Code_Aster ®
Version
6.4
Titrate:
Law CJS in géomechanics
Date:
18/11/03
Author (S):
C. CHAVANT, pH. Key AUBERT
:
R7.01.13-A Page
: 1/48
Organization (S): EDF-R & D/AMA, EDF-DIS/CNEPE
Handbook of Référence
R7.01 booklet: Modelings for Génie Civil and the géomatériaux ones
Document: R7.01.13
Law CJS in géomechanics
Summary:
One presents here the law CJS which applies to the soil mechanics. One specifies:
·
the description of the model,
·
the integration of the law in Code_Aster,
·
the description of the introduced routines.
Handbook of Référence
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HT-66/03/005/A
Code_Aster ®
Version
6.4
Titrate:
Law CJS in géomechanics
Date:
18/11/03
Author (S):
C. CHAVANT, pH. Key AUBERT
:
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: 2/48
Count
matters
1 Notations ................................................................................................................................................ 4
2 Introduction ............................................................................................................................................ 5
3 Description of law CJS ........................................................................................................................ 5
3.1 Partition of the deformations ............................................................................................................... 5
3.2 Elastic mechanism ....................................................................................................................... 5
3.3 Isotropic plastic mechanism ......................................................................................................... 5
3.4 Plastic mechanism déviatoire ...................................................................................................... 6
3.4.1 Isotropic work hardening .............................................................................................................. 7
3.4.2 Kinematic work hardening ....................................................................................................... 8
3.4.3 Law of evolution of the plastic mechanism déviatoire ................................................................ 9
3.4.4 Rough surface ................................................................................................................ 11
3.5 Hierarchisation of the model ............................................................................................................. 12
3.5.1 Summary description of three levels CJS ....................................................................... 12
3.5.2 Assessment of parameters CJS .................................................................................................... 12
3.5.3 Correspondence with the cohesion and the angle of friction ................................................. 13
4 Integration of law CJS ....................................................................................................................... 14
4.1 Choice of the variables intern ......................................................................................................... 14
4.2 Integration of the nonlinear elastic mechanism .......................................................................... 15
4.3 Isotropic integration of the mechanisms elastic nonlinear and plastic ..................................... 16
4.3.1 Initialization and solution of test ............................................................................................. 16
4.3.2 Iterations of Newton ............................................................................................................. 17
4.3.3 Test of convergence ............................................................................................................ 17
4.4 Integration of the mechanisms elastic nonlinear and plastic déviatoire .................................. 18
4.4.1 Initialization and solution of test ............................................................................................. 18
4.4.2 Iterations of Newton ............................................................................................................. 19
4.4.3 Test of convergence ............................................................................................................ 26
4.5 Integration of the mechanisms elastic nonlinear, plastic isotropic and plastic déviatoire….27
4.5.1 Initialization and solution of test ............................................................................................. 27
4.5.2 Iterations of Newton ............................................................................................................. 29
4.5.3 test of convergence ............................................................................................................. 29
4.6 Procedure of relieving based on an estimate of the normals on the surface of load
déviatoire ....................................................................................................................................... 29
4.7 Recutting of the step of time ..................................................................................................... 30
4.8 Various remarks ...................................................................................................................... 30
4.8.1 Calculation of the cos term (-
............................................................................................. 30
S
Q)
4.8.2 Calculation of R ........................................................................................................................ 31
R
4.8.3 Traction ................................................................................................................................. 31
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Code_Aster ®
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6.4
Titrate:
Law CJS in géomechanics
Date:
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Author (S):
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:
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5 tangent Operator ............................................................................................................................... 32
5.1 Tangent operator of the nonlinear elastic mechanism ............................................................. 32
5.2 Tangent operator of the mechanisms isotropic rubber band and plastic .......................................... 32
5.3 Tangent of the mechanisms rubber band and plastic operator déviatoire ......................................... 33
5.4 Tangent operator of the mechanisms rubber band, plastics isotropic and déviatoire ........................ 34
6 Sources Aster ...................................................................................................................................... 36
6.1 List modified and added routines ....................................................................................... 36
6.2 Top-level flowchart of the principal routines .......................................................................... 37
6.3 Details of the functionalities of developed routines FORTRAN ................................................ 38
6.3.1 Routine: CJSC3Q ................................................................................................................. 38
6.3.2 Routine: CJSCI1 ................................................................................................................. 38
6.3.3 Routine: CJSDTD ................................................................................................................. 38
6.3.4 Routine: CJSELA ................................................................................................................. 38
6.3.5 Routine: CJSIDE ................................................................................................................. 39
6.3.6 Routine: CJSIID ................................................................................................................. 39
6.3.7 Routine: CJSJDE ................................................................................................................. 40
6.3.8 Routine: CJSJID ................................................................................................................. 41
6.3.9 Routine: CJSJIS ................................................................................................................. 41
6.3.10
Routine: CJSMAT ..................................................................................................... 42
6.3.11
Routine: CJSMDE ..................................................................................................... 42
6.3.12
Routine: CJSMID ..................................................................................................... 43
6.3.13
Routine: CJSMIS ..................................................................................................... 43
6.3.14
Routine: CJSNOR ..................................................................................................... 44
6.3.15
Routine: CJSPLA ..................................................................................................... 44
6.3.16
Routine: CJSQCO ..................................................................................................... 45
6.3.17
Routine: CJSQIJ ..................................................................................................... 45
6.3.18
Routine: CJSSMD ..................................................................................................... 45
6.3.19
Routine: CJSSMI ..................................................................................................... 45
6.3.20
Routine: CJST ......................................................................................................... 46
6.3.21
Routine: CJSTDE ..................................................................................................... 46
6.3.22
Routine: CJSTEL ..................................................................................................... 46
6.3.23
Routine: CJSTID ..................................................................................................... 47
6.3.24
Routine: CJSTIS ..................................................................................................... 47
6.3.25
Routine: LCDETE ..................................................................................................... 47
6.3.26
Routine: NMCJS ....................................................................................................... 48
7 Bibliography ........................................................................................................................................ 48
Handbook of Référence
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Code_Aster ®
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6.4
Titrate:
Law CJS in géomechanics
Date:
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Author (S):
C. CHAVANT, pH. Key AUBERT
:
R7.01.13-A Page
: 4/48
1 Notations
The notations used here are the usual notations of the soil mechanics, to which are added
notations suitable for the writing of the parameters of law CJS.
One also gives the correspondence, if it takes place, between the parameters of the law and their notations
in Aster.
With
parameter of the model
A_CJS
B
parameter of the model
B_CJS
C
parameter of the model
C_CJS
N
parameter of the model
N_CJS
K
modulus of voluminal deformation elastic
E
K
parameter of the model
O
p
K
parameter of the model
KP
O
G
elastic modulus of rigidity
E
G
parameter of the model
O
D
G
function controlling the evolution of the plastic deformations déviatoires
S
diverter of the tensor of the constraints
I
first invariant of the constraints
1
p
pressure of initial criticism
PCO
Co
P
pressure of reference of the model
Pa
has
I
D
F, F
thresholds of the plastic mechanisms isotropic and déviatoire
Q
variable interns model corresponding to the acceptable limit of the plan
Iso
déviatoire
Q Q
,
tensors of the model
R, X
variables intern model corresponding to the average radius and the center
surface of load in the déviatoire plan
R
parameter of the model
RM
m
R
parameter of the model
RC
C
I
D
,
plastic multipliers of the mechanisms isotropic and déviatoire
E
IP
dp
,
, tensors of the respectively total, elastic, plastic deformations
isotropic and plastic déviatoires
voluminal deformations
v
parameter of the model
BETA_C
JS
parameter of the model
GAMMA_
CJS
angle of Lode
function limiting the evolution of X
µ
parameter of the model
MU_CJS
Q
parameter of the model
Q_INIT
init
Note:
Foreword: Contrary for the use of géomechanics, the convention of
sign reserve is that of the mechanics of the continuous mediums, i.e tractions are counted
positively.
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Code_Aster ®
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Titrate:
Law CJS in géomechanics
Date:
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C. CHAVANT, pH. Key AUBERT
:
R7.01.13-A Page
: 5/48
2 Introduction
Model CJS is an elastoplastic law of behavior adapted to the modeling of materials
granular. It was developed in École Centrale of Lyon ([bib1], [bib2], [bib3]).
Version CJS established in Code_Aster is a model arranged hierarchically including/understanding several
levels of complexity. In its most complete expression, the model has two surfaces of
charge: one is activated by the isotropic stresses, the other by the stresses déviatoires.
first undergoes an isotropic work hardening and the second a mixed work hardening (isotropic and kinematic).
The elastic law is of hypoelastic type nonlinear.
3
Description of law CJS
3.1
Partition of the deformations
The increment of total deformation breaks up into three parts, relating to each one of
mechanisms brought into play:
E
IP
dp
& = & + & +
ij
ij
ij
&ij
where E
&, IP
ij
& and dp
ij
ij
& are respectively the increments of elastic strain, of deformation
isotropic plastic and of plastic deformation déviatoire.
3.2 Mechanism
rubber band
The elastic part of the law is of hypoelastic type, whose general expression is:
s&
I&
E
ij
1
& =
+
ij
ij
G
2
9K
where I is the first invariant of the constraints: I = tr
, S is the déviatoire part tensor of
1
()
1
constraints, and where K and G are respectively the voluminal modulus of deformation and the module of
shearing rubber bands. Those depend on the state of stresses according to:
N
N
I + Q
I + Q
E
1
init
K = K
,
E
1
init
G = G
O
P
O
3
3P
has
has
E
K, G, P and N are parameters of the model. P is a pressure of reference equal to - 100
O
O
has
has
kPa.
3.3
Isotropic plastic mechanism
The surface of corresponding load I
F is, in the space of the principal constraints, a plan
perpendicular with the hydrostatic axis, is:
I + Q
I
F (Q
,
= - 1
+ Q
Iso)
(
init)
Iso
3
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Titrate:
Law CJS in géomechanics
Date:
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:
R7.01.13-A Page
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where Q is the thermodynamic force which depends on the variable interns Q according to:
Iso
N
Q
Q & = K p Q & = K p
Iso
Q
Iso
O
&
P
has
p
K, P and N are the parameters of the plastic mechanism déviatoire (P and N are identical to
O
has
has
those of the elastic mechanism). The rule of normality makes it possible to express the evolution of the deformation
plastic and of the variable of work hardening according to the evolution of the plastic multiplier I
:
I
F
1
I
F
IP
I
I
& = &
= - & and
I
I
Q & = - &
= - &
ij
ij
3
Q
ij
Iso
Taking into account the second equation, the law of work hardening can be also put in the form:
N
I
p
Iso
Q
&iso
Q
= - & KB
has
P
3.4
Plastic mechanism déviatoire
The surface of load of this second plastic mechanism is a convex surface with ternary symmetry
defined by the equation:
D
F (, R, X) = Q H + R I + Q
II
(Q) (1 init)
with Q = S - I X
ij
ij
1
ij
Q = Q Q
II
ij
ij
1/
det Q
H (
.
Q) = (1 +
cos (3 Q) 1/6
() 6
= 1+
54
3
qII
The scalar R and tensor X respectively represent the average radius and the center of surface
of load in the déviatoire plan.
S, Q and X are tensors déviatoires. is a parameter which translates the behavior
dissymmetrical of the grounds in compression and extension. is the angle of Lode.
This surface of load evolves/moves according to two types of work hardening: isotropic work hardening and work hardening
kinematics.
