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Code_Aster
®
Version
6.4
Titrate:
Law CJS in géomechanics
Date:
18/11/03
Author (S):
C. CHAVANT, pH. AUBERT
Key
:
R7.01.13-A
Page
:
1/48
Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/03/005/A
Organization (S):
EDF-R & D/AMA, EDF-DIS/CNEPE















Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering 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.
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Code_Aster
®
Version
6.4
Titrate:
Law CJS in géomechanics
Date:
18/11/03
Author (S):
C. CHAVANT, pH. AUBERT
Key
:
R7.01.13-A
Page
:
2/48
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R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/03/005/A
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 pitch of time ..................................................................................................... 30
4.8
Various remarks ...................................................................................................................... 30
4.8.1
Calculation of the term
(
)
Q
S
-
cos
............................................................................................. 30
4.8.2
Calculation of
R
R
........................................................................................................................ 31
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):
C. CHAVANT, pH. AUBERT
Key
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R7.01.13-A
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HT-66/03/005/A
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 main 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
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Code_Aster
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6.4
Titrate:
Law CJS in géomechanics
Date:
18/11/03
Author (S):
C. CHAVANT, pH. AUBERT
Key
:
R7.01.13-A
Page
:
4/48
Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/03/005/A
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
O
K
parameter of the model
p
O
K
parameter of the model
KP
G
elastic modulus of rigidity
E
O
G
parameter of the model
D
G
function controlling the evolution of the plastic deformations déviatoires
S
diverter of the tensor of the stresses
1
I
first invariant of the stresses
Co
p
pressure of initial criticism
PCO
has
P
pressure of reference of the model
AP
D
I
F
F
,
thresholds of the plastic mechanisms isotropic and déviatoire
Iso
Q
variable interns model corresponding to the acceptable limit of the plan
déviatoire
Q
Q,
tensors of the model
X
,
R
variables intern model corresponding to the average radius and the center
surface of load in the déviatoire plan
m
R
parameter of the model
RM
C
R
parameter of the model
RC
D
I
,
plastic multipliers of the mechanisms isotropic and déviatoire
dp
IP
E
,
,
,
tensors of the respectively total, elastic, plastic deformations
isotropic and plastic déviatoires
v
voluminal deformations
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
init
Q
parameter of the model
Q_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:
18/11/03
Author (S):
C. CHAVANT, pH. AUBERT
Key
:
R7.01.13-A
Page
:
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R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/03/005/A
2 Introduction
Model CJS is an elastoplastic law of behavior adapted to the modeling of materials
granular. It was developed at the Central School 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:
dp
ij
IP
ij
E
ij
ij
&
&
&
&
+
+
=
where
E
ij
&
,
IP
ij
&
and
dp
ij
&
are respectively the increments of elastic strain, 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:
ij
ij
E
ij
K
I
G
S
9
2
1
&
&
&
+
=
where
1
I
is the first invariant of the stresses:
()
tr
I
=
1
,
S
tensor is the déviatoire part of
stresses, 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
has
init
E
O
P
Q
I
K
K




+
=
3
1
,
N
has
init
E
O
P
Q
I
G
G




+
=
3
1
E
O
K
,
O
G
,
has
P
and
N
are parameters of the model.
has
P
is a pressure of reference equal to - 100
kPa.
3.3
Isotropic plastic mechanism
The surface of corresponding load
I
F
is, in the space of the main stresses, a plan
perpendicular with the hydrostatic axis, is:
(
)
(
)
Iso
init
Iso
I
Q
Q
I
Q
F
+
+
-
=
3
,
1
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Code_Aster
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6.4
Titrate:
Law CJS in géomechanics
Date:
18/11/03
Author (S):
C. CHAVANT, pH. AUBERT
Key
:
R7.01.13-A
Page
:
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HT-66/03/005/A
where
Iso
Q
is the thermodynamic force which depends on the internal variable
Q
according to:
Q
P
Q
K
Q
K
Q
N
has
Iso
p
O
p
Iso
&
&
&




=
=
p
O
K
,
has
P
and
N
are the parameters of the plastic mechanism déviatoire (
has
P
and
N
are identical to
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
:
ij
I
ij
I
I
IP
ij
F
&
&
&
3
1
-
=
=
and
I
Iso
I
I
Q
F
Q
&
&
&
-
=
-
=
Taking into account the second equation, the law of work hardening can be also put in the form:
N
has
Iso
p
O
I
Iso
P
Q
K
Q




-
=
&
&

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:
(
)
()
(
)
init
Q
II
D
Q
I
R
H
Q
R
F
+
+
=
1
,
,
X
with
ij
ij
ij
X
I
S
Q
1
-
=
ij
ij
II
Q
Q
Q
=
()
()
(
)
()
6
/
1
3
6
/
1
det
54
1
3
cos
1




+
=
+
=
II
Q
Q
Q
H
Q
.
The scalar
R
and the tensor
X
the average radius and the center of surface represent respectively
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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Code_Aster
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Titrate:
Law CJS in géomechanics
Date:
18/11/03
Author (S):
C. CHAVANT, pH. AUBERT
Key
:
R7.01.13-A
Page
:
7/48
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R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/03/005/A
Note:
The expression of the angle of Lode is found in the following way:
In a reference mark
()
J
I,
,
H
déviatoire plan the vector
HM
can be given from
outdistance
=
HM
and of the angle of Lode
S
(cf [Figure 3.4-a]). Co-ordinates of
HM
are:
(
)
S
S
cos
,
sin
=
HM
S
2
S
3
S
1
H
M
I
J
Appear 3.4-a: Angle of Lode in the déviatoire plan

The main components of the diverter are thus:
S
S
cos
1
=
,


-
=
S
S
3
4
cos
2
and


-
=
S
S
3
2
cos
3
Consequently, one a:
2
3
=
II
S
and
()
(
)
()
S
S
S
S
3
cos
4
1
sin
3
cos
cos
4
1
det
3
2
2
3
=
-
=
S
one deduces the relation then from it:
()
()
3
2
/
3
2
/
1
det
3
2
3
cos
II
S
S
S
=
The angle
Q
calculation in the same way.
3.4.1 Work hardening
isotropic
The isotropic law of work hardening is written as follows:
(
)
2
2
R
With
R
R
R
With
R
m
m
+
=
&
&
The thermodynamic force
R
is related to
R
whose evolution is given by:
(
)
5
.
1
1
1
5
.
1
1
3
3
-
-




+
+
-
=




+
-
=
has
init
init
D
has
init
D
D
P
Q
I
Q
I
P
Q
I
R
F
R
&
&
&
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Titrate:
Law CJS in géomechanics
Date:
18/11/03
Author (S):
C. CHAVANT, pH. AUBERT
Key
:
R7.01.13-A
Page
:
8/48
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HT-66/03/005/A
By direct integration of the law of work hardening, it comes:
R
With
R
R
R
With
R
m
m
+
=
, that is to say too
(
)
R
R
With
R
R
R
m
m
-
=
The law of work hardening can thus be also expressed by:
(
)
()
R
G
P
Q
I
Q
I
R
R
With
R
R
D
has
init
init
m
D
,
3
1
5
.
1
1
1
2
&
&
&
=




+
+




-
-
=
-
with
()
(
)
5
.
1
1
1
2
3
1
,
-




+
+




-
-
=
has
init
init
m
R
P
Q
I
Q
I
R
R
With
R
G
and where
m
R
(which is the average radius of the elastic range in rupture) and
With
are parameters of
model.
3.4.2 Work hardening
kinematics
The kinematic law of work hardening is given by:
ij
ij
B
X
&
&
1
=
The thermodynamic force
X
is a function of the variable
whose nonlinear evolution is given
by:
(
)
5
.
1
1
1
3
-




+




+
-




-
=
has
init
ij
init
ij
D
D
ij
P
Q
I
X
Q
I
X
F
Dev.
&
&
The term
(
)
X
init
Q
I
+
-
1
allows to obtain nonlinear kinematic work hardening, translating
limitation of the evolution of the surface of load.
By taking account of
(
)
ij
D
init
ij
kl
kl
D
ij
D
Q
F
Q
I
X
Q
Q
F
X
F
1
+
-
=
=
, and while posing:




=
ij
D
ij
Q
F
Dev.
Q
, it
comes finally for the law from work hardening:
(
)
(
)
()
X
,
3
1
5
.
1
1
1
X
ij
D
has
init
init
ij
ij
D
ij
G
P
Q
I
Q
I
X
Q
B
X
&
&
&
=




+
+
+
=
-
with
(
)
(
)
(
)
5
.
1
1
1
3
1
,
-




+
+
+
=
has
init
init
ij
ij
X
ij
P
Q
I
Q
I
X
Q
B
G
X
.
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:
()
()
()








+


+
=
ij
II
II
ij
ij
Q
Dev.
Q
Q
Q
H
Q
Q
det
6
54
3
cos
2
1
1
2
5
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Code_Aster
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Titrate:
Law CJS in géomechanics
Date:
18/11/03
Author (S):
C. CHAVANT, pH. AUBERT
Key
:
R7.01.13-A
Page
:
9/48
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R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/03/005/A
The preceding expression is obtained in the following way. One a:
()
()
ij
Q
II
ij
II
Q
ij
D
Q
H
Q
Q
Q
H
Q
F
+
=
where
ij
II
Q
Q
and
()
ij
Q
Q
H
are respectively given by:
II
ij
ij
II
Q
Q
Q
Q
=
()
()
()
ij
II
Q
II
ij
Q
Q
II
ij
Q
ij
Q
Q
Q
H
Q
Q
H
Q
Q
H
Q
H
+
-
=




+
=
)
det (
)
(
6
54
)
(
2
)
3
cos (
det
54
1
6
1
3
5
2
5
3
5
Q
Q
from where
()
()
()








