Wave motions in unbounded poroelastic solids infused with compressible fluids
ZZ. angew. Math. Phys. 53 (2002) 1110–11380044-2275/02/061110-29c Zeitschrift f¨ur angewandteMathematik und Physik ZAMP
Wave motions in unbounded poroelastic solids infused withcompressible fluids
S. Quiligotti, G.A. Maugin and F. dell’Isola
This paper is dedicated to the memory of Professor Eugen So´os
Abstract.
Looking at rational solid-fluid mixture theories in the context of their biomechanicalperspectives, this work aims at proposing a two-scale constitutive theory of a poroelastic solidinfused with an inviscid compressible fluid. The propagation of steady-state harmonic planewaves in unbounded media is investigated in both cases of unconstrained solid-fluid mixturesand fluid-saturated poroelastic solids. Relevant e ff ects on the resulting characteristic speed oflongitudinal and transverse elastic waves, due to the constitutive parameters introduced, arefinally highlighted and discussed. Mathematics Subject Classification (2000).
Keywords.
Solid-fluid mixtures, principle of virtual power, elastic waves.
1. Introduction
A solid-fluid mixture is customarily thought of as a couple of body manifolds em-bedded into the three-dimensional Euclidean space (see e.g. Atkin and Craine[2], Bowen [6], Krishnaswamy and Batra [21], Rajagopal and Tao [28], Truesdell[34, 35]), so as to occupy a common smooth region of the physical environmentwhile undergoing independent motions, here assumed to take place in a neighbor-hood of the initial solid configuration (section 2).If neither chemical reactions nor phase transitions occur, the mass of eachconstituent is conserved along the corresponding motion. Moreover, the local formof the fluid-mass conservation law can be advantageously written with respect tothe initial configuration, also regarded as a reference configuration (section 3).In order to take into consideration, at least coarsely, the most remarkablemicrostructural properties of the mixture (see e.g. Schrefler [30]), the concept ofvolume fraction (cf. Bowen [5, 7], de Boer [9, 10], Wilmanski [38]) is furthermoreintroduced within the framework of a first-order gradient theory (sections 5.1–5.2), assuming a linear constitutive dependence of microscopic mass densities on ol. 53 (2002) Wave motions in unbounded poroelastic solids 1111 macroscopic kinematical descriptors (section 4).If the saturation constraint is fulfilled (Klisch [20], Svendsen and Hutter [31]),i.e. the volume occupied by the constituents equals the volume available to themixture, then the stress response is determined by the motion except for an arbi-trary contribution, due to the saturation pressure which arises in the material soas to maintain each constituent in contact with the other one. Such a pressure canbe truly regarded as a Lagrangian multiplier in the expression of the strain-energydensity per unit volume of the mixture (dell’Isola et al. [11], Quiligotti et al. [13]),so as to deduce the splitting rule which governs the distribution of such a pressureamong the constituents as a result of the theory (section 5.2).As far as the power expended by inertial forces is concerned, the overall kineticenergy density per unit reference volume of the mixture is defined as the sum ofpeculiar kinetic energy densities. Its material derivative, following the motion ofthe mixture as a whole, is required to equal the power expended by inertial forces(section 5.3). This assumption, physically motivated although far from evident,seems to corroborate the importance of coupled inertial interactions (Biot [3], cf.de Boer [10]).The principle of virtual power (see e.g. Germain [16, 17]) is finally used as amain tool (Di Carlo [12], Maugin [23]) to deduce the set of balance equations andboundary conditions that governs the nonlinear dynamics of the mixture (Quilig-otti et al. [13]). With the aim of investigating the propagation of elastic planewaves in unbounded media (Achenbach [1], Gra ff [18]), these equations have beenlinearized (section 5.4), and their harmonic steady-state solutions have been foundin both cases of unconstrained solid-fluid mixtures and fluid-satured porous solids(section 5.5). A few relevant remarks, inherently based on a comparative analy-sis of emerging results, complete this study and implicitly outline some possiblebiomechanical applications that may be envisaged as a further development ofthe proposed two-scale constitutive theory (cf. Cowin [8]; see also Humphrey andRajagopal [19], Ehlers and Markert [15], Tao et al. [33], Taber [32]).
2. Kinematics
Let us consider a binary solid-fluid mixture, consisting of two smooth three-dimensional material manifolds, denoted by B S and B F (figure 1).By assumption (Noll and Virga [26]), a time-independent smooth embeddingof the solid body into the physical Euclidean space E , K S : B S → E (1) X S → X , (2)associates any material solid particle with a reference place.
112 S. Quiligotti, G.A. Maugin and F. dell’Isola ZAMP B S X χ F ( · , t ) χ S ( · , t ) χ − ( · , t ) X F X S K S BE E χ S ( B , t ) χ F ( B F , t ) B F x Figure 1. Kinematics of a binary solid-fluid mixture.
