## Abstract

We address Biot's equations governing the motion of an anisotropic fluid-saturated poroelastic material with certain properties. First, we investigate the uniqueness in solutions of the three-dimensional governing equations for the regular region of the poroelastic material and enumerate the conditions sufficient for the uniqueness. Next, by applying Hamilton's principle to the motion of the region, we obtain a variational principle that generates only the Biot–Newton equations and the associated natural boundary conditions. Then, by extending the variational principle for the region with an internal fixed surface of discontinuity through Legendre's transformation, we derive a six-field variational principle that operates on all the poroelastic field variables. The variational principle leads, as its Euler–Lagrange equations, to all the governing equations, including the jump conditions but the initial conditions, as a generalized version of the Hellinger–Reissner variational principle. Moreover, we consider the free vibrations of the region, and we discuss some basic properties of eigenvalues and present a variational formulation by Rayleigh's quotient. This work provides a standard tool with the features of variational principles when numerically solving the governing equations in heterogeneous media with finite element methods, treating the free vibrations and consistently deriving some one-dimensional/two-dimensional equations of the poroelastic region.

## 1. Introduction

Porous materials are biphasic, made of a solid skeleton containing interconnected fluid or air saturated pores. Most geomaterials and especially ceramics, due to the manufacturing process, have inherent porosity and anisotropy. To describe the physical response of porous materials under mechanical effects has been a fundamental research area for considerable time in applied natural sciences and engineering (e.g. Schrefler 2002; de Boer 2003). Investigations concerning the physical response under the mechanical as well as the electrical effect and alike in the elastic and inelastic states are also a rapidly developing field with a wide range of recent technological applications (Cheng *et al*. 1998; Gladkov 2003). Extensive works were performed to describe the physical response based on either a macromechanics viewpoint (the theory of poroelasticity: Biot 1956; Tolstoy 1992; the theory of mixtures: Bowen 1976) or a micromechanics viewpoint (the theory of homogenization: Auriault & Sanchez-Palencia 1977). Of the theories, Biot's theory of poroelasticity was long regarded as a phenomenological, universally accepted and most widely used one to investigate waves and vibrations in fluid-saturated porous materials. A comprehensive overview of the theories, including a historical development from Fick, Darcy, Fillunger and von Terzaghi to Biot and some recent progress can be found in the treatises (e.g. Selvadurai 1996; Cederbaum *et al*. 2000; de Boer 2000; Ehlers & Bluhm 2002; Coussy 2004; Romm 2004) and the review article by de Boer (2003).

We address Biot's theory of poroelasticity and consider its three-dimensional equations governing the motion of the regular region of an anisotropic fluid-saturated porous material in the elastic range (Biot 1955, 1956; Biot & Willis 1957). The region of the porous material is considered as two solid and fluid continua superimposed on each other and with a mechanical interaction between them. Accordingly, as an extension of the elasticity equations, the poroelasticity equations were established separately for the two continua in differential form. They consist of the divergence (Biot–Newton) equations of motion and the gradient equations for the fluid and solid phases, the constitutive relations that combine Hooke's law of elasticity and Darcy's law of fluid, and the boundary and initial conditions. Some experimental observations confirmed the poroelasticity equations for the isotropic case (Berryman 1980; Plona 1980). However, complete experimental data is scarce for the anisotropic case (Lo *et al*. 1986; Aoki *et al*. 1993). The poroelasticity equations were employed in numerous applications of various types of waves and vibrations in soil and rock mechanics, biomechanics, geophysics and acoustics (e.g. Sharma & Cogna 1991; Ben-Menahem & Gibson 1993; Sun *et al*. 1993; Carcione 1996, 2001; Gelinsky & Shapiro 1997; Wilmanski 1999; Kumar *et al*. 2003, 2004; Sharma 2003; Rovazzoli *et al*. 2004; Vashishth & Khurana 2004; Liu *et al*. 2005 and references therein). On the other hand, the internal consistency of the poroelasticity equations, that is, the existence and uniqueness of solutions, is of particular importance in mathematical modelling of the physical response. Deresiewicz & Skalak (1963) derived a set of boundary conditions sufficient for a unique solution of the poroelastic equations at an interface between two dissimilar statistically isotropic poroelastic media. The uniqueness of solutions was established with the sufficient boundary and initial conditions for a regular region of poroelastic isotropic materials with the help of the logarithmic convexity argument (Altay & Dökmeci 1998). We deal with the uniqueness in solutions of anisotropic poroelasticity equations using the energy argument due to its physical nature.

