elinge, B

Accepted 18 April 2015

Keywords:Contact mechanicsInterfaces

ical

method. Fluid ow is discretised through a three-node scheme, discretising the inner ow by additional

[46]. The modelling of limit states or post-failure behaviours ofthese foundations requires specic numerical tools able to takeinto account large relative displacements between the foundationand the surrounding soil.

oes not fail evenl and the ccaisson aallenges o

production wells was given much attention [15,16]. Disturbancescreated by such a process may affect the environment in triggeringmicro-earthquakes or inducing settlements. Recently the possibil-ity of carbon dioxide geological storage in reservoirs has given anew impetus to this topic [17]. The fault opening may create aleakage path from the storage, fracture the caprock [18] or triggerearthquakes [19]. Corresponding author.

E-mail address: f.collin@ulg.ac.be (F. Collin).

Computers and Geotechnics 69 (2015) 124140

Contents lists availab

Computers and

lseto driving [13]. Soil-foundation friction is also a major componentof the resistance of anchors or pile foundations to pull loading

modelling [13,14].The behaviour of geological faults in the vicinity of hydrocarbonfor engineers.Assessing the behaviour of foundations requires a deep under-

standing of the interface mechanisms. Prediction of the frictionalstrength of a pile is crucial to estimate and model its resistance

the foundation. It also ensures the foundation dafter full mobilisation of friction between the soiCorrectly representing the large uplifting of themobilisation of friction are among the main chhttp://dx.doi.org/10.1016/j.compgeo.2015.04.0160266-352X/ 2015 Elsevier Ltd. All rights reserved.aisson.nd thef theirThe role of interfaces and discontinuities is crucial in manyelds of geotechnical engineering and engineering geology. Theycover a wide range of scales from soil-structure interaction to geo-logical faults. In all cases, the interface delineates two distinctmedia and has a very thin width with respect to them. They oftenconstitute preferential paths for uid ows, deformation and fail-ure. Therefore the modelling of their behaviour is a major issue

structures [79]. They consist of steel cylinders open towards thebottom. They are installed within the soil by suction [10,11], i.e.the water inside the caisson is pumped out creating a uid owfrom outside. This creates a differential of water pressure betweeninside and outside, digging the caisson into the soil. This suctioneffect is also mobilised during the loading of the foundation espe-cially in traction [4,12]. It increases the total transient resistance ofFinite elementsOffshore engineeringHydro-mechanical couplings

1. Introductionnodes. The element is able to reproduce the contact/loss of contact between two solids as well as shear-ing/sliding of the interface. Fluid ow through and across the interface can be modelled. Opening of a gapwithin the interface inuences the longitudinal transmissivity as well as the storage of water inside theinterface. Moreover the computation of an effective pressure within the interface, according to theTerzaghis principle creates an additional hydro-mechanical coupling. The uplifting simulation of a suc-tion caisson embedded in a soil layer illustrates the main features of the element. Friction is progressivelymobilised along the shaft of the caisson and sliding nally takes place. A gap is created below the top ofthe caisson and lled with water. It illustrates the storage capacity within the interface and the transver-sal ow. Longitudinal uid ow is highlighted between the shaft of the caisson and the soil. The uid owdepends on the opening of the gap and is related to the cubic law.

2015 Elsevier Ltd. All rights reserved.

Suction caissons or bucket foundations are a particular case ofanchors. They may be used as permanent foundations for offshoreReceived 16 January 2015Received in revised form 1 April 2015

This paper presents the formulation of a 3D fully coupled hydro-mechanical nite element of interface.The element belongs to the zero-thickness family and the contact constraint is enforced by the penaltyResearch Paper

3D zero-thickness coupled interface nitand application

B. Cerfontaine a, A.C. Dieudonn a,b, J.P. Radu a, F. ColaUniversity of Liege, Geomechanics and Engineering Geology, Chemin des chevreuils, Lieb F.R.I.A., Fonds de la Recherche Scintique FNRS, Brussels, Belgium

a r t i c l e i n f o

Article history:

a b s t r a c t

In many elds of geotechn

journal homepage: www.eelement: Formulation

a,, R. Charlier aelgium

engineering, the modelling of interfaces requires special numerical tools.