+


+
=
ij
II
II
ij
Q
Q
ij
D
Q
Q
Q
Q
H
Q
F
Q
det
6
54
3
cos
2
1
1
2
5

The function
, as for it is given by:
()
II
S
O
Q
H
=
where
ij
ij
II
Q
Q
Q
=
and
()
()
(
)
()
6
/
1
3
6
/
1
det
54
1
3
cos
1




+
=
+
=
II
S
S
S
H
S
. The term
O
express yourself in
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:

The potential function is defined starting from the following kinematic condition:
II
dp
ij
ij
C
II
II
dp
v
S
E
S
S
S
&
&




-
-
=
1
where
is a parameter of the model and
C
II
S
represent the characteristic state of stress. A surface,
from form identical to the surface of load in the space of the stresses, separates the contracting states
dilating states. This surface, known as characteristic, has as an equation:
()
(
)
init
C
S
C
II
C
Q
I
R
H
S
F
+
+
=
1
D
ij
D
dp
ij
G
&
& =
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Code_Aster
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Version
6.4
Titrate:
Law CJS in géomechanics
Date:
18/11/03
Author (S):
C. CHAVANT, pH. AUBERT
Key
:
R7.01.13-A
Page
:
10/48
Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/03/005/A
where
C
R
is a parameter corresponding to the average radius of this characteristic surface. The condition
kinematics can be also put in the form:
0
1
1
=
+
=
+
=




-
+
=




-
+
dp
ij
ij
II
dp
v
dp
ij
ij
II
dp
v
II
dp
ij
ij
dp
ij
ij
dp
ij
ij
C
II
II
dp
v
II
dp
ij
ij
C
II
II
dp
v
S
S
E
S
S
S
E
S
E
S
E
S
S
S
S
E
S
S
S
&
&
&
&
&
&
&
&
&
&
where
(
)
dp
ij
ij
C
II
II
S
sign
S
S
&




-
=
1
.
It is then possible to seek to express this kinematic condition starting from a tensor
N
under
form:
0
=
ij
dp
ij
N
&
i.e., after decomposition of each term in déviatoire parts and hydrostatic:
(
)
0
3
1
2
1
2
1
=
+
=
+


+
=
dp
v
dp
ij
ij
ij
ij
ij
dp
v
dp
ij
ij
dp
ij
N
E
S
N
N
S
N
E
N
&
&
&
&
&
One deduces the relation from it
II
S
N
N
=
2
1
, which added to the condition of standardization
1
:
=
N
N
, led to
expressions:
3
2
1
+
=
II
S
N
and
3
1
2
2
+
=
N
, that is to say
3
2
+
+
=
ij
II
ij
ij
S
S
N
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
, that is to say:
D
ij
D
ij
kl
kl
D
ij
D
D
dp
ij
G
N
N
F
F
&
&
&
=








-
=
with
ij
kl
kl
D
ij
D
D
ij
N
N
F
F
G




-
=
.
In addition, for the calculation of the potential, one can note that:
(
)
ij
kl
kl
ij
ij
kl
kl
kl
kl
ij
kl
jl
ik
mm
D
kl
kl
kl
kl
ij
jl
ik
kl
ij
kl
kl
ij
jl
ik
kl
mm
D
kl
D
ij
ij
kl
kl
D
ij
D
R
X
Q
Q
R
X
Q
F
X
Q
Q
Q
R
X
Q
F
Q
F
Dev.
R
Q
Q
F
F
-
-
=
+




+
-
+


+
-
=
+




+
-




+




=
+
=
3
1
3
1
3
1
3
1
3
1
background image
Code_Aster
®
Version
6.4
Titrate:
Law CJS in géomechanics
Date:
18/11/03
Author (S):
C. CHAVANT, pH. AUBERT
Key
:
R7.01.13-A
Page
:
11/48
Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/03/005/A
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
. Limit of
R
, noted
m
R
, is reached when
R
tends towards the infinite one. Limit of
ij
X
is reached when
ij
X
&
becomes null.
Under these conditions:
ij
ij
X
Q
=
and
()
S
O
lim
II
lim
II
II
H
X
X
Q
1
=
=
In the state of rupture one thus has [Figure 3.4.4-a]:
(
)
Q
S
lim
II
II
II
X
I
S
Q
-
+
=
cos
cos
1
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:
()
(
)
0
1
=
+
+
=
init
R
S
II
R
Q
I
R
H
S
F
with
()
()
(
)
Q
S
m
Q
S
O
R
R
H
H
R
-
+
=
cos
cos
, average radius of the envelope, which is determined from
mechanical characteristics with the rupture of material. The value of
O
can then be deduced from it:
()
()
(
)
Q
S
m
Q
S
R
O
R
H
H
R
-
-
=
cos
cos
with
(
)
II
II
II
II
II
X
I
S
X
I
S
Q
1
2
1
2
2
2
cos
-
-
=
S
2
S
3
S
1
Q
2
Q
3
Q
1
rough surface
caracterisic surface
surface of load to the rupture
Q
S
Appear 3.4.4-a: Representation of the rough surfaces, characteristic and of load
in the déviatoire plan
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Code_Aster
®
Version
6.4
Titrate:
Law CJS in géomechanics
Date:
18/11/03
Author (S):
C. CHAVANT, pH. AUBERT
Key
:
R7.01.13-A
Page
:
12/48
Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/03/005/A
In addition,
R
R
is related to the maximum angle of friction and depends on the mean stress and on
relative density. To take into account the dependence of the maximum angle of friction in function
mean stress and relative density, the relation is considered:




+
+
=
init
C
C
R
Q
I
p
R
R
1
3
ln
µ
where
C
R
and
µ
are parameters of the model.
C
p
is the mean stress criticizes, i.e.
minimal mean stress (it is negative with our convention of sign) known by
material during its history. It depends on the initial relative density according to the conventional concept
of straight line criticizes in the plan
(
)
p
E ln
,
:
(
)
v
Co
C
C
p
p
-
=
exp
where
Co
p
is the initial critical pressure and
C
1
is the critical line slope of state in the plan
(
)
p
v
ln
,
.
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
isotropic
Plastic mechanism
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
O
K
E
O
G
p
K
C
R
With
B
m
R
µ
Co
p
C
has
p
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
O
K
and
O
G
) are directly taken in
count in the elastic characteristics of material, i.e. through the Young modulus
E
and
the Poisson's ratio
NAKED
.
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Code_Aster
®
Version
6.4
Titrate:
Law CJS in géomechanics
Date:
18/11/03
Author (S):
C. CHAVANT, pH. AUBERT
Key
:
R7.01.13-A
Page
:
13/48
Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/03/005/A
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
0
=
N
then level CJS1,
·
if (
0
N
and
0
With
) then level CJS2,
·
if (
0
N
and
0
=
With
) then level CJS3.
Note:
The user must fix the value of P
has
equalize with - 100 kPa according to the selected units. In
in addition to, for CJS3, the value of p
Co
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 Cohésion cohesion
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:
()
()
+
-
=




+
-
sin
3
sin
3
1
1
6
/
1
() (
)
()
-
-
=
sin
3
1
sin
3
2
2
6
/
1
m
R
()
-
=
cotan
.
3c
Q
init
()
()
-
-
=
sin
3
sin
6
2
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Code_Aster
®
Version
6.4
Titrate:
Law CJS in géomechanics
Date:
18/11/03
Author (S):
C. CHAVANT, pH. AUBERT
Key
:
R7.01.13-A
Page
:
14/48
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R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/03/005/A
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
0
I
F
and
0
D
F
, the elastic prediction is regarded as new state of stress,
·
if
0
>
I
F
and
0
D
F
, one makes the integration of the mechanisms elastic nonlinear and plastic
isotropic,
·
if
0
I
F
and
0
>
D
F
, one makes the integration of the mechanisms elastic nonlinear and plastic
déviatoire,
·
if
0
>
I
F
and
0
>
D
F
, one makes the integration of the mechanisms elastic nonlinear, plastic
isotropic and plastic déviatoire.
At exit 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
Variables
Q
,
R
and
are equivalent to the associated thermodynamic forces
Iso
Q
,
R
and
X
.
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
Iso
Q
,
R
and
X
.
In addition, we add to the number of the internal variables:
·
the sign of the product
dp
ij
ij
S
·
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
background image
Code_Aster
®
Version
6.4
Titrate:
Law CJS in géomechanics
Date:
18/11/03
Author (S):
C. CHAVANT, pH. AUBERT
Key
:
R7.01.13-A
Page
:
15/48
Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/03/005/A
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
=
Iso
Q
Iso
Q
Iso
Q
2 2
m
R
R
=
R
m
R
R
=
3
3 0
0
11
X
4 4
0
0
22
X
5 5
0
0
33
X
6.6
0
0
12
2X
7
0 0
13
2X
8
0 0
23
2X
9 7
(
)
init
m
Q
II
Q
I
R
H
Q
+
1
)
(
(
)
init
Q
II
Q
I
R
H
Q
+
1
)
(
(
)
init
m
Q
II
Q
I
R
H
Q
+
1
)
(
10 8
m
R
R
lim
II
II
X
X
11 9
init
Q
I
Q
+
1
3
init
Q
I
Q
+
1
3
12 10
Numbers
iterations
interns
Iteration count
interns
Iteration count
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
)
(
dp
ij
ij
S
sign
)
(
dp
ij
ij
S
sign
)
(
dp
ij
ij
S
sign
16
14
0,1,2,3 state of
material
0,1,2,3 state of
material
0,1,2,3 state of
material

4.2
Integration of the nonlinear elastic mechanism
In the elastic case, the new state of stress
+
, checks simply:
()
kl
ijkl
ij
ij
D
+
=
+
-
+
The dependence of the nonlinear tensor of elasticity according to the state of stresses is summarized in
fact with:
()
N
has
init
linear
ijkl
ijkl
P
Q
I
D
D




