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

Abstract

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.

Keywords:

• piezothermoelastic
• porous
• variational principle
• uniqueness
• reciprocity

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.

1. 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 (1978, 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 (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 materisals 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 BaTiO₃-CoFe₂O₄ 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.

2. Basic equations

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:

Constitutive equations:(2.1) $${\mathit{\sigma }}_{ij}=c_{ijkl}{\mathit{\epsilon }}_{kl}-e_{ijk}{E}_k-{\mathit{\alpha }}_{ij}\mathit{\theta }+m_{ij}{\mathit{\epsilon }}^{\ast }-{\mathit{\zeta }}_{kij}{E}_k^{\ast }-q_{ijk}{H}_k-q_{ijk}^f{H}_k^{\ast },$$(2.1)

(2.2) $${\mathit{\sigma }}^{\ast }=m_{ij}{\mathit{\epsilon }}_{ij}-{\mathit{\zeta }}_i{E}_i-{\mathit{\alpha }}^f\mathit{\theta }+R{\mathit{\epsilon }}^{\ast }-e_i^{\ast }{E}_i^{\ast }-l_i{H}_i-l_i^{\ast }{H}_i^{\ast },$$(2.2)

(2.3) $$-q_{i,i}=\mathit{\rho }{T}_0\dot{S},$$(2.3) (2.4) $$\mathit{\rho }S={\mathit{\alpha }}_{ij}{\mathit{\epsilon }}_{ij}+{\mathit{\tau }}_i{E}_i+r\mathit{\theta }+{\mathit{\alpha }}^f{\mathit{\epsilon }}^{\ast }+{\mathit{\tau }}_i^f{E}_i^{\ast }+m_i{H}_i+m_i^f{H}_i^{\ast },$$(2.4) (2.5) $$D_i={\mathit{\xi }}_{ij}{E}_j+e_{ijk}{\mathit{\epsilon }}_{jk}+{\mathit{\tau }}_i\mathit{\theta }+{\mathit{\zeta }}_i{\mathit{\epsilon }}^{\ast }+A_{ij}{E}_j^{\ast }+f_{ij}^f{H}_j^{\ast }+f_{ij}{H}_j,$$(2.5) (2.6) $$D_i^{\ast }={\mathit{\zeta }}_{ijk}{\mathit{\epsilon }}_{jk}+{\mathit{\tau }}_i^f\mathit{\theta }+A_{ij}{E}_j+{\mathit{\xi }}_{ij}^{\ast }{E}_j^{\ast }+e_i^{\ast }{\mathit{\epsilon }}^{\ast }+f_{ij}^f{H}_j+{\mathit{\gamma }}_{ij}{H}_j^{\ast },$$(2.6) (2.7) $$B_i=f_{ij}{E}_j+q_{ijk}{\mathit{\epsilon }}_{jk}+m_i\mathit{\theta }+l_i{\mathit{\epsilon }}^{\ast }+f_{ij}^f{E}_j^{\ast }+{\mathit{\beta }}_{ij}^f{H}_j^{\ast }+{\mathit{\beta }}_{ij}{H}_j,$$(2.7)

(2.8) $$B_i^{\ast }=q_{ijk}^f{\mathit{\epsilon }}_{jk}+{\mathit{\tau }}_i^f\mathit{\theta }+f_{ij}^f{E}_j+{\mathit{\gamma }}_{ij}{E}_j^{\ast }+l_i^{\ast }{\mathit{\epsilon }}^{\ast }+{\mathit{\beta }}_{ij}^f{H}_j+{\mathit{\gamma }}_{ij}^{\ast }{H}_j^{\ast },$$(2.8)

(2.9) $$E_i=-{\mathit{\varphi }}_{,i},$$(2.9)

(2.10) $$E_i^{\ast }=-{\mathit{\varphi }}_{,i}^{\ast },$$(2.10) (2.11) $$H_i=-{\mathit{\psi }}_{,i},$$(2.11) (2.12) $$H_i^{\ast }=-{\mathit{\psi }}_{,i}^{\ast },(i,j,k,l=1,2,3)$$(2.12)

Equations of motion:(2.13) $${\mathit{\sigma }}_{ij,j}+{\mathit{\rho }}_1{F}_i-{\mathit{\rho }}_{11}{\ddot{\text{u}}}_i-{\mathit{\rho }}_{12}{\ddot{\text{u}}}_i^{\ast }=0,$$(2.13)

(2.14) $${\mathit{\sigma }}_{,i}^{\ast }+{\mathit{\rho }}_2{f}_i-{\mathit{\rho }}_{12}{\ddot{\text{u}}}_i-{\mathit{\rho }}_{22}{\ddot{\text{u}}}_i^{\ast }=0,$$(2.14)

Equation of heat conduction:(2.15) $$-K_{ij}{\mathit{\theta }}_{,j}=\left(1+{\mathit{\tau }}_0\frac{\mathit{\partial }}{\mathit{\partial }t}\right)q_i$$(2.15)

Gauss equations:(2.16) $$D_{i,i}=0$$(2.16) (2.17) $$D_{i,i}^{\ast }=0$$(2.17) (2.18) $$B_{i,i}=0$$(2.18)

(2.19) $$B_{i,i}^{\ast }=0(i,j=1,2,3)$$(2.19)

In the Equations, (2.1)–(2.11), $c_{ijkl}(=c_{klij}=c_{jikl}=c_{ijlk}),m_{ij}(=m_{ji})$ are the tensors of elastic constants. The elastic constant R measures the pressure to be exerted on fluid, ρ₁₁ is the mass density for solid, ρ₂₂ is the mass density for fluid, ρ₁₂ is the mass coupling parameter and ${\mathit{\rho }}_1={\mathit{\rho }}_{11}+{\mathit{\rho }}_{12},{\mathit{\rho }}_2={\mathit{\rho }}_{12}+{\mathit{\rho }}_{22}\text{and}\mathit{\rho }$ is the mass density of the material, q_i are the components of heat flux vector q, respectively, $F_i\text{and}{f}_i$ are components of the external forces per unit mass for the solid and fluid phases, ${\text{u}}_i\text{and}{\text{u}}_i^{\ast }$ are the components of displacement vectors $\mathbf{u}\text{and}{\mathbf{u}}^{\ast }$, ${\mathit{\sigma }}_{ij}(={\mathit{\sigma }}_{ji})\text{and}{\mathit{\sigma }}^{\ast }$ are the components of the stress tensors for the solid and fluid phases, ${\mathit{\epsilon }}_{ij}=\frac{1}{2}\left({\text{u}}_{i,j}+{\text{u}}_{j,i}\right)\text{and}{\mathit{\epsilon }}^{\ast }={\text{u}}_{i,i}^{\ast }$ are the components of the strain tensors for the solid and fluid phases, K_ij(=K_ji) are, respectively, the components of thermal conductivity tensors, S is the entropy per unit mass, respectively, $E_i,E_i^{\ast }$ are the electric field intensities, $D_i,D_i^{\ast }$ are the electric displacements, ϕ, ϕ* are the electric potentials for the solid and fluid phases, $H_i,H_i^{\ast }$ are the magnetic field intensities, $B_i,B_i^{\ast }$ are the magnetic displacements, ψ and ψ^* are the magnetic potentials, θ is the absolute temperature of the medium, T₀ is the reference temperature of the body, r is the coefficient describing the measure of thermal effect, $e_{ijk},{\mathit{\zeta }}_{ijk},A_{ij},{\mathit{\xi }}_{ij},{\mathit{\alpha }}_{ij},{\mathit{\alpha }}^f,{\mathit{\tau }}_i^f,{\mathit{\tau }}_i,e_i^{\ast },{\mathit{\zeta }}_i,{\mathit{\xi }}_{ij}^{\ast },q_{ijk},q_{ijk}^f,l_i,l_i^{\ast },m_i,m_i^f,f_{ij}^f,f_{ij},{\mathit{\gamma }}_{ij},{\mathit{\gamma }}_{ij}^{\ast },{\mathit{\beta }}_{ij}^f,{\mathit{\beta }}_{ij}$ are tensors of porous magneto-piezothermal moduli, respectively, τ₀ is the thermal relaxation time, which ensures that the heat conduction equation predicts finite speeds of heat propagation speeds of matter from one medium to other. The symbol “^*” indicates the parameters for pore-fluid phase.

3. Variational principle

The principle of virtual work with variation in displacements for the elastic deformable body is written as(3.1) $$\underset{V}{\int }\left({\mathit{\rho }}_1{F}_i-{\mathit{\rho }}_{11}{\ddot{\text{u}}}_i-{\mathit{\rho }}_{12}{\ddot{\text{u}}}_i^{\ast }\right)\mathit{\delta }{u}_i\text{d}V+\underset{V}{\int }\left({\mathit{\rho }}_2{f}_i-{\mathit{\rho }}_{12}{\ddot{\text{u}}}_i-{\mathit{\rho }}_{22}{\ddot{\text{u}}}_i^{\ast }\right)\mathit{\delta }{u}_i^{\ast }\text{d}V+\underset{A}{\int }\left(h_i\mathit{\delta }{u}_i+h_i^{\ast }\mathit{\delta }{u}_i^{\ast })\text{d}A\right.+\underset{A}{\int }\left(c_0\mathit{\delta }\mathit{\varphi }+c_0^{\ast }\mathit{\delta }{\mathit{\varphi }}^{\ast }\right)\text{d}A+\underset{A}{\int }\left(b_0\mathit{\delta }\mathit{\psi }+b_0^{\ast }\mathit{\delta }{\mathit{\psi }}^{\ast }\right)\text{d}A=\underset{A}{\int }\left({\mathit{\sigma }}_{ij}{n}_j\mathit{\delta }{u}_i+{\mathit{\sigma }}^{\ast }{n}_i\mathit{\delta }{u}_i^{\ast }\right)\text{d}A+\underset{A}{\int }\left(D_i{n}_i\mathit{\delta }\mathit{\varphi }+D_i^{\ast }{n}_i\mathit{\delta }{\mathit{\varphi }}^{\ast }\right)\text{d}A+\underset{A}{\int }\left(B_i{n}_i\mathit{\delta }\mathit{\psi }+B_i^{\ast }{n}_i\mathit{\delta }{\mathit{\psi }}^{\ast }\right)\text{d}A,$$(3.1)

where $h_i={\mathit{\sigma }}_{ij}{n}_j,h_i^{\ast }={\mathit{\sigma }}^{\ast }{n}_i,c_0=D_i{n}_i,c_0^{\ast }=D_i^{\ast }{n}_i,b_0=B_i{n}_i\text{and}{b}_0^{\ast }=B_i^{\ast }{n}_i.$

On the left hand side, we have the virtual work of body forces $F_i,f_i$, inertial forces ${\mathit{\rho }}_1{\ddot{\text{u}}}_i,{\mathit{\rho }}_2{\ddot{\text{u}}}_i^{\ast }$, surface forces $h_i,h_i^{\ast }$, whereas on the right hand side, we have the virtual work of internal forces. We denote the outward normal of ∂V by n_i. $c_0,c_0^{\ast }$ are the surface charge densities and ϕ, ϕ* are the electric potentials, $b_0,b_0^{\ast }$ 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(3.2) $$\underset{V}{\int }({\mathit{\rho }}_1{F}_i-{\mathit{\rho }}_{11}{\ddot{\text{u}}}_i-{\mathit{\rho }}_{12}{\ddot{\text{u}}}_i^{\ast })\mathit{\delta }{u}_i\text{d}V+\underset{V}{\int }({\mathit{\rho }}_2{f}_i-{\mathit{\rho }}_{12}{\ddot{\text{u}}}_i-{\mathit{\rho }}_{22}{\ddot{\text{u}}}_i^{\ast })\mathit{\delta }{u}_i^{\ast }\text{d}V+\underset{A}{\int }(h_i\mathit{\delta }{u}_i+h_i^{\ast }\mathit{\delta }{u}_i^{\ast })\text{d}A+\underset{A}{\int }(c_0\mathit{\delta }\mathit{\varphi }+c_0^{\ast }\mathit{\delta }{\mathit{\varphi }}^{\ast })\text{d}A+\underset{A}{\int }(b_0\mathit{\delta }\mathit{\psi }+b_0^{\ast }\mathit{\delta }{\mathit{\psi }}^{\ast })\text{d}A=\underset{V}{\int }({\mathit{\sigma }}_{ij}\mathit{\delta }{u}_{i,j}+{\mathit{\sigma }}^{\ast }\mathit{\delta }{\text{u}}_{i,i}^{\ast })\text{d}V+\underset{V}{\int }(D_i\mathit{\delta }{\mathit{\varphi }}_{,i}+D_i^{\ast }\mathit{\delta }{\mathit{\varphi }}_{,i}^{\ast })\text{d}V+\underset{A}{\int }(B_i\mathit{\delta }{\mathit{\psi }}_{,i}+B_i^{\ast }\mathit{\delta }{\mathit{\psi }}_{,i}^{\ast })\text{d}A,$$(3.2)

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 (3.3) $$\underset{V}{\int }({\mathit{\rho }}_1{F}_i-{\mathit{\rho }}_{11}{\ddot{\text{u}}}_i-{\mathit{\rho }}_{12}{\ddot{\text{u}}}_i^{\ast })\mathit{\delta }{u}_i\text{d}V+\underset{V}{\int }({\mathit{\rho }}_2{f}_i-{\mathit{\rho }}_{12}{\ddot{\text{u}}}_i-{\mathit{\rho }}_{22}{\ddot{\text{u}}}_i^{\ast })\mathit{\delta }{u}_i^{\ast }\text{d}V+\underset{A}{\int }(h_i\mathit{\delta }{u}_i+h_i^{\ast }\mathit{\delta }{u}_i^{\ast })\text{d}A+\underset{A}{\int }(c_0\mathit{\delta }\mathit{\varphi }+c_0^{\ast }\mathit{\delta }{\mathit{\varphi }}^{\ast })\text{d}A+\underset{A}{\int }(b_0\mathit{\delta }\mathit{\psi }+b_0^{\ast }\mathit{\delta }{\mathit{\psi }}^{\ast })\text{d}A=\underset{V}{\int }(c_{ijkl}{\mathit{\epsilon }}_{kl}-e_{ijk}{E}_k-{\mathit{\alpha }}_{ij}\mathit{\theta }+m_{ij}{\mathit{\epsilon }}^{\ast }-{\mathit{\zeta }}_{kij}{E}_k^{\ast }-q_{ijk}{H}_k)\mathit{\delta }{\mathit{\epsilon }}_{ij}\text{d}V+\underset{V}{\int }(m_{ij}{\mathit{\epsilon }}_{ij}-{\mathit{\zeta }}_i{E}_i-{\mathit{\alpha }}^f\mathit{\theta }+R{\mathit{\epsilon }}^{\ast }-e_i^{\ast }{E}_i^{\ast }-l_i{H}_i-l_i^{\ast }{H}_i^{\ast })\mathit{\delta }{\mathit{\epsilon }}^{\ast }\text{d}V-\underset{V}{\int }{q}_{ijk}^f{H}_k^{\ast }\mathit{\delta }{\mathit{\epsilon }}_{ij}\text{d}V-\underset{V}{\int }(D_i\mathit{\delta }{E}_i+D_i^{\ast }\mathit{\delta }{E}_i^{\ast })\text{d}V-\underset{V}{\int }(B_i\mathit{\delta }{H}_i+B_i^{\ast }\mathit{\delta }{H}_i^{\ast })\text{d}V=\mathit{\delta }W-\underset{V}{\int }{e}_{ijk}{E}_k\mathit{\delta }{\mathit{\epsilon }}_{ij}\text{d}V-\underset{V}{\int }{\mathit{\alpha }}_{ij}\mathit{\theta }\mathit{\delta }{\mathit{\epsilon }}_{ij}\text{d}V-\underset{V}{\int }{\mathit{\zeta }}_{kij}{E}_k^{\ast }\mathit{\delta }{\mathit{\epsilon }}_{ij}\text{d}V-\underset{V}{\int }{q}_{ijk}{H}_k\mathit{\delta }{\mathit{\epsilon }}_{ij}\text{d}V-\underset{V}{\int }{e}_i^{\ast }{E}_i^{\ast }\mathit{\delta }{\mathit{\epsilon }}^{\ast }\text{d}V-\underset{V}{\int }{\mathit{\zeta }}_i{E}_i\mathit{\delta }{\mathit{\epsilon }}^{\ast }\text{d}V-\underset{V}{\int }{\mathit{\alpha }}^f\mathit{\theta }\mathit{\delta }{\mathit{\epsilon }}^{\ast }\text{d}V-\underset{V}{\int }{D}_i\mathit{\delta }{E}_i\text{d}V-\underset{V}{\int }{B}_i\mathit{\delta }{H}_i\text{d}V-\underset{V}{\int }{D}_i^{\ast }\mathit{\delta }{E}_i^{\ast }\text{d}V-\underset{V}{\int }{q}_{ijk}^f{H}_k^{\ast }\mathit{\delta }{\mathit{\epsilon }}_{ij}\text{d}V-\underset{V}{\int }{B}_i^{\ast }\mathit{\delta }{H}_i^{\ast }\text{d}V-\underset{V}{\int }{l}_i\mathit{\delta }{H}_i\text{d}V-\underset{V}{\int }{l}_i^{\ast }\mathit{\delta }{H}_i^{\ast }\text{d}V,$$(3.3)

where

$$W=\frac{1}{2}\underset{V}{\int }(c_{ijkl}{\mathit{\epsilon }}_{kl}{\mathit{\epsilon }}_{ij}+R{\mathit{\epsilon }}^{\ast }{\mathit{\epsilon }}^{\ast }+2m_{ij}{\mathit{\epsilon }}^{\ast }{\mathit{\epsilon }}_{ij})\text{d}V,\mathit{\delta }{u}_{i,j}=\mathit{\delta }{\mathit{\epsilon }}_{ij},\mathit{\delta }{u}_{i,i}^{\ast }=\mathit{\delta }{\mathit{\epsilon }}^{\ast },\mathit{\delta }{\mathit{\varphi }}_{,j}=-\mathit{\delta }{E}_j\text{and}\mathit{\delta }{\mathit{\varphi }}_{,j}^{\ast }=-\mathit{\delta }{E}_j^{\ast }\mathit{\delta }{\mathit{\psi }}_{,j}=-\mathit{\delta }{H}_j\text{and}\mathit{\delta }{\mathit{\psi }}_{,j}^{\ast }=-\mathit{\delta }{H}_j^{\ast }.$$

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.