In Biot's poroelasticity, the governing equations were established on the basis of a macroscopic thermodynamics in differential form. Some of the governing equations can be alternatively expressed in variational form through certain variational principles with their well-known advantages. The variational principles of poroelasticity, with or without an explicit functional (i.e. the integral and differential types), are only a few, and they were contrived almost always by use of either a general principle of physics (Tolstoy 1962; Berryman *et al*. 1988; Göransson 1998; Panneton *et al*. 1998; Lopatnikov & Cheng 2004) or a purely mathematical method of convolution (Sandhu & Hong 1987; Huang *et al*. 1990). The use of a general principle of physics always leads to some two-field variational principles that generate only the Biot–Newton equations and the associated natural boundary conditions, as the Euler–Lagrange equations. The rest of the governing equations remain as the constraint conditions, which prevent a free and simple choice of approximating (or trial) fields and a simultaneous approximation upon all the field variables, and hence, usually undesirable in computation. Nevertheless, the constraint conditions can be removed, for instance, through Legendre's transformation as in this paper. Thus, the counterpart of the Hellinger–Reissner variational principle in elasticity is obtained in poroelasticity.

In this paper, we consider the dynamic governing equations for the regular region of an anisotropic fluid-saturated porous material, and we investigate the uniqueness of solutions, the variational form and the free vibrations. In §2, we present the governing equations in differential form, and then we study the uniqueness in their solutions and enumerate the boundary and initial conditions sufficient for the uniqueness in §3. In §4, we state Hamilton's principle for the poroelastic region and obtain a two-field variational principle that operates on the fluid and solid displacements. In §5, we extend the variational principle by applying Legendre's transformation and find a generalized variational principle that operates on all the field variables for the region with an internal fixed surface of discontinuity. The variational principle generates all the governing equations, including the jump conditions across the surface of discontinuity except the initial conditions. In §6, we deal with the free vibrations of the regular region, including the basic properties of eigenvalues and a variational formulation by Rayleigh's quotient. Finally, in §7, we summarize our results and indicate some extensions and further needs of research.

*Notation*. In this paper, we use standard indicial notation in a three-dimensional Euclidean space . We imply Einstein's summation convention for all repeated Latin indices with the range 1, 2 and 3, and Greek indices with the range 1 and 2, unless indices are enclosed with parentheses. We denote by a superposed dot the time differentiation, by a comma before an index the partial differentiation with respect to the indicated space coordinate, by an asterisk the prescribed boundary and initial quantities and by an overbar the time harmonic field variables. Also, we indicate by the symbol the finite and bounded, regular poroelastic region, in Kellogg's (1946) sense, with its boundary surface and closure at time *t*, and by the Cartesian product of the region and the time-interval where may be infinity. We refer to the poroelastic region by a fixed, right-handed system of the Cartesian coordinates of the space . We adopt the conventional notation, the bold faced-brackets, , introduced by Christoffel (1877), to indicate the jump of a quantity across a surface of discontinuity *S*, by the mean value of , and by the inner product of the vectors and . Further, we indicate by the class of continuous functions with its continuous derivatives of order up to and including (*α*) and (*β*) with respect to the space coordinate and time *t*, respectively.

## 2. Governing equations of poroelasticity in differential form

We begin to express, following Biot (1956), the time-domain governing equations without dissipation for the finite and bounded, regular region of an anisotropic fluid-saturated poroelastic material, with its boundary surface and closure at the time-interval as follows.

