Variational principle , uniqueness and reciprocity theorems in the theory of generalized thermoelastic diffusion material

The equations of generalized thermoelastic diffusion with four relaxation times are given. The variational principle is derived. Using Laplace transforms, a uniqueness theorem for these equations is proved. Also, a reciprocity theorem is obtained.


INTRODUCTION
In recent years increasing attention has been directed towards the generalized theory of thermoelasticity, which was found to give more realistic results than the coupled or uncoupled theories of thermoelasticity, especially when short-time effects or step temperature gradients are considered.The absence of any elasticity term in the heat conduction equation for uncoupled thermoelasticity appears to be unrealistic, since due to the mechanical loading of an elastic body, the strain so produced causes variation in the temperature field.Moreover, the parabolic type of heat conduction equation results in an infinite velocity of the thermal wave propagation, which also contradicts the actual physical phenomena.Introducing the strain-rate term in the uncoupled heat conduction equation, Biot 1 extended the analysis to incorporate coupled thermoelasticity.In this way, although the first shortcoming was over, there remained the parabolic-type partial differential equation of heat conduction, which leads to the paradox of the infinite velocity of the thermal wave.
To take care of the paradox in Biot's theory, Kaliski 2 proposed a possible physical model, involving a finite velocity of heat propagation as actually required in nature.Lord and Shulman 3 developed a theory in which they modified the Fourier's law of heat conduction with the introduction of a thermal relaxation time parameter.The theory of generalized thermoelasticity with two relaxation times was first introduced by Mu ¨ller. 4A more explicit version was then introduced by Green and Laws, 5 Green and Lindsay, 6 and independently by Suhubi. 7In this theory, the temperature rates are considered among the constitutive variables.This theory also predicts finite speeds of propagation as in Lord and Shulman's theory of generalized thermoelasticity with one relaxation time. 3It differs from the latter in that Fourier's law of heat conduction is not violated if the body under consideration has a center of symmetry.
Diffusion can be defined as the random walk of an assemble of particles from a region of high concentration to that of low concentration.Nowadays, there is a great deal of interest in the study of this phenomenon due to its application in geophysics and the electronic industry.In an integrated circuit fabrication, diffusion is used to introduce dopants in controlled amounts into the semiconductor substance.In particular, diffusion is used to form the base and emitter in bipolar transistors, integrated resistors, the source/drain regions in metal oxide semiconductor (MOS) transistors, and dope polysilicon gates in MOS transistors.In most applications, the concentration is calculated using what is known as Fick's law.This is a simple law which does not take into consideration the mutual interaction between the introduced substance and the medium into which it is introduced, or the effect of temperature on this interaction.Study of the phenomenon of diffusion is used to improve the conditions of oil extraction (seeking ways of more efficiently recovering oil from oil deposits).These days, oil companies are interested in the process of thermodiffusion for more efficient extraction of oil from oil deposits.
The phenomena of diffusion in two phase systems are well known, for example, a gas diffusing through a porous solid medium or a liquid diffusing through a solid.But until recently, the phenomena of diffusion accompanied with temperature change, that is, thermodiffusion in solids, particularly metals, was considered to be a phenomena that is independent of body deformation.It is, however, clear that such an assumption is not valid, since the processes of thermodiffusion could have a very considerable influence upon the deformation of the solid phase, and vice-versa.If the body is considered to be an ideal elastic, we may assume the validity of the linear generalized Hooke's law, as modified by the terms representing the changes in the temperature and concentration of the diffusing phase at any given point of the body.If we also assume that during the processes of thermodiffusion, no chemical reactions take place between the two phases, then the law of conservation of mass can be taken to be valid within any given volume element.By analogy with thermoelasticity, such a phenomena of deformation will be called thermodiffusive elasticity.Of course, if the diffusion phenomena are absent, the theory will be reduced to the theory of thermoelasticity.
In the case of the thermodiffusive elasticity of a Hookean body the deformation, which is a reversible process, occurs coupled with the irreversible processes of heat conduction and diffusion.The work in this area started with three papers by Podstrigach, 8 and Podstrigach and Pavlina, 9,10 who gave the relationships between the deformation, temperature and concentration based upon the thermodynamics of irreversible processes.Podstrigach and Shvechuk 11 presented a variational equation equivalent to the system of governing equations of a model which allows description of the interconnection between the deformation, heat and mater diffusion processes.Nowacki, 12 -16 in a series of papers, presented the theorem of virtual work, fundamental energy theorem, theorem of reciprocity of works, generalized Maxwell reciprocity relations and theorems of Somiglina and Maysel type.Fichera 17 proved the uniqueness, existence and the estimation of the solution in the dynamical problems of thermodiffusion in an elastic solid.Nowacki 18 studied dynamic problems of thermodiffusion in solids.Herrera and Billok 19 discussed the dual variational principles for diffusion equations.Naerlovic ´-Veljkovic ´20,21 obtained the constitutive equations for thermodiffusion in elastic, magnetically saturated, current conducting media.
Shvets and Dasyuk 22 proved the variational theorems of thermodiffusion in deformed solid bodies.Gawinecki and Sierpin ˜ski 23,24 proved the existence, uniqueness and regularity of the solution of the first initial-boundary-value problem for quasistatic and dynamic equations of thermodiffusion in solid bodies, respectively.Kubik 25 studied the correspondence between equations of thermodiffusion and theory of mixtures.He studied the balance equations of mass, momentum, energy and entropy.He compared these equations and equations describing conjugate thermodiffusion flows in solids.Wro ´bel 26,27 investigated variational theorems for the problems of coupled thermoviscoelastic diffusion with finite velocities of heat and mass propagation, and for the problems of thermodiffusion flows coupled with the stress field, respectively.Gawinecki et al. 28 proved a theorem about existence, uniqueness and regularity of the solution to an initial-boundary value problem for a nonlinear coupled parabolic system.They used an energy method, method of Sobolev spaces, semigroup theory and Banach fixed point theorem to prove the theorem.Gawinecki and Szymaniec 29 proved a theorem about global existence of the solution to the initial-value problem, for a nonlinear hyperbolic parabolic system of coupled partial differential equation of second order, describing the process of thermodiffusion in solid body.
Sherief, Hamza and Saleh 30 proved the uniqueness and reciprocity theorems for the equations of generalized thermoelastic diffusion problem, in isotropic media, on the basis of the variational principle equations, under restrictive assumptions on the elastic coefficients.Due to the inherit complexity of the derivation of the variational principle equations, Aouadi 31 proved the theorem as given by Sherief, Hamza and Saleh 30 in the Laplace transform domain, under the assumption that the functions of the problem are continuous and the inverse Laplace transform of each is also unique.Aouadi 32 derived the uniqueness and reciprocity theorems in anisotropic media, under the restriction that the elastic, thermal conductivity and diffusion tensors are positive and definite.Aouadi 33 established the field equations of the linear theory of micropolar thermoelastic diffusion bodies.Aouadi 34 developed a theory of thermoelastic diffusion materials with voids and derived the uniqueness, reciprocity, continuous dependence and existence theorems.Kuang 35 derived the basic equations and proved variational principles for generalized thermodiffusion theory in pyroelectricity.Kumar, Kothari and Mukhopadhyay 36 established a convolutional type variational principle and a reciprocity theorem for the linear theory of generalized thermoelastic diffusion for isotropic elastic solids.Ezzat and Fayik 37 developed a new theory of thermodiffusion in elastic solids using the methodology of fractional calculus.
Kumar and Kansal 38 derived the constitutive relations and field equations for anisotropic generalized thermoelastic diffusion and studied the propagation of Lamb waves in transversely isotropic thermoelastic diffusive plate.Kumar and Kansal 39 discussed three-dimensional free vibration analysis of a transversely isotropic thermoelastic diffusive cylindrical panel.Kumar and Kansal 40 studied the propagation of plane waves and a fundamental solution in the generalized theories of thermoelastic diffusion.Kumar and Kansal 41 discussed the reflection and refraction phenomenon of plane waves at the interface of an elastic solid half-space and a thermoelastic diffusive solid half-space.Kumar and Kansal 42 studied the propagation of plane waves in anisotropic thermoelastic diffusive medium.