Following Biot (1956), we define a vector $\mathbf{J}$ connected with the entropy through the relation(3.4) $$\mathit{\rho }S=-J_{i,i}$$(3.4)

Equations (2.3), (2.4), (2.15) and (3.4) combined together yield(3.5) $$T_0{L}_{ij}\left(\frac{\mathit{\partial }}{\mathit{\partial }t}+{\mathit{\tau }}_0\frac{{\mathit{\partial }}^2}{\mathit{\partial }{t}^2}\right)J_i+{\mathit{\theta }}_{,j}=0$$(3.5) (3.6) $$-J_{i,i}={\mathit{\alpha }}_{ij}{\mathit{\epsilon }}_{ij}+{\mathit{\tau }}_i{E}_i+r\mathit{\theta }+{\mathit{\alpha }}^f{\mathit{\epsilon }}^{\ast }+{\mathit{\tau }}_i^f{E}_i^{\ast }+m_i{H}_i+m_i^f{H}_i^{\ast },$$(3.6)

where L_ij, the resistivity matrix, is the inverse of the thermal conductivity tensor K_ij.

Multiplying both sides of the equation (3.5) by δJ_j and integrating over the region occupied by the body gives(3.7) $$\underset{V}{\int }\left[{\mathit{\theta }}_{,j}+T_0{L}_{ij}\left(\frac{\mathit{\partial }{J}_i}{\mathit{\partial }t}+{\mathit{\tau }}_0\frac{{\mathit{\partial }}^2{J}_i}{\mathit{\partial }{t}^2}\right)\right]\mathit{\delta }{J}_j\text{d}V=0.$$(3.7)

Now(3.8) $$\underset{V}{\int }{\mathit{\theta }}_{,j}\mathit{\delta }{J}_j\text{d}V=\underset{V}{\int }{\left(\mathit{\theta }\mathit{\delta }{J}_j\right)}_{,j}\text{d}V-\underset{V}{\int }\mathit{\theta }\mathit{\delta }{J}_{j,j}\text{d}V,$$(3.8)

Applying the divergence theorem defined by,(3.9) $$\underset{V}{\int }{\left(\mathit{\theta }\mathit{\delta }{J}_j\right)}_{,j}\text{d}V=\underset{A}{\int }\left(\mathit{\theta }\mathit{\delta }{J}_j\right)n_j\text{d}A,$$(3.9)

in the Equation (3.8), yields(3.10) $$\underset{V}{\int }{\mathit{\theta }}_{,j}\mathit{\delta }{J}_j\text{d}V=\underset{A}{\int }\left(\mathit{\theta }\mathit{\delta }{J}_j\right)n_j\text{d}A-\underset{V}{\int }\mathit{\theta }\mathit{\delta }{J}_{j,j}\text{d}V.$$(3.10)

Substituting Equation (3.10) in the Equation (3.7), we obtain(3.11) $$\underset{A}{\int }\left(\mathit{\theta }\mathit{\delta }{J}_j\right)n_j\text{d}A-\underset{V}{\int }\mathit{\theta }\mathit{\delta }{J}_{j,j}\text{d}V+T_0\underset{V}{\int }{L}_{ij}\left(\frac{\mathit{\partial }{J}_i}{\mathit{\partial }t}+{\mathit{\tau }}_0\frac{{\mathit{\partial }}^2{J}_i}{\mathit{\partial }{t}^2}\right)\mathit{\delta }{J}_j\text{d}V=0.$$(3.11)

Making use of Equation (3.6) in the Equation (3.11), yields the second variational equation(3.12) $$\underset{A}{\int }\mathit{\theta }\mathit{\delta }{J}_j{n}_j\text{d}A+\underset{V}{\int }{\mathit{\alpha }}_{ij}\mathit{\theta }\mathit{\delta }{\mathit{\epsilon }}_{ij}\text{d}V+\underset{V}{\int }\mathit{\theta }{\mathit{\tau }}_j\mathit{\delta }{E}_j\text{d}V+\underset{V}{\int }{\mathit{\alpha }}^f\mathit{\delta }{\mathit{\epsilon }}^{\ast }\mathit{\theta }\text{d}V+\underset{V}{\int }{\mathit{\tau }}_i^f\mathit{\delta }{E}_i^{\ast }\mathit{\theta }\text{d}V+\underset{V}{\int }{m}_i^f\mathit{\delta }{H}_i^{\ast }\text{d}V+\underset{V}{\int }{m}_i\mathit{\delta }{H}_i\mathit{\theta }\text{d}V+\mathit{\delta }(M+R)=0,$$(3.12)

where the function of thermal potential M is defined by(3.13) $$M=\frac{r}{2}\underset{V}{\int }{\mathit{\theta }}^2\text{d}V,\mathit{\delta }M=r\underset{V}{\int }\mathit{\theta }\mathit{\delta }\mathit{\theta }\text{d}V,$$(3.13)

and the function of thermal dissipation R is defined by(3.14) $$R=\frac{T_0}{2}\underset{V}{\int }{L}_{ij}\left(\frac{\mathit{\partial }{J}_i}{\mathit{\partial }t}+{\mathit{\tau }}_0\frac{{\mathit{\partial }}^2{J}_i}{\mathit{\partial }{t}^2}\right)J_j\text{d}V,\mathit{\delta }R=T_0\underset{V}{\int }{L}_{ij}\left(\frac{\mathit{\partial }{J}_i}{\mathit{\partial }t}+{\mathit{\tau }}_0\frac{{\mathit{\partial }}^2{J}_i}{\mathit{\partial }{t}^2}\right)\mathit{\delta }{J}_j\text{d}V.$$(3.14)

Eliminating integrals

$\underset{V}{\int }{\mathit{\alpha }}^f\mathit{\delta }{\mathit{\epsilon }}^{\ast }\mathit{\theta }\text{d}V\text{and}\underset{V}{\int }{\mathit{\alpha }}_{ij}\mathit{\theta }\mathit{\delta }{\mathit{\epsilon }}_{ij}\text{d}V$ from Equations (3.3) and (3.12) with the aid of Equation (2.7) and (2.8), we obtain the variational principle in the following form:(3.15) $$\mathit{\delta }\left(W+M+R\right)=\underset{V}{\int }({\mathit{\rho }}_1{F}_i-{\mathit{\rho }}_{11}{\ddot{\text{u}}}_i-{\mathit{\rho }}_{12}{\ddot{\text{u}}}_i^{\ast })\mathit{\delta }{u}_i\text{d}V+\underset{V}{\int }({\mathit{\rho }}_2{f}_i-{\mathit{\rho }}_{12}{\ddot{\text{u}}}_i-{\mathit{\rho }}_{22}{\ddot{\text{u}}}_i^{\ast })\mathit{\delta }{u}_i^{\ast }\text{d}V+\underset{A}{\int }(h_i\mathit{\delta }{u}_i+h_i^{\ast }\mathit{\delta }{u}_i^{\ast })\text{d}A+\underset{A}{\int }(b_0\mathit{\delta }\mathit{\psi }+b_0^{\ast }\mathit{\delta }{\mathit{\psi }}^{\ast })\text{d}A+\underset{A}{\int }(c_0\mathit{\delta }\mathit{\varphi }+c_0^{\ast }\mathit{\delta }{\mathit{\varphi }}^{\ast })\text{d}A+\underset{V}{\int }{D}_i\mathit{\delta }{E}_i\text{d}V-\underset{A}{\int }\mathit{\theta }\mathit{\delta }{J}_j{n}_{j}dA+\underset{V}{\int }{e}_{ijk}{E}_k\mathit{\delta }{\mathit{\epsilon }}_{ij}\text{d}V+\underset{V}{\int }{\mathit{\zeta }}_{kij}{E}_k^{\ast }\mathit{\delta }{\mathit{\epsilon }}_{ij}\text{d}V+\underset{V}{\int }{l}_i\mathit{\delta }{H}_i\text{d}V+\underset{V}{\int }{D}_i^{\ast }\mathit{\delta }{E}_i^{\ast }\text{d}V+\underset{V}{\int }{B}_i\mathit{\delta }{H}_i\text{d}V+\underset{V}{\int }{B}_i^{\ast }\mathit{\delta }{H}_i^{\ast }\text{d}V+\underset{V}{\int }{l}_i^{\ast }\mathit{\delta }{H}_i^{\ast }\text{d}V+\underset{V}{\int }{e}_i^{\ast }{E}_i^{\ast }\mathit{\delta }{\mathit{\epsilon }}^{\ast }\text{d}V+\underset{V}{\int }{\mathit{\zeta }}_i{E}_i\mathit{\delta }{\mathit{\epsilon }}^{\ast }\text{d}V-\underset{V}{\int }{\mathit{\tau }}_i^f\mathit{\delta }{E}_i^{\ast }\mathit{\theta }\text{d}V+\underset{V}{\int }{q}_{ijk}{H}_k\mathit{\delta }{\mathit{\epsilon }}_{ij}\text{d}V+\underset{V}{\int }{q}_{ijk}^f{H}_k^{\ast }\mathit{\delta }{\mathit{\epsilon }}_{ij}\text{d}V-\underset{V}{\int }\mathit{\theta }{\mathit{\tau }}_j\mathit{\delta }{E}_j\text{d}V+\underset{V}{\int }{m}_i\mathit{\delta }{H}_i\mathit{\theta }\text{d}V+\underset{V}{\int }{m}_i^f\mathit{\delta }{H}_i^{\ast }\mathit{\theta }\text{d}V.$$(3.15)

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 ρ₁₂ = ρ₂₂ = 0, and then we obtain the similar results as obtained by Ieşan (1990).

4. Uniqueness theorem

We assume that the virtual displacements $\mathit{\delta }{u}_i,\mathit{\delta }{u}_i^{\ast }$, the virtual increment of the temperature δθ etc. correspond to the increments occurring in the body. Then(4.1) $$\mathit{\delta }{u}_i=\frac{\mathit{\partial }{u}_i}{\mathit{\partial }t}dt={\dot{\text{u}}}_i\text{d}t,\mathit{\delta }{u}_i{=}^{\ast }{\frac{\mathit{\partial }{u}_i}{\mathit{\partial }t}}^{\ast }\text{d}t={\dot{\text{u}}}_i^{\ast }dt,\mathit{\delta }\mathit{\theta }=\frac{\mathit{\partial }\mathit{\theta }}{\mathit{\partial }t}\text{d}t=\dot{\mathit{\theta }}\text{d}t,\text{etc.}$$(4.1)

and Equation (3.15) reduces to the following relation(4.2) $$\frac{\text{d}}{\text{d}t}\left(W+M+R\right)=\underset{V}{\int }({\mathit{\rho }}_1{F}_i-{\mathit{\rho }}_{11}{\ddot{\text{u}}}_i-{\mathit{\rho }}_{12}{\ddot{\text{u}}}_i^{\ast }){\dot{\text{u}}}_i\text{d}V+\underset{V}{\int }({\mathit{\rho }}_2{f}_i-{\mathit{\rho }}_{12}{\ddot{\text{u}}}_i-{\mathit{\rho }}_{22}{\ddot{\text{u}}}_i^{\ast }){\dot{\text{u}}}_i^{\ast }\text{d}V+\underset{A}{\int }(h_i{\dot{\text{u}}}_i+h_i^{\ast }{\dot{\text{u}}}_i^{\ast })\text{d}A+\underset{A}{\int }(b_0\dot{\mathit{\psi }}+b_0^{\ast }{\dot{\mathit{\psi }}}^{\ast })\text{d}A+\underset{A}{\int }(c_0\dot{\mathit{\varphi }}+c_0^{\ast }{\dot{\mathit{\varphi }}}^{\ast })\text{d}A+\underset{V}{\int }{D}_i{\dot{E}}_i\text{d}V-\underset{A}{\int }\mathit{\theta }{\dot{J}}_j{n}_{j}dA+\underset{V}{\int }{e}_{ijk}{E}_k{\dot{\mathit{\epsilon }}}_{ij}\text{d}V+\underset{V}{\int }{\mathit{\zeta }}_{kij}{E}_k^{\ast }{\dot{\mathit{\epsilon }}}_{ij}\text{d}V+\underset{V}{\int }{D}_i^{\ast }{\dot{E}}_i^{\ast }\text{d}V+\underset{V}{\int }{e}_i^{\ast }{E}_i^{\ast }{\dot{\mathit{\epsilon }}}^{\ast }\text{d}V+\underset{V}{\int }{\mathit{\zeta }}_i{E}_i{\dot{\mathit{\epsilon }}}^{\ast }\text{d}V-\underset{V}{\int }{\mathit{\tau }}_i^f{\dot{E}}_i^{\ast }\mathit{\theta }\text{d}V-\underset{V}{\int }\mathit{\theta }{\mathit{\tau }}_j{\dot{E}}_j\text{d}V+\underset{V}{\int }{l}_i{\dot{H}}_i\text{d}V+\underset{V}{\int }{B}_i{\dot{H}}_i\text{d}V+\underset{V}{\int }{B}_i^{\ast }{\dot{H}}_i^{\ast }\text{d}V+\underset{V}{\int }{l}_i^{\ast }{\dot{H}}_i^{\ast }\text{d}V+\underset{V}{\int }{m}_i\mathit{\delta }{H}_i\mathit{\theta }\text{d}V+\underset{V}{\int }{m}_i^f\mathit{\delta }{H}_i^{\ast }\mathit{\theta }\text{d}V.$$(4.2)

Now(4.3) $$\underset{V}{\int }\left({\mathit{\rho }}_{11}{\ddot{\text{u}}}_i{\dot{\text{u}}}_i+{\mathit{\rho }}_{12}{\ddot{\text{u}}}_i^{\ast }{\dot{\text{u}}}_i+{\mathit{\rho }}_{12}{\ddot{\text{u}}}_i{\dot{\text{u}}}_i^{\ast }+{\mathit{\rho }}_{22}{\ddot{\text{u}}}_i^{\ast }{\dot{\text{u}}}_i^{\ast }\right)\text{d}V=\frac{\mathit{\partial }K}{\mathit{\partial }t},$$(4.3)

where $K=\frac{1}{2}\underset{V}{\int }({\mathit{\rho }}_{11}{\dot{\text{u}}}_i{\dot{\text{u}}}_i+2{\mathit{\rho }}_{12}{\dot{\text{u}}}_i^{\ast }{\dot{\text{u}}}_i+{\mathit{\rho }}_{22}{\dot{\text{u}}}_i^{\ast }{\dot{\text{u}}}_i^{\ast })\text{d}V$, is the kinetic energy of the body enclosed by the volume V. We also have(4.4) $$M=\frac{1}{2}\underset{V}{\int }r{\mathit{\theta }}^2\text{d}V.$$(4.4)

