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Titrate:
Law of behavior of Laigle


Date:
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Author (S):
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Key:
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: 1/34

Organization (S): EDF-R & D/AMA
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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 this one. 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 constraint 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
principal constraints was rewritten with invariants of constraints 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 constraints ...................... 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 constraints ..................................................................................... 14
4.1.1.1 Derived intermediate compared to the diverter ....................................................... 14
4.1.1.2 Derived intermediate compared to the constraints ................................................... 14
4.1.1.3 Final expression of derived from the criterion compared to the constraints ................... 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 step of time ............................................................................................ 24
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The 7 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 constraints in small disturbances, noted in the form of
following vector:
11


22




33
212


213


2 23

One notes:

I = tr

first invariant of the constraints
1
()
I

S
1
= -
I
tensor of the constraints déviatoires
3

S = S S
.
second invariant of the tensor of the constraints déviatoires
II

major principal constraint
1

minor principal constraint
3
Tr ()

E = -
I
diverter of the deformations
3
= Tr
voluminal deformation
v
()
(
S

cos 3)
1/2 3/2 det ()
= 2 3

being the angle of Lode
3
software house

p
2 p p
=
E E
cumulated plastic deviatoric deformations
ij ij
3
N
normal of the hypersurface of deformation
G
function controlling the evolution of the plastic deformations and describing
direction of flow
~
Tr (G)

G = G -
I
diverter of G
3
G = Tr (G)
trace G
~
~ ~
~
G = G G
.
G normalizes
II

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 (p', Q) for the very strong constraints (function of
mineralogical nature of the rock)
S
Cohesion of the medium. Representative of the damage of the rock.
has
Characterization of the concavity 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 Valor
of
m of the ultimate criterion reached in ult
me Valeur
of
m of the intermediate criterion reached in E
ae Valeur
of
has intermediate criterion reached in E
m peak
Value of m of the criterion of peak reached with the peak of constraint
peak has
Value of A of the criterion of peak reached with the peak of constraint

Exhibitor 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
1p
Intersection of the intermediate criterion and the criterion of peak
p2
Intersection of the intermediate criterion and the ultimate criterion
Pa
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 principal 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 constraint, 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 principal 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 constraints. Non-linearities of the behavior are
likely to appear the peak of resistance before, in the case of tender rocks,
for a level of constraint 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 constraint
of surconsolidation is reached (course-model). Under very low constraints 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 constraints describing a criterion of rupture,
generally curve in the plan of Mohr or the plan of the principal constraints
major and minor. The assumption of a linear criterion of Mohr-Coulomb is thus only one
simplifying assumption, having tendency, for low constraints 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 constraint
of containment is weak. This decrease is related to a damage more or less
located rock, according to the level of containment. Whatever this constraint,
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 zone 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 constraint of containment
will increase, to become non-existent beyond of a certain constraint 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 this one.

· 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 constraint, which one can
to find on a Cape-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 constraints and the level of damage
rock, by cohesion, dilatancy or friction.

Cohesion defines the resistance of material as long as this one 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 freezing,…).

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.
S & = µ (
p
2nd & - E &)








éq 3.1-1
1
I = 3K (
p
v - v)
&
& & 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 constraints

1
2 A (p
)



1
1

p
3
m



F =
(-
1
3)
()
(p has
) - (c)

p


has (p)
(- 3) + S ()
éq
3.2.1.1-1


C








C




3.2.1.2 Expression
general

One transforms the preceding expression according to the first invariant and of the diverter of
constraints, by a retiming of the criterion on triaxial in compression, to obtain:
1
G (S) has (p
)
F =

- U (, p
) 0


éq
3.2.1.2-1
0
C C
H
with:
1/
S
H () = (


1+ cos


éq 3.2.1.2-2
cjs
(3) 1/6
det () 6
= 1+
54


cjs
3



software house
0


= = = -
C
H
H
(1
) 1/6

3
cjs

0 = +
T
H
(1 cjs) 1/6
G (S) = S H

éq
3.2.1.2-3
II ()



S


U (,)
m (p
p
) K (p) G () m (p) K (p)
= -
-
I + S

éq 3.2.1.2-4
0
1
(p
) K (p
)

6
H

3
C
C
C
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Note:

· One shows [Annexe 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.


S
Threshold with the peak
2
1
S
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).



p

p
p

S () = 1
if
< E


E


éq
3.2.2-1


S (p
) =
p

0
if
E


If p
> ult (
3
1 10
-
) # one chooses to take an epsilon of 3
10 - to avoid the errors
# numerical during division by in the equation [éq 3.2.2-2]
ult

= 1 has
m = ult
m
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If not

()
p
has - has -
p
E
peak
ult
E
=

éq
3.2.2-2
p

1 - has
-
E
E
ult

+
has (
has
p
)
(p
peak
)
=










éq 3.2.2-3
1+ (p
)



apic



p

p
C
p1
has

m ()
()
=
m
+1
-
peak
S (p
)
p
if
<




E

p1
C







éq
3.2.2-4


ae


2
m (
p
p
)
C
p has (
)
p

=
m
if

E



E



p2
C




1
(p 2 2

K)
(p
has
)
=
.