BASIC EQUATIONS
The law of conservation of energy for an arbitrary material volume V bounded by a surface A, at time t, can be written as where U is the internal energy per unit mass, r is the density, q i are the components of heat flux vector q, F i are components of the external forces per unit mass, u i are the components of the displacement vector u, s ij ð¼ s ji Þ are the components of the stress tensor, n j are the components of the outward unit normal vector n to the surface A.
Using the divergence theorem and the equations of motion the equation ( 1) can be written as where e ij ¼ 1 2 ðu i;j þ u j;i Þ are the components of the strain tensor.In what follows, we shall restrict our attention to the linear theory of homogeneous body.The balance of entropy can be written as where F is a strictly positive function, S, P, are entropy and chemical potential per unit mass, respectively.h i are the components of mass diffusion flux vector h.The right-hand side of the above equation is the entropy source In view of ( 5), we can write equation ( 4) in the form of an inequality called the Clausius Duhem inequality The equation of conservation of mass is where C is the concentration of the diffusion material in the elastic body.Using equations ( 3) and ( 7) in equation ( 6), we obtain If we introduce the Helmholtz free energy function c, defined by then from equation ( 8), we obtain We assume that where b is a constant small enough for squares and highest powers to be neglected and u 1 i ; T 1 ; C 1 are independent of b and T is the temperature measured from the constant absolute temperature T 0 of the body in its reference configuration.
We denote We consider thermoelastic diffusion solids having the constitutive equations c ¼ ĉðIÞ; s ij ¼ ŝij ðIÞ; q i ¼ q i ðIÞ; S ¼ ŜðIÞ; P ¼ PðIÞ; ð11Þ the constitutive functionals being of class C 2 and consistent with the assumption of the linear theory.We assume that It follows from equation ( 10) that where Thus, we obtain and the inequality ( 16) reduces to Using equation ( 9) in the equation (3), we obtain In view of equations ( 13), ( 17)-( 19), the energy equation ( 22) can be written in the form We assume that Assuming that the initial body is free from stress, heat flux and mass flux, in the context of linear theory, we have and From equations ( 18)-( 20) and ( 25), the expressions for s ij , S, P and q i are obtained as The entropy inequality (21) implies the following restrictions on the constitutive coefficients where Thus, we obtain the following constitutive equations In the context of the linear theory, the energy equation ( 23) reduces to If the material has centre of symmetry, then the constitutive equations ( 33)-( 36) take the form where we introduce Analogous to equation ( 41) for the heat flux vector, we assume a similar equation for the mass flux vector In the following sections, we will use the chemical potential as a state variable instead of the concentration.The equations ( 38)-( 40) are rewritten as: where The equation (41) with the aid of equations ( 37) and (45) yields Making use of equations ( 7) and (46) in the equation ( 43), we obtain