Using Equations (4.3) and (4.4) in the Equation (4.2), we obtain(4.5) $$\frac{\text{d}}{\text{d}t}\left(W+R+K+\frac{1}{2}\underset{V}{\int }r{\mathit{\theta }}^2\text{d}V\right)=\underset{V}{\int }{\mathit{\rho }}_1{F}_i{\dot{\text{u}}}_i\text{d}V+\underset{V}{\int }{\mathit{\rho }}_2{f}_i{\dot{\text{u}}}_i^{\ast }\text{d}V+\underset{A}{\int }(h_i{\dot{\text{u}}}_i+h_i^{\ast }{\dot{\text{u}}}_i^{\ast })\text{d}A+\underset{A}{\int }(b_0\dot{\mathit{\psi }}+b_0^{\ast }{\dot{\mathit{\psi }}}^{\ast })\text{d}A+\underset{A}{\int }(c_0\dot{\mathit{\varphi }}+c_0^{\ast }{\dot{\mathit{\varphi }}}^{\ast })\text{d}A+\underset{V}{\int }{D}_i{\dot{E}}_i\text{d}V-\underset{A}{\int }\mathit{\theta }{\dot{J}}_j{n}_j\text{d}A+\underset{V}{\int }{e}_{ijk}{E}_k{\dot{\mathit{\epsilon }}}_{ij}\text{d}V+\underset{V}{\int }{\mathit{\zeta }}_{kij}{E}_k^{\ast }{\dot{\mathit{\epsilon }}}_{ij}\text{d}V+\underset{V}{\int }{D}_i^{\ast }{\dot{E}}_i^{\ast }\text{d}V+\underset{V}{\int }{e}_i^{\ast }{E}_i^{\ast }{\dot{\mathit{\epsilon }}}^{\ast }\text{d}V+\underset{V}{\int }{\mathit{\zeta }}_i{E}_i{\dot{\mathit{\epsilon }}}^{\ast }\text{d}V-\underset{V}{\int }{\mathit{\tau }}_i^f{\dot{E}}_i^{\ast }\mathit{\theta }\text{d}V-\underset{V}{\int }\mathit{\theta }{\mathit{\tau }}_j{\dot{E}}_j\text{d}V+\underset{V}{\int }{l}_i{\dot{H}}_i\text{d}V+\underset{V}{\int }{B}_i{\dot{H}}_i\text{d}V+\underset{V}{\int }{B}_i^{\ast }{\dot{H}}_i^{\ast }\text{d}V+\underset{V}{\int }{l}_i^{\ast }{\dot{H}}_i^{\ast }\text{d}V+\underset{V}{\int }{m}_i\mathit{\delta }{H}_i\mathit{\theta }\text{d}V+\underset{V}{\int }{m}_i^f\mathit{\delta }{H}_i^{\ast }\mathit{\theta }\text{d}V,$$(4.5)

The above equation is the basis for the proof of the following uniqueness theorem.

Theorem

There is only one solution of the Equations (2.13)–(2.19), subjected to the boundary conditions on the surface A$$h_i={\mathit{\sigma }}_{ij}{n}_j=h_{i1},\mathit{\theta }={\mathit{\theta }}_1,c_0=D_i{n}_i=c_{01},h_i^{\ast }={\mathit{\sigma }}^{\ast }{n}_i=h_{i1}^{\ast },c_0^{\ast }=D_i^{\ast }{n}_i=c_{01}^{\ast },b_0=B_i{n}_i=b_{01},\text{and}{b}_0^{\ast }=B_i^{\ast }{n}_i=b_{01}^{\ast }$$

and the initial conditions on the surface at t = 0$${\text{u}}_i={\text{u}}_i^0,{\dot{\text{u}}}_i={\dot{\text{u}}}_i^0,{\text{u}}_i^{\ast }={\text{u}}_i^{\ast 0},{\dot{\text{u}}}_i^{\ast }={\dot{\text{u}}}_i^{\ast 0},\mathit{\theta }={\mathit{\theta }}^0,\dot{\mathit{\theta }}={\dot{\mathit{\theta }}}^0,\mathit{\varphi }={\mathit{\varphi }}^0,\dot{\mathit{\varphi }}={\dot{\mathit{\varphi }}}^0,{\mathit{\varphi }}^{\ast }={\mathit{\varphi }}^{\ast 0},{\dot{\mathit{\varphi }}}^{\ast }={\dot{\mathit{\varphi }}}^{\ast 0},\mathit{\psi }={\mathit{\psi }}^0,\dot{\mathit{\psi }}={\dot{\mathit{\psi }}}^0,{\mathit{\psi }}^{\ast }={\mathit{\psi }}^{\ast 0},{\dot{\mathit{\psi }}}^{\ast }={\dot{\mathit{\psi }}}^{\ast 0},$$

where $h_{i1},h_{i1}^{\ast },{\mathit{\theta }}_1,c_{01},c_{01}^{\ast },b_{01},b_{01}^{\ast },{\text{u}}_i^0,{\dot{\text{u}}}_i^0,{\text{u}}_i^{\ast 0},{\dot{\text{u}}}_i^{\ast 0},{\mathit{\theta }}^0,{\dot{\mathit{\theta }}}^0,{\mathit{\varphi }}^0,{\dot{\mathit{\varphi }}}^0,{\mathit{\varphi }}^{\ast 0},{\dot{\mathit{\varphi }}}^{\ast 0},{\mathit{\psi }}^0,{\dot{\mathit{\psi }}}^0,{\mathit{\psi }}^{\ast 0},{\dot{\mathit{\psi }}}^{\ast 0}$ are known functions. We assume that the material parameters satisfy the inequalities(4.6) $$T_0>0,{\mathit{\tau }}_0>0,{\mathit{\rho }}_{11}>0,{\mathit{\rho }}_{22}>0,\mathit{\rho }>0,$$(4.6)

$c_{ijkl},L_{ij}\text{and}{m}_{ij}$ are positive definite.

Proof

Let ${\text{u}}_i^{(1)},{\mathit{\theta }}^{(1)},{\text{u}}_i^{\ast (1)},{\mathit{\varphi }}^{(1)},{\mathit{\varphi }}^{\ast (1)},{\mathit{\psi }}^{(1)},{\mathit{\psi }}^{\ast (1)},$ and ${\text{u}}_i^{(2)},{\mathit{\theta }}^{(2)},{\text{u}}_i^{\ast (2)},{\mathit{\varphi }}^{(2)},{\mathit{\varphi }}^{\ast (2)},{\mathit{\psi }}^{(2)},ps{i}^{\ast (2)}$, be two solutions sets of Equations (2.1)–(2.19). Let us take(4.7) $$u_i={\text{u}}_i^{(1)}-{\text{u}}_i^{(2)},{\text{u}}_i^{\ast }={\text{u}}_i^{\ast (1)}-{\text{u}}_i^{\ast (2)},\mathit{\theta }={\mathit{\theta }}^{(1)}-{\mathit{\theta }}^{(2)},\mathit{\varphi }={\mathit{\varphi }}^{(1)}-{\mathit{\varphi }}^{(2)},\mathit{\psi }={\mathit{\psi }}^{(1)}-{\mathit{\psi }}^{(2)},\text{and}{\mathit{\varphi }}^{\ast }={\mathit{\varphi }}^{\ast (1)}-{\mathit{\varphi }}^{\ast (2)},{\mathit{\psi }}^{\ast }={\mathit{\psi }}^{\ast (1)}-{\mathit{\psi }}^{\ast (2)}.$$(4.7)

The functions ${\text{u}}_i,{\text{u}}_i^{\ast },\mathit{\theta },\mathit{\varphi },{\mathit{\varphi }}^{\ast },\mathit{\psi }$ 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,(4.8) $$\frac{\text{d}}{\text{d}t}\left(W+R+K+\frac{1}{2}\underset{V}{\int }r{\mathit{\theta }}^2\text{d}V\right)=0.$$(4.8)

Since, we have$$L_{ij}=L_{ji}$$

Therefore, from Equation (3.14), we obtain(4.9) $$\frac{\text{d}R}{\text{d}t}=T_0\underset{V}{\int }{L}_{ij}{\dot{J}}_i{\dot{J}}_j\text{d}V+\frac{\text{d}}{\text{d}t}\left[\frac{T_0{\mathit{\tau }}_0}{2}\underset{V}{\int }{L}_{ij}{\dot{J}}_i{\dot{J}}_j\text{d}V\right],$$(4.9)

Substitution of Equation (4.9) in the Equation (4.8) yields(4.10) $$\frac{\text{d}}{\text{d}t}\left(W+K+\frac{1}{2}\underset{V}{\int }r{\mathit{\theta }}^2\text{d}V+\frac{T_0{\mathit{\tau }}_0}{2}\underset{V}{\int }{L}_{ij}{\dot{J}}_i{\dot{J}}_j\text{d}V\right)+T_0\underset{V}{\int }{L}_{ij}{\dot{J}}_i{\dot{J}}_j\text{d}V=0.$$(4.10)

Using the inequalities (4.6) in Equation (4.10), we obtain(4.11) $$\frac{\text{d}}{\text{d}t}\left(W+K+\frac{1}{2}\underset{V}{\int }r{\mathit{\theta }}^2\text{d}V+\frac{T_0{\mathit{\tau }}_0}{2}\underset{V}{\int }{L}_{ij}{\dot{J}}_i{\dot{J}}_j\text{d}V\right)\le 0.$$(4.11)

We thus see that the expression(4.12) $$W+K+\frac{1}{2}\underset{V}{\int }r{\mathit{\theta }}^2\text{d}V+\frac{T_0{\mathit{\tau }}_0}{2}\underset{V}{\int }{L}_{ij}{\dot{J}}_i{\dot{J}}_j\text{d}V,$$(4.12)

is a decreasing function of time. We also note that the expression $\underset{V}{\int }r{\mathit{\theta }}^2\text{d}V$ occurring in the expression (4.12) is always positive, due to the laws of thermodynamics Nowacki (1974)(4.13) $$0<r.$$(4.13)

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.

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$$\mathit{\varphi }={\mathit{\varphi }}^{\ast }=\mathit{\psi }={\mathit{\psi }}^{\ast }={\text{u}}_i={\text{u}}_i^{\ast }=\mathit{\theta }={\mathit{\epsilon }}_{ij}={\mathit{\epsilon }}^{\ast }={\mathit{\sigma }}_{ij}={\mathit{\sigma }}^{\ast }=0.$$

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 ρ₁₂ = ρ₂₂ = 0, then we obtain the similar results as obtained by Ieşan (1990).

5. 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 ${\mathit{\sigma }}_{ij},{\mathit{\sigma }}^{\ast }$ and the strains ${\mathit{\epsilon }}_{ij},{\mathit{\epsilon }}^{\ast }$ are continuous together with their first order derivatives, whereas the displacements ${\text{u}}_i,{\text{u}}_i^{\ast }$, temperature θ and the electrical potentials ϕ, ϕ^*, magnetic potentials ψ, ψ^* are continuous and have continuous derivatives up to second order, for $x\in V+A,t>0.$ The components of surface tractions, the normal component of the heat flux and electric displacements at regular points of ∂V are given by(5.1) $$h_i={\mathit{\sigma }}_{ij}{n}_j,h_i^{\ast }={\mathit{\sigma }}^{\ast }{n}_i,q=q_i{n}_i,c_0^{\ast }=D_i^{\ast }{n}_i,c_0=D_i{n}_i,b_0=B_i{n}_i,b_0^{\ast }=B_i^{\ast }{n}_i$$(5.1)

respectively.

To the system of field equations, we must adjoin boundary conditions and initial conditions. We consider the following boundary conditions:(5.2) $${\text{u}}_i(x,t)={\text{U}}_i(x,t),{\text{u}}_i^{\ast }(x,t)={\text{U}}_i^{\ast }(x,t),\mathit{\varphi }(x,t)=e_0(x,t),{\mathit{\varphi }}^{\ast }(x,t)=e_0^{\ast }(x,t),\mathit{\psi }(x,t)=e_1(x,t){\mathit{\psi }}^{\ast }(x,t)=e_1^{\ast }(x,t),\mathit{\theta }(x,t)=\mathit{\eta }(x,t),$$(5.2)

for all $x\in A,t>0$

and the homogeneous initial conditions(5.3) $${\text{u}}_i(x,0)={\dot{\text{u}}}_i(x,0)=0,\mathit{\theta }(x,0)=\dot{\mathit{\theta }}(x,0)=0,{\text{u}}_i^{\ast }(x,0)={\dot{\text{u}}}_i^{\ast }(x,0)=0,\text{and}$$(5.3)

$\mathit{\varphi }(x,0)=\dot{\mathit{\varphi }}(x,0)=0,{\mathit{\varphi }}^{\ast }(x,0)={\dot{\mathit{\varphi }}}^{\ast }(x,0)=0,\mathit{\psi }(x,0)=\dot{\mathit{\psi }}(x,0)=0,{\mathit{\psi }}^{\ast }(x,0)={\dot{\mathit{\psi }}}^{\ast }(x,0)=0,$ for all $x\in V,t=0.$

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 $h_i(x,t),h_i^{\ast }(x,t)$, the heat flux q(x, t), the magnetic densities $b_0(x,t),b_0^{\ast }(x,t)$ and the surface charge densities $c_0(x,t),c_0^{\ast }(x,t)$.

We define the Laplace transform as(5.4) $$\overline{f}(x,s)=L(f(x,t))=\underset{0}{\overset{\infty }{\int }}f(x,t)e^{-st}\text{d}t,$$(5.4)

