Code_Aster
®
Version
7.4
Titrate:
Law of behavior of the porous environments: model of Barcelona
Date
:
31/03/05
Author (S):
J. EL GHARIB, G. DEBRUYNE
Key
:
R7.01.17-A
Page
:
1/44
Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
Organization (S):
EDF-R & D/AMA
Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
Document: R7.01.14
Law of behavior of the porous environments: model
from Barcelona
Summary:
The model of Barcelona [bib1] described the behavior soil mechanics unsaturated coupled with
hydraulic behavior (this model must thus be used in an environment THHM [bib7]). In the case
private individual of a ground completely saturated with water, it is reduced to the model Camwood modified Clay, also
implemented in Code_Aster [bib5]. He is particularly adapted to the study of the behavior of clays.
Code_Aster
®
Version
7.4
Titrate:
Law of behavior of the porous environments: model of Barcelona
Date
:
31/03/05
Author (S):
J. EL GHARIB, G. DEBRUYNE
Key
:
R7.01.17-A
Page
:
2/44
Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
1 Notations
T
indicate the tensor of the total stresses in small disturbances, noted in the shape of the vector
according to:
31
23
12
33
22
11
2
2
2
T
T
T
T
T
T
The behavior is described in a space of stresses to two variables:
I
p
gz
T
+
=
and
lq
gz
C
p
p
p
-
=
,
with
C
gz
lq
p
p
p
,
,
respectively pressure of fluid, gas pressure, capillary pressure (or
suction)
One notes:
I
the tensor unit of command 2 whose indicielle notation is
ij
4
I
the tensor unit of command 4 whose indicielle notation is
ijkl
()
tr
P
3
1
-
=
stress of containment
Pi
S
+
=
diverter of the stresses
()
S
S
tr
I
.
2
1
2
=
second invariant of the stresses
2
3I
Q
eq
=
=
equivalent stress
(
)
U
U
T
+
=
2
1
total deflection
HT
p
E
+
+
=
partition of the deformations (elastic, plastic, thermal)
()
(
)
0
3
T
T
tr
v
-
+
-
=
voluminal total deflection
()
p
p
v
tr
-
=
voluminal plastic deformation
I
v
3
1
~
+
=
diverter of the deformations
p
E
~
~
~
-
=
deviatoric elastic strain
Code_Aster
®
Version
7.4
Titrate:
Law of behavior of the porous environments: model of Barcelona
Date
:
31/03/05
Author (S):
J. EL GHARIB, G. DEBRUYNE
Key
:
R7.01.17-A
Page
:
3/44
Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
I
p
v
p
p
3
1
~
+
=
deviatoric plastic deformation
(
)
E
E
E
eq
tr
~
.
~
3
2
=
equivalent elastic strain
(
)
p
p
p
eq
tr
~
.
~
3
2
=
equivalent plastic deformation
E
index of the vacuums of the material (report/ratio of the volume of the pores on the volume of the solid matter constituents)
0
E
initial index of the vacuums
porosity (report/ratio of the volume of the vacuums on total volume (pores plus grains))
p
lq
E
lq
lq
,
,
content of total, elastic and plastic fluid
coefficient of swelling (elastic slope in a hydrostatic test of compression)
S
elastic coefficient of rigidity in a test of variation of suction
)
1
(
0
0
E
K
+
=
S
S
E
K
)
(
0
0
1
+
=
)
(
C
p
coefficient of compressibility (plastic slope in a hydrostatic test of compression)
*
coefficient of compressibility in conditions of saturation
S
coefficient of compressibility plastic in a test of variation of suction
)
(
)
1
(
0
-
+
=
E
K
)
(
)
(
S
S
S
E
K
-
+
=
0
1
M
critical line slope of state
coefficient of correction of the normality of the plastic flow
)
(
C
idiots
p
P
pressure of consolidation
)
(
C
Cr
p
P
critical pressure, variable interns model, equal to half of the pressure of consolidation
*
Cr
P
pressure criticizes in conditions of saturation
S
P
cohesion (hydrostatic traction limits to suction given)
Code_Aster
®
Version
7.4
Titrate:
Law of behavior of the porous environments: model of Barcelona
Date
:
31/03/05
Author (S):
J. EL GHARIB, G. DEBRUYNE
Key
:
R7.01.17-A
Page
:
4/44
Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
0
P
confining pressure of reference generally equal to the atmospheric pressure
has
P
C
K
slope of cohesion according to suction
parameter controlling the increase in
)
(
C
p
with
C
p
R
parameter defining the peak of
)
(
C
p
with
C
p
µ
elastic coefficient of shearing (coefficient of Lamé)
1
F
surface of load in space
)
,
(Q
P
2
F
surface of load in
C
p
0
C
p
threshold of irreversibility of suction
plastic multiplier
lq
S
water saturation,
lq
lq
S
=
p
vp
voluminal plastic deformation due to a loading in hydrostatic pressure
p
vs
voluminal plastic deformation due to a loading in suction
p
p
~
deviatoric plastic deformation due to a loading in hydrostatic pressure
B
coefficient of Biot
Code_Aster
®
Version
7.4
Titrate:
Law of behavior of the porous environments: model of Barcelona
Date
:
31/03/05
Author (S):
J. EL GHARIB, G. DEBRUYNE
Key
:
R7.01.17-A
Page
:
5/44
Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
2 Introduction
The concepts of plasticity used for the water-logged soils are extended on the unsaturated ground. The model
original of Barcelona is described using the variables
C
p
,
, which distinguishes it from the models from
mechanics coupled to a thermohydraulic behavior which is described using only one
effective stress (stress of Bishop). One can notice that this model is rewritten within a framework
poroplastic with the introduction of an additional poroplastic variable which is the water content
[bib2], allowing to collect the phenomena of hystereses which appear on the drying cycles
damping. This phenomenon is not taken into account in the here exposed original model.
2.1
Phenomenology of the behavior of the unsaturated grounds
2.1.1 Curve of retention of water
In addition to the main common mechanical aspects with the water-logged soils [bib3], the porous environments
comprising liquid and gas phases (grounds unsaturated with water) have as a characteristic
specific to be very sensitive to the phenomena of capillarity. The latter correspond to
localization of meniscuses of fluid (increasingly small as ground désature)
in which the water pressure is lower than the pressure of air (and all the more low as it
meniscus is small and thus the désaturé ground). One thus sees appearing the concept of pressure capillary or
suction
)
(
lq
gz
C
p
p
p
-
=
. While drying, a ground unsaturated has a water content
lq
weaker what
corresponds to a higher suction. The correspondence between these two sizes is the curve of
retention of water (cf [Figure 2.1.1-a]). The aforementioned is obtained by drying of a ground initially saturated (
suction is then null) and damping starting from the dry state.
lq
C
p
Appear Curved 2.1.1-a: of retention of water
2.1.2 Extension of the definition of the effective stresses on the unsaturated ground
The behavior soil mechanics unsaturated is primarily observed in laboratory with aid
aircraft with controlled suction (oedometers and triaxial). The modeling of this behavior
mechanics was initially tried by extending the concept of effective stress to the unsaturated mediums.
The aforementioned is a function of the total stress and intersticielle pressure:
)
,
(
'
lq
T
p
F
=
. In
the saturated case, one has simply additivity of the pressure and the stress:
I
p
lq
T
-
=
'
because
pressure of water acts in the same way in water and the solid in all the directions.
The widening of this concept in the mediums unsaturated in the years 1950 (holding account with
Code_Aster
®
Version
7.4
Titrate:
Law of behavior of the porous environments: model of Barcelona
Date
:
31/03/05
Author (S):
J. EL GHARIB, G. DEBRUYNE
Key
:
R7.01.17-A
Page
:
6/44
Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
pressures of the two liquid phases) brought the following form of the effective stress:
)
(
)
(
'
C
gz
T
p
G
I
p
+
-
=
there remained the stress suggested in the form:
C
lq
gz
T
dp
bS
p
D
D
+
-
=
)
(
'
where
lq
S
is the degree of saturation out of water and B the coefficient of Biot [bib7].
As of the years 1960, the experimental observation clarifies certain limitations of the concept of
effective stress extended on the unsaturated ground. In particular, the test of hull slamming to the oedometer
at fault the stress of Bishop puts: This test consists in consolidating a sample unsaturated with
maintaining suction constant, then to moisten again it with constant loading. One then is observed
hull slamming of the ground. If the consolidation is continued, the curve corresponds to a standard test in
saturated. However, if one refers to the effective stress, the aforementioned decreases during the remoistening (since
suction
)
(
lq
gz
C
p
p
p
-
=
cancel yourself) and as it is supposed to control the deformation, it would owe y
to have swelling what is contradictory with the experimental observation. Majority of
mechanics of the grounds agree now on impossibility of describing it completely
behavior of the grounds unsaturated using only one stress and note the need for using
two independent variables (stress and suction).
3
Description of the original model of Barcelona
In this model, the curve of retention of water does not have hysteresis, and it is not modified
by the mechanical deformation as it is the case in the presentation made by Dangla and coll [bib2]. It
exist nevertheless a threshold in capillary pressure
0
C
p
with beyond which unrecoverable deformations
appear. In this paragraph one distinguishes a mechanical part which treats deformations
mechanics induced by a mechanical loading and a hydro-mechanical part which treats effect
suction on mechanics before writing the equations of the complete behavior.
3.1
Purely mechanical behavior
One makes the assumption that suction
C
p
remain constant during the mechanical transformation.
deformations resulting from the variation of the stress are subscripted
p
.
One examines the behavior, under successively spherical and deviatoric loading, it
behavior being considered isotropic.
3.1.1 Spherical loading
3.1.1.1 Elasticity
The mechanical state of a ground unsaturated under hydrostatic stress is determined by tests
oedometric with controlled suction. As for the water-logged soils, volume
v
sample varies
logarithmiquement with the load with a slope
in a reversible way until a pressure of
consolidation
()
C
idiots
p
P
. One will choose
independent of
C
p
, the experiment showing weak
dependence of the elastic slope with respect to
C
p
.
The elastic component of the voluminal deformation varies then like:
)
(
if
0
1
C
p
idiots
P
P
P
P
E
E
vp
<
+
=
&
&
éq 3.1.1.1-1
Code_Aster
®
Version
7.4
Titrate:
Law of behavior of the porous environments: model of Barcelona
Date
:
31/03/05
Author (S):
J. EL GHARIB, G. DEBRUYNE
Key
:
R7.01.17-A
Page
:
7/44
Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
The preceding expression is in fact derived from a test oedometric with constant suction where one
measure the variation of the index of the vacuums according to the loading, from where the following elastic law:
(
)
[
]
p
vp
vp
K
P
P
-
=
0
0
exp
éq 3.1.1.1-2
with
(
)
0
0
1
E
K
+
=
, where
0
P
is the value of reference corresponding to
0
=
E
vp
and
0
E
E
=
, index of
vacuums initial.
3.1.1.2 Plasticity
Beyond the pressure of consolidation, the behavior of the ground is plastic and the slope
()
C
p
is
dependant on suction (cf [Figure 3.1.1.2-a]), this dependence being estimated way
semi-empirical following:
() () (
) (
)
[
]
R
p
R
p
C
C
+
-
-
=
exp
1
0
where
(
)
()
0
=
C
p
R
is a constant connected to the maximum of the rigidity of the ground and
a parameter which
control the evolution of rigidity according to suction.
The voluminal rate of deformation is then:
()
P
P
E
p
C
vp
&
&
0
1
+
=
if
idiots
P
P
>
,
from where the plastic component:
()
(
)
P
P
E
p
C
p
vp
&
&
0
1
+
-
=
.
The expression of
P
is thus written:
()
[
]
p
vp
K
P
P
exp
0
=
éq
3.1.1.2-1
with
(
)
-
+
=
0
1
E
K
Note:
The two expressions [éq 3.1.1.1-2] and [éq 3.1.1.2-1] are similar to those of the model of
Camwood-Clay [bib5] with the parameter
(or
K
) depend on the capillary pressure.
compressibility of the ground decreases with suction.
Code_Aster
®
Version
7.4
Titrate:
Law of behavior of the porous environments: model of Barcelona
Date
:
31/03/05
Author (S):
J. EL GHARIB, G. DEBRUYNE
Key
:
R7.01.17-A
Page
:
8/44
Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
P
ln
v
(
)
2
C
C
p
p
=
(
)
1
C
C
p
p
=
(
)
2
1
C
C
p
p
>
()
2
C
p
()
1
C
p
2
0
P
()
2
C
idiots
p
P
()
1
C
idiots
p
P
1
0
P
Appear 3.1.1.2-a: Variation of specific volume under loading oedometric
3.1.2 Loading
triaxial
3.1.2.1 Elasticity
The elastic component of the deviatoric deformation is proportional to the loading:
µ
2
~
S
E
=
éq 3.1.2.1-1
µ
is independent of suction.