*Biot–Newton equations* (divergence equations; the equations of motion for the solid and fluid phases)(2.1)(2.2)where are the total stress components acting on both the fluid and solid phases, measured per unit area of the poroelastic material, is the pore-fluid pressure measured per unit area of the fluid phase, *u*_{i} and *U*_{i} are the components of the averaged displacements of the solid and fluid phases and *f*_{i} and *F*_{i} are the components of the solid and fluid body forces, respectively, and stands for the alternating tensor. The densities are the mass coefficients, is the composite density, with the solid and fluid densities *ρ*_{s} and *ρ*_{f} and the porosity *p*, and represents a mass coupling parameter between the fluid and solid phases, together with the inequalities of the form(2.3)Equations (2.1) and (2.2), respectively, describe the motion of the solid phase and Darcy's law of the fluid flow that include the inertial coupling between the solid and fluid phases. However, the memory effects in the inertial terms are taken to be negligible, for instance, as in the cases of low frequency and/or large permeability.

*Gradient equations* (the solid and the fluid strain–displacement relations)(2.4)(2.5)Here, *e*_{ij} and *E*_{ij} are the components of linear strain tensors of the solid and fluid phases, respectively, and the fluid dilatation is given by(2.6)in terms of the displacement components of the fluid phase.

*Constitutive equations* (for a statistically anisotropic poroelastic material)(2.7)(2.8)where the strain energy density *W* is given by(2.9a)with the symmetry properties of the material coefficients by(2.9b)for an anisotropic poroelastic material. The connection between the dynamic and kinematic field variables is characterized by the material coefficients, (drained elastic modulus tensor), (Biot's effective stress coefficient tensor) and *M* (Biot's modulus). The material coefficients were micromechanically analysed, expressed in terms of the properties of the grain, pore-fluid and frame and measured in the field as well as in the lab (Rice & Cleary 1976; Carroll 1980; Thompson & Willis 1991; Hart & Wang 1995; Cheng 1997; Loret *et al*. 2001). The physical response of poroelastic materials is described using the 28 independent coefficients for the most general anisotropic case, and the 13, 8 and 4 independent coefficients for an orthotropic, transversely isotropic and isotropic poroelastic material, respectively.

*Boundary conditions*(2.10)(2.11)Here, and are the solid and fluid boundary traction vectors and is the unit outward vector normal to ∂*Ω*. The surfaces and denote the complementary regular sub-surfaces of the boundary surface . Equations (2.10) and (2.11) represent the poroelastic–poroelastic interface, and other types of boundary conditions (e.g. poroelastic–solid or poroelastic–fluid interface) readily follow from them as special cases.

*Initial conditions*(2.12)(2.13)Here, we denote by , , and the prescribed functions of the space coordinates in the regular poroelastic region.

The foregoing equations governing the physical response of the regular poroelastic region are deterministic, that is, they comprise the 26 equations accounting for the 26 dependent variables, for the solid phase and for the fluid phase. In the linear governing equations, the mass conservation equations are not needed. The fluid and solid mass densities are considered to be specified properties of the region at the natural state. The governing equations define an initial-mixed boundary value problem for the region. We will show in §3 that they always have a unique solution under the prescribed boundary and initial conditions, (2.10)–(2.13).

## 3. Uniqueness of solutions

We are now concerned with a theorem of uniqueness in Biot's poroelasticity as a counterpart of Neumann's uniqueness theorem in classical (non-polar) elasticity. Neumann's theorem (1885) relied on the positive-definiteness of stored energies that is one of the irrevocable axioms in mechanics. By using the energy argument, a number of authors (for instance, Weiner 1957; Deresiewicz & Skalak 1963; Mindlin 1968; Altay & Dökmeci 2004) investigated the uniqueness in solutions of the three-dimensional (or one-dimensional/two-dimensional) governing equations of certain materials. Apart from the energy argument, various methods that rely on a mathematical argument as fundamental (the method of analyticity, Holmgren's theorem, Protters's technique, the logarithmic convexity argument and the like) were used in studying the uniqueness of solutions (Knops & Payne 1972). Of these methods, the logarithmic convexity argument that does not impose the definiteness condition on the material elasticity was used in enumerating the boundary and initial conditions sufficient for the uniqueness in solutions of a statistically isotropic fluid-saturated poroelastic material (Altay & Dökmeci 1998). We prefer the energy argument with its physical nature in proving a theorem of uniqueness for the equations governing the regular region of an anisotropic fluid-saturated poroelastic material that are recorded in §2. To begin with, we state the theorem as follows.