+
=
+
+
3
1
where
linear
ijkl
D
is the tensor of isotropic linear elasticity conventional, obtained from
E
O
K
and
O
G
or by
equivalence from
E
and
Naked
.
From this relation, one deduces in particular that the first invariant of the stresses satisfied:
()
0
3
3
1
1
1
=




+
-
-
+
-
+
tr
P
Q
I
K
I
I
N
has
init
E
O
background image
Code_Aster
®
Version
6.4
Titrate:
Law CJS in géomechanics
Date:
18/11/03
Author (S):
C. CHAVANT, pH. AUBERT
Key
:
R7.01.13-A
Page
:
16/48
Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/03/005/A
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
()
+
=
-
+
tr
K
I
I
E
O
3
1
1
In the general case, the knowledge of
+
1
I
and thus of the term
N
has
init
P
Q
I




+
+
3
1
allows to define
the nonlinear operator of elasticity
()
+
ijkl
D
. Obtaining the new state of stress is then direct.
4.3 Integration of the mechanisms elastic nonlinear and plastic
isotropic
In this case, the new state of stress
+
, checks:
() (
)
IP
kl
kl
ijkl
ij
ij
D
-
+
=
+
-
+
Being given the simple form, plastic deformations of the isotropic plastic mechanism:
ij
I
IP
ij
-
=
3
1
the nonlinear system to solve is composed of:
·
ij
: the elastic law state:
()
0
3
1
=


+
-
-
+
-
+
kl
I
kl
ijkl
ij
ij
D
·
LQ
: the law of work hardening of the internal variable
Iso
Q
:
()
0
=
-
-
+
-
+
Iso
Q
I
Iso
Iso
Q
G
Q
Q
Iso
·
FI
: the equation of the isotropic surface of load:
0
3
1
=
+
+
-
+
+
Iso
init
Q
Q
I
Schematically, one thus seeks to solve the system
()
0
=
Y
R
, where the unknown factor
Y
is given
by
(
)
I
Iso
ij
Q
Y
+
+
,
,
=
and where
(
)
FI
LQ
R
,
,
ij
=
. The resolution of
()
0
=
Y
R
is done by the method
of Newton:
·
initialization and calculation of a solution of test
0
Y
·
iterations of Newton: resolution of
()
()
p
1
+
p
p
- Y
R
DY
Y
DY
DR.
=
·
test of convergence: if convergence
p
Y
=
Y
; if not
1
+
p
p
1
+
p
=
DY
Y
Y
+
and
1
+
= p
p
We detail these three stages below.
4.3.1 Initialization and solution of test
We take simply for
(
)
0
0
0
,
,
I
Iso
ij
Q
Y
=
0
, following values:
elas
ij
ij
=
0
: stresses given by the elastic prediction,
-
=
Iso
Iso
Q
Q
0
: variable interns with T
0
0
=
I
: plastic multiplier no one
Contrary to the other elastoplastic mechanisms, here a solution of test is not calculated.
background image
Code_Aster
®
Version
6.4
Titrate:
Law CJS in géomechanics
Date:
18/11/03
Author (S):
C. CHAVANT, pH. AUBERT
Key
:
R7.01.13-A
Page
:
17/48
Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/03/005/A
4.3.2 Iterations of Newton
The resolution of
()
()
p
1
+
p
p
- Y
R
DY
Y
DY
DR.
=
naturally require the calculation of derived from
ij
,
LQ
and
FI
compared to each component of
Y
. One a:












=
I
Iso
kl
I
Iso
kl
I
ij
Iso
ij
kl
ij
FI
Q
FI
FI
LQ
Q
LQ
LQ
Q
DY
DR.
with:
kl
N
has
init
has
Mn
I
Mn
linear
ijmn
jl
ik
Mn
I
Mn
kl
ijmn
jl
ik
kl
P
Q
I
P
N
D
D
1
1
3
3
3
1
3
1
-




+


+
-
=


+
-
=
ij
0
=
Iso
Q
ij
Mn
ijmn
I
D
3
1
-
=
ij
0
=
kl
LQ
1
1
1
-




+
=
-
=
N
has
Iso
has
p
O
I
Iso
Q
I
Iso
P
Q
P
K
N
Q
G
Q
LQ
Iso
Iso
Q
I
G
LQ
-
=
kl
kl
FI
3
1
-
=
1
=
Iso
Q
FI
0
=
I
FI
4.3.3 Test of convergence
The iterations of Newton are continued as much as the relative error
0
1
+
p
1
+
p
Y
Y
DY
-
remain higher than
tolerance allowed by the user and defined by the key word
RESI_INTE_RELA
. The standard used here is
the vectorial standard:
=
I
I
X
X
2
.
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Code_Aster
®
Version
6.4
Titrate:
Law CJS in géomechanics
Date:
18/11/03
Author (S):
C. CHAVANT, pH. AUBERT
Key
:
R7.01.13-A
Page
:
18/48
Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/03/005/A
4.4 Integration of the mechanisms elastic nonlinear and plastic
déviatoire
In this case, the new state of stress
+
, checks:
() (
)
dp
kl
kl
ijkl
ij
ij
D
-
+
=
+
-
+
The plastic deformations of the plastic mechanism déviatoire are given by the potential
D
G
:
D
ij
D
dp
ij
G
=
One deduces from it that the nonlinear system to solve is composed of:
·
ij
: the elastic law state:
()
(
)
(
)
0
,
,
=
-
-
-
+
+
+
+
-
+
X
R
G
D
D
kl
D
kl
ijkl
ij
ij
·
LR
: the law of work hardening of the variable
R
:
(
)
0
,
=
-
-
+
+
-
+
R
G
R
R
R
D
·
ij
LX
: the law of work hardening of the variable
ij
X
:
(
)
0
,
=
-
-
+
+
-
+
X
X
D
ij
ij
G
X
X
·
FD
: the equation of the surface of load déviatoire:
()
(
)
0
1
=
+
+
+
+
+
+
init
Q
II
Q
I
R
H
Q
As in the preceding paragraph one solves by the method of Newton the system
()
0
=
Y
R
, where
the unknown factor
Y
is given by
(
)
D
ij
ij
X
R
Y
=
+
+
+
,
,
,
and where
(
)
FD
LX
LR
R
,
,
,
ij
ij
=
.
4.4.1 Initialization and solution of test
Starting from the state at the moment T
(
)
-
-
-
ij
ij
X
R,
,
, we seek a solution of test which brings us closer
the final solution. For that we solve the following equation:
(
)
(
)
0
,
,
=
+
+
-
+
-
-
-
-
-
-
-
X
ij
D
ij
R
D
D
kl
D
kl
ijkl
ij
D
G
X
G
R
G
D
F
with
()
-
-
=
ijkl
ijkl
D
D
,
(
)
-
-
-
-
=
X
,
,
R
G
G
D
kl
D
kl
,
(
)
-
-
-
=
R
G
G
R
R
,
,
(
)
-
-
-
=
X
,
X
ij
X
ij
G
G
and where
the unknown factor is the plastic multiplier
D
, by only one iteration of Newton, i.e.
finally of we let us have:
0
0
=
=
-
=
D
D
D
D
D
D
F
F
that is to say still
0
0
=
=
-
=
D
D
D
D
D
D
F
F
with:
()
()
(
)
D
D
init
D
Q
II
D
II
Q
D
D
I
R
R
Q
I
H
Q
Q
H
F
+
+
+
+
=
1
1
Moreover,
one a:
()
()
(
)
-
-
-
-
+
=
D
D
tr
tr
K
I
I
G
3
1
1
then:
()
-
-
-
=
D
D
tr
K
I
G
3
1
one a:
-
-
+
=
R
D
G
R
R
then:
-
=
R
D
G
R
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Code_Aster
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Titrate:
Law CJS in géomechanics
Date:
18/11/03
Author (S):
C. CHAVANT, pH. AUBERT
Key
:
R7.01.13-A
Page
:
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Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/03/005/A
one a:
(
)
()
()
(
)
[
]


+
+
-
+
-
-
+
=
-
-
-
-
-
-
-
-
X
ij
D
ij
ij
D
D
D
kl
D
kl
ijkl
ij
ij
G
X
tr
tr
K
I
G
D
Q
3
1
3
1
G
then:
()
()
(
)
+
-


+
+
-
=
-
-
-
-
-
-
-
-
=
tr
K
I
G
X
tr
K
G
D
Q
X
ij
ij
ij
D
D
kl
ijkl
D
ij
D
3
3
1
3
1
0
G
one a:
D
ij
II
ij
D
ij
ij
II
D
II
Q
Q
Q
Q
Q
Q
Q
=
=
and
()
()
D
ij
ij
Q
D
Q
Q
Q
H
H
=
Ultimately, we take for the solution of test:
(
)
0
0
0
0
,
,
,
D
ij
ij
X
R
Y
=
0
, with the values
following:
0
D
: the value found according to the preceding formulation.
(
)
-
-
-
-
+
=
D
kl
D
kl
ijkl
ij
ij
G
D
0
0
-
-
+
=
R
D
G
R
R
0
0
-
-
+
=
X
ij
D
ij
ij
G
X
X
0
0
4.4.2 Iterations of Newton
DY
DR.
is given here by:


















=
D
ij
kl
D
ij
ij
ij
ij
kl
ij
D
ij
kl
D
ij
ij
ij
ij
kl
ij
FD
X
FD
R
FD
FD
LX
X
LX
R
LX
LX
LR
X
LR
R
LR
LR
X
R
DY
DR.
with:
(
)
kl
D
Mn
D
ijmn
kl
N
has
init
has
D
Mn
D
Mn
linear
ijmn
jl
ik
kl
G
D
P
Q
I
P
N
G
D
+




