of the model of speed
!y = F (y, T)
Choice of a diagram of integration
Writing of the system discretized R
(y) = 0
Writing of the routines specific to the model:
· recovery of the data materials,
· evaluation of the function threshold,
· evaluation of the operator of tangent behavior
· routine for the resolution of the system R
(y) = 0
(the algorithm proposes a method of Newton for one
implicit nonlinear system).
+ Modification of the routines of shunting of the algorithm
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Élasto-viscoplastic relation of behavior of the LMARC
Date:
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Key:
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Page:
10/18
3.3
Discretization of the equations of the model
If the increment of time corresponds to a loading elastoviscoplastic, a diagram is used
implicit of Euler whom one solves by a method of Newton.
In discretized form, the system of equations is written:
(
F
G) - H
-
v
= 0
t+t
t+t
(
2
F
L) X
p
Y (v) NR
(
Q X X1)
v
3
-
-
T +t
m
X
X
+ R
sinh
NR R
T
m
= 0
X0
X
T
+t
(
2
F
I) X1-
p
Y
1
(v) NR
Q (X X
1
2)
v
=
0
3
-
-
t+ T
(
2
F
J) X2-
p
Y
2
(v) NR
Q X
v
2
0
3
-
=
T + T
N
(
F
K) v
-
0! sinh
T
= 0
K
with Y (v) = Y
-
+ (Y0 - Y) E bv
3
T
X =
X R X
2
In more contracted way, one poses:
G (y
)
L
(y
)
X
F L (y
) = 0 = I (y
)
with
y
= X
1
J
(y
)
X2
K
(y
)
v
One solves this system by the method of Newton proposed in environment PLASTI, that is to say:
Fl
D (y
L
K) = - F (y
K)
y
K
y
+1 = y
+ D
K
K
(y
K)
While reiterating in K until convergence.
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Élasto-viscoplastic relation of behavior of the LMARC
Date:
11/07/96
Author (S):
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Key:
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Page:
11/18
The resolution requires the calculation of Jacobien of the local system F L. His general expression is given
hereafter; analytical calculations block per block are given in appendix (cf [§An1]).
G
G
G
G
G
X X
1
X2 v
L
L
L
L
L
X X1 X2 v
I
I
I
I
I
J =
X X1 X2 v
J
J
J
J
J
X X
1
X2 v
kT
kT
kT
kT
K
X X
1
X2 v
3.4
Operator of tangent behavior
The formed system of the equations of the model written in discretized form (Fl (y
) =)
0 are checked in
end of increment. For a small variation of F L, by regarding this time as variable and not
like parameter, the system remains with balance and one checks dF L = 0, i.e.:
Fl
Fl
Fl
Fl
Fl
Fl
+
+
X+
X1+
X2 +
v
= 0
X
X1
X2
v
This system can be still written:
H
X
0
F L
(y
) = X, with y
= X and
1
X = 0
y
X2
0
v
0
By successive substitution and elimination (cf [§An2]), one deduces from it that:
K = H
from where the required tangent operator:
C
- 1
M
K H
=
=
t+t
The preceding equations show that one is led to re-use the same matrix jacobienne J
that previously to evaluate the tangent operator. This operator is known as coherent (insinuation
with the system of integration) and still noted MC.
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Élasto-viscoplastic relation of behavior of the LMARC
Date:
11/07/96
Author (S):
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Key:
R5.03.10-A
Page:
12/18
In the case of the elastoplastic models, one can also calculate the tangent operator said of speed
(MV) starting from the equations of the model of speed [bib7].
4 Bibliography
[1]
TAP P.: Experimental study and modeling of the viscoplastic behavior
anisotropic of Zircaloy 4 in two metallurgical states, Thèse of Université de Franche
County, 1995
[2]
DELOBELLE P., TAP P.: Study of the behavior and viscoplastic modeling
of Zircaloy 4 recristallized under loadings monotonous and cyclic plain and multiaxis, J. Phys.