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(5.5) $${\mathit{\sigma }}_{ij}=c_{ijkl}{\mathit{\epsilon }}_{kl}-e_{ijk}{E}_k-{\mathit{\alpha }}_{ij}\mathit{\theta }+m_{ij}{\mathit{\epsilon }}^{\ast }-{\mathit{\zeta }}_{kij}{E}_k^{\ast }-q_{ijk}{H}_k-q_{ijk}^f{H}_k^{\ast },$$(5.5) (5.6) $${\mathit{\sigma }}^{\ast }=m_{ij}{\mathit{\epsilon }}_{ij}-{\mathit{\zeta }}_i{E}_i-{\mathit{\alpha }}^f\mathit{\theta }+R{\mathit{\epsilon }}^{\ast }-e_i^{\ast }{E}_i^{\ast }-l_i{H}_i-l_i^{\ast }{H}_i^{\ast },$$(5.6) (5.7) $$-q_{i,i}=\mathit{\rho }{T}_{0}sS,$$(5.7) (5.8) $$\mathit{\rho }S={\mathit{\alpha }}_{ij}{\mathit{\epsilon }}_{ij}+{\mathit{\tau }}_i{E}_i+r\mathit{\theta }+{\mathit{\alpha }}^f{\mathit{\epsilon }}^{\ast }+{\mathit{\tau }}_i^f{E}_i^{\ast }+m_i{H}_i+m_i^f{H}_i^{\ast },$$(5.8) (5.9) $$D_i={\mathit{\xi }}_{ij}{E}_j+e_{ijk}{\mathit{\epsilon }}_{jk}+{\mathit{\tau }}_i\mathit{\theta }+{\mathit{\zeta }}_i{\mathit{\epsilon }}^{\ast }+A_{ij}{E}_j^{\ast }+f_{ij}^f{H}_j^{\ast }+f_{ij}{H}_j,$$(5.9) (5.10) $$D_i^{\ast }={\mathit{\zeta }}_{ijk}{\mathit{\epsilon }}_{jk}+{\mathit{\tau }}_i^f\mathit{\theta }+A_{ij}{E}_j+{\mathit{\xi }}_{ij}^{\ast }{E}_j^{\ast }+e_i^{\ast }{\mathit{\epsilon }}^{\ast }+f_{ij}^f{H}_j+{\mathit{\gamma }}_{ij}{H}_j^{\ast },$$(5.10) (5.11) $$B_i=f_{ij}{E}_j+q_{ijk}{\mathit{\epsilon }}_{jk}+m_i\mathit{\theta }+l_i{\mathit{\epsilon }}^{\ast }+f_{ij}^f{E}_j^{\ast }+{\mathit{\beta }}_{ij}^f{H}_j^{\ast }+{\mathit{\beta }}_{ij}{H}_j,$$(5.11) (5.12) $$B_i^{\ast }=q_{ijk}^f{\mathit{\epsilon }}_{jk}+m_i^f\mathit{\theta }+f_{ij}^f{E}_j+{\mathit{\gamma }}_{ij}{E}_j^{\ast }+l_i^{\ast }{\mathit{\epsilon }}^{\ast }+{\mathit{\beta }}_{ij}^f{H}_j+{\mathit{\gamma }}_{ij}^{\ast }{H}_j^{\ast },$$(5.12) (5.13) $$E_i=-{\mathit{\varphi }}_{,i},$$(5.13) (5.14) $$E_i^{\ast }=-{\mathit{\varphi }}_{,i}^{\ast },$$(5.14) (5.15) $$H_i=-{\mathit{\psi }}_{,i},$$(5.15) (5.16) $$H_i^{\ast }=-{\mathit{\psi }}_{,i}^{\ast },(i,j,k,l=1,2,3)$$(5.16) (5.17) $${\mathit{\sigma }}_{ij,j}+{\mathit{\rho }}_1{F}_i-{\mathit{\rho }}_{11}{s}^2{\text{u}}_i-{\mathit{\rho }}_{12}{s}^2{\text{u}}_i^{\ast }=0,$$(5.17) (5.18) $${\mathit{\sigma }}_{,i}^{\ast }+{\mathit{\rho }}_2{f}_i-{\mathit{\rho }}_{12}{s}^2{\text{u}}_i-{\mathit{\rho }}_{22}{s}^2{\text{u}}_i^{\ast }=0,$$(5.18) (5.19) $$-K_{ij}{\mathit{\theta }}_{,j}=(1+{\mathit{\tau }}_{0}s)q_i,$$(5.19) (5.20) $$D_{i,i}=0,$$(5.20) (5.21) $$D_{i,i}^{\ast }=0,$$(5.21) (5.22) $$B_{i,i}=0,$$(5.22) (5.23) $$B_{i,i}^{\ast }=0,$$(5.23)

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 ${\text{u}}_i^{(1)},{\text{u}}_i^{\ast (1)},{\mathit{\epsilon }}_{ij}^{(1)},{\mathit{\epsilon }}^{\ast (1)},{\mathit{\sigma }}_{ij}^{(1)},{\mathit{\sigma }}^{\ast (1)},{\mathit{\theta }}^{(1)},{\mathit{\varphi }}^{(1)},{\mathit{\varphi }}^{\ast (1)},{\mathit{\psi }}^{(1)},{\mathit{\psi }}^{\ast (1)}$ for the first problem and ${\text{u}}_i^{(2)},{\text{u}}_i^{\ast (2)},{\mathit{\epsilon }}_{ij}^{(2)},{\mathit{\epsilon }}^{\ast (2)},{\mathit{\sigma }}_{ij}^{(2)},{\mathit{\sigma }}^{\ast (2)},{\mathit{\theta }}^{(2)},{\mathit{\varphi }}^{(2)},{\mathit{\varphi }}^{\ast (2)},{\mathit{\psi }}^{(2)},{\mathit{\psi }}^{\ast (2)}$ for the second problem. Each set of variables satisfies the Equations (5.5)–(5.23).

Using the assumption ${\mathit{\sigma }}_{ij}={\mathit{\sigma }}_{ji}$, we obtain(5.24) $$\underset{V}{\int }{\mathit{\sigma }}_{ij}^{(1)}{\mathit{\epsilon }}_{ij}^{(2)}\text{d}V+\underset{V}{\int }{\mathit{\sigma }}^{\ast (1)}{\mathit{\epsilon }}^{\ast (2)}\text{d}V=\underset{V}{\int }{\mathit{\sigma }}_{ij}^{(1)}{\text{u}}_{i,j}^{(2)}\text{d}V+\underset{V}{\int }{\mathit{\sigma }}^{\ast (1)}{\text{u}}_{i,i}^{\ast (2)}\text{d}V=\underset{V}{\int }{\left({\mathit{\sigma }}_{ij}^{(1)}{\text{u}}_i^{(2)}\right)}_{,j}\text{d}V+\underset{V}{\int }{\left({\mathit{\sigma }}^{\ast (1)}{\text{u}}_i^{\ast (2)}\right)}_{,i}\text{d}V-\underset{V}{\int }{\mathit{\sigma }}_{ij,j}^{(1)}{\text{u}}_i^{(2)}\text{d}V-\underset{V}{\int }{\mathit{\sigma }}_{,i}^{\ast (1)}{\text{u}}_i^{\ast (2)}\text{d}V.$$(5.24)

Using the divergence theorem in the first term of the right-hand side of Equation (5.24) yields(5.25) $$\underset{V}{\int }{\mathit{\sigma }}_{ij}^{(1)}{\mathit{\epsilon }}_{ij}^{(2)}\text{d}V+\underset{V}{\int }{\mathit{\sigma }}^{\ast (1)}{\mathit{\epsilon }}^{\ast (2)}\text{d}V=\underset{A}{\int }{\mathit{\sigma }}_{ij}^{(1)}{\text{u}}_i^{(2)}{n}_j\text{d}A+\underset{A}{\int }{\mathit{\sigma }}^{\ast (1)}{\text{u}}_i^{\ast (2)}{n}_i\text{d}A-\underset{V}{\int }{\mathit{\sigma }}_{ij,j}^{(1)}{\text{u}}_i^{(2)}\text{d}V-\underset{V}{\int }{\mathit{\sigma }}_{,i}^{\ast (1)}{\text{u}}_i^{\ast (2)}\text{d}V.$$(5.25)

Equation (5.25) with the aid of Equations (5.1) and (5.17) gives(5.26) $$\underset{V}{\int }\left({\mathit{\sigma }}_{ij}^{(1)}{\mathit{\epsilon }}_{ij}^{(2)}+{\mathit{\sigma }}^{\ast (1)}{\mathit{\epsilon }}^{\ast (2)}\right)\text{d}V=\underset{A}{\int }\left(h_i^{(1)}{\text{u}}_i^{(2)}+h_i^{\ast (1)}{\text{u}}_i^{\ast (2)}\right)\text{d}A+\underset{V}{\int }\left({\mathit{\rho }}_1{F}_i^{(1)}{\text{u}}_i^{(2)}-{\mathit{\rho }}_{11}{s}^2{\text{u}}_i^{(1)}{\text{u}}_i^{(2)}-{\mathit{\rho }}_{12}{s}^2{\text{u}}_i^{\ast (1)}{\text{u}}_i^{(2)}\right)\text{d}V+\underset{V}{\int }\left({\mathit{\rho }}_2{f}_i^{(1)}{\text{u}}_i^{\ast (2)}-{\mathit{\rho }}_{12}{s}^2{\text{u}}_i^{(1)}{\text{u}}_i^{\ast (2)}-{\mathit{\rho }}_{22}{s}^2{\text{u}}_i^{\ast (1)}{\text{u}}_i^{\ast (2)}\right)\text{d}V.$$(5.26)

A similar expression is obtained for the integral $\underset{V}{\int }\left({\mathit{\sigma }}_{ij}^{(2)}{\mathit{\epsilon }}_{ij}^{(1)}+{\mathit{\sigma }}^{\ast (2)}{\mathit{\epsilon }}^{\ast (1)}\right)\text{d}V$, from which together with the Equation (5.26), it follows that(5.27) $$\underset{V}{\int }\left({\mathit{\sigma }}_{ij}^{(1)}{\mathit{\epsilon }}_{ij}^{(2)}-{\mathit{\sigma }}_{ij}^{(2)}{\mathit{\epsilon }}_{ij}^{(1)}+{\mathit{\sigma }}^{\ast (1)}{\mathit{\epsilon }}^{\ast (2)}-{\mathit{\sigma }}^{\ast (2)}{\mathit{\epsilon }}^{\ast (1)}\right)\text{d}V=\underset{A}{\int }\left(h_i^{(1)}{\text{u}}_i^{(2)}-h_i^{(2)}{\text{u}}_i^{(1)}+h_i^{\ast (1)}{\text{u}}_i^{\ast (2)}-h_i^{\ast (2)}{\text{u}}_i^{\ast (1)}\right)\text{d}A+\underset{V}{\int }{\mathit{\rho }}_1\left(F_i^{(1)}{\text{u}}_i^{(2)}-F_i^{(2)}{\text{u}}_i^{(1)}\right)\text{d}V+\underset{V}{\int }{\mathit{\rho }}_2\left(f_i^{(1)}{\text{u}}_i^{\ast (2)}-f_i^{(2)}{\text{u}}_i^{\ast (1)}\right)\text{d}V.$$(5.27)

Now multiplying Equations (5.5), (5.6) by ${\mathit{\epsilon }}_{ij}^{(2)},{\mathit{\epsilon }}^{\ast (2)}$ and ${\mathit{\epsilon }}_{ij}^{(1)},{\mathit{\epsilon }}^{\ast (1)}$ for the first and second problems, respectively, subtracting and integrating over the region V, we obtain$$\underset{V}{\int }\left({\mathit{\sigma }}_{ij}^{(1)}{\mathit{\epsilon }}_{ij}^{(2)}-{\mathit{\sigma }}_{ij}^{(2)}{\mathit{\epsilon }}_{ij}^{(1)}+{\mathit{\sigma }}^{\ast (1)}{\mathit{\epsilon }}^{\ast (2)}-{\mathit{\sigma }}^{\ast (2)}{\mathit{\epsilon }}^{\ast (1)}\right)\text{d}V=\underset{V}{\int }{c}_{ijkl}\left({\mathit{\epsilon }}_{kl}^{(1)}{\mathit{\epsilon }}_{ij}^{(2)}-{\mathit{\epsilon }}_{kl}^{(2)}{\mathit{\epsilon }}_{ij}^{(1)}\right)\text{d}V-\underset{V}{\int }{\mathit{\alpha }}_{ij}\left({\mathit{\theta }}^{(1)}{\mathit{\epsilon }}_{ij}^{(2)}-{\mathit{\theta }}^{(2)}{\mathit{\epsilon }}_{ij}^{(1)}\right)\text{d}V-\underset{V}{\int }{\mathit{\zeta }}_i\left({\mathit{\varphi }}_{,k}^{(2)}{\mathit{\epsilon }}^{\ast (1)}-{\mathit{\varphi }}_{,k}^{(1)}{\mathit{\epsilon }}^{\ast (2)}\right)\text{d}V-\underset{V}{\int }{\mathit{\zeta }}_{ijk}\left({\mathit{\varphi }}_{,k}^{\ast (2)}{\mathit{\epsilon }}_{ij}^{(1)}-{\mathit{\varphi }}_{,k}^{\ast (1)}{\mathit{\epsilon }}_{ij}^{(2)}\right)\text{d}V-\underset{V}{\int }{e}_i^{\ast }\left({\mathit{\varphi }}_{,i}^{\ast (2)}{\mathit{\epsilon }}^{\ast (1)}-{\mathit{\varphi }}_{,i}^{\ast (1)}{\mathit{\epsilon }}^{\ast (2)}\right)\text{d}V-\underset{V}{\int }{\mathit{\alpha }}^f\left({\mathit{\theta }}^{(1)}{\mathit{\epsilon }}^{\ast (2)}-{\mathit{\theta }}^{(2)}{\mathit{\epsilon }}^{\ast (1)}\right)\text{d}V-\underset{V}{\int }{e}_{ijk}\left({\mathit{\varphi }}_{,k}^{(2)}{\mathit{\epsilon }}_{ij}^{(1)}-{\mathit{\varphi }}_{,k}^{(1)}{\mathit{\epsilon }}_{ij}^{(2)}\right)\text{d}V-\underset{V}{\int }{q}_{ijk}\left({\mathit{\psi }}_{,k}^{(2)}{\mathit{\epsilon }}_{ij}^{(1)}-{\mathit{\psi }}_{,k}^{(1)}{\mathit{\epsilon }}_{ij}^{(2)}\right)\text{d}V-\underset{V}{\int }{q}_{ijk}^f\left({\mathit{\psi }}_{,k}^{\ast (2)}{\mathit{\epsilon }}_{ij}^{(1)}-{\mathit{\psi }}_{,k}^{\ast (1)}{\mathit{\epsilon }}_{ij}^{(2)}\right)\text{d}V-\underset{V}{\int }{l}_i\left({\mathit{\psi }}_{,i}^{(2)}{\mathit{\epsilon }}^{\ast (1)}-{\mathit{\psi }}_{,i}^{(1)}{\mathit{\epsilon }}^{\ast (2)}\right)\text{d}V-\underset{V}{\int }{l}_i^f\left({\mathit{\psi }}_{,i}^{\ast (2)}{\mathit{\epsilon }}^{\ast (1)}-{\mathit{\psi }}_{,i}^{\ast (1)}{\mathit{\epsilon }}^{\ast (2)}\right)\text{d}V.$$

Using the symmetry properties of c_ijkl, we obtain(5.28) $$\underset{V}{\int }\left({\mathit{\sigma }}_{ij}^{(1)}{\mathit{\epsilon }}_{ij}^{(2)}-{\mathit{\sigma }}_{ij}^{(2)}{\mathit{\epsilon }}_{ij}^{(1)}+{\mathit{\sigma }}^{\ast (1)}{\mathit{\epsilon }}^{\ast (2)}-{\mathit{\sigma }}^{\ast (2)}{\mathit{\epsilon }}^{\ast (1)}\right)\text{d}V=-\underset{V}{\int }{e}_{ijk}\left({\mathit{\varphi }}_{,k}^{(2)}{\mathit{\epsilon }}_{ij}^{(1)}-{\mathit{\varphi }}_{,k}^{(1)}{\mathit{\epsilon }}_{ij}^{(2)}\right)\text{d}V-\underset{V}{\int }{\mathit{\alpha }}_{ij}\left({\mathit{\theta }}^{(1)}{\mathit{\epsilon }}_{ij}^{(2)}-{\mathit{\theta }}^{(2)}{\mathit{\epsilon }}_{ij}^{(1)}\right)\text{d}V-\underset{V}{\int }{\mathit{\zeta }}_i\left({\mathit{\varphi }}_{,k}^{(2)}{\mathit{\epsilon }}^{\ast (1)}-{\mathit{\varphi }}_{,k}^{(1)}{\mathit{\epsilon }}^{\ast (2)}\right)\text{d}V-\underset{V}{\int }{\mathit{\zeta }}_{ijk}\left({\mathit{\varphi }}_{,k}^{\ast (2)}{\mathit{\epsilon }}_{ij}^{(1)}-{\mathit{\varphi }}_{,k}^{\ast (1)}{\mathit{\epsilon }}_{ij}^{(2)}\right)\text{d}V-\underset{V}{\int }{e}_i^{\ast }\left({\mathit{\varphi }}_{,i}^{\ast (2)}{\mathit{\epsilon }}^{\ast (1)}-{\mathit{\varphi }}_{,i}^{\ast (1)}{\mathit{\epsilon }}^{\ast (2)}\right)\text{d}V-\underset{V}{\int }{\mathit{\alpha }}^f\left({\mathit{\theta }}^{(1)}{\mathit{\epsilon }}^{\ast (2)}-{\mathit{\theta }}^{(2)}{\mathit{\epsilon }}^{\ast (1)}\right)\text{d}V-\underset{V}{\int }{q}_{ijk}\left({\mathit{\psi }}_{,k}^{(2)}{\mathit{\epsilon }}_{ij}^{(1)}-{\mathit{\psi }}_{,k}^{(1)}{\mathit{\epsilon }}_{ij}^{(2)}\right)\text{d}V-\underset{V}{\int }{q}_{ijk}^f\left({\mathit{\psi }}_{,k}^{\ast (2)}{\mathit{\epsilon }}_{ij}^{(1)}-{\mathit{\psi }}_{,k}^{\ast (1)}{\mathit{\epsilon }}_{ij}^{(2)}\right)\text{d}V-\underset{V}{\int }{l}_i\left({\mathit{\psi }}_{,i}^{(2)}{\mathit{\epsilon }}^{\ast (1)}-{\mathit{\psi }}_{,i}^{(1)}{\mathit{\epsilon }}^{\ast (2)}\right)\text{d}V-\underset{V}{\int }{l}_i^f\left({\mathit{\psi }}_{,i}^{\ast (2)}{\mathit{\epsilon }}^{\ast (1)}-{\mathit{\psi }}_{,i}^{\ast (1)}{\mathit{\epsilon }}^{\ast (2)}\right)\text{d}V.$$(5.28)

