Acta Mech 217, 287–296 (2011) DOI 10.1007/s00707-010-0401-y

Roushan Kumar · Shweta Kothari · Santwana Mukhopadhyay

Some theorems on generalized thermoelastic diffusion

Received: 8 March 2010 / Revised: 24 September 2010 / Published online: 14 November 2010 © Springer-Verlag 2010

Abstract The objective of the present work is to establish a convolutional type (Gurtin in Arch. Rat. Mech. Anal. 16:34–50, 1964) variational theorem and a reciprocity theorem for the linear theory of generalized thermoelastic diffusion for homogeneous and isotropic elastic solids.

1 Introduction Diffusion can be regarded as the phenomenon of random walk, of an ensemble of particles, from regions of high concentration to regions of low concentration. This phenomenon has aroused much interest in recent years because of its several applications in geophysics and other industrial applications. Particularly, diffusion is used in integrated circuit fabrication to introduce “dopants” in controlled amounts into the semiconductor substrate and to form the base and emitter in bipolar transistors. It is used to form integrated resistors, form the source/drain regions in MOS transistors and dope poly-silicon gates in MOS transistors. Study of the phenomenon of diffusion has also application to improve the conditions of oil extractions (seeking ways of more efficiently recovering oil from oil deposits). In recent years, the process of thermoelastic diffusion is used for more efficient extraction of oil from oil deposits. In most of these applications, Fick’s law is used for the calculation of the concentration. However, this law does not take into consideration the mutual interactions between the introduced substance and the medium into which it is introduced or the effect of the temperature on this interaction. Nowacki [1–4] developed the theory of thermoelastic diffusion for the first time. In this theory, the coupled thermoelastic model is used in which infinite speed of propagation is involved. Gawinecki et al. [5] proved a theorem on existence, uniqueness and regularity of the solution for a nonlinear parabolic thermoelastic diffusion problem. A theorem about global existence of the solution for the same problem was established by Gawinecki and Szymaniec [6]. Recently, Sherief et al. [7] developed the generalized theory of thermoelastic diffusion by introducing one thermal relaxation time parameter and one diffusion relaxation parameter, which allows the finite speeds of propagation of waves. The variational principle in thermoelasticity is an alternative method for determining the state of dynamics of a thermoelastic system, by identifying it as an extremum of a function or functional. By the reciprocity theorem, it is possible to deduce the various methods of integrating the elasticity equations by means of Green’s function. Variational and reciprocal principles are not only of mathematical interest but also of practical utility [8–10]. An exhaustive treatment of the variational principles in thermoelasticity is elaborated in the books R. Kumar · S. Kothari · S. Mukhopadhyay (B) Department of Applied Mathematics, Institute of Technology, Banaras Hindu University, 221005 Varanasi, India E-mail: [email protected]; [email protected] R. Kumar Birla Institute of Technology, Mesra, Deoghar Campus, Jasidih 714142, India

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by Carlson [11], Gurtin [12] and Lebon [13]. In classical elasticity, the variational principle was given by Ignaczak [14] and Gurtin [15,16]. They established an alternative characterization of the solution to the boundary-initial value problem in which the initial conditions are incorporated into the field equations. The derivation of the thermoelasticity variational theorems was introduced for the first time by Biot [17,18]. In his two papers, he presented these theorems and explained their applications with the help of several examples. A generalized version of the variational principle due to Biot [18] was given by Hermann [19]. Subsequently, the most generalized version of variational principle of coupled thermoelasticity was reported by Ben-Amoz [20]. However, the first variational principle of Gurtin type in thermoelastodynamics was established by Iesan [21] and later on by Nickell and Sackman [22]. The uniqueness theorem and variational principles for coupled linear thermoelasticity theory have also been reported by many other authors. The uniqueness theorem had been proved by Weiner [23] for the coupled thermoelastic problem. The variational theorem for two-temperature thermoelasticity in case of a classical, homogeneous and isotropic solid has been provided by Iesan [24]. The uniqueness theorem, variational principle and reciprocity theorem on different generalized theories of thermoelasticity have been reported by Ignaczak [25], Sherief and Dhaliwal [26], Chandrasekharaiah and Srikantaiah [27], Wang et al. [28], Chandrasekharaiah [29] and Kumar et al. [30]. Sherief et al. [7] established the uniqueness theorem, a variational principle of Biot type and a reciprocity theorem for the generalized theory of thermoelastic diffusion. Aouadi [31] discussed the uniqueness and reciprocity theorems in the theory of generalized thermoelastic diffusion. Subsequently, Aouadi [32] also derived the uniqueness and reciprocity theorems for the generalized thermoelastic diffusion problem in anisotropic media. In this paper, a convolutional type variational principle and a reciprocity theorem are established for the linear theory of generalized thermoelastic diffusion for isotropic elastic solids. The work is organized as follows: Sect. 2 is devoted to basic equations and constitutive relations for generalized thermoelastic diffusive materials, supposed to be isotropic and homogeneous. Necessary assumptions on the constitutive constants and initial and boundary conditions are made. In Sect. 3, an alternative formulation of the mixed boundary-initial value problem is derived to incorporate the initial conditions into the field equations and the functionals arising in the variational formulation. In Sect. 4, a convolutional type (Gurtin [15]) variational principle is established, and finally by following Iesan [24] a reciprocal principle is also derived. This should permit to model the behavior of several kinds of thermoelastic material under generalized thermoelastic diffusion.

