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Titrate:
Law of behavior of Laigle
Date:
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Author (S):
R. FERNANDES, C. CHAVANT
Key:
R7.01.15-A
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Organization (S):
EDF-R & D/AMA















Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
Document: R7.01.15



Law of behavior of Laigle




Summary:

The rheological model of Laigle makes it possible to analyze the rock mechanics behavior.
development of this model of behavior was initiated following the difficulty in apprehending correctly
response of the solid mass during the excavation of an underground cavity, with an aim:
· to define the need and the nature of possible supportings to implement;
· to determine the extent of the ground around a work influenced by the digging.

The implementation of this elastoplastic model was mainly focused on the simulation of
behavior post-peak of the rock. It is supposed, accordingly, that there is no work hardening of the rock
before the rupture of the aforementioned. That results in a linear elastic behavior to the peak of
resistance (it can nevertheless y have damage of the rock whereas the material is not yet in
rupture). The definite criterion of plasticity is of type generalized Hoek and Brown and accounts for the influence of
level of stress on the shear strength. Radoucissement of material is associated one
progressive reduction in the properties of cohesion and angle of friction accompanied by a change by
volume. It is controlled by the plastic deformation déviatoire cumulated considered as only variable
of work hardening.

To facilitate the integration of this model in Code_Aster, the law initially developed in the formalism
main stresses was rewritten with invariants of stresses on a basis of the model
Cambou-Jafari-Sidoroff (CJS). The numerical formulation is implicit compared to the criterion and explicit by
report/ratio with the direction of flow.

The convention of sign used for the formulation of the equations, within the framework of this note, is that of
mechanics of the continuous mediums.
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Count
matters
1
Notations ................................................................................................................................................ 4
1.1
General information ...................................................................................................................................... 4
1.2
Parameters of the model .................................................................................................................... 5
2
Introduction ............................................................................................................................................ 6
2.1
Phenomenology of the behavior of the grounds .................................................................................. 6
2.2
Context of study and simplifying assumptions of the model ........................................................... 7
3
The continuous model .................................................................................................................................. 8
3.1
Elastic behavior ................................................................................................................. 8
3.2
Criterion of plasticity .......................................................................................................................... 8
3.2.1
Surface of load .................................................................................................................. 8
3.2.1.1
Expression of the criterion of Laigle in major and minor stresses ...................... 8
3.2.1.2
General expression .................................................................................................. 8
3.2.1.3
Pace of the thresholds ......................................................................................................... 9
3.2.2
Work hardening ............................................................................................................................ 9
3.2.3
Law of dilatancy .................................................................................................................... 10
3.2.3.1
Generalized writing ................................................................................................. 10
3.2.3.2
Determination of the intersection of the intermediate criterion and the ultimate criterion ............ 12
3.2.4
Plastic flow ........................................................................................................... 12
4
Calculation of derived the ............................................................................................................................. 14
4.1
Derived from the criterion .......................................................................................................................... 14
4.1.1
Derived compared to the stresses ..................................................................................... 14
4.1.1.1
Derived intermediate compared to the diverter ....................................................... 14
4.1.1.2
Derived intermediate compared to the stresses ................................................... 14
4.1.1.3
Final expression of derived from the criterion compared to the stresses ................... 15
4.1.2
Derived compared to the variable from work hardening .................................................................. 15
4.2
Total derivative of the criterion compared to the plastic multiplier ................................................... 15
4.3
Derived from the parameters compared to the variable of work hardening ................................................ 16
5
Tangent operator of speed .............................................................................................................. 17
6
Digital processing adapted to the nonregular models .................................................................. 18
6.1
Projection at the top of the cone ................................................................................................. 18
6.1.1
Definition of the jetting angle ........................................................................................ 18
6.1.2
Existence of projection .................................................................................................... 19
6.1.3
Rules of projection ............................................................................................................. 23
6.1.3.1
Case where the parameter of dilatancy is negative .......................................................... 23
6.1.3.2
Case where the parameter of dilatancy is positive ............................................................ 23
6.1.3.3
Graphic interpretation ........................................................................................... 23
6.1.3.4
Equations of flow ........................................................................................... 24
6.2
Local Recutting of the pitch of time ............................................................................................ 24
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7
The variables intern .......................................................................................................................... 25
7.1
V1: the plastic deformation déviatoire cumulated ......................................................................... 25
7.2
V2: the cumulated plastic voluminal deformation ........................................................................ 25
7.3
V3: fields of behavior of the rock .......................................................................... 25
7.4
V4: the state of plasticization .............................................................................................................. 26
8
Detailed presentation of the algorithm ................................................................................................. 26
8.1
Calculation of the elastic solution ....................................................................................................... 26
8.2
Calculation of the elastic criterion ............................................................................................................. 26
8.3
Algorithm ..................................................................................................................................... 27
9
Alternative on the expression of the criterion of plasticity ................................................................................. 29
9.1
General formulation .................................................................................................................... 29
9.2
Pace of the thresholds ............................................................................................................................ 30
10
Bibliography .................................................................................................................................. 31
Appendix 1
Retiming of the criterion on the triaxial one in compression .................................................. 32
Appendix 2
Standardization of Q .................................................................................................. 33
Appendix 3
Framing of the jetting angle ..................................................................... 34
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1 Notations
1.1 General
indicate the tensor of the effective stresses in small disturbances, noted in the form of
following vector:














23
13
12
33
22
11
2
2
2
One notes:
()
tr
I
=
1
first invariant of the stresses
I
S
3
1
I
-
=
tensor of the stresses déviatoires
S
S.
=
II
S
second invariant of the tensor of the stresses déviatoires
1
major main stress
3
minor main stress
()
I
E
3
Tr
-
=
diverter of the deformations
()
Tr
v
=
voluminal deformation
()
()
3
2
/
3
2
/
1
det
3
2
3
cos
II
S
S
=
being the angle of Lode
p
ij
p
ij
p
E
E
3
2
=