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Titrate:
Law CJS in géomechanics
Date:
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:
R7.01.13-A Page
: 7/48
Note:
The expression of the angle of Lode is found in the following way:
In a reference mark (H, I,) J of the déviatoire plan vector HM can be given from
outdistance HM = and angle of Lode (cf [Figure 3.4-a]). Co-ordinates of HM
S
are:
HM = (sin, cos
S
S)
s1
M
J
I
H
s2
s3
Appear 3.4-a: Angle de Lode in the déviatoire plan
The principal components of the diverter are thus:
4
2
S =
cos, S =
and S =
3
cos
- S
2
cos
- S
1
S
3
3
3
Consequently, one a: S =
and
II
2
(S) 1
det
= 3
cos
cos2
2
-
1
sin
3
= 3cos
3
S (
S
S)
(S)
4
4
one deduces the relation then from it:
(
S
cos 3 =
S)
1/2 3/2 det ()
2 3
3
software house
The angle calculation in the same way.
Q
3.4.1 Work hardening
isotropic
The isotropic law of work hardening is written as follows:
2
AR R
R
m
=
&
& (
R + R
With
m
) 2
The thermodynamic force R is related to R whose evolution is given by:
- 5
.
1
1
- 5
.
D
F I Q
I
Q
D
+
init
D
+
1
R & = - &
= - & (I + Q
init)
1
init
1
R
3
P
3
P
has
has
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By direct integration of the law of work hardening, it comes:
AR R
RR
R
m
=
, that is to say also R
m
=
R + R
With
(
WITH R - R
m
)
m
The law of work hardening can thus be also expressed by:
2
1
- .5
+
D
R
R & = - & A 1
(I +Q
1
= &
,
1
init) I
Qinit
D G R (R)
R
3P
m
has
2
1
- 5
.
R
I
Q
R
+
with G (, R) = - A 1 -
(I + Q
init)
1
init
1
R
3
P
m
has
and where R (which are the average radius of the elastic range in rupture) and A are parameters of
m
model.
3.4.2 Work hardening
kinematics
The kinematic law of work hardening is given by:
1
X & =
ij
&ij
B
The thermodynamic force X is a function of the variable whose nonlinear evolution is given
by:
1
- 5
.
D
F
& = -
I
Q
D
& Dev.
I
Q
X
ij
- (+ init)
+
1
ij
init
1
X
P
ij
3
has
The term - (I + Q
allows to obtain nonlinear kinematic work hardening, translating
1
init) X
limitation of the evolution of the surface of load.
D
D
D
F
F Q
F
D
F
By taking account of
kl
=
= - (I + Q
, and while posing: Q = Dev.
, it
init1)
X
Q X
Q
ij
Q
ij
kl
ij
ij
ij
comes finally for the law from work hardening:
1
- 5
.
I + Q
D 1
X & = &
Q + X
I + Q
= & G
ij
(ij
ij) (
init)
1
init
D
X
1
ij (, X)
B
3P
has
- 5
.
1
I
Q
X
1
+
with G
X
Q
X I
Q
.
ij (,
) = (+
ij
ij) (
+ init) 1
init
1
B
3
Pa
where a function which limits the evolution of X and is a parameter of the model.
The tensor Q is calculated according to the formula:
1
Q
54
det Q
Q =
1
cos
3
Dev.
ij
+
() ij
()
+
H ()
5
2
2
Q
6q
Q
II
II
ij
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Titrate:
Law CJS in géomechanics
Date:
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Author (S):
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:
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The preceding expression is obtained in the following way. One a:
D
F
= H (
Q
H
+ Q
Q)
(
II
Q)
II
Q
Q
Q
ij
ij
ij
Q
H (Q)
where
II and
are respectively given by:
Q
Q
ij
ij
Q
Q
II
ij
=
Q
Q
ij
II
H (
1
det Q
-
3
cos () Q
Q)
()
Q
ij
54
=
Q
1+ 54
=
+
det ()
Q
H
6
5
Q
q3
2h () 5
q2
6h () 5 q3
Q
ij
(Q) ij
II
Q
II
Q
II
ij
from where
D
F
1
Q
54 det Q
=
ij
1
cos
3
Q
H 5
2
Q
6q2
Q
ij
(Q) +
(Q)
()
+
II
II
ij
The function, as for it is given by:
= H Q
O
(S) II
1/
det S
where Q = Q Q and H (
. The term is expressed in
S) = (1 +
cos (3 S) 1/6
() 6
= 1+
54
II
ij
ij
3
S
O
II
function of characteristic to the rupture of material.
3.4.3 Law of evolution of the plastic mechanism déviatoire
In granular materials, a variation of volume can occur for a loading
purely déviatoire. This variation of volume is related to the discontinuous aspect of material and on
conditions kinematics which result during the loading. This particular phenomenon does not allow
to define the plastic deformations déviatoires starting from the only rule of normality. This is why it
plastic mechanism déviatoire is nonassociated. There is thus a potential function controlling
evolution of the deformations:
dp
D
D
& = & G
ij
ij
The potential function is defined starting from the following kinematic condition:
dp
S.E.
S
ij &ij
dp
II
& = -
- 1
v
Cs S
II
II
where is a parameter of the model and C
S represents the characteristic state of stress. A surface,
II
from form identical to the surface of load in the space of the constraints, separates the contracting states
dilating states. This surface, known as characteristic, has as an equation:
C
C
F = S H + R I + Q
II
(S) C (1 init)
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where R is a parameter corresponding to the average radius of this characteristic surface. The condition
C
kinematics can be also put in the form:
S
dp
S.E.&
S.E.
ij ij
S
dp
dp
&
S.E.&
dp
& +
II
- 1
= dp
& +
II
- 1 ij ij
ij ij
v
C
v
C
S
S
S
S.E.
S
II
II
II
dp
ij &ij
II
= dp
& +
dp
S.E.
v
ij &ij
software house
= dp
& +
dp
S & = 0
v
ij
ij
software house
S
where
II
=
- 1 sign (
dp
S
&.
C
ij
ij)
S
II
It is then possible to seek to express this kinematic condition starting from a tensor N under
form:
dp
& N = 0
ij
ij
i.e., after decomposition of each term in déviatoire parts and hydrostatic:
dp
dp 1
& N = E & + dp
& N S
N
N S.E.&
N
ij
ij
ij
v
ij (
+
=
dp +
dp
& =
1 ij
2
ij)
0
3
1 ij ij
2
v
n1
One deduces the relation from it
=
, which added to the condition of standardization N:N = 1, led to
N
S
2
II
expressions:
S
ij +
S
1
ij
S
N =
II
and N =
, N is =
II
1
2
ij
2
+ 3
2
+ 3
2
+ 3
The law of evolution of dp
ij & must be such as the kinematic condition is satisfied. It is thus proposed
to take the projection of dp
ij & on the hypersurface of deformation of normal N, is:
D
D
F
F
dp
D
D
D
& = &
-
N N = & G
ij
kl
ij
ij
ij
kl
D
D
F
F
with
D
G =
-
N N.
ij
kl
ij
ij
kl
In addition, for the calculation of the potential, one can note that:
D
D
F
F qkl
=
+ R ij
Q
ij
kl
ij
D
D
F 1 F
1
= Dev.
+
- + X + R
kl
ik jl
ij
kl
kl
ij
Q
3 Q
3
kl
mm
D
1
1 F
1
= Q - Q + Q X +
- + X + R
kl
ik
jl
ij
kl
kl
kl
kl
ik jl kl
ij
kl
kl
kl
kl
ij
3
3 Q
3
mm
= Q - Q X - R
ij
(kl kl
) ij
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Titrate:
Law CJS in géomechanics
Date:
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Author (S):
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:
R7.01.13-A Page
: 11/48
3.4.4 Rough surface
The state of rupture results from the nonlinear nature of the laws of work hardening and the existence of values
limits associated with the variables with work hardening R and X. The limit of R, noted R, is reached when
m
R tends towards the infinite one. The limit of X is reached when X & becomes null.
ij
ij
Under these conditions:
Q = X and Q =
1
X
X
=
ij
ij
II
II lim
II lim
H
O
(S)
In the state of rupture one thus has [Figure 3.4.4-a]:
S + I X
cos
II
Q =
1
II
cos (IIlim
-
S
Q)
By replacing this expression and the value of R in rupture, in the equation of the surface of load
in rupture, one obtains the equation of a limiting envelope for surfaces of load:
R
F = S H
R I
Q
II
(S) + R (+ init) = 0
1
cos
H (S)
with R =
+
R cos -, average radius of the envelope, which is determined from
R
H
O
(Q) m (S Q)
mechanical characteristics with the rupture of material. The value of can then be deduced from it:
O
=
cos
O
H (S)
R -
R cos -
R
H (Q) m
(S Q)
q2 - S 2 - I X
2
II
II
(1 II)
with cos =
2S I X
II 1
II
s1
rough surface
caracterisic surface
q1
S
s2
s3
Q
surface of load to the rupture
q2
q3
Appear 3.4.4-a: Représentation of the rough surfaces, characteristic and of load
in the déviatoire plan
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Titrate:
Law CJS in géomechanics
Date:
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:
R7.01.13-A Page
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In addition, R is related to the maximum angle of friction and depends on the average constraint and on
R
relative density. To take into account the dependence of the maximum angle of friction in function
average constraint and relative density, the relation is considered:
3p
R = R + µln
R
C
C
I + Q
1
init
where R and µ are parameters of the model. p is the average constraint criticizes, i.e.
C
C
minimal average constraint (it is negative with our convention of sign) known by
material during its history. It depends on the initial relative density according to the traditional concept
of straight line criticizes in the plan (E ln
, p):
p = p exp - C
C
Co
(
v)
where p is the initial critical pressure and 1 C is the critical line slope of state in the plan
Co
(ln
, p.
v
)
3.5
Hierarchisation of the model
3.5.1 Summary description of three levels CJS
Starting from the complete description of the model given above, one deduces three levels from complexity
increasing whose characteristics are summarized in the following table:
Elastic mechanism
Plastic mechanism
Plastic mechanism
isotropic
déviatoire
CJS1
linear
not activated
activated, perfect plasticity
CJS2
nonlinear
activated
activated, isotropic work hardening
CJS3
nonlinear
activated
activated, work hardening
kinematics
Table 3.5.1-1: Various mechanisms used by the various levels of model CJS
3.5.2 Assessment of parameters CJS
In addition, one can also summarize the correspondence between the various levels of the model and
parameters associated with each one of them:
N
E
K
E
µ
O
Go
p
K
C
R With B
Rm
pco C
Pa
CJS1
CJS2
CJS3
Table 3.5.2-1: Assessment of the various parameters according to levels CJS
In Code_Aster, the elastic parameters of model CJS (
E
K and G) are directly taken in
O
O
count in the elastic characteristics of material, i.e. through the Young modulus E and
the NAKED Poisson's ratio.
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:
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In Code_Aster, the user explicitly does not indicate selected level CJS only it. It is in
effect the choice of the various parameters which determines the corresponding level. We have for
to summarize the following logical tests which are integrated in the code:
·
if N = 0 then level CJS1,
·
if (N 0 and A 0) then level CJS2,
·
if (N 0 and A = 0) then level CJS3.
Note:
The user must fix the value of Pa equal to - 100 kPa according to the selected units. In
in addition to, for CJS3, the value of pco must be negative.
