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Titre : Loi de comportement des milieux poreux : modèle de[...] Date : 09/02/2011 Page : 1/44Responsable : PLESSIS Sarah Clé : R7.01.17 Révision :

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Law of behavior of the porous environments: modelof Barcelona

Summary:

The model of BARCELONA [bib1] described the behavior soil mechanics unsaturated coupled with the hydraulicbehavior (this model must thus be used in an environment THHM [bib7]). In the typical case of a groundcompletely saturated with water, it is reduced to the model CAM_CLAY modified, also implemented inCode_Aster [bib5]. It is particularly adapted under investigation behavior of clays.

Warning : The translation process used on this website is a "Machine Translation". It may be imprecise and inaccurate in whole or in partand is provided as a convenience.Copyright 2017 EDF R&D - Licensed under the terms of the GNU FDL (http://www.gnu.org/copyleft/fdl.html)



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Contents1Notations............................................................................................................................................... 4

2Introduction............................................................................................................................................ 7

2.1Phenomenology of the behavior of the unsaturated grounds..........................................................7

2.1.1Curve of Water retention........................................................................................................7

2.1.2Extension of the definition of the effective constraints on the unsaturated ground.................7

3Description of the original model of Barcelona......................................................................................8

3.1Purely mechanical behavior............................................................................................................8

3.1.1Spherical loading...................................................................................................................8

3.1.1.1Elasticity....................................................................................................................8

3.1.1.2Plasticity....................................................................................................................9

3.1.2Triaxial loading..................................................................................................................... 10

3.1.2.1Elasticity.................................................................................................................. 10

3.1.2.2Plasticity.................................................................................................................. 10

3.2Hydro-mechanical coupling or effect of suction on mechanics......................................................11

3.2.1Reversible part..................................................................................................................... 12

3.2.2Irreversible part.................................................................................................................... 12

3.3Complete behavior (mechanical and hydrous loading).................................................................14

3.3.1Reversible behavior.............................................................................................................14

3.3.2Thresholds of flow................................................................................................................15

3.3.3Laws of flow......................................................................................................................... 16

3.3.4Laws of work hardening.......................................................................................................16

3.3.5Inventory of the configurations of mechanical and hydrous coupling...................................16

3.3.5.1Total reversibility......................................................................................................16

3.3.5.2Elastoplastic behavior..............................................................................................17

3.3.5.3Hydrous behavior generating of the unrecoverable deformations............................18

3.4Data of the model of Barcelona.....................................................................................................19

4Digital integration of the relations of behavior......................................................................................20

4.1Recall of the problem....................................................................................................................20

4.2Incremental relations..................................................................................................................... 20

4.3Calculation of the constraints and the internal variables...............................................................21

5Tangent operator................................................................................................................................. 22

5.1Nonlinear elastic tangent operator................................................................................................23

5.2Plastic tangent operator of speed. Option RIGI_MECA_TANG.....................................................24

5.3Tangent operator into implicit. Option FULL_MECA......................................................................28

5.3.1If the mechanical criterion is reached...................................................................................28

5.3.1.1Treatment of the deviatoric part...............................................................................28

5.3.1.2Treatment of the hydrostatic part.............................................................................33

5.3.1.3Tangent operator......................................................................................................36Warning : The translation process used on this website is a "Machine Translation". It may be imprecise and inaccurate in whole or in partand is provided as a convenience.Copyright 2017 EDF R&D - Licensed under the terms of the GNU FDL (http://www.gnu.org/copyleft/fdl.html)



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5.3.2Tangent operator at the critical point....................................................................................37

5.3.2.1Treatment of the deviatoric part...............................................................................37

5.3.2.2Treatment of the hydrostatic part.............................................................................38

5.3.2.3Tangent operator......................................................................................................38

5.3.3If the hydrous criterion is reached........................................................................................39

6Summary of the model of Barcelona...................................................................................................41

7Implementation of the model...............................................................................................................42

7.1Data material................................................................................................................................. 42

7.2Initialization of calculation.............................................................................................................42

7.3Internal variables at exit................................................................................................................42

8Developmental perspectives of the model...........................................................................................42

9Bibliography......................................................................................................................................... 44

10Checking............................................................................................................................................ 44

11Description of the versions of the document......................................................................................44




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1 Notations

T indicate the tensor of the total constraints in small disturbances, noted in the shape of the

following vector:

T 11

T 22

T 33

2T 12

2T 23

2T 31

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

σ=σTp gz I and pc= pgz− plq ,

with p lq , pgz , pc respectively pressure of liquid, gas pressure, capillary pressure (or suction)

One notes:

I the tensor unit of order 2 whose indicielle notation is δ ij

I 4 the tensor unit of order 4 whose indicielle notation is δ ijkl

P=−13

tr σ constraint of containment

s=σ +PI diverter of the constraints

I 2=12

tr s . s second invariant of the constraints

Q=σeq= 3I2 equivalent constraint

ε=12∇ u∇T u total deflection

ε=εeε pεth partition of the deformations (elastic, plastic, thermal)

εv=−tr ε 3α T−T 0 voluminal total deflection

ε vp=−tr ε p voluminal plastic deformation

ε=ε13

εv I

diverter of the deformations

εe=ε−ε p deviatoric elastic strain

εp=ε p +

13

εvp I deviatoric plastic deformation




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ε eqe = 2

3tr εe . ε e equivalent elastic strain

ε eqp= 2

3tr ε p . ε p equivalent plastic deformation

e index of the vacuums of the material (report of the volume of the pores on the volume of the solidmatter constituents)

e0 initial index of the vacuums

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

φlq , φlqe , φlq

p content of total, elastic and plastic liquid

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

κ s elastic coefficient of rigidity in a test of variation of suction

k 0=1 + e0

κ

k 0s=1+ e0

κ s

λ pc coefficient of compressibility (plastic slope in a hydrostatic test of compression)

λ* coefficient of compressibility in conditions of saturation

λs coefficient of compressibility plastic in a test of variation of suction

k=1e0

λ−κ

k s=1e0

λs−κ s

M slope of the right-hand side of critical condition

α coefficient of correction of the normality of the plastic flow

P cons pc pressure of consolidation

Pcr pc critical pressure, variable interns model, equal to half of the pressure of consolidation

Pcr ¿ pressure criticizes in conditions of saturation

P s cohesion (hydrostatic traction limits to suction given)




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P0 confining pressure of reference generally equal to the atmospheric pressure Pa

k c slope of cohesion according to suction

β parameter controlling the increase in λ pc with pc

r parameter defining the peak of λ pc with pc

μ elastic coefficient of shearing (coefficient of Lamé)

f 1 surface of load in space P ,Q f 2 surface of load in pc

pc0 threshold of irreversibility of suction

Λ plastic multiplier

S lq water saturation, S lq=φlq

ϕε vp

p voluminal plastic deformation due to a loading in hydrostatic pressure

ε vsp

voluminal plastic deformation due to a loading in suction

ε pp

deviatoric plastic deformation due to a loading in hydrostatic pressure

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. Theoriginal model of Barcelona is described using the variables σ , pc , which distinguishes it from themodels of mechanics coupled to a thermohydraulic behavior which are described using one onlyeffective constraint (forced of Bishop). One can notice that this model is rewritten within a frameworkporoplastic with the introduction of an additional poroplastic variable which is the water content [bib2],making it possible to collect the phenomena of hystereses which appear on the cycles of drying-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 Water retention

In addition to the principal common mechanical aspects with the water-logged soils [bib3], the porousenvironments comprising of the liquid and gas phases (grounds unsaturated with water) have as aspecific characteristic to be very sensitive to the phenomena of capillarity. The latter correspond to thelocalization of meniscuses of liquid (increasingly small as ground désature) in which the water pressureis weaker than the air pressure (and all the more weak as the meniscus is small and thus désaturé

ground). One thus sees appearing the concept of pressure capillary or suction pc= pgz− plq .

While drying, a ground unsaturated has a water content φlq weaker what corresponds to a higher

suction. The correspondence between these two sizes is the curve of Water retention (cf [Figure 2.1.1-a]). This one is obtained by drying of a ground initially saturated (suction is then worthless) anddamping starting from the dry state.

lq

cp

Figure 2.1.1-a: Curve of Water retention

2.1.2 Extension of the definition of the effective constraints on the unsaturatedground

The behavior soil mechanics unsaturated is primarily observed in laboratory using devices withcontrolled suction (oedometers and triaxial). The modeling of this mechanical behavior was initiallytried by extending the concept of effective constraint in the unsaturated mediums. This one is a

function of the total constraint and intersticielle pressure: σ '= f σT , plq . In the saturated case,

there is simply additivity of the pressure and the constraint: σ '= σT−p lq I because the pressure of

water acts in the same way in water and the solid in all the directions. The widening of this concept in




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the mediums unsaturated in the years 1950 (taking account of the pressures of the two liquid phases)

brought the following form of the effective constraint: σ '= σT−p gz I g pc there remained the

constraint suggested in the form:

dσ '= d σT

− pgz bS lq dpc

where S lq 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 ofeffective constraint extended on the unsaturated ground. In particular, the test of collapse to theoedometer puts at fault the constraint of Bishop: This test consists in consolidating a sampleunsaturated by maintaining suction constant, then to moisten again it with constant loading. A collapseof the ground then is observed. If the consolidation is continued, the curve corresponds to a standardtest with of saturated. However, if one refers to the effective constraint, this one decreases during the

remoistening (since suction pc= pgz− plq cancel yourself) and as it is supposed to control the

deformation, there should be swelling what is contradictory with the experimental observation. Mostmechanics of the grounds agree now on impossibility of completely describing the behavior of thegrounds unsaturated using one only 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 Water retention does not have hysteresis, and it is not modified by themechanical deformation as it is the case in the presentation made by Dangla and collar. [bib2]. Thereexists nevertheless a threshold in capillary pressure pc0 with beyond which unrecoverabledeformations appear. In this paragraph one distinguishes a mechanical part which treat inducedmechanical deformations by a mechanical loading and a hydro-mechanical part which treats effect ofsuction on mechanics before writing the equations of the complete behavior.

