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Digital Object Identiﬁer (DOI) 10.1007/s001610100053

Continuum Mech. Thermodyn. (2001) 13: 287–306

A solid-ﬂuid mixture model allowing for solid dilatation

under external pressure

Giulio Sciarra1, Francesco dell’Isola1, Kolumban Hutter2

1Dipartimento di Ingegneria Strutturale e Geotecnica, Universit`

a di Roma La Sapienza Via Eudossiana 18, 00184 Roma, Italia

2Department of Mechanics, Darmstadt University of Technology, 64289 Darmstadt, Germany

Received January 26, 2000 / Published online August 16, 2001 – c

Springer-Verlag 2001

A sponge subjected to an increase of the outside ﬂuid pressure expands its volume but nearly

mantains its true density and thus gives way to an increase of the interstitial volume. This

behaviour, not yet properly described by solid-ﬂuid mixture theories, is studied here by using

the Principle of Virtual Power with the most simple dependence of the free energy as a

function of the partial apparent densities of the solid and the ﬂuid. The model is capable of

accounting for the above mentioned dilatational behaviour, but in order to isolate its essential

features more clearly we compromise on the other aspects of deformation. Speciﬁcally, the

following questions are addressed: (i) The boundary pressure is divided between the solid and

ﬂuid pressures with a dividing coefﬁcient which depends on the constituent apparent densities

regarded as state parameters. The work performed by these tractions should vanish in any cyclic

process over this parameter space. This condition severely restricts the permissible constitutive

relations for the dividing coefﬁcient, which results to be characterized by a single material

parameter. (ii) A stability analysis is performed for homogeneous, pressurized reference states

of the mixture by postulating a quadratic form for the free energy and using the afore mentioned

permissible constitutive relations. It is shown that such reference states become always unstable

if only the external pressure is sufﬁciently large, but the exact value depends on the interaction

terms in the free energy. The larger this interaction is, the smaller will be the critical (smallest

unstable) external pressure. (iii) It will be shown that within the stable regime of behaviour an

increase of the external pressure will lead to a decrease of the solid density and correspondingly

an increase of the speciﬁc volume, thus proving the wanted dilatation effects. (iv) We close by

presenting a formulation of mixture theory involving second gradients of the displacement as a

further deformation measure (Germain 1973); this allows for the regularization of the otherwise

singular boundary effects (dell’Isola and Hutter 1998, dell’Isola, Hutter and Guarascio 1999).

1 Introduction

In many engineering applications of binary mixture models of solid-ﬂuid interactions the pore space or the

permeabilities are prescribed functions of the spatial coordinates, but they do not evolve together with the

temporal changes of the other ﬁeld variables: the engineering models based on this assumption stem from

the studies of Terzaghi [23]. One example to the contrary is for instance the slow creeping deformation and

the percolation of brine through a salt formation from a pressurized cavern ﬁlled with a liquid (for a detailed

discussion of these phenomena see for instance [5]). Observations indicate that an increase in cavern pressure

will not only result in a very slow creeping deformation of the salt dome but equally enhance the diffusion

of liquid through the salt. This increase in percolation is not only due to an increase of the partial pressure

288 G. Sciarra et al.

of the liquid as a reaction to an increase of the cavern pressure but also because of grain boundary openings

in the vicinity of the cavern walls.

Many of the presently existing mixture theories treating solid-ﬂuid interactions may in principle be able

to cope with the ﬂuid dilatation mechanism: Bowen [1], [2], M¨

uller [16], Morland [15], Rajagopal & Tao

[17], Svensdsen & Hutter [22]; scrutiny has shown, however, that very particular constitutive behaviour must

be assumed to achieve it. Terzaghi and Fillunger were aware that their models were unable to describe this

dilatancy (see for instance de Boer & Ehlers [7]), Svendsen & Hutter’s [22] mixture theory allows for it,

but as shown by [10],[11] rather singular functional forms of the thermodynamic free energies ψ(·,n)as

functions of the porosity nare required if an appreciable space opening of the pores is to be achieved. It

was then thought that introducing density gradients as independent constitutive variables would regularize

the formulation [9], and indeed it did so.

However, in the problem stated above and in the mentioned papers we were confronted with a further

difﬁculty: the ﬂux boundary conditions between the single constituent body and the mixture. As shown by

Hutter et al. [14] the jump condition of momentum requires non-vanishing momentum production terms

to exist on the singular surface (see also Svendsen & Gray [21]). An alternative, simpler than this, and

not requiring surface balances is to postulate a parametrization how the traction on the single-constituent

side of the surface is distributed between the tractions on the mixture-side [18]. This parametrization can

be expressed as a scalar parameter (akin to surface fraction) depending on a number of variables, say the

constituent densities. Obviously, the parametrization must be such that the work done by these boundary

tractions in a simply connected closed circuit in this parameter space is zero for otherwise a perpetuum

mobile of the ﬁrst kind would result (Seppecher, personal communication). The construction of the potential

for the boundary tractions restricts the parametrization in our case to the extent that the functional dependence

is ﬁxed except for a single constant.

A mixture of two constituents one of which is a ﬂuid can only exist in equilibrium when it is conﬁned,

i.e., when a pre-stress is exerted on it; this means that reference states with non-vanishing ﬂuid densities

are always pre-stressed in such media. This fact gives rise to the question how the free energy describing

the interior behaviour of the mixture must be structured that perturbations about such pre-stressed states are

stable. For a quadratic dependence of the free energy upon the constituent densities the analysis shows that the

stability condition depends strongly on the coupling term involving the two densities. For isotropic stresses

and a homogeneous reference state we will prove that there is always an upper bound of the external pressure

beyond which such states become unstable, corroborating Fillunger and Terzaghi’s explosion under pressure

(de Boer [8]).

One of the signiﬁcant results determined in this paper is the fact that, depending on the coefﬁcients of

the parabolic representation of the free energy, a stable perturbation of a homogeneous pre-stressed reference

state can give rise to a decrease in the apparent solid density with an increase of the external pressures. If

the true density of the ﬂuid is essentially constant this corresponds to a dilatation of the space occupied by

the ﬂuid. This is essentially how a sponge responds to the absorption of water from the outside; however,

this property is exactly what is needed to achieve an increase of permeability without phase changes of the

salt in the salt cavern problem mentioned at the beginning. The spatially dependent pore opening close to

the cavern space is manifest as a boundary layer (in what is called Disturbed Rock Zone) and for this reason

we close this paper with a presentation of the higher gradient model corresponding to the analysis of the

earlier sections. A one-dimensional linear-elastic problem is ﬁnally presented as a ﬁrst application of the

newly introduced model: we prove that close to unstable pre-stressed reference conﬁgurations the thickness

of the boundary layer (where the apparent mass densities are not constant) at the external interface of the

solid-ﬂuid mixture tends to inﬁnity. The regularizing properties of the proposed second gradient model are

thus established.