3.5.3 Correspondence with the cohesion and the angle of friction
The mechanics of the grounds are accustomed to using the concepts of cohesion Cohésion C, of angle of
friction and of angle of dilatancy: . These parameters are used in the law of Mohr Coulomb.
Level 1 of law CJS makes it possible to find a very nearby behavior by making the following choice
parameters:
1/6
1 -
3 - sin ()
=
1+
3 + sin ()
2
2
sin () (1 -) 1/6
=
3
m
R
3 - sin ()
Q
= - 3.
C cotan
init
()
2 6 sin ()
= -
3 - sin ()
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Law CJS in géomechanics
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:
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4
Integration of law CJS
We detail below the integration of law CJS according to or of the activated mechanisms:
·
nonlinear rubber band,
·
nonlinear rubber band and isotropic plastic
·
nonlinear rubber band and plastic déviatoire
·
nonlinear rubber band, isotropic plastic and plastic déviatoire.
In each case, the goal is to calculate, starting from the fields known with the state less -
, -
and of
the increment of deformation
, the new state of stress +
.
In the sequence of calculations, one starts by making the assumption that only the elastic mechanism
nonlinear intervenes. An elastic prediction is thus carried out. This prediction is then used
to calculate the functions of load I
F and D
F, one seeks to know if one goes then beyond
thresholds:
·
if I
F 0 and D
F 0, the elastic prediction is regarded as new state of stress,
·
if
I
F > 0 and D
F 0, one makes the integration of the mechanisms elastic nonlinear and plastic
isotropic,
·
if
I
F 0 and D
F > 0, one makes the integration of the mechanisms elastic nonlinear and plastic
déviatoire,
·
if
I
F > 0 and D
F > 0, one makes the integration of the mechanisms elastic nonlinear, plastic
isotropic and plastic déviatoire.
At output of elastoplastic calculation, when only one plastic threshold was initially exceeded, one
recompute each function of load. Indeed, it is possible that while seeking to bring back itself on
one of the thresholds, one then exceeds the other threshold not activated initially by the elastic prediction. In
this case, one solves then by integrating all the mechanisms.
4.1
Choice of the internal variables
The variables Q, R and are equivalent to the associated thermodynamic forces Q, R and X.
Iso
For this reason and since their geometrical significance is more obvious, we will retain like
variables intern for the integration of law CJS, the sizes Q, R and X.
Iso
In addition, we add to the number of the internal variables:
·
the sign of the product
dp
S
ij
ij
·
the elastic or elastoplastic state of material, while noting:
- 0: elastic state
- 1: elastoplastic state, isotropic plastic mechanism
- 2: elastoplastic state, plastic mechanism déviatoire
- 3: elastoplastic state, plastic mechanisms isotropic and déviatoire
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:
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Finally, the variables intern are stored in a vector VI in the following order:
Internal index of variable
CJS1
CJS2
CJS3
3D 2D CJS1
CJS2
CJS3
1 1 Q =
Q
Q
Iso
Iso
Iso
2 2 R = R
R
R = R
m
m
3
3 0
0
X
11
4 4
0
0
X
22
5 5
0
0
X
33
6.6
0
0
2X
12
7
0 0 2X
13
8
0 0 2X
23
9 7 Q H ()
Q H ()
Q H ()
II
Q
II
Q
II
Q
R I + Q
R (I + Q
R I + Q
m (1
init)
1
init)
m (1
init)
10 8
R
X
II
R
lim
X
m
II
11 9
Q
3
Q
3
I + Q
I + Q
1
init
1
init
12 10
Numbers
iterations
Iteration count
Iteration count
interns
interns
interns
13
11
local test reached
local test reached
local test reached
14
12
no. of recutting no. of recutting no. of recutting
15 13
sign (
dp
S)
sign (
dp
S)
sign (
dp
S)
ij ij
ij ij
ij ij
16
14
0,1,2,3 state of
0,1,2,3 state of
0,1,2,3 state of
material
material
material
4.2
Integration of the nonlinear elastic mechanism
In the elastic case, the new state of stress +
, checks simply:
+ = - + D
+
ij
ij
ijkl (
) kl
The dependence of the nonlinear tensor of elasticity according to the state of stresses is summarized in
fact with:
I +
+
+ Q
D
= D
1
ijkl (
)
N
linear
init
ijkl
3P
has
where
linear
D
is the tensor of isotropic linear elasticity traditional, obtained from
E
K and G or by
ijkl
O
O
equivalence starting from E and Naked.
From this relation, one deduces in particular that the first invariant of the constraints satisfied:
+
N
+
-
E
I + Q
I - I - 3
1
K
init
tr =
1
1
O
() 0
3
Pa
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Law CJS in géomechanics
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:
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This nonlinear equation is solved by a method of the secant for CJS2 and CJS3, in
differentiating the cases following the sign from tr (). With regard to the model CJS1, for which
parameter N is null, the explicit resolution is immediate, since one has then
+
I = -
I + K E
3
tr
1
1
O
()
N
I + + Q
In the general case, the knowledge of +
I and thus of term 1
init
allows to define
1
P
3 A
the nonlinear operator of elasticity D (+
. Obtaining the new state of stress is then direct.
ijkl
)
4.3 Integration of the mechanisms elastic nonlinear and plastic
isotropic
In this case, the new state of stress +
, checks:
+ = - + D
+
-
ij
ij
ijkl (
) (
IP
kl
kl)
Being given the simple form, plastic deformations of the isotropic plastic mechanism:
IP
1
I
= -
ij
ij
3
the nonlinear system to solve is composed of:
+
-
+
1
I
·
: the elastic law state: - - D
ij
ij
ijkl (
) +
kl
kl = 0
ij
3
·
LQ: the law of work hardening of the variable interns Q: +
Q - -
Q - I Q
G Iso
Q
Iso
Iso
(+iso) = 0
Iso
+
I + Q
·
FI: the equation of the isotropic surface of load:
1
-
init + +
Q = 0
3
Iso
Schematically, one thus seeks to solve the system R (Y) = 0, where the unknown factor Y is given
by Y = (+
+
I
Q
,
,
and where R = (IT, LQ, FI. The resolution of R (Y) = 0 is done by the method
ij
)
ij
Iso
)
of Newton:
·
0
initialization and calculation of a solution of test Y
DR.
·
iterations of Newton: resolution of
(p
Y)
1
+
p
DY
= - R (p
Y)
DY
·
test of convergence: if convergence
p
Y = Y; if not
1
+
p
p
1
+
p
Y
= Y + DY
and p = p + 1
We detail these three stages below.
4.3.1 Initialization and solution of test
Y 0 = (
0
0
0
, Q, I
, following values:
ij
Iso
)
We take simply for
0
elas
=
: constraints given by the elastic prediction,
ij
ij
-
Q0 = Q: variable interns with T
Iso
Iso
0
I
= 0: plastic multiplier no one
Contrary to the other elastoplastic mechanisms, here a solution of test is not calculated.
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4.3.2 Iterations of Newton
DR.
The resolution of
(p
Y)
1
+
p
DY
= - R (p
Y) naturally requires the calculation of derived from,
DY
ij
LQ and FI compared to each component of Y. One a:
ij
ij
ij
I
Q
kl
Iso
DR.
LQ
LQ
LQ
=
I
DY
Q
kl
Iso
FI
FI
FI
Q
I
kl
Iso
with:
N
D
1
-
ij
+
ijmn
1
1
N
I
Q
I
linear
I
1
init
= -
+
= - D
+
ik
jl
mn
mn
ik
jl
ijmn
mn
mn
kl
3
3
P
3
P
3
kl
kl
has
has
LEij = 0
Qiso
LEij
1
= - D
I
ijmn
mn
3
LQ = 0
kl
N 1
-
Q
p
LQ
G Iso
nK Q
= 1 - I
= 1+ I
O
Iso
Q
Q
P
P
Iso
Iso
has
has
LQ
Iso
Q
= G
-
I
FI
1
= -
kl
3
kl
FI =1
Qiso
FI = 0
I
4.3.3 Test of convergence
1
+
p
DY
The iterations of Newton are continued as much as the relative error
remain higher than
1
+
p
0
Y
- Y
tolerance allowed by the user and defined by key word RESI_INTE_RELA. The standard used here is
the vectorial standard: X = x2.
I
I
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4.4 Integration of the mechanisms elastic nonlinear and plastic
déviatoire
In this case, the new state of stress +
, checks:
+ = - + D
+
-
ij
ij
ijkl (
) (
dp
kl
kl)
The plastic deformations of the plastic mechanism déviatoire are given by the potential
D
G:
dp
D
D
= G
ij
ij
One deduces from it that the nonlinear system to solve is composed of:
· : the elastic law state: +
- -
- D
ij
ij
ijkl (+) (
- D Gd
kl
kl (+,
+
R, +
X) = 0
ij
· LR: the law of work hardening of the variable R: +
R - -
R - D
GR. (+
, +
R) = 0
· LX: the law of work hardening of variable X:
+
X -
-
X - D X
G
ij
ij
(+
, +
X) = 0
ij
ij
· FD: the equation of the surface of load déviatoire: +
Q H
R I
Q
II
(+q) + + (+ + init) = 0
1
As in the preceding paragraph one solves by the method of Newton the system R (Y) = 0, where
the unknown factor Y is given by Y = (+
+
+
D
, R, X,
and where R = (IT, LR, LX, FD.
ij
ij
)
ij
ij
)
4.4.1 Initialization and solution of test
Starting from the state at the moment T (-
-
-
, R, X, we seek a solution of test which brings us closer
ij
ij)
the final solution. For that we solve the following equation:
D
F (-
+ -
D
G
R
G
X
G
ij
ijkl (
- D D
kl
kl),
- + D R, - + D X -
ij
ij
) = 0
with
-
D
= D
,
D -
G
= Gd
,
R
G
= G R (- -
, R), X
G
= X
G
and where
ij
ij (-
-
, X)
kl
kl (-
-
-
, R, X)
ijkl
(-
ijkl
)
the unknown factor is the plastic multiplier
D
, by only one iteration of Newton, i.e.
finally of we let us have:
D
F
D
D
F
=
D
= - D
0
F
D
D
=0 is still = -
D
D
F
D
=0
D
D =0
with:
D
F
= H () Q
H (Q)
R
I
II
+ Q
+ I + Q
+ R
1
D
Q
D
II
(
D
1
init)
D
D
Moreover,
I
one a:
-
-
I = I + K
3
then:
1
-
= - 3K tr G
D
(D)
1
1
(tr () - dtr (D
G)
R
one a:
-
D
R
R = R + G then:
R
= G
D
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-
-
1
D
D -
-
-
D
D -
-
D
X -
one a: Q = + D
G
I
3K tr
tr G
X
G
ij
ij
ijkl (
-
kl
kl) - [
+
1
(() - ()] + +
ij
ij
ij
3
qij
-
1
D -
-
D -
-
then:
= - D G + 3K tr G
3
D
ijkl
kl
() + X
ij
ij -
X -
Gij (-
I +
-
K tr
1
(
)
3
D
=0
Q
Q Q
Q Q
H (
H Q
Q)
(Q)
one a:
II
II
ij
ij
ij
=
=
and
ij
=
D
D
D
Q
Q
D
D
Q
ij
II
ij
Ultimately, we take for the solution of test: Y 0 = (0
0
0
D 0
, R, X,
, with the values
ij
ij
)
following:
D 0
: the value found according to the preceding formulation.