A smooth S -motion † may be described as a time sequence of mappings, χ S ( · , t ) : B → E (3) X → x , (4)which carry any solid particle from its reference place to its current one.In particular, so as to linearize the equations that govern the dynamics of themixture, we shall focus our attention on a smooth 1-parameter family of S -motions, χ S ( X , t ) := X + ε u S ( X , t ) , (5) † For the sake of conciseness, we shall refer to any motion of the body manifold B α (with α ∈ { S , F } ) as an α -motion .ol. 53 (2002) Wave motions in unbounded poroelastic solids 1113 whose gradient straightforwardly results in:Grad χ S = I + ε Grad u S . (6)Similarly, a smooth F -motion can be described as a time sequence of embed-dings which map the fluid-body manifold onto its current shape, χ F ( · , t ) : B F → E (7) X F → x , (8)so that, at time t , any given place in the current shape of the mixture, x = χ S ( K S ( X S ) , t ) = χ F ( X F , t ) ∈ B ( t ) , (9)with B ( t ) := χ S ( B , t ) χ F ( B F , t ) and B = K S ( B S ) , (10)is simultaneously occupied by a pair of di ff erent material particles, X S ∈ B S and X F ∈ B F .As the reference shape B does not depend on time, the velocity of any α -particle, X α ∈ B α , is just given by the material derivative following the α -motion, v S ( x , t ) := ∂ χ S ( X , τ ) ∂τ τ = t (11) v F ( x , t ) := ∂ χ F ( X F , τ ) ∂τ τ = t , (12)where X = K S ( X S ) and x = χ S ( X , t ) = χ F ( X F , t ) .In particular, the kinematical assumption (5) leads the Eulerian velocity fieldof the solid body (11) to read v S ( x , t ) = ε ∂ u S ( X , τ ) ∂τ τ = t =: ε v S ( X , t ) . (13)Moreover, as we exclude a priori the possibility that any three-dimensionalregion of the reference shape collapses under the motion χ S ,J S ( x , t ) = det F S ( X , t ) > , with F S := Grad χ S , (14)there exists a smooth inverse mapping, χ − ( · , t ) : χ S ( B , t ) → E , (15)that satisfies the trivial identity X = χ − χ S ( X , t ) , t , ∀ X ∈ B , (16)yielding the following property at any time t and place X ,grad χ − ( x , t ) = Grad χ S ( X , t ) − . (17)
114 S. Quiligotti, G.A. Maugin and F. dell’Isola ZAMP
In order to describe the motion of the fluid constituent through the referenceshape of the solid, we also notice that any fluid particle interacts with a smooth1-parameter family of solid ones, moving along the curve χ − χ F ( X F , · ) , · : t → X , (18)at the velocity w F ( X , t ) , defined by the relation v F ( x , t ) = F S ( X , t ) w F ( X , t ) + v S ( x , t ) , (19)where, as usual, x = χ F ( X F , t ) = χ S ( X , t ) .With the aim of linearizing the equations that govern the nonlinear dynamicsof the mixture, we shall consider a smooth 1-parameter family of relative velocityfields, w F := w FO + γ w F , (20)assuming, for the sake of simplicity, that the two independent perturbation pa-rameters introduced so far are of the same order of magnitude, o ( γ ) = o ( ε ) , andthe unperturbed velocity field w FO is identically equal to the null vector, w FO ( X , t ) = , ∀ X ∈ B , (21)so as to get the following expression for the Eulerian fluid velocity (19): v F ε v S + ε w F =: ε v F . (22)
3. Mass conservation
By definition, the α -mass content of any smooth region of the current shape ofthe solid-fluid mixture, V ⊂ B ( t ) , is given by the measure M α ( V ) = V α , with α ∈ { S,F } . (23)If mixture constituents undergo neither chemical reactions nor phase transi-tions, the time derivative of the α -mass content of V has to vanish followingthe α -motion of the migrating material surface that coincides, at the given time τ = t , with the boundary of the chosen smooth region V , i.e. dd τ V α ( τ ) α τ = t = V ∂ α ∂ t + div ( α v α ) = 0 , (24)leading to the local mass conservation law ∂ α ∂ t + div ( α v α ) = 0 . (25)Introducing an α -mass density per unit reference volume of the mixture, α ( X , t ) = J S ( x , t ) α ( x , t ) , ∀ x = χ S ( X , t ) ∈ B ( t ) , (26) ol. 53 (2002) Wave motions in unbounded poroelastic solids 1115 as the scalar field on B that satisfies the integral equality V α := V α , ∀ V = χ − ( V , t ) , (27)we may also evaluate, at any given time τ = t , the material derivative of the α -mass content of V , following the motion of the α -th constituent through thereference shape B , and taking into account that V α ( τ ) = χ − ( V α ( τ ) , τ ) , with V α ( t ) = V , dd τ V α ( τ ) α τ = t = V α ( t ) ∂ α ∂ t + Div ( α w α ) = 0 , (28)so as to get an alternative form of the local mass conservation law, ∂ α ∂ t + Div ( α w α ) = 0 . (29)Expanding in power series the reciprocal of the smooth functionJ S ( x , t ) = det F S ( X , t ) , ∀ x = χ S ( X , t ) ∈ B ( t ) (30)in a neighborhood of the reference shape,1det F S = 1 − ε Div u S + o ( ε ) , (31)we deduce that, by virtue of definition (26), the value of the current mass densityof any α -particle that occupies the given place x at time t , α ( x , t ) = α ( X , t ) (1 − ε Div u S ( X , t )) + o ( ε ) , with x = χ S ( X , t ) , (32)equals the value of the reference α -mass density α ( X , t ) if the displacement fieldof the solid body results to be divergence-free at the same given time and place,i.e. Div u S ( X , t ) = 0 .As the reference shape does not depend on time, when α = S the integralconservation laws (24) and (28) lead to the trivial conclusion that the scalar field S is independent of time ( w S = ) as well, ∂ S ∂ t = 0 , (33)and thus, bearing in mind the equality (32), S ( x , t ) − S ( X ) − S ( X ) Div ε u S ( X , t ) , ∀ x = χ S ( X , t ) ∈ B ( t ) . (34)Because of the overlapping between the two constituents, any smooth region ofthe current shape of the mixture can also be associated with a fluid subbody. Inparticular, for any choice of V S ( t ) , there exists a fluid subbody, P F ⊂ B F , suchthat V F ( τ ) = χ F ( P F , τ ) , with V F ( t ) = V S ( t ) at time τ = t .In order to linearize the local expression of the α -mass balance (29) in the caseof α = F , we shall assume that the fluid motion takes place in a neighborhood of