éq 3.2.2-5
3

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

apic

p



lim
m
(p
)

C
p1
has ()

=
m
+1
- S

peak
(p)
p







E
p1
C




that is to say:
apic


has

1
E
C
p
m =
m
+1

éq
3.2.2-6
E
peak






p1
C


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:

- m -
sin = sin ((')
'
1
=
ult


éq
3.2.3.1-1
'+m +1
ult
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with:
~
“=” (
-
I, G S,
T
=

éq
3.2.3.1-2
1
() t0) 1
0
~
-
3
T 0

2
2
2
2
2
S =
S cos (
);
S =
S cos (+

);
S =
S cos (-

);
1
where
is
3 II
2
3 II
3
3
3 II
3
the angle of Lode
I
I
I
1
1
1
=
+ S
; =
+ S
;
=
+ S
1
1
2
2
3
3
3
3
3

~
= with I


that

such
= max
1
I
I
(, j=12,
J
)

3


~
= with I


that

such
= min
3
I
I
(, j=12,
J
)

3

Note:


A condition to respect is that the report/ratio
remain lower than 1. In the case of rocks

hard very resistant, subjected to constraints 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 t0.

Characterization of t0 according to the parameters (has, m, S) characterizing the rock

· Case where (p
S) = 0
Disappearance of cohesion, one poses
0
0 =
T


· Case where (p
S) 0
1 - sin
t0 = 0 (0, C
T
0)
0
= 2C0

éq
3.2.3.1-3
1 + sin0
with:


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


2


has


C =
S
C
,
m S,
C
has
0
0 (
) =


1+
a-1
AMS

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3.2.3.2 Determination of the intersection of the intermediate criterion and the ultimate criterion

By writing the continuity of m in ult one obtains the following relation:


ae




C
p2 has

m (
m

ult)
()
=
ult
E





p2
C



ae


has

2 ult

C
p
m =
m

ult
E






p2
C
1 has
-

has
p E

2
m =
E
m

ult
E




C
1
1 has
-
m
= ult E
éq
3.2.3.2-1
p2
C
ae
me

3.2.4 Flow
plastic

The adopted formalism is rewritten on the basis of model CJS [R7.01.13]. When constraints
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

F
F
G =
-
N N
.
éq
3.2.4-1





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

2 6 sin
= ()
()
= -

3 - sin ()





éq 3.2.4-3

=
p

if

0
> (1 - -

3
10
ult
)
Note:

is positive when P=0 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:
. p
N = 0
& éq
3.2.4-4
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After decomposition of each term in déviatoire parts and hydrostatic, one finds the expression:

(
p
p

N S + N
e&

N S.E.
N

ij
ij
+
ij
v
ij =
p +
p =
1
2
)
1
.
0

3
1 ij ij
2
&
&

v
&
N
'
One deduces the relation from it 1 =
who added to the condition of standardization of tensor N leads to
N
S
2
II
the expression:
S + I
S
N =
II
éq
3.2.4-5
2
+3

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:

p
F
F
= G
=
& &


-
N
&




One also deduces the condition from it relating to the plastic voluminal deformation:

p
= G éq
3.2.4-6
v
& &

Handbook of Référence
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4
Calculation of the derivative

4.1
Derived from the criterion

4.1.1 Derived compared to the constraints
4.1.1.1 Derived intermediate compared to the diverter

G
S
H
One leaves:
= H () II
()
+ S

II
S
S
S
ij
ij
ij
S
H ()
where
II and
are respectively given by:
S
S
ij
ij
S
S
II
ij
=

S
S
ij
II
H ()
1

det (S)
=
1+
54

S
5
3
ij
6h ()

cjs

sij
software house

- cjs cos (
3)

54
cjs
det (S)
=
S +


2h ()
ij
5 s2
6h 5 s3
S
II
()



ij
II

Finally:

G
1

cjs
S

54
ij
cjs
det (S)
=
1+
cos (
3)
+


S
5
2
ij
H ()


2
S

6s
S


II
II
ij




And consequently:

G
1

cjs
S
cjs 54 det (S)
=
1+
cos (
3)
+

éq 4.1.1.1-1
S
H () 5


2





2
software house
6s

II

S



4.1.1.2 Derived intermediate compared to the constraints
G
One poses by definition: Q = Dev.

ij


sij
G
G S

G 1 G

1
kl


=
= Dev.
+








kl
-
ik
jl
ij
kl

S
S
3 S
3
ij
kl
ij


kl
mm


G
1
1 G
1

= Q - Q +

kl ik
jl
ij
kl kl
-
ik
jl
kl
ij kl kl


3
3q
3
ij
mm

G = Q ij

ij
G
It is then enough to take the deviatoric part of
to obtain:
sij
G
G

1

cjs
S

54

ij
cjs
det (S)
= Q = Dev.
=
1+
ij
cos (
3)
+
Dev.