*Given the regular region* *of an anisotropic fluid-saturated poroelastic material with the boundary surface* *at the time-interval T under a prescribed initial data, and it is set in a motion that is maintained by an application of the assigned surface tractions and also the prescribed deformation field over an appropriate portion of the boundary surface*. *Now, let* *be an admissible state of solutions given by*(3.1)*which is the functions of the space coordinates and time that satisfies the Biot–Newton equations,* *(2.1) and (2.2),* *the gradient equations,* *(2.4) and (2.5),* *the linear version of the constitutive equations,* *(2.7) and (2.8)* *and the boundary and initial conditions,* *(2.10)–(2.13)*. *Then,* *there exist at most one admissible state of solution* *(3.1)* *that satisfies the aforementioned governing equations of the regular poroelastic region*.

To prove theorem 3.1, we assume, as usual, that two distinct states of solutions exist to the linear governing equations of the poroelastic region for the same boundary and initial data and the same body forces. Also, we assume that the two states of solutions together with their derivatives exist and are continuous functions of the space coordinates and time in the interval . We indicate by the difference state of solutions, namely(3.2)Apparently, each state of solutions satisfies the governing equations, and also, by virtue of their linearity, the difference state of solutions satisfies the homogeneous Biot–Newton equations with zero body forces, (2.1) and (2.2), by(3.3)(3.4)the linearized version of the constitutive relations, (2.7) and (2.8), by(3.5)and the homogeneous boundary and initial conditions, (2.10)–(2.13), by(3.6)(3.7)(3.8)and the gradient equations, (2.4)–(2.6). We can show that the homogeneous governing equations have a trivial solution, , i.e. the two solutions are equal, , and the uniqueness of solutions is ensured. To begin with, we recall the strain energy density *W* in equations (2.9a) and (2.9b), the kinetic energy density *k* and their total values, *Σ* and *K* for the poroelastic region , namely(3.9)where *U* represents the total energy stored by the deformation of the region. By the use of equations (3.8), (2.7), (2.8) and (2.9a), and the symmetry properties of the stress tensor (2.1) and those of the material coefficients (2.9*b*), we obtain the time rate of the energies in the form(3.10)

(3.11)

Now, we multiply equations (3.3) and (3.4) by non-zero (*u*_{i} and *U*_{i}) and integrate over the region and find the equations of the form(3.12)(3.13)We add one another and integrate over the time-interval with the result(3.14)After applying Green's theorem to the regular poroelastic region, we have(3.15)with the denotations by(3.16)(3.17)Here, the gradient equations, (2.4)–(2.6), and the relations between the stress tensor and the traction vectors are considered. In view of equations (3.10) and (3.11), we express equation (3.17) as(3.18)We integrate this equation with respect to time and substitute into equation (3.15), and we find the condition as(3.19)where the integrand of equation (3.16) is taken to vanish due to the boundary and initial conditions, (3.6)–(3.8). The kinetic and strain energy densities, *k* and *W*, are positive-definite, by definition, and initially zero. Thus, the total kinetic and strain energies *K* and calculated by an integration from the difference state of solution have the same properties for the poroelastic region , and we conclude . This implies a trivial solution for the difference state of solution , i.e. the linear governing equations always have a unique solution.

The theorem states that the prescribed boundary and initial conditions are sufficient for the uniqueness of solutions of the linear governing equations of the anisotropic poroelastic region . Conversely, a theorem that states the conditions for which there exists at least one solution to the governing equations is not treated herein. We refer the reader to the treatise by Leis (1986) for this type of problem. ▪

## 4. Hamilton's principle for poroelastic media

Hamilton (1834, 1835) presented a general principle in dynamics of a discrete mechanical system and, much later, Kirchhoff (1876) extended the principle to a continuous medium. The principle, as a powerful and elegant tool, was extensively used in deriving some variational principles of mechanics. The application of Hamilton's principle to the motion of a finite region of media always leads to a variational principle that generates only the divergence equations and the associated natural boundary conditions, while the rest of the equations governing the motion remain as its constraint (subsidiary) conditions. In the variational principle, the variations of each of the field variables are independent (unconstrained) within the region and are constrained to vanish at the time *t*_{0} and *t*_{1} throughout the region and its boundary surface. Hamilton's principle admits an explicit functional for the case when the non-conservative forces are absent in a medium. Accordingly, the principle leads to an integral or differential type of variational principles with and without an explicit functional. We apply Hamilton's principle in deriving certain variational principles for the regular region of anisotropic fluid-saturated poroelastic materials.