116 S. Quiligotti, G.A. Maugin and F. dell’Isola ZAMP the reference shape of the mixture, i.e. F ( X , t ) = F0 ( X ) + ε F1 ( X , t ) + o ( ε ) , (35) w F ( X , t ) = ε w F ( X , t ) . (36)As a consequence, it is possible to deduce that the F -mass density fields F0 and F1 have to fulfill, respectively, two independent requirements: ∂ F0 ∂ t = 0 , (37) ∂ F1 ∂ t + Div F0 w F = 0 . (38)Moreover, keeping in mind the expression (32), we can remark that the dif-ference between current and reference F -mass densities is finally given by therelation: F ( x , t ) − F0 ( X ) = ε F1 ( X , t ) − F0 ( X ) Div ε u S ( X , t ) + o ( ε ) . (39)
4. Fluid-saturated poroelastic continua
In order to develop a macroscopic theory of saturated poroelastic continua (seee.g. Bowen [7, 5], Svendsen and Hutter [31], Klisch [20]), we shall enrich a self-consistent mathematical theory of binary solid-fluid mixtures by introducing theconcept of volume fraction (Fillunger, see de Boer [10]; cf. Bluhm et al. [4],dell’Isola et al. [11]). In particular, we consider two independent scalar fields ν α (with α ∈ { S,F } ), which represent the dimensionless ratio of the macroscopic mass density α to the (constitutively prescribed) microscopic mass density ˆ α , ν α ( x , t ) := α ( x , t )ˆ α X , E S ( X , t ) , F ( X , t ) , ∀ x = χ S ( X , t ) ∈ B ( t ) , (40)denoting by E S the Lagrangian strain tensor, E S = 12 F T S F S − I = ε sym (Grad u S ) + o ( ε ) , (41)such that tr E S ε Div u S . (42)A poroelastic solid infused with a compressible fluid is saturated if its solidskeleton is perfectly permeated by the fluid, i.e. if the saturation constraint isidentically fulfilled: ν S + ν F − . (43)This implies, by virtue of definitions (40) and (26), that S ( X ) ˆ F ( X , t ) + F ( X , t ) ˆ S ( X , t ) = ˆ S ( X , t ) ˆ F ( X , t ) det F S ( X , t ) . (44) ol. 53 (2002) Wave motions in unbounded poroelastic solids 1117 In order to linearize the algebraic equation (43), we consider a linear constitu-tive prescription for microscopic mass densities,ˆ α X , E S ( X , t ) , F ( X , t ) ˆ α ( X ) + ε ˆ α ( X , t ) , (45)where † ˆ α ( t ) := − ˆ α λ α S Div u S ( t ) + λ α F F1 ( t )ˆ F0 , (46)denoting by λ αβ ( α , β ∈ { S , F } ) a smooth time-independent scalar field on B .If we assume that the reference state is saturated, ν S0 + ν F0 − ⇐⇒ S ˆ F0 + F0 ˆ S0 = ˆ S0 ˆ F0 , (47)with ν S0 := S ˆ S0 and ν F0 := F0 ˆ F0 , (48)then the algebraic equation (44) can be finally linearized, by virtue of expansions(35) and (45), resulting in: S ˆ F1 ( t ) + F0 ˆ S1 ( t ) + ˆ S0 F1 ( t ) = ˆ S0 ˆ F1 ( t ) + ˆ F0 ˆ S1 ( t ) + ˆ S0 ˆ F0 Div u S ( t ) , (49)and thus leading to the requirement:ˆ F0 β S Div u S ( t ) + β F F1 ( t ) = 0 , (50)with β S := ν F0 λ FS + ν S0 λ SS − , and β F := ν F0 λ FF + ν S0 λ SF + 1 . (51)It is worth mentioning (see Table 1) that if the coe ffi cients λ αβ satisfy the twoequations ν F0 λ FS + ν S0 λ SS − ν F0 λ FF + ν S0 λ SF + 1 = 0 (52)for all X ∈ B , then the constraint (50) is identically fulfilled by any possible fluidand solid motion. Otherwise, bearing in mind the expansion (35) and the equations(37)-(38), if the former of (52) is identically satisfied and the latter is not, thenonly F -motions characterized by the conditionDiv F0 w F ( t ) = 0 (53)can be realized; analogously, if the latter of (52) is identically satisfied and theformer is not, only isochoric S -motions are allowed. In the general case of β S = 0and β F = 0 , the saturation constraint (50) just establishes a relation between theevolution of the fluid mass-density per unit reference volume of the mixture andthe solid motion, that may also be rewritten in the alternative form: F ( t ) F0 − β S ν F0 β F Div ε u S ( t ) . (54)
118 S. Quiligotti, G.A. Maugin and F. dell’Isola ZAMP β F = 0 β F = 0all S -motion all S -motion β S = 0 all F -motion F1 ( t ) = 0Div u S ( t ) = 0 β S = 0 ˆ F0 β S Div u S ( t ) + β F F1 ( t ) = 0all F -motionTable 1. The saturation constraint. Consistently, the expansions of volume fractions result in: ν S ( t ) ν S0 + εν S0 ( λ SS −
1) Div u S ( t ) + ελ SF ν S0 F1 ( t )ˆ F0 (55) ν F ( t ) ν F0 + εν F0 ( λ FS −
1) Div u S ( t ) + ε ( ν F0 λ FF + 1) F1 ( t )ˆ F0 . (56)In order to highlight the role played by the coe ffi cients λ αβ ( α , β ∈ { S , F } )within the framework of the propounded constitutive theory, we observe in passingthat, by virtue of definition (46) and hypothesis (45), it is possible to rewrite theexpressions postulated for microscopic mass densities,ˆ S ( t ) = ˆ S0 − λ SS ˆ S0 Div ε u S ( t ) − λ SF ˆ S0 ν F0 ε F1 ( t ) F0 (57)ˆ F ( t ) = ˆ F0 − λ FS ˆ F0 Div ε u S ( t ) − λ FF ε F1 ( t ) , (58)in a slightly di ff erent form, so as to emphasize their constitutive dependence onmacroscopic (current) mass densities S ( t ) and F ( t ), namely:ˆ S ( t ) − ˆ S0 ˆ S0 = λ SS S ( t ) − S S − ν F0 λ SF F ( t ) − F0 F0 (59)ˆ F ( t ) − ˆ F0 ˆ F0 = λ FS S ( t ) − S S − ν F0 λ FF F ( t ) − F0 F0 . (60) † For the sake of conciseness, from now on we shall drop the reference place X in any list ofarguments. For instance, we shall write ˆ α + ε ˆ α ( t ) in place of the right-hand side offirst-order expansion (45).ol. 53 (2002) Wave motions in unbounded poroelastic solids 1119