le at ScienceDirect

Geotechnics

vier .com/ locate/compgeo

Mathematical symbols

andFrom the numerical point of view, the problem of contactbetween two solids are early developed. The rst purely mechani-cal nite element of contact between two solids was early devel-oped [20]. It allows these solids to get into contact or to loosecontact during a simulation. The main concepts of this eld areestablished during the eighties [2124] and consolidate duringthe nineties [2529]. Many authors developed these elements inthe mechanical eld of research and especially metal forming[3032].

Rock and soil mechanics largely contribute to constitutive mod-

Nomenclature

Roman symbolse11; e12; e13 local system of coordinates dened on mortar sideE1;E2;E3 global system of coordinatesf wl longitudinal uid ux within the interfacef wt transversal uid ux across the interfaceFE;FI;FOB external, internal and out of balance energetically

equivalent nodal forcesgN ; _gN gap function, variation of this function_gT variation of tangential displacementJ Jacobian of the transformation from actual to isopara-

metric elementk intrinsic permeabilityK stiffness matrixKN ;KT penalty coefcientspN ;p0N contact pressure, effective contact pressureR rotation matrix_S storaget local contact stress vectorTwt transversal conductivityu vector of generalised coordinates x; y; z;pwW Gauss weight

B. Cerfontaine et al. / Computerselling of interfaces [3335]. The rst improvement is the develop-ment of non-linear mechanical constitutive laws characterisingrock joints or soil-structure interface. Criteria dening the maxi-mum friction available and stressstrain relations are developedin [34,3638]. A special attention is paid to the characterisationof shear-induced dilatancy [34,39,40]. The second improvementis the denition of experimental relations characterising the uidow within the rock joints [41,42]. Coupled nite elements com-bine these two ingredients. They include hydro-mechanical [4346] or multi-phase couplings [47]. They take into account the uidor multiphasic ow across and within the interface and its effect onthe normal pressure acting on the joint.

The purpose of this paper is to present a versatile formulation ofa fully coupled hydro-mechanical nite element of interface appli-cable to 3D simulations. It allies a mechanical large displacementformulation of a zero-thickness interface element with the mod-elling of uid ow using a three-node strategy. This strategy dis-cretises the eld of uid pressure on each side of the interfaceand inside it. Thence, the transversal uid ow creates a drop ofpressure across the interface. The element is hydro-mechanicallycoupled through the denition of an effective contact pressure,the uid storage due to the gap opening and the variation of theinterface longitudinal permeability with gap variation.

The originality lies in the coupling of the longitudinal andtransversal ows within the interface to a classical formulationof mechanical contact in large displacements. Particularly this owproblem is also tackled in case of contact loss and large tangentialdisplacements. Moreover both mechanical and ow problems aretreated within a unique nite element code LAGAMINE developedat the university of Liege [48,49]. This paper focuses on the generalframework of the nite element of interface. However the formu-lation is very versatile and any constitutive law describing bothmechanical and ow behaviours can be introduced instead of theproposed ones. An original application to the large uplift simula-tion of a suction caisson is provided to illustrate the capacities ofthe nite element of interface.

This paper is subdivided into four main parts. The rst partdescribes the basics of interface nite elements. It explains the dif-ferent ways to tackle and discretise mechanical contact and uid

$ gradient operator: tensor contraction scalar productT transpose operator1 inverse operatork k normdij Kronecker deltaGreek symbolsC1c area of contactC1~q area of the non-classical uid boundary conditiond _x virtual eld of velocitiesdpw virtual eld of uid pressures deformation tensorl friction coefcientqw uid densityr stress tensors tangential contact shear stress/in;g interpolation function related to node i, in the isopara-

metric system of coordinatesXi porous medium nb. i, solid nb. i

Geotechnics 69 (2015) 124140 125ow within interfaces. The second part sets out the governingequations of the coupled problem and its continuum formulation.The third part displays the discretisation of this continuum formu-lation into nite elements. It consists of the denition of energeti-cally equivalent nodal forces and stiffness matrix. Finally the lastpart describes the pull simulation of a suction caisson embeddedin a soil layer. This application illustrates all the features of theinterface element.