VARIATIONAL PRINCIPLE
The principle of virtual work with variation of displacements for the elastic deformable body is written as On the left hand side, we have the virtual work of body forces F i , internal forces r€ u i , surface forces h i ¼ s ji n j , whereas on the right hand side, we have the virtual work of internal forces.
Using the symmetry of the stress tensor and the definition of the strain tensor, the equation ( 50) can be written in the alternative form as ð The equation (51) with the aid of equation (44) yields where d ijlm e ij e lm dV: The equation ( 52) would be complete for the uncoupled problem of diffusion where the temperature T and the concentration C are known functions.When we take into account the coupling of the deformation field with the temperature and concentration, there arises the necessity of considering two additional relations characterizing the phenomena of thermal conductivity and of diffusion.
We define a vector J(Biot 1 ) connected with the entropy through the relation Combining equations ( 37), ( 41), ( 45) and (53), we obtain where L ij is the resistivity matrix, the inverse of the thermal conductivity tensor K ij .
Multiplying both sides of the equation ( 54) by dJ j and integrating over the region of the body, we get Applying the divergence theorem defined by, in the equation (57), we obtain Equation (56) with the aid of equation (59) yields Substituting the value of J i;i from equation (55) in the equation (60) gives the second variational equation In order to obtain the last of the variational equations, we now introduce the vector function N defined as follows Combining equations ( 7), ( 43), ( 46) and (62), we obtain where a ij is the inverse of the diffusion tensor a * ij .Multiplying equation ( 63) by dN j and integrating over the region of the body, we obtain We know that Substituting the value of Ð V ðPdN j Þ ;j dV from equation (67) in the equation (66) gives ð Making use of equation (68) in the equation (65) yields Using equation (64) in the equation (69), we obtain the third variational equation Eliminating integrals Ð V s ij T de ij dV and 52), ( 61) and (70), we obtain the final variational principle in the following form On the right-hand side of equation ( 71), we find all the causes, the mass forces, inertial forces, the surface forces, the heating and the chemical potential on the surface A bounding the body.
Therefore, the Laplace transforms of all the difference functions (75) are zeros, and, since they are continuous functions, the inverse Laplace transform of each is unique.This proves the uniqueness of the solution of the system of equations ( 2), (44), ( 48) and (49).

RECIPROCITY THEOREM
We shall consider a homogeneous anisotropic generalized thermoelastic diffusion body occupying the region V and bounded by the surface A. We assume that the stresses s ij and the strains e ij are continuous together with their first derivatives, whereas the displacements u i , temperature T, concentration C and the chemical potential P are continuous and have continuous derivatives up to the second order, for x 1 V þ A; t .0.
The components of surface traction, the normal component of heat flux and the normal component of chemical flux at regular points of ›V, are given by respectively.
To the system of field equation, we must adjoin boundary conditions and initial conditions.We consider the following boundary conditions: u i ðx; tÞ ¼ U i ðx; tÞ; T ðx; tÞ ¼ hðx; tÞ; Pðx; tÞ ¼ 6ðx; tÞ; ð115Þ for all x 1 A, t .0. and the homogeneous initial conditions We derive the dynamic reciprocity relationship for a generalized thermoelastic diffusion bounded body V, which satisfies equations ( 2), (44), ( 48) and (49), the boundary conditions (115) and the homogeneous initial conditions (116), and are subjected to the action of body forces F i ðx; tÞ, surface traction h i ðx; tÞ, the heat flux qðx; tÞ and the chemical flux pðx; tÞ.
Applying the Laplace transform defined by equation (82) on equations ( 2), (44), ( 48) and (49) and omitting the bars for simplicity, we obtain We now consider two problems where applied body forces, chemical 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 u ð1Þ i ; e ð1Þ ij ; s ð1Þ ij ; T ð1Þ ; P ð1Þ ; . . ., for the first problem and u ð2Þ i ; e ð2Þ ij ; s ð2Þ ij ; T ð2Þ ; P ð2Þ ; . . . .for the second problem.Each set of variables satisfies the equations ( 2), (44), ( 48) and (49).
Using the assumption s ij ¼ s ji , we obtain ð Using the divergence theorem in the first term of the right hand side of equation ( 121) yields Equation (122) with the use of equations ( 114) and (117) yields From equations (124) and (125), we get the first part of the reciprocity theorem which contains the mechanical causes of motion F i and h i .