3.1 Purely mechanical behavior

One makes the assumption that suction pc remain constant during the mechanical transformation.

The deformations resulting from the variation of the constraint are subscripted p .One examines the behavior, under successively spherical and deviatoric loading, this behavior beingconsidered isotropic.

3.1.1 Spherical loading

3.1.1.1 Elasticity

The mechanical state of a ground unsaturated under hydrostatic request is determined by testsoedometric with controlled suction. As for the water-logged soils, volume v sample varieslogarithmiquement with the load with a slope κ in a reversible way until a pressure of consolidation

P cons pc . One will choose κ independent of pc , the experiment showing a weak dependence of

the elastic slope with respect to pc .

The elastic component of the voluminal deformation varies then like:

εvpe=

κ1e0

PP

si PP cons pc éq 3.1.1.1 -1

The preceding expression is in fact derived from a test oedometric with constant suction where onemeasures the variation of the index of the vacuums according to the loading, from where the followingelastic law:

P=P0 exp [ k0 ε vp−εvpp ] éq 3.1.1.1 - 2




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with k 0=1+e0

κ, where P0 is the value of reference corresponding to ε vp

e=0 and e=e0 , initial

index of the vacuums.

3.1.1.2 Plasticity

Beyond the pressure of consolidation, the behavior of the ground is plastic and the slope λ pc is

dependent on suction (cf [3.1.1.2 figure - has]), this dependence being estimated way semi - empiricalfollowing:

λ pc =λ 0 [ 1−r exp −βpc + r ]

where r=λ pc∞

λ 0 is a constant connected to the maximum of the rigidity of the ground and β a

parameter which controls the evolution of rigidity according to suction.

The voluminal rate of deformation is then: εvp=λ pc 1e0

PP

if PP cons,

from where the plastic component : εvpp= λ pc −κ

1e0

PP

.

The expression of P is thus written:

P=P0 exp [ k ε vpp ] éq 3.1.1.2 - 1

with k=1 +e0

λ - κ

Note:

The two expressions [éq 3.1.1.1 - 2] and [éq 3.1.1.2 - 1] are similar to those of the Camwoodmodel - Clay [bib5] with the parameter λ (or k ) depending on the capillary pressure. Thecompressibility of the ground decreases with suction.




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Pln

v

2cc pp

1cc pp

21 cc pp

2cp

1cp

20P 2cconspP 1cconspP

10P

3.1.1.2 figure - has: Variation of specific volume under loading oedometric

3.1.2 Triaxial loading

3.1.2.1 Elasticity

The elastic component of the deviatoric deformation is proportional to the loading:

εe=

s2μ 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=σ1−σ3 (one will beable to extend the formulation who follows with the 3D by using the standard within the meaning of vonMises of the constraint). When suction becomes worthless (saturated medium), the model is supposedto be reduced to modified the Cam_Clay model [bib5]: the threshold of plasticity is then an ellipse of

center P cr* ,0 who cuts the axis of the hydrostatic constraints into zero and a value of pressure of

consolidation P cons*=2Pcr

* . The surface of load associated with a suction pc nonworthless is also an

ellipse of center P cr pc −P s

2,0 (cf [3.1.2.2 Figure - has]) which cuts the hydrostatic axis in

P cons pc =2Pcr pc and - P s , P s representing a cohesion varying linearly with suction:

P s=k c pc . line representing the critical conditions (worthless voluminal variation) preserves the

same slope M that in saturated but shifted condition P s . The equation of the surface of load in the

diagram P ,Q for pc data is written:

Q2−M 2 PP s 2Pcr−P =0 éq 3.1.2.2 - 1




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The plastic flow in the plan P ,Q thus with pc constant is not associated with the surface of load. If

it were the case, one would have:

εvpp=Λ

∂ f 1

∂Pε p= Λ

∂ f 1

∂ s

and the following report:εeq p

ε vpp =

2Q

M 2 2P + P s -2Pcr , éq 3.1.2.2 - 2

similar to the report obtained in the Camwood-Clay model (with P s=0 ). In fact in this model, one

introduces a parameter of correction α who destroys the character of normality, so that:εeq p

ε vpp =

2Qα

M 2 2P + P s -2Pcr . α is given by the authors of the model [bib1] as being:

α=M M−9 M−3

9 6−M { 1

[ 1− κλ 0 ] } éq 3.1.2.2 - 3

This corrector allows to better take into account the experimental results, and to better estimate inparticular the coefficient of pushed grounds.

M

M

Q

sP

P

0cp

0cp

*consP )( ccons pP

3.1.2.2 figure - has: Criterion 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 s

) reversible when pcpc0 and irreversible when suction exceeds the threshold pc0 .




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3.2.1 Reversible part

The tests oedometric with constant constraint and controlled suction give us the variation of the indexof the vacuums according to suction [Figure 3.2.1-a] reversible below threshold in suction:

e -e0=- κ s Lnpc

patm

si pc pc0 ,

with s independent of the state of containment.

Deformation being able to be written: ε v - εv0=-e−e0

1+ e0

, one a:

εvse=

κ s

1+ e0

pc

pc + patm

=1k 0s

pc

pc + patm éq 3.2.1-1

The evolution of suction is written then:

pc=patm exp k 0s εvse−ε v0

e , with k 0s=1+e0

κ s

éq 3.2.1-2

cLnp

e

s

s

0cLnp

0e

aLnp

Figure 3.2.1-a: Evolution of suction

3.2.2 Irreversible part

Beyond the threshold pc0 , unrecoverable deformations appear, the slope in the test oedometric

becoming λs . This slope can actually depend on the hydrostatic constraint applied to the sample, but

she is considered constant in the original model of Barcelona. As one can note it on [Figure 3.2.2], thepressure of consolidation increases with suction. [Figure the 3.2.2 (A)]) watch two compression tests in

condition saturated pc=0 and unsaturated pc0 . A relation enters P cons* (point 3) value of the

preconsolidation with of saturated and P cons (point 2) the pressure of preconsolidation in unsaturatedis established by comparing the specific volumes obtained on ways according to the items 1,2,3 [Figure

3.2.2 (a)] which describe a discharge of P cons with P cons* with constant suction followed by a

remoistening of a value pc to 0 with constant pressure Pcons* , from where the following equation:

v1 + Δv pression + Δv succion=v3




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One makes the assumption that the reduction of suction 2 3 be accompanied by recoverable

deformations. The elastic relation is the following one: dv=-κ s

dpc

pc + patm, where patm is the

atmospheric pressure. One writes for item 1 and 3 the expression of volume as follows:

v=N P0− λ pc lnPP0

where P0 is a pressure of reference corresponding to an initial volume N P0 . One combines this

expression and the elastic relations:

N P0 - λ pc lnP cons

P0

+ κ lnP cons

P cons* +κ s ln

pc + patm

patm

=N 0 - λ 0 lnPcons

*

P0

By eliminating initial volumes by the elastic relation:

Δv P0 ∣pc

0=N 0 −N P0 =κ s ln

pc patm

patm

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

P cons

P0= P cons

*

P0[

λ 0 −κλ p c−κ ]

Like P cons=2Pcr ,

One finds:

P cr=P0

2 2Pcr*

P0[

λ 0 −κλ p c−κ ]

éq 3.2.2-

[Figure 3.2.2] way 1-2-3 in the plan visualizes P , pc .




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12

3

g o n f l e m e n t

e f f o n d r e m e n t

refP *consP consP LnP

v

P

cp

*consP

12

3

consP

1,0 Sp c

Sp c ,

a b

L C

cpN

0N

1v2v3v

pressionv

succionv

S I

Figure 3.2.2: (A) Curves of compression for water-logged soils and not-saturated

(b) criterion in the diagram P , pc

The total component of the voluminal deformation due to the evolution of suction is:

εvs=λs

1+e0

pc

pc + patm

si pc patm éq 3.2.2-

from where the plastic component which is written:

εvsp=

λs -κ s 1+e0

pc

pc + patm

=1k s

pc

pc + patm

éq 3.3.2-

Note:

The variation of suction does not generate deviatoric deformations.