+
-
-
=
- 1
1
3
3
ij
R
G
D
R
D
Mn
D
ijmn
=
ij
kl
D
Mn
D
ijmn
kl
X
G
D
X
=
ij
D
Mn
ijmn
D
G
D
=
ij
kl
has
init
m
D
kl
R
D
kl
P
Q
I
R
R
With
G
LR
5
,
1
1
2
3
1
2
-




+




-
-
=
-
=
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Titrate:
Law CJS in géomechanics
Date:
18/11/03
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C. CHAVANT, pH. AUBERT
Key
:
R7.01.13-A
Page
:
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Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/03/005/A
(
)
5
,
1
1
1
3
1
2
1
1
-




+
+




-
-
=
-
=
has
init
init
m
m
D
R
D
P
Q
I
Q
I
R
R
R
With
R
G
R
LR
0
=
kl
X
LR
R
D
G
LR
-
=
kl
X
ij
D
kl
G
LX
-
=
ij
0
=
R
LX
ij
kl
X
ij
D
jl
ik
kl
X
G
X
LX
-
=
ij
X
ij
D
G
LX
-
=
ij
(
)
kl
Mn
Mn
kl
kl
D
kl
R
X
Q
Q
F
FD
-
-
=
=
1
I
R
FD =
kl
D
kl
X
F
X
FD
=
0
=
D
FD
In addition, the calculation of the terms
kl
D
Mn
G
,
R
G
D
Mn
,
kl
D
Mn
X
G
,
kl
X
ij
G
,
kl
X
ij
X
G
and
kl
D
X
F
is detailed
hereafter, as well as the calculation of useful intermediate terms:
·
calculation of
kl
D
X
F
:
() ()
kl
II
Q
kl
Q
II
kl
D
X
Q
H
X
H
Q
X
F
+
=
()
()
kl
Mn
Mn
II
Q
kl
Mn
Mn
Q
II
X
Q
Q
Q
H
X
Q
Q
H
Q
+
=
()
()
kl
Mn
II
Mn
Q
kl
Mn
Mn
Q
II
X
Q
Q
Q
H
X
Q
Q
H
Q
+
=
() ()
kl
Mn
II
Mn
Q
Mn
Q
II
X
Q
Q
Q
H
Q
H
Q




+
=
() ()
nl
mk
II
Mn
Q
Mn
Q
II
Q
Q
H
Q
H
Q
I




+
-
=
1
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Titrate:
Law CJS in géomechanics
Date:
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C. CHAVANT, pH. AUBERT
Key
:
R7.01.13-A
Page
:
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Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/03/005/A




-
=
kl
D
Q
F
I
1
One will notice for the continuation that:
kl
kl
D
Q
I
X
F
Dev.
1
-
=




·
calculation of
kl
ij
D
F




:
kl
Mn
Mn
ij
D
kl
ij
D
Q
Q
F
F




=




(
)
(
)
kl
Mn
Mn
ij
rs
rs
ij
Q
Q
R
X
Q
Q




-
-
=
kl
Mn
ij
rs
Mn
rs
Mn
ij
Q
X
Q
Q
Q
Q








-
=






+
-








-
=
Mn
Mn
kl
nl
mk
ij
rs
Mn
rs
Mn
ij
X
X
Q
Q
Q
Q
3
·
calculation of
Mn
ij
Q
Q
:
Au préalable, one definite the tensor
T
and its déviatoire part
D
T
while posing:
()
ij
ij
Q
T
Q
det
=
and
()




=
ij
D
ij
Q
Dev.
T
Q
det
One has as follows:














-
-
-
-
-
-
=














=
11
23
13
12
22
13
23
12
33
12
23
13
12
12
22
11
13
13
33
11
23
23
33
22
23
13
12
33
22
11
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
T
T
T
T
T
T
T
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Titrate:
Law CJS in géomechanics
Date:
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Author (S):
C. CHAVANT, pH. AUBERT
Key
:
R7.01.13-A
Page
:
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Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/03/005/A
()
()
() ()
()
()
()
()
() ()
()
()
() ()
() () ()
()
()
































-
+
-
+
-
+
+
+
+
-
=
+
+
+
+
+
+
-
=
2
2
2
54
6
5
1
2
det
3
4
54
2
5
1
3
3
cos
2
1
5
1
2
6
54
3
cos
2
1
6
5
2
6
54
5
1
3
cos
2
5
1
3
cos
2
1
5
1
2
6
54
3
cos
2
1
6
5
II
Q
Mn
Q
D
ij
T
Mn
Q
D
ij
T
II
Q
Q
H
Mn
Q
II
Q
Q
Mn
T
II
Q
ij
Q
Q
H
II
Q
Mn
Q
ij
Q
II
Q
jn
im
Q
Q
H
Mn
Q
Q
H
ij
T
Dev.
II
Q
II
Q
ij
Q
Q
Q
H
Mn
Q
II
Q
D
ij
T
Q
H
Mn
Q
Q
II
Q
ij
Q
Q
H
Mn
Q
II
Q
ij
Q
Q
Q
H
Mn
Q
Q
H
ij
T
Dev.
II
Q
II
Q
ij
Q
Q
Q
H
Mn
Q
ij
Q
The expression of
Mn
D
ij
Q
T
clarify yourself as follows:
(
)
(
)
(
)


















-
-
+
-
+
-
=
23
33
22
33
22
33
22
11
0
0
2
3
1
2
3
1 3
1
Q
Q
Q
Q
Q
Q
Q
Q
D
T
,
(
)
(
)
(
)


















-
-
+
-
+
-
=
0
0
2
3
1 3
1
2
3
1
13
33
11
33
11
33
11
22
Q
Q
Q
Q
Q
Q
Q
Q
D
T
,
(
)
(
)
(
)


















-
+
-
-
+
-
=
0
0
3
1
2
3
1
2
3
1
12
22
11
22
11
22
11
33
Q
Q
Q
Q
Q
Q
Q
Q
D
T
,


















-
-
=
13
23
33
12
12
12
12
3
4
3
2
3
2
Q
Q
Q
Q
Q
Q
Q
D
T
,


















-
-
=
12
22
23
13
13
13
13
3
23
4
3
2
Q
Q
Q
Q
Q
Q
Q
D
T
,


















-
-
=
11
12
13
23
23
23
23
3
2
3
23
4
Q
Q
Q
Q
Q
Q
Q
D
T
·
calculation of
kl
D
Mn
G
:
One a:
Mn
kl
rs
rs
D
rs
kl
rs
D
kl
Mn
rs
rs
D
kl
Mn
D
kl
D
Mn
N
N
F
N
F
N
N
F
F
G








+




-




-




=
background image
Code_Aster
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Version
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Titrate:
Law CJS in géomechanics
Date:
18/11/03
Author (S):
C. CHAVANT, pH. AUBERT
Key
:
R7.01.13-A
Page
:
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Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/03/005/A
One definite the tensor
N
~
by
ij
II
ij
ij
S
S
N
+
=
~
i.e. that
N
is then given by
II
ij
ij
N
N
N
~
~
=
with
3
~
2
+
=
II
N
In practice, for the calculation of
, one uses
ij
in the place of
dp
ij
, i.e. one a:
(
)
ij
ij
C
II
II
S
sign
S
S




-
=
1
One has then for
kl
D
Mn
G
:
kl
II
Mn
rs
rs
D
II
kl
Mn
rs
rs
D
II
Mn
kl
rs
rs
D
Mn
rs
kl
rs
D
kl
Mn
D
kl
D
Mn
N
N
N
F
N
N
N
F
N
N
N
F
N
N
F
F
G








-




-




-












-




=
2
2
2
~1
~
~
~1
~
~
~
~
~
with:
(
)
(
) ()
(
)
kl
C
II
II
C
II
II
kl
kl
kl
II
S
S
S
S
N




+




-
-
=
+
-
=




+
=




2
2
2
2
2
2
2
2
3
1
2
3
1
3
1
~1
·
calculation of
kl
C
II
II
S
S




:
()
()
kl
C
II
C
II
II
kl
II
C
II
kl
C
II
II
S
S
S
S
S
S
S
2
1
-
=




()
(
)
()
kl
S
init
C
C
II
II
kl
Mn
Mn
II
C
II
H
Q
I
R
S
S
S
S
S
S




+
-
-
=
1
2
1
()
(
)
()
()




+
+
-
-


-
=
kl
S
S
init
C
kl
S
C
C
II
II
kl
Mn
nl
mk
II
Mn
C
II
H
H
Q
I
R
I
H
R
S
S
S
S
S
2
1
1
2
3
1
1
()
(
)
()
()




+
+
-
-


-
=
kl
rs
rs
S
S
init
C
kl
S
C
C
II
II
kl
Mn
nl
mk
II
Mn
C
II
S
S
H
H
Q
I
R
H
R
S
S
S
S
S
2
1
2
3
1
1
background image
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Titrate:
Law CJS in géomechanics
Date:
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C. CHAVANT, pH. AUBERT
Key
:
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Page
:
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Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/03/005/A
·
calculation of
kl
Mn
N
~
:
(
)
(
)












-




+




-
=
kl
II
kl
C
II
Mn
ij
ij
kl
Mn
ij
ij
II
C
II
kl
Mn
S
S
S
S
sign
S
S
sign
S
S
N
1
1
1
1
~
(
)
(
)
()




-
+


-




-
=
kl
C
II
C
II
kl
II
II
Mn
ij
ij
kl
Mn
nl
mk
ij
ij
II
C
II
S
S
S
S
S
S
sign
S
sign
S
S
2
2
1
1
3
1
1
1
·
calculation of
R
G
D
Mn
:
Mn
rs
rs
D
rs
rs
D
Mn
rs
rs
D
Mn
D
D
Mn
N
R
N
F
N
R
F
R
N
N
F
R
F
R
G