Equating Equations (5.27) and (5.28), we get the first part of the reciprocity theorem(5.29) $$\underset{A}{\int }\left(h_i^{(1)}{\text{u}}_i^{(2)}-h_i^{(2)}{\text{u}}_i^{(1)}+h_i^{\ast (1)}{\text{u}}_i^{\ast (2)}-h_i^{\ast (2)}{\text{u}}_i^{\ast (1)}\right)\text{d}A+\underset{V}{\int }{\mathit{\rho }}_1\left(F_i^{(1)}{\text{u}}_i^{(2)}-F_i^{(2)}{\text{u}}_i^{(1)}\right)\text{d}V+\underset{V}{\int }{\mathit{\rho }}_2\left(f_i^{(1)}{\text{u}}_i^{\ast (2)}-f_i^{(2)}{\text{u}}_i^{\ast (1)}\right)\text{d}V=-\underset{V}{\int }{e}_{ijk}\left({\mathit{\varphi }}_{,k}^{(2)}{\mathit{\epsilon }}_{ij}^{(1)}-{\mathit{\varphi }}_{,k}^{(1)}{\mathit{\epsilon }}_{ij}^{(2)}\right)\text{d}V-\underset{V}{\int }{\mathit{\alpha }}_{ij}\left({\mathit{\theta }}^{(1)}{\mathit{\epsilon }}_{ij}^{(2)}-{\mathit{\theta }}^{(2)}{\mathit{\epsilon }}_{ij}^{(1)}\right)\text{d}V-\underset{V}{\int }{\mathit{\zeta }}_i\left({\mathit{\varphi }}_{,i}^{(2)}{\mathit{\epsilon }}^{\ast (1)}-{\mathit{\varphi }}_{,i}^{(1)}{\mathit{\epsilon }}^{\ast (2)}\right)\text{d}V-\underset{V}{\int }{\mathit{\zeta }}_{ijk}\left({\mathit{\varphi }}_{,k}^{\ast (2)}{\mathit{\epsilon }}_{ij}^{(1)}-{\mathit{\varphi }}_{,k}^{\ast (1)}{\mathit{\epsilon }}_{ij}^{(2)}\right)\text{d}V-\underset{V}{\int }{e}_i^{\ast }\left({\mathit{\varphi }}_{,i}^{\ast (2)}{\mathit{\epsilon }}^{\ast (1)}-{\mathit{\varphi }}_{,i}^{\ast (1)}{\mathit{\epsilon }}^{\ast (2)}\right)\text{d}V-\underset{V}{\int }{\mathit{\alpha }}^f\left({\mathit{\theta }}^{(1)}{\mathit{\epsilon }}^{\ast (2)}-{\mathit{\theta }}^{(2)}{\mathit{\epsilon }}^{\ast (1)}\right)\text{d}V-\underset{V}{\int }{q}_{ijk}\left({\mathit{\psi }}_{,k}^{(2)}{\mathit{\epsilon }}_{ij}^{(1)}-{\mathit{\psi }}_{,k}^{(1)}{\mathit{\epsilon }}_{ij}^{(2)}\right)\text{d}V-\underset{V}{\int }{q}_{ijk}^f\left({\mathit{\psi }}_{,k}^{\ast (2)}{\mathit{\epsilon }}_{ij}^{(1)}-{\mathit{\psi }}_{,k}^{\ast (1)}{\mathit{\epsilon }}_{ij}^{(2)}\right)\text{d}V-\underset{V}{\int }{l}_i\left({\mathit{\psi }}_{,i}^{(2)}{\mathit{\epsilon }}^{\ast (1)}-{\mathit{\psi }}_{,i}^{(1)}{\mathit{\epsilon }}^{\ast (2)}\right)\text{d}V-\underset{V}{\int }{l}_i^{\ast }\left({\mathit{\psi }}_{,i}^{\ast (2)}{\mathit{\epsilon }}^{\ast (1)}-{\mathit{\psi }}_{,i}^{\ast (1)}{\mathit{\epsilon }}^{\ast (2)}\right)\text{d}V.$$(5.29)

Equation (5.29) contains the mechanical causes of motion F_i, f_i and $h_i,h_i^{\ast }$.

Using Equation (5.8), Equation (5.7) reduces to(5.30) $$-q_{i,i}=T_{0}s\left({\mathit{\alpha }}_{ij}{\mathit{\epsilon }}_{ij}+{\mathit{\tau }}_i{E}_i+r\mathit{\theta }+{\mathit{\alpha }}^f{\mathit{\epsilon }}^{\ast }+{\mathit{\tau }}_i^f{E}_i^{\ast }+m_i{H}_i+m_i^f{H}_i^{\ast }\right).$$(5.30)

Now, taking the divergence on both sides of Equation (5.19) and using Equation (5.30), we arrive at the equation of heat conduction, namely(5.31) $$\frac{\mathit{\partial }}{\mathit{\partial }{x}_i}\left(K_{ij}{\mathit{\theta }}_{,j}\right)=\left(s+{\mathit{\tau }}_0{s}^2\right)T_0\left({\mathit{\alpha }}_{ij}{\mathit{\epsilon }}_{ij}+{\mathit{\tau }}_i{E}_i+r\mathit{\theta }+{\mathit{\alpha }}^f{\mathit{\epsilon }}^{\ast }+{\mathit{\tau }}_i^f{E}_i^{\ast }+m_i{H}_i+m_i^f{H}_i^{\ast }\right)$$(5.31)

To derive the second part, multiplying Equation (5.31) by θ⁽²⁾ and θ⁽¹⁾ for the first and the second problems, respectively, subtracting and integrating over V, we get(5.32) $$\underset{V}{\int }\left({\left(K_{ij}{\mathit{\theta }}_{,j}^{(1)}\right)}_{,i}{\mathit{\theta }}^{(2)}-{\left(K_{ij}{\mathit{\theta }}_{,j}^{(2)}\right)}_{,i}{\mathit{\theta }}^{(1)}\right)\text{d}V=\left(s+{\mathit{\tau }}_0{s}^2\right)T_0\underset{V}{\int }{\mathit{\alpha }}_{ij}\left({\mathit{\epsilon }}_{ij}^{(1)}{\mathit{\theta }}^{(2)}-{\mathit{\epsilon }}_{ij}^{(2)}{\mathit{\theta }}^{(1)}\right)\text{d}V+\left(s+{\mathit{\tau }}_0{s}^2\right)T_0\underset{V}{\int }{\mathit{\tau }}_i\left(E_i^{(1)}{\mathit{\theta }}^{(2)}-E_i^{(2)}{\mathit{\theta }}^{(1)}\right)\text{d}V+\left(s+{\mathit{\tau }}_0{s}^2\right)T_0\underset{V}{\int }{\mathit{\alpha }}^f\left({\mathit{\epsilon }}^{\ast (1)}{\mathit{\theta }}^{(2)}-{\mathit{\epsilon }}^{\ast (2)}{\mathit{\theta }}^{(1)}\right)\text{d}V+\left(s+{\mathit{\tau }}_0{s}^2\right)T_0\underset{V}{\int }{\mathit{\tau }}_i^f\left(E_i^{\ast (1)}{\mathit{\theta }}^{(2)}-E_i^{\ast (2)}{\mathit{\theta }}^{(1)}\right)\text{d}V+\left(s+{\mathit{\tau }}_0{s}^2\right)T_0\underset{V}{\int }{m}_i\left(H_i^{(1)}{\mathit{\theta }}^{(2)}-H_i^{(2)}{\mathit{\theta }}^{(1)}\right)\text{d}V+\left(s+{\mathit{\tau }}_0{s}^2\right)T_0\underset{V}{\int }{m}_i^f\left(H_i^{\ast (1)}{\mathit{\theta }}^{(2)}-H_i^{\ast (2)}{\mathit{\theta }}^{(1)}\right)\text{d}V.$$(5.32)

Now(5.33) $${\left(K_{ij}{\mathit{\theta }}_{,j}^{(1)}\right)}_{,i}{\mathit{\theta }}^{(2)}={\left(K_{ij}{\mathit{\theta }}_{,j}^{(1)}{\mathit{\theta }}^{(2)}\right)}_{,i}-K_{ij}{\mathit{\theta }}_{,j}^{(1)}{\mathit{\theta }}_{,i}^{(2)}\text{and}{\left(K_{ij}{\mathit{\theta }}_{,j}^{(2)}\right)}_{,i}{\mathit{\theta }}^{(1)}={\left(K_{ij}{\mathit{\theta }}_{,j}^{(2)}{\mathit{\theta }}^{(1)}\right)}_{,i}-K_{ij}{\mathit{\theta }}_{,j}^{(2)}{\mathit{\theta }}_{,i}^{(1)}$$(5.33)

Equation (5.32) with the help of Equations (5.1), (5.2), (5.33) and the divergence theorem is written as(5.34) $$\underset{A}{\int }\left(q^{(1)}{\mathit{\eta }}^{(2)}-q^{(2)}{\mathit{\eta }}^{(1)}\right)\text{d}A=-\left(s+{\mathit{\tau }}_0{s}^2\right)T_0\underset{V}{\int }{\mathit{\alpha }}_{ij}\left({\mathit{\epsilon }}_{ij}^{(1)}{\mathit{\theta }}^{(2)}-{\mathit{\epsilon }}_{ij}^{(2)}{\mathit{\theta }}^{(1)}\right)\text{d}V-\left(s+{\mathit{\tau }}_0{s}^2\right)T_0\underset{V}{\int }{\mathit{\tau }}_i\left(E_i^{(1)}{\mathit{\theta }}^{(2)}-E_i^{(2)}{\mathit{\theta }}^{(1)}\right)\text{d}V-\left(s+{\mathit{\tau }}_0{s}^2\right)T_0\underset{V}{\int }{\mathit{\alpha }}^f\left({\mathit{\epsilon }}^{\ast (1)}{\mathit{\theta }}^{(2)}-{\mathit{\epsilon }}^{\ast (2)}{\mathit{\theta }}^{(1)}\right)\text{d}V-\left(s+{\mathit{\tau }}_0{s}^2\right)T_0\underset{V}{\int }{\mathit{\tau }}_i^f\left(E_i^{\ast (1)}{\mathit{\theta }}^{(2)}-E_i^{\ast (2)}{\mathit{\theta }}^{(1)}\right)\text{d}V-\left(s+{\mathit{\tau }}_0{s}^2\right)T_0\underset{V}{\int }{m}_i\left(H_i^{(1)}{\mathit{\theta }}^{(2)}-H_i^{(2)}{\mathit{\theta }}^{(1)}\right)\text{d}V-\left(s+{\mathit{\tau }}_0{s}^2\right)T_0\underset{V}{\int }{m}_i^f\left(H_i^{\ast (1)}{\mathit{\theta }}^{(2)}-H_i^{\ast (2)}{\mathit{\theta }}^{(1)}\right)\text{d}V.$$(5.34)

The Equation (5.34) constitutes the second part of reciprocity theorem which contains the thermal causes of motion η and q.

To derive the third part, multiplying Equation (5.9), (5.10) by $E_i^{(2)},E_i^{\ast (2)}$ and $E_i^{(1)},E_i^{\ast (1)}$ for the first and the second problems, respectively, subtracting and integrating over V, we get(5.35) $$\underset{V}{\int }\left(D_i^{(1)}{E}_i^{(2)}-D_i^{(2)}{E}_i^{(1)}+D_i^{\ast (1)}{E}_i^{\ast (2)}-D_i^{\ast (2)}{E}_i^{\ast (1)}\right)\text{d}V=\underset{V}{\int }{e}_{ijk}\left({\mathit{\epsilon }}_{jk}^{(1)}{E}_i^{(2)}-{\mathit{\epsilon }}_{jk}^{(2)}{E}_i^{(1)})\text{d}V+\underset{V}{\int }{\mathit{\tau }}_i^f({\mathit{\theta }}^{(1)}{E}_i^{\ast (2)}-{\mathit{\theta }}^{(2)}{E}_i^{\ast (1)})\text{d}V+\underset{V}{\int }{\mathit{\tau }}_i({\mathit{\theta }}^{(1)}{E}_i^{(2)}-{\mathit{\theta }}^{(2)}{E}_i^{(1)}\right)\text{d}V+\underset{V}{\int }{\mathit{\zeta }}_i\left({\mathit{\epsilon }}^{\ast (1)}{E}_i^{(2)}-{\mathit{\epsilon }}^{\ast (2)}{E}_i^{(1)}\right)\text{d}V+\underset{V}{\int }{e}_i^{\ast }\left({\mathit{\epsilon }}^{\ast (1)}{E}_i^{\ast (2)}-{\mathit{\epsilon }}^{\ast (2)}{E}_i^{\ast (1)}\right)\text{d}V+\underset{V}{\int }{\mathit{\zeta }}_{ijk}\left({\mathit{\epsilon }}_{jk}^{(1)}{E}_i^{\ast (2)}-{\mathit{\epsilon }}_{jk}^{(2)}{E}_i^{\ast (1)}\right)\text{d}V+\underset{V}{\int }{f}_{ij}^f\left(H_j^{(1)}{E}_i^{\ast (2)}-H_j^{(2)}{E}_i^{\ast (1)}\right)\text{d}V+\underset{V}{\int }{\mathit{\gamma }}_{ij}\left(H_j^{\ast (1)}{E}_i^{\ast (2)}-H_j^{\ast (2)}{E}_i^{\ast (1)}\right)\text{d}V+\underset{V}{\int }{f}_{ij}^f\left(H_j^{\ast (1)}{E}_i^{(2)}-H_j^{\ast (2)}{E}_i^{(1)}\right)\text{d}V+\underset{V}{\int }{f}_{ij}\left(H_j^{(1)}{E}_i^{(2)}-H_j^{(2)}{E}_i^{(1)}\right)\text{d}V.$$(5.35)

Equation (5.35) with the aid of Equations (5.13)–(5.16) yields(5.36) $$\underset{V}{\int }\left(D_i^{(1)}{E}_i^{(2)}-D_i^{(2)}{E}_i^{(1)}+D_i^{\ast (1)}{E}_i^{\ast (2)}-D_i^{\ast (2)}{E}_i^{\ast (1)}\right)\text{d}V=-\underset{V}{\int }{e}_{ijk}\left({\mathit{\epsilon }}_{jk}^{(1)}{\mathit{\varphi }}_{,i}^{(2)}-{\mathit{\epsilon }}_{jk}^{(2)}{\mathit{\varphi }}_{,i}^{(1)}\right)\text{d}V-\underset{V}{\int }{\mathit{\tau }}_i^f\left({\mathit{\theta }}^{(1)}{\mathit{\varphi }}_{,i}^{\ast (2)}-{\mathit{\theta }}^{(2)}{\mathit{\varphi }}_{,i}^{\ast (1)}\right)\text{d}V-\underset{V}{\int }{\mathit{\tau }}_i\left({\mathit{\theta }}^{(1)}{\mathit{\varphi }}_{,i}^{(2)}-{\mathit{\theta }}^{(2)}{\mathit{\varphi }}_{,i}^{(1)}\right)\text{d}V-\underset{V}{\int }{\mathit{\zeta }}_i\left({\mathit{\epsilon }}^{\ast (1)}{\mathit{\varphi }}_{,i}^{(2)}-{\mathit{\epsilon }}^{\ast (2)}{\mathit{\varphi }}_{,i}^{(1)}\right)\text{d}V-\underset{V}{\int }{e}_i^{\ast }\left({\mathit{\epsilon }}^{\ast (1)}{\mathit{\varphi }}_{,i}^{\ast (2)}-{\mathit{\epsilon }}^{\ast (2)}{\mathit{\varphi }}_{,i}^{\ast (1)}\right)\text{d}V-\underset{V}{\int }{\mathit{\zeta }}_{ijk}\left({\mathit{\epsilon }}_{jk}^{(1)}{\mathit{\varphi }}_{,i}^{\ast (2)}-{\mathit{\epsilon }}_{jk}^{(2)}{\mathit{\varphi }}_{,i}^{\ast (1)}\right)\text{d}V+\underset{V}{\int }{f}_{ij}\left({\mathit{\psi }}_{,j}^{(1)}{\mathit{\varphi }}_{,i}^{(2)}-{\mathit{\psi }}_{,j}^{(2)}{\mathit{\varphi }}_{,i}^{(1)}\right)\text{d}V+\underset{V}{\int }{f}_{ij}^f\left({\mathit{\psi }}_{,j}^{\ast (1)}{\mathit{\varphi }}_{,i}^{(2)}-{\mathit{\psi }}_{,j}^{\ast (2)}{\mathit{\varphi }}_{,i}^{(1)}\right)\text{d}V+\underset{V}{\int }{f}_{ij}^f\left({\mathit{\psi }}_{,j}^{(1)}{\mathit{\varphi }}_{,i}^{\ast (2)}-{\mathit{\psi }}_{,j}^{(2)}{\mathit{\varphi }}_{,i}^{\ast (1)}\right)\text{d}V+\underset{V}{\int }{\mathit{\gamma }}_{ij}\left({\mathit{\psi }}_{,j}^{\ast (1)}{\mathit{\varphi }}_{,i}^{\ast (2)}-{\mathit{\psi }}_{,j}^{\ast (2)}{\mathit{\varphi }}_{,i}^{\ast (1)}\right)\text{d}V.$$(5.36)