Now, we express Hamilton's principle (e.g. Lanczos 1966) for the regular poroelastic region as follows:(4.1)Here, we define by *L* the Lagrangian function, and by and the virtual works done by the body forces and surface tractions, respectively, in the form(4.2)(4.3)In these equations, we place an asterisk upon to distinguish it from the variation operator . Besides, here and henceforth, we assume that all the field variables together with their derivatives exist and continuous functions of the space coordinates and time, and the limits of integrals are constant. Thus, we commute the operation of variation with that of integration as well as that of differentiation, and also keep in mind the axiom of conservation of mass in taking the variations. Accordingly, we find the variation of the kinetic energy (3.9) for the poroelastic region by(4.4)and after integrating by parts with respect to time, by(4.5)This may be expressed by(4.6)under the constraint conditions of the form(4.7)as is customary in the use of Hamilton's principle.

Likewise, we recall the constitutive relations, (2.7) and (2.8), and then, we express the variation of the strain energy as(4.8)Inserting equations (2.7) and (2.8) into this equation and using the gradient equations, (2.4)–(2.6), we find(4.9)which may be appropriately expressed by(4.10)Applying Green's theorem to the regular region, we obtain the variation of the strain energy in the form(4.11)

Next, we substitute equation (4.3) and the variations of the kinetic and strain energies, (4.6) and (4.10), into equation (4.1), and combine the terms in the surface and volume integrals, we finally arrive at a two-field variational principle of the form(4.12)with the notations(4.13a)(4.13b)Since the variations of the admissible state are arbitrary and independent inside the volume and on the boundary surface , we readily obtain from equation (4.12) the divergence equations by(4.14)(4.15)and the natural boundary conditions by(4.16)(4.17)as the Euler–Lagrange equations. Conversely, if the divergence equations and the boundary conditions are met, the variational principle is obviously satisfied.

The admissible state is subjected to the rest of the governing equations for the poroelastic region and the condition (4.7), as the constraint conditions of the two-field variational principle (4.12). We can remove the constraint conditions, since the variational principles as few constraints as possible are almost always desirable in computation. There are available a number of methods so as to remove all the constraint conditions of a non-polar media except the symmetry of the stress tensor (e.g. Altay & Dökmeci 2004). We apply Legendre's transformation (e.g. Dökmeci 1988 and references therein) that is applicable to holonomic as well as non-holonomic constraint conditions in removing some of the constraint conditions of the variational principle (4.12). Thus, we extend the two-field variational principle in §5.

## 5. Generalized variational principles

Now, to remove some of the constraint conditions in the variational principle (4.12), we introduce a dislocation potential for each constraint condition, i.e. each constraint as zero times a Lagrange multiplier, namely(5.1)for the volume constraint conditions (2.4) and (2.6), and(5.2)for the boundary constraint conditions (2.10) and (2.11). Next, we add the dislocation potential , (5.1) and (5.2), into Hamilton's principle (4.1) as(5.3)Here, the variations of the field variables and Lagrange multipliers are treated as free (unconstrained).

As before in equation (4.1), we carry out the variations indicated in equation (5.3), integrate by parts with respect to time, apply Green's theorem to the regular poroelastic region, and then rearrange the volumetric and boundary terms of each variation, we finally arrive at a unified variational equation of the form(5.4)with the denotations by(5.5)for the solid and fluid phases, respectively. Here, the symmetry of the stress tensor (2.1) and the condition (4.7) are considered. We express the Euler–Lagrange equations of the variational equation (5.4), and then we identify the Lagrangian undetermined multipliers in the form(5.6)By substituting equation (5.7) into equation (5.4), we obtain a six-field variational principle in the form(5.7)where(5.8)From equation (5.8), we write the Euler–Lagrange equations as(5.9)for the solid phase and(5.10)for the fluid phase, in the denotations of equation (4.14)–(4.17).