+




-




-




=
Mn
rs
rs
D
Mn
D
N
N
R
F
R
F












-




=
(
)
Mn
rs
rs
Mn
N
N
-
=
3
3
2
2
+
-
=
II
Mn
Mn
S
S
·
calculation of
kl
D
Mn
X
G
:
Mn
kl
rs
rs
D
rs
kl
rs
D
kl
Mn
rs
rs
D
kl
Mn
D
kl
D
Mn
N
X
N
F
N
X
F
X
N
N
F
X
F
X
G








+




-




-




=
Mn
rs
kl
rs
D
kl
Mn
D
N
N
X
F
X
F












-




=
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Law CJS in géomechanics
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C. CHAVANT, pH. AUBERT
Key
:
R7.01.13-A
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:
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Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/03/005/A
·
calculation of
kl
Mn
D
X
F




:
Mn
kl
rs
rs
rs
kl
rs
kl
Mn
kl
Mn
D
X
X
Q
X
X
Q
X
Q
X
F




+
-
=




Mn
ls
Kr
rs
rs
kl
ij
ij
rs
kl
ij
ij
Mn
Q
X
X
Q
Q
Q
X
Q
Q
Q




+




-
=
Mn
ls
Kr
rs
rs
jl
ik
ij
rs
jl
ik
ij
Mn
Q
X
Q
Q
I
Q
Q
I




+




-
-
-
=
1
1
·
calculation of
kl
X
ij
G
:
(
)
(
)
5
,
1
1
1
1
5
,
1
1
3
1
3
2
1
-
-




+
+




+
+




+
+
-
=
has
init
init
ij
kl
kl
ij
kl
has
init
ij
ij
kl
X
ij
P
Q
I
Q
I
X
Q
B
I
P
Q
I
X
Q
B
G
(
)
(
)
()
()
()
(
)
5
,
1
1
1
5
,
1
1
1
5
,
1
1
3
1
3
1
3
2
1
-
-
-




+
+




+
+
+




+
+
+




+
+
-
=
has
init
init
ij
kl
II
S
O
kl
S
II
O
kl
O
II
S
has
init
init
kl
Mn
Mn
ij
kl
has
init
ij
ij
P
Q
I
Q
I
X
Q
H
H
Q
Q
H
B
P
Q
I
Q
I
Q
Q
Q
B
P
Q
I
X
Q
B
·
calculation of
()
kl
S
H
:
()
()
kl
Mn
Mn
S
kl
S
S
S
H
H
=
()
()


-




-
=
kl
Mn
nl
mk
Mn
II
S
Q
Mn
II
S
S
Q
H
T
Q
H
3
1
2
)
3
cos (
6
54
2
5
3
5
·
calculation of
kl
II
Q
:
kl
Mn
Mn
rs
rs
II
kl
II
Q
Q
Q
Q
Q
Q




=






+
-




=
kl
kl
Mn
nl
mk
Mn
rs
II
rs
X
Q
Q
Q
Q
3
1
background image
Code_Aster
®
Version
6.4
Titrate:
Law CJS in géomechanics
Date:
18/11/03
Author (S):
C. CHAVANT, pH. AUBERT
Key
:
R7.01.13-A
Page
:
26/48
Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/03/005/A
·
calculation of
kl
O
:
()
()
(
)
()
(
)
()
()
()
(
)
() ()
()
(
)
()
()
(
)
2
2
cos
cos
cos
cos
1
cos
cos
cos
1




-
-




-
-
-
+
-
-
-
-
-
=
Q
S
m
Q
S
R
kl
Q
S
m
Q
S
kl
S
Q
S
m
Q
S
kl
S
Q
S
m
Q
kl
R
kl
Q
S
m
Q
S
R
kl
O
R
H
H
R
R
H
H
H
R
H
H
H
R
H
R
R
H
H
R
with:
(
)




+
-
-
-




-
-
=
kl
II
II
kl
II
II
II
II
II
II
II
kl
II
II
kl
II
kl
II
II
II
II
kl
S
X
I
I
X
S
X
I
S
S
X
I
Q
S
S
I
X
I
Q
Q
X
I
S
1
1
1
2
2
1
2
1
2
1
1
2
2
2
2
1
cos
(
)
(
)
(
)






+
-
-
-
-
-
=
II
kl
II
kl
II
II
II
II
II
kl
kl
II
kl
II
II
S
S
X
I
X
S
S
X
I
Q
S
X
I
Q
X
I
S
1
2
2
1
2
2
1
1
1
kl
init
kl
R
Q
I
R
µ
+
-
=
1
(
)
(
)




-
-
-
=
-
kl
Q
kl
S
Q
S
kl
Q
S
sin
cos
· calculation of
kl
X
ij
X
G
:
(
)
5
,
1
1
1
3
1
-




+
+




+
=
has
init
init
kl
ij
kl
ij
kl
X
ij
P
Q
I
Q
I
X
X
X
Q
B
X
G
(
)
5
,
1
1
1
3
1
-




+
+




+
=
has
init
init
jl
ik
kl
Mn
Mn
ij
P
Q
I
Q
I
X
Q
Q
Q
B
(
)
5
,
1
1
1
1
3
1
-




+
+




+
-
=
has
init
init
jl
ik
nl
mk
Mn
ij
P
Q
I
Q
I
Q
Q
I
B
4.4.3 Test of convergence
The criterion of convergence remains
0
1
+
p
1
+
p
Y
Y
DY
-
RESI_INTE_RELA
.
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Titrate:
Law CJS in géomechanics
Date:
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Author (S):
C. CHAVANT, pH. AUBERT
Key
:
R7.01.13-A
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:
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HT-66/03/005/A
4.5 Integration of the mechanisms elastic nonlinear, plastic
isotropic and plastic déviatoire
In this case, the new state of stress
+
, checks:
() (
)
dp
kl
IP
kl
kl
ijkl
ij
ij
D
-
-
+
=
+
-
+
Taking into account what precedes, one deduces from it that the nonlinear system to solve is composed of:
·
ij
: the elastic law state:
()
(
)
0
,
,
3
1
=


-
+
-
-
+
+
+
+
-
+
X
R
G
D
D
kl
D
kl
I
kl
ijkl
ij
ij
·
LQ
: the law of work hardening of the internal variable
Iso
Q
:
()
0
=
-
-
+
-
+
Iso
Q
I
Iso
Iso
Q
G
Q
Q
Iso
·
LR
: the law of work hardening of the variable
R
:
(
)
0
,
=
-
-
+
+
-
+
R
G
R
R
R
D
·
ij
LX
: the law of work hardening of the variable
ij
X
:
(
)
0
,
=
-
-
+
+
-
+
X
X
ij
D
ij
ij
G
X
X
·
FI
: the equation of the isotropic surface of load:
0
3
1
=
+
+
-
+
+
Iso
init
Q
Q
I
·
FD
: the equation of the surface of load déviatoire:
()
(
)
0
1
=
+
+
+
+
+
+
init
Q
II
Q
I
R
H
Q
As in the preceding paragraphs one solves by the method of Newton the system
()
0
=
Y
R
, where
the unknown factor
Y
is given by
(
)
D
I
ij
Iso
ij
X
R
Q
Y
=
+
+
+
+
,
,
,
,
,
and where
(
)
FD
FI
LX
LR
LQ
R
,
,
,
,
,
ij
ij
=
.
4.5.1 Initialization and solution of test
Starting from the state at the moment T
(
)
-
-
-
-
ij
Iso
ij
X
R
Q
,
,
,
, we seek a solution of test which us
bring closer the final solution. For that we solve the system of equations according to:




=




+
+


-
+
+
=




+


-
+
+
-
-
-
-
+
-
-
-
+
-
0
,
,
3
1
0
,
3
1
X
ij
D
ij
R
D
D
kl
D
kl
I
kl
ijkl
ij
D
Q
I
Iso
D
kl
D
kl
I
kl
ijkl
ij
I
G
X
G
R
G
D
F
G
Q
G
D
F
Iso
with:
()
-
-
=
ijkl
ijkl
D
D
,
(
)
-
-
-
-
=
X
,
, R
G
G
D
kl
D
kl
,
()
-
-
=
Iso
Q
Q
Q
G
G
Iso
Iso
(
)
-
-
-
=
R
G
G
R
R
,
,
(
)
-
-
-
=
X
,
X
ij
X
ij
G
G
and where the unknown factors are the plastic multipliers
I
and
D
, by one
only iteration of Newton, i.e. finally that we have:
0
,
0
0
,
0
0
,
0
0
,
0
0
,
0
0
,
0
=
=
=
=
=
=
=
=
=
=
=
=
-
=
+
-
=
+
D
I
D
I
D
I
D
I
D
I
D
I
D
D
D
D
I
I
D
I
D
D
I
I
I
I
F
F
F
F
F
F
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Law CJS in géomechanics
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:
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HT-66/03/005/A
that is to say still:
I
D
D
I
D
D
I
I
I
D
D
D
D
I
I
F
F
F
F
F
F
F
F
-
-
=
and
I
D
D
I
D
D
I
I
D
I
I
I
I
D
D
F
F
F
F
F
F
F
F
-
-
=
with:
(
)
-
-
+
-
=
p
I
I
K
K
F
()
-
-
=
D
D
I
tr
K
F
G
()
()
(
)
I
I
init
I
Q
II
I
II
Q
I
D
I
R
R
Q
I
H
Q
Q
H
F
+
+
+
+
=
1
1
()
()
(
)
D
D
init
D
Q
II
D
II
Q
D
D
I
R
R
Q
I
H
Q
Q
H
F
+
+
+
+
=
1
1
It is known that
D
D
F
is calculated in the same way that previously when only the mechanism
plastic déviatoire was activated. In addition, one has, for the calculation of
I
D
F
and when
0
=
I
and
0
=
D
, following relations:


+
-
=
-
-
-
ij
ij
E
kl
ijkl
I
ij
X
K
D
Q
3
1
3
3
1






+
-
=
=
-
-
-
ij
ij
E
kl
ijkl
II
ij
I
ij
ij
II
I
II
X
K
D
Q
Q
Q
Q
Q
Q
3
1
3
3
1
()
()
I
ij
ij
Q
I
Q
Q
Q
H
H
=
0
=
I
R
-
=
K
I
I
3
1
Ultimately, we take for the solution of test:
(
)
0
0
0
0
0
0
,
,
,
,
,
D
I
ij
Iso
ij
X
R
Q
Y
=
0
, with
following values:
0
I
: the value found according to the preceding formulation.
0
D
: the value found according to the preceding formulation.