2. Review of interface nite elements

Coupled interface elements involve two distinct but relatedissues: the mechanical and the ow problems. The formerdescribes the detection or the loss of contact between two bodies,the shearing of this contact zone. . . The ow problem describes theuid ow within the interface created by the vicinity of uid owswithin porous media. These two problems are coupled since theuid ow inuences the opening of the discontinuity and its trans-missivity. Moreover the uid ow across the interface creates atransversal drop of pressure between two porous media.

Numerically, two approaches exist within the framework of thenite element method to manage the mechanical contact betweentwo bodies as shown in Fig. 1. In the former approach, the interfacezone is represented by a very thin layer of elements speciallydesigned for large shear deformation [5052]. The secondapproach, adopted in the following, involves special boundary ele-ments. These elements have no thickness and are termedzero-thickness nite elements. They discretise the probable zone

The penalty method [32] regularises the constraint by authoris-ing an interpenetration of the solids in contact independently onthe roughness of the surfaces. The related pressure is a functionof the interpenetration through the penalty coefcient. Thereforethe stress-displacement relation looses its physical basements[54]. Both Lagrangian and penalty solutions are identical for in-nite penalty coefcient [55]. The main advantage is the simplicityof the method. The inconvenient is the risk of ill conditioning of thestiffness matrix. Both techniques are compared in Fig. 3.

The available maximum friction may also evolve with the rela-tive tangential displacement. In this case, a constitutive law ruling

-thickness approaches in case of Hertzian contact.

126 B. Cerfontaine et al. / Computers and Geotechnics 69 (2015) 124140of contact and are activated only in that case. These elements aresuitable for the modelling of large displacement and no remeshingtechnique is necessary. They are quite common in mechanics[20,30,35,32,53].

Basically three ingredients are necessary to develop such anapproach

a scheme to enforce the normal contact constraint; a technique to discretise the contact area between solids and tocompute a gap function gN;

a constitutive law to rule the normal/tangential behaviour.

The normal contact constraint ensures two solids in contact can-not overlap each other, the gap function is null, i.e. gN 0. Thiscontact gives birth to normal pressure on each side of the interfacepN and both solids deform. A physical constitutive law can rule thisnormal behaviour. The macroscopic relation between normalstress and deformation of the contact area depends on the micro-scopic geometry. For instance, in rock mechanics, thestress-displacement relation is non-linear [36] and depends onthe deforming asperities as shown in Fig. 2. In such a case, theinterpenetration of the solids in contact have a physical meaning,i.e. gN < 0.

On the other hand, the normal constraint condition can beensured on a purely geometrical basis, namely the interpenetrationof the two solids is not allowed. This is physical only in case of per-

Fig. 1. Comparison between thin layer and zerofectly smooth surfaces. The Lagrange multiplier method exactlyensures this condition [53]. It introduces additional variables, theLagrange multipliers, corresponding to the contact pressures.

Fig. 2. Constitutive law describing the normal behaviour of a rough rock joint. Normal pressure pN depends on the deformation of asperities and closing of the gap gN .

Fig. 3. Comparison of Lagrange multiplier and penalty methods on deformation anddistribution of contact pressures.

friction angle within the interface is also necessary. Dilatancy ofthe interface is also a crucial issue. This was extensively studiedin case of rock joints [56,57] and soil-structure interfaces [39,40].