0
-
-
= + D
0
G
ij
ij
ijkl (
D
D -
-
kl
kl)
0
-
D 0
R
R = R + G
0
-
D 0
X -
X = X + G
ij
ij
ij
4.4.2 Iterations of Newton
DR. is given here by:
DY
ij
ij
ij
ij
D
R
X
kl
ij
LR
LR
LR
LR
D
DR.
R
X
=
kl
ij
DY
LX
LX
LX
LX
ij
ij
ij
ij
D
R
X
kl
ij
FD
FD
FD
FD
R
X
D
kl
ij
with:
- 1
ij
N I + Q
G
linear
= - D
-
G
1
+ D
ik
jl
ijmn
(
D
D
mn
mn)
N
D
init
D
mn
kl
ijmn
P
3
P
3
kl
has
has
kl
LEij
D
G D
= D
mn
R
ijmn
R
D
LEij
G
D
mn
= D
ijmn
X
X
kl
kl
LEij
D
= D G
D
ijmn
mn
R
2
-,
1 5
LR
G
With
R I + Q
D
D
1
init
= -
= -
1
kl
2
R
P
3
kl
kl
m
has
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-,
1 5
R
LR
G
With
R
I
Q
D
D 2
+
= 1 -
= 1 -
1
(I + Q
init)
1
init
1
R
R
R
R
3
P
m
m
has
LR = 0
X kl
LR
R
= G
-
D
X
LX
G
ij
D
ij
= -
kl
kl
LX ij = 0
R
X
LX
G
ij
D
ij
= -
ik
jl
X
X
kl
kl
LX ij
X
= G
-
D
ij
D
FD F
=
= Q - Q X - R
kl
(mn mn
) kl
kl
kl FD = I 1
R
D
FD
F
=
X
X
kl
kl
FD = 0
D
D
G
Gd
D
G
X
G
X
G
D
F
In addition, the calculation of the terms
mn,
mn,
mn,
ij
,
ij
and
is detailed
R
X
X
X
kl
kl
kl
kl
kl
hereafter, as well as the calculation of useful intermediate terms:
D
F
·
calculation of
:
X kl
D
F
H (
Q)
Q
= Q
+ H
II
() II
Q
X
X
X
kl
kl
kl
H (
Q)
Q
Q
Q
mn
= Q
+ H
II
(Q) II mn
Q
X
Q X
mn
kl
mn
kl
H (
Q)
Q
Q
Q
mn
= Q
+ H
II
(Q) mn mn
Q
X
Q X
mn
kl
II
kl
H (
Q)
Q
Q
= Q
+ H
II
(Q) mn
mn
Q
Q X
mn
II
kl
H (
Q)
Q
= - I Q
+ H
1
II
() mn
Q
mk
nl
Q
Q
mn
II
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D
F
= - I
1
qkl
One will notice for the continuation that:
D
F
Dev.
= - I Q
1
kl
X
kl
D
F
·
ij
calculation of
:
kl
D
D
F
F
ij
ij qmn
=
Q
kl
mn
kl
(Q - Q X - R
ij
(rs rs
) ij) qmn
=
Q
mn
kl
Q
ij
Q
Q
rs
mn
=
-
X
rs
ij
Q
Q
mn
mn
kl
Q
ij
Q
rs
mn
=
-
X
X
rs ij
-
mk
nl
kl
+ mn
Q
Q
3
mn
mn
Q
·
calculation of
ij:
qmn
Au préalable, one definite the tensor T and its part déviatoire D
T while posing:
det (Q)
det Q
D
()
T =
and T = Dev.
ij
Q
ij
Q
ij
ij
One has as follows:
T
Q Q
Q Q
11
-
22
33
23
23
T
Q Q
Q Q
22
-
11 33
13 13
T Q Q - Q Q
T = 33 = 11 22
12 12
T
Q Q
Q Q
12
-
13
23
12
33
T Q Q - Q Q
13 12 23
13
22
T
Q Q
Q Q
23
-
12 13
23 11
Handbook of Référence
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Titrate:
Law CJS in géomechanics
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Author (S):
C. CHAVANT, pH. Key AUBERT
:
R7.01.13-A Page
: 22/48
ij
Q
- 5
qij 54
(H Q)
=
qmn
H (
Q) 6 1 + cos (3 Q)
+
2 Dev. (tij)
(
2
Q
6q
II
II
qmn
Q
D
T
ij
ij
1
2
Q
II
1
Q
cos
ij
(3 Q)
1
54 qII
+
H (
Q) 5 1 + cos (3 Q)
+
+
2
Q
H
mn
(Q) 5 2q Q
H
II
mn
(Q) 5 6 qmn
- 5
qij 54
(H Q)
1
Q Q
im jn
ij mn
=
H (
Q) 6 1 + cos (3 Q)
+
2 Dev. (tij)
()
+
1 + cos 3
-
Q
2
Q
6q
II
II
Q
H
mn
(Q) 5
()
3
2
Q
Q
II
II
D
1
Q
54
ij
det Q
1
54 tij
Q
D
+
H (Q)
mn
5
T-3
4
mn
2 qmn +
- 2tij
2 qII
qII
H ()
5
2
2
6 Q
Q
II Q
Q
mn
II
D
T
The expression of
ij
clarify yourself as follows:
qmn
- 1 (
1
1
Q + Q
(- Q + 2q
(- Q + 2q
11
22)
11
33)
22
33)
3
3
3
1 (-
1
1
Q + 2q
- (Q + Q
(2q - Q
11
22)
11
33)
22
33)
D
3
D
3
D
3
T
= 1 (
T
1
T
1
2q - Q
,
= (2q - Q
,
= - (Q + Q
,
11
22)
11
33)
22
33)
Q
Q
q33
22
11
3
3
3
0
0
- Q
12
0
- Q
0
13
- Q
0
0
23
2
2
4
Q
Q
- Q
12
13
23
3
3
3
2
4
2
Q
- Q
Q
13
23
D
12
3
D
3
D
3
T
= 4
T
T
,
= 2
,
= 2
Q
- Q
Q
Q
12 Q
13 Q
23
12
3
13
3
23
3
- Q
Q
Q
33
23
13
Q
- Q
Q
23
22
12
Q
Q
- Q
13
12
11
D
G
·
calculation of
mn:
kl
One a:
D
D
F
F
D
D
D
G
F
N
F N
mn
mn
mn
rs
rs
=
-
N
-
N +
N
rs
rs
mn
kl
kl
rs
kl
kl
rs
kl
Handbook of Référence
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Titrate:
Law CJS in géomechanics
Date:
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Author (S):
C. CHAVANT, pH. Key AUBERT
:
R7.01.13-A Page
: 23/48
S
One definite tensor N
~ by
ij
n~ =
+
ij
ij
software house
n~
i.e. that N is then given by
ij
N =
with ~
2
N = + 3
ij
n~
II
II
In practice, for the calculation of, one uses
in the place of
dp
, i.e. one a:
ij
ij
S
II
=
- 1 sign (S
C
ij
ij)
S
II
D
G
One has then for
mn:
kl
D
D
F F
1
D
D
~
~
D
~
D
G
F N N
F
N
1
F
n~ 2
mn
mn
rs
rs
mn
~
mn
~ ~
II
=
-
N N -
-
N
-
N
N
rs
mn
~ 2
rs
~ 2
rs
mn
N
N
kl
kl
kl
rs
kl
II
rs
kl
II
rs
kl
with:
1
1
S
S
2 2
- 1
n~2
2
+ 3
1
2
S
S
II
(
)
()
II
II
C
C
II
II
=
= -
= -
+ 2
2
3
+ 2
2
3
kl
kl
(
)
kl
(
)
kl
S
II
Cs
·
calculation of
II
:
kl
S
II
Cs
1
II
(S
S
S
II)
II
(cII)
=
-
C
C
S
2
kl
II
kl
S
kl
II
R I + Q
C (1
init)
-
1 (
S S
S
H
II)
(
mn
II
S)
=
-
C
C
S S
2
II
mn
kl
S
kl
II
1 S
1
S
R
I
R I
Q
H
mn
II
C
1
C (
+
1
init)
(S)
=
-
C
mk
nl
mn
kl -
C 2
-
S S
3
S
H
2
II
II
(S)
+
H
II
kl
(S)
kl
1 S
1
S
R
R I
Q
H S
mn
II
C
C (
+
1
init)
(S)
=
-
C
mk
nl
mn
kl -
rs
C 2
-
S S
3
S
H
H2
S
II
II
(S) +
kl
II
(S)
rs
kl
Handbook of Référence
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HT-66/03/005/A
Code_Aster ®
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Titrate:
Law CJS in géomechanics
Date:
18/11/03
Author (S):
C. CHAVANT, pH. Key AUBERT
:
R7.01.13-A Page
: 24/48
n~
·
calculation of
mn:
kl
1
1
n~
1
1
C
S
S
S
mn =
-
sign (S
sign S S
C
ij
ij)
mn +
(
ij
ij)
II
II
mn
-
S
S
kl
II
II
kl
kl
kl
1
1
C
1
1
1
=
-
sign (S
S
S
sign S S
C
ij
ij)
-
mk
nl
mn
kl +
(
ij
ij)
(II)
II
mn
S
S
3
-
2
C
S
2
II
II
II
kl
S
kl
II
Gd
·
calculation of
mn:
R
D
D
F
F
D
D
D
G
F
N
F N
mn
mn
mn
rs
rs
=
-
N
-
N +
N
rs
rs
mn
R
R
R
R
R
rs
rs
D
D
F F
mn
rs
=
-
N N
rs
mn
R
R
= - N N
mn
(rs rs) mn
S
2
- 3 mn
mn
S
=
II
2
+ 3
D
G
·
calculation of
mn:
X kl
D
D
F
F
D
D
D
G
F
N
F N
mn
mn
mn
rs
rs
=
-
N
-
N +
N
rs
rs
mn
X
X
X
X
X
kl
kl
rs
kl
kl
rs
kl
D
D
F F
mn
rs
=
-
N N
rs mn
X
X
kl
kl
Handbook of Référence
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Titrate:
Law CJS in géomechanics
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Author (S):
C. CHAVANT, pH. Key AUBERT
:
R7.01.13-A Page
: 25/48
D
F
·
calculation of
mn
:
X kl
D
F
mn
Q
Q
X
mn
rs
rs
=
-
X + Q
rs
rs
mn
X
X
X
X
kl
kl
kl
kl
Q Q
Q Q
mn
ij
rs
ij
=
-
X + Q
rs
rs
Kr
ls
mn
Q X
Q X
ij
kl
ij
kl
Q
Q
mn
rs
= - I
- - I
X + Q
1
ik
jl
1
ik
jl
rs
rs
Kr
ls
mn
Q
Q
ij
ij
X
G
·
calculation of
ij
:
kl
-
-
X
G
I
Q
I
Q
ij
1
= -
(Q +
I
Q
X
X
I
Q
ij
ij)
,
1 5
,
1 5
+
1
init
ij
+
1
1 +
+
ij (
+ init) 1
init
1
2b
3
P
B
3
P
kl
has
kl
kl
kl
has
-
-
1
I
Q
Q
= -
(Q +
Q
I
Q
X
I
Q
ij
ij)
,
1 5
,
1 5
+
1
init
ij
+
1
+
mn
kl
(+ init) 1
init
1
2b
3
P
B Q
3
P
has
mn
kl
has
-,
1 5
1
+
H (
H
Q
I
Q
Q
Q
H
X I
Q
S)
O
(S)
+
+
II
O
II
O (
)
II
S
ij (
+ init) +
1
init
1
B
3
P
kl
kl
kl
has
H (S)
·
calculation of
:
kl
H (
H S
S)
(S) mn
=
S
kl
mn
kl
54
3
cos (
)
Q
1
=
T -
S
5
mn
3
5
mn
mk
nl
mn kl
H
6 (Q
2h
q2
3
S)
II
(S)
-
II
Q
·
calculation of
II:
kl
Q
Q Q Q
II
II
rs
mn
=
Q Q
kl
rs
mn
kl
Q Q
1
rs
rs
=
-
X
mk
nl
mn
+
kl
kl
Q Q
3
II
mn
Handbook of Référence
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Titrate:
Law CJS in géomechanics
Date:
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C. CHAVANT, pH. Key AUBERT
:
R7.01.13-A Page
: 26/48
·
calculation of
O:
kl
O
1
cos
=
H (
kl
S)
R -
R
R
H (Q) cos
m
(-
S
) kl
Q
R
H
H
H
H
R
1
cos (-
S
S
S
S
S
Q)
-
R
-
+
R
-
-
R
H
H
H
kl
(Q) cos
m
(S Q) () ()
kl
(Q) cos
m
(S Q) () () m
2
kl
(Q)
kl
- cos
H (S)
2
R -
R
-
R
H (Q) cos
m
(S Q)
with:
cos
1
Q
2 I1
S
=
II
II
2q
- 2I X
- 2s
II
1
II
II
2S I X
kl
II 1
II
kl
kl
kl
Q2 - I X
2
s2
1
I1
S
II
(
II)
-
-
II
II
S X
+ I X
II
II
1
II
S I X
II 1
II
kl
kl
1
2
2
2
S
=
(Q - I X - S
Q
I X
S 2 S X
I X
kl
1
II
kl
kl) - (
-
II
(1 II) - II)
kl
+
II
II
kl
1
II
S I X
S
II 1
II
II
R
µ
R = -
kl
I + Q
kl
1
init
cos (-
S
Q) = - sin (-
S
Q)
S
Q
-
kl
kl
kl
X
G
· calculation of
ij
:
X kl
-,
1 5
X
G
Q
X
ij
1
ij
ij
=
+
(I +
I
Q
Q
init)
+
1
init
1
X
B X
X
3
P
kl
kl
kl
has
-,
1 5
1 Q
ij
Q
=
I
Q
mn +
I
Q
ik
jl (
+ init) +
1
init
1
B Q X
3
P
mn
kl
has
-,
1 5
1
Qij
I + Q
=
- I
+
I +
init
Q
1
mk
nl
ik
jl (
init)
1
1
B
Q
3
P
mn
has
4.4.3 Test of convergence
1
+
p
DY
The criterion of convergence remains
RESI_INTE_RELA.