cumulated plastic deviatoric deformations
N
normal of the hypersurface of deformation
G
function controlling the evolution of the plastic deformations and describing
direction of flow
()
I
G
G
G
3
~
Tr
-
=
diverter of
G
()
G
Tr
G
=
trace
G
G
G ~
.
~
~ =
II
G
normalizes
G
~
angle of dilatancy
angle of friction
F
surface of load
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1.2
Parameters of the model
Notation Description
m
Slope of the criterion in the plan
(
)
Q
p
, '
for the very strong stresses (function of
mineralogical nature of the rock)
S
Cohesion of the medium. Representative of the damage of the rock.
has
Characterization of the dishing of the criterion, function of the level of deterioration of the rock. It
the influence of the component of dilatancy in the behavior defines in large
deformations.
ult
Plastic deformation déviatoire corresponding to the ultimate criterion
E
Plastic deformation déviatoire corresponding to disappearance supplements cohesion
ult
m
Value
of
m
ultimate criterion reached in
ult
E
m
Value
of
m
intermediate criterion reached in
E
E
has
Value
of
has
intermediate criterion reached in
E
peak
m
Value of
m
criterion of peak reached with the peak of stress
peak
has
Value of
has
criterion of peak reached with the peak of stress
Exponent controlling work hardening
C
Compressive strength simple
First parameter regulating dilatancy
Second parameter regulating dilatancy
cjs
Parameter of form of the criterion of plasticity in the déviatoire plan
E
Young modulus
Poisson's ratio
1
p
Intersection of the intermediate criterion and the criterion of peak
2
p
Intersection of the intermediate criterion and the ultimate criterion
AP
Atmospheric pressure
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2 Introduction
The object of this note is to present the rheological model to analyze the behavior
rock mechanics, adapted to the simulation of the underground works, introduced into Code_Aster
and developed by the CIH [bib1]. The finality of this model is to be able to be implemented, in manner
rapid and industrial in order to answer the main interrogations that is posed the engineer at the time of
analysis and of the design of an underground cavity. The rheological law must for that remain
relatively simple, as well during the identification of the parameters as in its implementation and
during the interpretation of the results.
2.1
Phenomenology of the behavior of the grounds
One of the characteristics of a rock, compared to a ground, is that its mechanical behavior is, on
a range of important stress, controlled by cohesion. This cohesion is associated one
cementing of the medium, induced during the geological history of the solid mass, and is primarily of
epitaxic nature. On the contrary, the resistance of a ground is more particularly governed by the term of
friction and/or of dilatancy. Cohesion, of primarily capillary origin, does not have an influence then
that for very weak states of stresses of containment.
This distinction between a ground and a rock is important because it directs the choice and the assumptions of
base model of behavior.
The main rheological phenomena associated this context are as follows:
· In the field of the small deformations, the response of a rock, in particular under
weak states of containment, can be comparable with a linear elastic behavior,
slightly depend on the state of the stresses. Non-linearities of the behavior are
likely to appear the peak of resistance before, in the case of tender rocks,
for a level of stress of about 70 to 80% of the maximum value. This threshold decreases
with the increase in the average pressure for almost cancelling itself when the stress
of surconsolidation is reached (course-model). Under very low stresses of containment
representative of those reigning near the underground works, these non-linearities
are generally weak, more especially as cementing is important, and thus the level
of surconsolidation of the rock high.
· Dilatancy (increase in volume) is initiated when non-linearities appear on
stress-strain curve. This dilatancy increases until there is localization with
center of the sample. At this time, the rate of dilatancy (or the angle of dilatancy
) is
maximum, for then gradually decreasing and cancelling themselves with the very great deformations.
· The peak of resistance is reached for stresses describing a criterion of rupture,
generally curve in the plan of Mohr or the plan of the main stresses
major and minor. The assumption of a linear criterion of Mohr-Coulomb is thus only one
simplifying assumption, having tendency, for low stresses of containment, with
to raise the cohesion of the medium.
· Once maximum resistance reached, the resistance of the rock decreases. It
radoucissement post-peak is all the more fast and important (in intensity) the stress
of containment is weak. This decrease is related to a damage more or less
located rock, according to the level of containment. Whatever this stress,
beyond the peak, the rock cannot be regarded any more as continuous. Its behavior
is then controlled by the conditions of deformation and strength to the level of the area of
localization of the deformations.
· The appearance of one or more discontinuities kinematics within the rock is associated
a loss of cohesion. The behavior post-peak is then governed by the conditions of
friction and of dilatancy along the plans of discontinuity or within a tape of
localization of the deformations. It comes out from this reasoning that for the very large ones
deformations, the behavior of the comparable rock to a “structure”, is only
rubbing, and is characterized by an ultimate angle of friction
. This angle is a data
intrinsic of material, function of minerals constitutive of the rock. It thus does not depend
directly conditions of cohesion, and it can especially be regarded as independent
dimensions of the sample.
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· When the behavior only becomes rubbing, it is associated no deformation
voluminal. Dilatancy was thus cancelled, and does not exist any more with the great deformations.
· The evolution enters the resistance of peak and the state criticizes corresponding to large
deformations, is more or less progressive according to the state of the pressures applied.
For a state of null containment (simple compression), the behavior is only controlled
by cohesion, and the rupture results in an immediate and brutal loss of any resistance.
Radoucissement will be more progressive as the stress of containment
will increase, to become non-existent beyond of a certain stress of limiting containment
ductile and fragile fields of behavior.
2.2
Context of study and simplifying assumptions of the model
The will to develop a model easy to implement is necessarily accompanied by
simplifications, resulting from a compromise enters the awaited objectives, the conditions of use of
model (quality of the data input, times and cost available…) and means implemented for
to ensure these developments. These compromises are primarily the following:
· A linear elastic behavior to the peak of resistance. This amounts supposing
that there is no work hardening of the rock before the rupture of the aforementioned.
· Only a criterion of rupture in shearing is retained. This means that if the rock is
crushed in an isotropic way, the behavior remains elastic, and that there is not
damage and work hardening of material under this type of path. During the phases
of excavation of an underground work with implementation of a light supporting,
average pressure in the solid mass located in the vicinity can only decrease (or remain constant
in the ideal case of a circular cavity subjected to an isotropic stress, for one
linear elastic behavior). Plasticization under isotropic stress, which one can
to find on a Heading-Model or a law of the Camwood-Clay type did not seem to us
essential taking into account the sought objectives, and in the case of a stress
isotherm and short-term.
During the development of this model, we voluntarily focused ourselves on the study and
simulation of the behavior post-peak of the rock. In this field of behavior, the resistance of
material is supposed to be controlled, according to the state of the stresses and the level of damage
rock, by cohesion, dilatancy or friction.
Cohesion defines the resistance of material as long as the aforementioned remains continuous. It is active until
peak of resistance, and has only little influence on the radoucissant behavior, unless
cohesion is representative of a ductile “adhesive” (case of the grounds injected by silicate gel,…).
As cohesion worsens by damage, dilatancy increases, for
to reach its maximum value at the time of the loss of continuity of the medium. At this time, under the effect of
shearing of induced discontinuity, this dilatancy is degraded gradually and slowly.
rheology of the rock evolves then to a behavior purely rubbing.
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3
The continuous model
3.1 Behavior
rubber band
The elastic behavior is controlled by a linear law, with a constant module independent of
the state of stresses. The 2 parameters characterizing this behavior are the modulus of elasticity E and
the Poisson's ratio
.
(
)
p
2
E
E
S
&
&
&
-
=
µ
éq 3.1-1
(
)
p
v
v
1
K
3
I
-
=
&
&
&
éq 3.1-2
3.2
Criterion of plasticity
The adopted formulation is that of [bib2].
3.2.1 Surface of load

3.2.1.1 Expression of the criterion of Laigle in major and minor stresses
()
(
)
()
()
()
()
(
)
()








+
-
-
-






=
p
C
p
has
C
has
has
C
S
m
F
p
p
p
3
1
1
3
1
1
3
2
éq
3.2.1.1-1

3.2.1.2 Expression
general
One transforms the preceding expression according to the first invariant and of the diverter of
stresses, by a retiming of the criterion on triaxial in compression, to obtain:
()
()
()
0
,
1
0
-




=
p
has
C
C
U
H
G
F
p
S
éq
3.2.1.2-1
with:
()
()
(
)
()
6
/
1
3
6
/
1
det
54
1
3
cos
1




+
=
+
=
II
cjs
cjs
S
H
S
éq 3.2.1.2-2
(
)
6
/
1
0
1
3
cjs
C
H
H
-
=


=
=
(
)
6
/
1
0
1
cjs
T
H
+
=
()
()
H
S
G
II
=
S
éq
3.2.1.2-3
()
() ()
()
() ()
() ()
p
p
C
p
p
C
C
p
p
p
K
S
I
K
m
H
G
K
m
U
+
-
-
=
1
0
3
6
,
S
éq 3.2.1.2-4
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Note:
· One shows [Appendix 1] the equivalence of the two expressions
· One shows that a second formulation of the criterion with a retiming on triaxial in
compression and in extension is possible but we do not choose it. It is
however presented at the chapter [§9].