3.1.2.2 Plasticity
Into a triaxial compression test of revolution, one introduces the shear stress
3
1
-
=
Q
(one will be able
to extend the formulation which follows with the 3D by using the standard within the meaning of von Mises of the stress).
When suction becomes null (saturated medium), the model is supposed to be reduced to the Cam_Clay model
modified [bib5]: the threshold of plasticity is then an ellipse of center
)
,
(
*
0
Cr
P
who cuts the axis of
hydrostatic stresses in zero and a value of pressure of consolidation
*
*
2
Cr
idiots
P
P
=
.
surface of load associated with a suction
C
p
nonnull is also an ellipse of center
)
0
,
2
)
(
(
S
C
Cr
P
p
P
-
(cf [Figure 3.1.2.2-a]) which cuts the hydrostatic axis in
)
(
2
)
(
C
Cr
C
idiots
p
P
p
P
=
and
S
P
-
,
S
P
representing a cohesion varying linearly with suction:
C
C
S
p
K
P
=
. The line
representing the critical states (null voluminal variation) the same slope preserves
M
that that in
saturated but shifted condition S
P
. The equation of the surface of load in the diagram
(
)
Q
P,
for
C
p
data is written:
(
) (
)
0
2
2
2
=
-
+
-
P
P
P
P
M
Q
Cr
S
éq
3.1.2.2-1
Code_Aster
®
Version
7.4
Titrate:
Law of behavior of the porous environments: model of Barcelona
Date
:
31/03/05
Author (S):
J. EL GHARIB, G. DEBRUYNE
Key
:
R7.01.17-A
Page
:
9/44
Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
The plastic flow in the plan
(
)
Q
P,
thus with
C
p
constant is not associated the surface of
charge. If it were the case, one would have:
S
F
P
F
p
p
vp
=
=
1
1
~
&
&
&
&
and the following report/ratio:
(
)
Cr
S
p
vp
p
eq
P
P
P
M
Q
2
2
2
~
2
-
+
=
&
&
,
éq
3.1.2.2-2
similar to the report/ratio obtained in the Camwood-Clay model (with
0
=
S
P
). In fact in this model, one
introduced a parameter of correction
who destroys the character of normality, so that:
(
)
Cr
S
p
vp
p
eq
P
P
P
M
Q
2
2
2
~
2
-
+
=
&
&
.
is given by the authors of the model [bib1] as being:
(
) (
)
(
)
()
-
-
-
-
=
0
1
1
6
9
3
9
M
M
M
M
éq
3.1.2.2-3
This corrector allows to better take into account the experimental results, and in particular of
to better estimate the coefficient of thorough grounds.
M
M
Q
S
P
-
P
(
)
0
>
C
p
(
)
0
=
C
p
*
idiots
P
)
(
C
idiots
p
P
Appear 3.1.2.2-a: Criterion in space
(
)
Q
P,
3.2
Hydro-mechanical coupling or effect of suction on mechanics
The variations of suction (with constant load) involve deformations (those will be then
subscripted by
S
) reversible when
0
C
C
p
p
<
and irreversible when suction exceeds the threshold
0
C
p
.
Code_Aster
®
Version
7.4
Titrate:
Law of behavior of the porous environments: model of Barcelona
Date
:
31/03/05
Author (S):
J. EL GHARIB, G. DEBRUYNE
Key
:
R7.01.17-A
Page
:
10/44
Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
3.2.1 Part
reversible
The tests oedometric with constant stress and controlled suction give us the variation of
the index of the vacuums according to suction [Figure 3.2.1-a] reversible in lower part of the threshold in suction:
0
if
0
C
p
C
p
atm
C
S
p
p
Ln
E
E
<
-
=
-
,
with
S
independent of the state of containment.
Deformation being able to be written:
0
0
0
1
E
E
E
v
v
+
-
-
=
-
, one a:
(
)
atm
C
C
S
atm
C
C
S
E
vs
p
p
p
K
p
p
p
E
+
=
+
+
=
&
&
&
0
0
1
1
éq
3.2.1-1
The evolution of suction is written then:
(
)
E
v
E
vs
S
atm
C
K
p
p
0
0
(
exp
-
=
, with
S
S
E
K
0
0
1
+
=
éq
3.2.1-2
C
Lnp
E
S
S
0
C
Lnp
0
E
has
Lnp
Appear 3.2.1-a: Evolution of suction
3.2.2 Part
irreversible
Beyond the threshold
0
C
p
, of the unrecoverable deformations appear, the slope in the test oedometric
becoming
S
. This slope can actually depend on the hydrostatic stress applied to
the sample, but it is considered constant in the original model of Barcelona. Like one
can note it on [Figure 3.2.2], the pressure of consolidation increases with suction.
[Figure 3.2.2 (A)]) shows two compression tests in condition saturated
(
)
0
=
C
p
and unsaturated
(
)
0
>
C
p
. A relation enters
*
idiots
P
(point 3) value of the preconsolidation with of saturated and
idiots
P
(point 2)
pressure of preconsolidation in unsaturated is established by comparing specific volumes obtained
on paths according to the items 1, 2, 3 [Figure 3.2.2(a)] which describe a discharge of
idiots
P
with
*
idiots
P
with constant suction followed by a remoistening of a value
C
p
to 0 with constant pressure
*
idiots
P
, from where
the following equation:
3
1
v
v
v
v
suction
pressure
=
+
+
Code_Aster
®
Version
7.4
Titrate:
Law of behavior of the porous environments: model of Barcelona
Date
:
31/03/05
Author (S):
J. EL GHARIB, G. DEBRUYNE
Key
:
R7.01.17-A
Page
:
11/44
Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
One makes the assumption that the reduction of suction
3
2
be accompanied by recoverable deformations.
The elastic relation is as follows:
(
)
atm
C
C
S
p
p
dp
FD
+
-
=
, where
atm
p
is the atmospheric pressure.
One writes for item 1 and 3 the expression of volume as follows:
()
0
0
ln
)
(
P
P
p
P
NR
v
C
-
=
where
0
P
is a pressure of reference corresponding to an initial volume
)
(
0
P
NR
. One combines this
expression and elastic relations:
() ()
() ()
0
*
*
0
0
ln
0
0
ln
ln
ln
P
P
NR
p
p
p
P
P
P
P
p
P
NR
idiots
atm
atm
C
S
idiots
idiots
idiots
C
-
=
+
+
+
-
By eliminating initial volumes by the elastic relation:
()
()
()
atm
atm
C
S
p
p
p
p
P
NR
NR
P
v
C
+
=
-
=
ln
0
0
0
0
one then determines the following evolution of the threshold of consolidation in unsaturated condition:
()
()
-
-
=
C
p
idiots
idiots
P
P
P
P
0
0
*
0
Like
Cr
idiots
P
P
2
=
,
One finds:
()
()
-
-
=
C
p
Cr
Cr
P
P
P
P
0
0
*
0
2
2
éq
3.2.2-1
[Figure 3.2.2] path 1-2-3 in the plan visualizes
)
,
(
C
p
P
.
Code_Aster
®
Version
7.4
Titrate:
Law of behavior of the porous environments: model of Barcelona
Date
:
31/03/05
Author (S):
J. EL GHARIB, G. DEBRUYNE
Key
:
R7.01.17-A
Page
:
12/44
Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
1
2
3
G O N fl EM in T
effo N D R EM in T
ref.
P
*
idiots
P
idiots
P
LnP
v
C
p
*
idiots
P
1
2
3
idiots
P
(
)
1
,
0
=
=
S
p
C
(
)
S
p
C
,
()
has
()
B
L C
()
C
p
NR
()
0
NR
1
v
2
v
3
v
pressure
v
suction
v
S I
Figure 3.2.2: (A) Curves of compression for water-logged soils and not-saturated
(b) criterion in the diagram
(
)
C
p
P,
The total component of the voluminal deformation due to the evolution of suction is:
atm
p
C
p
atm
C
C
S
vs
p
p
p
E
>
+
+
=
if
)
1
(
0
&
&
éq
3.2.2-2
from where the plastic component which is written:
(
)
atm
C
C
S
atm
C
C
S
S
p
vs
p
p
p
K
p
p
p
E
+
=
+
+
-
=
&
&
&
1
)
1
(
0
éq
3.3.2-3
Note:
The variation of suction does not generate deviatoric deformations.
3.3
Complete behavior (mechanical and hydrous loading)
3.3.1 Behavior
reversible
Under spherical loading, the evolution of the total voluminal elastic component is thus written:
(
)
atm
C
C
S
E
vs
E
vp
E
v
p
p
p
K
P
P
K
+
+
=
+
=
&
&
&
&
&
0
0
1
1
éq 3.3.1-1
Evolutions of the parts hydrostatic and deviatoric of the stress
are thus written:
(
)
atm
C
C
S
E
v
p
p
p
K
K
K
P
P
+
-
=
&
&
&
0
0
0
,
éq 3.3.1-2
E
ij
ij
S
µ
&
&
~
2
=
,
éq 3.3.1-3
Code_Aster
®
Version
7.4
Titrate:
Law of behavior of the porous environments: model of Barcelona
Date
:
31/03/05
Author (S):
J. EL GHARIB, G. DEBRUYNE
Key
:
R7.01.17-A
Page
:
13/44
Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
3.3.2 Thresholds of flow
The two thresholds of the reversible field are such as:
Mechanical criterion:
(
) (
)
0
)
(
2
)
),
(
,
,
(
2
2
1
-
+
+
=
C
Cr
C
C
C
C
Cr
p
P
P
p
K
P
M
Q
p
p
P
Q
P
F
éq 3.3.2-1
Hydrous criterion:
(
)
0
,
0
0
2
-
=
C
C
C
C
p
p
p
p
F
éq 3.3.2-2
The three-dimensional field of reversibility in space
)
,
,
(
C
p
Q
P
is represented on
[Figure 3.3.2-a].
These two criteria are reduced in the plan
)
,
(
C
p
P
with curves called LLC (loading collapse) and IF
(suction increase) (cf [Figure 3.3.2-b]).
P
Q
LLC
IF
*
idiots
P
C
p
1
F
1
F
2
F
2
F
Appear 3.3.2-a: Surfaces of load in space
(
)
C
p
Q
P
,
,
idiots
P
S
P
-
P
C
p
0
C
p
LLC
IF
Appear 3.3.2-b: Surfaces of load in space
(
)
C
p
P,
Code_Aster
®
Version
7.4
Titrate:
Law of behavior of the porous environments: model of Barcelona
Date
:
31/03/05
Author (S):
J. EL GHARIB, G. DEBRUYNE
Key
:
R7.01.17-A
Page
:
14/44
Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
3.3.3 Laws of flow
The plastic flow is controlled by the two criteria in pressure and suction:
(
)
C
C
Cr
p
v
p
K
P
P
M
P
F
+
-
=
=
2
2
2
1
&
&
&
éq 3.3.3-1
S
S
Q
Q
F
S
F
p
=
=
=
&
&
&
&
3
~
1
1
éq 3.3.3-2
or
0
~
,
1
=
+
=
p
atm
C
C
S
p
v
p
p
p
K
&
&
&
éq 3.3.3-3
3.3.4 Laws
of work hardening
The evolution of surfaces of load is controlled by the forces of work hardening:
Cr
P
and
0
C
p
.
The laws of work hardening of each surface are:
On
1
F
,
p
v
Cr
Cr
K
P
P
&
&
=
éq 3.3.4-1
On
2
F
,
p
v
S
atm
C
C
K
p
p
p
&
&
=
+
0
0
éq 3.3.4-2
3.3.5 Inventory
configurations
of mechanical and hydrous coupling
One examines the various configurations of loading in space
)
,
(
C
p
P
.
3.3.5.1 Reversibility
total
The loading represented by the point
M
(cf [Figure 3.3.5.1-a]) is located inside the field of
reversibility: elasticity, and hydrous reversibility. That results in:
0
1
<
F
, or (
0
0
1
1
<
=
F
F
&
,
), and
0
C
C
p
p
<
, or (
0
C
C
p
p
=
,
0
<
C
p&
).
The relations expressing this reversibility are:
(
)
atm
C
C
S
v
p
p
p
K
K
K
P
P
+
-
=
&
&
&
0
0
0
i.e.:
(
)
S
K
K
atm
atm
C
v
v
p
p
p
K
P
P
0
0
0
0
0
/
)
(
exp
+
-
=
,
éq 3.3.5.1-1
Code_Aster
®
Version
7.4
Titrate:
Law of behavior of the porous environments: model of Barcelona
Date
:
31/03/05
Author (S):
J. EL GHARIB, G. DEBRUYNE
Key
:
R7.01.17-A
Page
:
15/44
Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
and
µ
~
2
=
S
éq 3.3.5.1-2
P
C
p
0
C
p
field of reversibility
M
lq
S
0
C
p
C
p
Appear 3.3.5.1-a: Field of reversibility in the plan
(
)
C
p
P,
- curve of retention of water
3.3.5.2 Elastoplastic behavior
The point
M
touch the criterion of mechanics alone (cf [Figure 3.3.5.2-a]):
0
1
=
F
,
0
1
=
f&
, and
0
C
C
p
p
<
(or
0
C
C
p
p
=
and
0
<
C
p&
)
The elastic evolution is thus written:
(
)
atm
C
C
S
E
v
p
p
p
K
K
K
P
P
+
-
=
&
&
&
0
0
0
,
i.e.:
(
)
S
K
K
atm
atm
C
E
v
E
v
p
p
p
K
P
P
0
0
0
0
0
/
)
(
exp
+
-
=
éq 3.3.5.2-1
and
µ
~
2
=
S
éq
3.3.5.2-2
The evolution of the components of the plastic deformation is:
S
p
=
3
~&
[
]
C
C
Cr
p
v
p
K
P
P
M
+
-
=
2
2
2
&
The evolution of the mechanical threshold is written:
[
]
C
C
Cr
Cr
p
vp
Cr
Cr
p
K
P
P
M
P
K
P
K
P
+
-
=
=
2
2
2
&
&
.
Code_Aster
®
Version
7.4
Titrate:
Law of behavior of the porous environments: model of Barcelona
Date
:
31/03/05
Author (S):
J. EL GHARIB, G. DEBRUYNE
Key
:
R7.01.17-A
Page
:
16/44
Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
A specificity of the original model of Barcelona is the assumption that mechanical work hardening is
completely coupled with the hydrous work hardening (cf [Figure 3.3.5.2-a]) from where the relation:
Cr
Cr
S
atm
C
C
P
P
K
K
p
p
p
&
&
=
+
0
0
éq 3.3.5.2-3
P
C
p
0
C
p
1
2
M
Appear 3.3.5.2-a: Coupling of mechanical work hardening in hydrous work hardening
3.3.5.3 Hydrous behavior generating of the unrecoverable deformations
The point M reaches the threshold in suction (cf [Figure 3.3.5.2-a]):
0
C
C
p
p
=
and
0
>
C
p&
The mechanical behavior is elastic:
(
)
S
K
K
atm
atm
C
E
v
E
v
p
p
p
K
P
P
0
0
0
0
0
/
)
(
exp
+
-
=
,
µ
&
&
~
2
=
S
éq
3.3.5.3-1
but as the mechanical threshold is coupled with that of suction, there is also mechanical work hardening:
atm
C
C
S
Cr
Cr
p
p
p
K
K
P
P
+
=
0
0
&
&
éq
3.3.5.3-2
The rate of plastic deformation is written:
atm
C
C
S
p
v
p
p
p
K
+
==
&
&
1
0
~
=
p
&
Code_Aster
®
Version
7.4
Titrate:
Law of behavior of the porous environments: model of Barcelona
Date
:
31/03/05
Author (S):
J. EL GHARIB, G. DEBRUYNE
Key
:
R7.01.17-A
Page
:
17/44
Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
P
C
p
0
C
p
1
2
M
Appear 3.3.5.3-a: Coupling of hydrous work hardening in mechanical work hardening
3.4
Data of the model of Barcelona
The model requires the following parameters:
1) Elastic parameters provided under the key word
ELAS
:
The thermal expansion factor
, two elastic coefficients
,
E
provided in
given from which the coefficient of Lamé is calculated
µ
.
2) Under the key word
CAM_CLAY
:
·
0
P
Initial hydrostatic pressure equal to noted atmospheric pressure AP under the key word
CAM_CLAY
·
instead of giving the initial index of the vacuums
0
E
one gives the initial porosity which must be of
value equalizes with that given under the key word
THM_INIT
, noted
PORO
.
·
Parameters associated with surface threshold LLC (forced isotropic):
*
Cr
P
, equalizes with half
pressure of preconsolidation
*
idiots
P
noted
PRES_CRIT
,
*
, the coefficient of
compressibility for a saturated state and
the elastic coefficient of compressibility, noted
LAMBDA and KAPA.
·
The critical slope
M
,
3) Under the key word
BARCELONA
:
·
R
and
, coefficients allowing to calculate
()
C
p
, noted R and BETA.
·
parameters related to a variation of suction:
S
, coefficient of compressibility related to one
variation of suction in the plastic range,
S
coefficient associated with the change with
suction in elastic range, noted LABDAS and KAPAS.
·
C
K
the parameter which controls the increase in cohesion with suction
·
the initial threshold of suction
0
C
p
, noted PC0_INIT
·
the coefficient of normality, noted ALPHAB.
Here a set of values of some of these parameters, resulting from [bib1]:
()
6
.
0
;
1
;
10
;
008
.
0
;
08
.
0
;
10
.
0
;
5
.
12
;
75
.
0
;
02
.
0
;
2
.
0
0
0
1
=
=
=
=
=
=
=
=
=
=
-
C
S
S
K
M
MPa
G
MPa
P
MPa
R
Code_Aster
®
Version
7.4
Titrate:
Law of behavior of the porous environments: model of Barcelona
Date
:
31/03/05
Author (S):
J. EL GHARIB, G. DEBRUYNE
Key
:
R7.01.17-A
Page
:
18/44
Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
4
Numerical integration of the relations of behavior
4.1
Recall of the problem
The numerical integration of the model is similar to that carried out for the Camwood-Clay law [bib5], in
operating a translation on the axis of the capillary pressures.
This model is obligatorily coupled with the hydraulic behavior, contrary to Camwood-Clay which
can be used within a purely mechanical framework (one simulates a drained behavior then).
The model of Barcelona is thus usable only within the framework of the behaviors
THHM
established
in Code_Aster [bib7] and [bib8]. It will be more particularly employed with modelings
KIT_HHM
and
KIT_THHM
(in this last case, there is not for the moment of dependence of
mechanical characteristics specific to the model of Barcelona with the temperature).
The variables of input of the model are
U
or
and
P1,
P2, P1 and P2 being equal in
modelings quoted with
C
p
,
gz
p
,
C
p
and
gz
p
that it is with hydrous modelings
LIQU_VAPE_GAZ
or
LIQU_GAZ
.
The variables of exit of the model are:
'
,
Cr
P
,
0
C
p
,
S
P
.
The following notations are employed:
With
With
With
-
,
,
respectively for the quantity evaluated at the moment
known T, at the moment
T
T
+
and its increment. The equations are discretized in an implicit way,
i.e. expressed according to the unknown variables at the moment
T
T
+
.
One will note:
-
Cr
P
quantity
)
(
-
-
C
Cr
p
P
,
)
(
C
Cr
p
P
-
quantity
()
()
-
-
-
-
C
C
p
p
Cr
P
P
P
0
0
2
2
and
(
)
p
v
C
Cr
C
Cr
K
p
P
p
P
=
-
exp
)
(
)
(
4.2 Relations
incremental
The rules of flow and the condition of consistency give the following relations of flow:
If the threshold
1
F
is reached, the increment of plastic deformation voluminal is written:
(
)
(
)
-
-
+
+
-
+
=
C
Cr
C
C
C
Cr
Cr
C
C
p
v
p
P
P
K
Q
M
Q
P
p
K
P
P
P
p
K
P
K
)
2
(
2
2
2
2
1
2
éq 4.2-1
The increment of the standard of the equivalent plastic deformation is then:
(
)
(
)
(
)
+
-
-
-
+
-
+
+
=
C
C
C
Cr
Cr
C
C
C
Cr
C
C
Cr
p
eqp
p
p
K
P
P
M
P
P
Q
K
Q
p
K
P
P
M
Q
P
M
Q
p
K
P
kP
2
2
)
2
(
2
2
2
2
4
2
2
éq 4.2-2
and the tensor deviatoric is written:
p
v
S
Cr
p
P
P
P
M
S
+
-
=
)
2
2
(
3
~
2
éq
4.2-3
Code_Aster
®
Version
7.4
Titrate:
Law of behavior of the porous environments: model of Barcelona
Date
:
31/03/05
Author (S):
J. EL GHARIB, G. DEBRUYNE
Key
:
R7.01.17-A
Page
:
19/44
Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
If
2
F
is reached, the increment of plastic deformation voluminal is determined by:
)
(
1
0
0
atm
C
atm
C
S
p
v
p
p
p
p
Ln
K
+
+
=
-
éq
4.2-4
plastic deformation being purely voluminal (
0
~
=
p
)
p
v
will be the main of the problem and given unknown factor while solving
0
)
,
),
(
,
,
(
1
=
-
-
-
p
v
C
C
Cr
p
p
P
Q
P
F
, or
0
)
,
(
0
2
=
C
C
p
p
F
, the increment of plastic deformation
voluminal being obtained from
0
C
p
.On then deduces the evolution from it from the stresses and the thresholds.
4.3
Calculation of the stresses and the internal variables
The elastic prediction of the deviatoric stress is written:
µ
~
2
+
=
-
S
S
E
éq 4.3-1
The elastic prediction is chosen
E
P
in the following way:
(
)
S
K
K
atm
C
atm
C
v
E
p
p
p
p
K
P
P
0
0
/
0
exp
+
+
=
-
-
éq 4.3-2
If
0
1
<
F
and
0
2
<
F
, then
()
()
0
,
2
2
,
0
,
,
0
0
0
=
=
=
=
=
-
-
-
-
C
p
p
Cr
Cr
p
E
E
p
P
P
P
P
S
S
P
P
C
C
,
If not:
p
E
S
S
µ
~
2
-
=
éq 4.3-3
[
]
p
v
E
K
P
P
-
=
0
exp
éq
4.3-4
()
()
[
]
p
v
p
p
Cr
Cr
K
P
P
P
P
C
C
=
-
-
-
-
exp
2
2
0
0
éq 4.3-5
[
]
p
v
S
atm
C
atm
C
K
p
p
p
p
+
=
+
-
exp
)
(
)
(
0
0
éq
4.3-6
The main unknown factor is thus
p
v
.
If
0
1
>
F
, then
While replacing
p
~
by its expression according to
p
v
[éq 4.2-3] one obtains:
(
)
C
C
Cr
p
v
E
p
K
P
P
M
S
S
+
-
+
=
2
2
6
1
2
µ
éq 4.3-7
Code_Aster
®
Version
7.4
Titrate:
Law of behavior of the porous environments: model of Barcelona
Date
:
31/03/05
Author (S):
J. EL GHARIB, G. DEBRUYNE
Key
:
R7.01.17-A
Page
:
20/44
Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
and:
[
]
p
v
E
K
P
P
-
=
0
exp
éq 4.3-8
The unknown factor is given while solving
0
)
,
),
(
,
,
(
)
,
,
,
(
1
1
=
=
-
p
v
C
C
Cr
E
E
C
Cr
p
p
P
Q
P
F
p
P
Q
P
F
,
I.e.:
(
) (
) (
)
Cr
C
C
C
C
Cr
p
v
E
P
P
p
K
P
p
K
P
P
M
M
Q
2
2
2
6
1
2
2
2
2
-
+
+
-
+
-
=
µ
,
or:
(
)
()
(
)
(
)
[
]
(
)
(
)
[
]
p
v
C
Cr
p
v
E
C
C
p
v
E
C
C
p
v
C
Cr
p
v
E
p
v
E
K
p
P
K
P
p
K
K
P
p
K
K
p
P
K
P
M
M
Q
µ
-
-
-
+
-
+
-
-
+
-
=
-
-
exp
)
(
2
exp
exp
exp
)
(
2
exp
2
6
1
0
0
2
0
2
2
2
éq
4.3-9
If
0
2
>
F
, then:
C
C
p
p
=
0
, the unknown factor is immediately given by:
)
(
1
0
0
atm
C
atm
C
S
p
v
p
p
p
p
Ln
K
+
+
=
-
, éq
4.3-10
from where
E
S
S
=
and
[
]
p
v
E
K
P
P
-
=
0
exp
éq 4.3-11
One has moreover
()
()
[
]
p
v
p
p
Cr
Cr
K
P
P
P
P
C
C
=
-
-
-
-
exp
2
2
0
0
.
