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Law of behavior of the porous environments: model of Barcelona Date
:

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:
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Organization (S): EDF-R & D/AMA
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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 of Cam Clay modified, also
implemented in Code_Aster [bib5]. He is particularly adapted to the study of the behavior of clays.

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1 Notations

T
indicate the tensor of the total constraints in small disturbances, noted in the shape of the vector
according to:

T
11



T
22

T

33


2 T



12
2 T
23

T

2 31

The behavior is described in a space of constraints to two variables:

T
= + p I
=
gz and p
-
C
pgz plq,

with plq, pgz, PC respectively pressure of liquid, gas pressure, capillary pressure (or
suction)

One notes:

I the tensor unit of command 2 whose indicielle notation is
ij
I

4 the tensor unit of command 4 whose indicielle notation is

ijkl

1

P = - tr ()
constraint of containment
3

S = + pi
diverter of the constraints

1

I = tr

second invariant of the constraints
2
(S.S)
2

Q = = 3I
equivalent constraint
eq
2


= 1

(U T
+ U)
total deflection
2

= + +
partition of the deformations (elastic, plastic, thermal)
E
p
HT

= - tr + -


v
() 3 (T T0)
voluminal total deflection

p
= -

tr
v
(p)
voluminal plastic deformation


= 1
~
+ I
diverter of the deformations
v
3
~e ~ ~p
= -
deviatoric elastic strain

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p
p
1
~
p
= + I
deviatoric plastic deformation
v
3


E

2
=
tr ~ ~
.

equivalent elastic strain
eq
(E E)
3


p

2
=
tr ~ ~
.
equivalent plastic deformation
eq
(p p)
3

E index of the vacuums of the material (report/ratio of the volume of the pores on the volume of the solid matter constituents)

E initial index of the vacuums
0

porosity (report/ratio of the volume of the vacuums on total volume (pores plus grains))

E
p
,
lq
lq
lq content of total, elastic and plastic liquid

coefficient of swelling (elastic slope in a hydrostatic test of compression)

S elastic coefficient of rigidity in a test of variation of suction
1
(+ E)
0
K =

0

(1+ E)
K
0
=
0s

S

(p)
C 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

(-)
(1+ E)
K =
0
S

(-)
S
S

M slope critical line of state

coefficient of correction of the normality of the plastic flow

P
(
)
idiots PC pressure of consolidation

P (
)
critical Cr PC 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)
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P confining pressure of reference generally equal to the atmospheric pressure
0
has
P

kc slope of cohesion according to suction

parameter controlling the increase in (p)
C with PC

R parameter defining the peak of (p)
C with PC

µ elastic coefficient of shearing (coefficient of Lamé)

f1 surface of load in space (P, Q)
f2 surface of load out of PC
pc0 threshold of irreversibility of suction
plastic multiplier
S
lq
water saturation, S
=

lq
lq

p
voluminal plastic deformation due to a loading in hydrostatic pressure
vp
p
voluminal plastic deformation due to a loading in suction
vs
p
~ deviatoric plastic deformation due to a loading in hydrostatic pressure
p
B coefficient of Biot
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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, PC, which distinguishes it from the models from
mechanics coupled to a thermohydraulic behavior which is described using only one
effective constraint (constraint 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 principal 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 liquid (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 p = (p - p). While drying, a ground unsaturated has a water content lower what
C
gz
lq
lq
corresponds to a higher suction. The correspondence between these two sizes is the curve of
retention of water (cf [Figure 2.1.1-a]). This one is obtained by drying of a ground initially saturated (
suction is then null) and damping starting from the dry state.
PC
lq

Appear 2.1.1-a: Courbe of retention of water

2.1.2 Extension of the definition of the effective constraints on the unsaturated ground

The behavior soil mechanics unsaturated is primarily observed in laboratory with the assistance
apparatuses with controlled suction (oedometers and triaxial). The modeling of this behavior
mechanics was initially tried by extending the concept of effective constraint to the unsaturated mediums.
This one is a function of the total constraint and intersticielle pressure: '
= F (T
, p)
lq. In
the saturated case, one has simply additivity of the pressure and the constraint:
T
'= - p I
lq 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
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pressures of the two liquid phases) brought the following form of the effective constraint:
'
= (T
- p I) + G (p)
gz
C of which there remained the constraint suggested in the form:
'
T
D = D (- pgz) + bSlq dpc
where S is the degree of saturation out of water and B the coefficient of Biot [bib7].
lq
As of the years 1960, the experimental observation clarifies certain limitations of the concept of
effective constraint extended on the unsaturated ground. In particular, the test of collapse to the oedometer
at fault the constraint of Bishop puts: This test consists in consolidating a sample unsaturated with
maintaining suction constant, then with the remouil er with constant loading. One then is observed
collapse of the ground. If the consolidation is continued, the curve corresponds to a standard test in
saturated. However, if one refers to the effective constraint, this one decreases during the remoistening (since
suction p = (p - p) is cancelled) and as it is supposed to control the deformation, it would owe y
C
gz
lq
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 constraint and note the need for using
two independent variables (constraint 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 pc0 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

The assumption is made that suction p remains constant during the mechanical transformation.
C
deformations resulting from the variation of the constraint 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 of the sample varies
logarithmiquement with the load with a slope of way reversible until a pressure of
consolidation
idiots
P
(PC). One will choose independent of p, the experiment showing weak
C
dependence of the elastic slope with respect to p.
C

The elastic component of the voluminal deformation varies then like:

&
E
P
& =
if P
P
(p)
vp
<




éq 3.1.1.1-1
1
idiots
C
+ e0 P
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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 = P0
[
exp k0 (
p
-
vp
vp)]





éq 3.1.1.1-2
(1+ e0)
with K =
, where P is the value of reference corresponding to E

and E = E, index of
vp = 0
0

0
0
vacuums initial.

3.1.1.2 Plasticity

Beyond the pressure of consolidation, the behavior of the ground is plastic and the slope (p is
c)
dependant on suction (cf [Figure 3.1.1.2-a]), this dependence being estimated way
semi-empirical following: (p = 0 1 - exp - +
c)
() ([R) (PC) R]
(p)
where R =
C
is a constant connected to the maximum of the rigidity of the ground and a parameter which
(0)
control the evolution of rigidity according to suction.

(PC) P&
The voluminal rate of deformation is then: vp
& =
if P > P,
1+ E P
idiots
0
p
((PC) -) P&
from where the plastic component: &vp =
.
1 + E
P
0
The expression of P is thus written:
P = P

éq
3.1.1.2-1
0
[
exp K (p
vp)]
(1+ E)
with K =
0

-

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.
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v


(pc1)
(pc2)
(p = p
C
c1)
(p > p
c1
c2)
(p = p
C
c2)
2
1
ln P
P
P0 P
p
P
p
idiots (
1
c)
idiots (c2)
0

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:
~e
S
=









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 Q = - (one will be able
1
3
to extend the formulation which follows with the 3D by using the standard within the meaning of von Mises of the constraint).
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 constraints in zero and a value of pressure of consolidation *
*
P
= 2P.
idiots
Cr
surface of load associated with a nonnull suction p is also an ellipse of center
C
P
(P (p)
S
-
)
0
,
Cr
C
(cf [Figure 3.1.2.2-a]) which cuts the hydrostatic axis out of P
(p) = 2P (p)
2
idiots
C
Cr
C
and - P, P representing a cohesion varying linearly with suction: P = K p. the line
S
S
S
C
C
representing the critical states (null voluminal variation) the same slope M preserves as that in
saturated but shifted condition S
P. The equation of the surface of load in the diagram (P, Q)
for p given is written:
C
2
2
Q - M (P + P
éq
3.1.2.2-1
S) (2Pcr - P) = 0
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The plastic flow in the plan (P, Q) thus with p constant is not associated the surface of
C
charge. If it were the case, one would have:

1
~

p
F
p
F
& = &
& = 1
&

vp
P

S

and the following report/ratio:
~ p
&eq
Q
2
=
,
éq
3.1.2.2-2
p
2
&vp
M (2P + S
P - 2 Cr
P)
similar to the report/ratio obtained in the Camwood-Clay model (with S
P = 0). In fact in this model, one
introduced a parameter of correction which destroys the character of normality, so that:
~ p
&eq
Q
2
=

. is given by the authors of the model [bib1] as being:
p
2
&vp
M (2P + S
P - 2 Cr
P)


M (M - 9) (M - 3)


1

=
(
éq
3.1.2.2-3
9 6 - M)




1


(0)

This corrector allows to better take into account the experimental results, and in particular of
to better estimate the coefficient of thorough grounds.