5. Dynamics
With the aim of describing local interactions exchanged by overlapped α -points(figure 1) within the framework of a first-order gradient theory (Germain [17],Williams [36]), we assume the stress power per unit reference volume to be givenby the expression: α ∈ { S,F } (J S π α · v α + T α · Grad v α ) , (61)where, respectively, T α denotes the partial Piola-Kirchho ff stress tensor asso-ciated with the α -th constituent of the mixture (see, e.g. Bowen [7], Wilman-ski [37, 38]), whereas π α represents a general zeroth-order interaction (see e.g.dell’Isola et al. [11], Quiligotti et al. [27]).According to the principle of material frame-indi ff erence, the internal powerdensity, expended on any rigid-body velocity field v RS ( x , t ) = v RF ( x , t ) = ω ( t ) + Ω ( t ) ( x − x ) , Ω ( t ) ∈ Skw , (62)needs to vanish for any choice of spatially uniform ω ( t ) and Ω ( t ) ∈ Skw . As aconsequence, admissible constitutive assumptions have to satisfy the preliminaryrequirements: skw ( T S + T F ) F T S = O , (63) π S + π F = . (64) Let us consider a material surface ∂ V ( τ ) ⊂ B , which migrates through the refer-ence shape following the motion of the mixture as a whole. As S-points are fixedin B (namely, w S = in (29)), we shall assume that any material point of themixture moves through the reference shape at the velocity w := ξ F w F , (65)denoting by ξ F the fluid-mass fraction, ξ F ( t ) := F ( t ) S + F ( t ) . (66)Moreover, we assume that there exists an overall strain-energy density per unitreference volume of the mixture, W ( X , t ) , such that the time derivative of thestrain-energy content of V ( τ ) ⊂ B , dd τ V ( τ ) W τ = t = V ( t ) ∂ W ∂ t + Div ( W w ) , (67)
120 S. Quiligotti, G.A. Maugin and F. dell’Isola ZAMP equals the stress power expended on the velocity pair ( v S , v F ), dd τ V ( τ ) W τ = t = α ∈ { S,F } V ( t ) (J S π α · v α + T α · Grad v α ) , (68)for any choice of V ( t ) ⊂ B . This leads to the local expression: ∂ W ∂ t + Div ( W w ) = α ∈ { S,F } (J S π α · v α + T α · Grad v α ) , (69)whose right-hand side may be alternatively rewritten, by taking into account therelation (19) and the preliminary requirement (64), as T · Grad v S + τ · w F + T · Grad w F , (70)with T := T S + T F (71) τ := (Grad F S ) T T F + J S F T S π F (72) T := F T S T F . (73) Unconstrained solid-fluid mixture
In order to investigate the physical meaning of local first-order interactions (71)-(73), let us consider an overall strain-energy density per unit reference volume ofthe mixture, W ( X , t ) = W X , C S ( X , t ) , F ( X , t ) , (74)whose value, at any time t and (reference) place X ∈ B , depends on:(a) the reference place itself, i.e. the solid particle steadily associated with it(namely, X S ∈ B S such that X = K S ( X S ) ∈ B ; see figure 1);(b) the corresponding value of the right Cauchy-Green tensor ( C S = F T S F S );(c) the value of the fluid mass density per unit reference volume of the corre-sponding fluid particle ( X F ∈ B F ) which is placed, at the given time t , atthe same current position occupied by the solid particle X S ∈ B S (namely, x = χ S ( X , t ) = χ F ( X F , t ) ∈ E ).With the aim of linearizing the equations that govern the nonlinear dynamics ofthe mixture, we may expand † in power series the strain-energy function in a neigh- † For the sake of conciseness, we omit the arguments of C S and F .ol. 53 (2002) Wave motions in unbounded poroelastic solids 1121 borhood of a pre-stressed (saturated) reference state ( C S = I , and F = F0 ), W X , C S , F W o + 12 A · ( C S − I ) + a F − F0 ++ 18 { B ( C S − I ) } · ( C S − I ) + (75)+ 12 b F − F0 + 12 F − F0 B · ( C S − I ) , where W o = W X , I , F0 .As ( C S − I ) is a symmetric tensor, only the symmetrical part of A , B and B is responsible for any contribution to the strain-energy density. Thus, we shallassume these coe ffi cients to be symmetrical. Moreover, as the strain-energy issupposed to be inhomogeneous, a, b, A , B and B will generally depend on X , aswell as W o . Recalling that12 ( C S − I ) ε sym (Grad u S ) + 12 ε (Grad u S ) T Grad u S , (76) F − F0 ε F1 , (77)we finally obtain the expression: W W o + A · ε Grad u S + ε a F1 ++ 12 ε { (Grad u S ) A } · Grad u S + 12 b ε F1 + (78)+ ε F1 B · Grad u S + 12 ε { B (Grad u S ) } · Grad u S . As it will be useful later on, we notice, in passing, that the considered first-order approximation for the F-mass density per unit reference volume (35) leadsthe definition (66) of the F-mass fraction ξ F ( t ) to yield: ξ F ( t ) F0 + ε F1 ( t ) S + F0 + ε F1 ( t ) ξ F0 + εξ F1 ( t ) , (79)where ξ F0 and ξ F1 ( t ) are given, respectively, by the expressions: ξ F0 := F0 S + F0 , and ξ F1 ( t ) := (1 − ξ F0 ) F1 ( t ) S + F0 . (80)The analogous first-order approximation for the S-mass fraction ξ S ( t ) reads: ξ S ( t ) S S + F0 + ε F1 ( t ) ξ S0 + εξ S1 ( t ) , (81)with ξ S0 := S S + F0 , and ξ S1 ( t ) := − ξ S0 F1 ( t ) S + F0 . (82)
122 S. Quiligotti, G.A. Maugin and F. dell’Isola ZAMP