III, 4, 1994, 1347
[3]
The PICHON I., GEYER P.: Modeling of the anisotropic viscoplastic behavior of
tubes of sleeving of the fuel pin, notes EDF-DER, HT-B2/95/018/A, 1995
[4]
LEMAITRE J., CHABOCHE J.L.: Mechanics of solid materials, ED. Dunod, 1985
[5]
MIALON P., LEFEBVRE J.P., quasi-static nonlinear Algorithme, note EDF-DER,
HI-75/7832 [R5.03.01]
[6]
SHOENBERGER P., Introduire a new relation of nonlinear behavior,
[D5.05.01], to appear
[7]
SHOENBERGER P., Intégration of the relation of behavior of Chaboche [R5.03.04],
to appear
Handbook of Référence
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Code_Aster ®
Version
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Titrate:
Élasto-viscoplastic relation of behavior of the LMARC
Date:
11/07/96
Author (S):
P. GEYER
Key:
R5.03.10-A
Page:
13/18
Appendix
1
Expression of Jacobien of the equations
elastoviscoplastic integrated
That is to say thus to evaluate the terms of the hypermatrice jacobienne J at the moment T + T
G
G
G
G
G
X X
1
X2 v
L
L
L
L
L
X X1 X2 v
I
I
I
I
I
J =
X X1 X2 v
J
J
J
J
J
X X
1
X2 v
kT
kT
kT
kT
K
X X
1
X2 v
G
2 F
G
2 F
= I + H
v
= H
v
2
X
X
t+t
t+ v
G
G
G
F
=
= 0
=
H
X1 X2
v
t+ T
L
2
2 F
= - p Y (v)
NR
v
3
2 t+ T
L
m
2
2 F
X
NR R
= I p Y (v)
NR
v
+ (pQ)
v
+ R
sinh
T
X
3
X
t+t
m
X0
X
t+t
t+ T
t+
T
m
T
X 1
m
m
R X
m 1
X
3
-
(
)
+ - R sinh
+ R
cosh
m
X
NR R X
T
X
m
2
0
X
m
X
X
2
0
0
X
T +t
t+ T
L
= - (
L
p Q)
v
= 0
X
t+ T
1
X2
L
2
F
Y
= - p NR
Y
T +t (v)
v + p Q (X X1)
v
3
+
v
t+ T
t+ T
T + T
Handbook of Référence
R5.03 booklet: Nonlinear mechanics
HT-B2/96/026/A
Code_Aster ®
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Titrate:
Élasto-viscoplastic relation of behavior of the LMARC
Date:
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Author (S):
P. GEYER
Key:
R5.03.10-A
Page:
14/18
I
2
2 F
= - p Y
1
(v)
NR
v
3
2 t+ T
I
2
2 F
= - p Y
1
(v)
NR
X
3
X
v
t+ T
I = I + (p1Q)
v
X
t+t
1
I = - (p1Q) v
X2
I
2
F
Y
= - p NR
Y
1 Q (1
X - X2)
1
t+ T (v)
v
+ p
v
3
+
t+ T
+
v
T T
t+t
2
2
J
2
F
J
2
F
= - p Y v
v
p Y v
2
() NR
= -
() NR
v
2
2
3
X
3
X
t+ T
t+ T
J
J
= 0
= I + (2
p Q)
v
1
X
X2
t+t
J
2
F
Y
= -
+ (p2 Q X2)
2
p NR
T
Y T (v)
v
v
3
+
+ v
t+
t+t
T + T
T
K
F n-1 1
F
F
K
= -
N 0!
sinh
cosh
T
K
K
K
= - X
T +t
K
K
K
=
= 0
= 1
X1 X2
v
with I the matrix of identity.
It appears in the preceding expressions the derivative first and seconds of the expression of
equipotential surfaces F compared to and X. Their evaluations hereafter are given:
F
3
(~ - X)
= M
2
~ - X
2 F
1
F
F T3
=
M
X
~ - X
-
2
2 F
1
3
F
F T
=
2
M D -
~ - X 2
Handbook of Référence
R5.03 booklet: Nonlinear mechanics
HT-B2/96/026/A
Code_Aster ®
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Titrate:
Élasto-viscoplastic relation of behavior of the LMARC
Date:
11/07/96
Author (S):
P. GEYER
Key:
R5.03.10-A
Page:
15/18
with D tensor unit in the space of the diverters:
2
1
1
-
-
0 0
0
3
3
3
1
2
1
-
-
0 0
0
3
3
3
D = 1
1
2
-
-
0 0 0
3
3
3
0
0
0
1 0 0
0
0
0
0 1 0
0
0
0
0 0 1
Note:
For detailed examples of calculations of these expressions, one will refer to the reference
[bib7].