-
+
+
=
-
-
-
D
kl
D
kl
I
kl
ijkl
ij
ij
G
D
0
0
0
3
1
-
-
+
=
Iso
Q
I
Iso
Iso
G
Q
Q
0
0
-
-
+
=
R
D
G
R
R
0
0
-
-
+
=
X
ij
D
ij
ij
G
X
X
0
0
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Titrate:
Law CJS in géomechanics
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Key
:
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:
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R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/03/005/A
4.5.2 Iterations of Newton
DY
DR.
is given here by:




























=
D
I
kl
Iso
kl
D
I
kl
Iso
kl
D
ij
I
ij
kl
ij
ij
Iso
ij
kl
ij
D
I
kl
Iso
kl
D
I
kl
Iso
kl
D
ij
I
ij
kl
ij
ij
Iso
ij
kl
ij
FD
FD
X
FD
R
FD
Q
FD
FD
FI
FI
X
FI
R
FI
Q
FI
FI
LX
LX
X
LX
R
LX
Q
LX
LX
LR
LR
X
LR
R
LR
Q
LR
LR
LQ
LQ
X
LQ
R
LQ
Q
LQ
LQ
X
R
Q
DY
DR.
where the new terms are null:
0
=
R
LQ
,
0
=
kl
X
LQ
,
0
=
D
LQ
,
0
=
Iso
Q
LR
,
0
=
I
LR
,
0
=
Iso
Q
LX
ij
,
0
=
I
LX
ij
,
0
=
R
FI
,
0
=
kl
X
FI
,
0
=
D
FI
,
0
=
Iso
Q
FD
,
0
=
I
FD
and where the already definite terms remain unchanged, except for
kl
ij
who becomes:
kl
D
Mn
D
ijmn
kl
N
has
init
has
D
Mn
D
Mn
I
Mn
linear
ijmn
jl
ik
kl
G
D
P
Q
I
P
N
G
D
+




+


-
+
-
=
- 1
1
3
3
3
1
ij
4.5.3 test of convergence
The criterion of convergence remains
0
1
+
p
1
+
p
Y
Y
DY
-
RESI_INTE_RELA
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. The aforementioned makes it possible to avoid certain problems of oscillation in
the calculation of the solution
1
+
p
Y
who lead finally to nonthe convergence of integration
numerical.
Thus, with the iteration
1
+
p
, instead of bringing up to date the unknown factor
1
+
p
Y
by a complete increment
1
+
p
Y
1
1
+
+
+
=
p
p
p
Y
Y
Y
one poses
1
1
+
+
+
=
p
m
p
p
m
Y
Y
Y
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Titrate:
Law CJS in géomechanics
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:
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:
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HT-66/03/005/A
and one seeks, by carrying out a loop on under-iterations
m
, to determine an optimal value
scalar
m
. This value is required by considering the rotation of the normal, in the plan
déviatoire, on surface
D
F
, during under-iterations. This normal, noted
m
N
~
, expresses itself with
to leave the stresses contained in the term
1
+
p
m
Y
by
()
()
(
)
ij
II
ij
Q
ij
D
II
Q
m
Q
Q
Q
Q
F
Q
H
+
+
=
=
)
det (
6
3
cos
2
2
~
5
Q
N
Starting from the initial value
0
.
1
0
=
, the process set up consists of the following stages:
·
calculation of the normals
1
~
-
m
N
and
m
N
~
·
calculation of the swing angle
m
between these normals:
m
m
m
m
m
N
N
N
N
~
~
~
:
~
cos
1
1
-
-
=
·
test on the evolution
m
cos
:
if
TOLROT
m
cos
then
m
m
DECREL
=
+1
and
1
+
= m
m
if not end of the under-iterations and
1
1
+
+
=
p
m
p
Y
Y

4.7
Recutting of the pitch 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 pitch of
time in order to facilitate numerical integration. This possibility is managed by the operand
ITER_INTE_PAS of the 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
recutting is positive is automatic, if it is negative recutting is not taken in
count that in the event of nonconvergence with the pitch 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 term
(
)
Q
S
-
cos
The term
(
)
Q
S
-
cos
appears in the expression of
O
. We adopted for his calculation
even method that that used with the ECL. I.e. we determine the angles
S
and
Q
of
the manner which follows:
()
()




=
S
S
S
3
cos
3
cos
Arctan
3
1
2
-
1
and
()
()




=
Q
Q
Q
3
cos
3
cos
Arctan
3
1
2
-
1
then we take the cosine of the difference.
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Titrate:
Law CJS in géomechanics
Date:
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Author (S):
C. CHAVANT, pH. AUBERT
Key
:
R7.01.13-A
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:
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R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/03/005/A
These expressions of
S
and
Q
are also useful for calculation of:
(
)
(
)




-
-
-
=
-
kl
Q
kl
S
Q
S
kl
Q
S
sin
cos
with




-
-
-
=
kl
II
kl
II
S
kl
S
Q
Q
Q
T
Q
2
3
2
)
det (
3
54
)
3
(
cos
1
3
1
4.8.2 Calculation
of
R
R
The radius of rupture introduced into model CJS3 is given by the formula




+
+
=
init
C
C
R
Q
I
p
R
R
1
3
ln
µ
In fact, when
C
init
p
Q
I
>
+
3
1
, one must lock
R
R
with the value of
C
R
. The field of dilatence
disappears and one does not only admit
R
R
can decrease in on this side
C
R
. Consequently, one introduces, with
place preceding formulation, the following expression






+
+
=
init
C
C
R
Q
I
p
R
R
1
3
ln
,
0
max
µ
4.8.3 Traction
Non-cohesive, the field of traction which corresponds to positive stresses is inadmissible
for the grounds. From the point of view of the integration of model CJS, when the state of the stresses tends towards
the node of the cone of the surface of load, the numerical risk to tilt 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 value of
1
I
positive. This value poses then
problem at the time to evaluate certain quantities like
5
.
1
1
3
-
+




+
has
init
P
Q
I
; in addition it
would generate from a theoretical point of view a value
II
Q
negative according to the equation of the surface of
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 stresses in the elastic range on the hydrostatic axis in
posing:
0
1
23
13
12
33
22
11
=
=
=
-
=
=
=
kPa
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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Law CJS in géomechanics
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:
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HT-66/03/005/A
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:
()
kl
linear
ijkl
N
has
init
kl
ijkl
ij
D
P
Q
I
D
&
&
&




+
=
=
3
1
from where immediately the tangent operator:
linear
ijkl
N
has
init
nl
elas
ijkl
D
P
Q
I
H




+
=
3
1
.
5.2
Tangent operator of the mechanisms isotropic rubber band and plastic
In this case, we have the following relation:
()
(
)
()


+
=
-
=
kl
I
kl
ijkl
IP
kl
kl
ijkl
ij
D
D
&
&
&
&
&
3
1
it comes:
(
)
I
v
K
I
&
&
&
+
= 3
1
By taking account of this relation and the law of work hardening of
Iso
Q
, the condition
0
=
I
f&
becomes:
(
)
0
3
1
=
-
+
-
=
+
-
=
I
p
I
v
Iso
I
K
K
Q
I
F
&
&
&
&
&
&
that is to say:
v
p
I
K
K
K
&
&
+
-
=
By deferring this result in the expression of
ij
&
, one finds:
kl
kl
Mn
ijmn
p
ijkl
kl
mm
p
kl
ijkl
ij
D
K
K
K
D
K
K
K
D
&
&
&
&


+
-
=


+
-
=
3
1
3
1
from where the tangent operator:
kl
Mn
ijmn
p
ijkl
IP
ijkl
D
K
K
K
D
H
+
-
=
3
1
background image
Code_Aster
®
Version
6.4
Titrate:
Law CJS in géomechanics
Date:
18/11/03
Author (S):
C. CHAVANT, pH. AUBERT
Key
:
R7.01.13-A
Page
:
33/48
Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/03/005/A
One can also write in matric form:














+
-
-
-
-
+
-
-
-
-
+
-




=
µ
µ
µ
µ
µ
µ
2
0
0
0
0
0
0
2
0
0
0
0
0
0
2
0
0
0
0
0
0
2
0
0
0
2
0
0
0
2
3
1
N
has
IP
P
I
H
where for this formula only
and
µ
are the coefficient of Lamé and
p
O
E
O
E
O
K
K
K
+
=
2
.
5.3
Tangent of the mechanisms rubber band and plastic operator déviatoire
The condition
0
=
D
f&
is written:
0
=
+
+
=
+
+
=
X
ij
D
ij
D
R
D
D
ij
ij
D
ij
ij
D
D
ij
ij
D
D
G
X
F
G
R
F
F
X
X
F
R
R
F
F
F
&
&
&
&
&
&
&
The tensor
X
G
being purely déviatoire, the product
X
ij
ij
D
G
X
F
is reduced to:
X
ij
ij
X
ij
ij
D
X
ij
ij
D
G
Q
I
G
X
F
Dev.
G
X
F
1
-
=