3.3 Complete behavior (mechanical and hydrous loading)

3.3.1 Reversible behavior

Under spherical loading, the evolution of the total voluminal elastic component is thus written:

εve= εvp

e + ε vse=

1k 0

PP

+1

k 0s

pc

pc + patm éq 3.3.1-1

Evolutions of the parts hydrostatic and deviatoric of the constraint σ are thus written:

PP=k 0 εv

e -k 0

k 0s

pc

pc + patm, éq 3.3.1-2

s ij=2μ εije , éq 3.3.1-3




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3.3.2 Thresholds of flow

The two thresholds of the reversible field are such as:

Mechanical criterion : f 1 P ,Q , P cr pc , pc =Q2M 2 Pk c pc P−2Pcr pc ≤0 éq 3.3.2-1

Hydrous criterion: f 2 pc , pc0 = pc−pc0≤0 éq 3.3.2-2

The three-dimensional field of reversibility in space P ,Q , pc is represented on [Figure 3.3.2 - has].

These two criteria are reduced in the plan P , pc with curves called LC (loading collapse) and IF(suction increase) (cf [Figure 3.3.2-b]).

P

Q

LCSI

*consP

cp

1f

1f

2f

2f

Figure 3.3.2-a: Surface of load in space P ,Q , pc

consPsP P

cp

0cp

LC

SI

Figure 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:

εvp= Λ

∂ f 1

∂P= Λ M 2 2P - 2Pcr +kc pc éq 3.3.3-1

ε p=α Λ

∂ f 1

∂ s=α Λ

∂ f 1

∂Q∂Q∂ s=3α Λ s éq 3.3.3-2

or εvp=

1k s

pc

pc + patm

, εp=0 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: Pcr and pc0 .

The laws of work hardening of each surface are:

On f 1 , Pcr

Pcr

=k vp

éq 3.3.4-1

On f 2 , pc0

pc0 + patm

=k s vp

éq 3.3.4-2

3.3.5 Inventory of the configurations of mechanical and hydrous coupling

One examines the various configurations of loading in space P , pc .

3.3.5.1 Total reversibility

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

f 10 , or ( f 1=0, f 10 ), and pc pc0 , or ( pc= pc0 , pc0 ).

The relations expressing this reversibility are:

PP=k 0 εv -

k 0

k 0s

pc

pc + patm

i.e.:

P=P0

exp k 0 εv−εv0

pc + patm

patm

k0 / k0s, éq 3.3.5.1 - 1




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and s=2μ ε éq 3.3.5.1 - 2

P

cp

0cp

domaine de réversibilité

M

lqS

0cp

cp

3.3.5.1 figure - has: Field of reversibility in the plan P , pc- curve of Water retention

3.3.5.2 Elastoplastic behavior

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

f 1=0 , f 1=0 , and pc pc0 (or pc= pc0 and pc0 )

The elastic evolution is thus written:

PP=k 0 εv

e -k 0

k 0s

pc

pc + patm ,

i.e.:

P=P0

exp k0 εve -ε v0

e

pc + patm

patm

k0 / k0séq 3.3.5.2 - 1

and s=2μ ε éq 3.3.5.2 - 2

The evolution of the components of the plastic deformation is:

ε p=3αΛs

εvp=ΛM 2 [ 2P -2Pcr + k c pc ]

The evolution of the mechanical threshold is written: Pcr=kPcr εvpp=kPcr ΛM 2 [ 2P -2Pcr + k c pc ] .




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A specificity of the original model of Barcelona is the assumption that mechanical work hardening iscompletely coupled with the hydrous work hardening (cf [3.3.5.2 Figure - has]) from where the relation:

pc0

pc0 + patm

=k s

kP cr

P cr

éq 3.3.5.2 - 3

P

cp

0cp

1

2

M

3.3.5.2 figure - has: Coupling of mechanical work hardening to hydrous work hardening

3.3.5.3 Hydrous behavior generating of the unrecoverable deformations

The point M reaches the threshold in suction (cf [3.3.5.2 Figure - has]):

pc= pc0 and pc0

The mechanical behavior is elastic:

P=P0

exp k0 εve−εv0

e

pc + patm

patm

k0 / k0s , s=2μ ε éq 3.3.5.3 - 1

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

P cr

P cr

=kk s

pc0

pc0patm

éq 3.3.5.3 - 2

The rate of plastic deformation is written:

εvp =

1k s

pc

pc + patm

ε p=0




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P

cp

0cp 1

2

M

3.3.5.3 figure - has: Coupling of hydrous work hardening to mechanical work hardening

3.4 Data of the model of Barcelona

The model requires the following parameters:

1) Elastic parameters provided under the keyword ELAS :The thermal dilation coefficient α , two elastic coefficients E , ν provide in data from which thecoefficient of Lamé is calculated μ .

2) Under the keyword CAM_CLAY :

1) P0 Initial hydrostatic pressure equal to noted atmospheric pressure Pa underthe keyword CAM_CLAY

2) instead of giving the initial index of the vacuums e0 one gives the initial porosity

which must be of value equal to that given under the keyword THM_INIT, notedPORO.

3) Parameters associated with surface threshold LC (forced isotropic): P cr* ,

equalizes with half of the pressure of preconsolidation P cons* noted

PRES_CRIT, λ* , the coefficient of compressibility for a saturated state and κthe elastic coefficient of compressibility, noted LAMBDA and KAPA.

4) The critical slope M ,

3) Under the keyword BARCELONA :

1) r and β , coefficients allowing to calculate λ pc , noted R and BETA.

2) parameters related to a variation of suction: λ s , coefficient of compressibility

related to a variation of suction in the plastic range, κ s coefficient associated

with the change with suction in elastic range, noted LABDAS and KAPAS.3) k c the parameter which controls the increase in cohesion with suction

4) the initial threshold of suction pc0 , noted PC0_INIT

5) α the coefficient of normality, noted ALPHAB.

Here a set of values of some of these parameters, resulting from [bib1]:




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λ 0 =0 .2 ; κ=0 .02 ;r=0 .75 ; β=12 .5 MPa−1 ; P0=0 .10 MPa ;λs=0.08 ; κ s=0.008 ;G=10 MPa ;M=1 ; kc=0 .6

4 Digital integration of the relations of behavior

4.1 Recall of the problem

The digital integration of the model is similar to that carried out for the Camwood-Clay law [bib5], byoperating a translation on the axis of the capillary pressures.This model is obligatorily coupled with the hydraulic behavior, contrary to Camwood-Clay which can beused 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 inCode_Aster [bib7] and [bib8]. It will be more particularly employed with modelings KIT_HHM andKIT_THHM (in this last case, there is not for the moment of dependence of the mechanicalcharacteristics specific to the model of Barcelona with the temperature).The variables of entry of the model are u or and P1 , P2 , P1 and P2 being equal

in modelings quoted to pc , pgz , pc and p gz that it is with hydrous modelingsLIQU_VAPE_GAZ or LIQU_GAZ.

The variables of exit of the model are: σ ' , Pcr , pc0 , P s .

The following notations are employed: A- , A , ΔA respectively for the quantity evaluated at the known

moment t , at the moment t + t and its increment. The equations are discretized in an implicit way,it is - with - to say expressed according to the unknown variables to the moment t + t .

One will note:

Pcr−

quantity P cr− pc

−,

Pcr− pc quantity

P0

2 2Pcr-

P0[

λ pc- -κ

λ pc - κ ] and

P cr pc =P cr− pc exp kΔε v

p

4.2 Incremental relations

The rules of flow and the condition of consistency give the following relations of flow:If the threshold f 1 is reached, the increment of plastic deformation voluminal is written:

Δε vp=

1k P + k c pc P cr

[ 2P -2Pcr + k c pc 2

ΔP +QM 2 ΔQ -

k c

22Pcr - P Δpc] éq 4.2-1

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

Δε eqpp=

αkPcr P + kc pc [

QM 2 ΔP +

2Q2

M 4 2P -2Pcr + k c pc ΔQ -

k c Q 2Pcr - P

M 2 2P- 2Pcr + k c pcΔpc] éq 4.2-2

and the tensor deviatoric is written:

Δ εp=

3αs

M 22P - 2Pcr + P s

Δε vp

éq 4.2-3

If f 2 is reached, the increment of plastic deformation voluminal is determined by:




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Δεvp=

1k s

Lnpc0 + patm

pc0- + patm

éq 4.2-4

plastic deformation being purely voluminal ( p=0 )

ε vpwill be the principal of the problem and given unknown factor while solving

f 1 P- ,Q- , P cr

- pc , pc , Δεv

p=0 , or f 2 pc , Δpc0 =0 , the increment of plastic deformation

voluminal being obtained from Δpc0 . One from of then deduced the evolution 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:

se=s−2 éq 4.3-1

The elastic prediction is chosen t t in the following way:

P e=P−

exp k 0 Δε v

pc +atm

pcpatm

k0 /k 0s éq 4.3-2

If f 10 and f 20 , then P=P e , s=se , Δε p=0,P cr=

P0

2 2Pcr-

P0[

λ pc- - κ

λ pc -κ ], Δpc0=0 ,

If not:

s=se−2μΔ ε p

éq 4.3-3

P=P e exp [−k0 Δεvp ] éq 4.3-4

P cr=P0

2 2Pcr-

P0[

λ pc- -κ

λ pc -κ ]exp [ kΔεv

p ] éq 4.3-5

pc0 + patm = pc0- + patm exp [ k s Δε v

p ] éq 4.3-6

The principal unknown factor is thus Δεvp

.