Consistently, we can verify that the independent requirements: ξ F0 + ξ S0 = 1 , (83) ξ F1 ( t ) + ξ S1 ( t ) = 0 , (84)are identically fulfilled at any time t .With the aim of deducing from (69) an admissible set of constitutive prescrip-tions for the interactions (71)-(73), consistent with the assumption (78), we cannow write down the resulting expression for the partial time derivative of theoverall strain-energy density per unit reference volume, ∂ W ∂ t [ I + Grad ( ε u S )] A + ε F1 B + B Grad ( ε u S ) · Grad ( ε v S ) + − F0 a + b ε F1 + B · Grad ( ε u S ) I · Grad ( ε w F ) + (85) − a + b ε F1 + B · Grad ( ε u S ) Grad F0 · ε w F , keeping in mind equation (38) and definitions (13) and (20).Moreover, recalling that w = ξ F w F and considering a first-order approxima-tion for the F-mass fraction ξ F (79), we obtain:Div ( W w ) = w F · Grad ( ξ F W ) + ξ F W I · Grad w F , (86)where w F · Grad ( ξ F W ) ε w F · Grad ( ξ F0 W o ) + Grad ( εξ F1 W o ) ++ Grad ξ F0 A · Grad ( ε u S ) + ξ F0 a ε F1 , (87)and ξ F W I · Grad w F Grad ( ε w F ) · ξ F0 W o + εξ F1 W o + ξ F0 a ε F1 ++ ξ F0 A · Grad ( ε u S ) I . (88)Henceforth, the coupled interactions (71)-(73) finally result in: T ( t ) T + ε T ( t ) (89) τ ( t ) τ + ε τ ( t ) (90) T ( t ) T + ε T ( t ) , (91)with: T = A (92) T ( t ) = F1 ( t ) B + { Grad u S ( t ) } A + B Grad u S ( t ) , (93) τ = − a Grad F0 + Grad ( ξ F0 W o ) (94) τ ( t ) = − b F1 ( t ) + B · Grad u S ( t ) Grad F0 + Grad ξ F1 ( t ) W o ++ Grad ξ F0 A · Grad u S ( t ) + a ξ F0 F1 ( t ) , (95) ol. 53 (2002) Wave motions in unbounded poroelastic solids 1123 T = ξ F0 W o − a F0 I (96) T ( t ) = ξ F1 ( t ) W o + a ξ F0 F1 ( t ) + ξ F0 A · Grad u S ( t ) I + − F0 b F1 ( t ) + B · Grad u S ( t ) I . (97) Constrained solid-fluid mixture
As constraints are naturally associated with reactive actions, if the solid con-stituent is required to be perfectly permeated by the fluid, a saturation pressure p ( X , t ) arises in the mixture so as to maintain each constituent in contact with theother one. Taking as granted that such a pressure does not expend power in anymotion compatible with the constraint (43), we may regard the additional scalarfield p as a Lagrangian multiplier in the expression of the overall strain-energydensity per unit reference volume, W ( X , t ) = W c X , C S ( X , t ) , F ( X , t ) , p ( X , t ) = (98)= W X , C S ( X , t ) , F ( X , t ) + p ( X , t ) { ν S ( X , t ) + ν F ( X , t ) − } , such that, expanding in power series the strain-energy function in a neighborhoodof a pre-stressed (saturated) reference state ( C S = I , F = F0 , p = p ), andbearing in mind the expressions (50) and (51), the second-order approximation ofthe last term of the strain-energy function (98) results † in: p ( t ) { ν S ( t ) + ν F ( t ) − } { p + ε p ( t ) } β S Div ε u S ( t ) + β F ε F1 ( t )ˆ F0 , (99)so as to get (compare with (78)), in the end, W W o + ˜ A · ε Grad u S + ε ˜ a F1 + 12 b ε F1 ++ 12 ε { (Grad u S ) A } · Grad u S + ε F1 B · Grad u S + (100)+ 12 ε { B Grad u S } · Grad u S , with ˜ A ( t ) := A + p β S I + ε p ( t ) β S I , (101)˜ a ( t ) := a + β F p ˆ F0 + ε β F p ( t )ˆ F0 . (102)Following the same procedure as that outlined in the former section, it is pos-sible to deduce from (100) a suitable expression for the generalized coupled forces † For the sake of conciseness, we shall drop the reference place X in any list of arguments.124 S. Quiligotti, G.A. Maugin and F. dell’Isola ZAMP (71)-(73), so as to describe a first-order interaction between the fluid-saturatedporoelastic solid and the filling fluid, namely (compare with (92)-(97)): T = A + p β S I (103) T ( t ) = F1 ( t ) B + { Grad u S ( t ) } A ++ B Grad u S ( t ) + p ( t ) β S I , (104) τ = − a + β F p ˆ F0 Grad F0 + Grad ( ξ F0 W o ) (105) τ ( t ) = − b F1 ( t ) + B · Grad u S ( t ) + β F p ( t )ˆ F0 Grad F0 ++ Grad { ξ F0 ( A + p β S I ) · Grad u S ( t ) } ++ Grad ξ F0 a + β F p ˆ F0 F1 ( t ) + ξ F1 ( t ) W o , (106) T = ξ F0 W o − a F0 − ν F0 β F p I (107) T ( t ) = ξ F1 ( t ) W o + ξ F0 a + β F p ˆ F0 F1 ( t ) I ++ { ξ F0 ( A + p β S I ) · Grad u S ( t ) } I + − F0 b F1 ( t ) + B · Grad u S ( t ) + β F p ( t )ˆ F0 I . (108) We assume (cf. Biot [3], de Boer [10], Edelman and Wilmanski [14] ) that thekinetic energy density per unit reference volume of the mixture is given by thesum α ∈ { S , F } α v α · v α = 12 v · v − ( d S · d F ) (109)where and v are, respectively, the overall mass density per unit referencevolume and the mean velocity of mixture particles, := S + F , (110) v := ξ S v S + ξ F v F = v S + ξ F F S w F , (111)whereas d α denotes the di ff usion velocity of the α -th constituent, d S := v S − v = ξ F ( v S − v F ) , (112) d F := v F − v = ξ S ( v F − v S ) . (113) ol. 53 (2002) Wave motions in unbounded poroelastic solids 1125 The time derivative of the kinetic energy associated with any smooth region ofthe reference shape, V ( τ ) ⊂ B , enveloped by a migrating surface which followsthe motion of the mixture as a whole, is required to equal, at any time t , theintegral over V ( t ) of the power expended by inertial forces, α ∈ { S , F } dd τ V ( τ ) α v α · v α τ = t = V ( t ) ( v S · f S + w F · f F ) , (114)whose local expression is given † by f S := D v D t , (115) f F := 12 F T S D v D t − ξ S D v S D t + ξ F D v F D t . (116)In order to linearize the equations that govern the nonlinear dynamics of themixture, we notice, by taking into account a first-order approximations of , F T S , v , v S , and v F (see sections 2 and 3), that linearized expressions for (115) and(116) result in: f F f F0 + ε f F1 = ε F0 ∂ u S ∂ t + F0 ∂ w F ∂ t , (117) f S f S0 + ε f S1 = ε ( S + F0 ) ∂ u S ∂ t + F0 ∂ w F ∂ t , (118)with f F0 = f S0 = . The required set of balance equations and boundary conditions, that governs thenonlinear dynamics of the mixture, can be straightforwardly deduced from a gen-eral statement of the principle of