Variational principle, uniqueness and reciprocity theorems in porous magneto-piezothermoelastic medium

The basic governing equations for an anisotropic porous magneto-piezothermoelastic medium are presented. The variational principle, uniqueness theorem and theorem of reciprocity in this model are established under the assumption of positive definiteness of magnetic and piezoelectric fields. Particular cases of interest are also deduced and compared with the known results. Subjects: Applied Mathematics; Mathematics & Statistics; Science


Introduction
With the increase in use of advanced composites as important structural components in speedy aircrafts, mobiles, missiles, ceramic plates as transducers, marine vehicles, aerospace structures and various other such applications has inspired the research activities. One such composite materials is porous magneto-piezothermoelastic material.
The theory of thermopiezoelectric material was first proposed by Mindlin (1974) who derived the governing equations of a thermopiezoelectric plate. The physical laws for the thermopiezoelectric material have been explored by Nowacki (1978Nowacki ( , 1979. Chandrasekharaiah (1984) used generalised Mindlin's theory of thermopiezoelectricity to account for the finite speed of propagation of thermal disturbances. Rao and Sunar (1993) pointed out the temperature variation in the piezoelectric media. Majhi (1995) studied the transient thermal response of the semi-infinite piezoelectric rod subjected to the heat source. Chen (2000) derived the general solution for transversely isotropic piezothermoelastic media. In this general solution, all components of the coupled field are expressed by four harmonic functions. Sharma and Kumar (2000) discussed the plane harmonic waves in piezothermoelastic material. Sharma, Pal, and Chand (2005) studied the propagation characteristics of Rayleigh waves in transversely isotropic piezothermoelastic materials. Sharma and Walia (2007) investigated Rayleigh waves in transversely isotropic piezothermoelastic materials. Sharma