The six-field variational principle (5.8) generates all the governing equations of the poroelastic region except the initial conditions, and it is the poroelastic counterpart of the Hellinger–Reissner variational principle in elasticity (e.g. Washizu 1983). We express the variational principle in a compact form(5.11a)with its explicit functional of the form(5.11b)The variational principle (5.11a) and (5.11b) with its explicit functional represents an integral type of variational principle with all the features of classical variational principles. Executing the variations indicated in equations (5.11a) and (5.11b), the validation of the variational principle can be readily shown, i.e. its Euler–Lagrange equations coincide with the Biot–Newton equations, (2.1) and (2.2), the gradient equations, (2.4)–(2.6), the constitutive relations, (2.7) and (2.8), and the boundary conditions, (2.10) and (2.11), of the poroelastic region. On the other hand, noteworthy is the fact that the variational principle is expressed in a mathematically, though not physically, appropriate form, due to its terms involving the virtual work associated with the surface tractions and body forces.

*Discontinuous field variables*. Whenever the regular region contains an internal fixed surface of discontinuity *S*, the field variables undergo a jump across the surface of discontinuity *S*, namely(5.12)for the solid and fluid phases. The surface of discontinuity splits the region into the subregions with the boundary surface . We denote by the unit vector normal to and by the unit vector normal to *S* and pointing (see figure 1), and also, we assume that the subregions have dissimilar poroelastic materials. Now, we state the variational principle (5.7) as(5.13)for both of the regions . The 12-field variational principle (5.13) evidently leads to all the poroelastic equations of each region under the constraint conditions of the principle (5.7) for each region and the jump conditions (5.12). To adjoin the jump conditions into the variational principle (5.13), we introduce the dislocation potentials of the form(5.14)and add them into equation (5.3) as(5.15)Here, and are the Lagrangian multipliers to be determined.

Now, we execute the variations indicated in equation (5.15), as we did in going from equation (5.3) to equation (5.7), and apply the generalized version of Green's theorem as(5.16)and then we collect the volumetric terms in and the surface terms on and *S*. After some manipulation, we obtain the variational equation as(5.17)We have, as the Euler–Lagrange equations, the poroelastic equations for each region, the jump conditions (5.14), and equation (5.6) and the multipliers of the form(5.18)Thus, we find a generalized variational principle of the form(5.19a)with the admissible state of the form(5.19b)where(5.20)Also, the variational principle is expressed by an explicit functional of the form(5.21)The 24-field variational principle yields, as its Euler–Lagrange equations, all the poroelastic equations for each region and the jump conditions across the internal fixed surface of discontinuity, except the symmetry of the stress tensor and the initial conditions. We state the principle as follows.

*Unified variational principle. Let* *denote a regular, finite and bounded poroelastic region with its piecewise smooth boundary surface* *with* *and its internal fixed surface of discontinuity S that splits the region into two dissimilar poroelastic regions* *, in the Euclidean three-dimensional space* *, and define the functional* *for each* . *Then the admissible states* *that satisfy the usual continuity, differentiability and existence conditions of the field variables as well as the symmetry of the stress tensor* *(2.1)* *and the initial conditions* *(2.12) and (2.13)* *for each region, are only those that admit* *, if and only if,* *they satisfy the Biot–Newton equations* *(2.1) and (2.2),* *the gradient equations* *(2.4) and (2.5),* *the constitutive relations* *(2.7) and (2.8),* *and the traction and displacement boundary conditions* *(2.10) and (2.11)* *for each region* *, including the jump conditions* *(5.12)**, as the appropriate Euler–Lagrange equations. Conversely, if all the governing equations are identically met, the 24-field variational principle* *(5.21)* *is evidently satisfied*.