virtual power (cf. Maugin [22]).As the thorny question of splitting the overall applied boundary traction amongthe constituents still stands as one of the greatest challenges that have to be facedup in order to put mixture theories into use (see e.g. Rajagopal and Tao [28],Reynolds and Humphrey [29]), it is worth emphasizing that, within the frameworkof variational principles, boundary conditions are derived as a result of the theory,as well as governing equations. † The di ff erential operator D ( ) D t := ∂ ( ) ∂ t + [Grad ( )] w denotes the material derivative of a given vector field ( ) , defined on the referenceconfiguration B , following the motion of the mixture as a single body.126 S. Quiligotti, G.A. Maugin and F. dell’Isola ZAMP If the total power expended on any conceivable pair of smooth test velocityfields ˆ v s and ˆ w f (in absence of applied body forces) is required to vanish, ∂ B t · ˆ v s + ξ F F T S t · ˆ w f − B (ˆ v s · f S + ˆ w f · f F ) + − B ( T · Grad ˆ v s + τ · ˆ w f + T · Grad ˆ w f ) = 0 , (119)the resulting set of nonlinear balance equations and boundary conditions is:Div T = f S Div T − τ = f F on B ⊂ E Tn = t T n = ξ F F T S t on ∂ B ⊂ E , (120)where t represents the overall applied boundary traction.These equations may be linearized by taking into account first-order expansionsof all vector and tensor fields involved. In particular, recalling that f S0 = f F0 = (section 5.3), it is possible to deduce the set of balance equations and boundaryconditions that characterize the reference state,Div T = Div T − τ = on B ⊂ E T n = t T n = ξ F0 t on ∂ B ⊂ E , (121)whereas the perturbed state is governed by the equations:Div T = S + F0 ∂ u S ∂ t + F0 ∂ w F ∂ t Div T − τ = F0 ∂ u S ∂ t + F0 ∂ w F ∂ t on B ⊂ E T n = t T n = ξ F0 t + ξ F0 (Grad u S ) T t + ξ F1 t on ∂ B ⊂ E . (122) A plane harmonic wave (see e.g. Achenbach [1], Gra ff [18]), propagating withphase velocity c in a direction defined by the unit vector q , can be generallyrepresented by the real (or the imaginary) part of a complex function, ψ ( X , t ) = ˜ ψ ( ω ) e ik ( X · q − ct ) , (123) ol. 53 (2002) Wave motions in unbounded poroelastic solids 1127 whose amplitude ˜ ψ ( ω ) is independent of ( X , t ) .By definition, the characteristic wavenumber k is related to the circular (orradial) frequency ω by the algebraic relation ω = kc , while the wavelength Λ isgiven by tha ratio Λ = 2 π /k .With the aim of investigating the propagation of harmonic displacement wavesin unbounded media, we shall assume homogeneity and isotropy of constitutiveprescriptions, F1 ( t ) B · Grad u S ( t ) := F1 ( t ) β I · Grad u S ( t ) = F1 ( t ) β Div u S ( t ) (124) B Grad u S ( t ) := λ S Div u S ( t ) I + µ S Grad u S ( t ) + µ S Grad T u S ( t ) (125) A := α I , (126)where α , β , µ S and λ S are assumed to be constant in the chosen referencestate, furthermore characterized by a uniform (macroscopic) fluid-mass density( Grad F0 = ), and a constant (macroscopic) fluid compressibility (namely,Grad b = ). Moreover, for the sake of simplicity, we focus our attention onuniform (microscopic) mass densities per unit reference volume ( Grad ˆ S0 = ,Grad ˆ F0 = ), so as to deal with a constant reference porosity ( Grad ν F0 = ). Unconstrained solid-fluid mixture
At first, we investigate the problem of the propagation of harmonic plane wavesin unbounded and unconstrained solid-fluid mixtures, whose linearized dynamicsis governed by the fluid-mass conservation law (38) and the set of field equations(122) . Recalling that, by assumption,Grad F0 = (127)and T ( t ) = β F1 ( t ) + λ S Div u S ( t ) I ++ 2 µ S sym (Grad u S ( t )) + α Grad u S ( t ) (128) τ ( t ) = Grad ξ F1 ( t ) W o + ξ F0 A · Grad u S ( t ) + a ξ F0 F1 ( t ) (129) T ( t ) = ξ F1 ( t ) W o + a ξ F0 F1 ( t ) + ξ F0 A · Grad u S ( t ) I + − F0 b F1 ( t ) + β Div u S ( t ) I , (130)
128 S. Quiligotti, G.A. Maugin and F. dell’Isola ZAMP it is possible to formulate the problem in terms of the field triplet u S , w F , F1 , ∂ F1 ∂ t + F0 Div w F = 0 (131) S + F0 ∂ u S ∂ t + F0 ∂ w F ∂ t − ( λ S + µ S ) Grad (Div u S ) + − ( µ S + α ) Div (Grad u S ) − β Grad F1 = (132) F0 ∂ u S ∂ t + F0 ∂ w F ∂ t + F0 β Grad (Div u S ) + b F0 Grad F1 = , (133)whereas the (unperturbed) reference state satisfies the set of linear field equationsdeducible from (121) , namely Div T = Div A = Grad α = Div T − τ = − F0 Grad a = . (134)Looking for steady-state solutions in the form: u S ( X , t ) = ˜ u e ik ( X · q − ct ) (135) w F ( X , t ) = − ikc ˜ w e ik ( X · q − ct ) (136) F1 ( X , t ) = ˜ e ik ( X · q − ct ) , (137)we find out, by virtue of equation (131), a relation between ˜ and ˜ w ,˜ = − ik F0 ( ˜ w · q ) , (138)which may be used to uncouple the subset of equations (132)-(133) from the fluidmass-conservation law (131). Henceforth, we can at first focus our attention onthe resulting subset of (algebraic) equations (Meirovitch [24, 25]) that governs thesteady-state linearized dynamics of the mixture, K − c M X = O , (139)where M , K and X denote, respectively: { M } := S + F0 I F0 I F0 I F0 I (140) { K } := S c TS I + S c LS − c TS ( q ⊗ q ) − β F0 ( q ⊗ q ) − β F0 ( q ⊗ q ) F0 c LF ( q ⊗ q ) (141) { X } := ˜ u ˜ w , (142)with c LS = λ S + 2 µ S + α S , c TS = µ S + α S , and c LF = F0 b . (143) ol. 53 (2002) Wave motions in unbounded poroelastic solids 1129 We notice in passing that the mass matrix M is symmetrical and positivedefinite for any referential value of mass fractions belonging to the open set ( 0 , ξ F0 + ξ S0 ± ξ + ξ =: h ± ( ξ S0 ) , (144)characterized