3.2.1.3 Pace of the thresholds
One traces the pace of the thresholds to the criterion of peak and the ultimate criterion.
1
S
2
S
Threshold with the peak
Ultimate threshold
3.2.2 Work hardening
To translate radoucissement post-peak of the rock laws of variations of the parameters are defined
m, S and have criterion according to the internal variable of work hardening
p
(it is about the deformation
déviatoire plastic cumulated, proportional to the second invariant of the tensor of the deformations
déviatoires, corresponding to the plastic distortion).
()
()




=
<




-
=
E
p
p
E
p
E
p
p
S
S
if
if
0
1
éq
3.2.2-1
If
(
)
3
10
1
-
-
>
ult
p
# one chooses to take an epsilon of
3
10
-
to avoid the errors
# numerical during division by
ult
in the equation [éq 3.2.2-2]
1
=
has
ult
m
m
=
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If not
()
p
ult
E
ult
E
peak
E
E
p
p
has
has
has
-
-
-
-




=
1
éq
3.2.2-2
()
()
()
p
p
peak
p
1
has
has
+
+
=
éq 3.2.2-3
()
()
()
()
()














=
<




-




+
=
E
p
has
has
C
p
E
p
C
p
E
p
p
has
has
C
p
peak
p
C
p
p
E
p
peak
m
m
S
m
m
if
if
2
2
1
1
1
éq
3.2.2-4
()
()
p
has
p
K


=
2
1
3
2
.
éq 3.2.2-5
These laws of evolutions for each of the 3 parameters are dependant from/to each other and
observe the conditions of intersection of the criteria during the phase of work hardening [bib1].
Note:
The condition of coherence to respect gate on the continuity of the parameter
m
in
E
:
()
()
()




-




+
=
p
has
has
C
p
peak
p
C
p
S
m
m
p
peak
E
p
1
lim
1
1
that is to say:
E
peak
has
has
C
p
peak
p
C
E
m
m




+
=
1
1
1
éq
3.2.2-6

3.2.3 Law of dilatancy

3.2.3.1 Writing
generalized
The law of dilatancy (one admits that the value of dilatancy is inversely proportional to that of
cohesion) can be generalized while writing:
()
(
)
1
'
1
'
'
sin
sin
+
+
-
-
=
=
ult
ult
m
m
éq
3.2.3.1-1
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with:
()
(
)
0
3
0
1
0
1
~
~
,
,
'
'
T
T
T
G
I
-
-
=
=
S
éq
3.2.3.1-2
);
3
2
cos (
3
2
);
3
2
cos (
3
2
);
cos (
3
2
3
2
1
-
=
+
=
=
II
II
II
S
S
S
S
S
S
where
is
the angle of Lode
3
1
3
2
1
2
1
1
1
3
;
3
;
3
S
I
S
I
S
I
+
=
+
=
+
=
(
)
(
)


=
=
=
=
=
=
3
2
1
J
,
min
~
3
2
1
J
,
max
~
3
1
,
,
I
,
,
I
J
I
I
J
I
I
that
such
with
that
such
with
Note:
A condition to respect is that the report/ratio
remain lower than 1. In the case of rocks
hard very resistant, subjected to stresses of containment relatively low, the law
of dilatancy can thus tend towards this report/ratio. If the two parameters are unit one
find the expression of the law of Rowe describing the law of dilatancy for grounds
powders. This approach amounts preserving the same expression as for a rock
strongly damaged, by comparing the effect of cohesion to that of a containment
additional of value
0
T
.
Characterization of
0
T
according to the parameters (has, m, S) characterizing the rock
· Case where
0
)
(
=
p
S
Disappearance of cohesion, one poses
0
0
=
T
· Case where
0
)
(
p
S
(
)
0
0
0
0
0
0
0
sin
1
sin
1
2
,
+
-
=
=
C
C
T
T
éq
3.2.3.1-3
with:
(
)
(
)
(
)




+
=
=
-
+
=
=
-
-
1
0
0
1
0
0
1
,
,
2
1
arctan
2
,
,
has
has
C
has
AMS
S
has
S
m
C
C
AMS
has
S
m

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Titrate:
Law of behavior of Laigle
Date:
09/09/05
Author (S):
R. FERNANDES, C. CHAVANT
Key:
R7.01.15-A
Page
:
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HT-66/05/002/A
3.2.3.2 Determination of the intersection of the intermediate criterion and the ultimate criterion
By writing the continuity of m in
ult
the following relation is obtained:
()
()








=
ult
E
has
has
C
p
E
p
C
ult
m
m
2
2
ult
E
has
has
C
p
E
p
C
ult
m
m




=
2
2
1
2
-




=
E
E
has
C
p
has
E
ult
m
m
1
1
2
-




=
E
E
has
has
E
ult
C
p
m
m
éq
3.2.3.2-1
3.2.4 Flow
plastic
The adopted formalism is rewritten on the basis of model CJS [R7.01.13]. When stresses
reach the edge of the field of reversibility, plastic deformations develop. For
to calculate, there is a function potential controlling the evolution of the deformations and defined by the relation
G
=
&
&
p
where
&
is the plastic multiplier and
N
N
G


-
=
F
F
.
éq
3.2.4-1
The potential function is obtained starting from the following kinematic condition:
II
p
p
v
S
S
&
&
.
-
=
éq
3.2.4-2
The parameter of dilatancy
is calculated starting from the angle of dilatancy
(defined by [éq 3.2.3.1-1])
by the formula:
()
()
()
(
)


-
>
=
-
-
=
=
-
3
ult
p
10
1
if
0
sin
3
sin
6
2
éq 3.2.4-3

Note:
is positive when
0
=
P
and in compression, then it becomes negative when plasticity
develop. It is always negative in traction


It is then possible to seek to express the kinematic condition [éq 3.2.4-2] starting from a tensor
N
in the form:
0
.
=
p
N
&
éq
3.2.4-4
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Titrate:
Law of behavior of Laigle
Date:
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Author (S):
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Key:
R7.01.15-A
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:
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HT-66/05/002/A

After decomposition of each term in déviatoire parts and hydrostatic, one finds the expression:
(
)
0
3
1
.
2
1
2
1
=
+
=


+
+
p
v
p
ij
ij
ij
p
v
p
ij
ij
ij
N
E
S
N
E
N
S
N
&
&
&
&
One deduces the relation from it
II
S
N
N
'
2
1
=
who added to the condition of standardization of the tensor
N
conduit with
the expression:
3
2
+
+
=
I
S
N
II
S
éq
3.2.4-5
The law of evolution of
p
&
must be such as the kinematic condition is satisfied. It is thus proposed
to take the projection of
p
&
on
N
(normal of the hypersurface of deformation), that is to say:






-
=
=
N
N
G
F
F
p
&
&
&
One also deduces the condition from it relating to the plastic voluminal deformation:
G
p
v
=
&
&
éq
3.2.4-6
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Law of behavior of Laigle
Date:
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Key:
R7.01.15-A
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:
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HT-66/05/002/A
4
Calculation of the derivative
4.1
Derived from the criterion
4.1.1 Derived compared to the stresses

4.1.1.1 Derived intermediate compared to the diverter
One leaves:
()
()
ij
II
ij
II
ij
S
H
S
S
S
H
S
G
+
=
where
ij
II
S
S
and
()
ij
S
H
are respectively given by:
II
ij
ij
II
S
S
S
S =
()
()
()
()
()
()
()




+
-
=




+
=
ij
II
cjs
ij
II
cjs
II
cjs
ij
ij
S
S
H
S
S
H
S
S
H
S
H
S
S
det
6
54
2
3
cos
det
54
1
6
1
3
5
2
5
3
5
Finally:
()
()
()








+




+
=
ij
II
cjs
II
ij
cjs
ij
S
S
S
S
H
S
G
S
det
6
54
3
cos
2
1
1
2
5
And consequently:
()
()
()








+




+
=
S
S
S
H
G
II
cjs
II
cjs
S
S
S
det
6
54
3
cos
2
1
1
2
5
éq 4.1.1.1-1

4.1.1.2 Derived intermediate compared to the stresses
One poses by definition:




=
ij
ij
S
G
Dev.
Q


-


+




=
=
kl
ij
jl
ik
kl
mm
kl
ij
kl
kl
ij
S
G
S
G
Dev.
S
S
G
G
3
1
3
1


-
+
-
=
kl
kl
ij
kl
jl
ik
mm
kl
kl
ij
jl
ik
kl
ij
Q
G
Q
Q
G
3
1
3
1
3
1
ij
ij
Q
G =
It is then enough to take the deviatoric part of
ij
S
G
to obtain:
()
()
()