If f 10 , then

While replacing p by its expression according to Δε v

p [éq 4.2-3] one obtains:

s=se

16αμ Δεv

p

M 2 2P−2Pcrk c pc

éq 4.3-7




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and:

P=P e exp [ -k 0 Δε vp ] éq 4.3-8

The unknown factor is given while solving f 1 P , Q , Pcr , pc = f 1 Pe , Qe , P cr

- pc , pc , Δεv

p=0 ,

I.e.:

Qe2

=−M 2[ 1 6αμ Δεvp

M 2 2P−2Pcrk c pc ]2

Pkc pc P−2Pcr ,

or:

Q e2

=−M 2[16 αμ Δε vp

M 2 2Pe exp - k0 Δεvp - 2Pcr

- pc exp kΔεv

p + k c pc ]2

[P e exp -k 0 Δεvp + k c pc ]

[P e exp -k 0 Δεvp - 2Pcr

- pc exp - kΔεv

p ]

éq 4.3-9

If f 20 , then: Δpc0=Δpc , the unknown factor is immediately given by:

Δεvp=

1k s

Lnpc0 + patm

p- c0 + patm

, éq 4.3-10

from where s=se and P=Pe exp [−k0 Δεvp ] éq 4.3-

11

One has moreover P cr=P0

2 2Pcr-

P0[

λ pc- -κ

λ pc -κ ]exp [ kΔεv

p ] . éq 4.3-12

²

5 Tangent operator

If the option is: RIGI_MECA_TANG, option used at the time of the prediction, the tangent operatorcalculated in each point of Gauss is known as of speed:

σ ij=Dijklelp ε kl

,

i.e. Dijklelp

is calculated starting from the not discretized equations.

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

dσ ij=Aijkl dε kl

In this case, Aijkl is calculated starting from the implicitly discretized equations.

The tangent operator of the generalized constraints is implemented in THHM under the nameDΣ DE 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 thetangent operator in elasticity, the operator of speed and the coherent operator.




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5.1 Nonlinear elastic tangent operator

The elastic relation of speed of the model of Barcelona is written:

σ ij=−P δ ij sij=k 0 Ptr { ε ¿ δij2μ εijk 0

k 0s

Ppc

pc patm

δ ij éq 5.1-1

σ ij= k0 P−23

μ tr ε δij2μ εijk 0

k 0s

Ppc

pc patm

δ ij éq 5.1-2

The tensor of the constraints used in the model of Barcelona (and the tests determining the data of themodel) is function of the total constraint and of the gas pressure and is written:

σ=σTp gz I éq 5.1-3

The tensor of the constraints of Bishop σ ' used in Code_Aster is such as: σT=σ' σ P I with

σ P=−b pgz−S lq pc éq 5.1-4

From where the expression of the constraint of Bishop according to the constraint of the model ofBarcelona:

σ '=σ b−1 pgz−bS lq pc I éq 5.1-5

Note:

The constraint of Bishop is generally regarded as an effective constraint (controlled only bythe deformation). It is not the case of the model of Barcelona where two constraints areneeded ( σ , pc to describe the behavior. Consequently, in the tangent operator, the term

∂ σ '

∂ pc

does not summarize itself with −∂ σ p

∂ pc .

Part [DMECDE] of the matrix DΣ DE correspondent with∂σ '

∂ ε is such as:

[σ 11

'

σ 22'

σ 33'

2 σ 12'

2 σ 23'

2 σ 31']=[

k 0 P43

μ k 0 P−23

μ k 0 P−23

μ 0 0 0

k 0 P−23

μ k 0 P43

μ k 0 P−23

μ 0 0 0

k 0 P−23

μ k 0 P−23

μ k0 P43

μ 0 0 0

0 0 0 2μ 0 00 0 0 0 2μ 00 0 0 0 0 2μ

] [ε11

ε22

ε33

2 ε12

2 ε23

2 ε31

]De

éq 5.1-6

Part [DMECP1] of the matrix DΣ DE is reduced to ∂ σ '

∂ p1with p1=pc who is such as:




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{σ 11

'

σ 22'

σ 33'

2 σ 12'

2 σ 23'

2 σ 31'}=[ k0

k 0s

Ppc patm

−bS lq k0

k 0s

Ppc patm

−bS lq k0

k 0s

Ppc patm

−bS lq 0 0 0 ] { p1 }

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 i−1

(the option RIGI_MECA_TANG called with the first iteration of a new increment of load).

So with t i−1 the border of the field of reversibility is reached, one writes the condition: f=0 who must

be checked jointly with the condition f =0 . So with t i−1 one is strictly inside the field, f 0 , then the

tangent operator is the operator of elasticity.

If the mechanical criterion is reached:f 1=0

f 1= ∂ f 1

∂ σ σ∂ f 1

∂ P cr

P cr∂ f 1

∂ pc

pc=0 éq 5.2-1

like Pcr=∂Pcr

∂ εvp εv

p∂Pcr

∂ pc

pc , then:

f 1= ∂ f 1

∂ σ σ∂ f 1

∂ P cr

∂P cr

∂ εvp εv

p∂ Pcr

∂ pc

pc ∂ f 1

∂ pc

pc=0 éq 5.2-2

One has in addition: σ ij=Dijkle ε eklk 0 P

pc

k 0s pc patm δij éq 5.2-3

i.e.:

ij=Dijklekl− D ijkl

e ∂ f 1

∂ skl

−13

∂ f 1

∂P klk 0 P

pc

k 0s pcpatmij éq 5.2-4

By writing the plastic module of work hardening:

H p=-∂ f 1

∂P cr

∂Pcr

∂ εvp

∂ f 1

∂ P, éq 5.2-5

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

∂ f 1

∂σ ij σ ij−ΛH p∂ f 1

∂ pc

∂ f 1

∂ P cr

∂Pcr

∂ pc

pc=0 éq 5.2-6

Multiplication of the equation [éq 5.2-4] by ∂ f 1

∂σ ij give:




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∂ f 1

∂σ ij σ ij= ∂ f 1

∂σ ij Dijkle ε kl - Λ ∂ f 1

∂ σ ij Dijkle α ∂ f 1

∂ skl

−13

∂ f 1

∂Pδ kl + ∂ f 1

∂ σ ij k 0 Pδ ij

pc

k 0s pc + patm éq 5.2-7

The two preceding equations make it possible to find:

H p Λ= ∂ f 1

∂σ ij Dijkle εkl−Λ ∂ f 1

∂σ ij D ijkle α ∂ f 1

∂ skl

−13

∂ f 1

∂Pδ klkl

∂ f 1

∂ σ ij k 0 Pδ ij

pc

k 0s pc patm ∂ f 1

∂ pc

∂ f 1

∂ P cr

∂ P cr

∂ pc

pc éq 5.2-8

from where and to deduce the expression from it from the plastic multiplier:

Λ= ∂ f 1

∂ σ ij Dijkle εkl[ ∂ f 1

∂σ ij k 0 Pδ ij1

k0s pc patm∂ f 1

∂ pc

∂ f 1

∂Pcr

∂Pcr

∂ pc

] pc

∂ f 1

∂ σ ij D ijkle α ∂ f 1

∂ skl

−13∂ f 1

∂Pδkl H p

éq 5.2-9

That is to say H the definite elastoplastic module like:

H= ∂ f 1

∂ σ ij Dijkle α ∂ f 1

∂ skl

−13

∂ f 1

∂Pδ klH p éq 5.2-10

The plastic multiplier is written:

Λ= ∂ f 1

∂ σ ij Dijkle ε kl[ ∂ f 1

∂ σ ij

k0 Pδij1

k 0s pcpatm∂ f 1

∂ pc

∂ f 1

∂ Pcr

∂ Pcr

∂ pc

] pc

H

éq 5.2-11

While replacing Λ by his expression in the equation [éq 5.2-4], one obtains:

σ ij=Dijkle ε kl -

1H [ ∂ f 1

∂ σ mn

Dmnope ε op] . Dijkl

e α∂ f 1

∂ skl

-13

∂ f 1

∂Pδ kl -

[ 1H [ ∂ f 1

∂ σ mn

k 0 P1k0s pc + patm

δmn + ∂ f 1

∂ pc

∂ f 1

∂P cr

∂P cr

∂ pc

]D ijkle α ∂ f 1

∂ skl

-13∂ f 1

∂ Pδ kl - k 0 P

k 0s pc + patmδij ] pc

éq 5.2-12One from of thus deduced the elastoplastic operator Delp

=De−D p :