éq 4.3-12
Code_Aster
®
Version
7.4
Titrate:
Law of behavior of the porous environments: model of Barcelona
Date
:
31/03/05
Author (S):
J. EL GHARIB, G. DEBRUYNE
Key
:
R7.01.17-A
Page
:
21/44
Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
5 Operator
tangent
If the option is:
RIGI_MECA_TANG
, option used at the time of the prediction, the tangent operator calculated in
each point of Gauss is known as of speed:
kl
elp
ijkl
ij
D
&
&
=
,
i.e.
elp
ijkl
D
is calculated starting from the not discretized equations.
If the option is:
FULL_MECA
, option used when one reactualizes the tangent matrix while updating
internal stresses and variables:
kl
ijkl
ij
D
With
D
=
In this case,
ijkl
With
is calculated starting from the implicitly discretized equations.
The tangent operator of the generalized stresses is implemented in THHM under the name
OF
D
and
partitionné in several blocks. The blocks concerned with the model are [DMECDE], [DMECP1] [bib8].
One calculates the contribution of the model to each one of these blocks for the tangent operator in elasticity,
the operator of speed and the coherent operator.
5.1
Nonlinear elastic tangent operator
The elastic relation of speed of the model of Barcelona is written:
ij
atm
C
C
S
ij
ij
ij
ij
ij
p
p
p
P
K
K
Ptr
K
S
P
µ
+
+
+
=
+
-
=
&
&
&
&
&
&
0
0
0
~
2
éq
5.1-1
ij
atm
C
C
S
ij
ij
ij
p
p
p
P
K
K
tr
P
K
µ
µ
+
+
+
-
=
&
&
&
&
0
0
0
2
)
3
2
(
éq
5.1-2
The tensor of the stresses used in the model of Barcelona (and the tests determining them
given model) is a function of the total stress and of the gas pressure and is written:
I
p
gz
T
+
=
éq 5.1-3
The tensor of the stresses of Bishop
'
used in Code_Aster is such as:
I
P
T
&
&
&
+
=
'
with
(
)
C
lq
gz
P
p
S
p
B
&
&
&
-
-
=
éq
5.1-4
From where the expression of the stress of Bishop according to the stress of the model of Barcelona:
(
)
I
p
bS
p
B
C
lq
gz
&
&
&
&
-
-
+
=
)
1
(
'
éq
5.1-5
Note:
The stress of Bishop is generally regarded as an effective stress
(controlled only by the deformation). It is not the case of the model of Barcelona where it
is necessary two stresses (
)
,
(
C
p
to describe the behavior. Consequently, in
the tangent operator, the term
C
p
'
&
does not summarize itself with
C
p
p
-
&
.
Code_Aster
®
Version
7.4
Titrate:
Law of behavior of the porous environments: model of Barcelona
Date
:
31/03/05
Author (S):
J. EL GHARIB, G. DEBRUYNE
Key
:
R7.01.17-A
Page
:
22/44
Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
Part [DMECDE] of the matrix
OF
D
agent with
'
is such as:
+
-
-
-
+
-
-
-
+
=
31
23
12
33
22
11
0
0
0
0
0
0
0
0
0
'
31
'
23
'
12
'
33
'
22
'
11
2
2
2
2
0
0
0
0
0
0
2
0
0
0
0
0
0
2
0
0
0
0
0
0
3
4
3
2
3
2
0
0
0
3
2
3
4
3
2
0
0
0
3
2
3
2
3
4
2
2
2
µ
µ
µ
µ
µ
µ
µ
µ
µ
µ
µ
µ
&
&
&
&
&
&
4
4
4
4
4
4
4
4
4
4
3
4
4
4
4
4
4
4
4
4
4
2
1
&
&
&
&
&
&
E
D
P
K
P
K
P
K
P
K
P
K
P
K
P
K
P
K
P
K
éq 5.1-6
Part [DMECP1] of the matrix
OF
D
is reduced to
1
'
p
with
(
)
C
p
p
=
1
who is such as:
{}
1
0
0
0
0
0
0
'31
'23
'12
'33
'22
'11
0
0
0
2
2
2
p
bS
p
p
P
K
K
bS
p
p
P
K
K
bS
p
p
P
K
K
lq
atm
C
S
lq
atm
C
S
lq
atm
C
S
&
&
&
&
&
&
&
-
+
-
+
-
+
=
éq 5.1-7
5.2
Plastic tangent operator of speed. Option
RIGI_MECA_TANG
The total tangent operator is in this case obtained starting from the results known at the moment
1
-
I
T
(the option
RIGI_MECA_TANG
called with the first iteration of a new increment of load).
If with
1
-
I
T
the border of the field of reversibility is reached, one writes the condition:
0
=
f&
who must be
checked jointly with the condition
0
=
F
. If with
1
-
I
T
one is strictly inside the field,
0
<
F
, then the tangent operator is the operator of elasticity.
If the mechanical criterion is reached:
0
1
=
f&
0
1
1
1
1
=
+
+
=
C
C
Cr
Cr
p
p
F
P
P
F
F
F
&
&
&
&
éq 5.2-1
like
C
C
Cr
p
v
p
v
Cr
Cr
p
p
P
P
P
&
&
&
+
=
, then:
0
)
(
1
1
1
1
=
+
+
+
=
C
C
C
C
Cr
p
v
p
v
Cr
Cr
p
p
F
p
p
P
P
P
F
F
F
&
&
&
&
&
éq
5.2-2
Code_Aster
®
Version
7.4
Titrate:
Law of behavior of the porous environments: model of Barcelona
Date
:
31/03/05
Author (S):
J. EL GHARIB, G. DEBRUYNE
Key
:
R7.01.17-A
Page
:
23/44
Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
One has in addition:
(
)
ij
atm
C
S
C
kl
E
E
ijkl
ij
p
p
K
p
P
K
D
+
+
=
0
0
&
&
&
éq 5.2-3
i.e.:
(
)
ij
atm
C
S
C
kl
kl
E
ijkl
kl
E
ijkl
ij
p
p
K
p
P
K
P
F
S
F
D
D
+
+
-
-
=
0
0
1
1
3
1
&
&
&
éq
5.2-4
By writing the plastic module of work hardening:
,
1
1
P
F
P
P
F
H
p
v
Cr
Cr
p
-
=
éq 5.2-5
The equations [éq 5.2-2] and [éq 5.2-5] give:
0
)
(
1
1
1
=
+
+
-
C
C
Cr
Cr
C
p
ij
ij
p
p
P
P
F
p
F
H
F
&
&
éq 5.2-6
Multiplication of the equation [éq 5.2-4] by
ij
F
1
give:
(
)
atm
C
S
C
ij
ij
kl
kl
E
ijkl
ij
kl
E
ijkl
ij
ij
ij
p
p
K
p
P
K
F
P
F
S
F
D
F
D
F
F
+
+
-
-
=
0
0
1
1
1
1
1
1
3
1
&
&
&
éq 5.2-7
The two preceding equations make it possible to find:
(
)
C
C
Cr
Cr
C
atm
C
S
C
ij
ij
kl
kl
kl
E
ijkl
ij
kl
E
ijkl
ij
p
p
p
P
P
F
p
F
p
p
K
p
P
K
F
P
F
S
F
D
F
D
F
H
&
&
&
)
(
3
1
1
1
0
0
1
1
1
1
1
+
+
+
+
-
-
=
éq 5.2-8
from where and to deduce the expression from it from the plastic multiplier:
(
)
p
kl
kl
E
ijkl
ij
C
C
Cr
Cr
C
atm
C
S
ij
ij
kl
E
ijkl
ij
H
P
F
S
F
D
F
p
p
P
P
F
p
F
p
p
K
P
K
F
D
F
+
-
+
+
+
+
=
1
1
1
1
1
0
0
1
1
3
1
)
(
1
&
&
éq
5.2-9
That is to say
H
the definite elastoplastic module like:
p
kl
kl
E
ijkl
ij
H
P
F
S
F
D
F
H
+
-
=
1
1
1
3
1
éq 5.2-10
Code_Aster
®
Version
7.4
Titrate:
Law of behavior of the porous environments: model of Barcelona
Date
:
31/03/05
Author (S):
J. EL GHARIB, G. DEBRUYNE
Key
:
R7.01.17-A
Page
:
24/44
Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
The plastic multiplier is written:
(
)
H
p
p
P
P
F
p
F
p
p
K
P
K
F
D
F
C
C
Cr
Cr
C
atm
C
S
ij
ij
kl
E
ijkl
ij
&
&
+
+
+
+
=
)
(
1
1
1
0
0
1
1
éq
5.2-11
While replacing
by his expression in the equation [éq 5.2-4], one obtains:
(
)
(
)
C
ij
atm
C
S
kl
kl
E
ijkl
C
Cr
Cr
C
Mn
atm
C
S
Mn
kl
kl
E
ijkl
COp
emnop
Mn
kl
E
ijkl
ij
p
p
p
K
P
K
P
F
S
F
D
p
P
P
F
p
F
p
p
K
P
K
F
H
P
F
S
F
D
D
F
H
D
&
&
&
&
+
-
-
+
+
+
-
-
-
=
0
0
1
1
1
1
0
0
1
1
1
1
3
1
)
(
1
1
3
1
.
1
éq 5.2-12
One thus deduces the elastoplastic operator from it
p
E
elp
D
D
D
-
=
:
(
)
(
)
C
D
ij
atm
C
S
C
Cr
Cr
C
atm
C
S
Mn
Mn
E
ijop
COp
COp
kl
D
Mn
Mn
emnkl
E
ijop
COp
E
ijkl
ij
p
p
p
K
P
K
p
P
P
F
p
F
p
p
K
P
K
F
D
P
F
S
F
H
P
F
S
F
D
D
F
H
D
C
p
ij
p
ijkl
&
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
3
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
2
1
&
4
4
4
4
4
4
4
4
4
3
4
4
4
4
4
4
4
4
4
2
1
&
+
-
+
-
+
-
-
-
-
=
0
0
1
1
0
0
1
1
1
1
1
1
))
(
1
(
3
1
1
3
1
1
éq 5.2-13
with,
-
=
Mn
Mn
emnkl
E
ijop
COp
p
ijkl
P
F
S
F
D
D
F
H
D
1
1
1
3
1
1
and
(
)
(
)
ij
atm
C
S
C
Cr
Cr
C
atm
C
S
Mn
Mn
E
ijop
COp
COp
ij
p
p
p
K
P
K
p
P
P
F
p
F
p
p
K
P
K
F
D
P
F
S
F
H
D
C
+
+
+
+
+
-
-
=
0
0
1
1
0
0
1
1
1
)
(
1
3
1
1
éq 5.2-14
Calculation of
p
ijkl
D
:
(
)
ij
ij
C
C
Cr
ij
S
p
K
P
P
M
F
3
2
2
3
1
2
1
+
+
-
-
=
,
éq
5.2-15
Code_Aster
®
Version
7.4
Titrate:
Law of behavior of the porous environments: model of Barcelona
Date
:
31/03/05
Author (S):
J. EL GHARIB, G. DEBRUYNE
Key
:
R7.01.17-A
Page
:
25/44
Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
who is written in vectorial notation:
(
)
(
)
(
)
+
+
-
-
+
+
-
-
+
+
-
-
31
23
12
33
2
22
2
11
2
2
3
2
3
2
3
3
2
2
3
1
3
2
2
3
1
3
2
2
3
1
S
S
S
S
p
K
P
P
M
S
p
K
P
P
M
S
p
K
P
P
M
C
C
Cr
C
C
Cr
C
C
Cr
éq
5.2-16
from where the expression of:
(
)
(
)
(
)
+
+
-
-
+
+
-
-
+
+
-
-
31
23
12
33
2
0
22
2
0
11
2
0
2
6
2
6
2
6
6
2
2
6
2
2
6
2
2
:
S
S
S
S
p
K
P
P
P
M
K
S
p
K
P
P
P
M
K
S
p
K
P
P
P
M
K
F
D
C
C
Cr
C
C
Cr
C
C
Cr
kl
E
ijkl
µ
µ
µ
µ
µ
µ
éq
5.2-17
and
2
2
4
0
1
1
12
)
2
2
(
3
1
Q
p
K
P
P
P
M
K
P
F
S
F
D
F
C
C
Cr
kl
kl
E
ijkl
ij
µ
+
+
-
=
-
éq
5.2-18
However the plastic module
H
is written in the form:
p
kl
kl
E
ijkl
ij
H
P
F
S
F
D
F
H
+
-
=
1
1
3
1
(
)
(
)
(
)
[
]
2
0
4
12
2
2
2
2
2
Q
p
K
P
kP
p
K
P
P
P
K
p
K
P
P
M
H
C
C
Cr
C
C
Cr
C
C
Cr
µ
+
+
+
+
-
+
-
=
éq 5.2-19
While posing:
(
)
(
)
ij
ij
C
C
Cr
ij
ij
ij
C
C
Cr
ij
S
p
K
P
P
P
M
K
With
S
p
K
P
P
P
M
K
With
µ
µ
6
2
2
'
,
6
2
2
2
0
2
0
+
+
-
-
=
+
+
-
-
=
,
éq 5.2-20
with:
)
2
2
(
3
)
(
2
0
C
C
Cr
p
K
P
P
P
M
K
With
tr
+
-
-
=
Code_Aster
®
Version
7.4
Titrate:
Law of behavior of the porous environments: model of Barcelona
Date
:
31/03/05
Author (S):
J. EL GHARIB, G. DEBRUYNE
Key
:
R7.01.17-A
Page
:
26/44
Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
SYM
p
S
S
With
S
With
S
With
S
S
S
S
With
S
With
S
With
S
S
S
S
S
S
With
S
With
S
With
S
With
S
With
S
With
With
With
With
With
With
With
S
With
S
With
S
With
With
With
With
With
With
With
S
With
S
With
S
With
With
With
With
With
With
With
H
D
=
2
31
2
31
33
31
22
31
11
31
23
2
2
23
2
23
33
23
22
23
11
31
12
2
23
12
2
2
12
2
12
33
12
22
12
11
31
33
23
33
12
33
33
33
33
22
113
33
31
22
23
22
12
22
33
22
22
22
22
11
31
11
23
11
12
11
33
11
22
11
11
11
36
.