Also, using (5.13) and (5.14), we have(5.37) $$\underset{V}{\int }\left(D_i^{(1)}{E}_i^{(2)}-D_i^{(2)}{E}_i^{(1)}+D_i^{\ast (1)}{E}_i^{\ast (2)}-D_i^{\ast (2)}{E}_i^{\ast (1)}\right)\text{d}V=\underset{V}{\int }\left(D_i^{(2)}{\mathit{\varphi }}_{,i}^{(1)}-D_i^{(1)}{\mathit{\varphi }}_{,i}^{(2)}\right)\text{d}V+\underset{V}{\int }\left(D_i^{\ast (2)}{\mathit{\varphi }}_{,i}^{\ast (1)}-D_i^{\ast (1)}{\mathit{\varphi }}_{,i}^{\ast (2)}\right)\text{d}V.$$(5.37)

Now(5.38) $$D_i^{(2)}{\mathit{\varphi }}_{,i}^{(1)}={\left(D_i^{(2)}{\mathit{\varphi }}^{(1)}\right)}_{,i}-D_{i,i}^{(2)}{\mathit{\varphi }}^{(1)},D_i^{(1)}{\mathit{\varphi }}_{,i}^{(2)}={\left(D_i^{(1)}{\mathit{\varphi }}^{(2)}\right)}_{,i}-D_{i,i}^{(1)}{\mathit{\varphi }}^{(2)},D_i^{\ast (2)}{\mathit{\varphi }}_{,i}^{\ast (1)}={\left(D_i^{\ast (2)}{\mathit{\varphi }}^{\ast (1)}\right)}_{,i}-D_{i,i}^{\ast (2)}{\mathit{\varphi }}^{\ast (1)},\text{and}{D}_i^{\ast (1)}{\mathit{\varphi }}_{,i}^{\ast (2)}={\left(D_i^{\ast (1)}{\mathit{\varphi }}^{\ast (2)}\right)}_{,i}-D_{i,i}^{\ast (1)}{\mathit{\varphi }}^{\ast (2)}.$$(5.38)

Using Equations (5.16), (5.17), (5.38) and divergence theorem in Equation (5.37), we obtain(5.39) $$\underset{V}{\int }\left(D_i^{(1)}{E}_i^{(2)}-D_i^{(2)}{E}_i^{(1)}+D_i^{\ast (1)}{E}_i^{\ast (2)}-D_i^{\ast (2)}{E}_i^{\ast (1)}\right)\text{d}V=\underset{V}{\int }\left({\left(D_i^{(2)}{\mathit{\varphi }}^{(1)}\right)}_{,i}-{\left(D_i^{(1)}{\mathit{\varphi }}^{(2)}\right)}_{,i}\right)\text{d}V+\underset{V}{\int }\left(D_{i,i}^{(1)}{\mathit{\varphi }}^{(2)}-D_{i,i}^{(2)}{\mathit{\varphi }}^{(1)}\right)\text{d}V+\underset{V}{\int }\left({\left(D_i^{(2)}{\mathit{\varphi }}^{\ast (1)}\right)}_{,i}-{\left(D_i^{(1)}{\mathit{\varphi }}^{\ast (2)}\right)}_{,i}\right)\text{d}V+\underset{V}{\int }\left(D_{i,i}^{\ast (1)}{\mathit{\varphi }}^{\ast (2)}-D_{i,i}^{\ast (2)}{\mathit{\varphi }}^{\ast (1)}\right)\text{d}V=\underset{A}{\int }\left(D_i^{(2)}{\mathit{\varphi }}^{(1)}{n}_i-D_i^{(1)}{\mathit{\varphi }}^{(2)}{n}_i+D_i^{\ast (2)}{\mathit{\varphi }}^{\ast (1)}{n}_i-D_i^{\ast (1)}{\mathit{\varphi }}^{\ast (2)}{n}_i\right)\text{d}A.$$(5.39)

Equation (5.39) with the aid of Equation (5.1) gives(5.40) $$\underset{V}{\int }\left(D_i^{(1)}{E}_i^{(2)}-D_i^{(2)}{E}_i^{(1)}+D_i^{\ast (1)}{E}_i^{\ast (2)}-D_i^{\ast (2)}{E}_i^{\ast (1)}\right)\text{d}V=\underset{A}{\int }\left(c_0^{(2)}{\mathit{\varphi }}^{(1)}-c_0^{(1)}{\mathit{\varphi }}^{(2)}+c_0^{\ast (2)}{\mathit{\varphi }}^{\ast (1)}-c_0^{\ast (1)}{\mathit{\varphi }}^{\ast (2)}\right)\text{d}A.$$(5.40)

From Equations (5.36) and (5.40), we have(5.41) $$\underset{A}{\int }\left(c_0^{(1)}{\mathit{\varphi }}^{(2)}-c_0^{(2)}{\mathit{\varphi }}^{(1)}+c_0^{\ast (1)}{\mathit{\varphi }}^{\ast (2)}-c_0^{\ast (2)}{\mathit{\varphi }}^{\ast (1)}\right)\text{d}A=\underset{V}{\int }{e}_{ijk}\left({\mathit{\epsilon }}_{jk}^{(1)}{\mathit{\varphi }}_{,i}^{(2)}-{\mathit{\epsilon }}_{jk}^{(2)}{\mathit{\varphi }}_{,i}^{(1)}\right)\text{d}V+\underset{V}{\int }{\mathit{\tau }}_i^f\left({\mathit{\theta }}^{(1)}{\mathit{\varphi }}_{,i}^{\ast (2)}-{\mathit{\theta }}^{(2)}{\mathit{\varphi }}_{,i}^{\ast (1)}\right)\text{d}V+\underset{V}{\int }{\mathit{\tau }}_i\left({\mathit{\theta }}^{(1)}{\mathit{\varphi }}_{,i}^{(2)}-{\mathit{\theta }}^{(2)}{\mathit{\varphi }}_{,i}^{(1)}\right)\text{d}V+\underset{V}{\int }{\mathit{\zeta }}_i\left({\mathit{\epsilon }}^{\ast (1)}{\mathit{\varphi }}_{,i}^{(2)}-{\mathit{\epsilon }}^{\ast (2)}{\mathit{\varphi }}_{,i}^{(1)}\right)\text{d}V+\underset{V}{\int }{e}_i^{\ast }\left({\mathit{\epsilon }}^{\ast (1)}{\mathit{\varphi }}_{,i}^{\ast (2)}-{\mathit{\epsilon }}^{\ast (2)}{\mathit{\varphi }}_{,i}^{\ast (1)}\right)\text{d}V+\underset{V}{\int }{\mathit{\zeta }}_{ijk}\left({\mathit{\epsilon }}_{jk}^{(1)}{\mathit{\varphi }}_{,i}^{\ast (2)}-{\mathit{\epsilon }}_{jk}^{(2)}{\mathit{\varphi }}_{,i}^{\ast (1)}\right)\text{d}V-\underset{V}{\int }{f}_{ij}\left({\mathit{\psi }}_{,j}^{(1)}{\mathit{\varphi }}_{,i}^{(2)}-{\mathit{\psi }}_{,j}^{(2)}{\mathit{\varphi }}_{,i}^{(1)}\right)\text{d}V-\underset{V}{\int }{f}_{ij}^f\left({\mathit{\psi }}_{,j}^{\ast (1)}{\mathit{\varphi }}_{,i}^{(2)}-{\mathit{\psi }}_{,j}^{\ast (2)}{\mathit{\varphi }}_{,i}^{(1)}\right)\text{d}V-\underset{V}{\int }{f}_{ij}^f\left({\mathit{\psi }}_{,j}^{(1)}{\mathit{\varphi }}_{,i}^{\ast (2)}-{\mathit{\psi }}_{,j}^{(2)}{\mathit{\varphi }}_{,i}^{\ast (1)}\right)\text{d}V-\underset{V}{\int }{\mathit{\gamma }}_{ij}\left({\mathit{\psi }}_{,j}^{\ast (1)}{\mathit{\varphi }}_{,i}^{\ast (2)}-{\mathit{\psi }}_{,j}^{\ast (2)}{\mathit{\varphi }}_{,i}^{\ast (1)}\right)\text{d}V.$$(5.41)

The Equation (5.41) constitutes the third part of reciprocity theorem which contains the electric potentials ϕ, ϕ* and surface charge densities $c_0,c_0^{\ast }$.

To derive the last part, multiplying Equations (5.11) and (5.12) by $H_i^{(2)},H_i^{(1)}$ and $H_i^{\ast (2)},H_i^{\ast (1)}$ for the first and the second problems, respectively, subtracting and integrating over V, we get(5.42) $$\underset{V}{\int }\left(B_i^{(1)}{H}_i^{(2)}-B_i^{(2)}{H}_i^{(1)}+B_i^{\ast (1)}{H}_i^{\ast (2)}-B_i^{\ast (2)}{H}_i^{\ast (1)}\right)\text{d}V=\underset{V}{\int }{q}_{ijk}\left({\mathit{\epsilon }}_{jk}^{(1)}{H}_i^{(2)}-{\mathit{\epsilon }}_{jk}^{(2)}{H}_i^{(1)}\right)\text{d}V+\underset{V}{\int }{m}_i\left({\mathit{\theta }}^{(1)}{H}_i^{(2)}-{\mathit{\theta }}^{(2)}{H}_i^{(1)}\right)\text{d}V+\underset{V}{\int }{f}_{ij}\left(E_j^{(1)}{H}_i^{(2)}-E_j^{(2)}{H}_i^{(1)}\right)\text{d}V+\underset{V}{\int }{l}_i\left({\mathit{\epsilon }}^{\ast (1)}{H}_i^{(2)}-{\mathit{\epsilon }}^{\ast (2)}{H}_i^{(1)}\right)\text{d}V+\underset{V}{\int }{q}_{ijk}^f\left({\mathit{\epsilon }}_{jk}^{(1)}{H}_i^{\ast (2)}-{\mathit{\epsilon }}_{jk}^{(2)}{H}_i^{\ast (1)}\right)\text{d}V+\underset{V}{\int }{m}_i^f\left({\mathit{\theta }}^{(1)}{H}_i^{\ast (2)}-{\mathit{\theta }}^{(2)}{H}_i^{\ast (1)}\right)\text{d}V+\underset{V}{\int }{f}_{ij}^f\left(E_j^{(1)}{H}_i^{\ast (2)}-E_j^{(2)}{H}_i^{\ast (1)}\right)\text{d}V+\underset{V}{\int }{l}_i^{\ast }\left({\mathit{\epsilon }}^{\ast (1)}{H}_i^{\ast (2)}-{\mathit{\epsilon }}^{\ast (2)}{H}_i^{\ast (1)}\right)\text{d}V.$$(5.42)

Equation (5.42) with the aid of Equations (5.13)–(5.16) yields(5.43) $$\underset{V}{\int }\left(B_i^{(1)}{H}_i^{(2)}-B_i^{(2)}{H}_i^{(1)}+B_i^{\ast (1)}{H}_i^{\ast (2)}-B_i^{\ast (2)}{H}_i^{\ast (1)}\right)\text{d}V=-\underset{V}{\int }{q}_{ijk}\left({\mathit{\epsilon }}_{jk}^{(1)}{\mathit{\psi }}_{,i}^{(2)}-{\mathit{\epsilon }}_{jk}^{(2)}{\mathit{\psi }}_{,i}^{(1)}\right)\text{d}V-\underset{V}{\int }{m}_i\left({\mathit{\theta }}^{(1)}{\mathit{\psi }}_{,i}^{(2)}-{\mathit{\theta }}^{(2)}{\mathit{\psi }}_{,i}^{(1)}\right)\text{d}V+\underset{V}{\int }{f}_{ik}\left({\mathit{\varphi }}_{,k}^{(1)}{\mathit{\psi }}_{,i}^{(2)}-{\mathit{\varphi }}_{,k}^{(2)}{\mathit{\psi }}_{,i}^{(1)}\right)\text{d}V-\underset{V}{\int }{m}_i^f\left({\mathit{\theta }}^{(1)}{\mathit{\psi }}_{,i}^{\ast (2)}-{\mathit{\theta }}^{(2)}{\mathit{\psi }}_{,i}^{\ast (1)}\right)\text{d}V-\underset{V}{\int }{l}_i\left({\mathit{\epsilon }}^{\ast (1)}{\mathit{\psi }}_{,i}^{(2)}-{\mathit{\epsilon }}^{\ast (2)}{\mathit{\psi }}_{,i}^{(1)}\right)\text{d}V-\underset{V}{\int }{q}_{ijk}^f\left({\mathit{\epsilon }}_{jk}^{(1)}{\mathit{\psi }}_{,i}^{\ast (2)}-{\mathit{\epsilon }}_{jk}^{(2)}{\mathit{\psi }}_{,i}^{\ast (1)}\right)\text{d}V+\underset{V}{\int }{f}_{ij}^f\left({\mathit{\varphi }}_{,j}^{(1)}{\mathit{\psi }}_{,i}^{\ast (2)}-{\mathit{\varphi }}_{,j}^{(2)}{\mathit{\psi }}_{,i}^{\ast (1)}\right)\text{d}V-\underset{V}{\int }{l}_i^{\ast }\left({\mathit{\epsilon }}^{\ast (1)}{\mathit{\psi }}_{,i}^{\ast (2)}-{\mathit{\epsilon }}^{\ast (2)}{\mathit{\psi }}_{,i}^{\ast (1)}\right)\text{d}V.$$(5.43)

Also, using Equations (5.15) and (5.16), we have(5.44) $$\underset{V}{\int }\left(B_i^{(1)}{H}_i^{(2)}-B_i^{(2)}{H}_i^{(1)}+B_i^{\ast (1)}{H}_i^{\ast (2)}-B_i^{\ast (2)}{H}_i^{\ast (1)}\right)\text{d}V=\underset{V}{\int }\left(B_i^{(2)}{\mathit{\psi }}_{,i}^{(1)}-B_i^{(1)}{\mathit{\psi }}_{,i}^{(2)}\right)\text{d}V+\underset{V}{\int }\left(B_i^{\ast (2)}{\mathit{\psi }}_{,i}^{\ast (1)}-B_i^{\ast (1)}{\mathit{\psi }}_{,i}^{\ast (2)}\right)\text{d}V.$$(5.44)

