An Eshelbian approach to the nonlinear mechanics of constrained solid-fluid mixtures
AAn Eshelbian approach to the nonlinear mechanicsof constrained solid-fluid mixtures
S. Quiligotti and
G.A. Maugin , Paris, France, and
F. dell’Isola , Rome, Italy
Received July 26, 2001; revised May 15, 2002Published online: January 16, 2003 (cid:1)
Springer-Verlag 2003
Summary.
Looking at rational mixture theories within the context of a new perspective, this work aims toput forward a proposal for an Eshelbian approach to the nonlinear mechanics of a constrained solid-fluidmixture, made up of an inhomogeneous poroelastic solid and an inviscid compressible fluid, which do notundergo any chemical reaction.
A binary solid-fluid mixture is usually thought of as a couple of body manifolds, B S and B F ,embedded into the three-dimensional Euclidean space so as to occupy, in the course of theirindependent motions, a common smooth region of the physical environment (see, e.g., Atkinand Craine [1], Bowen [2], Rajagopal and Tao [24], Truesdell [28], [29]). If a smooth region of thespace is chosen as a reference shape, which need not ever be occupied by the solid constituent,then a motion of B S can be described as a time sequence of mappings which carry the solidmanifold from the reference to the current shape, whereas the motion of the fluid constituent canbe conceived as a time sequence of embeddings of B F into the three-dimensional physical space.By virtue of such a customary fundamental assumption, any place in the current shape of themixture is simultaneously occupied by a material point belonging to each constituent. Henceforth,the motion of the fluid-body manifold may be described by taking into account that any fluid pointis naturally associated with the 1-parameter family of reference places occupied by the solid pointsthat are currently overlapped with it (see Wilmanski [31]–[33]). Accordingly, both the Eulerianfluid and solid velocity fields can be pulled back to the linear vector space associated with thereference shape of the solid constituent (Sect. 2), and a referential description of all relevant fluidproperties can be furthermore introduced and motivated (Sect. 3).In order to derive the required number of field balance equations and boundary conditionswhich govern the dynamics of unconstrained solid-fluid mixtures (Sect. 4), a suitable expressionfor the power expended by internal and external actions on any admissible test velocity field ispostulated within the framework of a first-order gradient theory (Sects. 4.1–4.3), and the thornyissue of splitting the overall traction applied on the boundary of the mixture is briefly addressed(Sect. 4.3).Recalling the duality highlighted by d’Alembert, the principle of virtual power (see, e.g.Di Carlo [11], Germain [15], [16], Maugin [19]) is finally used as a main tool to introduce theactions which expend power on the kinematical descriptors admissible in the theory [23], Acta Mechanica (2003)DOI 10.1007/s00707-002-0968-z
ACTA MECHANICA
Printed in Austria eading to consistent straightforward definitions of peculiar Cauchy-like [28], Piola-like [31] andEshelby-like stress tensors. In this way we extend the notion of material action (see, e.g. Eshelby[12]–[14], Maugin [20], [21]) to any rational theory of solid-fluid mixtures.As the investigation of the historical development of the theory of porous media seems topoint out (de Boer [6], [8]), a mathematical theory of mixtures, enriched by the concept ofvolume fractions, also provides a suitable framework for the development of a consistentmacroscopic theory of porous solids saturated with fluids (Bowen [3], [4], de Boer [7], [8],Wilmanski [33]).If the saturation constraint is satisfied (Klisch [17], Svendsen and Hutter [26]), i.e. the volumeoccupied by the constituents equals the volume available to the mixture, then the stress re-sponse is determined by the motion except for an arbitrary contribution, due to the pressurereaction which arises in the material so as to maintain each constituent in contact with the otherone. Within the framework of a variational theory (Sects. 4.4–4.7), the saturation pressure canbe truly interpreted as a Lagrangian multiplier in the expression of the strain-energy density perunit volume of the mixture [9], so as to extend the concept of effective stress and derive, as aresult of the theory, the splitting rule which governs the distribution of such a pressure amongthe constituents of a saturated solid-fluid mixture.