Q
(PC > 0)
M
(PC = 0)
M
P
- P
*
S
idiots
P
P (p)
idiots C

Appear 3.1.2.2-a: Critère in space (P, Q)

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) reversible when p <
and irreversible when suction exceeds the threshold p.
S
C
pc0
c0
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3.2.1 Part
reversible

The tests oedometric with constant constraint 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:
PC
E - E = - Ln
if
0
p < p
S
,
C
c0
patm
with S independent of the state of containment.
E - E
Deformation being able to be written:
0
- = -
, one a:
v
v0
1+ e0

E
S
p&c
1
p&c
& =
=
vs

éq
3.2.1-1
1+ E p
0
+
0
+
C
patm
K S (PC patm)
The evolution of suction is written then:
1 + E
p = p exp K (-
, with K
0
=
éq
3.2.1-2
C
atm
(E E
0s
vs
v0)
0s
S
E
e0
S
S
Lnpc
Lnp c0
Lnpa

Appear 3.2.1-a: Evolution of suction

3.2.2 Part
irreversible

Beyond threshold PC, unrecoverable deformations appear, the slope in the test oedometric
0
becoming S. Cette slope can actually depend on the hydrostatic constraint 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 (p
and unsaturated
C = 0)
(p
. A relation between *
P
(point 3) value of the preconsolidation with of saturated and P
(point 2)
C > 0)
idiots
idiots
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 P
with *
P

idiots
idiots
with constant suction followed by a remoistening of a value p to 0 with constant pressure *
P
, from where
C
idiots
the following equation:
v + v

+ v

= v
1
pressure
suction
3
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The assumption is made that the reduction of suction 2 3 is accompanied by recoverable deformations.
dp
The elastic relation is as follows:
C
FD = -
, where p
is the atmospheric pressure.
S (p + p
atm
C
atm)
One writes for item 1 and 3 the expression of volume as follows:

P
v = NR (P) - p ln

0
(c) P0
where P is a pressure of reference corresponding to an initial volume NR (P). One combines this
0
0
expression and elastic relations:

+
NR (
P
P
p
p
P
P - p ln idiots + ln idiots + ln C
atm = NR 0 - 0 ln idiots
0)
(c)
S
() ()
*
*
P
P
p
P
0
idiots
atm
0

By eliminating initial volumes by the elastic relation:

0
+
v
(P


0)
= NR (0) - NR (P0)
p
p
C
atm
=
ln
S
PC
patm

one then determines the following evolution of the threshold of consolidation in unsaturated condition:

(0) -

P P * (PC)
-
idiots
=



idiots

P0 P0
Like P
= 2P,
idiots
Cr

One finds:
(0) -

P 2P * (PC)
-
P = 0 Cr

éq
3.2.2-1
Cr


2 P0

[Figure 3.2.2] visualizes path 1-2-3 in the plan (P, p)
C.
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v
NR (0)
G O N fl EM in T
NR (PC)
PC
3
v
v
3
suction
v
1
S I
2
v pressure
2
2
1
v1
(p, S
C
)
L C
effo N D R EM in T
(PC = 0, S = 1)
3
P
*
P
LnP
*
P
ref.
P
P
idiots
idiots
idiots
idiots
(A)
(b)

Figure 3.2.2: (A) Courbes of compression for water-logged soils and not-saturated
(b) criterion in the diagram (P, p
c)

The total component of the voluminal deformation due to the evolution of suction is:
S
p&c
& =
if p > p
vs

éq
3.2.2-2
C
atm
1
(+ E) p
0
+
C
patm
from where the plastic component which is written:

p
(-
S
S)
p&c
1
p&c
& =
=
vs
éq
3.3.2-3
1
(+ E) p
0
+
+
C
patm ks PC patm

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:
1 &
E
E
E
P
1
p&c
& = & + & =
+
v
vp
vs




éq 3.3.1-1
K P
K
0
0
+
S (PC
patm)
The evolutions of the parts hydrostatic and deviatoric of the constraint are thus written:

P&
K
p
E
0
&c
= K
,





éq 3.3.1-2
0 & -
v
P
K
p
0
+ p
S (C
atm)
~e
S & = 2
µ &,







éq 3.3.1-3
ij
ij
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3.3.2 Thresholds of flow

The two thresholds of the reversible field are such as:
Mechanical criterion: F (P, Q, P (p), p)
2
2
Q
MR. P K p P
P p
éq 3.3.2-1
Cr
C
C
=
+
+ C C
- Cr C
1
(
) (2 ()) 0
Hydrous criterion: F p p
p
p









éq 3.3.2-2
C
C
= C - C
2 (
,
0)
0
0
The three-dimensional field of reversibility in space (P, Q, p) is represented on
C
[Figure 3.3.2-a].
These two criteria are reduced in the plan (P, p) to curves called LLC (loading collapse) and IF
C
(suction increase) (cf [Figure 3.3.2-b]).

Q
f1
f1
*
Pcons
f2
PC
IF
LLC
F
P
2

Appear 3.3.2-a: Surface of load in space (P, Q, p
c)

PC
p
IF
c0
LLC
- PS
Pcons
P

Appear 3.3.2-b: Surface of load in space (P, PC)

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3.3.3 Laws of flow

The plastic flow is controlled by the two criteria in pressure and suction:

p
F
1
2
&
&
&
v =
= M (2P - 2 Cr
P + kc PC)




éq 3.3.3-1
P

~ p
F

F
1

Q
1
& = &
= &
=
3 &s





éq 3.3.3-2
S

Q

S

p
1
p&c
~
or &

v =
, p
& = 0






éq 3.3.3-3
ks PC + patm

3.3.4 Laws
of work hardening

The evolution of surfaces of load is controlled by the forces of work hardening: Cr
P and pc0.
The laws of work hardening of each surface are:
P&
On F, Cr
p
= K












éq 3.3.4-1
1
&v
C
P R
p&
On F,
c0
p
= K










éq 3.3.4-2
2
S &v
p 0 +
C
patm

3.3.5 Inventory

configurations
of mechanical and hydrous coupling

One examines the various configurations of loading in space (P, p)
C.

3.3.5.1 Reversibility
total

The loading represented by the point M (cf [Figure 3.3.5.1-a]) is inside the field of
reversibility: elasticity, and hydrous reversibility. That results in:

F < 0
F =, F & <
p&
).
C <
1
, or (
0
0
1
1
), and p <
C
pc0, or (p =
C
pc0,
0

The relations expressing this reversibility are:

P&
K
p
0
&c
= K

0 & -
v
P
K
p
0
+ p
S (C
atm)

i.e.:
(
exp K -
0 (v
v0))
P = P0
,





éq 3.3.5.1-1
K/K
0
0s
p +

C
patm


p



atm

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and
S =

µ~
2









éq 3.3.5.1-2
C
p
PC
C
p 0
pc0
M
field of reversibility
P
Slq

Appear 3.3.5.1-a: Domaine of reversibility in the plan (P, p - curve of retention of water
c)

3.3.5.2 Elastoplastic behavior

The point M touches the criterion of mechanics alone (cf [Figure 3.3.5.2-a]):

F = 0, F & = 0, and p < p (or p = p and p&
)
C < 0
1
1
C
c0
C
c0

The elastic evolution is thus written:

P&
K
p
E
0
&c
= K
,
0 & -
v
P
K
p
0
+ p
S (C
atm)
i.e.:
(E E
exp K -
0 (v
v0))
P = P0







éq 3.3.5.2-1
K/K
0
0s
p +

C
patm


p



atm

and
S =

µ~
2
éq
3.3.5.2-2

The evolution of the components of the plastic deformation is:

~ p
& =
3
S

p
2
&v = M [2P - 2 Cr
P + kc PC]

The evolution of the mechanical threshold is written:
p
2
Cr
P & = K Cr
P &vp = K Cr
P
M

[2P - 2 Cr
P + kc PC].
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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:

p&
&
C
K
0
S
Cr
P
=







éq 3.3.5.2-3
p 0 +
C
patm
K Cr
P

C
p
p
2
C 0
1 M
P

Appear 3.3.5.2-a: Couplage 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]):

p = p and p&

C > 0
C
c0

The mechanical behavior is elastic:

(E E
exp K -
0 (v
v0))
P = P0
, S =

µ &
&
~
2
éq
3.3.5.3-1
K/K
0
0s
p +

C
patm


p



atm

but as the mechanical threshold is coupled with that of suction, there is also mechanical work hardening:

&cr
P
K
p&c
=
0
éq
3.3.5.3-2
P
K p 0 + p
Cr
S
C
atm
The rate of plastic deformation is written:
p
1
PC
==
&
&v

K
+
S PC
patm
~ p
& = 0
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C
p
M
p
1
c0
2
P

Appear 3.3.5.3-a: Couplage 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 key word ELAS:
The thermal dilation coefficient, two elastic coefficients E
, provided in
given from which the coefficient of Lamé µ is calculated.