.
'
2
6
'
2
6
'
2
6
36
36
.
'
2
6
'
2
6
'
2
6
36
36
36
'
2
6
'
2
6
'
2
6
2
6
2
6
2
6
'
'
'
2
6
2
6
2
6
'
'
'
2
6
2
6
2
6
'
'
'
1
µ
µ
µ
µ
µ
µ
µ
µ
µ
µ
µ
µ
µ
µ
µ
µ
µ
µ
µ
µ
µ
µ
µ
µ
éq. 5.2-21
One can write the components
'
piece [DMECDE] of the matrix
OF
D
who are those of
the operator
p
E
elp
D
D
D
-
=
.
According to the equation [éq 5.2.14]. Components
1
'
p
with
(
)
C
p
p
=
1
piece [DMECP1] of
stamp
OF
D
are:
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
-
-
+
-
-
+
+
-
-
-
+
-
-
+
+
-
-
-
+
-
-
+
+
-
-
+
+
-
-
+
-
-
+
+
-
-
+
+
-
-
+
-
-
+
+
-
-
+
+
-
-
+
-
-
+
+
-
13
'
2
0
*
2
13
0
23
'
2
0
*
2
23
0
12
'
2
0
*
2
12
0
0
0
33
'
2
0
*
2
33
0
0
0
22
'
2
0
*
2
22
0
0
0
11
'
2
0
*
2
11
0
)
)
(
(
)
0
(
2
)
(
2
)
2
(
2
6
)
(
2
2
)
)
(
(
)
0
(
2
)
(
2
)
2
(
2
6
)
(
2
2
)
)
(
(
)
0
(
2
)
(
2
)
2
(
2
6
)
(
2
2
'
)
)
(
(
)
0
(
2
)
(
2
)
2
(
'
3
)
(
'
)
)
(
(
)
0
(
2
)
(
2
)
2
(
'
3
)
(
'
)
)
(
(
)
0
(
2
)
(
2
)
2
(
'
3
)
(
S
p
P
P
Ln
p
K
P
P
P
P
K
M
S
p
p
HK
With
tr
S
H
p
P
P
Ln
p
K
P
P
P
P
K
M
S
p
p
HK
With
tr
S
H
p
P
P
Ln
p
K
P
P
P
P
K
M
S
p
p
HK
With
tr
bS
p
p
K
P
K
With
H
p
P
P
Ln
p
K
P
P
P
P
K
M
With
p
p
HK
With
tr
bS
p
p
K
P
K
With
H
p
P
P
Ln
p
K
P
P
P
P
K
M
With
p
p
HK
With
tr
bS
p
p
K
P
K
With
H
p
P
P
Ln
p
K
P
P
P
P
K
M
With
p
p
HK
With
tr
C
Cr
C
C
Cr
Cr
C
atm
C
S
C
Cr
C
C
Cr
Cr
C
atm
C
S
C
Cr
C
C
Cr
Cr
C
atm
C
S
lq
atm
C
S
C
Cr
C
C
Cr
Cr
C
atm
C
S
lq
atm
C
S
C
Cr
C
C
Cr
Cr
C
atm
C
S
lq
atm
C
S
C
Cr
C
C
Cr
Cr
C
atm
C
S
µ
µ
µ
µ
µ
µ
éq 5.2-22
with
[
]
)
exp (
)
1
(
)
0
(
'
C
C
p
R
p
-
-
-
=
=
If the hydrous criterion is reached:
One leaves again the equation [éq 5.2.3] with this time
(
)
atm
C
S
C
p
p
p
K
p
+
=
&
&
,
One finds a relation direct enters
&
and
C
p&
&,
form:
(
)
(
)
I
p
p
K
p
p
p
K
p
P
K
D
atm
C
S
C
atm
C
S
C
E
)
(
0
0
+
+
+
+
=
&
&
&
&
éq
5.2-23
Code_Aster
®
Version
7.4
Titrate:
Law of behavior of the porous environments: model of Barcelona
Date
:
31/03/05
Author (S):
J. EL GHARIB, G. DEBRUYNE
Key
:
R7.01.17-A
Page
:
27/44
Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
One deduces then the stress from Bishop
(
)
(
)
C
lq
atm
C
S
atm
C
S
E
p
I
bS
p
p
K
P
K
p
p
K
P
K
D
&
&
&
-
+
+
+
+
=
)
(
0
0
0
'
éq
5.2-24
Components
'
piece [DMECDE] of the matrix
OF
D
are nothing other than those
matrix
E
D
.
Only components of piece [DMECP1] of the matrix
OF
D
are thus those of
1
'
p
with
(
)
C
p
p
=
1
:
(
)
(
)
(
)
{}
1
0
0
0
0
0
0
'31
'23
'12
'33
'22
'11
0
0
0
)
1
1
(
)
1
1
(
)
1
1
(
2
2
2
p
bS
K
K
p
p
P
K
bS
K
K
p
p
P
K
bS
K
K
p
p
P
K
lq
S
S
atm
C
lq
S
S
atm
C
lq
S
S
atm
C
&
&
&
&
&
&
&
-
+
+
-
+
+
-
+
+
=
éq 5.2-25
5.3
Tangent operator into implicit. Option
FULL_MECA
To calculate the tangent operator into implicit, one chose as for the model Cam Clay separating
initially processing of the deviatoric part of the hydrostatic part for then them
to combine in order to deduce the tangent operator connecting the disturbance from the total stress to
disturbance of the total deflection.
5.3.1 If the mechanical criterion is reached
5.3.1.1 Processing of the deviatoric part
It is considered here that the variation of loading is purely deviatoric
)
0
(
=
P
.
The increment of the deviatoric stress is written in the form:
(
)
p
ij
ij
ij
S
µ
~
~
2
-
=
éq
5.3.1.1-1
Around the point of balance
(
)
+
-
, a variation is considered
S
deviatoric part of
stress:
(
)
p
kl
kl
kl
S
µ
~
~
2
-
=
éq
5.3.1.1-2
Calculation of
p
kl
~
:
It is known that:
kl
p
kl
S
=
3
~
éq
5.3.1.1-3
Code_Aster
®
Version
7.4
Titrate:
Law of behavior of the porous environments: model of Barcelona
Date
:
31/03/05
Author (S):
J. EL GHARIB, G. DEBRUYNE
Key
:
R7.01.17-A
Page
:
28/44
Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
By deriving this equation compared to the deviatoric stress, one obtains:
kl
kl
p
kl
S
S
+
=
3
3
~
éq
5.3.1.1-4
Calculation of
:
One a:
(
)
(
)
[
]
[
]
C
Cr
C
C
Cr
C
C
C
Cr
Mn
Mn
p
C
C
Cr
Cr
C
Mn
Mn
p
C
C
Cr
Cr
C
Mn
Mn
p
p
P
p
K
P
P
P
K
M
P
p
K
P
P
M
S
S
H
p
p
P
P
F
p
F
P
P
F
S
S
F
H
p
p
P
P
F
p
F
F
H
+
+
-
-
+
-
+
=
+
+
+
=
+
+
=
'
)
(
2
2
2
2
3
1
)
(
1
)
(
1
2
2
éq 5.3.1.1-5
If one considers only the evolution of the deviatoric part of
)
0
(
=
P
, then:
[
]
(
)
[
]
C
C
C
C
Cr
Cr
C
C
Cr
C
C
C
Cr
Mn
Mn
Mn
Mn
p
p
p
p
p
p
K
P
P
P
P
K
M
p
P
K
M
p
P
K
M
P
P
M
S
S
S
S
H
H
H
))
(
(
'
2
2
2
2
3
3
)
(
2
2
2
2
+
+
+
-
-
-
+
-
+
=
+
=
éq. 5.3.1.1-6
with
C
Cr
Cr
p
P
P
=
'
However:
P
v
Cr
Cr
kP
P
=
.
Like
),
2
2
(
2
C
C
Cr
p
v
p
K
P
P
M
+
-
=
one a:
C
C
Cr
C
C
Cr
p
v
p
M
K
P
M
p
K
P
P
M
+
-
+
-
=
2
2
2
2
)
2
2
(
éq.