S
5
2
ij

ij
H ()


2
S

6s
S


II
II
ij




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: 15/34

And consequently:

G
1

cjs
S
cjs 54
det (S)
Q =
=
1+
cos (
3)
+
Dev.
éq 4.1.1.2-1

H () 5




2





2
software house
6s

II

S



4.1.1.3 Final expression of derived from the criterion compared to the constraints

The derivative of the criterion compared to the constraints is then:

1
F
1 1 A ()
-
1 A (p
p
)

=
U
G
p

éq
4.1.1.3-1

has
(p has
)

()
0
() Q -


H

C C
with
U
m (p) K (p) 1
1
= -


Q + I
éq
4.1.1.3-2
0



C
6h
3
C


4.1.2 Derived compared to the variable from work hardening

1
2
F

1 G (S) has (p
)
G (S) has

U

=
-



Log

-
éq
4.1.2-1
p



(p has
)
0

0
p
p

C C
H


C C
H






with
U

1
(km)
= -
(p) G 1 (km)
-
(p) (ks)
I +
éq
4.1.2-2
0
1
(p)
p
p
p
p


6


H

3




C
C
C

4.2
Total derivative of the criterion compared to the plastic multiplier

Let us consider the function:
~
2
*
F (
)

-
~
=
E
E
p
F S - 2

µ G, I - K
3

G, +

G
éq
4.2-1
1
II

3


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:

*
F

F

= -
(~

. 2µG + KGI)
F
2 ~
+
G
éq
4.2-2
p
II






3
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4.3
Derived from the parameters compared to the variable of work hardening
S = - 1
p
if

<


E
p



E











éq 4.3-1
S =
p
0
if


E
p

m

= - C
p
if
<

E
S


p1











éq 4.3-2
m
p

= 0
if


E
S

apic
m



has

C
p1
peak
p1
has

= -
Log m
+1
m
+
p
1
if
<


peak

2 peak



E
has


1
has


p

C


C

éq
4.3-3

ae
m




C
p2 has
E
p2 has
= -
p
Log m
m
if



E

2nd


E
has
p2

a.c.
C


(- has has



ult
E)
-

=
E
peak



éq
4.3-4
p

(p
p
) - 1 + (p)


1

()
1 - has
2


E
E

-
ult
(- p

ult
)
has
1 - has
=
peak













éq 4.3-5


(+) 2
1
m
m has
m S

=
+
p
if
< E
p

p has

S p

m
m has

=
if
3
ult (1 -
-
10) > p
E
éq
4.3-6
p

p has

m
- 3
p

= 0
if ult (1 - 10) <
p


1
K
2 2a
2 1

= -
has
Log
ult (1 - -
10 3)
p

>
p

3
3 2 2
p has


éq
4.3-7
K

= 0
if not
p

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5
Tangent operator of speed

The condition
F & = 0











éq 5-1
is written:
.


& =
F
F
& + F p
= 0
ij
p


ij
2
From the expression of the cumulated plastic deviatoric deformation p
p p
=
E E and of
ij ij
3
p
~
relation E
G

& = &, the condition then is found:

F
F
2 ~
F & =
& +
G = 0
ij
&
p


3
II
ij

What gives us for the plastic multiplier:

F &ij
ij
&= -

2 F ~
G
p
II
3

By then considering the relation forced/deformations:

F
F
F
F
F
& =
D & =
D =
D & - &
D G
ij
ijkl
kl
ijkl
ijkl
kl
ijkl
kl





ij
ij
ij
ij
ij

and by deferring it in the expression of & one can write:

F
F
D & - &
D G
ijkl
kl
ijkl
kl


ij
ij
&= -

2 F ~
G
p
II
3
That is to say:
F D
ijkl &kl
ij
= -
&







éq 5-2
2 F ~
F
G -
D G
p
II
ijkl
kl
3
ij
By deferring this result in the expression of & one finds:
ij

F


D

&

ijkl
kl
ij

& =D
&
G

éq
5-3
ab
abcd
+
Cd
Cd

2 F ~
F


G -
D G
p
II
ijkl
kl


3
ij

Handbook of Référence
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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 step 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:

S = S + µ
2 (
p
E
- E
) E
p
= S - 2nd

µ

éq
6.1.1-1
I = I - + 3K
éq
6.1.1-2
1
1
(
p

-

= I - 3K

v
v)
E
p
1
v

One can also express the kinematic condition starting from tensor N (cf paragraph [§3.2.4]):

.
N p
= 0
éq
6.1.1-3

By deferring the two equations translating the elastic behavior in the preceding expression one
find:
1
E
p =
(- S) éq
6.1.1-4

p
1
=
-

éq
6.1.1-5
v
(IE I
1
1)
3K
One then expresses the kinematic condition by the following relation:
S + I
1 E