+




+
=




=
=
ij
II
cjs
II
ij
cjs
ij
ij
ij
S
Dev.
S
S
S
H
S
G
Dev.
Q
G
S
det
6
54
3
cos
2
1
1
2
5
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Law of behavior of Laigle
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Key:
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:
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HT-66/05/002/A
And consequently:
()
()
()








+




+
=
=
S
Dev.
S
S
H
G
II
cjs
II
cjs
S
S
Q
det
6
54
3
cos
2
1
1
2
5
éq 4.1.1.2-1

4.1.1.3 Final expression of derived from the criterion compared to the stresses
The derivative of the criterion compared to the stresses is then:
()
()
()
()
()
Q
-




=
-
U
G
H
has
F
p
p
p
has
has
has
C
C
p
1
1
0
1
1
éq
4.1.1.3-1
with
() ()




+
-
=
I
Q
3
1
6
1
0
C
C
p
p
H
K
m
U
éq
4.1.1.3-2
4.1.2 Derived compared to the variable from work hardening
()
()
()
()
p
p
C
C
has
C
C
p
p
U
has
H
G
H
G
has
F
p
-












-
=
0
1
0
2
Log
1
S
S
éq
4.1.2-1
with
()
()
()
()
()
()
p
p
p
p
C
C
p
p
C
p
ks
I
km
H
G
km
U
+
-
-
=
1
0
3
1
6
1
éq
4.1.2-2
4.2
Total derivative of the criterion compared to the plastic multiplier
Let us consider the function:
()




+
-
µ
-
=
-
II
p
E
1
E
*
G
~
3
2
,
G
K
3
I
,
~
2
F
F
G
S
éq
4.2-1
Where
G
is a fixed tensor independent of
. It is of this function of which we seek the zero
to find the state of stress:
(
)
II
p
*
G
~
3
2
F
KG
~
2
.
F
F
+
+
µ
-
=
I
G
éq
4.2-2
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:
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4.3
Derived from the parameters compared to the variable of work hardening




=
<
-
=
E
p
p
E
p
E
p
S
S
if
if
0
1
éq 4.3-1




=
<
-
=
E
p
E
p
p
C
S
m
S
m
if
if
0
1
éq 4.3-2












-
=
<




+




+
-
=
E
p
has
has
C
p
E
E
C
p
E
p
C
E
p
has
has
C
p
peak
peak
C
p
peak
p
C
E
peak
m
has
has
m
has
m
m
has
has
m
has
m
if
if
2
2
2
2
1
2
1
1
Log
1
1
Log
éq
4.3-3
(
)
()
() (
)




-
+
-
-
-
-
=
-
2
1
1
1
)
(
p
ult
p
p
p
ult
E
peak
E
E
E
ult
p
has
has
has
éq
4.3-4
(
)
2
1
1
+
-
=
peak
has
has
éq 4.3-5
(
)
(
)




<
-
=
>
-
=
<
+
=
-
-
p
ult
p
E
p
ult
p
p
E
p
p
p
p
m
has
has
m
m
S
S
m
has
has
m
m
3
3
10
1
0
10
1
if
if
if
éq
4.3-6
(
)




=
>
-




-
=
-
if not
0
10
1
2
1
3
2
3
2
3
2
2
1
p
p
ult
p
has
p
K
has
has
Log
K
éq
4.3-7
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:
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HT-66/05/002/A
5
Tangent operator of speed
The condition
0
=
f&
éq 5-1
is written:
0
=
+
=
.p
p
ij
ij
F
F
F
&
&
From the expression of the cumulated plastic deviatoric deformation
p
ij
p
ij
p
E
E
3
2
=
and of
relation
G
E
~
&
&
=
p
, the condition then is found:
0
~
3
2
=
+
=
II
p
ij
ij
G
F
F
F
&
&
&
What gives us for the plastic multiplier:
II
p
ij
ij
G
F
F
~
3
2
&
&
-
=
By then considering the relation forced/deformations:
kl
ijkl
ij
kl
ijkl
ij
ijkl
ij
kl
ijkl
ij
ij
ij
G
D
F
D
F
D
F
D
F
F
&
&
&
&
-
=
=
=
and by deferring it in the expression of
&
one can write:
II
p
kl
ijkl
ij
kl
ijkl
ij
G
F
G
D
F
D
F
~
3
2
&
&
&
-
-
=
That is to say:
kl
ijkl
ij
II
p
kl
ijkl
ij
G
D
F
G
F
D
F
-
-
=
~
3
2
&
&
éq 5-2
By deferring this result in the expression of
ij
&
one finds:








-
+
=
Cd
kl
ijkl
ij
II
p
kl
ijkl
ij
Cd
abcd
ab
G
G
D
F
G
F
D
F
D
~
3
2
&
&
&
éq
5-3
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Law of behavior of Laigle
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:
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HT-66/05/002/A
6
Digital processing adapted to the nonregular models
The law of evolution of the plastic mechanism, defined in the chapter [§3], must satisfy the condition
kinematics [éq 3.2.4-2]. The projection suggested on the normal of the hypersurface of deformation can
to lead to a “not-solution” which results in a failure of the digital processing (see
the graphic interpretation of the chapter [§6.1.3.3]). One proposes in this chapter to define rules of
projection allowing to manage the models known as “not-regular” in their imposing projection known as
“at the node of the cone”.
Moreover, as for other relations of behavior, one adds the possibility of cutting out
locally (at the points of Gauss) the pitch of time to facilitate numerical integration.
6.1
Projection at the top of the cone
6.1.1 Definition of the jetting angle
One places oneself in this chapter within the framework of finished increase. Equations translating it
elastic behavior are written:
(
)
p
E
p
E
S
E
E
S
S
µ
-
=
-
µ
+
=
-
2
2
éq
6.1.1-1
(
)
p
v
E
p
v
v
K
I
K
I
I
-
=
-
+
=
-
3
3
1
1
1
éq
6.1.1-2
One can also express the kinematic condition starting from the tensor
N
(cf paragraph [§3.2.4]):
0
.
=
p
N
éq
6.1.1-3
By deferring the two equations translating the elastic behavior in the preceding expression one
find:
(
)
S
S
E
-
=
E
p
µ
2
1
éq
6.1.1-4
(
)
1
1
3
1
I
I
K
E
p
v
-
=
éq
6.1.1-5
One then expresses the kinematic condition by the following relation:
()
(
)
0
3
1
3
1
2
1
.
1
1
=