1
+
p
0
Y
- Y
Handbook of Référence
R7.01 booklet: Modelings for Génie Civil and the géomatériaux ones
HT-66/03/005/A
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Titrate:
Law CJS in géomechanics
Date:
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Author (S):
C. CHAVANT, pH. Key AUBERT
:
R7.01.13-A Page
: 27/48
4.5 Integration of the mechanisms elastic nonlinear, plastic
isotropic and plastic déviatoire
In this case, the new state of stress +
, checks:
+ = - + D
+
-
-
ij
ij
ijkl (
) (
IP
dp
kl
kl
kl)
Taking into account what precedes, one deduces from it that the nonlinear system to solve is composed of:
·
: the elastic law state:
ij
+
- -
- D
ij
ij
ijkl (+)
1
+ I
- D Gd
kl
kl
kl (+
+
+
, R, X) = 0
3
·
LQ: the law of work hardening of the variable interns Q: +
Q - -
Q - I Q
G Iso
Q
Iso
Iso
(+iso) = 0
Iso
·
LR: the law of work hardening of the variable R: +
R - -
R - D
GR. (+
, +
R) = 0
·
LX: the law of work hardening of variable X: +
X -
-
X - D X
G
ij
ij
ij (+
, +
X) = 0
ij
ij
+
I + Q
·
FI: the equation of the isotropic surface of load:
1
-
init + +
Q = 0
3
Iso
·
FD: the equation of the surface of load déviatoire: +
Q H
R I
Q
II
(+q) + + (+ + init) = 0
1
As in the preceding paragraphs one solves by the method of Newton the system R (Y) = 0, where
the unknown factor
Y is given by Y = (+ +
+
+
I
D
Q
,
, R, X,
,
and where
ij
Iso
ij
)
R = (IT, LQ, LR, LX, FI, FD.
ij
ij
)
4.5.1 Initialization and solution of test
Starting from the state at the moment T (-
-
-
-
Q
,
, R, X
, we seek a solution of test which us
ij
Iso
ij)
bring closer the final solution. For that we solve the system of equations according to:
I -
+
1
I
D
D
-
I
Q -
F + D
G
Q
G Iso
ij
ijkl
+
-
kl
kl
kl,
+
=
Iso
0
3
D -
+
1
I
D
D
-
D
R
-
D
X -
F
+ D
G
R
G
X
G
ij
ijkl
+
-
kl
kl
kl,
+
,
+
=
ij
ij
0
3
with:
-
D
= D
,
D -
G
= Gd
,
Q -
G Iso = Q
G Iso (-
Q
R
G
= G R (- -
, R),
Iso)
kl
kl (-
-
-
, R, X)
ijkl
(-
ijkl
)
X -
G
= X
G
and where the unknown factors are the plastic multipliers
I
and
D
, by one
ij
ij (-
-
, X)
only iteration of Newton, i.e. finally that we have:
I
I
F
F
I
+
D
= - I
F I =0, D =0
I
D
I
D
=0, =0
I
=0, D
=0
D
D
F
F
I
+
D
= - D
F I =0, D =0
I
D
I
D
=0, =0
I
=0, D
=0
Handbook of Référence
R7.01 booklet: Modelings for Génie Civil and the géomatériaux ones
HT-66/03/005/A
Code_Aster ®
Version
6.4
Titrate:
Law CJS in géomechanics
Date:
18/11/03
Author (S):
C. CHAVANT, pH. Key AUBERT
:
R7.01.13-A Page
: 28/48
that is to say still:
I
D
F
F
D
I
F
F
D
I
F -
F
I
D
F -
F
D
D
I
I
I
=
and
D
=
I
D
I
D
F F
F F
I
D
I
D
-
F
F
F
F
-
I
D
D
I
I
D
D
I
with:
I
F = - (-
p
K + K
)
I
I
F
-
= K tr G
D
(D)
D
F
= H (
Q
H
R
I
+ Q
+ I + Q
+ R
1
I
Q)
(
II
Q)
I
II
(
I
1
init)
I
I
D
F
= H (
Q
H
R
I
+ Q
+ I + Q
+ R
1
D
Q)
(
II
Q)
D
II
(
D
1
init)
D
D
D
F
It is known that
is calculated in the same way that previously when only the mechanism
D
D
F
plastic déviatoire was activated. In addition, one has, for the calculation of
and when I
= 0 and
I
D
= 0, following relations:
qij
1 -
E 1
-
= D - K
3
X
I
ijkl
kl
+
ij
ij
3
3
Q
Q Q
Q
ij
ij 1
1
II
II
-
E
-
=
=
D
K
3
X
I
I
-
ijkl
kl
+
ij
ij
Q
Q
3
3
ij
II
H (
H Q
Q)
(Q) ij
=
I
I
Q
ij
R = 0
I
I1
-
= 3K
I
Ultimately, we take for the solution of test: Y 0 = (0
0
0
0
i0
D 0
, Q, R, X,
,
, with
ij
Iso
ij
)
following values:
i0
: the value found according to the preceding formulation.
D 0
: the value found according to the preceding formulation.
1
0
-
-
i0
D
D -
= + D
0
G
ij
ij
ijkl
+
-
kl
kl
kl
3
0
-
i0
-
Q = Q +
Iso
Q
G
Iso
Iso
0
-
D 0
R
R = R + G
0
-
D 0
X -
X = X + G
ij
ij
ij
Handbook of Référence
R7.01 booklet: Modelings for Génie Civil and the géomatériaux ones
HT-66/03/005/A
Code_Aster ®
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Titrate:
Law CJS in géomechanics
Date:
18/11/03
Author (S):
C. CHAVANT, pH. Key AUBERT
:
R7.01.13-A Page
: 29/48
4.5.2 Iterations of Newton
DR. is given here by:
DY
ij
ij
ij
ij
ij
ij
I
D
Q
R
X
kl
Iso
kl
LQ
LQ
LQ
LQ
LQ
LQ
I
D
Q
R
X
kl
Iso
kl
LR
LR
LR
LR
LR
LR
DR.
Q
R
X
I
D
=
kl
Iso
kl
DY
LX
LX
LX
LX
LX
LX
ij
ij
ij
ij
ij
ij
Q
R
X
I
D
kl
Iso
kl
FI
FI
FI
FI
FI
FI
I
D
Q
R
X
kl
Iso
kl
FD
FD
FD
FD
FD
FD
I
D
Q
R
X
kl
Iso
kl
where the new terms are null:
LQ
LX
=
LQ
LQ
LR
LR
0,
= 0,
= 0,
= 0,
= 0,
ij = 0,
R
X
D
Q
I
Q
kl
Iso
Iso
LX
ij =
FI
FI
FI
FD
FD
0,
= 0,
= 0,
= 0,
= 0,
= 0
I
R
X
D
Q
I
kl
Iso
and where the already definite terms remain unchanged, except for
ij which becomes:
kl
n-1
D
LEij
1
N I + Q
G
linear
I
D
D
1
init
D
mn
= - D
+
-
G
+ D
ik
jl
ijmn
mn
mn
mn
kl
ijmn
3
P
3
P
3
kl
has
has
kl
4.5.3 test of convergence
1
+
p
DY
The criterion of convergence remains
RESI_INTE_RELA
1
+
p
0
Y
- Y
4.6
Procedure of relieving based on an estimate of the normals with
surface of load déviatoire
When the plastic mechanism déviatoire intervenes, a procedure of relieving inside
iterations of Newton is taken into account. This one makes it possible to avoid certain problems of oscillation in
the calculation of the solution p 1
+
Y
who lead finally to nonthe convergence of integration
numerical.
Thus, with the iteration p + 1, instead of bringing up to date the unknown factor p 1
+
Y
by a complete increment
p 1
+
Y
p 1
+
p
p 1
+
Y
= Y + Y
one poses
p 1
+
p
p 1
+
Y
= Y + Y
m
m
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Titrate:
Law CJS in géomechanics
Date:
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:
R7.01.13-A Page
: 30/48
and one seeks, by carrying out a loop on under-iterations m, to determine an optimal value
scalar. This value is required by considering the rotation of the normal, in the plan
m
déviatoire, on surface
D
F, during under-iterations. This normal, noted n~m, is expressed with
to leave the constraints contained in the term
p 1
+
Y
by
m
F
6
~
5
det (Q
Nm =
)
2 H (Q)
D
qII
= (2 + cos (3q) qij +
qij
qII
Q
ij
Starting from the initial value = 0
.