2) Under key word CAM_CLAY:
·
0
P Initial hydrostatic Pression equal to noted atmospheric pressure Pa under the key word
CAM_CLAY
· instead of giving the initial index of the vacuums E one gives the initial porosity which must be of
0
value equalizes with that given under key word THM_INIT, noted PORO.
· Parameters associated with surface threshold LLC (forced isotropic): *
P, equalizes with half
Cr
pressure of preconsolidation *
P
noted
idiots
PRES_CRIT,
*
, the coefficient of
compressibility for a saturated state and the elastic coefficient of compressibility, noted
LAMBDA and KAPA.
· The slope criticizes M,

3) Under the key word BARCELONA:
· R and, coefficients allowing to calculate (p, noted R and BETA.
c)
· parameters related to a variation of suction: , coefficient of compressibility related to one
S
variation of suction in the plastic range, coefficient associated with the change with
S
suction in elastic range, noted LABDAS and KAPAS.
· K the parameter which controls the increase in cohesion with suction
C
· the initial threshold of suction p, noted PC0_INIT
c0
· the coefficient of normality, noted ALPHAB.

Here a set of values of some of these parameters, resulting from [bib1]:
(0) =
;
2
.
0 =
;
02
.
0
R =
;
75
.
0
=
5
.
12
1
-
MPa; P = 10
.
0
MPa;
0



G
MPa M
K
S =
;
08
.
0
S =
;
008
.
0
= 10
;
=;
1 C = 6
.
0
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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 established behaviors THHM
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
quoted modelings with p
, p
, p and p that it is with hydrous modelings
C
gz
C
gz
LIQU_VAPE_GAZ or LIQU_GAZ.
The variables of output of the model are: '
, P, p, P.
Cr
c0
S
The following notations are employed: Has, A, A
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 the quantity P (-)
Cr PC,
(-
PC) -

-
P 2P (PC)
-
P (
)
0
Cr
Cr PC quantity




and
2 P0
P (p) = p (p) exp K


Cr
C
Cr
C
(statement)

4.2 Relations
incremental

The rules of flow and the condition of consistency give the following relations of flow:
If the threshold f1 is reached, the increment of plastic deformation voluminal is written:

p
1
(2P - 2 Cr
P + kc PC)
Q
kc

v =
(2
)

éq 4.2-1
K (
P
Q
P
P
p
P + K
2
C PC)

+
-
Cr -
C
Cr
P
2
M
2


The increment of the standard of the equivalent plastic deformation is then:

2

p

Q
Q
2
kcQ (2 Cr
P -

P)
eqp =
P
Q
p
2
4
2
Cr
kP (P + kc PC)
+
-
C

M
M (2P - 2 Cr
P + kc PC)
M (2P - 2 Cr
P + kc PC)


éq 4.2-2

and the tensor deviatoric is written:
~ p
3 S

p


=

v
éq
4.2-3
M2 (2P - 2 Cr
P + S
P)
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If f2 is reached, the increment of plastic deformation voluminal is determined by:

p + p
p
1
=
Ln (c0
atm)
éq
4.2-4
v
K
p + p
S
c0
atm
plastic deformation being purely voluminal (~
p
= 0)
p
will be the principal of the problem and given unknown factor while solving
v
F (-
P, -
Q, -
P (p), p,
Cr
C
C p
) 0
1
v
=, or F (p,
C p
)
C
= 0
2
0
, the increment of plastic deformation
voluminal being obtained starting from p
c0.On then deduces the evolution from it from the constraints and the thresholds.

4.3
Calculation of the constraints and the internal variables

The elastic prediction of the deviatoric constraint is written:
= -
S + µ ~
2








éq 4.3-1
One chooses the elastic prediction E
P in the following way:

-
exp K
E
(0 v)
P = P







éq 4.3-2
K K
0/0 S
p + p
C
atm

p + p
C
atm
(-
PC) -
P
-
P (PC)
E
E
p
2

-
If F < 0
F <
0
P = P S = S
= P =
Cr
p
,
Cr
C =
1
and
0
2
, then
,
,
,
0
,
0
2


0
P0
If not:
E
~ p
S = S - 2µ











éq 4.3-3
E
P = P exp [
p
- K

éq
4.3-4
0 v]
(p -
c)

-

2
(

PC)
P
P

-
0
Cr
P =
exp K





éq 4.3-5
Cr




[statement]
2
P
0
(p
éq
4.3-6
0 + p
) = (p-0 + p) exp K
C
atm
C
atm
[p
S
v]

The principal unknown factor is thus
p
.
v

If F > 0
1
, then

While replacing
p
~

by its expression according to
p
[éq 4.2-3] one obtains:
v
E
S
S =







éq 4.3-7
p
6

µ v
1+ m2 (2P-2 crP +kcpc)
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and:
E
P = P exp [
p
- K







éq 4.3-8
0 v]

The unknown factor is given by solving F (P, Q, P, p)
Cr
C = F (E
P, E
Q, -
P (p), p,
Cr
C
C p
)
v
= 0
1
1
,

I.e.:
p
2
2


µ

2
6
E
v
Q = - M 1
+

+
- 2
,
M2

(
P K p P
P
2P - 2P + K p
Cr
C
c)
(
C
c) (
Cr)


or:
p
2

2
6
2


E
µ
Q
= - M 1 +


M

2 (
v
E
2P exp (
p
- k0
-
v) - 2 Cr
P (PC) exp (
p
K v) + kc PC)
[eP exp (
p
- K
éq
4.3-9
0 v) + kc PC]
[eP exp (
p
- k0
-
v) - 2 Cr
P (PC) exp (
p
- K v)]

If F > 0
2
, then: p 0 =
C
PC, the unknown factor are immediately given by:

p
1
p
+ p
=
Ln (c0
atm)
v
, éq
4.3-10
K
-
S
PC 0 + patm
from where
E
S = S and
E
P = P exp [
p
- K








éq 4.3-11
0 v]
(p -
c)

-

2
(

PC)
P
P

-
One has moreover
0
Cr
P =
exp K.








éq 4.3-12
Cr




[statement]
2
P
0

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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:

elp
& = D
ij
ijkl kl
&,

i.e.
elp
D is calculated starting from the not discretized equations.
ijkl

If the option is: FULL_MECA, option used when one reactualizes the tangent matrix while updating
internal constraints and variables:

D = A D
ij
ijkl
kl

In this case, A is calculated starting from the implicitly discretized equations.
ijkl
The tangent operator of the generalized constraints is implemented in THHM under name DDE 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:
K
p
~
0
&c
&
&
2 &
ij = - P ij + sij
& = K Ptr
0
&ij + µ ij +
P
ij
éq
5.1-1
k0s
PC + patm
2
K
p
0
&c
&ij = (K P
0
- µ) tr & ij + µ
2 ij
& +
P
ij éq
5.1-2
3
k0s
PC + patm
The tensor of the constraints used in the model of Barcelona (and the tests determining them
given model) is a function of the total constraint and of the gas pressure and is written:
T
= + p I
gz








éq 5.1-3

The tensor of the constraints of Bishop '
used in Code_Aster is such as: & = & '+ & I with
T
P
&
= -
-
P
B (p&gz Slq p&c) éq
5.1-4
From where the expression of the constraint of Bishop according to the constraint of the model of Barcelona:
& '
= & + (B
(-)
1 p & - bS p
gz
lq & c) I
éq
5.1-5
Note:

The constraint of Bishop is generally regarded as an effective constraint
(controlled only by the deformation). It is not the case of the model of Barcelona where it
is necessary two constraints ((, p)
C to describe the behavior. Consequently, in
'
&
& p
the tangent operator, the term
does not summarize itself with -
.
p
C
p
C
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'
Part [DMECDE] of matrix DDE corresponding to is such as:



4
2
2

'


K P
µ K P
µ K P
µ 0
0
0
&

11
0 +
0
-
0
-
11
&



3
3
3
'




2
4
2
&
&
22
K P
µ K P
µ K P
µ 0
0
0
0
-
0
+
0
-
22





éq 5.1-6
'



3
3
3



&
&
33


33

2
2
4
'=

K P
µ K P
µ K P
µ 0
0
0
0
-
0
-
0
+


2
2


&
&
12

3
3
3

12

'

0
0
0

0
0

2
2



&
&
23


23

0
0
0
0

0
'


2
2


&
&

31

31



0
0
0
0
0

1
4
4
4
4
4
4
4
4
4
4
2
4
4
4
4
4
4
4
4
4
4
3
E
D

'

Part [DMECP1] of matrix DDE is reduced to
with (p
who is such as:
1 = PC)
p
1
'


11
&


'
&

22


'


&






33
k0
P
k0
P
k0
P

=
- bS
- bS
- bS
0 0 0


lq
lq
lq
{p


&}
1
'
2 K
0 p + p
K
S
C
atm
0 p + p
K
S
C
atm
0 p + p


12
S
C
atm
&


'
2

&
23

'
2

&
31
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 T (the option
I 1
-
RIGI_MECA_TANG called with the first iteration of a new increment of load).
If with T the border of the field of reversibility is reached, the condition is written: F & = 0 which must be
I 1
-
checked jointly with the condition F = 0. If with T one is strictly inside the field,
I 1
-
F < 0, then the tangent operator is the operator of elasticity.

If the mechanical criterion is reached:
F & = 0
1
f1
f1
f1
F & =
& +
Cr
P & +
p&
0
1
C =











éq 5.2-1

Cr
P
PC
P

P

like
Cr
p
Cr
Cr
P & =
&v +
p&c, then:
p

p

v
C
f1
f1 Cr
P
p
Cr
P

F & =
F

& +
(
&v +
p &)
1
C +
p&
0
1
C =
éq
5.2-2

Cr
P
p

p
p
v
C
C
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p&
One has in addition:
E
E
C
&ij = D & kl + 0

ijkl
K P







éq 5.2-3
k0s (PC + p
) ij
atm

i.e.:
E
E
F

1 F
1


p
1
&c
&ij =
& -

-

+ 0

ijkl
D
kl
ijkl
D


kl
K P
éq
5.2-4
S


kl
3 P

K

0s (PC + p
) ij
atm

By writing the plastic module of work hardening:

F
1 P

F
Cr 1
H = -
,
p















éq 5.2-5
P
p


P
Cr

v
The equations [éq 5.2-2] and [éq 5.2-5] give:

f1
F
F P

&ij - H p + (1
1
+
Cr) p&c = 0 éq 5.2-6


ij
PC Cr
P PC
F
1
Multiplication of the equation [éq 5.2-4] by
give:
ij
F
1
F
1
E
F
1
E
F

1 F
1


1
F
1
p&c

&
éq 5.2-7
ij =

& -


-

+

ijkl
D
kl
ijkl
D


kl

K P


0
ij
ij
ij
ij
S
kl
3 P

ij
k0s (PC + patm)
The two preceding equations make it possible to find:

F
1
E
F
1
E
F

1 F


F

p
1
1
1
&c
F

F
1
1 Cr
P
H


p =

-


-

+


+
+
ijkl
D
&kl
ijkl
D


kl
K P


0
ij
(
) p&
ij
ij
S
kl
3 P



K
p + p
p

P

p

kl
ij
0s (C
atm)
C
C
Cr
C
éq 5.2-8
from where and to deduce the expression from it from the plastic multiplier:

F


1
E
F
1
1
F

F
1
1 Cr
P


& +
0
+ (
+
)
ijkl
D
kl
K P ij
p&
ij

ij
k0s (PC + patm)
C
p
C Cr
P
p
C
=
éq
5.2-9
F
1
E
F

1 F
1





-
1
+
ijkl
D


kl
H


p
ij
S
kl
3 P



That is to say H the definite elastoplastic module like:
F
1
E
F

1 F
1


H =


-
1
+
ijkl
D


kl
H


p








éq 5.2-10
ij
S
kl
3 P


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The plastic multiplier is written:

F


1
E
F
1
1
F

F
1

P
1

D & +
K P
Cr
ijkl kl
0 ij
+ (
+
) &


k0
+



ij

ij
S (p
p
C
atm)
p
p
P
p
C
C
Cr
C
=
éq
5.2-11
H
While replacing by his expression in the equation [éq 5.2-4], one obtains:

E
1 F
1
E
E
F

1 F
1


&

1
ij =
-


.

-

-
ijkl
D
&kl
Dmnop &op
ijkl
D


kl
H



mn

S

kl
3 P



1 f1
1
F

F

1
1

Cr
P
E
F

1 F
1


K P


K P
0

+ (
+
) D

-
1
-
0
p




&
H mn
k0s (PC + patm) mn
ijkl
kl
p
C Cr
P
p
C
S

kl
3 P

K

0s (PC + patm) ij
C



éq 5.2-12
One thus deduces the elastoplastic operator from it
elp
E
p
D
= D - D:







E
1 F
1
E
E

F

1 F
1


&
1

ij =
-



-

-
ijkl
D
ijop
D
Dmnkl
mn
&kl
H
3

COp
S
mn
P



1
4
4
4
4
4
4
4
4
4
2
4
4
4
4
4
4
4
4
4
3

D p


ijkl











1
F

1 F

1
1
E
F
1
1
F

F
1
1


Cr
P
K P
0



-
COp
ijop
D

(

K P
0 mn
- (
+
)) -
p&
H
S

3

COp
P



mn
k0s (PC + patm)
p
C Cr
P
p
C
k0s (PC + patm) ij
C

1
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
2
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
3

D PC


ij

éq 5.2-13
with,
p
1 f1

E
E
f1
1 F

D
=



1
ijkl
ijop
D
Dmnkl
-
mn
H COp
smn 3 P

and


p
1
F

1 F
1
1
E
F
1
1
F

F
1
1



Cr
P
K P
D C
0

ij = -

-



0
+ (
+
)
COp
ijop
D
K P mn
+

H
S
3



COp
P



mn
k0s (PC + patm)
p
C Cr
P
p
C
K

0s (PC + p
) ij
atm
éq 5.2-14
Calculation of
p
D:
ijkl
F

1
1

= - m2 (2P - 2P + K p + S
3,
éq
5.2-15
Cr
C
c) ij
ij

3
ij
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who is written in vectorial notation:

- 1 2
M (2P - 2Pcr + K p
C
c)


+ 3s11
31
-
2
M (2P - 2Pcr + K p
C
c)

+ 3s22
3

- 1 2
M (2P - 2P

éq
5.2-16
Cr + K p
C
c)


+ 3s33
3




3 2s12


3 2s23




3 2s31


from where the expression of:
-
2
K MR. P 2P 2P
K p
6 S
µ
0
(- Cr + C c)+


11
-
2
K MR. P 2P 2P
K p
6 S
µ
0
(- Cr + C c)


+
22
µ
E
F



-
2
K MR. P 2P 2P
K p
6 S
0
(- Cr + C c)+

D
éq
5.2-17
ijkl
:
33

µ
kl

6
2s12


6µ 2s


23



6µ 2s31


and
F

E
F
1 1 1f

4
2
2

D

-

= K
-
+
+ µ
ijkl
kl
0M P (2P
2P
K p)
12
Q
éq
5.2-18


S

3 P
Cr
C C
ij

kl



However the plastic module H is written in the form:

F

E
F

1 F
1


H =


-
1
+
ijkl
D


kl
H


p
ij
S
kl
3 P



4
H = M (2P - 2P + K p
Cr
C c) [K P (2P - 2P
+ K p
Cr
C c) + 2kPcr (P + K p
C c)]
2
0
+12µQ éq 5.2-19

While posing:
2
2
ij
With = - K MR. P
0
(2P - 2 Cr
P + kc PC) ij + 6 S
µ ij, A' ij = - K MR. P
0
(2P - 2 Cr
P + kc PC) ij + 6 S
µ ij,
éq 5.2-20
with: tr ()
With = - 3
2
K MR. P (2P - 2P + K p)
0
Cr
C
C
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In A'
In A'
In A'
6 2
With
µ S
6 2
With
µ S
6 2
With
µ S

11
11
11
22
11
33
11 12
11 23
11 31
A' A
In A'
In A'
6 2
With
µ S
6 2
With
µ S
6 2
With
µ S

11
22
22
22
22
33
22 12
22 23
22 31
1 A A'
In A'
In A'
6 2
With
µ S
6 2
With
µ S
6 2
With
µ S
p
D =

33
113
22
33
33
33
33 12
33 23
33 31

H 6 2 A
µ 'S
6 2 A
µ 'S
6 2 A
µ 'S
2
36µ s2
2
36µ S S
2
36µ S S
11 12
22
12
33 12
12
12 23
12 31
6 2 A
µ 'S
6 2 A
µ 'S
6 2 A
µ 'S
2
.
36µ s2
2
36µ S S

11
23
22
23
33
23
23
23 31
6 2 A
µ 'S
6 2 A
µ 'S
6 2 A
µ 'S
2
.
.
36µ S

2
11
31
22
31
33
31
31

SYM
éq. 5.2-21

'
One can write the components
D
piece [DMECDE] of the matrix
Which is those of

the operator
elp
E
p
D
= D - D.
'

According to the equation [éq 5.2.14]. Components
with (p
piece [DMECP1] of
1 = PC)
p
1
stamp DDE are:




*



2


M
2P
()
0
Cr
-
K (2P
C
Cr - P) - 2P (P
Cr
+ K p) Ln
C C


'




- tr ()
With
P

((p)
C -
2
0
)






K P
A11
'+
'
With +
0
11
- bS
3Hk
lq
0s (PC + patm)
H
k0s (PC + patm)





*



2


M
2P
()
0
Cr
-
K (2P
C
Cr - P) - 2P (P
Cr
+ K p) Ln
C C


'




- tr ()
With
P

((p)
C -
2
0
)






K P
'
To 22+
'
With
+
0
22
- bS
3Hk
lq
0s (PC + patm)
H
k0s (PC + patm)






*

2
2P
()
0
Cr
-
'


M K (2P
C
Cr - P) - 2P (P
Cr
+ K p) Ln
C C



2

P


- tr ()
With
0

((p)
C -)




k0P


'
With
'
With
bS
3Hk
lq
0s (PC + patm)
33+
33+
H
k0s (p
p
C
atm) -
+






*

2
2P
()
0
Cr
-
'


6 2µM K (2P
C
Cr - P) - 2P (P
Cr
+ K p) Ln
C C



2



- 2 2µtr ()
With
0
P

((p)
C -)






S

S

Hk0s (PC + patm) 12 +
12


H




*

2
2P
(0)
Cr
-


6 2µM K (2P
C
Cr - P) - 2P (P
Cr
+ K p) Ln
C C


'




- 2 2µtr ()
With

P ((p)
C -
2
0
)






S

S

Hk0s (PC + patm) 23 +
23


H




*


- 2 2µtr ()
With
2
2P
()
0
Cr
-

13
S + 6 2µM K (2P
C
Cr - P) - 2P (P
Cr
+ K p) Ln
C C


'
13
S

Hk0s (PC + patm)



P ((p)
C -
2
0
)







éq 5.2-22



with '
=
= - ([)
0 1
(- R) exp (- p
)
C]
p
C

If the hydrous criterion is reached:
p
One leaves again the equation [éq 5.2.3] with this time p
C
=
&
&
,
ks (PC + patm)
One finds a relation direct enters & and &, p&c of the form:
E
p&
p
&
= D & + K P
C
&c
(
+
)
0

éq
5.2-23
ks (p + p
C
atm)
k0s (PC + atm) I
p
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One deduces then the constraint from Bishop
'
E

K P
K P

&
= D & + (
0
+
0
) - bS
p
I



& éq
5.2-24
K
S (PC + patm) k0s (PC + patm)
lq
C



'
Components
D
piece [DMECDE] of the matrix
OF are nothing other than those

matrix
E
D.
'

The only components of piece [DMECP1] of matrix DDE are thus those of
with
p
1
(p
:
1 = PC)
'


11
&


'
&

22


'


&
33
K

0P
1
1
k0P
1
1
k0P
1
1

=
(
+
) - bS
(
+
) - bS
(
+
) - bS
0 0 0 p

&
'
lq
lq
lq
2 (p + p
C
atm) K
K
S
0s
(p + p
C
atm) K
K
S
0s
(p + p
C
atm)
{} 1
K
K
S
0
12
S
&


'
2

&
23

'
2

&
31
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 constraint 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 (P =)
0.
The increment of the deviatoric constraint is written in the form:
S
= µ ~
2
- ~


éq
5.3.1.1-1
ij
(
p
ij
ij)
Around the point of balance (- +
), one considers a variation S of the deviatoric part of
constraint:
S = µ ~
2
- ~ éq
5.3.1.1-2
kl
(
p
kl
kl)
Calculation of ~ p
:
kl

It is known that:
~ p
=
3
kl
skl
éq
5.3.1.1-3
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By deriving this equation compared to the deviatoric constraint, one obtains:
~ p
=
3
+ 3
kl
skl
skl
éq
5.3.1.1-4
Calculation of:
One a:
1 F

F

F
Cr
P

1 F

F

F

F
Cr
P

=


mn + (
+
) p
C =


S
mn +
P
+ (
+
) p
C
H p mn
p
C Cr
P
p
C
H

p S
mn
P

p
C Cr
P
p
C

= 1 [S
3
2
2
mn S
mn + M (2P - 2 Cr
P + kc PC) P
- M [kc (2 Cr
P - P) + (
2 P + kc PC) P' Cr] p
C]
H p
éq 5.3.1.1-5
If one considers only the evolution of the deviatoric part of (P =)
0, then:

(H
2
p) = H p + H p = [
3 smnsmn + 3smnsmn] - 2M
P
Cr
P +



M2 K
2
2
C PC - 2M kc Cr
P PC - M [kc (2 Cr
P - P) + 2P' Cr (P + kc (PC + PC))]PC
éq. 5.3.1.1-6
P

with
Cr
P' Cr =

p
C
However:
P
P = kP.
Cr
Cr
v

Like
p
2
= M

(2P - 2P + K p), one a:
v
Cr
C
C
p

2
2
2
v = M (2P - 2 Cr
P + kc PC) - 2M Cr
P + kcM p
C
éq.
5.3.1.1-7
From where:



1
M2 (2P
2
2
- 2 Cr
P + kc PC) =
+ 2M


Cr
P - kcM PC
éq
5.3.1.1-8
Cr
kP

In addition,
H = 2kM 4P P + K p 2P - 2P + K p
p
Cr (
C
c) (
Cr
C
c)
and



H = 2kM 4 P + K p (2P - 4P + K p
) P + 2kP M 4k 3P - 2P + 2k p p
p
(
C
c)
Cr
C
C
Cr
Cr
C (
Cr
C
c)
C
éq 5.3.1.1-9

By injecting this last equation in the equation [éq 5.3.1.1-6], one obtains:

H
4
2
2
p + [2 km

(P + kc PC) (2P - 4 Cr
P + kc PC) + 2M
P
+ 2M kc p
C] P =
- [
Cr
2
4
2
Cr
kP M kc 3
(P + 2kc PC - 2 Cr
P) + M [kc (2 Cr
P - (P + P
) + 2P' Cr (PC + p
c)] + 2P' Cr P] PC +
[
3 smn S
mn + 3smnsmn]
éq 5.3.1.1-10
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While using the relation [éq 5.3.1.1-8], it comes then:

[3s S

+ 3s S
mn
mn
mn
mn]
Zpc
=
-

éq
5.3.1.1-11
(H +)
With
(H +)
With
p
p


2
M (2P 2P
K p)
with
With = [


K 4
M (P + K p
C c) (2P - 4Pcr + K p)
C C +
2
MR. P +
2
M kcpc]
- Cr + C C

1
2


+ M


2kPcr






2
2 kkc M (P + kc PC) (2P - 4P
+ K p)
Cr
C C

Z = M
+ 2
2
kk P M
3
(P - 2P + K p)