σ ij=[Dijkle -

1H ∂ f 1

∂ σ op

Dijope Dmnkl

e α ∂ f 1

∂ smn

-13∂ f 1

∂ Pδmn

Dijklp

] ε kl -

{1H α∂ f 1

∂ sop

-13∂ f 1

∂PδopD ijop

e∂ f 1

∂ σ mn

k 0 Pδ mn1k0s pc + patm

-∂ f 1

∂ pc

+∂ f 1

∂P cr

∂ Pcr

∂ pc

-k 0 P

k 0s pc + patm δij} pc

Dijp c

éq 5.2-13with,

Dijklp=

1H

∂ f 1

∂ σ op

Dijope Dmnkl

e α ∂ f 1

∂ smn

-13

∂ f 1

∂Pδmn

and




e6ed09610822

Dijpc=-

1H α ∂ f 1

∂ sop

-13∂ f 1

∂Pδ op Dijop

e ∂ f 1

∂ σ mn

k 0 Pδmn1

k 0s pc + patm+∂ f 1

∂ pc

+∂ f 1

∂Pcr

∂P cr

∂ pc

+ k0 P

k0s pc + patmδ ij

éq 5.2-14

Calculation of Dijklp

:

∂ f 1

∂σ ij=-13

M 2 2P−2Pcr +k c pc δij +3sij , éq 5.2-15

who is written in vectorial notation:

[-

13

M 2 2P -2Pcr + k c pc + 3s11

-13

M 2 2P -2Pcr +k c pc +3s22

-13

M 2 2P -2Pcr +k c pc +3s33

3 2 s12

3 2 s23

3 2 s31

] éq 5.2-16

from where the expression of:

Dijkle ∂ f∂σ kl

: [-k 0 M 2 P 2P - 2Pcr +k c pc +6μs11

-k 0 M 2 P 2P - 2Pcr +k c pc +6μs22

-k 0 M 2 P 2P - 2Pcr +k c pc +6μs33

6μ 2 s12

6μ 2 s23

6μ 2 s31

] éq 5.2-17

and

∂ f∂ σ ij

Dijkle α ∂ f 1

∂ skl

-13

∂ f 1

∂Pδ kl=k 0 M 4 P 2P - 2Pcr + kc pc

2 +12αμQ 2éq 5.2-18

However the plastic module H is written in the form:

H= ∂ f∂σ ij Dijkl

e α ∂ f 1

∂ s kl

-13

∂ f 1

∂ Pδkl + H p

H=M 4 2P−2Pcr + k c pc [ k 0 P 2P - 2Pcr + k c pc + 2kPcr P + k c pc ]+ 12αμ Q2éq 5.2-19

While posing:

Aij=-k 0 M 2 P 2P -2Pcr +k c pc δij + 6μsij , A' ij=- k 0 M 2 P 2P -2Pcr + k c pc δij + 6αμ s ij , éq 5.2-20

with: tr A =-3k0 M 2 P 2P - 2Pcr + k c pc




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D p=

1H [

A11 A' 11 A11 A ' 22 A11 A ' 33 6 2 μα A11 s12 6 2 μα A11 s23 6 2 μα A11 s31

A' 11 A22 A22 A ' 22 A22 A ' 33 6 2 μα A22 s12 6 2 μα A22 s23 6 2 μα A22 s31

A33 A' 113 A22 A ' 33 A33 A ' 33 6 2 μα A33 s12 6 2 μα A33 s23 6 2 μα A33 s31

6 2 μA' 11 s12 6 2 μA' 22 s12 6 2 μA' 33 s12 36αμ2 s122 36αμ2 s12 s23 36αμ2 s12 s31

6 2 μA' 11 s23 6 2 μA ' 22 s23 6 2 μA ' 33 s23 . 36αμ2 s232 36αμ2 s23 s31

6 2 μA' 11 s31 6 2 μA' 22 s31 6 2 μA' 33 s31 . . 36αμ2 s312

]SYM

éq. 5.2-21

One can write the components ∂σ '

∂ εpiece [DMECDE] of the matrix DΣ DE who are those of the

operator Delp=De - D p .

According to the equation [éq 5.2.14]. Components ∂ σ '

∂ p1

with p1=pc piece [ DMECP1 ] matrix

DΣ DE are:

[- tr A 3 Hk 0s pc + patm

A '11 +

M2[kc 2Pcr- P -2Pcr P +kc p c Ln[ 2Pcr*

P0][ 0 -

pc - 2

' ]]H

A '11+k0 P

k0s pc + patm - bS lq

- tr A 3 Hk0s pc + p atm

A '22+

M 2[k c 2Pcr - P - 2Pcr P+ kc pc Ln[ 2Pcr*

P0][ 0 - pc -

2' ]]

HA ' 22+

k0 P

k0s pcp atm −bS lq

- tr A 3 Hk0s pc + p atm

A '33+

M 2[k c 2Pcr - P−2Pcr P +k c pc Ln[ 2Pcr*

P0][ 0 -

pc -2 ']]

HA '33 +

k0 P

k0s pcpatm -bS lq

-2 2 tr A Hk 0s p c + patm

alphas12 +

6 2muM 2[k c 2Pcr -P - 2Pcr P +k c pc Ln[ 2Pcr*

P0][ 0 - pc -

2 ' ]]

Halphas12

-2 2 tr A Hk 0s pc + patm

alphas23 +

6 2muM 2[k c 2Pcr -P - 2Pcr P +k c pc Ln[ 2Pcr*

P0][ 0 - pc -

2 ' ]]

Halphas23

-2 2 tr A Hk 0s pc + patm

alphas13 +62 muM2[kc 2Pcr- P - 2Pcr P +kc pc Ln[ 2Pcr*

P0][ 0 -

pc - 2' ]]alphas13

] éq 5.2-22

with λ '=∂ λ∂ pc

=- βλ0 [1 -r exp - βpc ]

If the hydrous criterion is reached:

One leaves again the equation [éq 5.2.3] with this time ε p=

pc

k s pc + patm ,

One finds a direct relationship enters σ and ε , pc form:




e6ed09610822

σ=De ε + k 0 P pc

k s pc + patm+

pc

k 0s pc + patm I éq 5.2-23

One deduces then the constraint from Bishop

σ '=De ε + k 0 P

k s pc + patm +

k 0 P

k 0s pc + patm -bS lq I pc éq 5.2-24

Components ∂σ '

∂ ε piece [ DMECDE ] matrix DΣ DE are nothing other than those of the matrix

De .

Only components of the piece [ DMECP1 ] matrix DΣ DE are thus those of ∂ σ '

∂ p1

with p1=pc :

[11

'

22'

33'

2 12'

2 23'

2 31']=[ k0 P

p c + patm

1k s

+1k0s

-bS lq

k0 P

pc + patm

1k s

+1

k0s

- bSlq

k0 P

pc + patm

1k s

+1k0s

-bS lq 0 0 0 ]{ p1 } 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 Camwood Clay initiallyseparating the treatment from the deviatoric part of the hydrostatic part for then combining them inorder to deduce the tangent operator connecting the disturbance from the total constraint to thedisturbance of the total deflection.

5.3.1 If the mechanical criterion is reached

5.3.1.1 Treatment of the deviatoric part

It is considered here that the variation of loading is purely deviatoric δP=0.The increment of the deviatoric constraint is written in the form:

Δsij=2μ Δ εij - Δ εijp éq 5.3.1.1 - 1

Around the point of balance σ - + Δσ , a variation is considered δs deviatoric part of the constraint:

δs kl=2μ δ ε kl -δ εklp éq 5.3.1.1 - 2

Calculation of δ εklp :

It is known that:

Δ ε klp=3Λ αskl éq 5.3.1.1 - 3




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By deriving this equation compared to the deviatoric constraint, one obtains:

δ εklp= 3α δ Λ skl + 3 α Λ δ skl éq 5.3.1.1 - 4

Calculation of δ Λ :One a:

Λ=1H p[ ∂ f∂ σ mn

Δσmn + ∂ f∂ pc

+∂ f∂ P cr

∂Pcr

∂ pc

Δpc ]= 1H p[ ∂ f∂ s mn

Δsmn +∂ f∂ P

ΔP + ∂ f∂ pc

+∂ f∂ Pcr

∂ Pcr

∂ pc

Δpc]=

1H p

[3smn Δsmn + M 2 2P - 2Pcr + k c pc ΔP - M 2 [ k c 2Pcr - P + 2 P + k c pc P ' cr ] Δpc]éq 5.3.1.1 - 5

If one considers only the evolution of the deviatoric part of σ δP=0, then:

δ Λ H p =δ ΛH p + ΛδH p=[ 3δsmn Δsmn + 3smn δsmn ] - 2M2 ΔPδPcr +

M 2 k c ΔPδpc - 2M2 k c δP cr Δpc - M 2 [ k c 2Pcr - P +2P ' cr P + k c pc + Δpc ] δpc

éq. 5.3.1.1 - 6

with P ' cr=∂P cr

∂ pc

However: δ P cr=kPcr δ ε vP .