-
+
-
I
S
S
N
I
I
K
K
E
E
µ
with
3
2
+
+
=
I
S
N
II
S
Maybe by combining the two preceding relations where
N
indicate the normal of the hypersurface of
deformation:
(
)
(
)
()
0
.
9
1
3
2
1
1
1
2
=
-
+
-
+
+
N
S
S
I
S
Tr
I
I
K
S
E
E
II
µ
(
)
(
)
0
3
1
.
2
1
1
1
=
-
+
-
I
I
K
S
E
II
E
S
S
S
µ
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Titrate:
Law of behavior of Laigle
Date:
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Key:
R7.01.15-A
Page
:
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HT-66/05/002/A
This last equation defines the point
()
S
,
1
I
like a projection of the point
(
)
E
E
I S
,
1
on the criterion.
not
(
)
II
S
I,
1
will be the oblique projection of the point
(
)
E
E
II
S
I,
1
, projection whose direction varies with
. One
can give the chart of it of the chapter [§6.1.3.3].
The preceding relation can then be rewritten as follows:
(
)
II
E
E
S
K
I
I
S
S
S
-
µ
-
=
-
.
2
3
1
1
éq
6.1.1-6
The jetting angle then is defined
S
by the relation:
(
)
(
) (
)
S
S
S
S
S
S
S
-
-
-
=
E
E
II
E
S
S
.
cos
éq
6.1.1-7
By deferring the definition of the angle
S
in the relation of projection one finds the relation:
(
) (
)
S
E
E
E
K
I
I
µ
-
=
-
-
-
cos
2
3
1
1
S
S
S
S
éq
6.1.1-8
6.1.2 Existence of projection
The principle of this paragraph is to discuss on the question the existence the angle
S
such as
projection of the point
(
)
E
E
I S
,
1
always belongs to the surface of load. These problems appear
essential for projections around the node of the surface of load, in other words when
0
S
. There is by definition the relation:
(
)
(
) (
)
(
)
S
S
S
S
S
S
S
S
S
S
S
S
-
-
=
-
-
-
=
E
II
E
E
E
II
E
S
S
S
.
.
cos
éq
6.1.2-1
By combining this equation with the expression:
G
S
E
S
S
~
2
2
µ
µ
-
=
-
=
E
p
E
One obtains:
II
II
S
G
S ~
~
.
cos
G
S
=
éq
6.1.2-2
One seeks an estimate of
S
cos
.
Stage 1: estimate of
II
S
G
S ~
.
One places oneself in this paragraph under the conditions:
0
S
and
0
=
F
.
By definition of
G
~
and of
G
one a:
()
S
N
N
S
G
S
I
G
G
S
G
.
.
.
3
.
~






-
=
=


-
=
F
F
Tr
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:
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HT-66/05/002/A
For preoccupations with a simplification of calculation one brings back the resolution of
F
with the resolution of the equation:
()
()
()
()
()
()
0
,
0
,
0
2
1
0
=
-




=
=
-




=
p
p
has
p
C
C
p
has
C
C
U
H
G
F
U
H
G
F
S
S
éq 6.1.2.3
By derivation of this new function one finds the relation:
() ()
()
() ()
()
Q
-




=
-




=
-
-
U
U
has
H
U
U
has
G
H
F
p
p
has
p
p
C
C
has
p
p
C
C
1
0
1
0
2
,
1
,
1
with:
() ()




+
-
=
I
Q
3
1
6
1
0
C
C
p
p
H
K
m
U
Who gives after simplification:
I
Q
B
With
F
+
=
2
éq
6.1.2.4
Where:
() () () ()
()
() () () ()
()




=




+
=
-
-
1
0
1
0
,
3
,
6
1
1
p
p
has
p
C
C
p
p
p
has
p
p
p
p
C
C
U
H
K
m
has
B
U
K
m
has
H
With
éq
6.1.2.5
One has as follows:
(
)
3
3
.
3
3
.
.
2
2
2
2
+
+
+
=
+
+
+
=
B
S
With
S
B
With
F
II
II
S
Q
I
S
I
Q
N
And consequently:
S
N
N
S
G
.
.
~






-
=
F
F
S
I
S
S
Q
I
Q
.
3
.
3
3
.
3
2
2
2






+
+




+
+
+
-
+
=
II
II
S
B
S
With
B
With
II
S
B
With
3
3
.
3
3
2
2
+
-
+
=
S
Q
From where it is deduced that:
3
3
.
3
3
.
~
2
2
+
-
+
=
B
S
With
S
II
II
S
Q
S
G
éq
6.1.2.6
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Code_Aster
®
Version
7.4
Titrate:
Law of behavior of Laigle
Date:
09/09/05
Author (S):
R. FERNANDES, C. CHAVANT
Key:
R7.01.15-A
Page
:
21/34
Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
By definition of
Q
one a:
()
()
()
S
S
S
S
S
S
Q
.
det
6
54
3
cos
2
1
1
.
.
2
5








+




+
=


=
S
Dev.
S
S
H
G
Dev.
II
cjs
II
cjs
()
()
II
cjs
S
H




+
=
3
cos
2
1
1
5
()
II
S
H
=
One expresses finally:
()
3
3
3
3
.
~
2
2
+
-
+
=
B
H
With
S
II
S
G
éq
6.1.2.7
When
0
S
then
(
)
0
,
1
0
,
0
B
H
With
U
C
C
p
and
And thus:
When
0
S
then
()
(
)
3
3
.
~
2
0
+
C
C
II
H
H
S
0
S
S
G
éq
6.1.2.8
Stage 2: estimate of
II
G~
One places oneself in this paragraph under the conditions:
0
,
1
,
0
B
H
With
C
C
0
S






-
=
N
N
G
F
F
~






+
+




+
+
+
-
+
=
3
.
3
3
.
1
3
1
2
2
0
2
0
I
S
S
Q
I
Q
II
II
C
C
C
C
S
B
S
H
B
H
()
(
)
S
Q
II
C
C
C
C
S
H
H
H
0
2
2
0
3
1
+
-
=

(
)
()
(
)
(
)
()
(
)
(
)
2
0
2
2
2
2
2
0
2
2
2
2
4
2
0
2
2
3
2
3
~
.
~
~
C
C
II
C
C
II
C
C
II
II
H
H
S
H
S
H
H
Q
G
+
-
+
+
=
=
G
G
(
)
(
)
()
(
)




+
+
-
=
2
2
2
2
2
2
2
0
3
6
1
H
Q
H
II
C
C
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Code_Aster
®
Version
7.4
Titrate:
Law of behavior of Laigle
Date:
09/09/05
Author (S):
R. FERNANDES, C. CHAVANT
Key:
R7.01.15-A
Page
:
22/34
Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
It is shown [Appendix 2] that:
()
()
()
()








+
+
+




+
=
3
cos
2
1
3
cos
4
3
cos
2
1
1
2
2
10
2
cjs
cjs
cjs
cjs
II
H
Q
éq 6.1.2.9
and thus like
()
()
(
)
6
/
1
3
cos
1
cjs
H
+
=
:
(
)
()
()
()
(
)
()
(
)
()
(
)




+
+
-








+
-
+
+




+
=
2
2
2
2
2
6
6
2
2
6
10
2
0
2
3
6
2
2
1
1
4
2
2
1
1
1
~
H
H
H
H
H
H
G
cjs
C
C
II
(
)
()
()
()
(
)
()
(
)




+
+
-
-
+
+
=
2
2
2
2
2
10
2
4
2
2
0
2
3
6
4
1
2
1
4
3
1
~
H
H
H
H
H
G
cjs
C
C
II
()
()
()




-




+
+
-
+




=
4
1
3
3
4
1
2
1
~
2
2
12
2
6
2
0
2
H
H
H
H
G
cjs
C
C
II
And consequently:
()
()
()
12
2
6
2
2
0
4
1
2
1
4
1
3
3
~
H
H
H
H
G
cjs
C
C
II
-
+
+
-




+




=
éq
6.1.2.10
Stage 3: estimate of
S
cos
One deduces from the two paragraphs precedent the expression of the following jetting angle:
When
0
S
then:
(
)
(
)
(
)
(
)
(
)
2
2
2
2
2
3
cos
1
4
1
3
cos
1
2
1
4
1
3
3
3
3
cos
+
-
+
+
+
-




+
+
cjs
cjs
cjs
S
éq 6.1.2.11
It is noticed that
S
depends on the angle of Lode
, and that consequently limit of
the jetting angle when
0
S
do not exist. However a framing of
S
cos
us allows
to determine an area of projection at the top a priori (demonstration of the framing in
[Appendix 3]):
with
:
(
)
(
)




=
-
+




+
+
=
1
cos
1
4
3
3
3
3
cos
cos
cos
cos
max
2
2
2
2
2
min
max
min
S
cjs
cjs
S
S
S
S
éq 6.1.2.12
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Code_Aster
®
Version
7.4
Titrate:
Law of behavior of Laigle
Date:
09/09/05
Author (S):
R. FERNANDES, C. CHAVANT
Key:
R7.01.15-A
Page
:
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Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
6.1.3 Rules of projection
One calls
0
1
I
the intersection of the field of reversibility with the hydrostatic axis. One obtains:
()
()
p
p
C
m
S
I
=
.
3
0
1
éq
6.1.3-1
While deferring
0
1
I
and the framing of
S
cos
, when
0
S
, in the relation
(
) (
)
S
E
E
E
K
I
I
µ
-
=
-
-
-
cos
2
3
1
1
S
S
S
S
, one deduces the following rules of projection from them according to
sign parameter of dilatancy
, and for values of
E
I
1
and of
E
II
S
data:

6.1.3.1 Case where the parameter of dilatancy is negative
If
min
0
1
cos
2
3
1
S
K
S
I
I
E
II
E
µ
-
<
-
then projection will be regular;
If
max
0
1
cos
2
3
1
S
K
S
I
I
E
II
E
µ
-
>
-
then projection will be at the top.