The ultimate problem will have to incorporate visco-plastic components to also account for the creeping

deformation of the salt (see Cosenza [5]).

A solid-ﬂuid mixture model 289

2 Theoretical concepts

Consider a binary mixture of a solid matrix with connected pores which are ﬁlled with a liquid. This ar-

rangement can be thought of as a soil, rock or sponge. Let the two components be referred to as the solid

and the ﬂuid and indicate them by the sufﬁces sand f.Let, moreover ρs,ρ

fand vs,vfbe the solid and the

ﬂuid apparent densities and velocities, respectively, in the mixture. The mixture density and the barycentric

velocity are then given by

ρ=ρs+ρf⇒1=ρs

ρ+ρf

ρ=: ξs+ξf,(1)

v=ξsvs+ξfvf,(2)

in which ξsand ξfare the mass fractions of the solid and the ﬂuid, respectively.

We conceive this mixture to be non-reactive so that the balances of mass for the constituents reduce to

∂ρa

∂t+div (ρava)=0,(a=s,f).(3)

In the ensuing analysis we shall restrict ourselves to purely mechanical processes; temperature will play no

role, and so the constituent momentum equations are the only additional dynamical equations to be added

to (3). Instead of a direct application of these laws we shall use the Principle of Virtual Power (PVP)

applied to the appropriate energy functional to derive them. Let ψint be the energy volume density and, as we

limit ourselves to spherical states, assume it to be depending on the densities ρa:ψint =ψint (ρs,ρ

f).Other

dependencies could be incorporated, but will not be here for simplicity. The PVP states that the variation of

the total energy in the considered body (a mixture) related to its (barycentric) motion equals the power of the

external forces. If the exterior of the mixture body is a ﬂuid then the boundary traction exerted on the mixture

is a pressure, pext which must be distributed between the constituents via the phenomenological ansatz

pf=dfpext ,ps=dspext ,ds+df=1,(4)

in which dsand dfare surface fraction parameters which, like the energy itself, depend upon the densities ρs

and ρf.This parametrization is of constitutive nature and has been proposed already in the past (Morland [15],

Rajagopal & Tao [17]). Generally the areal fractions are identiﬁed with the volume fractions. This identiﬁcation

traces back in mining engineering to the law of Delesse (1848), as quoted by de Boer [8]. In this paper we

demonstrate the interrelation of the two via an argument concerning the ﬁrst law of therodynamics. Thus the

PVP yields the statement

d

dt

B

ψint dV =−

∂B

dapext n·vadA,(5)

where summation over doubly repeated indices ais understood and ndenotes the outward pointing unit

normal to ∂Bwhich is the boundary of the Eulerean region B, the actual placement of the mixture body.

We assume the external pressure pext to be a conservative ﬁeld. This means that the work performed by

pext on a cyclic quasi-static variation of the state parameters ρsand ρfmust be path independent for otherwise

a perpetuum mobile of the ﬁrst kind would emerge. This is avoided if a potential ψext exists such that1

d

dt

B

ψext dV =−

∂B

dapext n·vadA.(6)

Employing the Reynolds Transport Theorem on the LHS, the divergence theorem on the RHS of (6) and

using (3) yields

∂B−ρa∂ψext

∂ρa+ξaψext +dapextn·vadA +

B

ρa∇∂ψext

∂ρa·vadV =0.(7)

1We are grateful to P. Seppecher for drawing our attention to this fact.

290 G. Sciarra et al.

Since (7) must hold for all velocity ﬁelds va,using a localization procedure the following restrictions on

constitutive equations emerge:

∂ψext

∂ρa=1

ρψext +da

ρapext ,(8)

ρa∇∂ψext

∂ρa=0 =⇒∂ψext

∂ρa=pextca,(9)

where pext cαare constants and pext is inserted for convenience. Each of these statements constitutes two

equations and from their exploitation ψext and dacan be determined. They can be regarded as a vectorial

statement in the 2D state-variable space spanned by ρsand ρf.The second one implies that (cs,cfand c

being integration constants)

ψext (ρs,ρ

f)=pext csρs+cfρf+c,(10)

so that using this in (8) we obtain

ds=ξs+ρs

[ρ](1−ξs)=ξs1+ ρf

[ρ],c=−1,(11)

where

cs−cf=: 1

[ρ].(12)

Let us pause to interpret this result: the requirement that the external pressure does not perform work

along a closed trajectory in the state space ρs,ρ

fhas led to a restriction on the constitutive equations for

ds(and df), which divide the external pressure pext between the partial pressures psand pf,respectively. The

division pa=ξapext is an obvious and allowable possibility ([ρ]→∞), but it is not exhaustive. Any choice

obeying (8) and (9) must have the form (11) in which the quantity ρs/[ρ]is a scale parameter controlling

how the boundary pressure is divided between the constituents.

Recall that 0 <ds<1 and 0 <df<1 which imply

−1

ρf<1

[ρ]<1

ρs.(13)

Evidently the typical scale parameter for the division of the pressure pext between the constituent pressures

involves an inverse density.Obviously the mixture density in the reference conﬁguration is a permissible

choice for [ρ], in principle it may be inﬁnitely large; dsthen lies within the interval 0 <ds<1 and is a

linear function of ξsbetween these two values.

Observe that the condition da= 0 implies that either ρa=0orρ−ρa=[ρ]. The second of these conditions

is physically impossible as both ρaand ρmust be considered as variables. This means that when one part of

the external pressure pext acting on the ﬂuid (or on the solid) is equal to zero then the ﬂuid-solid mixture at

the boundary reduces to the solid skeleton (or to a pure ﬂuid) only.

3 Linearization in the neighbourhood of a pre-stressed reference state

Let B0be the mixture body in its reference state with positions labeled by X.Constituent densities and the

external pressure in this state are denoted by ρ0

a(a=s,f) and pext

0, respectively; they will be assumed to be

different from zero, and thus will give rise to constituent stress distributions p0

a.Such a nontrivial pre-stressed

situation is a necessary requirement for the mixture material introduced in Sect.2. Indeed, the model is only

meaningful when ρs>0 and ρf>0; thus ds>0 and df>0,and this necessarily implies, through the

parametrization of the constituent pressures that p0

s= 0 and p0

f=0,if pext

0=0.For the ﬂuid to exist at all, a

nonvanishing pressure is mandatory. So only nontrivial pre-stressed conditions are possible.

A solid-ﬂuid mixture model 291

Consider a pre-stressed reference conﬁguration of the solid and suppose the displacement of the material

particles of the solid constituent to be small so that geometric linearizations are justiﬁed throughout. Reference

values of ﬁeld quantities will carry the sufﬁx (·)0, perturbed ﬁelds will be denoted by a tilde,

(·); therefore,

in view of (10) we have

ψext =pext

0csρ0

s+cfρ0

f−1

+pext

0cs˜ρs+cf˜ρf+˜pext csρ0

s+cfρ0

f−1(14)

+˜pext cs˜ρs+cf˜ρf.