Let us focus our attention on the kinematics of a binary mixture B , consisting of two smooththree-dimensional material manifolds, B : ¼ f B S ; B F g : ð Þ In order to avoid confusion between particles which belong to any constituent of the mixture,we refer to material points X a B a (Fig. 1) as a - points , with a S ; F f g .By assumption [22], there exists a smooth embedding of the body manifold B S into the three-dimensional Euclidean space E , K S : B S ! E ; ð Þ which associates any material S -point with a reference place. As the embedding K S does notdepend on time, a smooth motion of B S may be simply thought of as a time sequence ofmappings, v S (cid:8) ; t ð Þ : B ! E ; ð Þ which carry the body manifold B S from the reference shape B (cid:9) E to the current shape v S ð B ; t Þ (cid:9) E . Similarly, a smooth motion of B F may be described by a time sequence ofembeddings, v F (cid:8) ; t ð Þ : B F ! E ; ð Þ which map the body manifold B F onto the current shape v F B F ; t ð Þ (cid:9) E .According to the classical theory of mixtures, any place x B t in the current shape B t : ¼ f v S ð K S ð B S Þ ; t Þg \ f v F ð B F ; t Þg ð Þ We definitely do not deal with Cantor dust (fluid drops or solid slivers) and fractals such as Mengersponges and Sierpinski gaskets. S. Quiligotti et al. s simultaneously occupied by a material point of each constituent, X S B S and X F B F , suchthat x ¼ v S K S ð X S Þ ; t ð Þ ¼ v F ð X F ; t Þ : ð Þ The Eulerian velocity fields v a ð(cid:8) ; t Þ : x v a x ; t ð Þ ; a S ; F f g ð Þ associate the velocity pair v S x ; t ð Þ ; v F x ; t ð Þ with the place currently occupied by X S and X F . Inparticular, as the reference shape B of the material manifold B S does not depend on time, thevelocity of any S -point can be easily obtained by taking the partial derivative of the motion v S with respect to time, v S x ; t ð Þ : ¼ D S D t v S (cid:8) ; t ð Þ (cid:11) K S ½ (cid:13) X S ð Þ ¼ @ v S @ t X ; t ð Þ ; ð Þ where X ¼ K S ð X S Þ and x ¼ v S X ; t ð Þ .As we exclude a priori the possibility that a three-dimensional region of the reference shapecan collapse under the motion v S , det F S > ; F S : ¼ Grad v S ¼ @ v S @ X ; ð Þ there exists a smooth inverse mapping, v (cid:14) S (cid:8) ; t ð Þ : B t ! E ; ð Þ which satisfies the trivial identity X ¼ v (cid:14) S v S X ; t ð Þ ; t ð Þ ; for any X B : ð Þ Denoting by V S the referential description of the partial derivative of v (cid:14) S with respect to time[20], B F B S X S K S E E x X F χ F (·, t ) χ S ( B , t ) χ S –1 (·, t ) B X χ S (·, t ) χ F ( B F , t ) B F B S X S K S E E x X F χ F (·, t ) χ S ( B , t ) χ S –1 (·, t ) B X χ S (·, t ) χ F ( B F , t ) Fig. 1.
Kinematics of a binary solid-fluid mixture
An Eshelbian approach to constrained solid-fluid mixtures S (cid:8) ; t ð Þ : X V S X ; t ð Þ : ¼ @ v (cid:14) S @ t x ; t ð Þ ; ð Þ it can be easily shown that the following property holds: I ¼ @ v (cid:14) S @ x (cid:11) v S (cid:2) (cid:3) @ v S @ X ¼) grad v (cid:14) S (cid:4)(cid:4) x ; t ¼ Grad v S ð Þ (cid:14) (cid:4)(cid:4)(cid:4) X ; t : ð Þ Moreover, we notice that the material S -derivative of the identity (11) leads to the furtherremarkable property F S V S þ @ v S @ t ¼ ; ð Þ which asserts that V S is just the opposite of the pull-back of v S .Bearing in mind the identity (14) and the following definition: Grad F S : ¼ Grad Grad v S ð Þ ; ð Þ we can furthermore deduce that Grad @ v S @ t ¼ (cid:14) Grad F S ð Þ V S (cid:14) F S Grad V S ð Þ : ð Þ In order to deal with a description of the motion of F -points through the reference shape ofthe solid constituent, we notice that any F -point which belongs to the mixture at time t , namely X F v (cid:14) F B t ; t ð Þ (cid:9) B F , interacts with a 1-parameter family of S -points, moving along the curve v (cid:14) S v F X F ; (cid:8)ð Þ ; (cid:8)ð Þ : t X ; ð Þ at the velocity w F X ; t ð Þ , defined by v F x ; t ð Þ ¼ F S X ; t ð Þ w F X ; t ð