E

S
.
N
(S - S) 1 1
+

(I
II
1 - I1) I = 0 with N =


3K 3


K

2
+ 3

Maybe by combining the two preceding relations where N indicates the normal of the hypersurface of
deformation:
S + I
1
software house
(- S) 1
+
(IE - I Tr N =
1
1).
() 0
2
2
µ
9
+ 3
K
1
S (. E
S - S)
1

+
(IE - I =
1
1)
0

S
3K
II
Handbook of Référence
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This last equation defines the point (I, S like a projection of the point (E E
I, S) on the criterion.
1
)
1
not (I, S
will be the oblique projection of the point (E E
I, S), projection whose direction varies with. One
1
II)
1
II
can give the chart of it of the chapter [§6.1.3.3].

The preceding relation can then be rewritten as follows:

3K S (E -
E
S
S)
I - I =
-
éq
6.1.1-6
1
1
.
µ
2
software house
One defines the jetting angle then S by the relation:

S (.se - S)
cos =

éq
6.1.1-7
S
software house (- S) (- S)

By deferring the definition of the angle S in the relation of projection one finds the relation:

E
I - I
3K
(1 1
=
-
cos
éq
6.1.1-8
E
S - S) (S - S)
S
E
µ
2

6.1.2 Existence of projection

The principle of this paragraph is to discuss on the question the existence the angle such as
S
projection of the point (E E
I, S) always belongs to the surface of load. These problems appear
1
essential for projections around the node of the surface of load, in other words when
S 0. There is by definition the relation:
S (.se - S)
S (.se - S)
cos =

éq
6.1.2-1
S
S
S
II
(
=
- S) (- S)
- S
II

E
p
E
~
By combining this equation with the expression: S = S - 2µ E

= S - 2µ G


One obtains:
~
.
S G
cos =

éq
6.1.2-2
S
~
S G
II
II
An estimate of coss is sought.
~
S G
.
Stage 1: estimate of

software house

One places oneself in this paragraph under the conditions: S 0 and F = 0.
~
~

Tr (G)
F


By definition of G and G one a: G S
. =
F
G -
I S. = G S. =
-
N N
S.

3











Handbook of Référence
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Key:
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For preoccupations with a simplification of calculation one brings back the resolution of F to the resolution of the equation:
1
G (S) has (p
)
p

F =

- U (
G

has

S
, p
)
()
= 0 F =

p
2
- U (,) () = 0

éq 6.1.2.3
0

0
C C
H
C C
H

By derivation of this new function one finds the relation:

F


1
G
p
has -
2
p
p
1 U
p
1
has -
=

-
1
() U has (



p
p
U
,) ()
=
Q
- () U has (,) ()




0
H




0
H

C C
C C
U
m (p
) K (p
) 1
1
with:
= -


Q + I
0




H
C
6
3
C

Who gives after simplification:

F
2 = Q
WITH + I
B
éq
6.1.2.4

Where:

1
(p has
) m (p
) K (p
)
p
- 1 has
With =
1+
U (, p
) ()

0

C C
H
6


éq
6.1.2.5

(p has
) m (p
) K (p
)
p
has -
B =
U (, p
) () 1


0
3 C C
H
S + I
F
software house

With
3B
One has as follows:
2 .N = (AQ + BI).
=
.
Q S +

2
+ 3
2
+ 3
2
software house
+ 3

And consequently:

~
F
F
G S
. =
-
N N
S.











S


+ I



With
3

=
B
S
Q
WITH + I
B -
Q S
. +
.

II
S.



2
+ 3
2
software house
+ 3
2
+ 3






3A
3B
=
.
Q S -
S
2
2
II
+ 3
+ 3

From where it is deduced that:
~
G.s
3A
.
Q S
3
B
=
-
éq
6.1.2.6
2
software house
+ 3
2
software house
+ 3

Handbook of Référence
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Titrate:
Law of behavior of Laigle


Date:
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Key:
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By definition of Q one a:
G
1
cjs
S

54
cjs
det (S)
Q S
. = Dev.
S. =
1+
cos (3)
+
Dev.
.

S

H ()
S
5
2







software house
6 2
S

II

S


1


cjs
=
1+
cos (
3)

H ()
S
5
II





2

= H () S
II

One expresses finally:
~
G.s
3A

B
=
H ()
3
-
éq
6.1.2.7
2
software house
+ 3
2
+ 3
1
When S 0 then U (, p
) 0
and
With
, B 0
0
H
C C
And thus:

~
G.s
3h ()
When S 0 then




éq
6.1.2.8
0
S
S
II
0
C C
H (2
+ 3)

~
Stage 2: estimate of G
II
1
One places oneself in this paragraph under the conditions: S,
0 A
, B 0
0
H
C C
~
F
F
G =
-
N