PUBLIC INTEREST STATEMENT
This paper contains the basic equations of thermodynamics depicting the combined effect of piezoelectric and magnetic fields in a porous thermoelastic medium. We established the basic theorems of thermoelasticity which are the part of the continuum mechanics and would help many researchers in establishing such more results.
(2010) discussed the propagation of inhomogeneous waves in anisotropic piezothermoelastic media. Alshaikh (2012) presented the mathematical model for studying the influence of the initial stresses and relaxation waves in piezothermoelastic half-space.
From the historical background, it is identified that the two theories namely the Biot Theory and Theory of Porous Media have been used nowadays to study multiphase continuum mechanics. On the basis of work done by Von Terzaghi, a theoretical description of porous material saturated by a viscous fluid was presented by Biot and then extended his theory to anisotropic and further poroviscoelastic cases. The dynamic behaviour of porous medium is important in the field of seismic exploration. The porosity and permeability are the basic and economic parameters for the field of oil production. Reservoir rocks also possess anisotropic behaviour in permeability of pores as a reservoir is a fluid-saturated porous solid medium pervaded by aligned cracks. Porosity is the geometrical property of the solid to hold the fluid. Biot developed the full dynamic theory for wave propagation in fluid-saturated porous media. Biot used Lagrange's equations to derive a set of coupled differential equations that govern the motions of solid and fluid phases. Biot (1962a) extended the acoustic propagation theory in the wider context of the mechanics of porous media. Biot (1962b) developed new features of the extended theory in more detail. On the other hand, Theory of Porous Media is based on the work done by Fillunger which further is preceeded from the assumption of immiscible and superimposed continua with internal interaction. Sharma and Gogna (1991) discussed wave propagation in porous solid with a viscoelastic frame filled with a viscous fluid. Sharma (2004a) used Biot's 1956 theory to study the phase velocities and attenuations of quasi-waves in a general anisotropic porous solid with anisotropic permeability controlling the flow of viscous fluid in its pores. Sharma (2004b) studied velocities and polarisation in anisotropic porous solid saturated with non-viscous fluid. Sharma (2005) studied the polarisations of quasi-waves in a general anisotropic porous solid saturated with viscous fluid. Sharma (2008) investigated the wave propagation in thermoelastic saturated porous medium. The boundary conditions for porous solids saturated with viscous fluid are described by Sharma (2009).
Porous piezoelectric materials are studied due to their applications such as low-frequency hydrophones, underwater sensing and actuation application (Arai et al., 1991;Hashimoto & Yamaguchi, 1986). It has high hydrostatic figures of merit and low sound velocity of these materials due to which the reduction in acoustic impedance and enhancement of coupling with water are possible. Some experimental studies (Hayashi et al., 1991;Xia, Ma, Qiu, Wu, & Wang, 2003) have been made for the characterisation of properties of porous piezoelectric materials. A number of authors (Banno, 1993;Gómez Alvarez-Arenas & Montero de Espinosa, 1996) developed theoretical models to study the effect of porosity on the elastic, piezoelectric and dielectric properties of porous piezoelectric materials. Vashishth and Gupta (2009) described the vibrations of porous piezoelectric ceramic plates.
With the development of active material systems, there is significant interest in coupling effects between elastic, electric, magnetic and thermal fields, for their applications in sensing and actuation. Although natural materials rarely show full coupling between elastic, electric, magnetic and thermal fields, some artificial materials do. Van Run, Terrell, and Scholing (1974) reported the fabrication of BaTiO 3 -CoFe 2 O 4 composite which had the magnetoelectric effect not existing in either the constituent. Li and Dunn (1998) quantitatively explained the magnetoelectric coupling created through the interaction between piezoelectric and piezomagnetic phases. Oatao and Ishihara (2013) analysed the laminated hollow cylinder constructed of isotropic elastic and magneto-electro-thermoelastic material. Pang and Li (2014) studied the SH interfacial waves between piezoelectric/piezomagnetic half-spaces with magneto-electro-elastic imperfect bonding. The effects of piezoelectric and piezomagnetic on the surface wave velocity of magneto-electro-elastic solids are studied by Li and Wei (2014).
A comprehensive work has been done on uniqueness, reciprocity theorems and variational principle by different authors in different media. Ignaczak (1979) studied the uniqueness theorem in generalised thermoelasticity, Sherie and Dhaliwa (1980) studied variational principle along with uniqueness theorem for generalised thermoelasticity, Ieşan (1990) discussed the reciprocity, uniqueness and minimum principles in the linear theory of piezoelectricity, Ezzat and El Karamany (2002) discussed uniqueness and reciprocity theorems for generalised thermoviscoelastic media, Li (2003) studied these results for for linear thermo-electro-magneto-elasticity. Similarly, Othman (2004) proved these results for thermoviscoelastic medium with thermal relaxation times and Aouadi (2007) proved it for thermoelastic diffusive medium. Kuang (2010) established variational principles for generalised thermodiffusive pyroelectric media, Vashishth and Gupta (2011) proved these results and solved eigenvalue problems in porous piezoelectric media, Kumar and Kansal (2013) proved these results for generalised thermoelastic diffusive medium and Kumar and Gupta (2013) discussed these results for generalised thermoelastic diffusive medium with fractional order derivative.
Inspite of these studies, not much work has been done in porous magneto-piezothermoelastic body. The main focus of the present investigation is to study the variational problem, reciprocity theorem and uniqueness of solutions in the considered model. These theorems will be helpful for the further investigation of the various problems. Following Li (2003), Kuang (2010) and Vashishth and Gupta (2011), the governing equations in a homogeneous, anisotropic porous magneto-piezothermoelastic medium in the absence of thermal and magnetic sources and independent of free charge densities and magnetic densities are:

Basic equations
Constitutive equations:

Variational principle
The principle of virtual work with variation in displacements for the elastic deformable body is written as (2.12) On the left hand side, we have the virtual work of body forces F i , f i , inertial forces 1üi , 2ü * i , surface forces h i , h * i , whereas on the right hand side, we have the virtual work of internal forces. We denote the outward normal of ∂V by n i . c 0 , c * 0 are the surface charge densities and ϕ, ϕ* are the electric potentials, b 0 , b * 0 are the magnetic densities and ψ, ψ* are the magnetic potentials for the solid and fluid phases.
Using the symmetry of the stress tensors, divergence theorem and the definition of the strain tensors, the Equation (3.1) is written in the alternative form as Substituting the value of σ ij and σ* from the relation (2.1) and (2.2) in the Equation (3.2) and using Equations (2.9)-(2.12), we obtain where The Equation (3.3) would be complete for the uncoupled problem of porous magneto-piezothermoelastic, where the temperature θ, the electric potentials ϕ, ϕ*, the magnetic potentials ψ, ψ* are known functions. In this case, when we take into account the coupling of the deformation field with the temperature, there arises the necessity of considering one additional relation characterising the phenomenon of the thermal conductivity.  (2.8), we obtain the variational principle in the following form: On the right-hand side of Equation (3.15), we find all the causes, the mass forces, inertial forces, the surface forces and the heating on the surface A bounding the body.