The contact constraint is a continuous condition over theboundary. Its discretisation in nite elements strongly impactsthe performance of the computation. The node-to-node discretisa-tion is the simplest one [23], as it is described in Fig. 4. In this case,the contact constraint is imposed on a nodal basis. The gap andcontact forces are computed between each pair of nodes. This for-mulation is dedicated to small relative displacements only.

The node-to-segment discretisation overcomes this drawback[58]. The contact constraint is applied between the nodes of oneside of the interface, termed slave surface and the segments ofthe other side, termedmaster surface. The gap function is computedthrough the projection of the slave node onto the master surface.Such discretisation is sensitive to sudden change in projectiondirection between two adjacent segments and is improved bysmoothing techniques [29].

The segment-to-segment discretisation [32,59,60] is based onthe mortar method developed in [61]. In this case, the contact con-straint is applied in a weak sense over the element. The gap func-tion is computed through the closest-point projection of a point ofthe non-mortar surface onto the mortar one which is given moreimportance. It is extrapolated over the element by the means ofinterpolation functions.

Finally, the contact domain discretisation does not involve anyprojection method [62,63]. The gap between the solids potentiallyin contact is discretised by a ctitious mesh. Thence the gap func-tion is continuous between them and avoids many discrepanciesand loss of unicity due to projection.

If the interface represents a discontinuity saturated with a uid,several additional ingredients are necessary:

a technique to discretise the ow within and through theinterface;

a law relating the ow to the gradient of pressure.

The single node discretisation of ow is the simplest one asshown Fig. 5. It simply superposes a discontinuity for uid owto a continuous porous medium [64]. In this case, there is nohydro-mechanical coupling and the opening of the discontinuityis constant and user-dened. It acts such as a pipe creating a pref-erential path for uid ow.

B. Cerfontaine et al. / Computers and Geotechnics 69 (2015) 124140 127Fig. 4. Comparison between the discretisation methods of the contact area.Fig. 5. Comparison between the discretisation methoThe double-node discretisation describes the uid ow withinthe interface as a function of the gradients pressure of each sideof the interface [43,16,65,66]. There is an hydro-mechanical cou-pling since the discontinuity is able to open. The ow throughthe interface depends on a transversal transmissivity and the gra-dient of pressure across the interface.

Another option is to discretise the eld of uid pressure insidethe interface by additional nodes. This method is termedtriple-node discretisation [16,47]. The underlying hypothesis is thatthe eld of pressure is homogeneous inside the interface. Howeverthere is a drop of pressure across the interface, between the twosolids in contact.

Boussinesq [67] rstly provides a mathematical law character-ising the laminar ow of a viscous incompressible uid betweentwo smooth parallel plates. The total uid ow is proven to be pro-portional to the cube of the aperture between the plates, and thisrelation is termed cubic law. In this case, the longitudinal perme-ability of the fault is a function of its opening gN

kl gN 2

12: 1

Its applicability to rock mechanics is proven [68,42,69] despiteimprovements are necessary due to the underlying strong hypoth-esis. The non smoothness of the rock edges of the interface is takeninto account by considering an hydraulic aperture rather than amechanical one [41].ds of the ow within and through the interface.

3. Governing equations of the interface problem

The developed nite element of interface is zero-thicknesswhich is more suitable for large displacements. It does not involveany remeshing technique. The contact constraint is enforced by apenalty method. Indeed, this approach is easy to implement andadditional unknowns are not required. Furthermore, the imple-mentation is based on an analogy with elastoplasticity. It is veryexible and complex constitutive laws can be introduced instead.The uid ow within and across the interface is discretised using

used. In case of contact, the relation between the pressure and the

128 B. Cerfontaine et al. / Computers and3.1.1. Denition of the mechanical problem and gap function

Let us consider two deformable porous mediaX1 and X2 in theircurrent congurations at time t. The global system of coordinates istermed E1;E2;E3. A 2D cross section of these bodies is illustratedin Fig. 6. Their evolution is assumed to be quasi-static. Theirboundaries in current congurations are denoted C1 and C2.Imposed displacement (Dirichlet) and traction (Neumann) bound-aries are respectively denoted Ciu and C

it .