S


+ I



1


1
3B

software house

=
Q + I
B -
Q S

0
H

. +
0
2
2
H S
C C


.
2


+3 C C II
+ 3
+


3


2
1
H ()
=
Q -

h0
2
+ 3 h0s
C C
(
)
S
C C
II

2
4
2
~
Q
H S

2
~ ~
H
II
() 2
2
2
II
()
G = G.G =
+
-

II
(H
+ H S
+ H
C c)
2
2
0
(
) 2
2
3 (C c)2
0
2
II
(2) 3 (C c)2
0

2
2
2
1


H
2
(+6) ()


= (
Q


II
H
C c)2
0

-
2

(2 +)
3

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It is shown [Annexe 2] that:

2
2
1

2



cjs

cjs

cjs

Q =
1+
cos
éq 6.1.2.9
II
(
3) +
+ cos
cjs
(
3) 1+
cos (
3)
H ()

10







2

4

2




and thus like H () = (1 + cos
:
cjs
(3) 1/6


2
~
H
cjs
H

H
2
1


1
1
() 6
2

6
1

6
2 2
2
G =

+
+
+
6
H
1
II
2
10
(
-)
()
(+) ()
(0
H
C c)
()
+
-

H ()






2
2
4
2
2
2
2


(+) 3




~
1
3h
1

1
cjs

H
2
() 2
2
2
2
2

-
(+6) ()
G =

II
(hcc) 2
0

+
4
2h () +
4
4h () -
10
2

(2 +)
3

2
2
~
H

2
()

1
2 - 1

cjs

3

G =

+
+
- 1
II


0
H
C C 2h () 6
4h ()

12


2


+ 3
4



And consequently:
~
H ()
2
2

3

1
1
- 1
G
cjs
=
- +
+
II


0
2
H





éq
6.1.2.10
+
C C

3
4

2h () 6
4h () 12

Stage 3: estimate of cos
S

One deduces from the two paragraphs precedent the expression of the following jetting angle:
When S 0 then:

3
cos

éq 6.1.2.11
S
2
2

1
-
(
3 1
1

cjs
2
+) 3
- +
+
2
3
+ 4
(
2 1+
(
cos 3))
cjs
(
4 1+
(
cos 3)) 2
cjs

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

cos min

cos cos max
S
S
S

min
3
cos
=
S
2
with
:

2




2
3

cjs

éq 6.1.2.12

(+3)
+

2


+ 3
4 (1 - 2
cjs)


cos max

= 1
S
Handbook of Référence
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6.1.3 Rules of projection

0 are called
I the intersection of the field of reversibility with the hydrostatic axis. One obtains:
1
0

3
S (p

C
)
I =
.

éq
6.1.3-1
1
m (p
)
By deferring 0
I and the framing of cos, when S 0, in the relation
1
S
E
I - I
3K
(1 1
=
-
cos, one deduces the following rules of projection from them according to
E
S - S) (S - S)
S
E
µ
2
sign parameter of dilatancy, and for values of E
I and of E
S given:
1
II

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

2
S
µ
II

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

2
S
µ
II

6.1.3.3 Interpretation
graph

(E E
I, S)

1
II

software house
3K
min

-
cos
3K
2
S
µ
max
-


cos
2
S
µ

Zone of regular projection

Intermediate zone

(E E
I, S)

1
II

Zone of projection at the top

0
I
I
1
1
Handbook of Référence
R7.01 booklet: Modelings for Génie Civil and the géomatériaux ones
HT-66/05/002/A

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

6.1.3.4 Equations
of flow

In the intermediate zone 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:

S = 0 éq
6.1.3.4-1
0

3
S
. (p

C
)
I
=

éq
6.1.3.4-2
1
m (p
)
p
1
2nd


=
S éq
6.1.3.4-3
II
2µ 3

6.2
Local Recutting of the step 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 step of
time in order to facilitate numerical integration. This possibility is managed by the operand
ITER_INTE_PAS of key word CONVERGENCE of operator STAT_NON_LINE. If 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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Titrate:
Law of behavior of Laigle


Date:
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Key:
R7.01.15-A Page
: 25/34

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.
2
p
p p
=
E E
ij ij
3
tr
p
p
(pij)
with E = -

ij
ij
ij
3

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: p
= G
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 p
= the 0 then material is in field 1;
- If
p
0 < < E then the material is in field 2;
- If
p
< < then the material is in field 3;
E
ult
- If p
> then the material is in field 4.
ult



State field of the rock
0 Intact
1 Damage
pre-peak
2 Damage
post-peak
3 Fissured
4 Fractured








Handbook of Référence
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Code_Aster ®
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7.4
Titrate:
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Date:
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Author (S):
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Key:
R7.01.15-A Page
: 26/34

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 step, whereas the direction of flow is that
calculated at the beginning of the step (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 E is deduced
and
;
v
· Constraints at the beginning of the step -
from where one deduces -
S and -
I;
1
· The values of the variables intern at the beginning of the step of time (only the plastic deformation
-
cumulated p is necessary).