2 Basic governing equation We employ a rectangular co-ordinate system x k and usual indicial notations throughout our paper. In threedimensional Euclidean space, let V˜ denote a regular region of space, whose boundary is ∂ V , enclosing a thermoelastic material. The interior of V˜ is V , and n i are the components of the outward unit normal to ∂ V , and ∂ Vi (i = 1, 2, 3, 4, 5, 6) denote subsets of ∂ V such that ∂ V1 ∪ ∂ V2 = ∂ V3 ∪ ∂ V4 = ∂ V5 ∪ ∂ V6 = ∂ V, ∂ V1 ∩ ∂ V2 = ∂ V3 ∩ ∂ V4 = ∂ V5 ∩ ∂ V6 = ∅.

We consider the motion relative to an undistorted stress-free reference state. The basic field equations and constitutive equations in case of a homogenous isotropic material under the linear theory of generalized thermoelastic diffusion in the presence of heat source, body force and mass diffusing source are therefore as follows [7,8]: Equations of motion: σi j, j + Fi = ρ u¨ i ;

(1)

Q − ρT0 S˙ = qi,i ;

(2)

σ − C˙ = ηi,i ;

(3)

Equation of energy:

Equation of conservation of mass:

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The constitutive equations: σi j = 2μei j + (λ0 ekk − γ1 θ − γ2 P) δi j , ρ S = cθ + γ1 ekk + d P, C = γ2 ekk + n P + dθ, qi + τ0 q˙i = −kθ,i , ηi + τ η˙ i = −D P,i ;

(4) (5) (6) (7) (8)

  2ei j = u i, j + u j,i ≡ 2u (i, j) on V × [0, ∞[ ,

(9)

Geometrical equations:

β2

a a 1 E where γ1 = β1 + βb2 a , γ2 = βb2 , λ0 = λ − b2 , c = ρc T0 + b , d = b , n = b . In the above equations, u i and qi are the components of displacement vector and heat flux vector, respectively. ei j and σi j are the components of the strain tensor and stress tensor, respectively. Fi are the components of the body force vector. T is the absolute temperature   and θ = T − T0 , where T0 is the temperature of the medium   in its natural state assumed to be such that  Tθ0   1. S is the entropy per unit mass, C is the concentration, P is the chemical potential per unit mass, and ηi is the component of the diffusing mass vector. ρ is the constant mass density, k is the thermal conductivity of the material, D is the diffusion coefficient and c E is the specific heat at constant strain. The τ0 and τ are thermal relaxation time and diffusion relaxation time, respectively. β1 = (3λ + 2μ) αt , β2 = (3λ + 2μ) αc where λ, μ are the Lame’s elastic constants, αt and αc are the coefficient of linear thermal expansion and coefficient of linear diffusion expansion, respectively. σ is the intensity of mass diffusing sources, and Q is the heat source per unit volume. The constants a and b are the measures of thermodiffusion effects and diffusive effects, respectively. A superposed dot denotes the partial differentiation with respect to time, and subscripts preceded by a comma refer to partial differentiation with respect to the corresponding Cartesian coordinates. We assume that the constitutive coefficients satisfy the inequalities 2

μ > 0, 3λ0 + 2μ > 0, ρ > 0, k > 0, τ0 > 0, τ > 0.D > 0, n > 0, c > 0, T0 > 0.