Þ þ v S x ; t ð Þ ; ð Þ with x ¼ v F X F ; t ð Þ ¼ v S X ; t ð Þ 2 B t .Finally, we introduce the velocity field V F , which identically meets the requirement v F x ; t ð Þ þ F S X ; t ð Þ V F X ; t ð Þ ¼ ; ð Þ such that V F (cid:14) V S ¼ (cid:14) w F : ð Þ With the aim to sketch out the role played in the theory by the velocity fields V S and V F [23], afew concluding remarks can be made.Let us consider a migrating surface which envelops, at time t , a smooth region of the currentshape of the mixture, V t ¼ c t ð Þ (cid:9) B t . If such a surface moves independently of the solidconstituent, the time derivative of the integral of any smooth Eulerian scalar field u , followingthe motion of the migrating surface, is given by the expression: dd s Z c s ð Þ u s ¼ t ¼ dd s Z V S s ð Þ u s ¼ t þ Z @ V t u v (cid:14) v S ð Þ (cid:8) n ; ð Þ where v represents the independent velocity of the moving boundary, V S s ð Þ the shape at time s of the solid subbody associated with the smooth fixed region of the reference shape V ? ¼ v (cid:14) S V t ; t ð Þ , and n the outward normal to the migrating surface.As the inverse mapping v (cid:14) S carries c s ð Þ onto c ? s ð Þ at any time s , denoting by w the velocity atwhich the boundary @ c ? s ð Þ moves through the reference shape B , S. Quiligotti et al. X ; t ð Þ ¼ F (cid:14) S X ; t ð Þ v x ; t ð Þ (cid:14) v S x ; t ð Þf g ; ð Þ and by u ? the referential description of the Eulerian scalar field u , u ? X ; t ð Þ ¼ det F S X ; t ð Þ u x ; t ð Þ ; ð Þ we may also consider the alternative expression: dd s Z c ? s ð Þ u ? s ¼ t ¼ dd s Z V ? S s ð Þ u ? s ¼ t þ Z @ V ? t u ? w (cid:8) N ; ð Þ where V ? S ð s Þ ¼ V ? for any s , and V ? t ¼ c ? t ð Þ ¼ V ? S t ð Þ . If we introduce the velocity field V ,which identically meets the requirement v x ; t ð Þ þ F S X ; t ð Þ V X ; t ð Þ ¼ ; ð Þ such that Z @ V t u v (cid:14) v S ð Þ (cid:8) n ¼ (cid:14) Z @ V ? t u ? V (cid:14) V S ð Þ (cid:8) N ¼ Z @ V ? t u ? w (cid:8) N ; ð Þ then we can deduce [10] that, at any time t , the integral of the partial time derivative of thesmooth scalar field u over V t is given by the expression: Z V t @ u @ s (cid:13) (cid:14) s ¼ t ¼ dd s Z c ð s Þ u s ¼ t (cid:14) Z @ V t u v (cid:8) n ¼ dd s Z c ? s ð Þ u ? s ¼ t þ Z @ V ? t u ? V (cid:8) N : ð Þ Moreover, if we think of a fixed Eulerian surface @ V (cid:9) B t , we notice that if the materialsolid surface currently overlapped with @ V expands, then the migrating surface associated with @ V by the inverse mapping v (cid:14) S shrinks; conversely, if the material solid surface currentlyoverlapped with @ V shrinks, then the associated surface expands, Z @ V v S (cid:8) n ¼ (cid:14) Z @ V ? t det F S ð Þ V S (cid:8) N ; V ? t ¼ v (cid:14) S ð V ; t Þ : ð Þ Let us consider a subset of the solid material manifold, P S (cid:9) B S , and denote by V ? S ¼ K S ð P S Þ its reference shape. At any given time t , the solid motion v S carries the material subbody P s from its reference shape V ? S to its current shape, V S ð t Þ ¼ v S ð V ? S ; t Þ (cid:9) B t .Focusing our attention on the solid constituent, we can properly introduce at least two differentmass densities per unit volume of the mixture, namely the smooth scalar fields . S and . ? S , definedin such a way that the solid-mass content of both V S ð t Þ (cid:9) B t and V ? S (cid:9) B equals the measure M S V t ð Þ ¼ Z V S ð t Þ . S ¼ Z V ? S . ? S ; . ? S ð X ; t Þ ¼ J S ð x ; t Þ . S ð x ; t Þ ; ð Þ where J S ð x ; t Þ ¼ det F S ð X ; t Þ and V t ¼ V S ð t Þ . An Eshelbian approach to constrained solid-fluid mixtures f no phase transition between the constituents of the mixture is allowed, at any given time s ¼ t , the value of the material S -derivative of M S ð V s Þ vanishes for any V S ð t Þ , dd s Z V S ð s Þ . S s ¼ t ¼ Z V S ð t Þ @ . S @ t þ div ð . S v S Þ (cid:2) (cid:3) ¼ ; ð Þ leading to the following expression of the local conservation law: @ . S @ t þ div ð . S v S Þ ¼ : ð Þ Recalling the definition of the referential mass density (29), we may also notice that thesmooth scalar map . ? S does not depend on time, dd s Z V S ð s Þ . S s ¼ t ¼ dd s Z V ? S . ? S s ¼ t ¼ Z V ? S @ . ? S @ s (cid:13) (cid:14) s ¼ t ¼ ; ð Þ and therefore the local conservation law (31) can be rewritten in the alternative form: @ . ? S @ t ¼ : ð Þ Because of the overlapping between the two constituents, any smooth region of the currentshape of the mixture can also be associated with a fluid subbody; in particular, there exists asubbody P F (cid:9) B F such that V F ð s Þ ¼ v F ð P F ; s Þ , with V F ð s Þ ¼ V S ð s Þ at time s ¼ t . As aconsequence, we can state the integral conservation law dd s Z V F ð s Þ . F s ¼ t ¼ Z V F ð t Þ @ . F @ t þ div ð . F v F Þ (cid:2) (cid:3) ¼ ; ð Þ which yields the local expression @ . F @ t þ div ð . F v F Þ ¼ : ð Þ As the fluid-mass content of any smooth region of the current shape of the mixture, V t ¼ V F ð t Þ (cid:9) B t , equals the measure [31] M F ð V t Þ ¼ Z V F ð t Þ . F ¼ Z V ? F ð t Þ . ? F ; ð Þ where . ? F is the fluid-mass density per unit reference volume, . ? F X ; t ð Þ ¼ J S x ; t ð Þ . F x ; t ð Þ ; ð Þ and V ? F ð t Þ ¼ v (cid:14) S V F ð t Þ ; t ð Þ (cid:9) B ; ð Þ by definition, at any given time t , the value of the material F -derivative of M F V s ð Þ vanishes forany V t ¼ V F ð t Þ , S. Quiligotti et al. d s Z V F ð s Þ . F s ¼ t ¼ Z V ? F ð t Þ @ . ? F @ t þ Div . ? F w F (cid:15) (cid:16)(cid:2) (cid:3) ¼ ; ð Þ leading to the following expression of the local conservation law: @ . ? F @ t þ Div . ? F w F (cid:15) (cid:16) ¼ : ð Þ Finally, let us think of both mass conservation laws (30) and (34) from a slightly different pointof view, referring to the motion of the mixture as a single body [29]. In particular, let us assume thatany material point of the mixture as a whole moves at the given velocity v : ¼ n S v S þ n F v F ; ð Þ where n a represents the mass fraction associated with the a -th constituent, n a : ¼ . a . ; a S ; F f g ; ð Þ i.e. the dimensionless ratio of the current mass density of the a -th constituent to the currentmass density of the mixture per unit volume, . : ¼ . S þ . F ; ð Þ such that, by definition, n S þ n F ¼ : ð Þ Taking into account the assumptions (30) and (34), we find out that the following integralconservation law of global mass holds for any smooth region of the current shape of themixture: dd s Z V S ð s Þ . S þ Z V F ð s Þ . F s ¼ t ¼ Z V t @ . @ t þ div . v ð Þ (cid:2) (cid:3) ¼ ; ð Þ where, as usual, V t ¼ V S ð t Þ ¼ V F ð t Þ . Similarly, referring to the reference shape of the solidconstituent, we can write that dd s Z V ? S . ? S þ Z V ? F ð s Þ . ? F s ¼ t ¼ Z V ? t @ . ? @ t þ Div . ? w ð Þ (cid:2) (cid:3) ¼ ; ð Þ where . ? is the mass density of the mixture per unit reference volume, . ? X ; t ð Þ : ¼ . ? S X ; t ð Þ þ . ? F X ; t ð Þ ¼ J S x ; t ð Þ . x ; t ð Þ ; ð Þ and w : ¼ n F w F : ð Þ As a general rule, we finally notice that the time derivative of the a -mass contentof any smooth region V s (cid:9) B s , enveloped by a migrating surface which follows themotion of the mixture as a whole, does not vanish at any time s ¼ t . This means that thequantity An Eshelbian approach to constrained solid-fluid mixtures d s Z V s . a s ¼ t ¼ Z V t . D n a D s (cid:13) (cid:14) s ¼ t ¼ (cid:14) Z @ V t . a v a (cid:14) v ð Þ (cid:8) n ; ð Þ where the difference v a (cid:14) v represents the velocity of diffusion of the a -th constituent throughthe mixture, may be non-vanishing. By assumption, any given place in the current shape of the mixture is simultaneously occupiedby both an S -point and an F -point (Fig. 1).In order to describe their local interactions within the framework of a first-order gradienttheory, we assume that the stress power x , expended on any pair of smooth velocity fields( v S ; v F ), is given by the expression [16]: x : ¼ X a S ; F f g p a (cid:8) v a þ r a (cid:8) grad v a ð Þ : ð Þ According to the principle of material frame-indifference, the stress power expended on anyrigid-body velocity field, v S x ; t ð Þ ¼ v F x ; t ð Þ ¼ w o t ð Þ þ W ð t Þ x (cid:14) x o ð Þ ; W ð t Þ 2