=
The plastic multiplier can thus be put in the form:
ij
ij
D
Dev.
D
F
H
&
&
1
=
while revealing the plastic module
Dev.
H
, given by:
(
)




+
+




-




+
=
-
ij
ij
ij
m
has
init
Dev.
X
Q
Q
B
R
R
With
P
Q
I
I
H
1
1
3
2
5
,
1
1
2
1
The relation stress-strains then makes it possible to write:
(
)
D
kl
ijkl
ij
D
D
kl
ijkl
ij
D
D
kl
D
kl
ijkl
ij
D
ij
ij
D
G
D
F
D
F
G
D
F
F
&
&
&
&
&
-
=
-
=
what gives finally for the plastic multiplier:
D
kl
ijkl
ij
D
Dev.
kl
ijkl
ij
D
D
G
D
F
H
D
F
+
=
&
&
By deferring this result in the expression of
ij
&
, one finds:








+
-
=
D
kl
D
you
rstu
rs
D
Dev.
Mn
pqmn
pq
D
kl
ijkl
ij
G
G
D
F
H
D
F
D
&
&
&
background image
Code_Aster
®
Version
6.4
Titrate:
Law CJS in géomechanics
Date:
18/11/03
Author (S):
C. CHAVANT, pH. AUBERT
Key
:
R7.01.13-A
Page
:
34/48
Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/03/005/A
from where the tangent operator:
D
you
rstu
rs
D
Dev.
pqkl
pq
D
D
Mn
ijmn
ijkl
dp
ijkl
G
D
F
H
D
F
G
D
D
H
+
-
=
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
ijkl
H
but
dp
ijkl
H~
who is given by:
2
~
dp
klij
dp
ijkl
dp
ijkl
H
H
H
+
=
with
ij
and
kl
taken in
(
)
23
,
13
,
12
,
33
,
22
,
11
5.4
Tangent operator of the mechanisms rubber band, plastics isotropic and
déviatoire
One must satisfy the two following conditions:
0
=
I
f&
and
0
=
D
f&
. Taking into account the relation
stress-strains which is written:


-
+
=
D
kl
D
kl
I
kl
ijkl
ij
G
D
&
&
&
&
3
1
the first condition gives:
(
)
0
=
-
-
+
-
=
I
p
D
v
D
I
v
I
K
G
K
F
&
&
&
&
&
where one posed
()
D
D
kk
D
v
tr
G
G
G
=
=
.
The second condition led to:
0
3
1
=
-
-
+
D
Dev.
D
kl
ijkl
ij
D
D
kl
ijkl
ij
D
I
kl
ijkl
ij
D
H
G
D
F
D
F
D
F
&
&
&
&
Thus, plastic multipliers
I
&
and
D
&
are obtained by solving the system:
(
)


=




+
+
-
=
+
+
-
kl
ijkl
ij
D
D
Dev.
D
kl
ijkl
ij
D
I
kl
ijkl
ij
D
v
D
D
v
I
p
D
F
H
G
D
F
D
F
K
G
K
K
K
&
&
&
&
&
&
3
1
that is to say:
(
)
xy
wvxy
VW
D
D
v
Dev.
D
you
rstu
rs
D
p
v
Dev.
D
pq
mnpq
Mn
D
kl
ijkl
ij
D
D
v
I
D
F
G
K
H
G
D
F
K
K
K
H
G
D
F
D
F
G
K
3
1
-




+
+




+
-
=
&
&
&
(
)
(
)
xy
vwxy
VW
D
D
v
Dev.
D
you
rstu
rs
D
p
v
pq
mnpq
Mn
D
kl
ijkl
ij
D
p
D
D
F
G
K
H
G
D
F
K
K
D
F
K
D
F
K
K
3
1
3
1
-




+
+
-
+
=
&
&
&
background image
Code_Aster
®
Version
6.4
Titrate:
Law CJS in géomechanics
Date:
18/11/03
Author (S):
C. CHAVANT, pH. AUBERT
Key
:
R7.01.13-A
Page
:
35/48
Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/03/005/A
These expressions are still written:
kl
kl
I
T
&
&
1
=
and
kl
kl
D
T
&
&
2
=
where tensors
1
T
and
2
T
are given by:
(
)
xy
vwxy
VW
D
D
v
Dev.
D
you
rstu
rs
D
p
kl
Dev.
D
pq
mnpq
Mn
D
ijkl
ij
D
D
v
kl
D
F
G
K
H
G
D
F
K
K
K
H
G
D
F
D
F
G
K
T
3
1
1
-




+
+




+
-
=
(
)
(
)
xy
vwxy
VW
D
D
v
Dev.
D
you
rstu
rs
D
p
kl
pq
mnpq
Mn
D
ijkl
ij
D
p
kl
D
F
G
K
H
G
D
F
K
K
D
F
K
D
F
K
K
T
3
1
3
1
2
-




+
+
-
+
=
By deferring the expressions
I
&
and
D
&
of in the formula of
ij
&
, one finds:


-
+
=
D
kl
pq
pq
kl
Nm
Nm
kl
ijkl
ij
G
T
T
D
&
&
&
&
2
1
3
1
from where the tangent operator:
kl
D
pq
ijpq
kl
Mn
ijmn
ijkl
idp
ijkl
T
G
D
T
D
D
H
2
1
3
1
-
+
=
This tangent operator not being symmetrical, we retain not
idp
ijkl
H
but
idp
ijkl
H~
who is given
by:
2
~
idp
klij
idp
ijkl
idp
ijkl
H
H
H
+
=
with
ij
and
kl
taken in
(
)
23
,
13
,
12
,
33
,
22
,
11
background image
Code_Aster
®
Version
6.4
Titrate:
Law CJS in géomechanics
Date:
18/11/03
Author (S):
C. CHAVANT, pH. AUBERT
Key
:
R7.01.13-A
Page
:
36/48
Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/03/005/A
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.
background image
Code_Aster
®
Version
6.4
Titrate:
Law CJS in géomechanics
Date:
18/11/03
Author (S):
C. CHAVANT, pH. AUBERT
Key
:
R7.01.13-A
Page
:
37/48
Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/03/005/A
6.2
Top-level flowchart of the main routines
Main routines FORTRAN for the integration of law CJS are connected in the following way:
nmcjs.f
nmcomp.f
cjsmat.f
cjsela.f
cjssmi.f
cjspla.f
cjsmis.f
cjsmde.f
cjsmid.f
cjssmd.f
cjstel.f
cjstis.f
cjstid.f
cjstde.f
background image
Code_Aster
®
Version
6.4
Titrate:
Law CJS in géomechanics
Date:
18/11/03
Author (S):
C. CHAVANT, pH. AUBERT
Key
:
R7.01.13-A
Page
:
38/48
Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/03/005/A
6.3
Details of the functionalities of developed routines FORTRAN
6.3.1 Routine
:
CJSC3Q
Objective: calculation of
()
Q
3
cos
Variables of input and exit
:
IN
SIG: STRESSES
X: VARIABLES HAMMER-HARDENED MOVIES
AP: 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:
resolution of the equation
()
0
3
3
1
1
1
=




-
-
+
-
+
tr
P
I
K
I
I
N
has
E
O
by the method of the secant,
for the nonlinear elastic behavior
Variables of input and exit
:
IN
CRIT: CRITERIA OF CONVERGENCE
MATER: COEFFICIENTS MATERIAL A T+DT
LIFO: INCREMENT OF DEFORMATION
SIGD: STRESS A T
OUT
I1: TRACE SIG A T+DT
LEAFLET: 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 exit
:
IN
MOD: MODELING
Q: TENSOR (6 COMPONENTS)
OUT
DTDDQ: TENSOR RESULT (6 COMPONENTS)

6.3.4 Routine
:
CJSELA
Objective:
nonlinear elastic design of the stresses
Variables of input and exit
:
IN
MOD: MODELING
CRIT: CRITERIA OF CONVERGENCE
MATERF: COEFFICIENTS MATERIAL A T+DT
SIGD: STRESS A T
LIFO: INCREMENT OF DEFORMATION
OUT
SIGF: STRESS A T+DT
background image
Code_Aster
®
Version
6.4
Titrate:
Law CJS in géomechanics
Date:
18/11/03
Author (S):
C. CHAVANT, pH. AUBERT
Key
:
R7.01.13-A
Page
:
39/48
Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/03/005/A
Organization of CJSELA
·
calculation of the first invariant of the stresses
I1
with t+dt:
- call of CJSCI1
·
calculation of the coefficients of the elastic matrix and assembly of the matrix
·
calculation of the increment of the stresses and the stresses 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 exit
:
IN
MOD: MODELING
MATER: COEFFICIENTS MATERIAL A T+DT
EPSD: DEFORMATION A T+DT
LIFO: 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
,
·
calculation of the threshold
D
F
, of its derivative
D
D
F
and of the plastic multiplier
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 exit
:
IN
MOD: MODELING
MATER: COEFFICIENTS MATERIAL A T+DT
EPSD: DEFORMATION A T+DT
LIFO: 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
background image
Code_Aster
®
Version
6.4
Titrate:
Law CJS in géomechanics
Date:
18/11/03
Author (S):
C. CHAVANT, pH. AUBERT
Key
:
R7.01.13-A
Page
:
40/48
Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/03/005/A
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
,
·
calculation of the thresholds
I
F
and
D
F
, of their derivative
I
I
F
,
D
I
F
,
I
D
F
and
D
D
F
, and of
plastic multipliers
I
and
D
,
·
calculation of the solution of test