Alternatively, we express the potential ψint as a quadratic form about the reference state ρ0

s,ρ

0

f; this will be

done in the form ψint =ρϕint , in which ϕint is the internal energy per unit mass. What obtains is as follows:

ψint =ρϕint ρs,ρ

f

(ρ0+˜ρ)γs˜ρs+γf˜ρf+1

2γss ˜ρ2

s+γsf ˜ρs˜ρf+1

2γff ˜ρ2

f+...(15)

=ρ0γs˜ρs+γf˜ρf+1

2(2γs+ρ0γss )˜ρ2

s+1

22γf+ρ0γff ˜ρ2

f

+γs+γf+ρ0γsf ˜ρs˜ρf,

where an arbitrary constant in ϕint is irrelevant and where the constants

γs:= ∂ϕint

∂ρsρ0

s,ρ0

f

,γ

f:= ∂ϕint

∂ρfρ0

s,ρ0

f

,

γss := ∂2ϕint

∂ρ2

sρ0

s,ρ0

f

,γ

ff := ∂2ϕint

∂ρ2

fρ0

s,ρ0

f

,γ

sf := ∂2ϕint

∂ρs∂ρfρ0

s,ρ0

f

(16)

are supposed known when the reference state is known. Depending upon the value of γsf relative to the

values of γss ,γ

ff coupling will be called weak or strong. Generally growing γsf increases this coupling.

Next, we wish to write down the equilibrium equations which follow from the localization of

d

dt

B

ψtot dV =d

dt

Bψint −ψext dV =0.(17)

For static conditions this yields

ρs∂ψint

∂˜ρs−ξsψint =ρs∂ψext

∂˜ρs−ξsψext ,

ρf∂ψint

∂˜ρf−ξfψint =ρf∂ψext

∂˜ρf−ξfψext .

(18)

Using (14) and (15) and expanding as illustrated above yields the following zeroth and ﬁrst order problems:

zeroth order

γs=pext

0

ρ2

0+ρ0

f

ρ2

0cs−cfpext

0,γ

f=pext

0

ρ2

0−ρ0

s

ρ2

0cs−cfpext

0.(19)

ﬁrst order

R11 R12

R21 R22 ˜ρs

˜ρf=

ξ0

s+ρ0

sρ0

f

ρ0cs−cf

ξ0

f−ρ0

sρ0

f

ρ0cs−cf

˜pext ,(20)

292 G. Sciarra et al.

where

R11 := ρ0

s(2γs+ρ0γss )+ρ0

fγs−pext

0ξ0

f2cs−cf+ξ0

f

ρ0,

R12 := ρ0

sγs+γf+ρ0γsf −ρ0

sγf−pext

0ξ0

s2cs−cf−ξ0

s

ρ0,

R21 := ρ0

fγs+γf+ρ0γsf −ρ0

fγs+pext

0ξ0

f2cs−cf+ξ0

f

ρ0,

R22 := ρ0

f2γf+ρ0γff +ρ0

sγf+pext

0ξ0

s2cs−cf−ξ0

s

ρ0.

(21)

Equations (19) relate γsand γfto one another. On the other hand, (20) could be inverted to obtain ˜ρsand ˜ρf

in terms of ˜pext

˜ρs

˜ρf=[R]−1

ξ0

s+ρ0

sρ0

f

ρ0cs−cf

ξ0

f−ρ0

sρ0

f

ρ0cs−cf

˜pext ,(22)

however, this inversion is only possible if the matrix [R]is invertible. We will show later that conditions of

invertibility agree with the requirements of the stability of the pre-stressed reference state. This is the problem

we shall address now.

4 Linear stability analysis of pre-stressed reference states

The scope of this section is not to embed the equilibrium properties of our system into a full non linear

dynamic stability analysis, we simply wish to know whether a particular given reference state is stable with

respect to small perturbations. In this spirit, the reference states described by the density ﬁelds ρ0

s,ρ

0

fand

the pressure pext

0will now be assumed to be spatially uniform. Our interest is in their stability against linear

perturbations of the densities and the pressures. It is expected that such stability properties depend on the

functional form of the total energy

ψtot =ψint −ψext (23)

and its properties around an equilibrium state. In a static situation stability of the equilibrium state requires the

function ψtot to assume its minimum in equilibrium (Dirichlet criterion), so that the Hessian matrix Hψtot

is positive deﬁnite in a neighbourhood of the equilibrium,

H=2γs+ρ0γss γs+γf+ρ0γsf

γs+γf+ρ0γsf 2γf+ρ0γff ,(24)

implying the Rouse-Hurwitz criteria

2γs+ρ0γss >0,detH>0.(25)

With the help of (19)1the ﬁrst inequality implies

2pext

0

ρ2

0+ρ0

f

ρ2

0cs−cf

>−1/ρ0

f

pext

0+ρ0γss >0,

requiring at worst that γss >0. We will therefore suppose that γss >0 for all cases. The second inequality

for stability can be written as

−ρ2

0cs−cf2pext

02+2ρ3

0βpext

0+ρ6

0γss γff −γ2

sf >0,(26)

A solid-ﬂuid mixture model 293

in which

β:= β0+cs−cfβ1,

β0:= γss +γff −2γsf ,(27)

β1:= ρ0

sγsf −γss +ρ0

fγff −γsf .

The LHS of (26) is a quadratic form P2pext

0.It represents a set of parabolas (see Fig. 1)2with a positive value

at the vertex and which are open in the downward direction; the two solutions of the equation P2pext

0∗=0,

pext

0∗=−1

cs−cf2−ρ0β±ρ2

0β2+ρ4

0cs−cf2γss γff −γ2

sf ,(28)

are positive and negative irrespective of whether β>0orβ<0.The stability region is pext

0<pext

0C,

where pext

0Cis the root of (28) with the negative square root sign (as the other root is negative for all

admissible choices of the involved parameters).

Fig. 1. Plot of P2(p∗)= 0 for a value of α=±1. Two parabolas are obtained if cs−cf=0.The parabolas have their vertices at

p∗=1/2αand are open for negative values of p∗.If p∗>p∗cthen the static equilibrium of the basic state is unstable. For cs−cf=0

the two parabolas reduce to the same pair of straight lines. The one with positive slope has an unbounded stability limit, while that with

negative slope has a ﬁnite stability limit. These limits are denoted by p∗c

When cs−cfis zero or [ρ]→∞then inequality (26) reduces to a linear statement in pext

0and the parabolas

become straight lines. For β0>0 the stability limit for pext

0is unbounded, while for β0<0 it is bounded

and is given by

pext

0C=ρ3

0

γss γff −γ2

sf

22γsf −γss −γff .(29)

2Figure 1 is a condensed graphical representation in which the transformation

p∗=pext

0

¯p,¯p:= ρ6

0γss γff −γ2

sf

2ρ3

0|β|

reduces (26) to

P2(p∗):= −αp2

∗±p∗+1>0,α:= ρ2

0cs−cf2ρ6

0γss γff −γ2

sf

2ρ3

0|β|2,

thus collapsing the family of parabolas to two single graphs.