by a triple multiplicity, are associated with the following set of lin-early independent eigenvectors, ξ F0 e j g ± ( ξ S0 ) e j ; j ∈ { , , } , (145)where e i · e j = δ ij , and (figure 2) g ± ( ξ S0 ) := ± ξ + ξ − ξ S0 . (146)Moreover, it can be shown that det M = F0 S (figure 3).The sti ff ness matrix K is also symmetrical. Furthermore, it is positive semidef-inite † if the (macroscopic) coupling coe ffi cient β , introduced in (75) by means ofassumption (124), meets the requirement: | β | ≤ β max , β max := S F0 c LS c LF = b ( λ S + 2 µ S + α ) . (147)General features of both eigenvectors and eigenvalues of K are summarized intables 2, 3 and 4. ˜ u k = ˜ u k = g ± ( β ) q ˜ u k · q = 0˜ w k = — — S c TS ˜ w k = g ± ( β ) q — h ± k ( β ) —˜ w k · q = 0 0 — —Table 2. Eigenvectors and eigenvalues of K ( β = 0 ). In particular, in order to investigate the role played by the macroscopic couplingparameter β within the framework of the constitutive theory proposed, we can † In fact, it is worth recalling that the strain-energy density per unit reference volume of themixture (75) depends on the macroscopic kinematics of the fluid constituent, by assumption,only through the trace of its velocity gradient. Consistently, no shear wave can be sustained inthe fluid.130 S. Quiligotti, G.A. Maugin and F. dell’Isola ZAMP˜ u k = ˜ u k = q ˜ u k · q = 0˜ w k = — S c LS S c TS ˜ w k = q F0 c LF — —˜ w k · q = 0 0 — —Table 3. Eigenvectors and eigenvalues of K ( β = 0 ).˜ u k = q ˜ u k = − g k q ˜ w k = g k q S c LS + F0 c LF —˜ w k = q — 0Table 4. Longitudinal eigenvectors and eigenvalues of K ( β = β m ). remark that the longitudinal coupled eigenvectors, g ± ( β ) q g ± ( β ) q , (148)and their corresponding eigenvalues, h ± k ( β ) := 12 S c LS + F0 c LF ± β F0 + S c LS − F0 c LF , (149)have to satisfy identically, for any admissible value of β (147), the following setof algebraic equations: S c LS − h ± k ( β ) g ± ( β ) − β F0 g ± ( β ) = 0 − β F0 g ± ( β ) + F0 c LF − h ± k ( β ) g ± ( β ) = 0 , (150)which yields, in the general case of β = 0 , g ± ( β ) g ± ( β ) = 12 β F0 S c LS − F0 c LF ∓ β F0 + S c LS − F0 c LF . (151)It is worth taking notice of the fact that, in the case of β = 0 (table 3), h + k (0) = S c LS , g +1 (0) q g +2 (0) q ∝ q0 ,h − k (0) = F0 c LF , g − (0) q g − (0) q ∝ , (152) ol. 53 (2002) Wave motions in unbounded poroelastic solids 1131 whereas the assumption β = β m = ( sgn β ) β max results in (table 4) h + k ( β m ) = S c LS + F0 c LF , g +1 ( β m ) q g +2 ( β m ) q ∝ q g k q ,h − k ( β m ) = 0 , g − ( β m ) q g − ( β m ) q ∝ − g k qq , (153)with g k := ( sgn β m ) S c LS F0 c LF = sgn β m F0 λ S + 2 µ S + α b ; sgn β m = β m | β m | . (154) -2-1.5-1-0.500.511.52 0 0.2 0.4 0.6 0.8 1 h + ( ξ S0 ) h − ( ξ S0 ) g + ( ξ S0 ) g − ( ξ S0 ) Referential S-mass fraction ξ S0 Figure 2. Eigenvectors and eigenvalues of M ( h ± ( ξ S0 ) and g ± ( ξ S0 )) .132 S. Quiligotti, G.A. Maugin and F. dell’Isola ZAMP Referential S-mass fraction ξ S0 Figure 3. det M = ( ξ F0 ξ S0 ) . We can now investigate the overall dynamical properties of the mixture, emerg-ing from the analysis of the algebraic compact form (139) of the set of di ff erentialequations (132)-(133), S c TS − c ˜ u c + S c LS − c TS ( q · ˜ u c ) q + − β F0 ( q · ˜ w c ) q − F0 c ˜ w c = 0 (155) − F0 c ˜ u c − F0 c ˜ w c − β F0 ( q · ˜ u c ) q + F0 c LF ( q · ˜ w c ) q = 0 , (156)whose solutions physically characterize all possible kinds of plane elastic waves thatcan be sustained in the medium, according to the proposed constitutive (macro-scopic) theory. Transverse waves.
As the strain-energy density per unit reference volume ofthe mixture is assumed to depend on the macroscopic kinematics of the fluid ol. 53 (2002) Wave motions in unbounded poroelastic solids 1133 constituent only through the trace of its velocity gradient, the eigenvectors ˜ u c ˜ w c ∝ ˜ w c , with ˜ w c · q = 0 , (157)are naturally associated with null eigenvalues. Moreover, it can be straightfor-wardly deduced from equations (155)-(156) that elastic transverse waves are asso-ciated with the eigenvectors that satisfy the requirements:˜ u c · q = ˜ w c · q = 0 , (158)˜ u c + ˜ w c = . (159)By virtue of constitutive prescriptions assumed for the coupling coe ffi cient B (124), the resulting phase velocity of transverse waves propagating in the mixture, c = c T , is exactly equal to the characteristic speed of transverse waves that wouldpropagate in the solid constituent, c T = c TS . Longitudinal waves.
Longitudinal coupled eigenvectors, ˜ u c ˜ w c ∝ λ ± ( β ) q λ ± ( β ) q , (160)are naturally associated with the longitudinal waves that can be sustained in themixture, whose phase velocities c L ± = 12 S h ( β ) ± h ( β ) + 4 S F0 β − β max , (161)with (cf. Biot [3], Wilmanski [38], Edelman and Wilmanski [14]) h ( β ) = S c LS + F0 + S c LF + 2 β F0 , (162)are given by the solutions of the characteristic equation ξ F0 c L + β − c L − c LF c L − ξ S0 c LS = 0 . (163)Accordingly, the coe ffi cients λ ± ( β ) and λ ± ( β ) in (160) have to satisfy identi-cally, for any admissible value of β (147), the set of algebraic equations: S c LS − S + F0 c L ± λ ± ( β ) − β F0 + F0 c L ± λ ± ( β ) = 0 − β F0 + F0 c L ± λ ± ( β ) + F0 c LF − F0 c L ± λ ± ( β ) = 0 . (164) Constrained solid-fluid mixture
Although it proves itself capable to catch the bare essentials of the mechanicalbehavior exhibited by a poroelastic solid infused with a Stokesian fluid (cf. Cowin[8]), the theoretical model presented so far is purely macroscopic .