6.3.7 Routine
:
CJSJDE
Objective:
calculation of
DRDY
and
R
for the resolution of
()
()
p
1
+
p
p
- Y
R
DY
Y
DY
DR.
=
(mechanism
plastic déviatoire)
Variables of input and exit
:
IN
MOD: MODELING
MATER: COEFFICIENTS MATERIAL A T+DT
EPSD: DEFORMATION A T
LIFO: 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
SIGN: 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 intermediates
·
calculation of the terms
ij
R
G
,
R
G
R
,
ij
X
Mn
G
,
ij
X
Mn
X
G
,
ij
D
Mn
G
,
R
G
D
Mn
,
ij
D
Mn
X
G
·
calculation of the components of
DRDY
and
R
·
assembly of
DRDY
and
R
background image
Code_Aster
®
Version
6.4
Titrate:
Law CJS in géomechanics
Date:
18/11/03
Author (S):
C. CHAVANT, pH. AUBERT
Key
:
R7.01.13-A
Page
:
41/48
Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/03/005/A
6.3.8 Routine
:
CJSJID
Objective:
calculation of
DRDY
and
R
for the resolution of
()
()
p
1
+
p
p
- Y
R
DY
Y
DY
DR.
=
(mechanisms
plastics isotropic and déviatoire)
Variables of input and exit
:
IN
MOD: MODELING
MATER: COEFFICIENTS MATERIAL A T+DT
EPSD: DEFORMATION A T
LIFO: 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
SIGN: 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 intermediates
·
calculation of the terms
Iso
Q
Q
G
Iso
,
ij
R
G
,
R
G
R
,
ij
X
Mn
G
,
ij
X
Mn
X
G
,
ij
D
Mn
G
,
R
G
D
Mn
,
ij
D
Mn
X
G
·
calculation of the components of
DRDY
and
R
·
assembly of
DRDY
and
R

6.3.9 Routine
:
CJSJIS
Objective:
calculation of
DRDY
and
R
for the resolution of
()
()
p
1
+
p
p
- Y
R
DY
Y
DY
DR.
=
(mechanism
isotropic plastic)

Variables of input and exit
:
IN
MOD: MODELING
MATER: COEFFICIENTS MATERIAL A T+DT
LIFO: 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
background image
Code_Aster
®
Version
6.4
Titrate:
Law CJS in géomechanics
Date:
18/11/03
Author (S):
C. CHAVANT, pH. AUBERT
Key
:
R7.01.13-A
Page
:
42/48
Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/03/005/A
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 exit
:
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,
·
reconnaissance of level CJS chosen according to the parameters given.

6.3.11 Routine: CJSMDE
Objective:
elastoplastic calculation of the stresses with the plastic mechanism deviatoire activated:
resolution by the method of Newton of
()
0
=
Y
R
Variables of input and exit
:
IN
MOD: MODELING
CRIT: CRITERIA OF CONVERGENCE
MATER: COEFFICIENTS MATERIAL A T+DT
NVI: NB INTERNAL VARIABLES
EPSD: DEFORMATIONS A T
LIFO: INCREMENT OF DEFORMATION
SIGD: STRESS A T
VIND: INTERNAL VARIABLES A T
STOPNC: STOP IN THE EVENT OF NOT CONVERGENCE
VAR
SIGF: STRESS 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
DY
YD
YF
+
=
- 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 stresses and internal variables
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Code_Aster
®
Version
6.4
Titrate:
Law CJS in géomechanics
Date:
18/11/03
Author (S):
C. CHAVANT, pH. AUBERT
Key
:
R7.01.13-A
Page
:
43/48
Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/03/005/A
6.3.12 Routine: CJSMID
Objective:
elastoplastic calculation of the stresses with the plastic mechanisms isotropic and deviatoire
activated: resolution by the method of Newton of
()
0
=
Y
R
Variables of input and exit
:
IN
MOD: MODELING
CRIT: CRITERIA OF CONVERGENCE
MATER: COEFFICIENTS MATERIAL A T+DT
NVI: NB INTERNAL VARIABLES
EPSD: DEFORMATIONS A T
LIFO: INCREMENT OF DEFORMATION
SIGD: STRESS A T
VIND: INTERNAL VARIABLES A T
STOPNC: STOP IN THE EVENT OF NOT CONVERGENCE
VAR
SIGF: STRESS 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
DY
YD
YF
+
=
- 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 stresses and internal variables
6.3.13 Routine: CJSMIS
Objective:
elastoplastic calculation of the stresses with the activated isotropic plastic mechanism:
resolution by the method of Newton of
()
0
=
Y
R
Variables of input and exit
:
IN
MOD: MODELING
CRIT: CRITERIA OF CONVERGENCE
MATER: COEFFICIENTS MATERIAL A T+DT
LIFO: INCREMENT OF DEFORMATION
SIGD: STRESS A T
VIND: INTERNAL VARIABLES A T
STOPNC: STOP IN THE EVENT OF NOT CONVERGENCE
VAR
SIGF: STRESS 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
DY
YD
YF
+
=
- 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 stresses and internal variables
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Code_Aster
®
Version
6.4
Titrate:
Law CJS in géomechanics
Date:
18/11/03
Author (S):
C. CHAVANT, pH. AUBERT
Key
:
R7.01.13-A
Page
:
44/48
Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/03/005/A
6.3.14 Routine: CJSNOR
Objective:
calculation of a vector parallel with
ij
D
Q
F
Variables of input and exit
:
IN
MATER: MATERIAL
SIG:
STRESSES
X:
VARIABLES INTERN KINEMATICS
OUT
NOR:
ESTIMATE OF THE DITRECTION OF THE NORMAL
ON SURFACE DEVIATOIRE IN PLAN DEVIATOIRE
PERPENDICULAR
With
TRISECTING
THE NOR VECTOR (1:NDT) NR IS NOT STANDARD
SA
NORMALIZES
EAST
NOR (NDT+1)
6.3.15 Routine: CJSPLA
Objective:
elastoplastic calculation of the stresses.
Variables of input and exit
:
IN
MOD: MODELING
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
LIFO: INCREMENT OF DEFORMATION
SIGD: STRESS A T
VIND: INTERNAL VARIABLES A T
VAR
SIGF: STRESS 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 the thresholds
I
F
and
D
F
calculated starting from the elastic prediction,
·
processing of the possible recutting of the pitch 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 stresses 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
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Code_Aster
®
Version
6.4
Titrate:
Law CJS in géomechanics
Date:
18/11/03
Author (S):
C. CHAVANT, pH. AUBERT
Key
:
R7.01.13-A
Page
:
45/48
Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/03/005/A
6.3.16 Routine: CJSQCO
Objective:
utility routine of CJS allowing the calculation of standard sizes listed below
Variables of input and exit
:
IN
GAMMA: PARAMETER MATERIAL
SIG: STRESSES
X: VARIABLES HAMMER-HARDENED MOVIES
PREF: CLOSE REF. FOR STANDARDIZATION
EPSSIG: DEVIATIVE EPSILON FOR NULLITY
I1: TRACE TENSOR OF THE STRESSES
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
ij
Q
Variables of input and exit:
IN
NR: DIMENSION OF S, X, Q
S: DIVERTER
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 exit
:
IN
SIG: STRESS
WINE: 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 exit
:
IN
SIG: STRESS
WINE: INTERNAL VARIABLES
OUT
SEUILI: THRESHOLD ELASTICITY OF THE ISOTROPIC MECHANISM
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Code_Aster
®
Version
6.4
Titrate:
Law CJS in géomechanics
Date:
18/11/03
Author (S):
C. CHAVANT, pH. AUBERT
Key
:
R7.01.13-A
Page
:
46/48
Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/03/005/A
6.3.20 Routine: CJST
Objective:
calculation of
T
=
S
S
det
.
Variables of input and exit
:
IN
S: STAMP
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 exit
:
IN
MOD: MODELING
MATER: COEFFICIENTS MATERIAL
NVI: NB INTERNAL VARIABLES
EPS: DEFORMATIONS
SIG: STRESSES
WINE: 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 exit
:
IN
MOD: MODELING
MATER: COEFFICIENTS MATERIAL
SIG: STRESSES
OUT
HOOK: ELASTIC OPERATOR RIGIDITY
Organization of CJSTEL
·
calculation of the elastic operator
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Code_Aster
®
Version
6.4
Titrate:
Law CJS in géomechanics
Date:
18/11/03
Author (S):
C. CHAVANT, pH. AUBERT
Key
:
R7.01.13-A
Page
:
47/48
Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/03/005/A
6.3.23 Routine: CJSTID
Objective:
calculation of the tangent matrix for the plastic mechanisms isotropic and déviatoire
Variables of input and exit
:
IN
MOD: MODELING
MATER: COEFFICIENTS MATERIAL
NVI: NB INTERNAL VARIABLES
EPS: DEFORMATIONS
SIG: STRESSES
WINE: 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 exit
:
IN
MOD: MODELING
MATER: COEFFICIENTS MATERIAL
SIG: STRESSES
WINE: INTERNAL VARIABLES
OUT
DSDE: STAMP TANGENT
Organization of CJSTEL
·
calculation of the tangent matrix

6.3.25 Routine: LCDETE
Objective:
calculation of a matrix determining 3
×3
Variables of input and exit
:
IN
A: STAMPS
OUT
LCDETE: DETERMINANT
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Code_Aster
®
Version
6.4
Titrate:
Law CJS in géomechanics
Date:
18/11/03
Author (S):
C. CHAVANT, pH. AUBERT
Key
:
R7.01.13-A
Page
:
48/48
Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/03/005/A
6.3.26 Routine: NMCJS
Objective:
realization of the integration of law CJS: calculation of the stresses with t+dt and/or the matrix
tangent, according to the selected option of calculation.
Variables of input and exit
:
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 stresses 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, “hierarchical Modeling of the behavior of
grounds ", to appear.
[2]
B. CAMBOU, K. JAFARI, “Models behavior of the non-cohesive soils”, Frank rev.
Géotech. N
°44, p. p 43-55, 1988.
[3]
K. ELAMRANI, “Contribution to the validation of model CJS for granular materials”,
Thesis of Doctorate of the Central School of Lyon, 1992.