Handbook of Référence
R5.03 booklet: Nonlinear mechanics
HT-B2/96/026/A
Code_Aster ®
Version
3.0
Titrate:
Élasto-viscoplastic relation of behavior of the LMARC
Date:
11/07/96
Author (S):
P. GEYER
Key:
R5.03.10-A
Page:
16/18
Appendix 2 Evaluation of the coherent tangent operator MC
While regarding as variable in the discretized system, one can write:
G
G
G
0
0
X
v
L
L
L
L
D
H D
0
X X1
v
D X
0
I
I
I
I
I
X 0
X X
1 =
1
X2
D
v
J
J
J
J
D X2
0
0
X
X
0
2
v
D v
K T
kT
K
0
0
X
v
While operating by successive eliminations and substitutions, the fourth block of the system of equation
give:
J
1
-
J
J
J
D X = -
D
2
+
D X+
D v
X2
X
v
While posing:
I J -
1
B =
X
2 X
2
and by replacing D X
2 in the second and third block of the system of equations, one obtains:
L
L
L
L
D
D X
D X1+
D v
0
+
+ X1
v
=
X
éq An2-1
I
J
I
J
I
- B
D
B
D X
D
X
+
-
X
X
+ X
1
1
éq An2-2
I
J
+
- B
D v
0
v
v
=
the equation [éq An2-2] gives:
I
1
-
I
J
I
J
I
J
D X = -
- B
D
1
B
D X
B
D v
X
+
-
+
-
1
X
X
v
D v
Handbook of Référence
R5.03 booklet: Nonlinear mechanics
HT-B2/96/026/A
Code_Aster ®
Version
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Titrate:
Élasto-viscoplastic relation of behavior of the LMARC
Date:
11/07/96
Author (S):
P. GEYER
Key:
R5.03.10-A
Page:
17/18
While posing:
L I -
1
C =
X X
1
1
and by replacing D X
1 in the equation [éq An2-1], one obtains:
L
I
J
- C
- B
D
L
I
J
+
- C
- B
D X
éq An2-3
X
X
X
L
I
J
+
- C
- B
D v
0
v
v
v
=
The fifth block of the system of equation gives:
K - 1 K T
K -
1 kT
D v
= -
D
D
X
v
-
v
X
While posing:
L
I
J K
F =
- C
- B
v
v
v
v
and by replacing D v
in the equation [éq An2-3], one obtains:
L
I
J
kT
- C
- B
F
D
-
éq An2-4
L
I
J
kT
+
- C
- B
F
D X = 0
X
X
X -
Handbook of Référence
R5.03 booklet: Nonlinear mechanics
HT-B2/96/026/A
Code_Aster ®
Version
3.0
Titrate:
Élasto-viscoplastic relation of behavior of the LMARC
Date:
11/07/96
Author (S):
P. GEYER
Key:
R5.03.10-A
Page:
18/18
While posing:
L
J
I
J
kT
D =
- A
- C
- B
F
-
and
L
J
I
J
kT
E =
- A
- C
- B
F
X
X
X
X - X
the equation [éq An2-4] is written:
D D + E D X = 0
from where by replacing D X
and D in the first block of the system of equation, one obtains:
G
G K - 1 kT G
G K - 1 kT
- 1
-
-
-
E D D
= H D
v
v
X v
v
X
Finally, the coherent tangent operator is written:
- 1
T
- 1
T
- 1
G
G K
K
G
G K
K
- 1
M =
-
E D H
C
v
v
-
-
X v
v
X
Handbook of Référence
R5.03 booklet: Nonlinear mechanics
HT-B2/96/026/A
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