134 S. Quiligotti, G.A. Maugin and F. dell’Isola ZAMP
To take into account the most relevant microstructural properties of the mix-ture, we may better enrich such a model by introducing two independent scalarfields of volume fraction (see sections 4 and 5.2), constrained by the kinematicalrequirement (43). As kinematical constraints are naturally associated with reac-tive actions, a saturation pressure p ( X , t ) needs to arise in the mixture so as tomaintain each constituent in contact with the other one. Regarding the saturationpressure as a Lagrangian multiplier in the expression of the overall strain-energydensity per unit reference volume (100), and recalling that, by assumption,Grad F0 = (165)and (compare with expressions (128)-(130)), T ( t ) = β F1 ( t ) + λ S Div u S ( t ) + p ( t ) β S I ++ 2 µ S sym (Grad u S ( t )) + α Grad u S ( t ) (166) τ ( t ) = Grad ξ F1 ( t ) W o + ξ F0 ( A + p β S I ) · Grad u S ( t ) ++ Grad ξ F0 a + β F p ˆ F0 F1 ( t ) (167) T ( t ) = ξ F1 ( t ) W o + ξ F0 a + β F p ˆ F0 F1 ( t ) I ++ { ξ F0 ( A + p β S I ) · Grad u S ( t ) } I + − F0 b F1 ( t ) + β Div u S ( t ) I , (168)it is possible to formulate the problem in terms of the field quadruplet { u S , w F , F1 , p } , ∂ F1 ∂ t + F0 Div w F = 0 (169) S + F0 ∂ u S ∂ t + F0 ∂ w F ∂ t − ( λ S + µ S ) Grad (Div u S ) + − ( µ S + α ) Div (Grad u S ) − β Grad F1 − β S Grad p = (170) F0 ∂ u S ∂ t + F0 ∂ w F ∂ t + F0 β Grad (Div u S ) + F0 b Grad F1 ++ ν F0 β F Grad p = (171)ˆ F0 β S Div u S ( t ) + β F F1 ( t ) = 0 , (172)whereas the (unperturbed) reference state is characterized by a further set of linearequations, deducible from (121) , Div T = Div ( α + p β S ) I = Div T − τ = − F0 Grad a + β F p ˆ F0 = . (173) ol. 53 (2002) Wave motions in unbounded poroelastic solids 1135 Looking for steady-state solutions in the form: u S ( X , t ) = ˜ u e ik ( X · q − ct ) (174) w F ( X , t ) = − ikc ˜ w e ik ( X · q − ct ) (175) F1 ( X , t ) = ˜ e ik ( X · q − ct ) (176) p ( X , t ) = ˜ p e ik ( X · q − ct ) , (177)we notice that, by virtue of the local fluid-mass conservation law (169) and thekinematical constraint (172),˜ = − ik F0 ( ˜ w · q ) (178)˜ = − ik β S β F ˆ F0 (˜ u · q ) , (179)the reduced set of equations (170)-(172) can be uncoupled from the fluid-massconservation law (169). Moreover, as the gradient of the pressure field p ( t ) isindeed parallel to the direction of wave propagation (defined by the unit vector q ), the set of equations that describes the longitudinal dynamics in terms of thescalar unknowns { ˜ , ˜ p, ˜ u L , ˜ w L } can be uncoupled from the one that describes thetransversal dynamics in terms of { ˜ u T , ˜ w T } , where˜ u := ˜ u L q + ˜ u T , ˜ u T · q = 0 (180)˜ w := ˜ w L q + ˜ w T , ˜ w T · q = 0 . (181)Accordingly, it is worth emphasizing that the transverse dynamics is una ff ected bythe saturation constraint (172). Longitudinal waves
Looking for longitudinal steady-state solutions of equations(169)-(172) and combining, respectively, the fluid-mass conservation law (169)with the kinematical constraint (172), and Cauchy’s law of motion (170) with theanalogous equation (171), we can at first focus our attention on a reduced set ofscalar (algebraic) equations in the field doublet { ˜ u L , ˜ w L } , D ˜ u L + D ˜ w L = 0D ˜ u L + D ˜ w L = 0 , (182)with D := β S D := − ν F0 β F D := ν F0 β F S c LS − β S β F0 − ν F0 β F S + F0 + β S F0 c D := β S F0 c LF − ν F0 β F β F0 − F0 ( β S + ν F0 β F ) c . (183)
136 S. Quiligotti, G.A. Maugin and F. dell’Isola ZAMP
As the coe ffi cients D and D do not depend on the longitudinal characteristicsquared speed c = c L , the equationD D − D D = 0 (184)is linear in c , and yields the unique solution c = c L = S c LS ( ν F0 β F ) − β S ( ν F0 β F ) β F0 + F0 c LF ( β S ) F0 ( ν F0 β F + β S ) + S ( ν F0 β F ) . (185)Henceforth, a dependence of the characteristic longitudinal speed on the macro-scopic coupling parameter β (124) can be taken into account in the case of fluid-saturated poroelastic solids (185) as well as in the case of unconstrained solid-fluidmixtures (161). In particular, recalling the requirement | β | ≤ β max , we may fi-nally point out, by virtue of the linear dependence of c L on β , that the square ofthe characteristic speed of the (unique) longitudinal plane wave (185) has to rangefrom c L min = ν F0 β F S c LS − β S F0 c LF F0 ( ν F0 β F + β S ) + S ( ν F0 β F ) ( β = β max ) , (186)to c L max = ν F0 β F S c LS + β S F0 c LF F0 ( ν F0 β F + β S ) + S ( ν F0 β F ) ( β = − β max ) , (187)whenever the saturation constraint is kinematically satisfied.In conclusion, we remark that the presence of such a constraint (43), that ob-viously reduces the degree of freedom of the longitudinal dynamics of the mixture,furthermore allows the microstructural constitutive parameters β S and β F tocontribute to the resulting characteristic speed c L (refer to definitions (51) and(51) , formerly introduced in section 4).Accordingly, a relevant dependence on the constitutive information associatedwith the definition of microscopic mass-density fields (45)-(46) may be broughtforth at the macroscopic level. Acknowledgments
This work has been developed within the framework of the TMR European Net-work on ”Phase Transitions in Crystalline Solids”. Fruitful discussions with Prof.Daniel Lhuillier are gratefully acknowledged.
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