1
C Cr
Cr
C C

+



2
2
km


Cr
P

2
(2
2
2
M kcP + kc M
p
c)
2
- M kc P
+
+ M kc (2 Cr
P - P) + 2 2
Mr. P' (
Cr P + kc (PC + PC))
1
+
2
2
km
C
P R

One then obtains immediately the variation of the deviatoric part of the plastic deformation:

p

= 9
~
(



+

+ 9


kl
smn smnskl smn smnskl)
smn smn skl
H p + A
H p

3
2

3
2

+
3
éq
5.3.1.1-12
M (
Z
2P - 2 Cr
P + kc PC) P
skl -
M kc (2 Cr
P - P) p
cskl -
pcskl
H p
H p
H p + A

- 6 m2 (P + kc PC) P' Cr p
cskl
H p
S is written then:
ij
18
~
µ
18µ
sij = 2µij -
([sklsijskl + sklsijskl)]-
skl S
klsij
(H p + A)
H p

2

2

-
éq
5.3.1.1-13
M (
Z
2P - 2 Cr
P + kc PC) P
sij +
M kc (2 Cr
P - P) p
csij +
sijpc
H p
H p
(H p + A)
12µ
+
M2 (P + kc PC) P' Cr p
csij
H p

i.e.:



2
18µ
18µ


ijkl + ijkl
M (2P - 2 Cr
P + kc PC) P +
(sklsij + sklsij) +
smnsmnijkl
H

p
H p + A
H p


S
kl =

2
12µ




-
2
- 2 -
2
+
'

ijkl
kcM (Cr
P
P) PC
M (P kc PC) P Cr PC

H


p
H p

6
~
µ Z
2µij +
sijpc
H p + A
éq 5.3.1.1-14
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or in tensorial writing:


D

2
18µ

I 1+
M (2P - 2
2

+
+

-
-

4
Cr
P
kc PC) P
S: S
M kc (2 Cr
P
P) PC

H


p
H p
H p



S =

18µ
12µ


éq 5.3.1.1-15
+
(S + S
) S -
M2 (P + K

C PC) P' Cr
p
C
(H


p + A)
H p

6
Z
~
µ
2µ +
spc
(H p + A)
that one can still write by symmetrizing the tensor (S + S
) S:



D

2

2
18µ
12µ
I 1+
M (2P - 2
2

+
-
- +

-
+


4
Cr
P
kc PC) P
kcM (2 Cr
P
P) PC
S: S
M (P kc PC) P' Cr PC

H


p
H p
H p
H p



S =

18µ

+

(H


p + A)

6
Z
~
µ
2µ +
spc
(H p + A)
éq 5.3.1.1-16
1
with: = [
T
((S + S
) S) + (S (S + S
)) ]
2
Calculation of, while posing:
=
+
ij
T
sij
sij
T S
T S
T S
2T S
2T S
2T S
11 11
11 22
11 33
11 12
11 23
11 31
T S
T S
T S
2T S
2T S
2T S
22 11
22 22
22 33
22 12
22 23
22 31


T
T S
T S
T S
2T S
2T S
2T S
S = 33 11
33 22
33 33
33 12
33 23
33 31
2T S
2T S
2T S
2T S
2T S
2T S
12 11
12 22
12 33
12 12
12 23
12 31


2T S
2T S
2T S
2T S
2T S
2T S
23 11
23 22
23 33
23 12
23 23
23 31

2T S
2T S
2T S
T S
2T S
2T S

31 11
31 22
31 33
31 12
31 23
31 31
1
= [
T
T
(S) + T
(S)]
2
That is to say:


D




C =
1
I
+ 3
2
kc
4
M (2P - Pcr + K p
C c) P + 9
2
S S - 3
:
2
M (2Pcr - P PC - 6
)
2
M (P + K p
C c)



P' Cr p

H
H
H
H
C

p
p
p
p


+
9

(H p +)
With

one poses:
9
C = (S
: S)
H p
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:

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:
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3
D =
M2 (2P - 2P + K p
Cr
C c) P

H p
- 3
G =
M2 K 2P - P p

C (
Cr
) C
H p

6
H = -
M2 (P + K p P' p
C
c)
Cr
C
H p

The symmetrical matrix C of dimensions (6,6) is too large to be presented whole, one
break up into 4 parts C, C, C and C:
1
2
3
4
C
C
1
2
C =

C
C
3
4
with


1
9
9
9

+ C + D + G + H +
11
S 11
T
(11
T s22 + 22
T
11
S)
(11
T 33
S + 33
T 11
S)


(H p +)
With
2 (H p +)
With
2 (H p +)
With


9
1
9
9

1
C =
(22
T
11
S + 11
T s22)
+ C + D + G + H +
22
T s22
(22
T
33
S + 33
T s22)


2 (H p +)
With

(H p +)
With
2 (H p +)
With


9
9
1
9


(33
T 11
S + 11
T 33
S)
(22
T
33
S + 33
T s22)
+ C + D + G + H +
33
T
33
S


2 (H p +)
With
2 (H p +)
With

(H p +)
With


éq 5.3.1.1-17


9 2
9 2
9 2

(11
T 12
S + S T)
(
11 12
11
T 23
S + S T)
(
11 23
11
T 13
S + S T)
11 13
(
2 H p +)
With
(
2 H p +)
With
(
2 H p +)
With







éq 5.3.1.1-18
C2 = 9
2
(22
T 12
S +
9
2
S T)
(
22 12
22
T
23
S +
9
2
S T)
(
22 23
22
T 13
S + S T)
(
2 H p +)
With
(
2 H p +)
With
(
2 H p +)
22 13
With



9 2


(33
T 12
S +
9
2
S T)
(
33 12
33
T
23
S +
9
2
S T)
(
33 23
33
T 13
S + S T)
(
2 H p +)
With
(
2 H p +)
With
(
2 H p +)
33 13
With



C =
3
C2
















éq 5.3.1.1-19


1
18
9
9

+ C + D + G + H +
12
S
12
T
(12
T s23 + 23
T 12
S)
(12
T s23 + 23
T
12
S)


(H p +)
With
(H p +)
With
(H p +)
With


9
1
18
9

C4 =
(23
T
12
S + 12
T s23)
+ C + D + G + H +
23
T s23
(23
T
13
S + 13
T s23)


(H p +)
With

(H p +)
With
(H p +)
With


9
9
1
18


(13
T 12
S + 12
T 13
S)
(13
T s23 + 23
T 13
S)
+ C + D + G + H +
13
T 13
S


(H p +)
With
(H p +)
With

(H p +)
With


éq 5.3.1.1-20
Calculation of the rate of variation of volume:
p


= m2 (2P - 2P + K p),
v
Cr
C
C
p
= M
2 (2P - 2P + K p) - 2M2 P + m2 K p éq 5.3.1.1-21
v
Cr
C
C
Cr
C
C
= B + Dpc
B
3
BZ
=
(S + S
). S + (D -

) PC
(H + A)
(H + A)
p
p
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2
M (2P - 2Pcr + K p)
with:
2
B = M (2P - 2P

Cr + K p)
2
C
C
- M
C
C.
1
2
+ M
2kPcr
2
K M
C

and
2
2
D = K M
C
- M
.
1
2
+ M
2kPcr
One thus has:
p
3B
BZ
v =
(S + S

). S - (
- D
) PC éq
5.3.1.1-22
(H p + A)
(H p + A)
and finally:
B
= (C -
(S + S)) S -
ij
ijkl
kl
ij
kl
(H + A)
p

éq
5.3.1.1-23
BZ
D
ij
3Z
(-
+ +
+
S
) p
ij
ij
ij
C
(
3 H + A)
3
K
3
(p + p) (H + A)
p
0s
C
atm
p

5.3.1.2 Processing of the hydrostatic part

It is considered now that the variation of loading is purely spherical (S = 0).
The increment of P is written in the form:




E
-
exp (k0v)


P = P
-
1
éq
5.3.1.2-1

K
K
PC + p
0/0s


atm

-

PC + p


atm


The derivation of this equation gives:
P = K P
0
(
p
k0
P
v - v) -
PC éq 5.3.1.2-2
k0s PC + patm
Calculation of
p
:
v
It is known that:
p