C1c and C2c denote both parts of the boundary where contact is

likely to happen. In that area, a local system of coordinatee11; e12; e13 is dened along the mortar side C1c as shown in Fig. 6,where e11 denotes the normal to the surface. The closest point pro-

jection x1 of a point of x2 of the boundary C2c onto C1c is dened

such that [53]

gN x2 x1 e11; 2

where e11; e12; e13 denotes the local system of coordinates at pointx1. This function gN is referred as the gap function, where the sub-script N stands for normal direction. If there is no contact betweenthe solids, gN is positive. The contact is termed ideal if there is nointerpenetration of the solids. For instance in Hertzian contact[70], the gap function is equal to zero. This can be enforced if theLagrange multiplier method is used. If the penalty method isemployed, interpenetration is necessary to generate contact pres-sure and the gap function becomes negative.

The denition of a relative tangential displacement betweentwo points in the plane of contact has no meaning in the eld oflarge displacement [32]. Instead normal (N) and tangential (T1and T2) velocities are dened in the local system of coordinates.They are gathered into the vector _g such that

_g _gN e11 _gT1 e12 _gT2 e13: 3

3.1.2. Normal contact constraintContact between two solids gives birth to non-zero stress vec-

tors t1 t2 along their common boundary as shown in Fig. 6.a three-node approach taking easily into account the storage andlongitudinal ow.

3.1. Mechanical problemFig. 6. Statement of the mechanical problem, cross-section of the 3D problem in theE1;E2 plane.gap function reads

_pN KN _gN ; 6where the minus sign ensures the contact pressure is positive wheninterpenetration increases, i.e. gN < 0 and _gN < 0.

3.1.3. Tangential contact constraintWhen solids are in contact, the ideal tangential behaviour of the

interface distinguishes between the stick and slip states [53]. In theformer state, two points in contact cannot move tangentially. Theykeep stuck together during the simulation, i.e. _gT1 0 and _gT2 0.The second state involves a relative tangential displacement in theplane of the interface. This is summarised in a condition similar toSections 3.1.2 and 3.1.4

_gslTi P 0; f t;q 6 0 and _gslTi f t;q 0 i 1;2 7

where _gslTi is the variation of the non-recoverable displacement ineach tangential direction. It is related to the variation of tangentialdisplacement

_gT sign _s1 _gslT1 e12 sign _s2 _gslT2 e13: 8

Stick and slip states are distinguished by the criterion f t;q. Itdepends on the stress state t and a set of internal variables q.The evolution of the stress state within the interface depends onthe constitutive law described hereafter.

The ideal stick state, _gT 0, is also regularised by the penaltymethod, i.e. a relative displacement is allowed. Thence the relationbetween the shear stress and the tangential variation of displace-ment reads

_si KT _gTi i 1;2: 9

3.1.4. Constitutive lawIt is shown that both rock joints and soil-structure interfaces

present a very complex mechanical behaviour [7173,57] inducingdilatancy, degradation of the friction angle, critical state. . . Thispaper focuses on the general formulation of the coupled nite ele-ment of interface. Therefore the constitutive law is kept as basic aspossible in order to highlight the couplings inherent to the formu-lation. The MohrCoulomb criterion is adopted for that purpose.However interested reader should refer to [74,38,40] for a deeperinsight into more accurate constitutive laws.

The constitutive law adopted only depends on the stress state twithin the interface and a single internal variable, the friction coef-cient l. Mathematically it readsqThese vectors are described in the corresponding local system ofcoordinates at each contacting point such that

t1 pN e11 s1 e12 s2 e13; 4where pN is the normal pressure, s1 and s2 are the shear stresses inboth directions in the plane of the interface. The ideal contact con-straint is summarised into the Hertz-Signorini-Moreau condition[53],

gN P 0; pN P 0 and pN gN 0: 5If there is no contact, the gap function gN is positive and the contactpressure pN is null. When contact arises, the gap function is null andthe contact pressure is positive.