Like Δε vp=ΛM 2

2P -2Pcr + k c pc , one a:

δεvp=δΛM 2

2P - 2Pcr + k c pc - 2M2 ΛδPcr + k c M 2 Λδpc éq. 5.3.1.1 - 7

From where:

δ ΛM 22P - 2Pcr + kc pc =[ 1kPcr

+2ΛM2] δPcr - kc ΛM 2 δpc éq 5.3.1.1 - 8

In addition,

H p=2 kM 4 P cr P + k c pc 2P−2Pcr + k c pcetδ H p=2 kM 4 P + k c pc 2P- 4Pcr + k c pc δ P cr + 2 kPcr M 4 k c 3P - 2Pcr + 2k c pc δpc

éq 5.3.1.1 - 9

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

δ ΛH p + [ 2Λ kM 4 P + k c pc 2P -4Pcr + k c pc + 2M2 ΔP + 2M2 kc Δpc ]δ Pcr=

- [ 2Λ kPcr M 4 k c 3P +2kc pc - 2Pcr + M 2 [ k c 2Pcr - PΔP + 2P ' cr pc + Δpc ] + 2P ' cr P ] δpc +

[ 3δsmn Δsmn + 3smn δsmn ]

éq

5.3.1.1 - 10

While using the relation [éq 5.3.1.1 - 8], it comes then:

δ Λ=[ 3δsmn Δsmn + 3smn δsmn ]

H p + A-

Zδpc

H p + Aéq 5.3.1.1 - 11

avec A=[ k ΛM 4 P +k c pc 2P - 4Pcr + kc pc + M 2 ΔP + M 2 kc Δpc ][M 22P - 2Pcr + k c pc

12kPcr

+ ΛM 2 ]




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Z=M2Λ

kk c ΛM 2 P + k c pc 2P - 4Pcr + k c pc

1

2 kM 2 P cr

+ Λ+ 2kk c P cr M

23P -2Pcr + k c pc

- M 2 kc ΔP +M 2 k c ΔPΛk c

2 ΛM 2 Δpc 12kM 2 P cr

+ Λ+ M 2 kc 2Pcr - P + 2M2 P ' cr P + k c pc + Δpc

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

δ εklp=

9αH p + A

Δsmn δsmn skl + smn δsmn skl +9αH p

smn Δsmn δskl

+3αH p

M 2 2P -2Pcr + k c pc ΔPδskl -3αH p

M 2 k c 2Pcr - P Δpc δskl -3αZH p + A

δpc skl

-6αH p

M 2 P + k c pc P ' cr Δpc δskl

éq 5.3.1.1 - 12

δsij is written then:

δsij=2 μδ { εij -18 μαH p + A [ Δskl s ij δskl + skl s ij δskl ] -

18 μαH p

skl Δskl δsij

-6 μαH p

M 2 2P - 2Pcr + kc pc ΔPδsij +6 μαH p

M 2 k c 2Pcr - P Δpc δsij

+6 μα ZH p + A

s ij δpc +12 μα

H p

M 2 P + k c pc P ' cr Δpc δsij

éq 5.3.1.1 - 13

i.e.:

ijkl +ijkl6 H p

M 2 2P - 2Pcr +kc pc P +18 H p + A

skl sij + skl s ij+18

H p

smn smnijkl

-ijkl6 H p

k c M 2 2Pcr -2P pc -12

H p

M 2 P +kc pc P 'cr pc skl=

2 ij +6 ZH p+ A

sij pc

éq 5.3.1.1 -

14




e6ed09610822

or in tensorial writing:

I 4d 1+

6 μαH p

M 2 2P -2Pcr + k c pc ΔP +18 μα

H p

Δs : s -6 μαH p

M 2 k c 2Pcr - P Δpc+

18 μαH p + A

s + Δs⊗ s -12 μα

H p

M 2 P + k c pc P ' cr Δpc δs=

2 μδ { ε ¿+6 μα ZH p + A

sδp c

éq 5.3.1.1 - 15

that one can still write by symmetrizing the tensor sΔs⊗ s :

{I 4d 1+

6H p

M 2 2P -2Pcr +kc pc P -6H p

k c M 2 2Pcr - P pc

+18

H p

s : s -12

H p

M2 P + kc pc P 'cr pc +18H p + A

ℵ } s=2 +6ZH p + A

s pc

éq 5.3.1.1 - 16

with: ℵ=12[ s + Δs ⊗ s + s⊗ s+ Δs T ]

Calculation of ℵ, while posing: T ij=sij + Δsij

T ⊗ s= [T 11 s11 T 11 s22 T 11 s33 2T 11 s12 2T 11 s23 2T 11 s31

T 22 s11 T 22 s22 T 22 s33 2T 22 s12 2T 22 s23 2T 22 s31

T 33 s11 T 33 s22 T 33 s33 2T 33 s12 2T 33 s23 2T 33 s31

2T 12 s11 2T 12 s22 2T 12 s33 2T12 s12 2T12 s23 2T12 s31

2T 23 s11 2T 23 s22 2T 23 s33 2T23 s12 2T23 s23 2T23 s31

2T 31 s11 2T 31 s22 2T 31 s33 T 31 s12 2T31 s23 2T31 s31

]

ℵ=12[T⊗ s + T ⊗ s T ]

That is to say:

C= I 4d 12μ

+3αH p

M 2 2P - 2Pcr + k c pc ΔP +9αH p

Δs : s -3αk c

H p

M 22Pcr - P Δpc -

6αH p

M 2 P + k c pc P ' cr Δpc+

9αH p + A

ℵ

one poses:

c=9αH p

Δs : s




e6ed09610822

d=3αH p

M 2 2P -2Pcr + k c pc ΔP

g=-3αH p

M 2 k c 2Pcr - P Δpc

h=-6αH p

M 2 P + k c pc P ' cr Δpc

The symmetrical matrix C dimensions (6.6) is too large to be presented whole, one breaks up it into 4

parts C1 , C2 , C3 and C4 :

C=[ C1 C2

C3 C4]

with

C1=[1

2μ+c + d + g +h +

9αH p + A

s11T 11

9α2 H p + A

T 11 s22T 22 s119α

2 H p + A T 11 s33+ T 33 s11

9α2 H p + A

T 22 s11 +T 11 s22 1

2μ+ c + d + g + h +

9α H p + A

T 22 s22

9α2 H p + A

T 22 s33+ T 33 s22

9α2H p+ A

T 33 s11 +T 11 s339α

2 H p + AT 22 s33 +T 33 s22

12μ

+c +d + g + h+ 9α H p + A

T 33 s33]

éq 5.3.1.1 - 17

C2=[9α 2

2H p + A T 11 s12 + s11T 12

9α 22 H p + A

T 11 s23 + s11T 239α 2

2H p + AT 11 s13+ s11T 13

9α 22H p + A

T 22 s12 + s22 T 129α 2

2H p + A T 22 s23 + s22T 23

9α 22 H p + A

T 22 s13+ s22T 13

9α 22H p + A

T 33 s12 + s33T 129α 2

2 H p + A T 33 s23 + s33T 23

9α 22H p + A

T 33 s13+ s33T 13 ]

éq 5.3.1.1 - 18

C3=C 2 éq 5.3.1.1 - 19

C 4=[1

2μ+c +d + g + h+

18αH p + A

s12T 12

9αH p + A

T 12 s23+T 23 s129α

H p + A T 12 s23+ T 23 s12

9αH p+ A

T 23 s12 +T 12 s231

2μ+c +d + g + h+

18αH p + A

T 23 s23

9αH p + A

T 23 s13+ T 13 s23

9αH p + A

T 13 s12 +T 12 s13 9α

H p + A T 13 s23+T 23 s13

12μ

+ c + d + g + h +18 αH p + A

T 13 s13]

éq 5.3.1.1 - 20

Calculation of the rate of variation of volume:

Δε vp=M 2 Λ2P - 2Pcr +k c pc ,

δε vp=M 2 δΛ2P - 2Pcr + kc pc - 2M2 Λδ Pcr +M 2 Λk c δpc

= Bδ Λ + Dδpc

=3B

H p + A s + Δs . δ s + D-

BZH p + A

δpc

éq 5.3.1.1 - 21




e6ed09610822

with: B=M 2

2P - 2Pcr +k c pc - M 2 ΛM 22P - 2Pcr + k c pc

12kP cr

+ M 2 Λ.

and D=k c M 2 Λ- M 2 Λ

k c M 2 Λ1

2 kPcr

+ M 2 Λ.

One thus has:

δεvp=

3BH p + A

s + Δs .δ s - BZ

H p + A - D δpc éq 5.3.1.1 - 22

and finally:

δ εij=C ijkl -BH p + A

s + Δs kl δij δ skl -

-BZ3H p + A

δij +D3

δij +δij

3k 0s pc + patm +

3ZH p + A

sij δpc

éq 5.3.1.1 - 23

5.3.1.2 Treatment of the hydrostatic part

It is considered now that the variation of loading is purely spherical ( δ s=0 ).