5.3.1.1-7
From where:
C
C
Cr
Cr
C
C
Cr
p
M
K
P
M
kP
p
K
P
P
M
2
2
2
2
1
)
2
2
(
-
+
=
+
-
éq
5.3.1.1-8
In addition,
(
) (
)
(
)
(
)
C
C
C
Cr
C
Cr
Cr
C
C
Cr
C
C
p
C
C
Cr
C
C
Cr
p
p
p
K
P
P
K
M
kP
P
p
K
P
P
p
K
P
km
H
p
K
P
P
p
K
P
P
km
H
2
2
3
2
)
4
2
(
2
2
2
2
4
4
4
and
+
-
+
+
-
+
=
+
-
+
=
éq 5.3.1.1-9
By injecting this last equation in the equation [éq 5.3.1.1-6], one obtains:
(
)
[
]
(
)
[
]
[
]
[
]
Mn
Mn
Mn
Mn
C
Cr
C
C
Cr
Cr
C
Cr
C
C
C
Cr
Cr
C
C
C
C
Cr
C
C
p
S
S
S
S
p
P
P
p
p
P
P
P
P
K
M
P
p
K
P
K
M
kP
P
p
K
M
P
M
p
K
P
P
p
K
P
km
H
3
3
'
2
)
(
'
2
(
2
)
2
2
3
(
2
2
2
)
4
2
(
2
2
4
2
2
4
+
+
+
+
+
+
-
+
-
+
-
=
+
+
+
-
+
+
éq 5.3.1.1-10
Code_Aster
®
Version
7.4
Titrate:
Law of behavior of the porous environments: model of Barcelona
Date
:
31/03/05
Author (S):
J. EL GHARIB, G. DEBRUYNE
Key
:
R7.01.17-A
Page
:
29/44
Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
While using the relation [éq 5.3.1.1-8], it comes then:
[
]
)
(
)
(
3
3
With
H
p
Z
With
H
S
S
S
S
p
C
p
Mn
Mn
Mn
Mn
+
-
+
+
=
éq
5.3.1.1-11
(
)
[
]
+
+
-
+
+
+
-
+
=
2
2
2
2
4
2
1
)
2
2
(
)
4
2
(
M
kP
p
K
P
P
M
p
K
M
P
M
p
K
P
P
p
K
P
M
K
With
Cr
C
C
Cr
C
C
C
C
Cr
C
C
with
(
)
(
)
(
)
(
)
)
(
'
2
2
2
1
)
2
3
(
2
2
1
)
4
2
(
2
2
2
2
2
2
2
2
2
2
2
C
C
C
Cr
Cr
C
Cr
C
C
C
C
C
C
Cr
Cr
C
Cr
C
C
Cr
C
C
C
p
p
K
P
P
M
P
P
K
M
P
km
p
M
K
P
K
M
P
K
M
p
K
P
P
M
P
kk
P
km
p
K
P
P
p
K
P
M
kk
M
Z
+
+
+
-
+
+
+
+
-
+
-
+
+
+
-
+
=
One then obtains immediately the variation of the deviatoric part of the plastic deformation:
(
)
(
)
(
)
(
)
kl
C
Cr
C
C
p
kl
C
p
kl
C
Cr
C
p
kl
C
C
Cr
p
kl
Mn
Mn
p
kl
Mn
Mn
kl
Mn
Mn
p
p
kl
S
p
P
p
K
P
M
H
S
p
With
H
Z
S
p
P
P
K
M
H
S
P
p
K
P
P
M
H
S
S
S
H
S
S
S
S
S
S
With
H
+
-
+
-
-
-
+
-
+
+
+
+
=
'
6
3
2
3
2
2
3
9
9
~
2
2
2
éq
5.3.1.1-12
ij
S
is written then:
(
)
[
]
(
)
(
)
(
)
ij
C
Cr
C
C
p
C
ij
p
ij
C
Cr
C
p
ij
C
C
Cr
p
ij
kl
kl
p
kl
ij
kl
kl
ij
kl
p
ij
ij
S
p
P
p
K
P
M
H
p
S
With
H
Z
S
p
P
P
K
M
H
S
P
p
K
P
P
M
H
S
S
S
H
S
S
S
S
S
S
With
H
S
µ
µ
µ
µ
µ
µ
µ
+
+
+
+
-
+
+
-
-
-
+
+
-
=
'
12
)
(
6
2
6
2
2
6
18
)
(
18
~
2
2
2
2
éq
5.3.1.1-13
i.e.:
(
)
(
)
(
)
(
)
C
ij
p
ij
kl
C
Cr
C
C
p
C
Cr
C
p
ijkl
ijkl
Mn
Mn
p
ij
kl
ij
kl
p
C
C
Cr
p
ijkl
ijkl
p
S
With
H
Z
S
p
P
p
K
P
M
H
p
P
P
M
K
H
S
S
H
S
S
S
S
With
H
P
p
K
P
P
M
H
µ
µ
µ
µ
µ
µ
µ
+
+
=
+
-
-
-
+
+
+
+
+
-
+
6
~
2
'
12
2
2
6
18
18
2
2
6
2
2
2
éq 5.3.1.1-14
Code_Aster
®
Version
7.4
Titrate:
Law of behavior of the porous environments: model of Barcelona
Date
:
31/03/05
Author (S):
J. EL GHARIB, G. DEBRUYNE
Key
:
R7.01.17-A
Page
:
30/44
Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
or in tensorial writing:
(
)
(
)
(
)
C
p
C
Cr
C
C
p
p
C
Cr
C
p
p
C
C
Cr
p
D
p
S
With
H
Z
S
p
P
p
K
P
M
H
S
S
S
With
H
p
P
P
K
M
H
S
S
H
P
p
K
P
P
M
H
I
µ
µ
µ
µ
µ
µ
µ
)
(
6
~
2
'
12
)
(
)
(
18
2
6
:
18
2
2
6
1
2
2
2
4
+
+
=
+
-
+
+
+
-
-
+
+
-
+
éq 5.3.1.1-15
that one can still write by symmetrizing the tensor
S
S
S
+
)
(
:
(
)
(
)
(
)
C
p
p
C
Cr
C
C
p
p
C
Cr
C
p
C
C
Cr
p
D
p
S
With
H
Z
S
With
H
p
P
p
K
P
M
H
S
S
H
p
P
P
M
K
H
P
p
K
P
P
M
H
I
µ
µ
µ
µ
µ
µ
µ
)
(
6
~
2
)
(
18
'
12
:
18
2
6
2
2
6
1
2
2
2
4
+
+
=
+
+
+
-
+
-
-
+
-
+
éq 5.3.1.1-16
with:
[
]
T
S
S
S
S
S
S
))
(
(
)
)
((
2
1
+
+
+
=
Calculation of
, while posing:
ij
ij
ij
S
S
T
+
=
31
31
23
31
12
31
33
31
22
31
11
31
31
23
23
23
12
23
33
23
22
23
11
23
31
12
23
12
12
12
33
12
22
12
11
12
31
33
23
33
12
33
33
33
22
33
11
33
31
22
23
22
12
22
33
22
22
22
11
22
31
11
23
11
12
11
33
11
22
11
11
11
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
=
S
S
T
S
T
S
T
S
T
S
T
S
T
S
T
S
T
S
T
S
T
S
T
S
T
S
T
S
T
S
T
S
T
S
T
S
T
S
T
S
T
S
T
S
T
S
T
S
T
S
T
S
T
S
T
S
T
S
T
S
T
S
T
S
T
S
T
S
T
S
T
S
T
T
[
]
T
S
T
S
T
)
(
)
(
2
1
+
=
That is to say:
(
)
(
)
+
+
+
-
-
-
+
+
-
+
=
)
(
9
'
6
)
2
(
3
:
9
2
2
3
2
1
2
2
2
4
With
H
p
P
p
K
P
M
H
p
P
P
M
H
K
S
S
H
P
p
K
P
P
M
H
I
C
p
C
Cr
C
C
p
C
Cr
p
C
p
C
C
Cr
p
D
µ
one poses:
(
)
S
S
H
C
p
:
9
=
Code_Aster
®
Version
7.4
Titrate:
Law of behavior of the porous environments: model of Barcelona
Date
:
31/03/05
Author (S):
J. EL GHARIB, G. DEBRUYNE
Key
:
R7.01.17-A
Page
:
31/44
Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
(
)
P
p
K
P
P
M
H
D
C
C
Cr
p
+
-
=
2
2
3
2
(
)
C
Cr
C
p
p
P
P
K
M
H
G
-
-
=
2
3
2
(
)
C
Cr
C
C
p
p
P
p
K
P
M
H
H
+
-
=
'
6
2
The symmetrical matrix
C
dimensions (6,6) is too large to be presented whole, one
break up into 4 parts
1
C
,
2
C
,
3
C
and
4
C
:
=
4
3
2
1
C
C
C
C
C
with
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
=
33
33
22
33
33
22
33
11
11
33
22
33
33
22
22
22
22
11
11
22
11
33
33
11
11
22
22
11
11
11
1
)
(
9
2
1
)
(
)
(
2
9
)
(
)
(
2
9
)
(
)
(
2
9
)
(
9
2
1
)
(
)
(
2
9
)
(
)
(
2
9
)
(
)
(
2
9
)
(
9
2
1
S
T
With
H
H
G
D
C
S
T
S
T
With
H
S
T
S
T
With
H
S
T
S
T
With
H
S
T
With
H
H
G
D
C
S
T
S
T
With
H
S
T
S
T
With
H
S
T
S
T
With
H
T
S
With
H
H
G
D
C
C
p
p
p
p
p
p
p
p
p
µ
µ
µ
éq 5.3.1.1-17
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
=
)
(
)
(
2
2
9
)
(
)
(
2
2
9
)
(
)
(
2
2
9
)
(
)
(
2
2
9
)
(
)
(
2
2
9
)
(
)
(
2
2
9
)
(
)
(
2
2
9
)
(
)
(
2
2
9
)
(
)
(
2
2
9
13
33
13
33
23
33
23
33
12
33
12
33
13
22
13
22
23
22
23
22
12
22
12
22
13
11
13
11
23
11
23
11
12
11
12
11
2
T
S
S
T
With
H
T
S
S
T
With
H
T
S
S
T
With
H
T
S
S
T
With
H
T
S
S
T
With
H
T
S
S
T
With
H
T
S
S
T
With
H
T
S
S
T
With
H
T
S
S
T
With
H
C
p
p
p
p
p
p
p
p
p
éq 5.3.1.1-18
2
3
C
C
=
éq 5.3.1.1-19
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
=
13
13
13
23
23
13
13
12
12
13
23
13
13
23
23
23
23
12
12
23
12
23
23
12
12
23
23
12
12
12
4
)
(
18
2
1
)
(
)
(
9
)
(
)
(
9
)
(
)
(
9
)
(
18
2
1
)
(
)
(
9
)
(
)
(
9
)
(
)
(
9
)
(
18
2
1
S
T
With
H
H
G
D
C
S
T
S
T
With
H
S
T
S
T
With
H
S
T
S
T
With
H
S
T
With
H
H
G
D
C
S
T
S
T
With
H
S
T
S
T
With
H
S
T
S
T
With
H
T
S
With
H
H
G
D
C
C
p
p
p
p
p
p
p
p
p
µ
µ
µ
éq 5.3.1.1-20
Calculation of the rate of variation of volume:
C
p
p
C
C
C
Cr
C
C
Cr
p
v
C
C
Cr
p
v
p
With
H
BZ
D
S
S
S
With
H
B
p
D
B
p
K
M
P
M
p
K
P
P
M
p
K
P
P
M
)
)
(
(
).
(
)
(
3
2
)
2
2
(
),
2
2
(
2
2
2
2
+
-
+
+
+
=
+
=
+
-
+
-
=
+
-
=
éq
5.3.1.1-21
Code_Aster
®
Version
7.4
Titrate:
Law of behavior of the porous environments: model of Barcelona
Date
:
31/03/05
Author (S):
J. EL GHARIB, G. DEBRUYNE
Key
:
R7.01.17-A
Page
:
32/44
Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
with:
.
2
1
)
2
2
(
)
2
2
(
2
2
2
2
+
+
-
-
+
-
=
M
kP
p
K
P
P
M
M
p
K
P
P
M
B
Cr
C
C
Cr
C
C
Cr
and
.
2
1
2
2
2
2
+
-
=
M
kP
M
K
M
M
K
D
Cr
C
C
One thus has:
C
p
p
p
v
p
D
With
H
BZ
S
S
S
With
H
B
)
)
(
(
).
(
)
(
3
-
+
-
+
+
=
éq
5.3.1.1-22
and finally:
C
ij
p
atm
C
S
ij
ij
ij
p
kl
ij
kl
p
ijkl
ij
p
S
With
H
Z
p
p
K
D
With
H
BZ
S
S
S
With
H
B
C
)
)
(
3
)
(
3
3
)
(
3
(
)
)
(
)
(
(
0
+
+
+
+
+
+
-
-
+
+
-
=
éq
5.3.1.1-23
5.3.1.2 Processing of the hydrostatic part
It is considered now that the variation of loading is purely spherical (
0
=
S
).