This condition is not veried any more if the penalty method is

Geotechnics 69 (2015) 124140f t;l s1 2 s2 2|{z}ksk

lpN: 10

3.2. Flow problem

3.2.1. Denition of the problemLet us consider a discontinuity of very thin width embedded in

a porous medium in its current conguration, as depicted in Fig. 8.This could represent for example an open fault within a rock mass.This discontinuity creates a preferential path for uid ow.Moreover there is a transversal uid ow between the rock massand the discontinuity.

There is a conceptual difference between the treatment of themechanical and ow contact problems. The mechanical contactconstraint consists of a non-zero pressure pN applied along thecontact zone Cc between the two solids X

1 and X2.On the other hand, the opening of the discontinuity creates a

gap gN lled with water. This gap creates a new volume X3 in

which uid ow takes place, as shown in Fig. 9. It is bounded bythe two porous media X1 and X2. Their boundary are termed C1~q

tates through the Mohr-Coulomb criterion.

and Geotechnics 69 (2015) 124140 129where ksk is the norm of the tangential stresses. The criterion isrepresented in Fig. 7b. In the absence of contact, the stress state lieson the apex of the criterion. Both normal pressure and tangentialstresses are null, i.e. t 0. If the combination of tangential and nor-mal stresses lies below the criterion (f < 0), the tangential state isconsidered stick. Otherwise, if the stress state lies on the criterion(f 0), the tangential state is considered as slip.

The evolution of the stresses lies within the framework ofelastoplasticity. Indeed the stick state is regularised and can becompared to an elastic state. Therefore the incremental relationbetween variations of stresses _t and variations of the gap function_g reads

_pN_s1_s2

264

375 KN 0 00 KT 0

0 0 KT

264

375

|{z}De

_gN_gT;1_gT;2

264

375; 11

where De is equivalent to the elastic compliance tensor. In this case,the penalty coefcients introduced on a purely numerical basis arecompared to elastic coefcients which are physical. When the inter-face reaches the slip state, an elastoplastic compliance tensor Dep isdened such that

_pN_s1_s2

264

375

KN 0 0lKN s1ksk KT 1 s1

2

ksk2

KT s1 s2ksk2lKN s2ksk KT s1 s2ksk2 KT 1

s2 2ksk2

26664

37775

|{z}Dep

_gN_gT;1_gT;2

264

375:

12This tensor was introduced in [30] and is based on a non-associatedow rule.

3.1.5. Continuum formulation

Each solid Xi veries the classic mechanical equilibrium equa-

Fig. 7. Differentiation of stick and slip s

B. Cerfontaine et al. / Computerstions in quasi-static conditions [75]. Solving the mechanical con-tact problem consists in nding the eld of displacement u for

all points x 2 Xi verifying these equations and subjected to thecontact constraints Eqs. (5) and (7).

Considering a eld of admissible virtual velocities d _x on Xi, theweak form of the principle of virtual power reads

X2i1

ZXir : d _x dX

X2i1

ZXi

f : d _xdXZCit

t : d _xdXZCic

Ti : d _xdC

" #;

13

where f are the body forces, u are the imposed displacements, t are

the imposed tractions, n is the normal to Cit and Ti is the projection

of the local stress tensor ti in global coordinates. The equality of Eq.(13) is enforced when the contact area Cic is known.Fig. 8. Denition of the ow problem (cross section of the 3D case in the E1;E2plane), porous medium, discontinuity and boundaries.Fig. 9. Denition of the equivalent interior porous medium X3 bounded by C1~q andC2~q .

and C2~q . Therefore C~q represents a boundary where the solids areclose enough, uid interaction hold and mechanical contact islikely to happen. It always includes the contact zone Cc .

X3 is modelled as an equivalent porous medium. The uid owwithin it is described by the cubic law. Fluid ows exist betweenthe inner volume X3 and both adjacent porous media X1 and X2.This ow is a function of the difference of pressure between them.This is a non-classical boundary condition since it is not a imposedux nor an imposed pressure.