Skw ; ð Þ vanishes for any choice of spatially uniform w o ð t Þ and W ð t Þ , w o (cid:8) X a S ; F f g p a þ W (cid:8) X a S ; F f g r a þ p a (cid:17) x (cid:14) x o ð Þf g ¼ : ð Þ As a consequence, only the constitutive assumptions which meet the following preliminaryrequirements, skw r S þ r F ð Þ ¼ O ; ð Þ p S þ p F ¼ ; ð Þ can be considered admissible in the theory; in particular, while the former restriction (53) statesthat the sum of peculiar Cauchy-like stress tensors has to be symmetric, the latter (54) statesthat the force exerted on any S -point by the overlapped F -point is just the opposite of the forceexerted on the F -point by the overlapped F -point. Keeping in mind both definitions (43) and (41), and denoting the diffusion velocity of the a -thconstituent by d a , S. Quiligotti et al. S : ¼ v S (cid:14) v ¼ n F v S (cid:14) v F ð Þ ; ð Þ d F : ¼ v F (cid:14) v ¼ n S v F (cid:14) v S ð Þ ; ð Þ we define the kinetic energy density per unit volume of the mixture, conceived as a single bodyin motion at the velocity v , by the expression K : ¼ . v (cid:8) v ; ð Þ which results in the sum: K ¼ . S . F . S þ . F v S (cid:8) v F þ X a S ; F f g . a v a (cid:8) n a v a ¼¼ (cid:14) . S . F . S þ . F v F (cid:14) v S ð Þ (cid:8) v F (cid:14) v S ð Þ þ X a S ; F f g . a v a (cid:8) v a ¼¼ . d S (cid:8) d F ð Þ þ X a S ; F f g . a v a (cid:8) v a : ð Þ Accordingly, the time derivative of the kinetic energy associated with any smooth region of thecurrent shape, enveloped by a migrating surface which follows the motion of the mixture as asingle body, equals the integral of the power expended by inertial forces on the mean velocityfield v , dd s Z V s . v (cid:8) v s ¼ t ¼ Z V t v (cid:8) . a ; a : ¼ D v D s (cid:13) (cid:14) s ¼ t ; ð Þ where the differential operator DD s denotes the material derivative following the motion of themixture as a single body.It may be pointed out that, while the velocity of the centre of mass is given by the sum ofpeculiar velocities v a weighted by mass fractions n a , v (cid:8) . a ¼ X a S ; F f g v a (cid:8) . a a ; ð Þ as a general rule [28], the material derivative of v , following the motion of the mixture as asingle body, does not equal the mean of peculiar accelerations a a , associated with the over-lapped a -points, a ¼ X a S ; F f g n a a a (cid:14) . div . a d a (cid:17) d a ð Þ (cid:13) (cid:14) ; a a : ¼ D a v a D s (cid:13) (cid:14) s ¼ t ; ð Þ where the differential operator D a D s denotes the material derivative following the a -motion.Moreover, denoting by (cid:2) rr the second-order tensor which takes into account the apparent stressdue to diffusive motions, (cid:2) rr : ¼ X a S ; F f g . a d a (cid:17) d a ; ð Þ and recalling the property Alternative forms (and the associated drawbacks) are discussed in de Boer [8].An Eshelbian approach to constrained solid-fluid mixtures a div (cid:2) rr ¼ div n a (cid:2) rr ð Þ (cid:14) (cid:2) rr grad n a ; a S ; F f g ; ð Þ the expression (61) finally results in: . S a ¼ . S a S (cid:14) div n S (cid:2) rr ð Þ þ . S . F . S þ . F a F (cid:14) a S ð Þ þ (cid:2) rr grad n S ; ð Þ . F a ¼ . F a F (cid:14) div n F (cid:2) rr ð Þ þ . S . F . S þ . F a S (cid:14) a F ð Þ þ (cid:2) rr grad n F : ð Þ So as to deduce the required number of local balance equations which govern the dynamics ofthe mixture, we state that the total power expended vanishes on any conceivable smooth test velocity field ^ vv a , X a S ; F f g Z B t . a f (cid:14) a ð Þ (cid:8) ^ vv a þ Z @ B t n a t (cid:8) ^ vv a (cid:13) (cid:14) (cid:14) X a S ; F f g Z B t p a (cid:8) ^ vv a þ r a (cid:8) grad ^ vv a ð Þ (cid:13) (cid:14) ¼ ; ð Þ where f is the applied external force per unit mass