294 G. Sciarra et al.

Noticing that

β0>0⇒γss +γff

2>γ

sf ,β

0<0⇒γss +γff

2<γ

sf (30)

we deduce that the ﬁrst case corresponds to a weak coupling of the solid and the ﬂuid phases via γsf , whilst

the second one is related to a strong coupling. So stability exists for all pressures in the presence of weak

coupling, whilst for strong coupling stability is restricted to small pre-stresses.

These conditions change qualitatively, when cs−cfis not equal to zero. The stability limit is now

bounded in both cases, β>0 and β<0.Under these circumstances instability is always reached if only pext

0

is sufﬁciently large. To be more exact equation (28) implies that3

pext

0C=1

cs−cf2ρ0β+ρ2

0β2+ρ4

0cs−cf2γss γff −γ2

sf (31)

is a positive function of cs−cf, irrespective of whether β>0orβ<0; it is increasing (decreasing)

for negative (positive) values of cs−cf,assumes its maximum at cs−cf= 0 and local minima at

the boundaries |cs−cf|= min(1/ρ0

s,1/ρ0

f).Denoting by cs−cfBthe point at which pext

0Cattains its

minimum, it follows from (31) that the minimum critical pressure pext

0C(cs−cf)B

depends on, see (27),

β|B:= 1−ρ0

scs−cfB

≥0γss −γsf +1+ρ0

fcs−cfB

≥0γff −γsf .(32)

Thus, β|Bis a weighted average of γss −γsf and γff −γsf with weights which depend on the division

of the external traction onto the solid and the ﬂuid phases. Moreover, β|Bis positive (negative) according to

whether γss −γsf and γff −γsf are positive (negative). Since strong coupling corresponds to large |γsf |

values, it is evident that it enhances the potential of instability.

Deﬁning

I=β|B

ρ0cs−cfBγss γff −γ2

sf (33)

for negative β|B, (31) takes the form

pext

0C(cs−cf)B

=ρ0β|B

cs−cf21+1+1/I2.

Therefore, the larger I2, or the more negative Iis, the closer to zero will be pext

0C(cs−cf)B

.Consequently

Ican be taken as a measure of instability.

5 Conditions for pressure induced dilatancy of the solid matrix

Notice that our intention is to ﬁnd pressurized conditions that yield dilatancy. It is plain, that such states

make only sense if they are stable. This was the reason why the stability analysis was presented in the ﬁrst

place. In this spirit, let us return to the system (20) determining the perturbations ˜ρsand ˜ρf,if ˜pext is given.

A formal inversion of this system of equations yields

˜ρs=det Rρs

det[R]=detRρs

ρ0

sρ0

fdet[H],(34)

˜ρf=det Rρf

det[R]=detRρf

ρ0

sρ0

fdet[H],

3We explicitly remark that when cs−cftends to zero expression (31) tends to inﬁnity if β>0 but to (29) if β<0.

A solid-ﬂuid mixture model 295

where

detRρs=˜pext ρ0

sρ0

f

ρ3

0pext

0ρ0cs−cf1−cs−cfρ0

s(35)

+ρ3

01+cs−cfρ0

fγff −ρ3

01−cs−cfρ0

sγsf ,

detRρf=˜pext ρ0

sρ0

f

ρ3

0−pext

0ρ0cs−cf1+cs−cfρ0

f

+ρ3

01−cs−cfρ0

sγss −ρ3

01+cs−cfρ0

fγsf .

These can be computed by using Cramer’s rule implemented in maple. For stability det [H]>0, so the

signs of ˜ρs,˜ρfare dictated by detRρsand detRρf,respectively.

To obtain conditions of dilatancy of the solid matrix, induced by an increase of pressure applied on its

boundary, one can remark that ˜ρsis negative if and only if det Rρs<0, and this requires that

1. if cs−cf>0,

pext

0<

ρ2

01−cs−cfρ0

sγsf −1+cs−cfρ0

fγff

cs−cf1−cs−cfρ0

s,(36)

2. if cs−cf<0,

pext

0>

ρ2

01+cs−cfρ0

fγff −1−cs−cfρ0

sγsf

|cs−cf|1−cs−cfρ0

s,(37)

3. if cs−cf=0,(35)1implies that det Rρsis independent of the pressure pext

0.Then

detRρs(cs−cf)=0 =˜pext ρ0

sρ0

f(γff −γsf ),(38)

and, therefore, ˜ρsis negative provided that

γff <γ

sf .(39)

If the RHS of (36) is negative then a solution with ˜ρs<0 does not exist for pext

0>0.Should the RHS of

(37) be negative, then dilatancy occurs for all pext

0>0.

The foregoing analysis shows that dilatancy under external pressure is possible in a material if only the

coupling coefﬁcient γsf is sufﬁciently large. This, however, does not yet demonstrate that a real material

exists such that dilatancy is indeed established.

6 Second gradient energy describing pore micro-deformations

The model equations derived so far enjoy the following properties: when a uniform external pressure is

applied to the mixture body, exhibiting purely spherical stress states, the constituent densities are equally

uniform. This is so because the model does not account for the possible formation of a boundary layer (in

which apparent densities are spatially variable) at the mixture external interface. From a physical point of

view one could state that this absence is a consequence of a lacking description of the microscopic pore

deformation. This is a singular behaviour of the ﬁrst gradient theory.

To cure the ﬁrst gradient model from such singular features we now develop a second gradient mixture

model. Such theories were preveously developped as non-simple mixture models (M¨

uller [16], Rajagopal &

Tao [17]). We are not aware of any second gradient PVP-approach for mixtures in the spirit pursued here, but

our approach essentially follows Gouin [13], Casal [3] and Seppecher [20] (for simple mixtures) and Germain

[12] who use the PVP for single constituent bodies.

296 G. Sciarra et al.

6.1 General balance equations

Let us now consider the PVP for the general situation in which (i) body forces baare present, (ii) the internal

energy ψint may also depend on the second deformation gradients and (iii) the action of the exterior to the

body is given by tractions taand double forces τa(see for more details [12] and [9]). For such a case equation

(5) is generalized in the form

d

dt

B

ψint dV =

B

ba·vadV +

∂Bta·va+τa·∂va

∂ndA; (40)

τsand τfare the double forces acting on the solid and on the ﬂuid, respectively. We select the simplest

gradient dependence of ψint ,

ψint =ε(ρa)+λs

2fss ,fss := ∇ρs·∇ρs,(41)

where λsis a constant. This corresponds to a gradient dependence for the solid but not for the ﬂuid, which

is special. The localization of (40) with λs= 0 is easily shown to be

∇pa−ma=ba,in B,pa=dapext ,on ∂B,(42)

where

pa:= ρa∂ε

∂ρa−ξaε,

ma:= ∂ε

∂ρa∇ρa−∇(ξaε),(no sum.over a)

(43)

The details of the derivation follow eqs.(7) (8) (9) and are e.g. also given in [9]. It therefore sufﬁces do deal

with the additional gradient dependent term in (40)

Iadd :=

d

dt

B

ψint dV

add

=λs

2

d

dt

B

∇ρs·∇ρsdV =

∂Btaadd ·va+τa·∂va

∂n.

