A solid-fluid mixture model allowing for solid dilatation under external pressure

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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 +dapextn·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)=ξs1+ 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
fhas 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
0csρ0
s+cfρ0
f−1
+pext
0cs˜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)
=ρ0γs˜ρs+γf˜ρf+1
2(2γs+ρ0γss )˜ρ2
s+1
22γf+ρ0γff ˜ρ2
f
+γs+γf+ρ0γsf ˜ρs˜ρf,
where an arbitrary constant in ϕint is irrelevant and where the constants
γs:= ∂ϕint
∂ρsρ0
s,ρ0
f
,γ
f:= ∂ϕint
∂ρfρ0
s,ρ0
f
,
γss := ∂2ϕint
∂ρ2
sρ0
s,ρ0
f
,γ
ff := ∂2ϕint
∂ρ2
fρ0
s,ρ0
f
,γ
sf := ∂2ϕint
∂ρs∂ρfρ0
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 
Bψint −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
0cs−cfpext
0,γ
f=pext
0
ρ2
0−ρ0
s
ρ2
0cs−cfpext
0.(19)
first order
R11 R12
R21 R22  ˜ρs
˜ρf=




ξ0
s+ρ0
sρ0
f
ρ0cs−cf
ξ0
f−ρ0
sρ0
f
ρ0cs−cf




˜pext ,(20)

292 G. Sciarra et al.
where
R11 := ρ0
s(2γs+ρ0γss )+ρ0
fγs−pext
0ξ0
f2cs−cf+ξ0
f
ρ0,
R12 := ρ0
sγs+γf+ρ0γsf −ρ0
sγf−pext
0ξ0
s2cs−cf−ξ0
s
ρ0,
R21 := ρ0
fγs+γf+ρ0γsf −ρ0
fγs+pext
0ξ0
f2cs−cf+ξ0
f
ρ0,
R22 := ρ0
f2γf+ρ0γff +ρ0
sγf+pext
0ξ0
s2cs−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
ρ0cs−cf
ξ0
f−ρ0
sρ0
f
ρ0cs−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 Hψtot 
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
2pext
0
ρ2
0+ρ0
f
ρ2
0cs−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
0cs−cf2pext
02+2ρ3
0βpext
0+ρ6
0γss γff −γ2
sf >0,(26)

A solid-fluid mixture model 293
in which
β:= β0+cs−cfβ1,
β0:= γss +γff −2γsf ,(27)
β1:= ρ0
sγsf −γss +ρ0
fγff −γsf .
The LHS of (26) is a quadratic form P2pext
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 P2pext
0∗=0,
pext
0∗=−1
cs−cf2−ρ0β±ρ2
0β2+ρ4
0cs−cf2γss γff −γ2
sf ,(28)
are positive and negative irrespective of whether β>0orβ<0.The stability region is pext
0<pext
0C,
where pext
0Cis 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−cfis 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
0C=ρ3
0
γss γff −γ2
sf
22γsf −γss −γff .(29)
2Figure 1 is a condensed graphical representation in which the transformation
p∗=pext
0
¯p,¯p:= ρ6
0γss γff −γ2
sf 
2ρ3
0|β|
reduces (26) to
P2(p∗):= −αp2
∗±p∗+1>0,α:= ρ2
0cs−cf2ρ6
0γss 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−cfis 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
0C=1
cs−cf2ρ0β+ρ2
0β2+ρ4
0cs−cf2γss 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−cfBthe point at which pext
0Cattains its
minimum, it follows from (31) that the minimum critical pressure pext
0C(cs−cf)B
depends on, see (27),
β|B:= 1−ρ0
scs−cfB
 
≥0γss −γsf +1+ρ0
fcs−cfB
 
≥0γff −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
ρ0cs−cfBγss γff −γ2
sf (33)
for negative β|B, (31) takes the form
pext
0C(cs−cf)B
=ρ0β|B
cs−cf21+1+1/I2.
Therefore, the larger I2, or the more negative Iis, the closer to zero will be pext
0C(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]=detRρs
ρ0
sρ0
fdet[H],(34)
˜ρf=det Rρf
det[R]=detRρf
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
detRρs=˜pext ρ0
sρ0
f
ρ3
0pext
0ρ0cs−cf1−cs−cfρ0
s(35)
+ρ3
01+cs−cfρ0
fγff −ρ3
01−cs−cfρ0
sγsf ,
detRρf=˜pext ρ0
sρ0
f
ρ3
0−pext
0ρ0cs−cf1+cs−cfρ0
f
+ρ3
01−cs−cfρ0
sγss −ρ3
01+cs−cfρ0
fγsf .
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 detRρsand detRρf,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
01−cs−cfρ0
sγsf −1+cs−cfρ0
fγff 
cs−cf1−cs−cfρ0
s,(36)
2. if cs−cf<0,
pext
0>
ρ2
01+cs−cfρ0
fγff −1−cs−cfρ0
sγsf 
|cs−cf|1−cs−cfρ0
s,(37)
3. if cs−cf=0,(35)1implies that det Rρsis independent of the pressure pext
0.Then
detRρs(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 +
∂Bta·va+τa·∂va
∂ndA; (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 =
∂Btaadd ·va+τa·∂va
∂n.
(44)
Applying the Reynolds Transport Theorem yields (see Appendix A)
Iadd =λs
B1
2(fss )I·∇(ξava)+∇ρs·∂
∂t∇ρs+∇⊗∇ρs(ξava)dV
=−λs
Bdiv 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 ρsρsI+1
2fss I−∇ρs⊗∇ρs=bs,
∇pf−mf=bf,
in B(46)

A solid-fluid mixture model 297
−psn+λsρsρsI+1
2fss ξsI−∇ρs⊗∇ρsn=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 +
∂Bta·va+τa·∂va
∂ndA.(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 vdV .(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·∇ρsI·∇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+ξfψextI,
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
divsΩTu=(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
∂nn+ρs∇s∂ψext
∂(∇ρs)·n,
tf=−∂ψext
∂ρfρf+ξfψextn,(57)
τs=−ρs∂ψext
∂(∇ρs)·nn,
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·ntr ∇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=ξs1+(cs−cf)ρf+ξfksdρs
dx ,
df=ξf1−(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+C2expx
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/λsαsC1+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+zA1Y.(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+zA1has 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+k11
ξ0exp(−ξ)+1+ 1
ξ0exp−ξ
ξ0,
rf=k3−k1a4
a61
ξ0exp(−ξ)+1+ 1
ξ0exp−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
√λsdetH
H22 ,(79)
where His defined in (24). Thus
ξ0=√λs
x0H22
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 <ρ0αsH22,(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 =
B1
2fss I·∇(ξava)+∇ρs·∂
∂t∇ρs+∇⊗∇ρs(ξava) dV
=
B1
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⊗∇ρsvs+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 ρsρsI+1
2fss I−∇ρs⊗∇ρs·vs
+∇pf−mf·vf#dV
+
∂B"−psn+λsρsρsI+1
2fss I−∇ρs⊗∇ρsn·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 Vthe collection of all linear endomorphisms on Vand LIN V,LIN Vthe 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)=−divsρs∂ψext
∂(∇ρs)·nI=−ρs∂ψext
∂(∇ρs)·ndivsI(86)
−∇sρs∂ψext
∂(∇ρs)·n
=−ρs∂ψext
∂(∇ρs)·n∇sn−∇
sρs∂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
∂nn+ρs∇s∂ψext
∂(∇ρs)·n,
tf=−∂ψext
∂ρfρf+ξfψextn,(87)
τs=−ρs∂ψext
∂(∇ρs)·nn,
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.
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