6.1.3.2 Case where the parameter of dilatancy is positive
If
max
0
1
cos
2
3
1
S
K
S
I
I
E
II
E
µ
-
<
-
then projection will be regular;
If
min
0
1
cos
2
3
1
S
K
S
I
I
E
II
E
µ
-
>
-
then projection will be at the top.

6.1.3.3 Interpretation
graph























0
1
I
II
S
(
)
E
E
II
S
I,
1
Area of regular projection
Area of projection at the top
(
)
E
E
II
S
I,
1
min
cos
2
3
S
K
µ
-
max
cos
2
3
S
K
µ
-
Intermediate area
1
I
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Code_Aster
®
Version
7.4
Titrate:
Law of behavior of Laigle
Date:
09/09/05
Author (S):
R. FERNANDES, C. CHAVANT
Key:
R7.01.15-A
Page
:
24/34
Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
6.1.3.4 Equations
of flow
In the intermediate area one solves the equations corresponding to a regular projection. If this
resolution does not give a solution one then solves the equations of flow of projection with
node.
In the case of projection at the top there are the relations:
0
S
=
éq
6.1.3.4-1
()
()
p
p
C
m
S
I
.
3
0
1
=
éq
6.1.3.4-2
E
p
II
S
3
2
2
1
µ
=
éq
6.1.3.4-3
6.2
Local Recutting of the pitch of time
As for other relations of behavior (model CJS for example) one added
possibility for the model of LAIGLE 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 the value of
ITER_INTE_PAS (itepas) is worth 0,1 or ­ 1 it N `has no recutting there (note: 0 are the value by
defect). If itepas is positive recutting is systematic, if it is negative recutting is taken
in account only in the event of nonnumerical convergence.
Recutting consists in carrying out the integration of the plastic mechanism with an increment of
deformation whose components correspond to the components of the increment of deformation
initial divided by the absolute value of itepas (cf Doc. STAT_NON_LINE [U4.51.03]).
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Code_Aster
®
Version
7.4
Titrate:
Law of behavior of Laigle
Date:
09/09/05
Author (S):
R. FERNANDES, C. CHAVANT
Key:
R7.01.15-A
Page
:
25/34
Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
7
Internal variables
For implementation the data-processing we retained the 4 following internal variables:
7.1
V1: the plastic deformation déviatoire cumulated
The variable of work hardening
p
is proportional to the second invariant of the tensor of the deformations
déviatoires.
p
ij
p
ij
p
E
E
3
2
=
with
()
ij
p
ij
p
ij
p
ij
3
tr
E
-
=
7.2
V2: cumulated plastic voluminal deformation
The plastic voluminal deformation is defined by the relation presented at the paragraph [§3.2.4] on
law of evolution of the plastic mechanism:
G
p
v
=
&
&
7.3
V3: fields of behavior of the rock
Five fields of behavior, numbered from 0 to 4 (cf appears), are identified to make it possible to have
a relatively simple representation of the state of damage of the rock, since the rock
intact to the rock in a residual state. These fields are a function of the deformation déviatoire
figure cumulated
p
and of the state of stress. Each increment of number of field defines it
passage in a field of higher damage.
· If the diverter is lower than 70% of the diverter of peak, then the material is in the field
0;
· If not:
- If
0
=
p
then the material is in field 1;
- If
E
p
<
<
0
then the material is in field 2;
- If
ult
p
E
<
<
then the material is in field 3;
- If
ult
p
>
then the material is in field 4.
State field of the rock
0 Intact
1 Damage
pre-peak
2 Damage
post-peak
3 Fissured
4 Fractured
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®
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7.4
Titrate:
Law of behavior of Laigle
Date:
09/09/05
Author (S):
R. FERNANDES, C. CHAVANT
Key:
R7.01.15-A
Page
:
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Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
7.4
V4: the state of plasticization
It is an internal indicator in Code_Aster. It is worth 0 if the point of gauss is in elastic load or in
discharge, and is worth 1 if the point of gauss is in plastic load.


8
Detailed presentation of the algorithm
One retains a formulation implicit compared to the criterion and explicit compared to the direction
of flow: the criterion will have to be checked at the end of the pitch, whereas the direction of flow is that
calculated at the beginning of the pitch (and thus the value of dilatancy will be also that calculated at the beginning of
no time).
One places oneself in a material point, and one considers that are given:
· The tensor of increase in deformations
from where one deduces
E
and
v
;
· Stresses at the beginning of the pitch
-
from where one deduces
-
S
and
-
1
I
;
· The values of the variables intern at the beginning of the pitch of time (only the plastic deformation
cumulated
-
p
is necessary).
It is a question of calculating:
· Stresses at the end of the pitch of time
;
· The variables intern in end of the pitch of time (
p
,
p
v
, fields of behavior);
· The tangent behavior at the end of the pitch:

8.1
Calculation of the elastic solution
v
E
E
E
K
I
I
T
µ
+
=
+
=
-
=
-
-
-
3
2
1
1
E
S
S
8.2
Calculation of the elastic criterion
Calculation of
()
E
E
E
H
S
G
II
=
Calculation of
()
-
-
=
p
m
m
,
()
-
-
=
p
S
S
,
()
-
-
=
p
has
has
and
()
-
-
= has
K
K
Calculation of
-
-
-
-
-
-
+
-
-
=
K
S
I
K
m
H
G
K
m
U
E
C
C
E
C
E
1
3
6
0
Calculation of
E
has
C
C
E
E
U
H
G
F
-




=
-
1
0
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Version
7.4
Titrate:
Law of behavior of Laigle
Date:
09/09/05
Author (S):
R. FERNANDES, C. CHAVANT
Key:
R7.01.15-A
Page
:
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Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
8.3 Algorithm
If
0
>
E
F
Calculation of:
()
(
)
(
)
(
)
(
)
()
()
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
=
=
=
=
=
=
=
=
;
'
;
,
,
,
;
,
,
;
,
,
;
.
3
0
1
0
0
0
0
0
1
'
0
0
0
0
T
T
G
I
C
has
S
m
C
C
has
S
m
G
G
m
S
I
T
C
S
Calculation a priori of projection at the top
0
S
=
; Calculation of
node
E
p
p
p
II
S
µ
=
+
=
-
3
2
2
1
and of
()
()
node
p
p
C
I
m
S
I
1
.
3
1
=
=
.
If
(
)
(
)




µ
-
<
-
<
µ
-
<
-
-
-
-
-
0
if
;
0
if
;
min
max
cos
2
3
cos
2
3
1
1
1
1
S
E
node
E
S
E
node
E
II
II
S
K
I
I
S
K
I
I
Projection at the top is not retained a priori. The regular solution is calculated.
()
()



=
=
-
-
-
-
0
Q
0
Q
Q
if
if
E
(
)
(
)