of the mixture, . f ¼ X a S ; F f g . a f ; ð Þ and t the overall external boundary traction.Keeping in mind that, for any a S ; F f g , r a (cid:8) grad ^ vv a ¼ div r T a ^ vv a (cid:15) (cid:16) (cid:14) ^ vv a (cid:8) div r a ; ð Þ the integral equation (66) leads to the set of local equations: div r a þ . a f (cid:14) p a ¼ . a a ; ð Þ r a n ¼ n a t : ð Þ Consistently [28], taking the sum over a and recalling the requirements (53) and (54), it ispossible to show that the dynamics of the mixture as a single body results to be governed by theequations div r þ . f ¼ . a ; ð Þ r n ¼ t ; ð Þ with r : ¼ X a S ; F f g r a ; r Sym : ð Þ As the tricky question of splitting the overall applied boundary traction t (72) among theconstituents still stands as one of the greatest challenges that have to be faced up in order to putmixture theories to use [24], it is worth recalling that, within the framework of variationalprinciples, boundary conditions are straightforwardly derived, as well as governing equations,as a result of the theory. All conclusions drawn here might be consistently extended to higher-order gradient theories, provided that a meaningful physical interpretation of further emergingboundary conditions can be taken for granted [9]. Nevertheless, such an approach is far fromgeneral (see, e.g. Reynolds and Humphrey [25]), as well as other available approaches [5], [24],[33] do not exhaust the list of possibilities that need to be considered, for instance, within theframework of biomechanical applications of mixture theories [27]. S. Quiligotti et al. inally, bearing in mind the expressions (64) and (65), and denoting by ~ rr a and ~ ff a , respec-tively, the a -th peculiar stress tensor [2], [29], [30] and the external force per unit mass of the a -th constituent, ~ rr a : ¼ r a þ n a (cid:2) rr ; ð Þ . a ~ ff a : ¼ . a f þ . a a a (cid:14) a ð Þ (cid:14) p a (cid:14) div n a (cid:2) rr ð Þ ; ð Þ Cauchy’s first law of motion (69) and the expression of the symmetric stress tensor (73)respectively, result in div ~ rr a þ . a ~ ff a ¼ . a a a ð Þ and r ¼ X a S ; F f g ~ rr a (cid:14) n a (cid:2) rr ð Þ ¼ X a S ; F f g ~ rr a (cid:14) . a d a (cid:17) d a ð Þ : ð Þ In order to investigate the configurational nature of hyperelastic interactions, let us assume thatthere exists a strain-energy density per unit volume of the current shape of the mixture, suchthat [9] the material time derivative of the strain-energy content, associated with any fit regionwhich follows the motion of the mixture, dd s Z V s W s ¼ t ¼ Z V t @ W @ t þ div W v ð Þ (cid:2) (cid:3) ; ð Þ is equal to the stress power expended on the velocity pair ( v S ; v F ), dd s Z V s W s ¼ t ¼ X a S ; F f g Z V t p a (cid:8) v a þ r a (cid:8) grad v a ð Þ : ð Þ As the previous integral definition holds for any choice of V t (cid:9) B t , we can properly localize iton the current shape of the mixture, @ W @ t þ W div v þ v (cid:8) grad W ¼ X a S ; F f g p a (cid:8) v a þ r a (cid:8) grad v a ð Þ : ð Þ Let us introduce a partial Piola-Kirchhoff stress tensor T a for each constituent of the mixture[31], such that, for any V (cid:9) B , Z V T a (cid:8) Grad v a ¼ Z V t r a (cid:8) grad v a ; V t ¼ v S V ; t ð Þ ; ð Þ i.e. T a ¼ J S r a F (cid:14) TS ; a S ; F f g : ð Þ An Eshelbian approach to constrained solid-fluid mixtures y assumption, the time derivative of the stored energy associated with any fit region of thereference shape, enveloped by a migrating surface which follows the motion of the mixture as awhole, equals the power expended by internal actions, dd s Z V s ? J S W s ¼ t ¼ X a S ; F f g Z V ? t J S p a (cid:8) v a þ T a (cid:8) Grad v a ð Þ : ð Þ Bearing in mind the identity (27), let us consider the possibility to define a further set ofdynamical descriptors, b a and s a , which satisfy the identity X a S ; F f g Z B J S p a (cid:8) ^ vv a þ T a (cid:8) Grad ^ vv a ð Þ þ Z @ B J S W ^ VV (cid:8) N ¼¼ X a S ; F f g Z