1, the process set up consists of the following stages:
0
·
calculation of the normals ~
Nm 1
- and N
~m
N
~
N
~
:
·
calculation of the swing angle
m 1
-
m
=
m between these normals: cos
m
N
~
~
m
N
1
-
m
·
test on the evolution cos:
m
if cos TOLROT then
= DECREL and m = m +1
m
m+1
m
if not end of the under-iterations and
p 1
+
1
+
Y
= p
Y
m
4.7
Recutting of the step of time
As for other relations of behavior (CHABOCHE, VISCOCHAB, TAHERI, LMARC), it was
introduced for model CJS the possibility of redécouper locally (at the points of Gauss) the step of
time in order to facilitate numerical integration. This possibility is managed by the operand
ITER_INTE_PAS of key word CONVERGENCE of operator STAT_NON_LINE. If itepas, the value of
ITER_INTE_PAS, is worth 0, 1 or - 1 it has no recutting there (note: 0 are the default value). If
itepas is positive recutting is automatic, if it is negative recutting is not taken in
count that in the event of nonconvergence with the step of initial time.
Recutting consists in realizing, after the phase of elastic prediction, the integration of
plastic mechanisms put in plays with an increment of deformation of which components
correspond to the components of the initial increment of deformation divided by the absolute value of
itepas.
4.8 Remarks
any other business
4.8.1 Calculation of the cos term (-
S
Q)
The cos term (- appears in the expression of. We adopted for his calculation
S
Q)
O
even method that that used with the ECL. I.e. we determine the angles and of
S
Q
the manner which follows:
1
1 cos2
-
(
3
1 cos
1
2
-
(3q)
S)
= Arctan
and = Arctan
S
3
(
cos
3
Q
3
(
cos
3 Q)
S)
then we take the cosine of the difference.
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Law CJS in géomechanics
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:
R7.01.13-A Page
: 31/48
These expressions of and are also useful for calculation of:
S
Q
cos (-
S
Q) = - sin (-
S
Q)
S
Q
-
kl
kl
kl
1
54
2
det (Q)
with
S = -
1 - cos 3
()
T
3
Q
S
3
-
kl
2
kl
3
Q
Q
kl
II
II
4.8.2 Calculation
of
R
R
The radius of rupture introduced into model CJS3 is given by the formula
3p
R = R + µln
R
C
C
I + Q
1
init
I + Q
In fact, when 1
init > p, one must block R with the value of R. the field of dilatence
C
3
R
C
disappears and one does not admit that R can decrease in on this side R. Par conséquent, one introduces, with
R
C
place preceding formulation, the following expression
3 p
R = R + µ max,
0 ln
R
C
C
I + Q
1
init
4.8.3 Traction
Non-cohesive, the field of traction which corresponds to positive constraints is inadmissible
for the grounds. From the point of view of the integration of model CJS, when the state of the constraints tends towards
the node of the cone of the surface of load, the numerical risk to rock in this prohibited field
increase. However when that one projects oneself or when one makes a prediction in a point of it
field, numerical calculation ends either in an erroneous result, or with a fatal error. Indeed,
traction appears numerically by a positive value of I. This value poses then
1
-.
1 5
+
I + Q
problem at the time to evaluate certain quantities like
1
init
; in addition it
3
Pa
would generate from a theoretical point of view a negative value Q according to the equation of the surface of
II
charge déviatoire.
Such a phenomenon was detected on several levels: in a particular way in the elastic prediction
with model CJS1, and in a general way in the local iterations of Newton utilizing it
mechanism déviatoire. The same answer was brought in order to free itself from this pathology: it
acts to virtually project the constraints in the elastic range on the hydrostatic axis in
posing:
= = = 1
- kPa
11
22
33
= = = 0
12
13
23
One thus repositions the state of stresses in the field of compression while moving away little of
inadmissible initial prediction considered, and by hoping that considerations of structures
will allow total calculation to converge.
Moreover internal variables do not evolve/move and one supposes being returned in the elastic range
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:
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5 Operator
tangent
The tangent operator called by option RIGI_MECA_TANG corresponds to the tangent operator deduced from
problem of speed and calculated starting from the results known at the moment T.
The tangent operator called by option FULL_MECA should correspond to the tangent operator with
discretized problem in an implicit way. Actually, we did not carry out this calculation. We take
then, when option FULL_MECA is retained, the tangent operator deduced from the problem of speed and
calculated starting from the results known at the moment t+dt.
We detail below the tangent operator deduced from the problem of it speed according to or of
mechanisms brought into play.
5.1
Tangent operator of the nonlinear elastic mechanism
We have simply the following nonlinear elastic relation:
N
I + Q
& = D
& = 1
D
ij
ijkl ()
init
linear
kl
ijkl
&kl
P
3 A
from where immediately the tangent operator:
N
.
+
elas nl
I
Q
1
init
linear
H
=
D
ijkl
ijkl
P
3 A
5.2
Tangent operator of the mechanisms isotropic rubber band and plastic
In this case, we have the following relation:
1
& = D &
&
D
&
&
ij
ijkl () (
- IP
kl
kl) =
ijkl ()
I
+
kl
kl
3
it comes: I & = K
3
1
(
I
& + &
v
)
By taking account of this relation and the law of work hardening of Q, condition I
F & = 0 becomes:
Iso
I&
I
1
F & = -
+ Q & = - K & &
K &
Iso
(+ I
v
) - p I = 0
3
K
that is to say: I
& = -
&
p
v
K + K
By deferring this result in the expression of &, one finds:
ij
1
K
1
K
& = D & -
& = D -
D
ij
ijkl
kl
&
p
mm
kl
ijkl
p
ijmn
mn
kl
kl
3 K + K
3 K + K
from where the tangent operator:
1
K
IP
H
= D -
D
ijkl
ijkl
p
ijmn
mn
kl
3 K + K
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:
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One can also write in matric form:
- + 2µ
-
-
0
0
0
-
- + 2µ
-
0
0
0
I N -
-
- + 2µ
0
0
0
IP
H =
1
3Pa
0
0
0
2µ
0
0
0
0
0
0
2µ
0
0
0
0
0
0
2µ
E
K 2
where for this formula only and µ are the coefficient of Lamé and
O
=
.
E
p
K + K
O
O
5.3
Tangent of the mechanisms rubber band and plastic operator déviatoire
The condition
D
F & = 0 is written:
D
D
D
D
D
D
F
F
F
F
F
F
D
F & =
& +
R & +
X & =
& +
D
R
& G +
D
X
& G = 0
ij
ij
ij
ij
R
X
R
X
ij
ij
ij
ij
D
F
The tensor
X
G being purely déviatoire, the product
X
G is reduced to:
ij
X ij
D
D
F
F
X
X
X
G = Dev.
G
= - I Q G
ij
ij
1
ij
ij
X
X
ij
ij
The plastic multiplier can thus be put in the form:
D
F
D
&
1
=
&
Dev.
ij
H
ij
while revealing the plastic module
Dev.
H
, given by:
-,
1 5
2
Dev.
2
I + Q
R
1
H
=
1
init
I
To 1
+ Q Q
X
1
ij (
+
ij
ij)
P
3
R
B
has
m
The relation stress-strains then makes it possible to write:
D
D
F
F
F
F
& =
D
& - & G =
D & - &
D G
ij
ijkl (
D
D
kl
kl)
D
D
D
D
ijkl
kl
ijkl
kl
ij
ij
ij
ij
what gives finally for the plastic multiplier:
D
F D
ijkl &kl
D
ij
& =
D
F
Dev.
D
H
+
D G
ijkl
kl
ij
By deferring this result in the expression of &, one finds:
ij
D
F
D
&
pqmn
mn
pq
& = D &
G
ij
ijkl
-
D
kl
D
kl
F
Dev.
H
+
D
D G
rstu
you
rs
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Titrate:
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:
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from where the tangent operator:
D
F Dpqkl
dp
D
pq
H
= D - D G
ijkl
ijkl
ijmn
mn
D
F
Dev.
D
H
+
D G
rstu
you
rs
The tangent operator thus obtained is not symmetrical. However for the moment law CJS rests on
finite elements which claim a symmetrical operator. Ultimately, we retain not
dp
H but
ijkl
~ dp
H which is given by:
ijkl
~
dp
dp
H
+ H
dp
ijkl
klij
H
=
with ij and kl taken in (,
11
,
13
,
12
,
33
,
22
23)
ijkl
2
5.4
Tangent operator of the mechanisms rubber band, plastics isotropic and
déviatoire
One must satisfy the two following conditions: I
F & = 0 and D
F & = 0. Taking into account the relation
stress-strains which is written:
1 I
D
D
& = D &
&
& G
ij
ijkl
+
-
kl
kl
kl
3
the first condition gives:
I
F & = - K (& + I
& - D D
& G
K &
v
v) -
p
I = 0
where one posed
D
D
G = G = tr G.
v
kk
(D)
The second condition led to:
D
F
1
D
D
D & +
F
F
I
&
D - D
D
&
D G -
Dev.
D
H
& = 0
ijkl
kl
3
ijkl
kl
ijkl
kl
ij
ij
ij
Thus, plastic multipliers I
& and D
& are obtained by solving the system:
- (K + p
K) I
& +
D
D
KG & = K&
v
v
D
1 F
D
F
D
F
-
I
D & +
D
D G +
Dev.
H
D
& =
D
ijkl
kl
ijkl
kl
ijkl &kl
3
ij
ij
ij
that is to say:
D
D
F
F
D
D
Dev.
KG
D & -
D
G + H
K
v
ijkl
kl
mnpq
pq
&v
I
ij
mn
& =
(
F
1
F
p
K + K)
D
D
D
Dev.
D
D G + H
- KG
D
rstu
you
v
wvxy
xy
3
rs
VW
(
F
1
F
p
K + K)
D
D
D & - K
D
ijkl
kl
mnpq
pq &v
3
D
ij
mn
& =
(
F
1
F
p
K + K)
D
D
D
Dev.
D
D G + H
- KG
D
rstu
you
v
vwxy
xy
3
rs
VW
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:
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These expressions are still written:
I
& = T and D
& = T
1kl &kl
2 kl &kl
where the tensors T and T are given by:
1
2
D
D
F
F
D
D
Dev.
KG
D
-
D
G + H
K
v
ijkl
mnpq
pq
kl
ij
mn
T
=
1kl
(
F
1
F
p
K + K)
D
D
D
Dev.
D
D G + H
- KG
D
rstu
you
v
vwxy
xy
3
rs
VW
(
F
1
F
p
K + K)
D
D
D
- K
D
ijkl
mnpq
pq
kl
3
ij
mn
T
=
2 kl
(
F
1
F
p
K + K)
D
D
D
Dev.
D
D G + H
- KG
D
rstu
you
v
vwxy
xy
3
rs
VW
By deferring expressions I
& and D
& of in the formula of &, one finds:
ij
1
D
& = D &
T
1
&
T
2
& G
ij
ijkl
+
-
kl
Nm
Nm
kl
pq
pq
kl
3
from where the tangent operator:
idp
1
D
H
= D + D T - D G T
ijkl
ijkl
ijmn
mn 1kl
ijpq
pq
2 kl
3
~
This tangent operator not being symmetrical, we retain not
idp
H
but
idp
H
who is given
ijkl
ijkl
by:
~
idp
idp
H
+ H
idp
ijkl
klij
H
=
with ij and kl taken in (,
11
,
13
,
12
,
33
,
22
23)
ijkl
2
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6 Sources
Aster
6.1
List modified and added routines
Only the routine nmcomp.f was modified. It makes it possible to call, when behavior CJS is
chosen, the routine nmcjs.f, starting point of the integration of the law.