Now(5.45) $$B_i^{(2)}{\mathit{\psi }}_{,i}^{(1)}={\left(B_i^{(2)}{\mathit{\psi }}^{(1)}\right)}_{,i}-B_{i,i}^{(2)}{\mathit{\psi }}^{(1)}{B}_i^{(1)}{\mathit{\psi }}_{,i}^{(2)}={\left(B_i^{(1)}{\mathit{\psi }}^{(2)}\right)}_{,i}-B_{i,i}^{(1)}{\mathit{\psi }}^{(2)}{B}_i^{\ast (2)}{\mathit{\psi }}_{,i}^{\ast (1)}={\left(B_i^{\ast (2)}{\mathit{\psi }}^{\ast (1)}\right)}_{,i}-B_{i,i}^{\ast (2)}{\mathit{\psi }}^{\ast (1)},\text{and}{B}_i^{\ast (1)}{\mathit{\psi }}_{,i}^{\ast (2)}={\left(B_i^{\ast (1)}{\mathit{\psi }}^{\ast (2)}\right)}_{,i}-B_{i,i}^{\ast (1)}{\mathit{\psi }}^{\ast (2)}.$$(5.45)

Using Equations (5.45), (5.20), (5.23) and divergence theorem in Equation (5.44), we obtain(5.46) $$\underset{V}{\int }\left(B_i^{(1)}{H}_i^{(2)}-B_i^{(2)}{H}_i^{(1)}+B_i^{\ast (1)}{H}_i^{\ast (2)}-B_i^{\ast (2)}{H}_i^{\ast (1)}\right)\text{d}V=\underset{V}{\int }\left({\left(B_i^{(2)}{\mathit{\psi }}^{(1)}\right)}_{,i}-{\left(B_i^{(1)}{\mathit{\psi }}^{(2)}\right)}_{,i}\right)\text{d}V+\underset{V}{\int }\left(B_{i,i}^{(1)}{\mathit{\psi }}^{(2)}-B_{i,i}^{(2)}{\mathit{\psi }}^{(1)}\right)\text{d}V+\underset{V}{\int }\left({\left(B_i^{(2)}{\mathit{\psi }}^{\ast (1)}\right)}_{,i}-{\left(B_i^{(1)}{\mathit{\psi }}^{\ast (2)}\right)}_{,i}\right)\text{d}V+\underset{V}{\int }\left(B_{i,i}^{\ast (1)}{\mathit{\psi }}^{\ast (2)}-B_{i,i}^{\ast (2)}{\mathit{\psi }}^{\ast (1)}\right)\text{d}V=\underset{A}{\int }\left(B_i^{(2)}{\mathit{\psi }}^{(1)}{n}_i-B_i^{(1)}{\mathit{\psi }}^{(2)}{n}_i+B_i^{\ast (2)}{\mathit{\psi }}^{\ast (1)}{n}_i-B_i^{\ast (1)}{\mathit{\psi }}^{\ast (2)}{n}_i\right)\text{d}A.=\underset{A}{\int }\left(B_i^{(2)}{\mathit{\psi }}^{(1)}{n}_i-B_i^{(1)}{\mathit{\psi }}^{(2)}{n}_i\right)\text{d}A.$$(5.46)

Equation (5.46) with the aid of Equation (5.1), gives(5.47) $$\underset{V}{\int }\left(B_i^{(1)}{H}_i^{(2)}-B_i^{(2)}{H}_i^{(1)}+B_i^{\ast (1)}{H}_i^{\ast (2)}-B_i^{\ast (2)}{H}_i^{\ast (1)}\right)\text{d}V=\underset{A}{\int }\left(b_0^{(2)}{\mathit{\psi }}^{(1)}-b_0^{(1)}{\mathit{\psi }}^{(2)}+b_0^{\ast (2)}{\mathit{\psi }}^{\ast (1)}-b_0^{\ast (1)}{\mathit{\psi }}^{\ast (2)}\right)\text{d}A.$$(5.47)

From Equations (5.43) and (5.47), we have(5.48) $$\underset{A}{\int }\left(b_0^{(1)}{\mathit{\psi }}^{(2)}-b_0^{(2)}{\mathit{\psi }}^{(1)}+b_0^{\ast (1)}{\mathit{\psi }}^{\ast (2)}-b_0^{\ast (2)}{\mathit{\psi }}^{\ast (1)}\right)\text{d}A=\underset{V}{\int }{q}_{ijk}\left({\mathit{\epsilon }}_{jk}^{(1)}{\mathit{\psi }}_{,i}^{(2)}-{\mathit{\epsilon }}_{jk}^{(2)}{\mathit{\psi }}_{,i}^{(1)}\right)\text{d}V+\underset{V}{\int }{m}_i\left({\mathit{\theta }}^{(1)}{\mathit{\psi }}_{,i}^{(2)}-{\mathit{\theta }}^{(2)}{\mathit{\psi }}_{,i}^{(1)}\right)\text{d}V-\underset{V}{\int }{f}_{ij}\left({\mathit{\varphi }}_{,j}^{(1)}{\mathit{\psi }}_{,i}^{(2)}-{\mathit{\varphi }}_{,j}^{(2)}{\mathit{\psi }}_{,i}^{(1)}\right)\text{d}V+\underset{V}{\int }{l}_i^{\ast }\left({\mathit{\epsilon }}^{\ast (1)}{\mathit{\psi }}_{,i}^{\ast (2)}-{\mathit{\epsilon }}^{\ast (2)}{\mathit{\psi }}_{,i}^{\ast (1)}\right)\text{d}V+\underset{V}{\int }{q}_{ijk}^f\left({\mathit{\epsilon }}_{jk}^{(1)}{\mathit{\psi }}_{,i}^{\ast (2)}-{\mathit{\epsilon }}_{jk}^{(2)}{\mathit{\psi }}_{,i}^{\ast (1)}\right)\text{d}V+\underset{V}{\int }{m}_i^f\left({\mathit{\theta }}^{(1)}{\mathit{\psi }}_{,i}^{\ast (2)}-{\mathit{\theta }}^{(2)}{\mathit{\psi }}_{,i}^{\ast (1)}\right)\text{d}V+\underset{V}{\int }{l}_i\left({\mathit{\epsilon }}^{\ast (1)}{\mathit{\psi }}_{,i}^{(2)}-{\mathit{\epsilon }}^{\ast (2)}{\mathit{\psi }}_{,i}^{(1)}\right)\text{d}V-\underset{V}{\int }{f}_{ij}^f\left({\mathit{\varphi }}_{,j}^{(1)}{\mathit{\psi }}_{,i}^{\ast (2)}-{\mathit{\varphi }}_{,j}^{(2)}{\mathit{\psi }}_{,i}^{\ast (1)}\right)\text{d}V.$$(5.48)

The Equation (5.48) constitutes the last part of reciprocity theorem which contains the magnetic potentials ψ, ψ^* and magnetic densities $b_0,b_0^{\ast }$.

Eliminating the integrals$$\underset{V}{\int }{\mathit{\alpha }}^f\left({\mathit{\epsilon }}^{\ast (1)}{\mathit{\theta }}^{(2)}-{\mathit{\epsilon }}^{\ast (2)}{\mathit{\theta }}^{(1)}\right)\text{d}V,\underset{V}{\int }{\mathit{\alpha }}_{ij}\left({\mathit{\epsilon }}_{ij}^{(1)}{\mathit{\theta }}^{(2)}-{\mathit{\epsilon }}_{ij}^{(2)}{\mathit{\theta }}^{(1)}\right)\text{d}V,\underset{V}{\int }{e}_{ijk}\left({\mathit{\epsilon }}_{jk}^{(1)}{\mathit{\varphi }}_{,i}^{(2)}-{\mathit{\epsilon }}_{jk}^{(2)}{\mathit{\varphi }}_{,i}^{(1)}\right)\text{d}V,\underset{V}{\int }{\mathit{\tau }}_i^f\left({\mathit{\theta }}^{(1)}{\mathit{\varphi }}_{,i}^{\ast (2)}-{\mathit{\theta }}^{(2)}{\mathit{\varphi }}_{,i}^{\ast (1)}\right)\text{d}V,\underset{V}{\int }{\mathit{\tau }}_i\left({\mathit{\theta }}^{(1)}{\mathit{\varphi }}_{,i}^{(2)}-{\mathit{\theta }}^{(2)}{\mathit{\varphi }}_{,i}^{(1)}\right)\text{d}V,\underset{V}{\int }{e}_i^{\ast }\left({\mathit{\epsilon }}^{\ast (1)}{\mathit{\varphi }}_{,i}^{\ast (2)}-{\mathit{\epsilon }}^{\ast (2)}{\mathit{\varphi }}_{,i}^{\ast (1)}\right)\text{d}V,\underset{V}{\int }{\mathit{\zeta }}_{ijk}\left({\mathit{\epsilon }}_{jk}^{(1)}{\mathit{\varphi }}_{,i}^{\ast (2)}-{\mathit{\epsilon }}_{jk}^{(2)}{\mathit{\varphi }}_{,i}^{\ast (1)}\right)\text{d}V,\underset{V}{\int }{\mathit{\zeta }}_i\left({\mathit{\epsilon }}^{\ast (1)}{\mathit{\varphi }}_{,i}^{(2)}-{\mathit{\epsilon }}^{\ast (2)}{\mathit{\varphi }}_{,i}^{(1)}\right)\text{d}V,\underset{V}{\int }{q}_{ijk}^f\left({\mathit{\epsilon }}_{jk}^{(1)}{\mathit{\psi }}_{,i}^{\ast (2)}-{\mathit{\epsilon }}_{jk}^{(2)}{\mathit{\psi }}_{,i}^{\ast (1)}\right)\text{d}V,\underset{V}{\int }{q}_{ijk}\left({\mathit{\epsilon }}_{jk}^{(1)}{\mathit{\psi }}_{,i}^{(2)}-{\mathit{\epsilon }}_{jk}^{(2)}{\mathit{\psi }}_{,i}^{(1)}\right)\text{d}V,\underset{V}{\int }{m}_i^f\left({\mathit{\theta }}^{(1)}{\mathit{\psi }}_{,i}^{\ast (2)}-{\mathit{\theta }}^{(2)}{\mathit{\psi }}_{,i}^{\ast (1)}\right)\text{d}V,\underset{V}{\int }{m}_i\left({\mathit{\theta }}^{(1)}{\mathit{\psi }}_{,i}^{(2)}-{\mathit{\theta }}^{(2)}{\mathit{\psi }}_{,i}^{(1)}\right)\text{d}V,\underset{V}{\int }{f}_{ij}\left({\mathit{\varphi }}_{,j}^{(1)}{\mathit{\psi }}_{,i}^{(2)}-{\mathit{\varphi }}_{,j}^{(2)}{\mathit{\psi }}_{,i}^{(1)}\right)\text{d}V,\underset{V}{\int }{f}_{ij}^f\left({\mathit{\varphi }}_{,j}^{(1)}{\mathit{\psi }}_{,i}^{\ast (2)}-{\mathit{\varphi }}_{,j}^{(2)}{\mathit{\psi }}_{,i}^{\ast (1)}\right)\text{d}V,\underset{V}{\int }{l}_i\left({\mathit{\epsilon }}^{\ast (1)}{\mathit{\psi }}_{,i}^{(2)}-{\mathit{\epsilon }}^{\ast (2)}{\mathit{\psi }}_{,i}^{(1)}\right)\text{d}V,\underset{V}{\int }{f}_{ik}^f\left({\mathit{\psi }}_{,k}^{(1)}{\mathit{\varphi }}_{,i}^{\ast (2)}-{\mathit{\psi }}_{,k}^{(2)}{\mathit{\varphi }}_{,i}^{\ast (1)}\right)\text{d}V,\underset{V}{\int }{\mathit{\gamma }}_{ij}\left({\mathit{\psi }}_{,j}^{\ast (1)}{\mathit{\varphi }}_{,i}^{\ast (2)}-{\mathit{\psi }}_{,j}^{\ast (2)}{\mathit{\varphi }}_{,i}^{\ast (1)}\right)\text{d}V,\underset{V}{\int }{l}_i^{\ast }\left({\mathit{\epsilon }}^{\ast (1)}{\mathit{\psi }}_{,i}^{\ast (2)}-{\mathit{\epsilon }}^{\ast (2)}{\mathit{\psi }}_{,i}^{\ast (1)}\right)\text{d}V,$$

from Equations (5.29), (5.34), (5.41) and (5.48) with the aid of Equations (5.13)–(5.16), we obtain(5.49) $$s\left(1+{\mathit{\tau }}_{0}s\right)T_0\left[\underset{A}{\int }\left(h_i^{(1)}{\text{u}}_i^{(2)}-h_i^{(2)}{\text{u}}_i^{(1)}+h_i^{\ast (1)}{\text{u}}_i^{\ast (2)}-h_i^{\ast (2)}{\text{u}}_i^{\ast (1)}\right)\text{d}A+\underset{V}{\int }{\mathit{\rho }}_1\left(F_i^{(1)}{\text{u}}_i^{(2)}-F_i^{(2)}{\text{u}}_i^{(1)}\right)\text{d}V\right]+s\left(1+{\mathit{\tau }}_{0}s\right)T_0\left[\underset{V}{\int }{\mathit{\rho }}_2\left(f_i^{(1)}{\text{u}}_i^{\ast (2)}-f_i^{(2)}{\text{u}}_i^{\ast (1)}\right)\text{d}V+\underset{A}{\int }\left(c_0^{(1)}{\mathit{\varphi }}^{(2)}-c_0^{(2)}{\mathit{\varphi }}^{(1)}+c_0^{\ast (1)}{\mathit{\varphi }}^{\ast (2)}-c_0^{\ast (2)}{\mathit{\varphi }}^{\ast (1)}\right)\text{d}A\right]+s\left(1+{\mathit{\tau }}_{0}s\right)T_0\underset{A}{\int }\left(b_0^{(1)}{\mathit{\psi }}^{(2)}-b_0^{(2)}{\mathit{\psi }}^{(1)}+b_0^{\ast (1)}{\mathit{\psi }}^{\ast (2)}-b_0^{\ast (2)}{\mathit{\psi }}^{\ast (1)}\right)\text{d}A+\underset{A}{\int }\left(q^{(1)}{\mathit{\eta }}^{(2)}-q^{(2)}{\mathit{\eta }}^{(1)}\right)\text{d}A=0.$$(5.49)

This is the general reciprocity theorem in the Laplace transform domain.