It is a question of calculating:

· Constraints at the end of the step of time;
· The variables intern in end of the step of time (p
, p
, fields of behavior);
v

· The tangent behavior at the end of the step:




8.1
Calculation of the elastic solution

E


=

- - T

E
S
=
-
S
+ 2µe
E
I
= I -
+ 3K

1
1
v

8.2
Calculation of the elastic criterion

Calculation of E
E
G = S H (E
)
II

m = (-
-
p
m
) S = (-
-
p
S
) has = (-
-
p
has
)
Calculation of
,
,
and -
K = K (-
has)

- -
E
- -
E
m K G
m K
Calculation of
E
-
-
U = -
-
I + S K
6 h0

1
3
C
C
C
1
E

has
G
-
Calculation of E
E
F =
- U



h0
C C
Handbook of Référence
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Code_Aster ®
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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 Page
: 27/34

8.3 Algorithm

If E
F > 0
Calculation of:
-
-
S
0

3.
C
-
I
=
;
G = G S
1
-
(-)
m
-
= m S.A.
C
C m S.A.
C
0 (
-
-
-
,
)
-
;
= 0 (- - -
,
)
-
;
= t0 (- -
,
)
0
0
T 0
0
0
-
= '(- - -
I, G,)
-
;
= (-
')
-
;

1
T 0
= (-
)
Calculation a priori of projection at the top
-
1
2
3 S
. (p

C
)
S = 0
node
; Calculation of p
p
E
p
=
+
S =
and of I =
= I
.
II
2µ 3
1
m ()
node
p
1
(I.E.(internal excitation) - node) < - 3K
I
- E
S
S
1
1
cosmax
-

if

;

II
<
0



µ
2
If (

E
I - node) < - 3K
I
- E
S
S
1
1
cosmin
-

if

;

II

0



µ
2
Projection at the top is not retained a priori. The regular solution is calculated.

Q (-


F

G (- -
,) -
if
F

N (- -
,) -
-
)
-
if 0
if 0
0
Q =
N
=
G
=


Q (E)
-
if = 0

N (E, E)
-
if = 0

G (E, E)
-
if = 0
-
If p

= 0
0
-
Initialization
0
p
p
0
E
0
E
0
E
= 0;
=
;S = S; I = I; F = F
1
1

1p
= 1 max E



10
ij
And
1

p

1

p

=
3

~

Bfr
G
2
II
If not
Calculation of the increase in the plastic multiplier
by Newton:
0
-
Initialization
0
p
p
0
E
0
E
0
E
= 0;
=
;S = S; I = I; F = F
1
1

U
0
U
-
m K -
-
- m

=
= -
Q - K
I



6 h0

3
C
C
C
U 0
1
(km)
E
= -
G
1
km
ks


I


p
(- p
p
)
()
-
0
(- p
p
) E ()
+
1
(- p
p
)


6

H

3


C
C
C
1
F 0
1 1
- -
-
1 A
has
0
E
U
F
-
-

-

=
G has

-
0
() Q -







has
H



C C
1
2
0
E
-
E
has
F

1 G
G has

0
-
U

F
-

=
-
log

-

-

p


0


0
(p
p
) p
p


H has
H






C C
C C
F 0 *

F 0

~
F 0

2 ~

= -
(F
F
. 2ΜG + KG I)
F
+
G
p
II






3
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Code_Aster ®
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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 Page
: 28/34

Loop iterations N
N
F

n+1
N
= - F


N 1
+
N
N 1
+


=
+
N 1
+
2 ~
N 1
p
N 1
F
p
p +
+
F


=

G
;


=

G
II
v
3
N 1
+
+
~
N 1
N 1
E
p
F
N 1
E
p +
+
F
S
= S - 2

µ
G
;
I
= I - K
3


G
1
1
N 1
If
+
p

< 0 Not convergence
Calculation
N 1
+
Q
N 1
+
G
= G (N 1+
S);

N
m
= m (N 1
1
+
+
p

); NS = S (N 1
1
+
+
p

); Na = has (N 1
1
+
+
p

); N 1+
K
= K (N 1+
has);

N 1
+
N 1
+
N 1
+
N 1
+
N 1
+
N 1
+
m K
G
m K
N 1
+
N 1
+
N 1
+
U
= -
-
I
+ S
K

0
1
6
H
3
C
C
C
1
N 1
+
N 1
G
+
has
N 1
+
N 1
+
F
=
-


U

0
H
C C
n+1
n+1
n+1
N
U

m
K
1
+1
+1 m +
= -
Qn
N
- K
I

6 h0

3
C
C
C
n+1
U
1
(km)
n+
n+
1
1
p
G
1


(km) n+1
p
n+



(ks)
= -
n+


-

I
+

1
1
p




p
6

p
0
H

C
C
C

p
1
3

p

1
N 1
N 1
+
N 1
F
1 1
- +
+
has
n+
has
n+
N
U
+
N

1
+
=
has
G
Q
-

N 1
+
0
(1) 1
1
1



has
H



C C
1
n+
2
1
n+1
n+1
N
has
F

1 G

G +1 has

+1
n+1
U
=
-
log
+


-
p
N 1


0


0
() N
p
p
p


H has
H




C C
C C
N 1 *
+
N
F

F 1
+
+
= -
(~
1

F
2 ~
. 2ΜG + KG I)
N
F
F
F
+
G
p
II






3
If
N
F +1/>

C
prec
n=n+1
If N > no. ite interns max

(I.E.(internal excitation) - node) > - 3K
I
E
S


1
1
cos min

if

;