The increment of
P
is written in the form:
(
)
-
+
+
=
-
-
1
exp
0
0
/
0
S
K
K
atm
C
atm
C
ev
p
p
p
p
K
P
P
éq
5.3.1.2-1
The derivation of this equation gives:
(
)
C
atm
C
S
p
v
v
p
p
p
P
K
K
P
K
P
+
-
-
=
0
0
0
éq 5.3.1.2-2
Calculation of
p
v
:
It is known that:
(
)
C
C
Cr
p
v
p
K
P
P
M
+
-
=
2
2
2
éq 5.3.1.2-3
By differentiating this equation, one obtains:
(
) (
)
(
)
C
C
Cr
C
C
Cr
p
v
p
K
P
P
p
K
P
P
M
+
-
+
+
-
=
2
2
2
2
2
éq 5.3.1.2-4
Code_Aster
®
Version
7.4
Titrate:
Law of behavior of the porous environments: model of Barcelona
Date
:
31/03/05
Author (S):
J. EL GHARIB, G. DEBRUYNE
Key
:
R7.01.17-A
Page
:
33/44
Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
One knows the expression of
:
(
)
[
]
p
p
C
Cr
C
C
Cr
C
C
C
Cr
H
B
H
p
P
p
K
P
P
P
K
M
S
S
P
p
K
P
P
M
=
+
+
-
-
+
+
-
=
'
)
(
2
)
2
(
3
2
2
2
2
éq 5.3.1.2-5
while posing
(
)
[
]
C
Cr
C
C
Cr
C
C
C
Cr
p
P
p
K
P
P
P
K
M
S
S
P
p
K
P
P
M
B
+
+
-
-
+
+
-
=
'
)
(
2
)
2
(
3
2
2
2
2
While differentiating
, it comes:
(
) (
)
[
]
(
)
(
)
+
-
+
+
+
-
-
+
+
-
-
+
-
+
-
-
-
-
-
+
-
+
+
-
=
C
C
C
Cr
Cr
C
C
C
C
C
C
C
Cr
Cr
Cr
C
C
Cr
Cr
p
C
Cr
C
C
C
Cr
C
C
C
Cr
C
C
Cr
C
C
C
Cr
C
C
Cr
p
p
p
K
P
P
P
K
p
K
p
Km No
p
K
P
PP
P
P
p
K
P
P
PP
H
B
km
p
P
p
K
P
p
P
p
K
P
p
P
P
K
p
P
P
K
P
p
K
P
P
P
p
K
P
P
H
M
2
2
3
3
4
4
2
2
3
2
2
2
'
)
(
2
'
)
(
2
)
2
(
)
2
(
2
2
2
2
2
2
2
2
4
2
éq 5.3.1.2-6
One seeks the expression of
Cr
P
according to
:
One a:
p
v
Cr
Cr
kP
P
=
éq
5.3.1.2-7
One can write:
(
)
(
)
C
C
Cr
C
C
Cr
Cr
Cr
p
K
P
P
M
p
K
P
P
M
kP
P
+
-
+
+
-
=
2
2
2
2
2
2
éq
5.3.1.2-8
(
)
C
C
C
C
Cr
Cr
Cr
Cr
p
K
M
P
M
p
K
P
P
M
kP
kP
M
P
2
2
2
2
2
2
2
2
1
+
+
+
-
=
+
(
)
C
Cr
Cr
C
Cr
Cr
Cr
Cr
C
C
Cr
Cr
p
M
kP
kP
K
M
P
M
kP
kP
M
M
kP
kP
p
K
P
P
M
P
+
+
+
+
+
+
-
=
2
2
2
2
2
2
2
1
2
1
2
2
1
2
2
éq 5.3.1.2-9
One poses
(
)
[
]
+
+
-
=
Cr
C
C
Cr
Cr
kP
M
p
K
P
P
kP
M
C
2
2
2
1
2
2
,
[
]
+
=
Cr
Cr
kP
M
kP
M
has
2
2
2
1
2
,
[
]
+
=
Cr
Cr
C
kP
M
kP
M
K
D
2
2
2
1
One has then:
C
Cr
p
D
C
P
has
P
+
+
=
éq 5.3.1.2-10
Code_Aster
®
Version
7.4
Titrate:
Law of behavior of the porous environments: model of Barcelona
Date
:
31/03/05
Author (S):
J. EL GHARIB, G. DEBRUYNE
Key
:
R7.01.17-A
Page
:
34/44
Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
By replacing the expression of
Cr
P
in
[éq 5.3.1.1-6], one finds:
(
)
(
) (
)
[
]
(
)
[
]
(
) (
)
[
]
C
C
C
C
C
Cr
C
C
Cr
Cr
C
p
C
C
C
C
Cr
p
C
C
C
C
Cr
C
C
Cr
Cr
p
p
C
C
C
C
Cr
C
C
Cr
C
C
C
C
Cr
C
C
C
C
C
C
Cr
p
p
K
P
p
K
P
P
D
p
K
P
P
P
K
H
B
km
p
K
P
p
K
P
P
C
H
B
km
P
p
K
P
p
K
P
P
has
p
K
P
P
P
H
B
km
H
M
p
p
p
K
P
P
p
K
P
D
P
P
P
K
p
K
P
C
P
p
P
p
ak
P
has
p
K
P
p
K
P
P
)
(
4
2
2
2
3
2
)
(
4
2
2
)
(
4
2
3
2
4
2
.
)))
(
(
'
2
)
(
2
)
2
(
(
)
(
2
'
2
2
2
2
2
2
2
4
2
4
2
4
2
+
+
-
+
+
-
-
+
+
-
-
+
+
-
+
+
-
-
+
+
-
+
-
-
+
+
+
-
-
-
-
+
+
+
-
=
éq 5.3.1.2-11
By gathering the terms in
and those in
P
, one finds:
C
p
E
H
P
E
F
+
=
éq 5.3.1.2-12
with,
[
]
(
)
(
)
[
]
2
2
2
2
4
2
3
4
4
2
3
2
4
2
'
2
2
2
)
)
2
1
(
(
2
2
C
C
C
C
C
C
Cr
Cr
Cr
C
C
Cr
p
C
Cr
C
C
C
Cr
p
p
K
p
Km No
p
K
P
PP
P
has
P
p
K
P
P
H
B
km
p
P
P
has
P
p
has
p
K
P
P
H
M
F
+
+
-
-
+
+
-
-
-
-
+
-
+
+
-
=
[
]
(
)
(
)
[
]
C
C
Cr
Cr
C
C
C
C
C
Cr
p
C
C
C
Cr
C
Cr
C
C
C
C
p
p
K
P
P
P
K
p
K
P
p
K
P
P
D
H
B
km
p
p
K
P
P
P
K
P
K
P
K
p
dk
P
D
H
M
H
2
2
3
)
(
4
2
2
))
(
(
'
2
2
2
2
2
4
2
+
-
+
+
+
-
-
+
+
-
+
-
+
-
-
=
(
)
2
2
2
2
4
2
3
4
4
2
2
)
(
2
1
C
C
C
C
C
C
Cr
Cr
p
p
C
C
p
K
p
Km No
p
K
P
PP
P
H
bckM
H
p
K
P
cm
E
+
+
-
-
+
+
+
=
The expression of
p
v
thus becomes:
C
p
v
p
Y
P
X
+
=
éq 5.3.1.2-13
with,
(
)
)
)
2
2
(
2
2
(
))
2
2
(
2
2
2
(
2
2
-
+
-
+
-
=
+
-
+
-
-
=
C
C
C
Cr
C
C
Cr
K
D
E
H
C
E
H
p
K
P
P
M
Y
p
K
P
P
E
F
E
F
C
has
M
X
from where the expression of
P
according to
v
and
C
p
:
)
)
(
1
(
)
1
(
0
0
0
C
atm
C
S
V
p
p
p
K
Y
P
K
PX
K
P
+
+
-
=
+
éq
5.3.1.2-14
Code_Aster
®
Version
7.4
Titrate:
Law of behavior of the porous environments: model of Barcelona
Date
:
31/03/05
Author (S):
J. EL GHARIB, G. DEBRUYNE
Key
:
R7.01.17-A
Page
:
35/44
Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
Calculus of the variation of deviatoric deformation:
ij
C
ij
p
ij
S
p
E
H
PS
E
F
S
3
3
3
~
~
+
=
=
=
éq 5.3.1.2-15
One thus has finally:
C
ij
ij
ij
p
K
P
F
+
=
éq 5.3.1.2-16
with
D
atm
C
S
D
p
p
K
P
K
PY
K
S
E
H
K
P
K
PX
K
S
E
F
F
1
)
)
(
3
3
(
3
,
1
3
1
3
0
0
0
0
0
+
+
-
=
+
-
=
éq
5.3.1.2-17
5.3.1.3 Operator
tangent
The tangent operator connects the variation of total stress to the variation of the deformation and of
suction. Since the increment of the total deflection under loading deviatoric is written:
Mn
klmn
ij
kl
p
ijkl
C
ij
ij
D
S
S
With
H
B
C
p
H
1
)
)
(
)
(
(
+
+
-
=
+
,
éq 5.3.1.3-1
with:
-
-
-
-
-
-
=
1
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
0
0
0
0
0
3
/
2
3
/
1
3
/
1
0
0
0
3
/
1
3
/
2
3
/
1
0
0
0
3
/
1
3
/
1
3
/
2
1
D
éq
5.3.1.3-2
projection in space deviatoric,
and that under spherical loading one a:
kl
kl
ij
C
ij
ij
D
F
p
K
2
=
-
éq
5.3.1.3-3
with:
-
-
-
=
0
0
0
3
/
1
3
/
1
3
/
1
2
D
éq
5.3.1.3-4
hydrostatic projection, one has then:
C
ij
kl
ijkl
ij
p
B
With
+
=
éq
5.3.1.3-5
Code_Aster
®
Version
7.4
Titrate:
Law of behavior of the porous environments: model of Barcelona
Date
:
31/03/05
Author (S):
J. EL GHARIB, G. DEBRUYNE
Key
:
R7.01.17-A
Page
:
36/44
Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
with:
1
2
1
)
)
(
)
(
(
-
+
+
+
-
=
kl
ij
mnkl
ij
Mn
p
ijmn
ijkl
D
F
D
S
S
With
H
B
C
With
éq
5.3.1.3-6
)
(
)
)
(
)
(
(
1
2
1
kl
kl
kl
ij
mnkl
ij
Mn
p
ijmn
ij
K
H
D
F
D
S
S
With
H
B
C
B
-
+
+
+
-
=
-
éq
5.3.1.3-7
The stress of Bishop is thus written:
(
)
C
lq
ij
kl
ijkl
ij
p
bS
B
With
-
+
=
'
5.3.2 Tangent operator at the critical point
As for the model
CAM_CLAY
one writes a tangent operator specific to the critical point. Like
for the general case, one makes a processing of the deviatoric part and another for the part
hydrostatic.
5.3.2.1 Processing of the deviatoric part
According to the equation [éq 4.3.3] one finds:
S
S
S
F
S
S
S
E
E
p
E
-
=
-
=
-
=
µ
µ
µ
6
2
~
2
éq
5.3.2.1-1
Expressions of the plastic multiplier
and of its derivation
are written in the following way:
µ
6
/
1
-
=
Q
Q
E
and
2
6
6
Q
Q
Q
Q
Q
E
E
µ
µ
-
=
éq
5.3.2.1-2
with,
E
E
E
E
Q
S
S
Q
2
3
=
and
Q
S
S
Q
2
3
=
from where the expression of
:
-
=
3
2
3
6
1
Q
S
S
Q
Q
Q
S
S
E
E
E
E
µ
éq
5.3.2.1-3
Let us point out in the same way the expression of
S
:
(
)
ij
ij
ij
ij
S
S
S
µ
-
-
=
3
3
~
2
While replacing
and
by their expressions, one can write:
ij
E
ij
kl
kl
E
ij
E
E
kl
E
kl
ij
ij
S
Q
Q
S
S
S
Q
Q
S
Q
Q
S
S
S
µ
-
-
+
-
=
1
1
2
3
2
3
~
2
3
éq
5.3.2.1-4
Code_Aster
®
Version
7.4
Titrate:
Law of behavior of the porous environments: model of Barcelona
Date
:
31/03/05
Author (S):
J. EL GHARIB, G. DEBRUYNE
Key
:
R7.01.17-A
Page
:
37/44
Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
kl
E
ij
E
kl
ijkl
ij
kl
E
ijkl
ijkl
E
ijkl
kl
Q
Q
S
S
S
S
Q
Q
Q
Q
S
µ
~
.
2
3
2
.
2
3
1
1
3
-
=
-
-
+
éq
5.3.2.1-5
or in tensorial writing:
µ
~
2
3
2
2
3
1
1
4
3
4
4
4
4
4 3
4
4
4 2
1
4
4
4
4
4
4
3
4
4
4
4
4
4
2
1
H
E
E
D
G
E
D
D
E
Q
Q
S
S
I
S
S
Q
Q
I
I
Q
Q
S
-
=
-
-
+
éq
5.3.2.1-6
Like
S
does not depend on
v
, one can confuse
~
with
.
By using the tensor of projection in the space of the deviatoric stresses
1
D
[éq 5.3.1.3-2], one
can write:
µ
.
2
.
.
1
1
-
=
H
G
D
éq
5.3.2.1-7
5.3.2.2 Processing of the hydrostatic part
In tensorial writing, there is according to the equation [éq 5.3.1.2-2] the following relation:
C
D
atm
C
S
v
D
p
I
p
p
P
K
K
P
K
P
I
+
-
=
0
0
0
éq
5.3.2.2-1
knowing that at the critical point,
0
=
p
v
.
Like
P
does not depend on
~
then one can confuse
v
with
.
C
D
atm
C
S
D
p
I
p
p
P
K
K
P
K
P
I
+
-
=
0
0
0
éq 5.3.2.2-2
By using the tensor of projection in the space of the hydrostatic stresses
2
D
[éq 5.3.1.3-3], one
can write:
C
D
atm
C
S
D
p
I
p
p
P
K
K
P
K
D
I
+
-
=
0
0
0
2
from where
(
)
C
atm
C
S
D
D
p
p
p
K
I
P
K
D
I
+
+
=
0
0
2
éq 5.3.2.2-3
5.3.2.3 Operator
tangent
By combining the contributions of the two parts deviatoric and hydrostatic, one finds the writing of
the tangent operator who connects the variation of the total stress to the variation of the total deflection to
not criticizes:
(
)
C
atm
C
S
D
D
p
p
p
K
I
P
K
D
I
H
G
D
µ
+
+
+
=
-
0
0
2
1
1
.