Particular case:
In the absence of magnetic effect and further if we put coupling coefficients of pore-fluid phase to zero with ρ 12 = ρ 22 = 0, and then we obtain the similar results as obtained by Ieşan (1990).

Uniqueness theorem
We assume that the virtual displacements u i , u * i , the virtual increment of the temperature δθ etc. correspond to the increments occurring in the body. Then and Equation (3.15) reduces to the following relation  and the initial conditions on the surface at t = 0 ̇0, 0 ,̇0, * 0 ,̇ * 0 , 0 ,̇0, * 0 ,̇ * 0 are known functions. We assume that the material parameters satisfy the inequalities c ijkl , L ij and m ij are positive definite.
, * (2) , (2) , psi * (2) , be two solutions sets of Equations (2.1)-(2.19). Let us take The functions u i , u * i , , , * , and ψ* satisfy the governing equations with zero body forces and homogeneous initial and boundary conditions. Thus, these functions satisfy an equation similar to the Equation (4.5) with zero right-hand side, that is, Since, we have Therefore, from Equation (3.14), we obtain Substitution of Equation (4.9) in the Equation (4.8) yields Using the inequalities (4.6) in Equation (4.10), we obtain We thus see that the expression is a decreasing function of time. We also note that the expression ∫ V r 2 dV occurring in the expression (4.12) is always positive, due to the laws of thermodynamics Nowacki (1974) Thus, the expression (4.12) vanishes for t = 0, due to the homogeneous initial conditions, and it must be always non-positive for t > 0. (4.12) (4.13) 0 < r.
Using inequalities (4.6) and (4.11), it follows immediately that the expression (4.12) must be identically zero for t > 0. We thus have This proves the uniqueness of the solution to the complete system of field equations subjected to the electric potential-magnetic potential-displacement-temperature initial and boundary conditions.

Particular case:
In the absence of magnetic effect and further if we put coupling coefficients of pore-fluid phase to zero with ρ 12 = ρ 22 = 0, then we obtain the similar results as obtained by Ieşan (1990).

Reciprocity theorem
We shall consider a homogeneous anisotropic porous magneto-piezothermoelastic elastic body occupying the region V and bounded by the surface A. We assume that the stresses ij , * and the strains ij , * are continuous together with their first order derivatives, whereas the displacements u i , u * i , temperature θ and the electrical potentials ϕ, ϕ*, magnetic potentials ψ, ψ* are continuous and have continuous derivatives up to second order, for x ∈ V + A, t > 0. The components of surface tractions, the normal component of the heat flux and electric displacements at regular points of ∂V are given by respectively.
To the system of field equations, we must adjoin boundary conditions and initial conditions. We consider the following boundary conditions: for all x ∈ A, t > 0 and the homogeneous initial conditions We derive the dynamic reciprocity relationship for a generalised porous magneto-piezothermoelastic bounded body V, which satisfies Equations (2.1)-(2.19), the boundary conditions (5.2) and the homogeneous initial conditions (5.3), and are subjected to the action of body forces F i (x, t), f i (x, t), surface tractions h i (x, t), h * i (x, t), the heat flux q(x, t), the magnetic densities b 0 (x, t), b * 0 (x, t) and the surface charge densities c 0 (x, t), c * 0 (x, t).
We define the Laplace transform as Applying the Laplace transform defined by the Equation (5.4) on the Equations (2.1)-(2.19) and omitting the bars for simplicity, we obtain = * = = * = u i = u * i = = ij = * = ij = * = 0. (5.1) We now consider two problems where applied body forces, electric potential and the surface temperature are specified differently. Let the variables involved in these two problems be distinguished by superscripts in parentheses. Thus, we have (5.5) Using the assumption ij = ji , we obtain Using the divergence theorem in the first term of the right-hand side of Equation ( ij , * (2) and (1) ij , * (1) for the first and second problems, respectively, subtracting and integrating over the region V, we obtain Using the symmetry properties of c ijkl , we obtain Equating Equations (5.27) and (5.28), we get the first part of the reciprocity theorem ,i * (2) dV. (5.28) ,i * (2) dV.