S
II
< 0



If

(I.E.(internal excitation) - node) > - 3K
I
E
S


1
1
cos max

if

;

S
II
0



node
One retains projection at the top:
node
p
p
S = 0; I = I
;
=
1

1
If not
Not convergence
If not
Not convergence
If not
Convergence
Handbook of Référence
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HT-66/05/002/A

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Titrate:
Law of behavior of Laigle


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Author (S):
R. FERNANDES, C. CHAVANT
Key:
R7.01.15-A Page
: 29/34

If FULL_MECA
Calculation of:
T
N
+1
F
F
H G
.
.
H
N
+1





= H


+

T
n+1
N

2 F

~
+1
F
F
F
G -
HG
p
II


3


Mechanical symmetrization:
n+
1

n+1
+ T
N 1
sym = 1

+


2




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 constraints, by a retiming on triaxial in compression and extension by
following relations:

9.1 Formulation
general
S.A. p
II
(1)
F =
-
p




U (,) 0 éq
9.1-1
C
Where the expression of (
p
U,) is:

If
0
cjs


+ 0 - 0
2



U (,)
m (p K
H
H
H
p
) (p) ()
m
K
T
C
(p) (p)
= -
-
I + S

éq 9.1-2
0
0
1
(p
) K (p
)



6
H - H



3
C

T
C

C
If
= 0
cjs


3 1



U (,)
m (p
p
) K (p)
= -
+ cos (
3)
m (p) K (p)
-
I + S
1
(p) K (p) éq
9.1-3
6
2 2


3
C
C
Handbook of Référence
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Code_Aster ®
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Titrate:
Law of behavior of Laigle


Date:
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Key:
R7.01.15-A Page
: 30/34

9.2
Pace of the thresholds

One places oneself if
=

;
7
.
0
m =

;
21 S =
;
1 A = 1, then one traces the pace of the thresholds in
cjs
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 I = 0 [Figure 9.2-a] and I = -
3

1
1
C
[Figure 9.2-b].
I1/3SIGC=0
0,1
0,05
)
T
has
(
T
E
0
Formulation
S
O
(a)
- 0,15
- 0,1
- 0,05
0
0,05
0,1
0,15
Version 2
)
*
C
Formulation
C
Version 1 (a)
compression
ig
- 0,05
II/S
(
S
- 0,1
- 0,15
(Software house/sigc) * Sin (teta)
Appear 9.2-a: Allure of the thresholds for a null containment

I1/3SIGC=-1
2
1,5
1
)
T
has
0,5
(you
Formulation
S
O
0
(a)
Version 2
Formulation
) * C - 2
- 1
0
1
2
C
- 0,5
Ve
Co rsion
mpre 1
ssibi
O S
N
I
G
I
I
/
S
- 1
(S
- 1,5
- 2
- 2,5
(Software house/sigc) * Sin (teta)
Appear 9.2-b: Allure 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.
Handbook of Référence
R7.01 booklet: Modelings for Génie Civil and the géomatériaux ones
HT-66/05/002/A

Code_Aster ®
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7.4
Titrate:
Law of behavior of Laigle


Date:
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Author (S):
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Key:
R7.01.15-A Page
: 31/34

10 Bibliography

[1]
F. LAIGLE: Downstream of the cycle ­ Ouvrages rheological undergrounds ­ Modèles for the analysis of
rock mechanics behavior. Note EDF-CIH IH.AVCY.01.003.A (2001).
[2]
PH. KOLMAYER: Downstream of the cycle ­ Ouvrages undergrounds ­ 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.
Handbook of Référence
R7.01 booklet: Modelings for Génie Civil and the géomatériaux ones
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Code_Aster ®
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7.4
Titrate:
Law of behavior of Laigle


Date:
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Key:
R7.01.15-A Page
: 32/34

Appendix 1 Recalage 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:

1
G (S) (p) m (p
has
) K (p
) G (S) m (p
) K (p
)
F =
- -
-
I + S
0
0
1
(p
) K (p
)






H

6
H

3
C C

C
C
C

1
2
(p has
)

2


- H


H
1
3
p
p
-

3

1 m () K ()
1
3
3
m (p
) K (p
)

=
+
+
(+
2

0
0
1
3) - S (p
) K (p
)