= m2

2P - 2P + K p



éq 5.3.1.2-3
v
(
Cr
C
c)

By differentiating this equation, one obtains:

p

2
v = M ((2P - 2 Cr
P + kc PC) + (
2 P -
2 Cr
P + kcpc)

éq 5.3.1.2-4
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One knows the expression of:

M2 (2P - 2
2
Cr
P + kc PC) P + 3ss - M [kc (2 Cr
P - P) + 2 (P + kc PC) P' Cr] PC
B
=
=

H p
H p
éq 5.3.1.2-5
while posing
B = m2 (2P - 2
2
Cr
P + kc PC) P + 3ss - M [kc (2 Cr
P - P) + 2 (P + kc PC) P' Cr] PC

While differentiating, it comes:
2

= M ([P
2 - P
2
K p P

2 P
2 P

K p

P K P
2
(
P) p

K 2
(P

P
) p
P
(
2
K p P
) 'p

(
2 P
K p
P
) '
p
Cr + C c)
+ (
- Cr + C c) - C Cr -
C - C
Cr -
C -
+ C C Cr C -
+ C C Cr C]
HP


4

3

km
2
b2 PP

P
2
P
K p
P

P
2 2
PP
4
P
4 K p
Km No
3
p k2p2
Cr
- Cr + C c+ Cr (- Cr - Cr C C + C C + C c)
-
2


2

HP +k P P3 P
2
K
2 p p

C Cr (
- Cr + C c)

C


éq 5.3.1.2-6
One seeks the expression of P according to:
Cr

One a:
p
P = kP
éq
5.3.1.2-7
Cr
Cr
v
One can write:
Cr
P = m2 (2P - 2
2
Cr
P + kc PC) + M (
2 P -
2 Cr
P + kcpc) éq
5.3.1.2-8
Cr
kP
1+ 2M2

Cr
kP

2
2
2


Cr
P
= M (2P - 2 Cr
P + kc PC) + 2M P + M

kcpc




Cr
kP

M2 (2P - 2 +

2 2

2

Cr
P
kc PC) Cr
kP
M
Cr
kP
M kc Cr
kP







Cr
P =
+
P +
PC


2


2

2

1 + 2 Cr
kP
M


1+ 2 Cr
kP
M


1+ 2 Cr
kP
M


éq 5.3.1.2-9
One poses

M2 kP 2P 2P
K p
2M2
K
2
C M
Cr
kP
Cr
kP
Cr (
- Cr + C c)
C =
[
, has =
, D =

1+ 2M2 kP
[1+2M2
[1+2M2 Cr
kP]
Cr
kP]
Cr]
One has then:
Cr
P = aP + C + dpc




éq 5.3.1.2-10
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By replacing the expression of P in [éq 5.3.1.1-6], one finds:
Cr

(2P - 2
2
Cr
P + kc PC + 2P + kcpc - 2aP - 2akcpc - 2P Cr
'PC) P - 2c (P + kcpc
) M
=
.
+ (K


C (P + P - 2 Cr
P) - 2D (P + kcpc) - 2P Cr
'(P + kc (PC + PC)))PC
H

p

2kM 4b
-
[
2kM 4b
Cr
P (4P - 2 Cr
P + 3kc PC) + has (2P - 4 Cr
P + kc PC) (P + kc PC)]P -
[C (2P - 4 Cr
P + kc PC) (P + kc PC)]
H2
2
p
H p
2kM 4b
-
[kc Cr
P (3P - 2 Cr
P + 2kc PC) + D (2P - 4 Cr
P + kc PC) (P + kc PC)]PC
H2 p
éq 5.3.1.2-11
By gathering the terms in and those out of P, one finds:
F
H
= P + PC éq 5.3.1.2-12
E
E
with,
2
M
F =
[2P - 2P + K (p + 1 (- 2a) p) + 2P - 2aP - 2P'
Cr
C
C
C
Cr PC]
H p

4
2kM B
-
(4P - 2P + 3k p P + has 2P - 4PP - 4P K p + 3Pk p + K p
2
[
Cr
C c) Cr
(2
2 2
Cr
Cr C C
C C
C c)]
H p
M2
H =
[- 2dP - 2dkcpc + kcP - 2kc Cr
P + kcP - 2P' Cr (P + kc (PC + PC))]
H p

2kM 4b
-
[D (2P - 4 Cr
P + kc PC) (P + kc PC) + kc Cr
P (3P - 2 Cr
P + 2kc PC)]
H2 p
2
4
2cM (P + K p)
C C
2bckM
E = 1 +
+
2P - 4PP - 4P K p + 3Pk p + K p
2
(2
2 2
Cr
Cr C C
C C
C
c)
H p
H p
The expression of
p
thus becomes:
v
p
v = XP + Ypc éq 5.3.1.2-13
with,
2
X =
F
F
M (2 - 2a - 2c
+ (2P - 2P
K p
Cr +
))
C
C
E
E
2
Y =
H
H
M ((2P - 2P
K p
C
D K

Cr +
C
c)
- (2
+ 2 -)
c)
E
E

from where the expression of P according to and p
:
v
C

1

P 1
(+ K PX) = K P (- Y
+
)
0
0
V
p

éq
5.3.1.2-14
K (p + p
)
C

0s
C
atm
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:

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Calculus of the variation of deviatoric deformation:
~
~ p
F
H
ij = =
3 S = 3 Psij + 3 pcsij

éq 5.3.1.2-15
E
E
One thus has finally:

= ij
F P + Kijpc







éq 5.3.1.2-16
ij
with
3 F
1 + K PX
0
D
F =
S -
1,
E
K
3 P
0

éq
5.3.1.2-17
H
3
K PY
K P
0
0
D
K =
S - (
+
1
)
E
3
K
3
(p + p)
0s
C
atm

5.3.1.3 Operator
tangent

The tangent operator connects the variation of total constraint to the variation of the deformation and of
suction. Since the increment of the total deflection under loading deviatoric is written:
B

1
ij + H ijpc = C
(ijkl -
(S + S) kl ij) Dklmn mn,

éq 5.3.1.3-1
(H p + A)
with:
2/3 - 1/3 - 1/3 0 0 0


- 1/3 2/3 - 1/3 0 0 0

1
- 1/3 - 1/3 2/3 0 0 0
D =

éq
5.3.1.3-2
0
0
0
1 0 0
0
0
0
0 1 0



0
0
0
0 0 1
projection in space deviatoric,

and that under spherical loading one a:

2
ij - K ijpc = ij
F Dkl kl éq
5.3.1.3-3
with:
- 1/
3


- 1/
3

2
- 1/
3
D =

éq
5.3.1.3-4
0
0



0
hydrostatic projection, one has then:
ij = ijkl
With kl + Bijpc éq
5.3.1.3-5
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with:
- 1

B

1
2

ijkl
With
= (Cijmn -
(S + S)
)
mn ij Dmnkl +
ij
F Dkl éq
5.3.1.3-6

(H p +)

With


1

-
B

B = (C
-
(S + S
))
1
2
D
+ F D (H - K)
ij
ijmn
éq
5.3.1.3-7
(H +)
mn ij
mnkl
ij
kl
kl
kl


p
With


The constraint of Bishop is thus written:

'= A + B - bS p
ij
ijkl
kl
(ij
lq)
C


5.3.2 Tangent operator at the critical point

As for 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:

~

E
p
E
F
S = S - 2
µ = S - 2µ
= - 6
µ S éq
5.3.2.1-1
S


The expressions of the plastic multiplier and its derivation are written in the following way:
Qe

E
E
=
-
Q
Q Q
1/6µ




and =
-
éq
5.3.2.1-2
Q

2
6
Q
µ
6
Q
µ
with,
E
E
3
3
E
S S

S S
Q =
and Q =

E
2 Q
2 Q
from where the expression of:
1 3 sese
Qess
=

-

éq
5.3.2.1-3
E
3

6Μ 2 Q Q
Q

Let us point out in the same way the expression of S:

S = µ ~
2
-
3 S -
3 S
ij
(ij
ij
ij)

While replacing and by their expressions, one can write:
E
E
E
E
3 S
~
S
3 Q
1 Q

kl
kl
S = 2µ -
S +
S S S -
- 1 S éq
5.3.2.1-4
ij
ij
E
ij
3
kl
kl ij



ij

2
Q Q

2 Q
Q


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E
E
E

1 Q
1
3 Q


3 S.S
kl
ij
S

+
- -
S.S = 2µ

-

~ éq 5.3.2.1-5
kl
ijkl
ijkl
ijkl
3
kl
ij
ijkl
E
kl

Q


2 Q


2 Q Q


or in tensorial writing:

E
E
E
Q





D
D
1
3 Q
S
S
S
I + I 1
D
- -
S S

=
I -
éq
5.3.2.1-6
4
4
µ
3
2
3
4

~
Q




2 Q



2
E
Q Q
1
4
4
4
4
4
4
2
4
4
4
4
4
4
3
1 4
4 2
4
4
4 3
G
H
As S does not depend on, one can confuse ~ with.
v

By using the tensor of projection in the space of the deviatoric constraints 1
D [éq 5.3.1.3-2], one
can write:
1
1
-
D.G.
=
H

.