The whole of routines FORTRAN developed within the framework of the integration of law CJS in
Code_Aster is as follows:
cjsc3q.f,
cjsci1.f,
cjsdtd.f,
cjsela.f,
cjside.f,
cjsiid.f,
cjsjde.f,
cjsjid.f,
cjsjis.f,
cjsmat.f,
cjsmde.f,
cjsmid.f,
cjsmis.f,
cjsnor.f,
cjspla.f,
cjsqco.f,
cjsqij.f,
cjssmd.f,
cjssmi.f,
cjst.f, cjstde.f,
cjstel.f,
cjstid.f,
cjstis.f,
lcdete.f,
nmcjs.f,
cjsinp.f,
cjsncn.f,
cjsncv.f,
cjsnvi.f,
cjsqq.f.
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6.2
Top-level flowchart of the principal routines
Principal routines FORTRAN for the integration of law CJS are connected in the following way:
nmcomp.f
nmcjs.f
cjsmat.f
cjsela.f
cjssmi.f
cjssmd.f
cjspla.f
cjsmis.f
cjsmde.f
cjsmid.f
cjstel.f
cjstis.f
cjstde.f
cjstid.f
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6.3
Details of the functionalities of developed routines FORTRAN
6.3.1 Routine
:
CJSC3Q
Objective: calculation of cos (3
Q)
Variables of input and output:
IN
SIG: CONTRAINTES
X: VARIABLES HAMMER-HARDENED MOVIES
Pa: CLOSE ATMOSPHERIC (DATA MATERIAL)
OUT
Q: DEV. (SIG) - TRACE (SIG) * X
QII: SQRT (QIJ * QIJ)
COS3TQ: SQRT (54) * DET (Q)/(QII ** 3)
6.3.2 Routine
:
CJSCI1
Objective:
+ N
E
I
resolution of the equation +
I - -
I - 3
1
K
tr = by the method of the secant,
1
1
O
() 0
3
Pa
for the nonlinear elastic behavior
Variables of input and output:
IN
CRIT: CRITERIA OF CONVERGENCE
MATER: COEFFICIENTS MATERIAL A T+DT
DEPS: INCREMENT OF DEFORMATION
SIGD: CONSTRAINT A T
OUT
I1: TRACE SIG A T+DT
TRACT: LOGICAL VARIABLE INDICATING TRACTION
6.3.3 Routine
:
CJSDTD
Objective:
calculation of derived from the tensor D
T compared to Q
Variables of input and output:
IN
MOD: MODELISATION
Q: TENSOR (6 COMPONENTS)
OUT
DTDDQ: TENSOR RESULT (6 COMPONENTS)
6.3.4 Routine
:
CJSELA
Objective:
nonlinear elastic design of the constraints
Variables of input and output:
IN
MOD: MODELISATION
CRIT: CRITERIA OF CONVERGENCE
MATERF: COEFFICIENTS MATERIAL A T+DT
SIGD: CONSTRAINT A T
DEPS: INCREMENT OF DEFORMATION
OUT
SIGF: CONSTRAINT A T+DT
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Organization of CJSELA
·
calculation of the first invariant of the I1 constraints to t+dt:
- call of CJSCI1
·
calculation of the coefficients of the elastic matrix and assembly of the matrix
·
calculation of the increment of the constraints and the constraints with t+dt:
- call of LCPRMV and LCSOVE
6.3.5 Routine
:
CJSIDE
Objective:
for the integration of the plastic mechanism déviatoire, calculation of a solution of test so
to start the local iterations of Newton then.
Variables of input and output:
IN
MOD: MODELISATION
MATER: COEFFICIENTS MATERIAL A T+DT
EPSD: DEFORMATION A T+DT
DEPS: INCREMENT OF DEFORMATION
YD: VARIABLES A T = (SIGD, VIND, LAMB)
VAR
GD: TENSOR OF THE LAW D FLOW PLASTIC DEV.
OUT
DY: SOLUTION D TEST
Organization of CJSIDE
·
calculation of the elastic operator,
·
calculation of laws of work hardening
R
G and X
G,
·
calculation of the law of flow of the plastic mechanism déviatoire
D
G,
D
F
·
calculation of the threshold
D
F, of its derivative
and of the plastic multiplier
D
,
D
·
calculation of the solution of test
6.3.6 Routine
:
CJSIID
Objective:
for the simultaneous integration of the plastic mechanisms isotropic and déviatoire, calculation of one
solution of test in order to start the local iterations of Newton then.
Variables of input and output:
IN
MOD: MODELISATION
MATER: COEFFICIENTS MATERIAL A T+DT
EPSD: DEFORMATION A T+DT
DEPS: INCREMENT OF DEFORMATION
YD: VARIABLES A T = (SIGD, VIND, LAMB)
VAR
GD: TENSOR OF THE LAW D FLOW PLASTIC DEV.
OUT
DY: SOLUTION D TEST
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Organization of CJSIID
·
calculation of the elastic operator,
·
calculation of laws of work hardening
R
G and X
G,
·
calculation of the law of flow of the plastic mechanism déviatoire
D
G,
I
F
I
F
D
F
D
F
·
calculation of thresholds I
F and D
F, of their derivative
,
,
and
, and of
I
D
I
D
plastic multipliers
I
and
D
,
·
calculation of the solution of test
6.3.7 Routine
:
CJSJDE
Objective:
DR.
calculation of DRDY and R for the resolution of
(p
Y)
1
+
p
DY
= - R (p
Y) (mechanism
DY
plastic déviatoire)
Variables of input and output:
IN
MOD: MODELISATION
MATER: COEFFICIENTS MATERIAL A T+DT
EPSD: DEFORMATION A T
DEPS: INCREMENT OF DEFORMATION
YD: VARIABLES A T = (SIGD, VIND, LAMBDD)
YF: VARIABLES A T+DT = (SIGF, VINF, LAMBDF)
VAR
GD: TENSOR OF THE LAW D FLOW PLASTIC DEV.
OUT
R: SECOND MEMBER
SIGNE: SIGN S:DEPSDP
DRDY: JACOBIEN
Organization of CJSJDE
·
calculation of the elastic operator,
·
calculation of laws of work hardening
R
G and X
G,
·
calculation of the law of flow of the plastic mechanism déviatoire
D
G,
·
calculation of multiple derivative intermediaries
R
G
G R
X
G
X
G
D
G
Gd
D
G
·
calculation of the terms
,
,
mn,
mn,
mn,
mn,
mn
R
X
R
X
ij
ij
ij
ij
ij
·
calculation of the components of DRDY and R
·
assembly of DRDY and R
Handbook of Référence
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Titrate:
Law CJS in géomechanics
Date:
18/11/03
Author (S):
C. CHAVANT, pH. Key AUBERT
:
R7.01.13-A Page
: 41/48
6.3.8 Routine
:
CJSJID
Objective:
DR.
calculation of DRDY and R for the resolution of
(p
Y)
1
+
p
DY
= - R (p
Y) (mechanisms
DY
plastics isotropic and déviatoire)
Variables of input and output:
IN
MOD: MODELISATION
MATER: COEFFICIENTS MATERIAL A T+DT
EPSD: DEFORMATION A T
DEPS: INCREMENT OF DEFORMATION
YD: VARIABLES A T = (SIGD, VIND, LAMBDD)
YF: VARIABLES A T+DT = (SIGF, VINF, LAMBDF)
VAR
GD: TENSOR OF THE LAW D FLOW PLASTIC DEV.
OUT
R: SECOND MEMBER
SIGNE: SIGN S:DEPSDP
DRDY: JACOBIEN
Organization of CJSJID
·
calculation of the elastic operator,
·
calculation of laws of work hardening
Iso
Q
G
,
R
G and X
G,
·
calculation of the law of flow of the plastic mechanism déviatoire
D
G,
·
calculation of multiple derivative intermediaries
Q
G Iso
R
G
G R
X
G
X
G
D
G
Gd
D
G
·
calculation of the terms
,
,
,
mn,
mn,
mn,
mn,
mn
Q
R
X
R
X
Iso
ij
ij
ij
ij
ij
·
calculation of the components of DRDY and R
·
assembly of DRDY and R
6.3.9 Routine
:
CJSJIS
Objective:
DR.
calculation of DRDY and R for the resolution of
(p
Y)
1
+
p
DY
= - R (p
Y) (mechanism
DY
isotropic plastic)
Variables of input and output:
IN
MOD: MODELISATION
MATER: COEFFICIENTS MATERIAL A T+DT
DEPS: INCREMENT OF DEFORMATION
YD: VARIABLES A T = (SIGD, VIND, LAMBDD)
YF: VARIABLES A T+DT = (SIGF, VINF, LAMBDF)
OUT
R: SECOND MEMBER
DRDY: JACOBIEN
Organization of CJSJIS
·
calculation of the elastic operator,
·
calculation of the components of DRDY and R
·
assembly of DRDY and R
Handbook of Référence
R7.01 booklet: Modelings for Génie Civil and the géomatériaux ones
HT-66/03/005/A
Code_Aster ®
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Titrate:
Law CJS in géomechanics
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18/11/03
Author (S):
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:
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6.3.10 Routine: CJSMAT
Objective:
recovery of data materials, the component count of the fields, the number of
internal variables and of selected level CJS.
Variables of input and output:
IN
IMAT: ADDRESS MATERIAL CODES
MOD: TYPE OF MODELING
TEMPF: TEMPERATURE A T+DT
OUT
MATERF: COEFFICIENTS MATERIAL A T+DT
NDT: NB TOTAL OF COMPONENTS TENSORS
NDI: NB DIRECT COMPONENTS TENSORS
NVI: NB INTERNAL VARIABLES
NIVCJS: LEVEL 1, 2 OR 3 OF LAW CJS
Organization of CJSMAT
·
recovery of the component count of the fields and the number of variables intern in
function of modeling chosen,
·
recovery of data materials,
·
recognition of level CJS chosen according to the parameters given.