The increment of P is written in the form:

ΔP=P -

[exp k 0 Δεv

e

pc + patm

p- c + patm

k0 /k 0s

-1] éq 5.3.1.2 - 1

The derivation of this equation gives:

P=k 0 P v - vp -

k 0

k 0s

Ppc + patm

pc éq 5.3.1.2 - 2

Calculation of δε vp

:

It is known that:

Δε vp=ΛM 2 2P - 2Pcr + kc pc éq 5.3.1.2 - 3

By differentiating this equation, one obtains:

δεvp=M 2 δ Λ 2P -2Pcr + k c pc + Λ 2δP- 2δPcr + k c δpc éq 5.3.1.2 - 4




e6ed09610822

One knows the expression of Λ :

Λ=M 2 2P - 2Pcr +k c pc ΔP +3sΔs - M 2 [ k c 2Pcr - P + 2 P + k c pc P ' cr ] Δpc

H p

=b

H p

éq 5.3.1.2 - 5

while posing

b=M 2 2P -2Pcr + k c pc ΔP + 3sΔs - M 2 [ k c2Pcr - P + 2 P + k c pc P ' cr ] Δpc

While differentiating ΔΛ, it comes:

δΛ=M 2

H p [ 2P -2Pcr + k c pc δP + 2δP- 2δPcr + k c δpc ΔP -k c 2Pcr - P δpc−k c 2δPcr - δP Δpc

- 2 P + k c pc P ' cr δpc - 2δP + k c δpc P ' cr Δpc ]-

2 kM 4 bH p

2 [2δ PPcr 2P - P cr +32

kc pc + δP cr 2P2- 4 PP cr - 4Pcr k c pc + 3 Pkc pc + kc2 pc

2

+ k c P cr 3P -2Pcr +2kc pc δpc]

éq 5.3.1.2 - 6One seeks the expression of δP cr according to δΛ :

One a:

δP cr=kPcr δεvp

éq 5.3.1.2 - 7

One can write: δPcr

kPcr

=δΛM 2 2P−2Pcrk c pc ΛM 2 2δP−2δPcrk c δpc éq 5.3.1.2 - 8

δP cr 1Λ2M 2 kPcr

kPcr=δΛM 2 2P−2Pcrk c pc Λ2M 2 δPΛM 2 k c δp c

δP cr= M 2 2P−2Pcrk c pc kPcr

12kP cr ΛM 2 δΛ 2ΛM2 kPcr

12kPcr ΛM 2 δP ΛM 2 k c kPcr

12kPcr ΛM 2 δpc éq 5.3.1.2 - 9

One poses

c=M 2 kP cr 2P−2Pcrkc pc

[12M2 kPcr Λ ], a=

2M2 kPcr Λ

[12M2 kPcr Λ], d=

k c M 2 kPcr Λ

[12M2 kPcr Λ ]

One has then:

δP cr=aδPcδΛdδpc éq 5.3.1.2 - 10

By replacing the expression of δP cr in δΛ [éq 5.3.1.1 - 6], one finds:




e6ed09610822

δΛ=[ 2P−2Pcrk c pc2ΔPk c Δpc−2aΔP−2ak c Δpc−2P ' cr Δpc δP−2c ΔPk c Δpc δΛk c PΔP−2Pcr −2d ΔPk c Δpc −2P ' cr Pk c pcΔpc δpc ] .M

2

H p

−2kM 4 bH p

2 [ P cr 4P−2Pcr3kc pc a 2P−4Pcrk c pc Pk c pc ] δP

−2kM 4 b

H p2 [ c 2P−4Pcrk c pc Pk c pc ] δΛ

−2kM 4 bH p

2 [ k c Pcr 3P−2Pcr2kc pcd 2P−4Pcrkc pc Pk c pc ]δpc

éq 5.3.1.2 - 11By gathering the terms in δΛ and those in δP , one finds:

δΛ=fe

δPhe

δpc éq 5.3.1.2 - 12

with,

f =M 2

H p[ 2P−2Pcrk c pc1−2a Δpc 2ΔP−2aΔP−2P' cr Δpc ]

−2kM 4 bH p

2 [ 4P−2Pcr3kc pc P cra 2P2−4 PP cr−4Pcr k c pc3 Pk c pck c

2 pc2 ]

h=M 2

H p[−2dΔP−2dk c Δpck c ΔP−2k c P crk c P−2P ' cr Pkc pcΔpc ]

−2kM 4 bH p

2 [d 2P−4P crk c pc Pkc pc kc P cr 3P−2Pcr2kc pc ]

e=12 cM 2

ΔPk c pc

H p

2 bckM 4

H p2 2P2

−4 PP cr−4Pcr k c pc3 Pk c pck c2 pc

2

The expression of δε vp

thus becomes:

δεvp=XδPYδpc éq 5.3.1.2 - 13

with,

X=M 22Λ−2aΛ−2Λc

fe

fe2P−2Pcrk c pc

Y=M 2 2P−2Pcrk c pc he−2c

he2d−k c Λ

from where the expression of δP according to δεv and δpc :

δP 1k 0 PX =k 0 P δεV−[ Y 1k0s pc patm ] δpc éq 5.3.1.2 - 14




e6ed09610822

Calculus of the variation of deviatoric deformation:

δ εij= δ εp= 3δ Λ s=3

fe

δ Psij3he

δ pc sij éq 5.3.1.2 - 15

One thus has finally: δε ij=F ij δPK ij δpc éq 5.3.1.2 - 16

with

F=3fe

s−1k 0 PX

3k0 P1d ,

K=3he

s−k0 PY3

k 0 P3k0s pcpatm

1d

éq 5.3.1.2 - 17

5.3.1.3 Tangent operator

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

δ εijH ij δpc=C ijkl−B

H pA sΔs kl δij D klmn

1 δ σ mn , éq 5.3.1.3 - 1

with:

D1=[

2/3 −1/3 −1/3 0 0 0−1 /3 2 /3 −1/3 0 0 0−1 /3 −1/3 2 /3 0 0 0

0 0 0 1 0 00 0 0 0 1 00 0 0 0 0 1

] éq 5.3.1.3 - 2

projection in space deviatoric,

and that under spherical loading one a:

δ εij−K ij δpc= F ij D kl2 δ σ kl éq 5.3.1.3 - 3

with:

D 2=[−1/3−1/3−1/3

000] éq 5.3.1.3 - 4

hydrostatic projection, one has then:δσ ij=Aijkl δεklBij δpc éq 5.3.1.3 - 5




e6ed09610822

with:

Aijkl=[C ijmn−B

H pA sΔs mnδ ij Dmnkl 1F ij D kl

2 ]−1

éq 5.3.1.3 - 6

Bij=[C ijmn−B

H pA sΔs mnδ ij Dmnkl 1F ij D kl

2 ]−1

H kl−K kl éq 5.3.1.3 - 7

The constraint of Bishop is thus written:

δσ ij'=Aijkl δεkl B ij−bS lq δpc

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. As for the casegeneral, one makes a treatment of the deviatoric part and another for the hydrostatic part.

5.3.2.1 Treatment of the deviatoric part

According to the equation [éq 4.3.3] one finds:

s=se−2μΔ ε

p=se

−2μΛ∂ f∂ s=se

−6 μα Λs éq 5.3.2.1 - 1

Expressions of the plastic multiplier Λand of its derivation δΛ are written in the following way:

Λ= Qe

Q−1 /6 μα and δΛ=

δQe

6 μαQ−

Qe δQ6 μαQ 2 éq 5.3.2.1 - 2

with,

δQe=

32

se δse

Q e and δQ=32

sδsQ

from where the expression of δΛ :

δΛ=1

6μ3

2α [ se δse

Q e Q−

Q e sδsQ3 ] éq 5.3.2.1 - 3

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

δsij=2μ δ εij−3δΛsij−3Λδsij

While replacing Λand δΛ by their expressions, one can write:

δsij=2 μδ { ε¿ ij−3

2αskl

e δs kle

Q e Qsij

32α

Qe

Q3 skl δskl sij−1α Q e

Q−1 δsij éq 5.3.2.1 - 4




e6ed09610822

δs kl[ δijkl1α

Qe

Qδijkl−

1α

δijkl−3

2αQ e

Q3 skl . sij ]=2μ [ δ ijkl−

32α

skle . sij

QeQ ] δ εkl éq 5.3.2.1 - 5

or in tensorial writing:

δs [ Qe

QαI 4

dI 4

d 1− 1α − 3

2αQe

Q3 s⊗ s] G

=2μ [ I 4d−

32α

se⊗ sQ e Q ]

H

δ ε éq 5.3.2.1 - 6

Like δs does not depend on δε v , one can confuse δ ε with δε .

By using the tensor of projection within the space of deviatoric constraints D1 [éq 5.3.1.3 - 2], one canwrite:

δε=D1 .G . H−1

2μ.δσ éq 5.3.2.1 - 7

5.3.2.2 Treatment of the hydrostatic part

In tensorial writing, there is according to the equation [éq 5.3.1.2 - 2] the following relation:

I d δP=k 0 P δεv−k 0

k 0s

Ppcpatm

I d δpc éq 5.3.2.2 - 1

knowing that at the critical point, δε vp=0 .

Like δPdoes not depend on δ ε then one can confuse δε v with δε .