*Certain special cases*. The 24-field variational principle (5.21) is quite general for the regular poroelastic region , and it is compatible with and contains, as special cases, some of earlier variational principles in the absence of the terms involving some of the field variables and/or the fluid phase as well as the internal surface of discontinuity. To begin with, from the principle, we have a three-field variational principle by(5.22)for the solid phase and its reciprocal principle by(5.23)for the fluid phase. Moreover, we have another two-field principle operating on the stresses of the solid and the pressure of the fluid by(5.24)and a four-field variational principle by(5.25)as its reciprocal principle. We deduce a two-field variational principle from equation (5.21) by(5.26)which is the so-called displacement–pressure variational principle. Another special case readily follows from equation (5.21) as a four-field variational principle, operating on the solid and fluid displacements of each region in the form(5.27)This variational principle leads, as its Euler–Lagrange equations, to the Biot–Newton equations and the natural displacement boundary conditions for each region and the displacement jump conditions across the surface of discontinuity, and it has the rest of the poroelastic equations for each region and the traction jump conditions across *S*, as the constraint conditions. Lastly, by neglecting all the acceleration terms or by considering the sufficiently smooth motions of the poroelastic region, we find the quasi-static, consolidation equations in variational form that were frequently used in soil mechanics, and we may similarly find some variational principles operating on other poroelastic fields.

## 6. Free vibrations of an anisotropic poroelastic region

*Free vibrations*. We are now concerned, following Tiersten (1969) and Yang & Batra (1994) in piezoelectricity, with the free vibrations of the regular poroelastic region , with some basic properties of eigenvalues and variational formulation by Rayleigh's quotient. For the time-harmonic free vibrations with a circular frequency , we express the field variables in the form(6.1)Inserting equation (6.1) into the governing equations of the region recorded in §2, we have(6.2)(6.3)(6.4)and(6.5)Here and henceforth, we suppress the overbars for brevity. We substitute equations (6.3) and (6.4) into equations (6.1) and (6.5), and then we express the governing equations of the free vibrations as follows:(6.6)(6.7)and the boundary conditions by(6.8)(6.9)in terms of the displacement components of the solid and fluid phases.

*An eigenvalue problem*. Equations (6.6) and (6.7) define an eigenvalue problem in seeking the values for the non-trivial displacement field under the boundary conditions, (6.8) and (6.9). Alternatively, we define the eigenvalue problem by(6.10)with(6.11)where we introduce the operators, ** P** and

**, related to the potential and kinetic energies of the poroelastic region as(6.12)(6.13)Also, the admissible function spaces**

*A***,**

*S***and**

*F**Γ*by(6.14)are defined. Moreover, we write equation (6.10) as(6.15)in terms of , the complex conjugate of

**, that corresponds to the eigenvalue as well.**

*U**Properties of operators*. We consider the inner product of ** P** and

**as(6.16)Applying Green's theorem to the regular region , we write equation (6.16) in the form(6.17)and then, using equations (2.4)–(2.6) and (6.14) in the form(6.18)In this equation, the first term of the right-hand side vanishes due to the boundary conditions (6.11) and the second term is equal to the potential energies (2.9a) and (2.9b), and hence, we write(6.19)Similarly, we have the inner product of**

*U***and**

*A***as(6.20)which may be expressed by(6.21)in view of equations (3.9) and (6.1). Equations (6.17) and (6.19) indicate that the operators**

*U***and**

*P***are positive-definite on the function space as already noted in §3.**

*A*We define a function space and consider the inner product of ** P** and

**as(6.22)After applying Green's theorem to the regular region, we express equation (6.22) as(6.23)in terms of equation (6.13). One more application of the theorem in equation (6.23) results in(6.24)We express this equation by(6.25)Here, the boundary conditions (6.11) are considered for**

*V***as well. Likewise, we write the inner product of**

*V***and**

*A***as(6.26)which may be expressed by(6.27)Equations (6.25) and (6.27) show that the operators**

*V***{**

*P***} and**

*U***{**

*A***} are self-adjoint on the function space**

*U**Γ*{

**}.**

*U**Properties of eigenvalues*. We consider an eigenpair (, ** U**) and its complex conjugate , and then write the inner product of both sides of equation (6.10) and its complex conjugate, as(6.28)Subtracting one from another, we obtain(6.29)due to the self-adjointness of the operators

**and**

*P***and equation (6.19). From equation (6.29), we conclude that the eigenvalue is real.**

*A*On the other hand, for an eigenpair (, ** U**), we write the inner product of equation (6.10) with the eigenvector as(6.30)and we obtain the eigenvalue in the form(6.31)In this equation, the numerator is the potential energy and the denominator is the kinetic energy and both are positive-definite due to equations (6.19) and (6.21). Thus, the eigenvalue is positive as well.