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:
K
D
P
0
D
I P = K P
éq
5.3.2.2-1
0
-
I p
v
C
K
p
0
+ p
S
C
atm
knowing that at the critical point,
p

.
v = 0

As P then does not depend on ~ one can confuse with.
v
K
D
P
0
D
I P = K P



éq 5.3.2.2-2
0
-
I PC
K
p
0
+ p
S
C
atm
By using the tensor of projection in the space of the hydrostatic constraints
2
D [éq 5.3.1.3-3], one
can write:
2
K
D
P
0
D
I D = K P

0
-
I PC
K
p
0
+ p
S
C
atm
from where
D
2
D
I D
I
=
+
p




éq 5.3.2.2-3
K P
K
p
0
0
+ p
S (C
) C
atm
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 constraint to the variation of the total deflection to
not criticizes:
1
- 1
D
2
D
D G
. .H
I D
I
=
+

.
+
p


K P
K
p
0

0
+ p
S (C
) C
atm

= A - B p
éq
5.3.2.3-1
ij
ijkl
kl
ij
C
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with
1
1
1
-
D
2
-
D.G.H
I D
With
éq
5.3.2.3-2
ijkl =
+
2


µ
K P
0

and
D
I
B = -










éq 5.3.2.3-3
ij
K
p
0
+ p
S (C
atm)

As it is necessary to deduce the variation from the constraint of Bishop, one finds:

D
I
B = -
- bS








éq 5.3.2.3-4
ij
K
p
0
+ p
S (C
)
lq
atm

5.3.3 If the hydrous criterion is reached

The variation of the elastic strain is written in the form:
E
E
1
~
E
= - éq
5.3.3-1
kl
kl
v
kl
3
that is to say:
S
P
p
E
kl
C
=
-
-
éq
5.3.3-2
kl
kl

K
3 P
K
3
p
0
0
+ p
S (C
) kl
atm
In this case the plastic deviatoric deformation is null thus the plastic deformation has
the following expression:
p
1
p
= - éq
5.3.3-3
kl
v
kl
3
that is to say:
p
p
C
= -

éq
5.3.3-4
kl
K
3
p 0 + p
S (C
) kl
atm
By combining each component rubber band and plastic one finds:

S

1
1
1

E
p
P
kl
= + =
-
-
+
p
éq 5.3.3-5
kl
kl
kl
kl

K
3 P


3 K
p
0
0
+ p
K p 0 + p


S (C
atm)
S (C
atm)
C kl

By using the matrices of projection in the space of the deviatoric and hydrostatic constraints one
leads to the following expression:

D1
D
2

1
1
1

ijkl
ij
kl
=
-


-
+
p
éq 5.3.3-6
kl
ij

K
3 P


3 K
p
0
0
+ p
K p 0 + p




S (C
atm)
S (C
atm)
C kl

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:
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thus one can write:

- 1
1
1
2
1
2
-




ijkl
D
ij
D kl
1
1
1

ijkl
D
ij
D kl






éq 5.3.3-7
ij =
-
kl +
+
-
p


K
3 P
3 K
+
+




0
0s (C
p
atm
p
) ks (cp0 atm
p
)
kl
C
2

µ
K
3 P
0




D1
D2
A is posed
= ijkl
ijkl
- ij kl




K
3 P
0

2

1

1



+
- +
- +
0 0 0
3 9k P
3 9k P
3 9k P
0
0
0

1

2

1


- +
+
- +
0 0 0
K P
K P
K P
1 3 9
3
9
3 9
0
0
0

or A
µ
µ
µ
ijkl =
- 1 + 2
- 1 + 2
2 + 2
0 0 0





éq 5.3.3-8
3 9k P
3 9k P
3
9k P


0
0
0
0
0
0
1 0 0



0
0
0
0 1 0



0
0
0
0 0 1

and by deducing the constraint from Bishop, one finds:

1
1
1


'= A-1 +
+
A-1 - bS




p éq 5.3.3-9
ij
ijkl
kl
3 K
p
0
+ p
K p 0 + p
S (C
atm)
S (C
atm)
ijkl
kl
lq
C



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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:

'-
, +
p, +
p, -
P, -
p,
C
gz
Cr
c0

, p
C and p
gz

Variables of output:

·
'

, more tangent operators (necessary to operator STAT_NON_LINE).
· Variables intern +
P, newer variables +
p: threshold in suction and +
P: pressure of
Cr
c0
S
cohesion, and indicators of mechanical work hardening I1 and hydrous I2.

Elastic prediction:
exp

E
-
(K
0
v)
P = P
~
, = -
S + 2µ

k0/ks0
p +

C
patm


p

C + p

atm

·
F < 0 and F < 0 (p <
): reversible behavior
1
2
C
pc0

E
P = P, S = E p
S, = 0
-
-
, Cr
P = Cr
P, p
p
c0 =
c0

·
F > 0
F >
1
or
0
2
plasticization and mechanical and hydrous work hardening

E
P = P exp [
p
- K
,
0 v]
E
S
S =
p
6

µ v
1 + m2 (2P - 2P + K p
Cr
C
c)
P = p exp K
Cr
Cr
[statement],
p 0 + p
= p 0 + p
exp K

C
atm
(C
atm)
[p
S
v]

p
Q
2


The single unknown factor is
p

F =
~ p
v
=
v determined by
0 (one have then:
)
1
M2 (2P +
-
S
P
2 Cr
P)
or F = 0
~
p
=
2
(and
0)

Note:

The constraint resulting from the data of the model of Barcelona east
D
= +
early
pgz1, it will be
thus the variable used in the routine describing the behavior, the constraint of output
provided to STAT_NON_LINE being the constraint of Bishop: '=
-
early
p.
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Tangent operators:

The tangent operator of the generalized constraints is implemented in THHM under name DDE and

'&
& '

partitionné in several blocks. The components concerned with the model are
and


p
&c

& '
'

'

&
&



p

&
p
C
&
blocks [DMECDE] and [DMECP1] correspondent with:
&,
C



.
&
'
'



p
& p
& p





& p&
p
C
&c

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 command DEFI_MATERIAU.

7.2
Initialization of calculation

It is necessary that the initial state of material is plastically acceptable (the constraint 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 constraint
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 (- K p) and pressure of consolidation (2P). The constraint used to describe it
C
C
Cr
behavior (forced total plus gas pressure) is different from the constraint to initialize in
ETAT_INIT (forced of Bishop '). The relation between the two types of constraint is:

&
'= & + [(B -) 1 p&gzI - bSlq p&c]

7.3
Variables intern at output

The model produces five internal variables:

V
: critical pressure
1 = Pcr
V2 = I1: mechanical indicator of irreversibility
V = p: hydrous threshold of irreversibility
3
c0
V4 = I2: hydrous indicator of irreversibility
V
: pressure of cohesion
5 = PS
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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 constraint according to the deformation and of the capillary pressure. By
example, the evolution in the field of reversibility is given by:

E
D = - NR (E
, p) dp + B (E
, p) dtr (E
)
lq
C
C
C

dP = B (E
, p) dp + K (E
, p) dtr (E
)
C
C
C

PC
Curve of drainage
Field of reversibility
Curve of imbibition

E
p
lq
lq lq

Appear 8-a

Where (NR, b) 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
constraint.

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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 behavior Cam_Clay, 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).

Handbook of Référence
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