(44)

Applying the Reynolds Transport Theorem yields (see Appendix A)

Iadd =λs

B1

2(fss )I·∇(ξava)+∇ρs·∂

∂t∇ρs+∇⊗∇ρs(ξava)dV

=−λs

Bdiv 1

2fss ξsI−fss I−∇ρs⊗∇ρs+div (ρs∇ρs⊗I)

+1

2∇fss −1

2∇(ξsfss )·vsdV

+λs

∂B"1

2fss ξsI−fss I−∇ρs⊗∇ρs+div (ρs∇ρs⊗I)n·vs

+1

2fss ξfn·vf−(ρs∇ρs⊗I)n·∇vs#dA.(45)

Consequently, localization of the complete equation (40) leads to the following boundary value problem:

∇ps−ms−λsdiv ρsρsI+1

2fss I−∇ρs⊗∇ρs=bs,

∇pf−mf=bf,

in B(46)

A solid-ﬂuid mixture model 297

−psn+λsρsρsI+1

2fss ξsI−∇ρs⊗∇ρsn=ts,

−pfn+λs

2fss ξfn=tf,

−λsρs(∇ρs·n)n=τs,

0=τf,

on ∂B. (47)

Notice that the gradient effects only enter the ﬁeld equations of the solid, this obviously because of our

restrictive assumption (41). The boundary conditions of the ﬂuid are, however, affected by the gradient

terms; they generate an additional pressure. The surface double forces only enter the boundary conditions of

the solid constituent, because the free energy does not depend on ∇ρf.

6.2 External action potential

We now address the problem of ﬁnding a potential ψext for the external action speciﬁed on RHS of (40). We

are looking for a potential ψext that depends on the state parameters ρs,ρ

f,and ∇ρs,such that

d

dt

B

ψext dV =

B

ba·vadV +

∂Bta·va+τa·∂va

∂ndA.(48)

Its existence assures that a cyclic quasi-static variation of these parameters is path-independent. Using the

Reynolds Transport Theorem the LHS of (48) becomes

d

dt

B

ψext dV =

B∂

∂tψext +div ψext vdV .(49)

Performing the differentiations term by term and using the constituent balances of mass yields

∂

∂tψext =−∂ψext

∂ρaρaI·∇va−∂ψext

∂ρa∇ρa·va−∂ψext

∂∇ρs·∇ρsI·∇vs

−∇⊗∇ρs∂ψext

∂∇ρs·vs+∂ψext

∂∇ρs⊗∇ρs·∇vs

−ρs∂ψext

∂∇ρs⊗I·∇⊗∇vs(50)

div ψext v=div ψext ξava=ξaψext div va+va·∇ξaψext.

Substituting these above allows to write (49) as

d

dt

B

ψext dV =

B

(βa·va+Ba·∇va+Bs·∇⊗∇vs)dV ,(51)

where the quantities βa,Ba(a=s,f) and Bsare deﬁned by the following expressions:

βs:= −∂ψext

∂ρs∇ρs−(∇⊗∇ρs)∂ψext

∂(∇ρs)+∇(ξsψext ),

βf:= −∂ψext

∂ρf∇ρf+∇(ξfψext),

Bs:= −∂ψext

∂ρsρs−∂ψext

∂(∇ρs)·∇ρs+ξsψext I−∂ψext

∂(∇ρs)⊗∇ρs,(52)

Bf:= −∂ψext

∂ρfρf+ξfψextI,

Bs:= −ρs∂ψext

∂(∇ρs)⊗I.

298 G. Sciarra et al.

The RHS of (48) and (51) agree with one another if ba,ta,and τaare given by

bs=βs−div (Bs−div Bs),bf=βf−div Bf,

ts=(Bs−div Bs)n−divs(Bsn),tf=Bfn,

τs=(Bsn)n,τf=0;(53)

in these formulas we consider the following decomposition for the gradient of an n-th order tensor ﬁeld Ω:

∇Ω=∇sΩ+n⊗∂Ω

∂n,(54)

∇sΩbeing the restriction of ∇Ωon ∂B, and we deﬁne the surface divergence of an n-th order tensor ﬁeld

Ω, the differential operator divssuch that

divsΩTu=(divsΩ)·u+Ω·∇

su,∀(n−1)-th order tensor ﬁeld u(55)

and ∂B

divsΩdA =∂∂B

Ων dS,(56)

where νis the outward normal to ∂∂Bthe line boundary of ∂B. For smooth surfaces ∂Bthe integral on

the RHS of (56) vanishes. In this case the contact action on the solid and the ﬂuid, the double forces on the

solid and the ﬂuid are given by (see Appendix B)

ts=−∂ψext

∂ρsρs+ξsψext −∂ψext

∂(∇ρs)·∇ρs+div ρs∂ψext

∂(∇ρs)

+ρs∂ψext

∂(∇ρs)·n(tr ∇sn)−∂ψext

∂(∇ρs)·n∂ρs

∂nn+ρs∇s∂ψext

∂(∇ρs)·n,

tf=−∂ψext

∂ρfρf+ξfψextn,(57)

τs=−ρs∂ψext

∂(∇ρs)·nn,

τf=0.

We distinguish the normal and the shear parts of tsand assume that the double force acting on the solid

depends linearly on the external pressure: τs=dDpext n; this is reasonable, since increasing the pressure

increases the pore space and the latter is opened by the action of the double force. In so doing we obtain

the following forms of the constitutive relations for the coefﬁcient dDand for the coefﬁcients da(a=s,f),

appearing in (4) valid in the case of second gradient solid matrices:

dD=−1

pext ρs∂ψext

∂(∇ρs)·n,

ds=1

pext ∂ψext

∂ρsρs−ξsψext −ρsdiv ∂ψext

∂∇ρs(58)

+ρs∂ψext

∂∇ρs·ntr ∇sn+∂ψext

∂∇ρs·n∇ρs·n,

df=1

pext ∂ψext

∂ρfρf−ξfψext.

These formulas simply emerge if one divides τs,and the components of tanormal to the surface by pext .

Assume that the body forces bsand bfvanish; then equations (53)1,2yield the conditions

ρs∇∂ψext

∂ρs−div ∂ψext

∂∇ρs=0,ρ

f∇∂ψext

∂ρf=0.(59)

Further investigations will be necessary to generalize the results, found in the §2, about ψext implied by

the condition df+ds= 1. We simply remark here that one can ﬁnd in the subsequent one-dimensional problem,

developed as an application of the introduced new model, a form for ψext verifying the above constraints.