6.3.11 Routine: CJSMDE
Objective:
elastoplastic calculation of the constraints with the plastic mechanism deviatoire activated:
resolution by the method of Newton of R (Y) = 0
Variables of input and output:
IN
MOD: MODELISATION
CRIT: CRITERIA OF CONVERGENCE
MATER: COEFFICIENTS MATERIAL A T+DT
NVI: NB INTERNAL VARIABLES
EPSD: DEFORMATIONS A T
DEPS: INCREMENT OF DEFORMATION
SIGD: CONSTRAINT A T
VIND: INTERNAL VARIABLES A T
STOPNC: STOP IN THE EVENT OF NOT CONVERGENCE
VAR
SIGF: CONSTRAINT A T+DT
VINF: INTERNAL VARIABLES A T+DT
NOCONV: NO CONVERGENCE
Organization of CJSMDE
·
initialization of YD by the state with T
·
calculation of a solution of test with CJSIDE
·
loop on the iterations of Newton
- incrementing YF = YD + DY
- calculation of DRDY and R: CJSJDE
- resolution of the system by the method of Gauss: MTGAUS
- updating of the solution DY
- test of convergence
·
update of the constraints and internal variables
Handbook of Référence
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Code_Aster ®
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Titrate:
Law CJS in géomechanics
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C. CHAVANT, pH. Key AUBERT
:
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6.3.12 Routine: CJSMID
Objective:
elastoplastic calculation of the constraints with the plastic mechanisms isotropic and deviatoire
activated: resolution by the method of Newton of R (Y) = 0
Variables of input and output:
IN
MOD: MODELISATION
CRIT: CRITERIA OF CONVERGENCE
MATER: COEFFICIENTS MATERIAL A T+DT
NVI: NB INTERNAL VARIABLES
EPSD: DEFORMATIONS A T
DEPS: INCREMENT OF DEFORMATION
SIGD: CONSTRAINT A T
VIND: INTERNAL VARIABLES A T
STOPNC: STOP IN THE EVENT OF NOT CONVERGENCE
VAR
SIGF: CONSTRAINT A T+DT
VINF: INTERNAL VARIABLES A T+DT
NOCONV: NO CONVERGENCE
Organization of CJSMID
·
initialization of YD by the state with T
·
calculation of a solution of test with CJSIID
·
loop on the iterations of Newton
- incrementing YF = YD + DY
- calculation of DRDY and R: CJSJID
- resolution of the system by the method of Gauss: MTGAUS
- updating of the solution DY
- test of convergence
·
update of the constraints and internal variables
6.3.13 Routine: CJSMIS
Objective:
elastoplastic calculation of the constraints with the activated isotropic plastic mechanism:
resolution by the method of Newton of R (Y) = 0
Variables of input and output:
IN
MOD: MODELISATION
CRIT: CRITERIA OF CONVERGENCE
MATER: COEFFICIENTS MATERIAL A T+DT
DEPS: INCREMENT OF DEFORMATION
SIGD: CONSTRAINT A T
VIND: INTERNAL VARIABLES A T
STOPNC: STOP IN THE EVENT OF NOT CONVERGENCE
VAR
SIGF: CONSTRAINT A T+DT
VINF: INTERNAL VARIABLES A T+DT
NOCONV: NO CONVERGENCE
Organization of CJSMIS
·
initialization of YD by the elastic prediction
·
loop on the iterations of Newton
- incrementing YF = YD + DY
- calculation of DRDY and R: CJSJIS
- resolution of the system by the method of Gauss: MTGAUS
- updating of the solution DY
- test of convergence
·
update of the constraints and internal variables
Handbook of Référence
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Code_Aster ®
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:
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6.3.14 Routine: CJSNOR
Objective:
D
F
calculation of a vector parallel with
qij
Variables of input and output:
IN
MATER: MATERIAU
SIG:
CONTRAINTES
X:
VARIABLES INTERN KINEMATICS
OUT
NOR:
ESTIMATE OF THE DITRECTION OF THE NORMAL
ON SURFACE DEVIATOIRE IN PLAN DEVIATOIRE
PERPENDICULAIRE
With
TRISECTRICE
THE NOR VECTOR (1:NDT) NR IS NOT STANDARD
SA
NORME
EST
NOR (NDT+1)
6.3.15 Routine: CJSPLA
Objective:
elastoplastic calculation of the constraints.
Variables of input and output:
IN
MOD: MODELISATION
CRIT: CRITERIA OF CONVERGENCE
MATER: COEFFICIENTS MATERIAL A T+DT
SEUILI: FUNCTION OF ISO LOAD. CALCULEE WITH PREDICT ELAS
SEUILD: FUNCTION OF LOAD DEV. CALCULEE WITH PREDICT ELAS
NVI: A NUMBER OF INTERNAL VARIABLES
EPSD: DEFORMATIONS A T
DEPS: INCREMENT OF DEFORMATION
SIGD: CONSTRAINT A T
VIND: INTERNAL VARIABLES A T
VAR
SIGF: CONSTRAINT A T+DT (IN - > ELAS, OUT - > PLASTI)
OUT
VINF: INTERNAL VARIABLES A T+DT
MECANI: MECHANISM (S) ACTIVATES (S)
Organization of CJSPLA
·
assumption on the plastic mechanisms activated according to the values of thresholds I
F
and
D
F calculated starting from the elastic prediction,
·
processing of the possible recutting of the step of time
·
back up elastic prediction,
·
elastoplastic calculation,
- isotropic plastic mechanism: CJSMIS
- plastic mechanism déviatoire: CJSMDE
- plastic mechanisms isotropic and déviatoire simultaneously: CJSMID
·
calculation of the thresholds starting from the constraints with t+dt
- call of CJSSMI and CJSSMD
- if (assumption of an isotropic mechanism and D
F positive) or (assumption of a mechanism
déviatoire and I
F positive): return to elastoplastic calculation with plastic mechanisms
isotropic and déviatoire simultaneously,
- if not end of routine
Handbook of Référence
R7.01 booklet: Modelings for Génie Civil and the géomatériaux ones
HT-66/03/005/A
Code_Aster ®
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Titrate:
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:
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6.3.16 Routine: CJSQCO
Objective:
utility routine of CJS allowing the calculation of standard sizes listed below
Variables of input and output:
IN
GAMMA: PARAMETER MATERIAL
SIG: CONTRAINTES
X: VARIABLES HAMMER-HARDENED MOVIES
PREF: CLOSE REF. FOR STANDARDIZATION
EPSSIG: DEVIATIVE EPSILON FOR NULLITY
I1: TRACE TENSOR OF THE CONSTRAINTS
OUT
S: DEV. (SIG)
Software house: SQRT (S:S)
SIIREL: SOFTWARE HOUSE/PREF
COS3TS: LODE (SIG)
HTS: FUNCTION H (TETHA_S)
DETS: DETERMINANT OF S
Q: Q (SIG-X)
QII: SQRT (Q:Q)
QIIREL: QII/PREF
COS3TQ
HTQ: FUNCTION H (TETHA_Q)
DETQ: DETERMINANT OF Q
6.3.17 Routine: CJSQIJ
Objective:
calculation of the tensor Q
ij
Variables of input and output:
IN
NR: DIMENSION OF S, X, Q
S: DEVIATEUR
I1: FIRST INV.
X: CENTER SURFACE OF LOAD DEVIATOIRE
OUT
Q: TENSOR RESULT
6.3.18 Routine: CJSSMD
Objective:
calculation of the threshold of the plastic mechanism déviatoire.
Variables of input and output:
IN
SIG: CONTRAINTE
VIN: INTERNAL VARIABLES
OUT
SEUILD: THRESHOLD ELASTICITY OF MECHANISM DEVIATOIRE
6.3.19 Routine: CJSSMI
Objective:
calculation of the threshold of the isotropic plastic mechanism.
Variables of input and output:
IN
SIG: CONTRAINTE
VIN: INTERNAL VARIABLES
OUT
SEUILI: THRESHOLD ELASTICITY OF THE ISOTROPIC MECHANISM
Handbook of Référence
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6.3.20 Routine: CJST
Objective:
det S
calculation of T =
.
S
Variables of input and output:
IN
S: MATRICE
OUT
T: T (IN VECTORIAL FORM WITH RAC2)
6.3.21 Routine: CJSTDE
Objective:
calculation of the tangent matrix for the plastic mechanism déviatoire
Variables of input and output:
IN
MOD: MODELISATION
MATER: COEFFICIENTS MATERIAL
NVI: NB INTERNAL VARIABLES
EPS: DEFORMATIONS
SIG: CONTRAINTES
VIN: INTERNAL VARIABLES
OUT
DSDESY: STAMP TANGENT SYMETRISEE
Organization of CJSTDE
·
calculation of the elastic operator,
·
calculation of laws of work hardening
R
G and X
G,
·
calculation of the law of flow of the plastic mechanism déviatoire
D
G,
·
calculation of intermediate terms
·
calculation of the tangent matrix
·
symmetrization of the tangent matrix
6.3.22 Routine: CJSTEL
Objective:
calculation of the tangent matrix for the elastic mechanism
Variables of input and output:
IN
MOD: MODELISATION
MATER: COEFFICIENTS MATERIAL
SIG: CONTRAINTES
OUT
HOOK: ELASTIC OPERATOR RIGIDITY
Organization of CJSTEL
·
calculation of the elastic operator
Handbook of Référence
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Code_Aster ®
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Titrate:
Law CJS in géomechanics
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:
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6.3.23 Routine: CJSTID
Objective:
calculation of the tangent matrix for the plastic mechanisms isotropic and déviatoire
Variables of input and output:
IN
MOD: MODELISATION
MATER: COEFFICIENTS MATERIAL
NVI: NB INTERNAL VARIABLES
EPS: DEFORMATIONS
SIG: CONTRAINTES
VIN: INTERNAL VARIABLES
OUT
DSDESY: STAMP TANGENT SYMETRISEE
Organization of CJSTEL
·
calculation of the elastic operator,
·
calculation of laws of work hardening
R
G and X
G,
·
calculation of the law of flow of the plastic mechanism déviatoire
D
G,
·
calculation of intermediate terms
·
calculation of the tangent matrix
·
symmetrization of the tangent matrix
6.3.24 Routine: CJSTIS
Objective:
calculation of the tangent matrix for the isotropic plastic mechanism
Variables of input and output:
IN
MOD: MODELISATION
MATER: COEFFICIENTS MATERIAL
SIG: CONTRAINTES
VIN: INTERNAL VARIABLES
OUT
DSDE: STAMP TANGENT
Organization of CJSTEL
·
calculation of the tangent matrix
6.3.25 Routine: LCDETE
Objective:
calculation of a determining matrix 3×3
Variables of input and output:
IN
A: STAMPS
OUT
LCDETE: DETERMINANT
Handbook of Référence
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Titrate:
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6.3.26 Routine: NMCJS
Objective:
realization of the integration of law CJS: calculation of the constraints with t+dt and/or the matrix
tangent, according to the selected option of calculation.
Variables of input and output:
IN
STANDARD TYPMOD OF MODELING
IMAT ADDRESSES MATERIAL CODES
COMP BEHAVIOR OF L ELEMENT
CRIT LOCAL CRITERIA
URGENT INSTAM T
URGENT INSTAP T+DT
TEMPM TEMPERATURE A T
TEMPF TEMPERATURE A T+DT
TREF TEMPERATURE OF REFERENCE
EPSD TOTAL DEFLECTION A T
LIFO INCREMENT OF TOTAL DEFLECTION
FORCED SIGD A T
VARIABLE VIND INTERN A T + INDICATING STATE T
OPT OPTION OF CALCULATION TO BE MADE
OUT
FORCED SIGF A T+DT
VARIABLE VINF INTERN A T+DT + INDICATING STATE T+DT
DSDE STAMPS TANGENT BEHAVIOR A T+DT OR T
Organization of NMCJS
·
recovery of data materials, the component count of the fields, the number of
internal variables and of selected level CJS:
- call of CJSMAT
·
blocking of variables intern according to selected level CJS
·
calculation of the constraints with t+dt
- elastic prediction: CJSELA
- isotropic calculation of the thresholds of the mechanisms and déviatoire: CJSSMI and CJSSMD
- if one of the thresholds is exceeded, elastoplastic calculation: CJSPLA
·
calculation of the tangent matrix according to the mechanism brought into play
- rubber band: CJSTEL
- isotropic plastic: CJSTIS
- plastic déviatoire: CJSTDE
- isotropic plastic and déviatoire: CJSTID
7 Bibliography
[1]
Mr. MALEKI, B. CAMBOU, P. DUBUJET, “Modélisation treated on a hierarchical basis of the behavior of
grounds ", to appear.
[2]
B. CAMBOU, K. JAFARI, “Modèle of behavior of the non-cohesive soils”, Frank Rev.
Géotech. n°44, p. p 43-55, 1988.
[3]
K. ELAMRANI, “Contribution with the validation of model CJS for granular materials”,
Thesis of Doctorat of École Centrale of Lyon, 1992.
Handbook of Référence
R7.01 booklet: Modelings for Génie Civil and the géomatériaux ones
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