Finally imposed ux and pressure boundaries on X1 and X2 arerespectively denoted Ciq and C

ipw .

two longitudinal uxes (f and f ) in the local tangential

~q

130 B. Cerfontaine et al. / Computers andwl1 wl2

directions (e12; e13) in the plane of the interface;

two transversal uxes (f wt1 and f wt2) in the local normal direc-tion (e11).

The generalised Darcys law is assumed to reproduce the locallongitudinal uid ows fwl1 and fwl2 in the plane of the interface.It reads in each local tangential direction e12; e13,

fwli1 kllwre1

ipw3 qw gre1i z

qw for i 2;3 14

where re1iis the gradient in the direction e1i ;lw is the dynamic vis-

cosity of the uid, g the acceleration of gravity, qw is the density ofthe uid and kl is the permeability.

Each transversal uid ux is a function of a transversal conduc-tivity Twi and the drop of pressure across C

i~q. They read

fwt1 qw Tw1 pw1 pw3 on C1~q ; 15fwt2 qw Tw2 pw3 pw2 on C2~q : 16

3.2.3. Continuum formulation

Each porous medium Xi i 1;2;3 veries the classic hydraulicequilibrium equations [76]. Solving the contact problem consists

in nding the pore water distribution on Xi verifying the equilib-rium equations and satisfying the non-classical boundary condi-tions Eqs. (15) and (16) over Ci . Considering a eld of admissible3.2.2. Fluid ow formulationA three-node formulation is adopted to describe the uid ow

through and within the interface, as described in Fig. 10.Therefore uid pressures on each side of the interface (pw1 andpw2) and inner uid pressure (pw3) are the uid variables. At eachpoint within the interface, four uxes are denedFig. 10. Denition of longitudinal and transversal ows.virtual pore water pressures dpw on X, the weak formulation ofthe virtual power principle reads

X3i1

ZXi

_Sdpw fw $ dpw dX

X3i1

ZXi

Q dpw dXZCiq

qdpw dCZCi~q

~qdpw dC

" #17

where fw is the uid ux at point x; _S is the storage term, Q is theimposed volume source, pw is the imposed uid pressure, i 1;2corresponds to the two porous media in contact and i 3 to the vol-ume of the interface. The uid ow ~q along the boundary corre-sponds to the transversal uid ows fwti dened in Eqs. (15) and(16). The source term Q associated to X3 is null.

The mechanical problem was given more importance to themortar side C1c . Similarly, the integral over X

3 is transformed into

a surface integral over C1~q. This hypothesis is valid since it isassumed the inner pressure is constant over the aperture gN ofthe interface. Thence, Eq. (17) for i 3 nally readsZC1~q

_Sdpw fwl1re12 dpw fwl2re13 dpw h i

gN dC

ZC1~q

qw Tw1 pw1 pw3 dpw qw Tw2 pw3 pw2 dpw dC; 18

where re1iis the gradient in the e1i direction.

In the porous media X1 and X2, the storage component _S is cou-pled with the deformation of the solid skeleton. The treatment ofthis component for X3 is different and treated hereafter.

3.3. Couplings between mechanical and ow problems

The ow problem within the interface intrinsically depends onthe mechanical problem. The gap function gN dened in themechanical problem directly inuences total uid ow withinthe interface since the cubic law is related to the mechanical open-ing gN .

However, it is worth noting hydraulic and mechanical aperturesshould sometimes be differentiated. If two perfectly smooth platesare in ideal contact, the gap function gN is equal to zero betweenthem. Thence the uid ow is null since the permeability is equalto zero. However, if the surfaces are rough, a uid ow is still pos-sible even if the solids are in contact. A residual hydraulic apertureD0 is considered. Hence, the permeability is computed according to[41,16]

kl D0 212 if gN 6 0D0gN 2

12 otherwise:

8