A solid-ﬂuid mixture model 299

7 A one dimensional application

Consider a one-dimensional problem in which the body forces on the solid and on the ﬂuid vanish. Assume

constant external pressure and suppose that the derivative of ψext with respect to ∇ρsis constant in Bi.e.

d

dx ∂ψext

∂(ρs,x)=0,(60)

where ∂ψext /∂(ρs,x):=∂ψext /∂∇ρs·e,where eis the unit vector deﬁning the xdirection. With this, eqs.

(57) reduce to

dspext =∂ψext

∂ρsρs−ξsψext +∂ψext

∂(ρs,x)dρs

dx ,dfpext =∂ψext

∂ρfρf−ξfψext,

tshear

s=0,tshear

f=0,

τs=−ρs∂ψext

∂(ρs,x)e,τ

f=0

(61)

and equations (59) become

d

dx ∂ψext

∂ρa=0,a=s,f.(62)

Eqs. (60) and (62) imply that ψext is a linear function of its arguments ρaand dρs/dx. Thus, with an

appropriate normalization, one has

ψext =pext csρs+cfρf+ksdρs

dx −1,(63)

whilst the constitutive relations deﬁning the coefﬁcients da(a=s,f) and dDare

ds=ξs1+(cs−cf)ρf+ξfksdρs

dx ,

df=ξf1−(cs−cf)ρs−ξfksdρs

dx ,(64)

dD=−ksρs(n·e),

as easily deducible from (63) and (58) or from (61). Note that constitutive relations for dsand dfdiffer

from relations (11), obtained by a ﬁrst gradient mixture model, by an additive quantity; this is due to the

assumption on the derivative of ψext with respect to ∇ρs:ψext simply depends linearly on ρsand dρs/dx.

Consider a linearized theory and assume that in the reference conﬁguration the constituent density of the

solid (and of the ﬂuid) is not uniform. This hypothesis is necessary to appreciate second gradient effects: if

the densities were uniform in the reference conﬁguration, then (47)3would imply τs= 0, i.e., ks=0,so that

there would be no possibility to have a non-vanishing double force on ∂B,acting on the solid skeleton.

We also assume that the coefﬁcient γsof the linear term of the internal potential energy εis not constant

but a linear function of ρ0

s,

γs=αsρ0

s,(65)

where αsis assumed uniform in B; a justiﬁcation for this will be given shortly. With these prerequisites we

may now perform a perturbation analysis in the vicinity of a pre-stressed reference state with ψext given by

(63) and ψint by (41). With an approach entirely analogous to that of §3 we then deduce from the balance

laws the following zeroth and ﬁrst order equations:

zeroth order problem

dρ0

dx αsρ0

s+ρ0αsdρ0

s

dx −λsd3ρ0

s

dx3=0,ρ

0

fdρ0

dx γf=0,(66)

300 G. Sciarra et al.

ﬁrst order problem

−λsd3˜ρs

dx3+2αsρ0

s+ρ0γss d˜ρs

dx +αsρ0

s+γf+ρ0γsf d˜ρf

dx +

+2αsdρ0

s

dx ˜ρs+αsdρ0

s

dx ˜ρf=0,(67)

2γf+ρ0γff d˜ρf

dx +αsρ0

s+γf+ρ0γsf d˜ρs

dx +αsdρ0

s

dx ˜ρs=0.

To these ODEs at each perturbation order four boundary conditions must be added: we suppose that the

tractions on the solid and on the ﬂuid and the double force are known at x=0,and we assume that the

double force (and therefore dρ0

s/dx) vanishes as x→∞.The condition (65) corresponds to the idea that the

apparent density of the solid (and of the ﬂuid) in the reference state is given by the sum of a constant and an

exponentially decreasing term, smaller than zero; if γswere constant in Bthen equation (66)1would imply

that ρ0

s=const because of the condition as x→∞.

The solution of the zeroth order problem is

ρ0

s(x)=C1+C2expx

x0+C3exp−x

x0,

ρ0(x)=C4,

where

x0:= λs/ρ0αs.(68)

This explicitly demonstrates that the double forces are responsable for the exponential decay of ρ0

s(x)asone

moves away from the mixture surrounding environment, since λs=0.The boundary condition at x→∞

implies that C2= 0; the boundary condition on the value of the double force at x= 0 implies that C3≤0,so

the apparent densities of the two constituents are given by

ρ0

s(x)=C1−|C3|exp −x

x0,(69)

ρ0

f(x)=(C4−C1)+|C3|exp−x

x0;

x0is the characteristic decay length of the zeroth order solution. We do not show the explicit expression

for C1,(C4−C1) and |C3|which are rather cumbersome; we simply recall that they depend on the external

actions and on the interface constitutive parameters (e.g. cs−cf) and can be interpreted respectively as the

apparent solid and ﬂuid mass densities far from the mixture-surrounding environment and their maximum

variations induced by applied external double forces.

To compute the solution of the ﬁrst order problem, consider the following non-dimensionalization of the

independent and dependent variables:

ξ:= x

x0,rs:= ˜ρs

C1,rf:= ˜ρf

C4−C1.

Equations (67) constitute a fourth order differential problem; so it can be expressed as a system of ﬁrst order

differential equations given in the following form:

dY

dξ=(R+ exp(−ξ)A0+ exp(−2ξ)A1)Y,(70)

where Y,R,A0,A1are deﬁned by

A solid-ﬂuid mixture model 301

Y:=

rs

rf

drs/dξ

d2rs/dξ2

,R:=

0010

00 −a4/a60

0001

00a1−a3a40

,

A0:=

00 00

−a5/a60a5/a60

00 00

2a2−a3a5a2a62(a3a5−a2)0

,(71)

A1:=

0000

0000

0000

a2a50−a2a50

,

and

a1:= x2

0/λs(2αsC1+C4γss ),

a2:= αs|C3|x2

0/λs,

a3:= x2

0/λsαsC1+γf+C4γsf ,

a4:= αsC1+γf+C4γsf /2γf+C4γff ,

a5:= αs|C3|/2γf+C4γff ,

a6:= (C4−C1)/C1.

(72)

Consider the following change of variable: z= exp(−ξ); the differential problem (70) becomes

dY

dz =−1

zR+A0+zA1Y.(73)

This change of variable maps the open set (0,∞) onto the open set (0,1); boundary conditions in ξ= 0 now

are given at z= 1 and conditions at ξ→∞at z=0.