B s a (cid:8) ^ VV a þ b a (cid:8) Grad ^ VV a (cid:24) (cid:25) ; ð Þ for any choice of smooth velocity fields ^ VV a and ^ vv a , such that ^ vv a þ F S ^ VV a ¼ for any a S ; F g ,and ^ VV ¼ n S ^ VV S þ n F ^ VV F . As a consequence, we obtain that b a ¼ n a J S W I (cid:14) F TS T a ; ð Þ s a ¼ Grad n a J S W ð Þ (cid:14) Grad F S ð Þ T T a (cid:14) J S F TS p a ; ð Þ where b a represents the peculiar Eshelby stress tensor (see, e.g. Eshelby [12]–[14], Maugin [20],[21]), associated with the a -th constituent of the mixture. Moreover, in order to meet therequirement (53), it may be pointed out that the sum of peculiar stress tensors (85) needs tobe symmetric with respect to the right Cauchy-Green tensor C S ¼ F TS F S (Quiligotti [23],cf. Maugin [19]), i.e., skw b S þ b F ð Þ C S ½ (cid:13) ¼ O : ð Þ So as to describe the macroscopic interactions exchanged by an inhomogeneous poroelasticsolid and a compressible inviscid fluid (Krishnaswamy and Batra [18], Rajagopal and Tao [24],Svensen and Hutter [26]), we assume that the value of the strain-energy density,
W x ; t ð Þ ,depends on the value of both the gradient of the S -motion, referred to the S -point whichoccupies the current place x at time t , and the fluid-mass density per unit volume of themixture, referred to the overlapped F -point. Finally, as the solid body is assumed to be in-homogeneous, a further dependence of the strain-energy density on reference places is takeninto account, i.e., W F S X ; t ð Þ ; . F x ; t ð Þ ; X ð Þ ; X ¼ v (cid:14) s x ; t ð Þ : ð Þ This hypothesis yields the following constitutive prescriptions for both p -like interactions: p ð u Þ S ¼ n S (cid:14) ð Þ grad F S ð Þ T @ W @ F S þ W grad n S þ n S @ W @ . F grad . F þ n S (cid:14) ð Þ F (cid:14) TS @ W @ X ; ð Þ p ð u Þ F ¼ n F grad F S ð Þ T @ W @ F S þ W grad n F þ n F (cid:14) ð Þ @ W @ . F grad . F þ n F F (cid:14) TS @ W @ X ; ð Þ and the r -like interactions S. Quiligotti et al. ð u Þ F ¼ (cid:14) . F @ W @ . F (cid:14) n F W (cid:2) (cid:3) I ; ð Þ r ð u Þ S ¼ @ W @ F S F TS þ n S W I : ð Þ It may be pointed out that, while the sum of p -like interactions satisfies the condition (54),the requirement (53) should be met by taking into account that the stress tensor r results in r ð u Þ S þ r ð u Þ F ¼ @ W @ F S F TS þ . F W . F (cid:14) @ W @ . F (cid:2) (cid:3) I : ð Þ Accordingly, the identity (85) leads to the following straightforward expressions for thepeculiar Eshelby stress tensors: b ð u Þ S ¼ (cid:14) J S F TS @ W @ F S ; and b ð u Þ F ¼ J S . F @ W @ . F I ; ð Þ whose definitions seem to corroborate the importance of the role played by partial chemicalpotentials within the context of solid-fluid mixture theories (see Bowen [2]–[4]). Finally, theidentity (86) yields the following expressions of s -like interactions: s ð u Þ S ¼ J S @ W @ X ; and s ð u Þ F ¼ @ W @ . F Grad J S . F ð Þ ; ð Þ for which the contributions given by partial derivatives of the strain-energy density with respectto the state variables . F and F S are uncoupled. So as to develop a consistent macroscopic theory of saturated poroelastic media (see, e.g.Bowen [3], [4], Svendsen and Hutter [26], Klisch [17]), let us enrich the mathematical theory ofbinary solid-fluid mixtures with the introduction of the concept of volume fractions, m a : ¼ . a ^ .. a ; given by the dimensionless ratio of the macroscopic mass density . a to the microscopic mass density ^ .. a , which depends on the usual state variables (Fillunger, see de Boer [6], dell’Isola et al. [9]), ^ .. a F S X ; t ð Þ ; . F x ; t ð Þ ; X ð Þ ; X ¼ v (cid:14) S x ; t ð Þ : ð Þ A poroelastic solid infused with a compressible fluid is saturated if the solid skeleton isperfectly permeated by the fluid, i.e. if the saturation constraint is satisfied, m S þ m F (cid:14) ¼ : ð Þ As constraints are naturally associated with reactive actions, a saturation pressure p , whichdoes not expend power on any motion compatible