In the admissible state function space {**U**}, we consider two different eigenpairs in equation (6.10), namely(6.32)We take the inner product of both sides of this equation with and subtract one from another with the result(6.33)Due to the self-adjointness of the operators ** P** and

**, we find from equation (6.33)(6.34a)This may be expressed by(6.34b)**

*A*This is the orthogonality condition for the free vibrations of the regular poroelastic region. By using equations (6.6) and (6.7), the orthogonality condition is expressed by(6.35)in terms of the material coefficients. Similar results were presented for a polar region (Dökmeci & Altay 2001).

*Rayleigh's quotient*. We now define a functional *R*{** U**} as a ratio of the quantities

*N*{

**} and**

*U**D*{

**} related to the potential and kinetic energies (6.19) and (6.21), namely(6.36)with(6.37)Taking the first variation of the functional**

*U**R*, we have the Euler–Lagrange equations as(6.38)which is reduced to(6.39)We readily express this condition(6.40)in view of the self-adjointness of the operators

**{**

*P***} and**

*U***{**

*A***}, and then we read(6.41)which simply indicates(6.42)due to equation (6.10). Thus, we conclude that the Euler–Lagrange equation of the functional**

*U**R*{

**} is equal to the eigenvalue with . We explicitly write the functional**

*U**R*by(6.43)Rayleigh's quotient, equations (6.36) and (6.43), with its well-known features is the counterpart of that given in elasticity. It has a stationary value in the neighbourhood of an eigenvalue

**. The functional**

*U**R*of equation (6.43) is positive-definite in view of equations (6.19) and (6.21), and hence, it is bounded from below by zero and the smallest eigenvalue is a minimum.

## 7. Conclusions

In this paper, we rigorously studied Biot's equations in differential form governing the motion of the regular region of an anisotropic fluid-saturated poroelastic material. Our study dealt with three features of the governing equations: (i) a theorem of uniqueness, including the boundary and initial conditions sufficient to the uniqueness in their solutions in §3; (ii) their variational form by certain variational principles deduced from Hamilton's principle by modifying it through Legendre's transformation in §§4 and 5 and (iii) their applications to the time-harmonic free vibrations, including Rayleigh's quotient in §6. The variational principle (5.21) operates on all the field variables and generates, as its Euler–Lagrange equations, all the governing equations of the regular poroelastic region with and without an internal fixed surface of discontinuity, including the jump conditions but excluding the initial conditions and the symmetry of the stress tensor. The initial conditions may be relaxed by way of relaxation (Tiersten 1969), but the symmetry of the stress tensor always remains as a constraint condition in all the variational principles of non-polar media. The variational principle is an extended version of the Hellinger–Reissner variational principle in Biot's poroelasticity.

The generalized variational principle (5.21) can be useful when solving numerically the poroelastic equations in heterogeneous media with finite element methods, and also, it is capable of establishing, consistently and systematically, some one-dimensional/two-dimensional equations of poroelastic structures, including laminated ones. Besides, the principle presents a standard tool in solving directly the initial and mixed boundary value problems defined by the poroelastic equations and, in particular, provides a simultaneous approximation to all the field variables. Additionally, Rayleigh's quotient is a powerful tool in treating the free vibrations of the regular poroelastic region.

In conclusion, we presented the results that seem to indicate many interesting features, and we will focus in our future works on involving some specific applications and extensions in time- and temperature-dependent porous media in elastic as well as plastic ranges which is of special importance.

## Acknowledgements

The authors are grateful to Professor José M. Carcione (Istituto Nazionale di Oceanografia e di Geofisica Sperimentale, Trieste) for his enlightening comments. M.C.D. would like to express his sincere thanks to Professor Oral Büyüköztürk (Department of Civil & Environmental Engineering, M.I.T.) for the funding and kind invitation to a 3-day workshop on engineering materials on 9–11 June, 2004, Cambridge. The financial support by TUBA is gratefully acknowledged.

## Footnotes

- Received September 30, 2005.
- Accepted January 6, 2006.

- © 2006 The Royal Society