=
=
-
-
-
-

0
N
0
N
N
if
if
E
E
F
,
,
(
)
(
)



=
=
-
-
-
-

0
G
0
G
G
if
if
E
E
F
,
,
If
0
=
-
p
Initialization
E
E
E
p
p
F
F
I
I
=
=
=
=
=
-
0
1
0
1
0
0
0
;
;
;
;
0
S
S
And




=
=
2
3
~
max
10
1
1
1
1
B
F
II
p
p
E
ij
p
G
If not
Calculation of the increase in the plastic multiplier
by Newton:
Initialization
E
E
E
p
p
F
F
I
I
=
=
=
=
=
-
0
1
0
1
0
0
0
;
;
;
;
0
S
S
I
Q
C
C
C
m
K
H
K
m
U
U
-
-
=
=
-
-
-
-
-
-
3
6
0
0
()
()
()
()
()
()
-
-
-
+
-
-
=
p
p
E
p
p
C
C
E
p
p
C
p
ks
I
km
H
G
km
U
1
3
1
6
1
0
0
()
Q
-




=
-
-
-
-
-
-
-
F
U
G
H
has
F
has
has
E
has
C
C
0
1
1
0
0
1
1
()
p
p
p
p
C
C
E
has
C
C
E
p
F
U
has
H
G
H
G
has
F
-










-
=
-
-
-
-
0
0
1
0
2
0
log
1
(
)
F
II
p
F
F
G
F
KG
F
F
~
3
2
~
2
.
0
0
0
*
µ
+
+
-
=
I
G
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Code_Aster
®
Version
7.4
Titrate:
Law of behavior of Laigle
Date:
09/09/05
Author (S):
R. FERNANDES, C. CHAVANT
Key:
R7.01.15-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-66/05/002/A

Loop iterations N
N
N
N
F
F
-
=
+1
1
N
N
1
N
+
+
+
=
F
1
N
p
p
v
F
II
1
N
1
N
p
G
;
G
~
3
2
+
+
+
=
=
F
1
N
p
E
1
1
N
1
F
1
N
p
E
1
N
G
K
3
I
I
;
~
2
+
+
+
+
-
=
µ
-
=
G
S
S
If
0
1
<
+
N
p
Not convergence
Calculation
1
+
N
Q
()
()
()
()
()
;
;
;
;
;
1
1
1
1
1
1
1
1
1
1
+
+
+
+
+
+
+
+
+
+
=
=
=
=
=
N
N
N
p
N
N
p
N
N
p
N
N
N
has
K
K
has
has
S
S
m
m
G
G
S
1
1
1
1
1
0
1
1
1
1
1
3
6
+
+
+
+
+
+
+
+
+
+
-
-
=
N
N
N
C
N
N
C
N
C
N
N
N
K
S
I
K
m
H
G
K
m
U
1
1
0
1
1
1
+
+
+
-




=
+
N
has
C
C
N
N
U
H
G
F
N
I
Q
C
N
N
N
C
N
C
N
N
m
K
H
K
m
U
-
-
=
+
+
+
+
+
+
3
6
1
1
1
0
1
1
1
()
()
()




+




-




-
=
+
+
+
+
+
+
1
1
1
0
1
1
1
1
3
1
6
1
N
p
p
N
N
p
p
C
C
N
N
p
p
C
N
p
ks
I
km
H
G
km
U
()
1
1
1
1
1
0
1
1
1
1
1
1
1
+
+
-
+
+
+
-




=
+
+
+
N
N
has
has
N
has
C
C
N
N
U
G
H
has
F
N
N
N
Q
()
p
N
p
p
C
C
N
has
C
C
N
N
p
N
U
has
H
G
H
G
has
F
N
N
-










-
=
+
+
+
+
+
+
+
1
0
1
1
0
1
2
1
1
1
1
log
1
(
)
F
II
p
N
F
F
N
N
G
F
KG
F
F
~
3
2
~
2
.
1
1
1
*
µ
+
+
-
=
+
+
+
I
G
If
prec
C
N
F
>
+
/
1
n=n+1
If N > no. ite interns max
If
(
)
(
)




-
>
-
<
-
>
-
0
if
;
cos
2
3
0
if
;
cos
2
3
max
min
1
1
1
1
µ
µ
S
node
E
S
node
E
E
E
II
II
S
K
I
I
S
K
I
I
One retains projection at the top:
node
p
p
node
I
I
=
=
=
;
;
1
1
0
S
If not
Not convergence
If not
Not convergence
If not
Convergence
background image
Code_Aster
®
Version
7.4
Titrate:
Law of behavior of Laigle
Date:
09/09/05
Author (S):
R. FERNANDES, C. CHAVANT
Key:
R7.01.15-A
Page
:
29/34
Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
If FULL_MECA
Calculation of:
F
T
N
F
II
p
N
T
N
F
N
F
G
F
F
HG
H
G
H
H




-




+
=
+
+
+
+
1
1
1
1
~
3
2
.
.
Mechanical symmetrization:




+
=
+
+
+
T
N
N
N
sym
1
1
1
2
1


9
Alternative on the expression of the criterion of plasticity
In this alternative proposal, one expresses the criterion of plasticity according to the first invariant and
diverter of the stresses, by a retiming on triaxial in compression and extension by
following relations:
9.1 Formulation
general
()
()
0
,
1
-




=
p
has
C
II
U
S
F
p
éq
9.1-1
Where the expression of
()
p
U
,
is:
If
0
cjs
(
)
() ()
()
() ()
() ()
p
p
C
p
p
C
T
C
T
C
p
p
p
K
S
I
K
m
H
H
H
H
H
K
m
U
+
-




-
-
+
-
=
1
0
0
0
0
3
2
6
,
éq 9.1-2
If
0
=
cjs
()
() ()
()
() ()
() ()
p
p
C
p
p
C
p
p
p
K
S
I
K
m
K
m
U
+
-


+
-
=
1
3
3
cos
2
1
2
3
6
,
éq
9.1-3
background image
Code_Aster
®
Version
7.4
Titrate:
Law of behavior of Laigle
Date:
09/09/05
Author (S):
R. FERNANDES, C. CHAVANT
Key:
R7.01.15-A
Page
:
30/34
Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
9.2
Pace of the thresholds
One places oneself if
1
;
1
;
21
;
7
.
0
=
=
=
=
has
S
m
cjs
, then one traces the pace of the thresholds in
the plan perpendicular to the hydrostatic axis (known as plan
), one standardizes compared to
C
and one
consider the two values of containments such as
0
1
=
I
[Figure 9.2-a] and
C
I
3
1
-
=
[Figure 9.2-b].
Appear 9.2-a: Pace of the thresholds for a null containment
Appear 9.2-b: Pace of the thresholds for a null containment in compression
One notes in these charts that the (a) formulation has the disadvantage of having one
nonconvex pace in the plan
.
I1/3SIGC=0
- 0,15
- 0,1
- 0,05
0
0,05
0,1
- 0,15
- 0,1
- 0,05
0
0,05
0,1
0,15
(Software house/sigc) * Sin (teta)
(
S
II/S
ig
C
)
*
C
O
S
(
T
E
T
has
)
Version 2
Version 1 (a)
Formulation
(a)
Formulation
compression
I1/3SIGC=-1
- 2,5
- 2
- 1,5
- 1
- 0,5
0
0,5
1
1,5
2
- 2
- 1
0
1
2
(Software house/sigc) * Sin (teta)
(S
I
I
/
S
I
G
C
) * C
O
S
(you
T
has
)
Version 2
Version 1 (a)
Formulation
(a)
Formulation
compression
background image
Code_Aster
®
Version
7.4
Titrate:
Law of behavior of Laigle
Date:
09/09/05
Author (S):
R. FERNANDES, C. CHAVANT
Key:
R7.01.15-A
Page
:
31/34
Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
10 Bibliography
[1]
F. LAIGLE: Downstream of the cycle ­ underground Ouvrages ­ rheological Modèles for the analysis of
rock mechanics behavior. Note EDF-CIH IH.AVCY.01.003.A (2001).
[2]
PH. KOLMAYER: Downstream of the cycle ­ underground Ouvrages ­ Ecriture of the law of behavior
CIH on a basis of the model Cambou-Jafari-Sidoroff (CJS) known of Code_Aster. Note
EDF-CIH.IH.AVCY.38.005.A (2002).
[3]
C. CHAVANT: Specifications for the introduction of a model of rock into Code_Aster.
Note EDF-I74/E27131.
[4]
C. CHAVANT, pH. AUBERT: Law CJS in géomechanics. Reference document of
Code_Aster R7.01.13.
[5]
PH. KOLMAYER, R. FERNANDES, C. CHAVANT, 2004: “Numerical implementation off has
new rheological law for mudstones ", Applied Clay Science 26, 499-510.
background image
Code_Aster
®
Version
7.4
Titrate:
Law of behavior of Laigle
Date:
09/09/05
Author (S):
R. FERNANDES, C. CHAVANT
Key:
R7.01.15-A
Page
:
32/34
Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
Appendix 1 Retiming of the criterion on the triaxial one in compression