with the constraint (97), arises in thematerial so as to maintain each constituent in contact with the other one. We think of such apressure as a Lagrangian multiplier in the expression of the strain-energy density [9], W þ p m S þ m F (cid:14) ð Þ ; ð Þ and generalize the effective stress principle, bearing in mind that @ m S @ F S ¼ (cid:14) m S F (cid:14) TS þ m S . S @ ^ .. S @ F S (cid:2) (cid:3) ; ð Þ An Eshelbian approach to constrained solid-fluid mixtures m F @ F S ¼ (cid:14) m F . F @ ^ .. F @ F S ; ð Þ @ m S @ . F ¼ (cid:14) m S . S @ ^ .. S @ . F ; ð Þ and @ m F @ . F ¼ m F . F (cid:14) m F @ ^ .. F @ . F (cid:2) (cid:3) : ð Þ Neglecting the dependence on X in (88) and (96) for the sake of simplicity, we find out thefollowing expressions of p -like interactions: p ð c Þ S ¼ ^ pp ð u Þ S þ p m F n S . F (cid:14) m F @ ^ .. F @ . F (cid:14) m S m F . F . S @ ^ .. S @ . F (cid:2) (cid:3) grad . F (cid:14) p m S n S (cid:14) ð Þ grad F S ð Þ T F (cid:14) TS þ m S . S @ ^ .. S @ F S þ m F m S . F @ ^ .. F @ F S (cid:2) (cid:3) ; ð Þ p ð c Þ F ¼ ^ pp ð u Þ F þ p n F (cid:14) ð Þ m F . F (cid:14) m F @ ^ .. F @ . F (cid:14) m S m F . F . S @ ^ .. S @ . F (cid:2) (cid:3) grad . F (cid:14) p m S n F grad F S ð Þ T F (cid:14) TS þ m S . S @ ^ .. S @ F S þ m F m S . F @ ^ .. F @ F S (cid:2) (cid:3) ; ð Þ where ^ pp ð u Þ S and ^ pp ð u Þ F are given by the general relations (89) and (90), assuming that the partialderivative of W with respect to X vanishes, i.e., that the value of the stored energy density (88)does not explicitly depend on X .Furthermore, the r -like interactions result in the following expressions: r ð c Þ F ¼ ^ rr ð u Þ F (cid:14) p m F (cid:14) m F @ ^ .. F @ . F (cid:14) m S m F . F . S @ ^ .. S @ . F (cid:2) (cid:3) I ; ð Þ r ð c Þ S ¼ ^ rr ð u Þ S (cid:14) p m S F (cid:14) TS þ m S . S @ ^ .. S @ F S þ m F m S . F @ ^ .. F @ F S (cid:2) (cid:3) F TS ; ð Þ where ^ rr ð u Þ S and ^ rr ð u Þ F are derived, respectively, by the relations (92) and (91).As a general result, it may be pointed out that the saturation pressure is distributed among theconstituents proportionally to their volume fractions only if they are microscopically incom-pressible [9], i.e., if the value of microscopic mass densities ^ .. a x ; t ð Þ is independent of the value ofboth the macroscopic fluid-mass density ( . F x ; t ð Þ ) and the gradient of the S -motion ( F S X ; t ð Þ ).Finally, we get the expressions of partial Eshelby stress tensors, b ð c Þ F ¼ ^ bb ð u Þ F þ p m F J S (cid:14) m F @ ^ .. F @ . F (cid:14) m S m F . F . S @ ^ .. S @ . F (cid:2) (cid:3) I ; ð Þ b ð c Þ S ¼ ^ bb ð u Þ S þ p m S J S F TS F (cid:14) TS þ m S . S @ ^ .. S @ F S þ m F m S . F @ ^ .. F @ F S (cid:2) (cid:3) ; ð Þ and s -like interactions, s ð c Þ S ¼ ^ ss ð u Þ S ¼ ; ð Þ S. Quiligotti et al. ð c Þ F ¼ ^ ss ð u Þ F þ p m F . F (cid:14) m F @ ^ .. F @ . F (cid:14) m S m F . F . S @ ^ .. S @ . F (cid:2) (cid:3) Grad J S . F ð Þ ; ð Þ where ^ bb ð u Þ a and ^ ss ð u Þ a are derived, respectively, by the relations (94) and (95). In order to put forward a proposal for an Eshelbian approach to a first-order gradient theory ofconstrained solid-fluid mixtures, the Eulerian velocity fields associated with each of the con-stituents have been pulled back to the reference shape of the solid one.Using the principle of virtual power as a main tool, further dynamical descriptors have beenintroduced, by duality, and the role that they play in the theory has been sketched out withinthe framework of a variational approach to the dynamics of both unconstrained inhomoge-neous and saturation-constrained homogeneous poroelastic solids, infused with compressibleinviscid fluids.
Acknowledgements
This work has been developed within the framework of the TMR European Network on ‘‘PhaseTransitions in Crystalline Solids’’. The anonymous referee’s advises are gratefully acknowledged.
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