The matrix A(z)=−z−1R+A0+zA1has at most a pole at z= 0 but it is analytic for 0 <|z|<a,

a>0 and the point z= 0 is a singular point of the ﬁrst kind for the system (73) (see [4]), so it fulﬁlls the

hypotheses of theorem 3.1, p. 117 and 4.1, p.119 in [4]. Therefore, the fundamental matrix of system (73) is

represented in terms of a series, convergent in the set 0 <|z|<a,

Φ(z)=&∞

'

i=0

Qizi(e(ln z)J(74)

where Jis the canonical form4of R, if and only if Rhas characteristic roots which do not differ by positive

integers, and

RQ0=Q0J,(75)

Qm+1 [J+(m+1)I]=RQm+1 +m

)

k=0AkQm−k,

where, in this context, Iis the (4×4)unit matrix. So it follows that the fundamental matrix given as a

function of ξis

4The canonical form of a matrix is deﬁned as its Jordan form. If the matrix admits linearly independent eigenvectors its canonical

form is diagonal.

302 G. Sciarra et al.

Φ(ξ)=(Q0+ exp (−ξ)Q1+ exp(−2ξ)Q2+...)e−ξJ.(76)

When truncating this series at the ﬁrst order term the non-dimensional solid and ﬂuid densities are given by

(see Fig.2)

rs=k2+k11

ξ0exp(−ξ)+1+ 1

ξ0exp−ξ

ξ0,

rf=k3−k1a4

a61

ξ0exp(−ξ)+1+ 1

ξ0exp−ξ

ξ0,

(77)

where ξ0:= (a1−a3a4)−1/2,kiare integration constants to be determined by imposing the boundary conditions

implied by (61) at the ﬁrst order; experiments must give information on their values.

Fig. 2. Scaled solid density rsplotted against dimensionless distance ξparameterized for various values of the e-folding distance ξ0.

All curves approach the asymptote as ξ→∞

With the above expressions for aj(j=1,3,4), we have

√a1−a3a4=x0

√λs*2γf+C4γff (2αsC1+C4γss )−αsC1+γf+C4γsf 2

2γf+C4γff .(78)

Considering that C4=ρ0and in a ﬁrst gradient theory C1=ρ0

s(compare with (69)) we may rewrite (78) on

using (65) as

√a1−a3a4=x0

√λsdetH

H22 ,(79)

where His deﬁned in (24). Thus

ξ0=√λs

x0H22

detH.(80)

The eigenvalues of Rare +±√a1−a3a4=±ξ−2

0,0,0,.

In the stability regime one has detH>0,and H22 >0; therefore, ξ0is real valued. The fundamental solution

of (73) corresponding to the negative eigenvalue of Rgrows with increasing ξ, so the boundary condition at

ξ→∞inforcing regularity requires this solution to be absent in (77).

A solid-ﬂuid mixture model 303

The parameter ξ0can be greater than unity, if

detH=H11H22 −H2

12 <ρ0αsH22,(81)

and it will eventually tend to inﬁnity when H2

12 →H11H22.This last condition occurs if the coupling γsf is

sufﬁciently large: the effect of the second-gradient-depending-deformation energy is an increasing widening

of the boundary layer near the external surface of the body when the instability conditions are approached.

8 Conclusions

In this paper a binary mixture model was presented which possesses the ingredients that an external pressure

may cause a dilatation of the pores, a phenomenon sometimes observed in heterogeneous porous materials.

Terzaghi and in particular Fillunger were aware of this phenomenon and knew that their models could

not predict this behaviour, and they dismissed the possibility after extensive search for evidence and own

experimentation5. Perhaps they were too ambitious, for their arguments seem to suggest that they were looking

for an explosion of the solid matrix of a pressurized solid-ﬂuid mixture if only the external pressure would

be sufﬁciently large.

The present paper showed within the context of a very simple mixture model −too simple to describe

the deformation of the solid accurately, but sufﬁciently complex to isolate this detail −how Terzaghi’s and

Fillunger’s search could be interpreted. To this end we assumed all constitutive quantities to depend on the

apparent densities of the solid and the ﬂuid (and eventually on their gradients) and no more. It turned out

that a dependence of the internal free energy on the interaction term ρsρf,i.e., on the product of the apparent

solid and ﬂuid densities, is important. Terzaghi’s and Fillunger’s explosion is interpreted here as the loss of

stability of a pre-stressed reference state. The critical external pressure, which causes this reference state to

become unstable, is dictated by two physically distinct properties: i) the coefﬁcients of the quadratic terms

of the free internal energy and in particular its interaction term and ii) the parameterization how the external

pressure is distributed between the solid and the ﬂuid normal boundary tractions. There are parameter sets

for which instability never arises and others for which instability can, in principle, always occur, if only the

external pressure is sufﬁciently large (§4).

The second question addressed by Terzaghi and Fillunger is the mentioned dilatancy phenomenon viewed

possibly by them to be the same as the explosive problem of the solid matrix. Our model also gives an answer

to this question. If the pressure corresponding to our reference state is increased then the new stable state

can exist only under the above mentioned stability conditions. This new state possesses a smaller or larger

apparent solid density provided that the interaction term of the free energy obeys certain equalities involving

the other coefﬁcients of the free energy and the parametrization of the constituent boundary tractions (§5).

We ﬁnd it most intriguing, that thermodynamics of bulk and boundary quantities provides the answer to this

subtle behaviour of the mixture.

While we do now understand how the above mentioned dilatational effect can be predicted by the model

equations, it does not have typical boundary layer structure in a ﬁrst gradient theoretical setting. This boundary

enhancement can be achieved by adding a density-gradient dependence of the solid phase to the free energy

[9]. This dependence will lead to an enhancement of the pore space close to the interface between the mixture

and the exterior world which dies out as one moves away from the interface, (§6).

The one-dimensional problem which we solve in the last section (§7) proves that second gradient regular-

ization is necessary if one wants to describe the explosion phenomenon imagined by Fillunger and Terzaghi.

Indeed if one interprets it as a loss of stability the transition from stable to unstable states can (in the frame-

work of the present model) be parameterized by the coupling coefﬁcient γsf ,ceteris paribus. When γsf is

increased, H2

12 also increases and detHtends to zero: the boundary layer at the interface between the mixture

body and the external world becomes wider and wider and eventually occupies the whole body before the

instability conditions arise. As the second gradient boundary layer is characterized by a lower value of the

apparent solid mass density one can state that instability is attained by a progressive dilatational process

which is induced by the pore ﬂuid pressure and initially arises at the boundary of the mixture body.

5As beautifully summarized by [7], Fillunger dismissed the fact that the pore pressure would affect the strength of the porous material.