For applying inverse Laplace transform on the equations (5.29), (5.34), (5.41), (5.48) and (5.49), we use the convolution theorem(5.50) $$L^{-1}(F(s)G(s))=\underset{0}{\overset{t}{\int }}f(t-\mathit{\xi })g(\mathit{\xi })\text{d}\mathit{\xi }=\underset{0}{\overset{t}{\int }}g(t-\mathit{\xi })f(\mathit{\xi })\text{d}\mathit{\xi },$$(5.50)

and the symbolic notation(5.51) $$Y(f)=1+{\mathit{\tau }}_0\frac{\mathit{\partial }f(x,\mathit{\xi })}{\mathit{\partial }\mathit{\xi }},$$(5.51)

Equations (5.29), (5.34), (5.41) and (5.48) with the aid of Equation (5.50) yield the first, second and last parts of the reciprocity theorem in the final form(5.52) $$\underset{A}{\int }\underset{0}{\overset{t}{\int }}\left(h_i^{(1)}(x,t-\mathit{\xi }){\text{u}}_i^{(2)}(x,\mathit{\xi })+h_i^{\ast (1)}(x,t-\mathit{\xi }){\text{u}}_i^{\ast (2)}(x,\mathit{\xi })\right)\text{d}\mathit{\xi }\text{d}A+\underset{V}{\int }\underset{0}{\overset{t}{\int }}{\mathit{\rho }}_1\left(F_i^{(1)}(x,t-\mathit{\xi }){\text{u}}_i^{(2)}(x,\mathit{\xi })\right)\text{d}\mathit{\xi }\text{d}V+\underset{V}{\int }\underset{0}{\overset{t}{\int }}{\mathit{\rho }}_2\left(f_i^{(1)}(x,t-\mathit{\xi }){\text{u}}_i^{\ast (2)}(x,\mathit{\xi })\right)\text{d}\mathit{\xi }\text{d}V-\underset{V}{\int }\underset{0}{\overset{t}{\int }}{e}_{ijk}\left({\mathit{\varphi }}_{,k}^{(1)}(x,t-\mathit{\xi }){\mathit{\epsilon }}_{ij}^{(2)}(x,\mathit{\xi })\right)\text{d}\mathit{\xi }\text{d}V+\underset{V}{\int }\underset{0}{\overset{t}{\int }}{\mathit{\alpha }}_{ij}\left({\mathit{\theta }}^{(1)}(x,t-\mathit{\xi }){\mathit{\epsilon }}_{ij}^{(2)}(x,\mathit{\xi })\right)\text{d}\mathit{\xi }\text{d}V+\underset{V}{\int }\underset{0}{\overset{t}{\int }}{\mathit{\alpha }}^f\left({\mathit{\theta }}^{(1)}(x,t-\mathit{\xi }){\mathit{\epsilon }}^{\ast (2)}(x,\mathit{\xi })\right)\text{d}\mathit{\xi }\text{d}V-\underset{V}{\int }\underset{0}{\overset{t}{\int }}{\mathit{\zeta }}_{ijk}{\mathit{\varphi }}_{,k}^{\ast (1)}(x,t-\mathit{\xi }){\mathit{\epsilon }}_{ij}^{(2)}(x,\mathit{\xi })\text{d}\mathit{\xi }\text{d}V-\underset{V}{\int }\underset{0}{\overset{t}{\int }}{e}_i^{\ast }{\mathit{\varphi }}_{,i}^{\ast (1)}(x,t-\mathit{\xi }){\mathit{\epsilon }}^{\ast (2)}(x,\mathit{\xi })\text{d}\mathit{\xi }\text{d}V-\underset{V}{\int }\underset{0}{\overset{t}{\int }}{q}_{ijk}\left({\mathit{\psi }}_{,k}^{(1)}(x,t-\mathit{\xi }){\mathit{\epsilon }}_{ij}^{(2)}(x,\mathit{\xi })\right)\text{d}\mathit{\xi }\text{d}V-\underset{V}{\int }\underset{0}{\overset{t}{\int }}{q}_{ijk}^f\left({\mathit{\psi }}_{,k}^{\ast (1)}(x,t-\mathit{\xi }){\mathit{\epsilon }}_{ij}^{(2)}(x,\mathit{\xi })\right)\text{d}\mathit{\xi }\text{d}V-\underset{V}{\int }\underset{0}{\overset{t}{\int }}{l}_i{\mathit{\psi }}_{,i}^{(1)}(x,t-\mathit{\xi }){\mathit{\epsilon }}^{\ast (2)}(x,\mathit{\xi })\text{d}\mathit{\xi }\text{d}V-\underset{V}{\int }\underset{0}{\overset{t}{\int }}{l}_i^{\ast }{\mathit{\psi }}_{,i}^{\ast (1)}(x,t-\mathit{\xi }){\mathit{\epsilon }}^{\ast (2)}(x,\mathit{\xi })\text{d}\mathit{\xi }\text{d}V=S_{21}^{12},$$(5.52) (5.53) $$\underset{A}{\int }\underset{0}{\overset{t}{\int }}{q}^{(1)}(x,t-\mathit{\xi }){\mathit{\eta }}^{(2)}(x,\mathit{\xi })\text{d}\mathit{\xi }\text{d}A-T_0\underset{V}{\int }\underset{0}{\overset{t}{\int }}{\mathit{\alpha }}_{ij}{\mathit{\theta }}^{(1)}(x,t-\mathit{\xi })\frac{\mathit{\partial }Y\left({\mathit{\epsilon }}_{ij}^{(2)}(x,\mathit{\xi })\right)}{\mathit{\partial }\mathit{\xi }}\text{d}\mathit{\xi }\text{d}V-T_0\underset{V}{\int }\underset{0}{\overset{t}{\int }}{\mathit{\alpha }}^f{\mathit{\theta }}^{(1)}(x,t-\mathit{\xi })\frac{\mathit{\partial }Y\left({\mathit{\epsilon }}^{\ast (2)}(x,\mathit{\xi })\right)}{\mathit{\partial }\mathit{\xi }}\text{d}\mathit{\xi }\text{d}V+T_0\underset{V}{\int }\underset{0}{\overset{t}{\int }}{\mathit{\tau }}_i{\mathit{\theta }}^{(1)}(x,t-\mathit{\xi })\frac{\mathit{\partial }Y\left({\mathit{\varphi }}_{,i}^{(2)}(x,\mathit{\xi })\right)}{\mathit{\partial }\mathit{\xi }}\text{d}\mathit{\xi }\text{d}V+T_0\underset{V}{\int }\underset{0}{\overset{t}{\int }}{\mathit{\tau }}_i^f{\mathit{\theta }}^{(1)}(x,t-\mathit{\xi })\frac{\mathit{\partial }Y\left({\mathit{\varphi }}_{,i}^{\ast (2)}(x,\mathit{\xi })\right)}{\mathit{\partial }\mathit{\xi }}\text{d}\mathit{\xi }\text{d}V+T_0\underset{V}{\int }\underset{0}{\overset{t}{\int }}{m}_i{\mathit{\theta }}^{(1)}(x,t-\mathit{\xi })\frac{\mathit{\partial }Y\left({\mathit{\psi }}_{,i}^{(2)}(x,\mathit{\xi })\right)}{\mathit{\partial }\mathit{\xi }}\text{d}\mathit{\xi }\text{d}V+T_0\underset{V}{\int }\underset{0}{\overset{t}{\int }}{m}_i^f{\mathit{\theta }}^{(1)}(x,t-\mathit{\xi })\frac{\mathit{\partial }Y\left({\mathit{\psi }}_{,i}^{\ast (2)}(x,\mathit{\xi })\right)}{\mathit{\partial }\mathit{\xi }}\text{d}\mathit{\xi }\text{d}V=S_{21}^{12},$$(5.53) (5.54) $$\underset{A}{\int }\underset{0}{\overset{t}{\int }}\left(c_0^{(1)}(x,t-\mathit{\xi }){\mathit{\varphi }}^{(2)}(x,\mathit{\xi })+c_0^{\ast (1)}(x,t-\mathit{\xi }){\mathit{\varphi }}^{\ast (2)}(x,\mathit{\xi })\right)\text{d}\mathit{\xi }\text{d}A+\underset{V}{\int }\underset{0}{\overset{t}{\int }}{e}_{ijk}{\mathit{\varphi }}_{,i}^{(1)}(x,t-\mathit{\xi }){\mathit{\epsilon }}_{jk}^{(2)}(x,\mathit{\xi })\text{d}\mathit{\xi }\text{d}V+\underset{V}{\int }\underset{0}{\overset{t}{\int }}{\mathit{\tau }}_i{\mathit{\varphi }}_{,i}^{(1)}(x,t-\mathit{\xi }){\mathit{\theta }}^{(2)}(x,\mathit{\xi })\text{d}\mathit{\xi }\text{d}V+\underset{V}{\int }\underset{0}{\overset{t}{\int }}{\mathit{\tau }}_i^f{\mathit{\varphi }}_{,i}^{\ast (1)}(x,t-\mathit{\xi }){\mathit{\theta }}^{(2)}(x,\mathit{\xi })\text{d}\mathit{\xi }\text{d}V+\underset{V}{\int }\underset{0}{\overset{t}{\int }}{e}_i^{\ast }{\mathit{\varphi }}_{,i}^{\ast (1)}(x,t-\mathit{\xi }){\mathit{\epsilon }}^{\ast (2)}(x,\mathit{\xi })\text{d}\mathit{\xi }\text{d}V+\underset{V}{\int }\underset{0}{\overset{t}{\int }}{\mathit{\zeta }}_{ijk}{\mathit{\varphi }}_{,i}^{\ast (1)}(x,t-\mathit{\xi }){\mathit{\epsilon }}_{jk}^{(2)}(x,\mathit{\xi })\text{d}\mathit{\xi }\text{d}V+\underset{V}{\int }\underset{0}{\overset{t}{\int }}{\mathit{\zeta }}_i{\mathit{\varphi }}_{,i}^{(1)}(x,t-\mathit{\xi }){\mathit{\epsilon }}^{\ast (2)}(x,\mathit{\xi })\text{d}\mathit{\xi }\text{d}V+\underset{V}{\int }\underset{0}{\overset{t}{\int }}{f}_{ij}{\mathit{\varphi }}_{,i}^{(1)}(x,t-\mathit{\xi }){\mathit{\psi }}_{,i}^{(2)}(x,\mathit{\xi })\text{d}\mathit{\xi }\text{d}V+\underset{V}{\int }\underset{0}{\overset{t}{\int }}{f}_{ij}^f{\mathit{\varphi }}_{,i}^{(1)}(x,t-\mathit{\xi }){\mathit{\psi }}_{,i}^{\ast (2)}(x,\mathit{\xi })\text{d}\mathit{\xi }\text{d}V+\underset{V}{\int }\underset{0}{\overset{t}{\int }}{f}_{ij}^f{\mathit{\varphi }}_{,i}^{\ast (1)}(x,t-\mathit{\xi }){\mathit{\psi }}_{,i}^{(2)}(x,\mathit{\xi })\text{d}\mathit{\xi }\text{d}V+\underset{V}{\int }\underset{0}{\overset{t}{\int }}{\mathit{\gamma }}_{ij}{\mathit{\varphi }}_{,i}^{\ast (1)}(x,t-\mathit{\xi }){\mathit{\psi }}_{,i}^{\ast (2)}(x,\mathit{\xi })\text{d}\mathit{\xi }\text{d}V=S_{21}^{12}.$$(5.54)

and(5.55) $$\underset{A}{\int }\underset{0}{\overset{t}{\int }}\left(b_0^{(1)}(x,t-\mathit{\xi }){\mathit{\psi }}^{(2)}(x,\mathit{\xi })+b_0^{\ast (1)}(x,t-\mathit{\xi }){\mathit{\psi }}^{\ast (2)}(x,\mathit{\xi })\right)\text{d}\mathit{\xi }\text{d}A+\underset{V}{\int }\underset{0}{\overset{t}{\int }}{q}_{ijk}{\mathit{\psi }}_{,i}^{(1)}(x,t-\mathit{\xi }){\mathit{\epsilon }}_{jk}^{(2)}(x,\mathit{\xi })\text{d}\mathit{\xi }\text{d}V+\underset{V}{\int }\underset{0}{\overset{t}{\int }}{m}_i{\mathit{\psi }}_{,i}^{(1)}(x,t-\mathit{\xi }){\mathit{\theta }}^{(2)}(x,\mathit{\xi })\text{d}\mathit{\xi }\text{d}V+\underset{V}{\int }\underset{0}{\overset{t}{\int }}{m}_i^f{\mathit{\psi }}_{,i}^{\ast (1)}(x,t-\mathit{\xi }){\mathit{\theta }}^{(2)}(x,\mathit{\xi })\text{d}\mathit{\xi }\text{d}V+\underset{V}{\int }\underset{0}{\overset{t}{\int }}{l}_i^{\ast }{\mathit{\psi }}_{,i}^{\ast (1)}(x,t-\mathit{\xi }){\mathit{\epsilon }}^{\ast (2)}(x,\mathit{\xi })\text{d}\mathit{\xi }\text{d}V+\underset{V}{\int }\underset{0}{\overset{t}{\int }}{q}_{ijk}^f{\mathit{\psi }}_{,i}^{\ast (1)}(x,t-\mathit{\xi }){\mathit{\epsilon }}_{jk}^{(2)}(x,\mathit{\xi })\text{d}\mathit{\xi }\text{d}V+\underset{V}{\int }\underset{0}{\overset{t}{\int }}{l}_i{\mathit{\psi }}_{,i}^{(1)}(x,t-\mathit{\xi }){\mathit{\epsilon }}^{\ast (2)}(x,\mathit{\xi })\text{d}\mathit{\xi }\text{d}V-\underset{V}{\int }\underset{0}{\overset{t}{\int }}{f}_{ij}{\mathit{\psi }}_{,i}^{(1)}(x,t-\mathit{\xi }){\mathit{\varphi }}_{,j}^{(2)}(x,\mathit{\xi })\text{d}\mathit{\xi }\text{d}V-\underset{V}{\int }\underset{0}{\overset{t}{\int }}{f}_{ij}^f{\mathit{\psi }}_{,i}^{\ast (1)}(x,t-\mathit{\xi }){\mathit{\varphi }}_{,j}^{(2)}(x,\mathit{\xi })\text{d}\mathit{\xi }\text{d}V=S_{21}^{12}.$$(5.55)

Here, $S_{21}^{12}$ indicates the same expression as on the left-hand side except that the superscripts (1) and (2) are interchanged. Finally, Equation (5.49) with the aid of Equation (5.50) gives the general reciprocity theorem in the final form(5.56) $$\underset{A}{\int }\underset{0}{\overset{t}{\int }}{h}_i^{(1)}(x,t-\mathit{\xi })\frac{\mathit{\partial }Y\left({\text{u}}_i^{(2)}(x,\mathit{\xi })\right)}{\mathit{\partial }\mathit{\xi }}\text{d}\mathit{\xi }\text{d}A+\underset{A}{\int }\underset{0}{\overset{t}{\int }}{h}_i^{\ast (1)}(x,t-\mathit{\xi })\frac{\mathit{\partial }Y\left({\text{u}}_i^{\ast (2)}(x,\mathit{\xi })\right)}{\mathit{\partial }\mathit{\xi }}\text{d}\mathit{\xi }\text{d}A+\underset{V}{\int }\underset{0}{\overset{t}{\int }}\left({\mathit{\rho }}_1{F}_i^{(1)}(x,t-\mathit{\xi })\frac{\mathit{\partial }Y\left({\text{u}}_i^{(2)}(x,\mathit{\xi })\right)}{\mathit{\partial }\mathit{\xi }}+{\mathit{\rho }}_2{f}_i^{(1)}(x,t-\mathit{\xi })\frac{\mathit{\partial }Y\left({\text{u}}_i^{\ast (2)}(x,\mathit{\xi })\right)}{\mathit{\partial }\mathit{\xi }}\right)\text{d}\mathit{\xi }\text{d}V+\underset{A}{\int }\underset{0}{\overset{t}{\int }}\left(c_0^{(1)}(x,t-\mathit{\xi })\frac{\mathit{\partial }Y\left({\mathit{\varphi }}^{(2)}(x,\mathit{\xi })\right)}{\mathit{\partial }\mathit{\xi }}+c_0^{\ast (1)}(x,t-\mathit{\xi })\frac{\mathit{\partial }Y\left({\mathit{\varphi }}^{\ast (2)}(x,\mathit{\xi })\right)}{\mathit{\partial }\mathit{\xi }}\right)\text{d}\mathit{\xi }\text{d}A+\underset{A}{\int }\underset{0}{\overset{t}{\int }}\left(b_0^{(1)}(x,t-\mathit{\xi })\frac{\mathit{\partial }Y\left({\mathit{\psi }}^{(2)}(x,\mathit{\xi })\right)}{\mathit{\partial }\mathit{\xi }}+b_0^{\ast (1)}(x,t-\mathit{\xi })\frac{\mathit{\partial }Y\left({\mathit{\psi }}^{\ast (2)}(x,\mathit{\xi })\right)}{\mathit{\partial }\mathit{\xi }}\right)\text{d}\mathit{\xi }\text{d}A+\frac{1}{T_0}\underset{A}{\int }\underset{0}{\overset{t}{\int }}\left(q^{(1)}(x,t-\mathit{\xi })\left({\mathit{\eta }}^{(2)}(x,\mathit{\xi })\right)\right)\text{d}\mathit{\xi }\text{d}A=S_{21}^{12}.$$(5.56)

Particular case:

In the absence of the magnetic effect and further if we put coupling coefficients of pore-fluid phase to zero with ρ₁₂ = ρ₂₂ = 0, then we obtain the similar results as obtained by Ieşan (1990).

6. Conclusion

In this paper, the governing equations for porous magneto-piezothermoelastic model are considered in the context of Biot’s theory of poroelasticity and Lord and Shulman’s generalised theory of thermoelasticity. The variational principle, reciprocity and uniqueness theorems are proved in the above proposed model. The deduced results in the above model are verified from the known results.

Additional information

Funding

Funding. The authors received no direct funding for this research.

Notes on contributors

Rajneesh Kumar

Dr Rajneesh Kumar has been involved in teaching for over three decades. He was a Professor of Mathematics with Kurukshetra University, Kurukshetra India. He is an enthusiastic researcher and has publications in several Indian and international journals. His area of expertise is Applied Mathematics.

Poonam Sharma

Poonam Sharma is a master of science in Mathematics. She is a PhD scholar in Applied Mathematics at Kurukshetra University. Her research domain is Thermodynamics and has special interest in Mechanics of solids.