I d δP=k 0 P δε−k 0

k 0s

Ppcpatm

I d δpc éq 5.3.2.2 - 2

By using the tensor of projection within the space of hydrostatic constraints D 2 [éq 5.3.1.3 - 3], one canwrite:

I d D2 δσ=k0 P −k 0

k 0s

Ppcpatm

I d δpc

from where

δε=I d D2

k 0 Pδσ

I d

k 0s pcpatmδpc éq 5.3.2.2 - 3

5.3.2.3 Tangent operator

By combining the contributions of the two parts deviatoric and hydrostatic, one finds the writing of thetangent operator who connects the variation of the total constraint to the variation of the total deflectionat the critical point:

δε=[ D1 .G . H−1

2μ

I d D2

k0 P ] . δσI d

k 0s pc patm δpc

δσ ij=Aijkl δεkl−Bij δpc éq 5.3.2.3 - 1




e6ed09610822

with

Aijkl=[ D1 .G . H−1

2μ

I d D2

k 0 P ]−1

éq 5.3.2.3 - 2

and

Bij=−I d

k 0s pc patm éq 5.3.2.3 - 3

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

Bij=−I d

k 0s pc patm−bS lq 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:

δε kle=δ ε kl

e−

13

δεve δkl éq 5.3.3-1

that is to say:

δε kle=

δs kl

2μ−

δP3k0 P

δ kl−δpc

3k0s pcpatm δkl éq 5.3.3-2

In this case the plastic deviatoric deformation is worthless thus the plastic deformation has thefollowing expression:

δε klp=−

13

δε vp δ kl éq 5.3.3-3

that is to say:

δε klp=−

δpc

3ks pc0 patmδ kl éq 5.3.3-4

By combining each one of the components rubber band and plastic one finds:

δε kl=δεkleδεkl

p=

δskl

2μ−

δP3k0 P

δ kl−13 1

k 0s pc patm

1k s pc0 patm δpc δkl éq 5.3.3-5

By using the matrices of projection within the space of deviatoric and hydrostatic constraints one leadsto the following expression:

δε kl= Dijkl1

2μ−

Dij2 δkl

3k0 P δσ ij−13 1

k 0s pc patm

1k s pc0 patm δpc δ kl éq 5.3.3-6




e6ed09610822

thus one can write:

δσ ij= Dijkl1

2μ-

Dij2 δ kl

3k0 P -1

δεkl +13 1

k 0s pc + patm +

1k s pc0 + patm

Dijkl1

2μ-

Dij2 δ kl

3k0 P -1

δkl δpc éq 5.3.3-7

one poses Aijkl= Dijkl1

2μ−

Dij2 δ kl

3k0 P

or Aijkl=12μ

23

2μ9k0 P

−13

2μ9k0 P

−13

2μ9k0 P

0 0 0

−13

2μ9k0 P

23

2μ9k0 P

−13

2μ9k0 P

0 0 0

−13

2μ9k0 P

−13

2μ9k0 P

23

2μ9k0 P

0 0 0

0 0 0 1 0 00 0 0 0 1 00 0 0 0 0 1

éq 5.3.3-8

and by deducing the constraint from Bishop, one finds:

δσ ij'=Aijkl

-1 δε kl +[ 13 1

k 0s pc + patm +

1k s pc0 + patm Aijkl

-1 δ kl - bS lq ] δpc éq 5.3.3-9




e6ed09610822

6 Summary of the model of Barcelona

Modelings THHM:

KIT_HHM and KIT_THHM (in this last case, there is no dependence of the mechanical characteristicswith the temperature).

Variables of entry:

σ '- , pc+ , pgz

+ ,P cr- , pc0

- , Δε , Δpcand Δp gz

Variables of exit:

1) Δσ ' , more tangent operators (necessary to operator STAT_NON_LINE).

2) Internal variables Pcr+ , newer variables pc0

+ : threshold in suction and P s+ :

pressure of cohesion, and indicators of mechanical work hardening I 1 and

hydrous I 2 .

Elastic prediction:

P e=P−

exp k0 Δεv

pc patm

p−cpatm

k0 / k s0

, se=s−2μΔ ε

1) f 10 and f 20 pc pc0 : reversible behavior

P=P e , s=se , ε p=0 , Pcr=Pcr

− , pc0= pc0−

1) f 10 or f 20 plasticization and mechanical and hydrous work hardening

P=P e exp [−k0 Δεvp ],

s=se

16αμ Δεv

p

M 2 2P−2Pcrk c pc

Pcr=Pcr− exp [ kΔεv

p ] ,

pc0patm= pc0patm exp [ k s Δε vp ]

The single unknown factor is vp determined by f 1=0 (one has then:

Δ εp=

2QαΔεvp

M 2 2PP s−2Pcr ) or f 2=0 (and Δ ε p

=0 )

Note:




e6ed09610822

The constraint resulting from the data of the model of Barcelona east σ=σ totpgz 1d

, it will be thus the variable used in the routine describing it behavior, the constraint of

exit provided to STAT_NON_LINE being the constraint of Bishop: σ '=σ tot−σ p .

Tangent operators:

The tangent operator of the generalized constraints is implemented in THHM under the name DΣ DE

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

∂ εand

∂ σ '

∂ pc

blocks [ DMECDE ] and [ DMECP1 ] correspondent with: [∂ σ '

∂ ε∂ σ p

∂ ε] ,[∂ σ '

∂ pc

∂ σ '

∂∇ pc

∂ σ p'

∂ pc

∂ σ p'

∂∇ pc

] .

7 Implementation of the model

7.1 Data material

The use of the model of BARCELONA require to enrich the data of the model by CAM_CLAY byadditional data specific to the unsaturated grounds. This is concretized by the simultaneous adoption ofthe two keywords CAM_CLAY and BARCELONA under the order DEFI_MATERIAU.

7.2 Initialization of calculation

It is necessary that the initial state of material either plastically acceptable (the constraint and thecapillary pressure are thus such as the point of initial loading or inside the surface of load). It isnecessary thus on the one hand that suction is lower than the hydrous threshold, and on the otherhand that the constraint is inside the ellipse defined in the plan of initial suction. In particular, if theinitial mechanical loading is purely hydrostatic, it must be understood between the terminalsrepresented by cohesion ( −k c pc ) and pressure of consolidation ( 2Pcr ). The constraint used todescribe the behavior (forced total plus gas pressure) is different from the constraint to initialize inETAT_INIT (constraint of Bishop σ '). The relation between the two types of constraint is:

'=[b−1 pgz I−bS lq pc ]

7.3 Internal variables at exit

The model produces five internal variables:

V 1=P cr : critical pressure

V 2=I 1 : mechanical indicator of irreversibility

V 3= pc0 : hydrous threshold of irreversibility

V 4=I 2 : hydrous indicator of irreversibility

V 5=PS : pressure of cohesion

8 Developmental perspectives of the model




e6ed09610822

One of the phenomena not studied in the original model of Barcelona is it not reversibility of thecapillary curve of pressure [Figure 8-a] and its dependence with the state of stress. This is treated byDangla and collar. [bib2] by integrating the model of Barcelona within a framework poroplastic with theintroduction of the water content like additional poroplastic variable, whose evolution is directlyconnected not only to the capillary variation of pressure via the curve of drainage-imbibition, but alsoto the mechanical evolution of the medium. It is necessary to distinguish two coupled distinct aspectsthere but nevertheless phenomenon. Nonthe reversibility of the curved drainage-imbibition is aphenomenon purely hydraulic and thus independent of the mechanical law adopted in a modelingTHHM, but this curve thus depends on the index of the vacuums of the mechanical state of themedium. The partition of the water content partly elastic and plastic and of the thermodynamicconsiderations [bib2] makes it possible to deduce the evolution at the same time from the water content(and thus of the degree of saturation) and from the constraint according to the deformation and thecapillary pressure. For example, the evolution in the field of reversibility is given by:

dφlqe=−N εe , pc dpcb ε e , pc dtr εe

dP=b ε e , pc dpcK εe , pc dtr ε e

lqplq

elq

cpCourbe de drainage

Courbe d’imbibition

Domaine de réversibilité

Figure 8-a

Where N ,b are the generalized coefficients of Biot [bib6]. To enrich the model by Barcelona in thisdirection thus implies two separate developments:

1) The introduction of a curve of drainage-imbibition into developments THHM.2) The completeness of the model of Barcelona by the calculation of the degree of

saturation besides the constraint.




e6ed09610822

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 forunsaturated soils: unified approach has. (1997). International Symposium one NumericalModels in Geomechanics. Pietruszczak & Pande (eds) Balkema, Rotterdam.

3) P. DELAGE: The behavior of the not-saturated grounds. (2000). ENPC. Course exemptedat DER, Clamart.

4) NR. TARDIEU, I. VAUTIER, E. LORENTZ: Quasi-static nonlinear algorithm. Referencematerial 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. 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-Mechanics THHM. General information andalogorithmes, Doc. [R7.01.10], Code_Aster (2001).

10 Checking

The law of behavior of BARCELONA is checked by the cases following tests:

WTNV123 Triaxial compression test with suction fixed with the model ofBarcelona

[V7.31.123]

WTNV124 Test of désaturation-consolidation with the model of Barcelona [V7.31.124]

WTNV126 Answer to mixed ways of saturation-consolidation with the modelof Barcelona

[V7.31.126]

11 Description of the versions of the document

Version Aster

Author (S) Organization (S)

Description of the modifications

7,4 G.Debruyne, J.El-Gharib EDF- R&D/AMA

Initial text


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