2
.
.
C
ij
kl
ijkl
ij
p
B
With
-
=
éq
5.3.2.3-1
Code_Aster
®
Version
7.4
Titrate:
Law of behavior of the porous environments: model of Barcelona
Date
:
31/03/05
Author (S):
J. EL GHARIB, G. DEBRUYNE
Key
:
R7.01.17-A
Page
:
38/44
Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
with
1
0
2
1
1
2
.
.
-
-
+
=
P
K
D
I
H
G
D
With
D
ijkl
µ
éq
5.3.2.3-2
and
(
)
atm
C
S
D
ij
p
p
K
I
B
+
-
=
0
éq 5.3.2.3-3
As it is necessary to deduce the variation from the stress of Bishop, one finds:
(
)
lq
atm
C
S
D
ij
bS
p
p
K
I
B
-
+
-
=
0
éq 5.3.2.3-4
5.3.3 If the hydrous criterion is reached
The variation of the elastic strain is written in the form:
kl
E
v
E
kl
E
kl
3
1
~
-
=
éq
5.3.3-1
that is to say:
(
)
kl
atm
C
S
C
kl
kl
E
kl
p
p
K
p
P
K
P
S
µ
+
-
-
=
0
0
3
3
2
éq
5.3.3-2
In this case the plastic deviatoric deformation is null thus the plastic deformation has
the following expression:
kl
p
v
p
kl
3
1
-
=
éq
5.3.3-3
that is to say:
(
)
kl
atm
C
S
C
p
kl
p
p
K
p
+
-
=
0
3
éq
5.3.3-4
By combining each component rubber band and plastic one finds:
(
)
(
)
kl
C
atm
C
S
atm
C
S
kl
kl
p
kl
E
kl
kl
p
p
p
K
p
p
K
P
K
P
S
µ
+
+
+
-
-
=
+
=
0
0
0
1
1
3
1
3
2
éq 5.3.3-5
By using the matrices of projection in the space of the deviatoric and hydrostatic stresses one
leads to the following expression:
(
)
(
)
kl
C
atm
C
S
atm
C
S
ij
kl
ij
ijkl
kl
p
p
p
K
p
p
K
P
K
D
D
µ
+
+
+
-
-
=
0
0
0
2
1
1
1
3
1
3
2
éq 5.3.3-6
Code_Aster
®
Version
7.4
Titrate:
Law of behavior of the porous environments: model of Barcelona
Date
:
31/03/05
Author (S):
J. EL GHARIB, G. DEBRUYNE
Key
:
R7.01.17-A
Page
:
39/44
Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
thus one can write:
(
)
(
)
C
kl
kl
ij
ijkl
atm
C
S
atm
C
S
kl
kl
ij
ijkl
ij
p
P
K
D
D
p
p
K
p
p
K
P
K
D
D
µ
µ
1
0
2
1
0
0
1
0
2
1
3
2
1
1
3
1
3
2
-
-
-
+
+
+
+
-
=
éq 5.3.3-7
one poses
-
=
P
K
D
D
With
kl
ij
ijkl
ijkl
0
2
1
3
2
µ
or
+
+
-
+
-
+
-
+
+
-
+
-
+
-
+
=
1
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
0
0
0
0
0
9
2
3
2
9
2
3
1
9
2
3
1
0
0
0
9
2
3
1
9
2
3
2
9
2
3
1
0
0
0
9
2
3
1
9
2
3
1
9
2
3
2
2
1
0
0
0
0
0
0
0
0
0
P
K
P
K
P
K
P
K
P
K
P
K
P
K
P
K
P
K
With
ijkl
µ
µ
µ
µ
µ
µ
µ
µ
µ
µ
éq 5.3.3-8
and by deducing the stress from Bishop, one finds:
(
)
(
)
C
lq
kl
ijkl
atm
C
S
atm
C
S
kl
ijkl
ij
p
bS
With
p
p
K
p
p
K
With
-
+
+
+
+
=
-
-
1
0
0
1
'
1
1
3
1
éq
5.3.3-9
Code_Aster
®
Version
7.4
Titrate:
Law of behavior of the porous environments: model of Barcelona
Date
:
31/03/05
Author (S):
J. EL GHARIB, G. DEBRUYNE
Key
:
R7.01.17-A
Page
:
40/44
Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
6
Summary of the model of Barcelona
Modelings THHM:
KIT_HHM
and
KIT_THHM
(in this last case, there is no dependence of the characteristics
mechanics with the temperature).
Variables of input:
,
,
,
,
,
0
'
-
-
+
+
-
C
Cr
gz
C
p
P
p
p
,
C
p
and
gz
p
Variables of exit:
·
'
, more tangent operators (necessary to operator STAT_NON_LINE).
·
Internal variables
+
Cr
P
, newer variables
+
0
C
p
: threshold in suction and
+
S
P
: pressure of
cohesion, and indicators of mechanical work hardening
1
I
and hydrous
2
I
.
Elastic prediction:
(
)
0
0
/
0
exp
S
K
K
atm
C
atm
C
v
E
p
p
p
p
K
P
P
+
+
=
-
-
,
µ
~
2
+
=
-
S
S
E
·
0
1
<
F
and
0
2
<
F
)
(
0
C
C
p
p
<
: reversible behavior
E
P
P
=
,
0
=
=
p
E
S
S
,
-
-
=
=
0
0
C
C
Cr
Cr
p
p
P
P
,
,
·
0
1
>
F
or
0
2
>
F
plasticization and mechanical and hydrous work hardening
[
]
p
v
E
K
P
P
-
=
0
exp
,
(
)
C
C
Cr
p
v
E
p
K
P
P
M
S
S
+
-
+
=
2
2
6
1
2
µ
[]
p
v
Cr
Cr
K
P
P
=
-
exp
,
(
)
[
]
p
v
S
atm
C
atm
C
K
p
p
p
p
+
=
+
exp
0
0
The single unknown factor is
p
v
determined by
0
1
=
F
(one has then:
(
)
Cr
S
p
v
p
P
P
P
M
Q
2
2
2
~
2
-
+
=
)
or
0
2
=
F
(and
0
~
=
p
)
Note:
The stress resulting from the data of the model of Barcelona east
D
gz
early
p 1
+
=
, it will be
thus the variable used in the routine describing the behavior, the stress of exit
provided to
STAT_NON_LINE
being the stress of Bishop:
p
early
-
=
'
.
Code_Aster
®
Version
7.4
Titrate:
Law of behavior of the porous environments: model of Barcelona
Date
:
31/03/05
Author (S):
J. EL GHARIB, G. DEBRUYNE
Key
:
R7.01.17-A
Page
:
41/44
Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
Tangent operators:
The tangent operator of the generalized stresses is implemented in THHM under the name
OF
D
and
partitionné in several blocks. The components concerned with the model are
'
&
and
C
p&
&
'
blocks [DMECDE] and [DMECP1] agent with:
C
p
C
p
C
C
p
p
p
p
p
&
&
&
&
&
&
&
&
&
&&
&
'
'
'
'
'
,
.
7
Implementation of the model
7.1 Data
material
The use of the model of Barcelona requires to enrich the data of the model by Camwood-clay by
additional data specific to the unsaturated grounds. This is concretized by the simultaneous adoption
of the two key words Camwood-clay and Barcelona under the control
DEFI_MATERIAU
.
7.2
Initialization of calculation
It is necessary that the initial state of material is plastically acceptable (the stress and the pressure
thin cable are thus such as the point of initial loading is inside the surface of load). It
is necessary thus on the one hand that suction is lower than the hydrous threshold, and on the other hand that the stress
maybe inside the ellipse defined in the plan of initial suction. In particular, if the loading
mechanics initial is purely hydrostatic, it must lie between the terminals represented by
cohesion (
C
C
p
K
-
) and pressure of consolidation (
Cr
P
2
). The stress
used to describe it
behavior (forced total plus gas pressure) is different from the stress to initialize in
ETAT_INIT
(stress of Bishop
'
). The relation between the two types of stress is:
[
]
C
lq
gz
p
bS
I
p
B
&
&
&
&
-
-
+
=
)
1
(
'
7.3
Variables intern at exit
The model produces five internal variables:
Cr
P
V
=
1
: critical pressure
1
2
I
V
=
: mechanical indicator of irreversibility
0
3
C
p
V
=
: hydrous threshold of irreversibility
2
4
I
V
=
: hydrous indicator of irreversibility
S
P
V
=
5
: pressure of cohesion
Code_Aster
®
Version
7.4
Titrate:
Law of behavior of the porous environments: model of Barcelona
Date
:
31/03/05
Author (S):
J. EL GHARIB, G. DEBRUYNE
Key
:
R7.01.17-A
Page
:
42/44
Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
8
Development prospects of the model
One of the phenomena not studied in the original model of Barcelona is nonthe reversibility of
capillary curve of pressure [Figure 8-a] and its dependence with the state of stress. This is treated by
Dangla and coll [bib2] by integrating the model of Barcelona within a framework poroplastic with
the introduction of the water content like additional poroplastic variable, whose evolution is
directly connected not only to the capillary variation of pressure via the curve
of drainage-imbibition, but also with the mechanical evolution of the medium. Two aspects should be distinguished there
distinct but nevertheless coupled phenomenon. Nonreversibility of the curve drainage-imbibition
is a phenomenon purely hydraulic and thus independent of the mechanical law adopted in one
modeling THHM, but this curve thus depends on the index of the vacuums of the mechanical state on
medium. The partition of the water content partly elastic and plastic and of the considerations
thermodynamic [bib2] allows to deduce the evolution at the same time from the water content (and thus from the degree
of saturation) and stress according to the deformation and of the capillary pressure. By
example, the evolution in the field of reversibility is given by:
)
(
)
,
(
)
,
(
E
C
E
C
C
E
E
lq
dtr
p
B
dp
p
NR
D
+
-
=
)
(
)
,
(
)
,
(
E
C
E
C
C
E
dtr
p
K
dp
p
B
dP
+
=
lq
p
lq
E
lq
C
p
Curve of drainage
Curve of imbibition
Field of reversibility
Appear 8-a
Where
)
,
(
B
NR
are the generalized coefficients of Biot [bib6]. To enrich the model by Barcelona in it
direction thus implies two separate developments:
1) The introduction of a curve of drainage-imbibition into developments THHM.
2) The complétude of the model of Barcelona by the calculation of the degree of saturation in addition to
stress.
Code_Aster
®
Version
7.4
Titrate:
Law of behavior of the porous environments: model of Barcelona
Date
:
31/03/05
Author (S):
J. EL GHARIB, G. DEBRUYNE
Key
:
R7.01.17-A
Page
:
43/44
Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
9 Bibliography
[1]
E.E. ALONSO, A. GENS, A. JOSA: With constitutive model for partially saturated soils.
Geotechnics 40. NO3, 405 430., 1990.
[2]
P. DANGLA, L. MALINSKY, O. COUSSY: Plasticity and imbibition drainage curves for
unsaturated soils: unified approach has. (1997). International Symposium one Numerical Models
in Geomechanics. Pietruszczak & Pande (eds) Balkema, Rotterdam.
[3]
P. DELAGE: The behavior of the not-saturated grounds. (2000). ENPC. Run exempted to
DER, Clamart.
[4]
NR. TARDIEU, I. VAUTIER, E. LORENTZ
: Quasi-static nonlinear algorithm.
Reference material Aster [R5.03.01].
[5]
J. EL GHARIB, G. DEBRUYNE: Law of Cam_Clay behavior, Doc. [R7.01.14],
Code_Aster (2002).
[6]
T. LASSABATERE: Hydraulic couplings in porous environment unsaturated with
phase shift: application to rtrait of dessication. Thesis of doctorate of the ENPC,
Paris (1994).
[7]
C. CHAVANT: Models of behavior THHM, Doc. [R7.01.11], Code_Aster (2001).
[8]
C. CHAVANT
: Modelings Thermo Hydro-mechanical THHM. General information and
alogorithmes, Doc. [R7.01.10], Code_Aster (2001).
Code_Aster
®
Version
7.4
Titrate:
Law of behavior of the porous environments: model of Barcelona
Date
:
31/03/05
Author (S):
J. EL GHARIB, G. DEBRUYNE
Key
:
R7.01.17-A
Page
:
44/44
Manual of Reference
R7.01 booklet: Modelings for the Civil Engineering and the géomatériaux ones
HT-66/05/002/A
Intentionally white left page.