304 G. Sciarra et al.

Appendix A

In this section we give the details of the calculation which permit us to obtain relation (45). Using the balance

of mass for the solid gives

1

λsIadd =

B1

2fss I·∇(ξava)+∇ρs·∂

∂t∇ρs+∇⊗∇ρs(ξava) dV

=

B1

2fss I·(ξa∇va+∇ξa⊗va)−∇ρs·ρs(∇⊗∇vs)TI

−fss I·∇vs−∇vs·∇ρs⊗∇ρs−∇ρs·(∇⊗∇ρs)vs

+∇ρs·∇⊗∇ρs(ξava)dV

(82)

Consider the following identities for a second order tensor ﬁeld A, a third order tensor ﬁeld A6and a vector

ﬁeld v

A·∇v=div ATv−v·divA,

A·∇⊗∇v=div AT∇v−∇v·divA(83)

=div AT∇v−div (divA)Tv+v·div (divA),

using these identities in (82) for v=vs,Aand Abeing the coefﬁcients of ∇vsand ∇⊗∇vsin (82)

respectively, the aforementioned equation takes the alternative form

1

λsIadd =

B"div 1

2fss ξsI−fss I−∇ρs⊗∇ρsvs+1

2fss ξfvf

−ρs(I⊗∇ρs)∇vs+[div (ρs∇ρs⊗I)]Tvs

−vs·1

2ξf∇fss −1

2fss ∇ξs+div 1

2fss ξsI−fss I−∇ρs⊗∇ρs

+div div (ρs∇ρs⊗I)#dV

(84)

Using the divergence theorem where appropriate yields formula (45).

Combining this result with the “ﬁrst gradient ” expression of (40) −using (42) and (43) yields now

d

dt

B

ψint dV =

B"∇ps−ms−λsdiv ρsρsI+1

2fss I−∇ρs⊗∇ρs·vs

+∇pf−mf·vf#dV

+

∂B"−psn+λsρsρsI+1

2fss I−∇ρs⊗∇ρsn·vs

+−pfn+λs

2fss ·vf−λsρs(∇ρs·n)I·∇vs#dA,

(85)

6We assume that a third order tensor Ais a linear map deﬁned as follows

A:V→LIN V,LIN V

Vbeing a linear space, LIN Vthe collection of all linear endomorphisms on Vand LIN V,LIN Vthe collection of all

the linear morphisms mapping Vinto LIN V.

The transpose of Ais assumed to fulﬁll the following relation

Au·U=u·ATU

for any u∈Vand any U∈LIN V.

A solid-ﬂuid mixture model 305

from which the local statements (46) and (47) are now readily deduced.

Appendix B

In this Appendix we derive formulas (57) using (52) and (53). To this end, one needs

divBs=−div ρs∂ψext

∂(∇ρs)⊗I=div ρs∂ψext

∂(∇ρs)I

divs(Bsn)=−divsρs∂ψext

∂(∇ρs)·nI=−ρs∂ψext

∂(∇ρs)·ndivsI(86)

−∇sρs∂ψext

∂(∇ρs)·n

=−ρs∂ψext

∂(∇ρs)·n∇sn−∇

sρs∂ψext

∂(∇ρs)·n.

Inserting these expressions in the formulas (53) one obtains

ts=−∂ψext

∂ρsρs+ξsψext −∂ψext

∂(∇ρs)·∇ρs+div ρs∂ψext

∂(∇ρs)+

+ρs∂ψext

∂(∇ρs)·n(tr ∇sn)−∂ψext

∂(∇ρs)·n∂ρs

∂nn+ρs∇s∂ψext

∂(∇ρs)·n,

tf=−∂ψext

∂ρfρf+ξfψextn,(87)

τs=−ρs∂ψext

∂(∇ρs)·nn,

τf=0,

which agrees with (57).

Acknowledgements. The authors wish to thank Prof. Pierre Seppecher from Universit´

e de Toulon et du Var for his constructive criticism

and long discussions about conservative boundary conditions in mixture theories. They also acknowledge the constructive reviews of

two referees.

References

1. Bowen RM (1980) Incompressible porous media models by use the theory of mixtures. Int. J. Engng. Sci. 18, 1129–1184

2. Bowen RM (1982) Compressible porous media models by use of the theory of mixtures. Int. J. Engng. Sci. 20, 697–734

3. Casal P (1961) La Capillarit´

e interne. Cahier du groupe Francais d’Etudes de Rheologie C.N.R.S. VI(3), 31–37

4. Coddington EA, Levinson N (1955) Theory of ordinary differential equations. New York: Mc Graw-Hill

5. Cosenza P (1996) Sur les couplages entre comportement m´

ecanique et processus de transfert de masse dans le sel gemme. Th`

ese

Universit´

e. PARIS VI

6. Coussy O (1995) Mechanics of porous media. New York: John Wiley and Sons

7. de Boer R, Ehlers W (1990) The development of the concept of effective stress. Acta Mechanica 83, 77–92

8. de Boer R (2000) Theory of porous media. Berlin Heidelberg New York: Springer

9. dell’Isola F, Guarascio M, Hutter K (2000) A variational approach for the deformation of a saturated porous solid. A second gradient

theory extending Terzaghi’s effective stress principle. Archive of Applied Mechanics 70, 323–337

10. dell’Isola F, Hutter K (1998) A qualitative analysis of the dynamics of a sheared and pressurized layer of saturated soil. Proc. R.

Soc. Lond.A 454, 3105–3120

11. dell’Isola F, Hutter K (1999) Variations of porosity in a sheared pressurized layer of saturated soil induced by vertical drainage of

water Proc. R. Soc. Lond.A 455, 2841–2860

12. Germain P (1973) La m´

ethode des puissances virtuelles en m´

ecanique des milieux continus Journal de M´

ecanique,12(2), 235–274

13. Gouin H (1991) Variational methods for ﬂuid mixtures of grade n Continuum Models and Discrete Systems 2, 243–252

14. Hutter K, J¨

ohnk K, Svendsen B (1994) On interfacial transition conditions in two phase gravity ﬂow. ZAMP, 45 746–762

15. Morland LW (1972) A simple constitutive theory for a ﬂuid saturated porous solid, J. Geoph. Res. 77, 890–900

16. M¨

uller I (1985) Thermodynamics, Pittman, Boston, etc.

306 G. Sciarra et al.

17. Rajagopal KR, Tao L (1995) Mechanics of Mixtures, World Scientiﬁc

18. Rajagopal KR, Wineman AS, Gandhi MV (1986) On boundary conditions for a certain class of problems in mixture theory,. Int. J.

Engng. Sci. 24, 1453–1463

19. Seppecher P (1987) Etude d’une modelisation des zones capillaires ﬂuides: interfaces et lignes de contact. Th`

ese Universit´

e. PARIS

VI

20. Seppecher P (1989) Etude des condition aux limites en th´

eorie du second gradient: cas de la capillarit´

e C. R. Acad. Sci. Paris 309

(S´

eries II) 597–502

21. Svendsen B, Gray JMNT (1996) Balance relations for classical mixtures containing a moving non-material surface. Continuum

Mechanics and Thermodynamics 8, 171–187

22. Svendsen B, Hutter K (1995) On the thermodynamics of a mixture of isotropic materials with constraints. Int. J. Engng. Sci. 33,

2021–2054

23. von Terzaghi K (1946) Theoretical Soil Mechanics. New York: John Wiley and Sons