6

3

H
H
C C

C
C
C
C





1
has (p)
2



-
1
3
(
m p) K (p) 2
(
m p) K (p)


=
3

+
- +
(+2) - S (p) K (p)



1
3
1
3




6
3
3
C




C
C



1
has (p)
2


1
m p K p
m p K p

=
3
(
-
)
(
) (
)
(
) (
)

has (p)

+

-
+
(+2) - S (p) K (p)

1
3
1
3
1
3
3
3
C




C
C


1
has (p)
2


1
m p K p
m p K p

=
3
(
-
)
(
) (
)

has (p)

+
(
-)
(
) (
)
+
(+2) - S (p) K (p)

1
3
3
1
1
3
3
3
C




C
C


1
has (p)
2


1
m p K p

=
3
(
-
)
(
) (
)

has (p)

+
() - S (p) K (p)

1
3
3
C



C


1
2 A (p
)


1
3
1
p
p
2


=
(-
1
3)
m
(p has
)
has () ()
-

(- 3) + S (p
)








C
3
C





1
2 A (p
)



1
1

p
3
m



=
(-
1
3)
()
(p has
) - (c)

p


has (p)
(- 3) + S ()


C








C




Handbook of Référence
R7.01 booklet: Modelings for Génie Civil and the géomatériaux ones
HT-66/05/002/A

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

Appendix 2 Normalization of Q

1

cjs
S

54
cjs
det (S)
Q =
1+
cos (
3)
+
Dev.

H () 5


2





2
software house
6.sII
S



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




S


1


3
54
2
cjs
2
2



cjs
D
D
cjs

cjs

Q =.
Q Q =
1+
cos 3
.
T .t
1
cos 3
.
S T

II
() +
+
+
() D
H ()

10


4
3





2

2 S
3 S
.
2
II
II






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

S
S
2 S - S S - S S
2 3
1 2
1 3
1
S


S
2 S - S S - S S
2
1 3
1 2
2 3
S
D
1 S
2 S - S S - S S
3
As follows: s=
1 2
1 3
2 3
and T =

0
3
0

0

0




0


0



By using the property of S: S + S + S = 0
4
2 2
2 2
2 2
1
2
3
, it is shown that S
= 4
II
(1ss2 + 1ss3 + s2s3) and by
consequent:
S
2 S - S S - S S
S
2 S - S S - S S
2 3
1 2
1 3
2 3
1 2
1 3
S
2 S - S S - S S
S
2 S - S S - S S
1 3
1 2
2 3
1 3
1 2
2 3
-
-
-
-
D
D
1
S
2 S
S S
S S
S
2 S
S S
S S
1 2
1 3
2 3
1 2
1 3
2 3
s4
T .t =


.

II
=

9
0
0
6
0
0
0
0

One also shows starting from the property S + S + S = 0
3
3
3
1
2
3
that S + S + S = 3s S S =.
3 det (S)
1
2
3
1 2 3
and by
consequent:

S
S
2 S - S S - S S
1
2 3
1 2
1 3
S
S
2 S - S S - S S
2
1 3
1 2
2 3
. 54
. 54
cjs
D
cjs
S
S
2 S - S S - S S
. 54
.
S T =
3
.
1 2
1 3
2 3 = cjs
det (S) = .cos

3
3
3
cjs
(
3)
S
.
3
S
.
9
0
0
S
II
II
II
0
0
0
0
One deduces some as follows:

2
2
1

2



cjs

cjs

cjs

Q =
1+
cos

II
(
3) +
+ cos
cjs
(
3) 1+
cos (
3)
H ()

10







2

4

2



Handbook of Référence
R7.01 booklet: Modelings for Génie Civil and the géomatériaux ones
HT-66/05/002/A

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

Appendix 3 Encadrement of the jetting angle

3
It is pointed out that coss





S 0
(

3

1
1

- 1
2
+)
2
2
3
cjs

- +
+

2
+ 3
4
2

(1+ cos
cjs
(3)) 4 (1+ cos
cjs
(3))2

2
1
- 1
One poses: X () =
+
where [,
0 2
[

2 (
cjs
1+ cos
+

cjs
()
(41
cos () 2
cjs

It is noted that: X (-) = X (), function X being even one restricts the interval of study at [,
0 [
.

dX
sin
cjs
()
The resolution of
= 0 give
.
+
=
3
cjs (
.
cos
cjs
) 0
D
2 (1+ cos
cjs
()
()

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

X (=) = 1
0


4


(cjs) =
1
X
4 (
where
cos

that

is such


1 - 2cjs)
cjs
(cjs) = -

cjs


One can thus give the framing of cos according to:
min
max
cos
cos cos
with:
S
S
S
S


min
3
cos
=
S
2
2




2
3

cjs

(+) 3
+


2


+ 3
4 (1 - 2
cjs)


cos max

= 1
S

Handbook of Référence
R7.01 booklet: Modelings for Génie Civil and the géomatériaux ones
HT-66/05/002/A

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