By taking the general expression of the criterion under the conditions of triaxial in compression, one finds:
()
()
() ()
()
() ()
() ()




+
-
-
-




=
p
p
C
p
p
C
C
p
p
has
C
C
K
S
I
K
m
H
G
K
m
H
G
F
p
1
0
1
0
3
6
S
S
()
() ()
() ()
(
)
() ()






-
+
+
-
+






-
=
p
p
C
p
p
C
C
p
p
C
has
C
C
K
S
K
m
H
H
K
m
H
H
p
3
1
0
3
1
1
0
3
1
2
3
3
2
6
1
3
2
(
)
(
) (
)
(
) (
)
(
)
(
) (
)




-
+
+
-
+




-
=
p
p
C
p
p
C
p
p
has
C
K
S
K
m
K
m
p
3
1
3
1
1
3
1
2
3
3
2
6
3
2
(
)
(
)
(
)
(
) (
)
(
) (
)
(
)
(
) (
)


-
+
+
-
+
-




=
p
p
C
p
p
C
p
p
has
has
C
K
S
K
m
K
m
p
p
3
1
3
1
1
3
1
1
2
3
3
3
2
(
)
(
)
(
)
(
) (
)
(
)
(
) (
)
(
)
(
) (
)


-
+
+
-
+
-




=
p
p
C
p
p
C
p
p
has
has
C
K
S
K
m
K
m
p
p
3
1
1
3
1
3
1
1
2
3
3
3
2
(
)
(
)
(
)
(
) (
)
(
)
(
) (
)


-
+
-




=
p
p
C
p
p
has
has
C
K
S
K
m
p
p
3
1
3
1
1
3
2
()
(
)
()
()
()
(
)
()




+
-




-
-






=
p
C
p
has
has
has
C
S
m
p
p
p
3
1
1
3
1
1
3
2
3
2
()
(
)
()
()
()
()
(
)
()








+
-
-
-






=
p
C
p
has
C
has
has
C
S
m
p
p
p
3
1
1
3
1
1
3
2
background image
Code_Aster
®
Version
7.4
Titrate:
Law of behavior of Laigle
Date:
09/09/05
Author (S):
R. FERNANDES, C. CHAVANT
Key:
R7.01.15-A
Page
:
33/34
Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
Appendix 2 Standardization of Q
()
()
()








+




+
=
S
S
S
Q
det
.
6
54
3
cos
2
1
1
2
5
Dev.
S
S
H
II
cjs
II
cjs
One poses
()
S
S
T
= det
and
()




=
S
S
T
det
Dev.
D
(cf reference document CJS R7.01.13)
()
()
()








+
+
+




+
=
=
D
cjs
3
II
cjs
D
D
4
II
2
cjs
2
cjs
10
2
II
.
3
cos
2
1
S
.
3
54
.
S
.
2
3
3
cos
2
1
H
1
.
Q
T
S
T
T
Q
Q

To evaluate this expression, one places oneself if
S
is diagonal by preoccupations with a simplification of
calculations.
As follows:












=
0
0
0
3
2
1
S
S
S
S
and














-
-
-
-
-
-
=
0
0
0
S
S
S
S
S
S
2
S
S
S
S
S
S
2
S
S
S
S
S
S
2
3
1
3
2
3
1
2
1
3
2
2
1
3
1
3
1
2
1
3
2
D
T
By using the property of
S
:
0
3
2
1
=
+
+
S
S
S
, it is shown that
(
)
2
3
2
2
2
3
2
1
2
2
2
1
4
4
S
S
S
S
S
S
S
II
+
+
=
and by
consequent:
6
S
0
0
0
S
S
S
S
S
S
2
S
S
S
S
S
S
2
S
S
S
S
S
S
2
.
0
0
0
S
S
S
S
S
S
2
S
S
S
S
S
S
2
S
S
S
S
S
S
2
9
1
.
4
II
3
2
3
1
2
1
3
2
2
1
3
1
3
1
2
1
3
2
3
2
3
1
2
1
3
2
2
1
3
1
3
1
2
1
3
2
D
D
=
-
-
-
-
-
-
-
-
-
-
-
-
=
T
T
One also shows starting from the property
0
3
2
1
=
+
+
S
S
S
that
)
det (
.
3
3
3
2
1
3
3
32
3
1
S
=
=
+
+
S
S
S
S
S
S
and by
consequent:
()
=
=
-
-
-
-
-
-
=
3
cos
.
)
det (
S
54
.
0
0
0
S
S
S
S
S
S
2
S
S
S
S
S
S
2
S
S
S
S
S
S
2
.
0
0
0
S
S
S
S
.
9
54
.
.
S
.
3
54
.
cjs
3
II
cjs
3
2
3
1
2
1
3
2
2
1
3
1
3
1
2
1
3
2
3
2
1
3
II
cjs
D
3
II
cjs
S
T
S
One deduces some as follows:
()
()
()
()








+
+
+




+
=
3
cos
2
1
3
cos
4
3
cos
2
1
1
2
2
10
2
cjs
cjs
cjs
cjs
II
H
Q
background image
Code_Aster
®
Version
7.4
Titrate:
Law of behavior of Laigle
Date:
09/09/05
Author (S):
R. FERNANDES, C. CHAVANT
Key:
R7.01.15-A
Page
:
34/34
Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
Appendix 3 Framing of the jetting angle
It is reminded the meeting that
(
)
()
(
)
()
(
)
2
2
2
2
2
3
cos
1
4
1
3
cos
1
2
1
4
1
3
3
3
3
cos
cjs
cjs
cjs
S
+
-
+
+
+
-




+
+
0
S
One poses:
()
()
(
)
()
(
)
2
cjs
2
cjs
cjs
cos
1
4
1
cos
1
2
1
X
+
-
+
+
=
where
[
[
2
,
0

It is noted that:
()
()
=
-
X
X
, the function
X
being even one restricts the interval of study at
[[
,
0
.
The resolution of
0
D
dX =
give
()
()
(
)
()
(
)
0
cos
.
.
cos
1
2
sin
cjs
cjs
3
cjs
cjs
=
+
+

One deduces from it that the limits lower and higher from the function
X
are:
(
)
() ()
()




-
=
-
=
=
=
cos
that
is such
where
1
4
1
X
4
1
0
X
cjs
cjs
cjs
2
cjs
cjs
One can thus give the framing of
S
cos
according to:
max
min
cos
cos
cos
S
S
S
with:
(
)
(
)




=
-
+




+
+
=
1
cos
1
4
3
3
3
3
cos
max
2
2
2
2
2
min
S
cjs
cjs
S