(10)

To the system of field equations, we adjoin the initial conditions on V˜ , u i (x, 0) = di (x), u˙ i (x, 0) = vi (x), S(x, 0) = S0 (x), C(x, 0) = C0 (x), qi (x, 0) = q0i , ηi (x, 0) = η0i

(11.1–6)

and the boundary conditions σi = σi j n j = σ˜ i on ∂ V2 × [0, ∞[ , u i = u˜ i on ∂ V1 × [0, ∞[ , q = qi n i = q˜ on ∂ V3 × [0, ∞[ , θ = θ˜ on ∂ V4 × [0, ∞[ , P = P˜ on ∂ V6 × [0, ∞[ . η = ηi n i = η˜ on ∂ V5 × [0, ∞[ ,

(12.1–6)

Here, di , vi , S0 , C0 , q0i and η0i are the prescribed initial displacements, velocity, entropy, concentration, ˜ θ˜ , η˜ and P˜ are the surface displacements, heat flux and flow of diffusing mass vector, respectively. u˜ i , σ˜ i , q, tractions, normal heat flow, temperature, normal flow of diffusing mass and chemical potential, respectively. We used the notation x = (x1 , x2 , x3 ). The following smoothness requirements and other regularity assumptions on the ascribable functions are introduced as hypotheses on data: (I) (II) (III) (IV) (V)

di is continuously differentiable on V˜ ; vi , S0 , C0 , q0i and η0i are continuous on V˜ ; Fi , Q and σ are continuously differentiable on V˜ × [0, ∞[; u˜ i , θ˜ and P˜ are continuous on ∂ V˜1 × [0, ∞[ , ∂ V˜4 × [0, ∞[ and ∂ V˜6 × [0, ∞[, respectively; q, ˜ σ˜ i and η˜ are piecewise continuous on ∂ V˜3 × [0, ∞[ , ∂ V˜2 × [0, ∞[ and ∂ V˜5 × [0, ∞[, respectively.

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We consider f as of position and time defined on V˜ × [0, ∞[.  an function  m ∂ f If ∂ xi ∂ x j∂.......∂ xk ∂t n exists and is continuous on V˜ × [0, ∞[, for m = 0, 1, . . . , M; n = 0, 1, . . . , N , and m + n ≤ max {M, N }, then we say that f ∈ C M,N .   We introduce the notation of admissible state, denoted as R = u i , qi , ei j , S, σi j , θ, ηi , C, P , which is an ordered array of functions u i , qi , ei j , S, σi j , θ, ηi , C, P defined on V˜ × [0, ∞[ with the properties: (a) u i ∈ C 1,2 , σi j ∈ C 1,0 , ei j ∈ C 0,0 , qi ∈ C 1,1 , ηi ∈ C 1,1 , S ∈ C 0,1 , C ∈ C 0,1 , θ ∈ C 1,0 , P ∈ C 1,0 , (b) ei j = e ji , σi j = σ ji on V˜ × [0, ∞[. We define the addition of admissible states and multiplication of a state by a scalar λ through   R + R = u i + u i , qi + qi , . . . , θ +θ , ηi +ηi , C + C , P + P , λR = {λu i , λqi , . . . , λθ, ληi , λC, λP}. Then we can say that the set of all admissible states is a linear space. By a solution of the mixed problem, we mean an admissible state, which satisfies the field Eqs. (1)–(9), the initial conditions (11) and boundary conditions (12). Let f 1 and g1 be functions of space and time defined on V˜ × [0, ∞[ such that both are continuous on [0, ∞[ for each x ∈ V . Then the convolution f 1 ∗ g1 of f 1 and g1 is defined by t [ f 1 ∗ g1 ] (x, t) =

f 1 (x, t − τ )g1 (x, τ )dτ

(x, t) ∈ V¯ × [0, ∞[,

0

and convolution has the following properties: (1) (2) (3) (4)

g1 ∗ f 1 = f 1 ∗ g1 , g1 ∗ ( f 1 ∗ h 1 ) = (g1 ∗ f 1 ) ∗ h 1 = g1 ∗ f 1 ∗ h 1 , g1 ∗ ( f 1 + h 1 ) = (g1 ∗ f 1 ) + (g1 ∗ h 1 ), f 1 ∗ g1 = 0 ⇒ f 1 = 0 or g1 = 0.

(13) (14) (15) (16)

3 Alternative formulation Following Nickell and Sackman [22], we will now formulate the mixed problem of the present consideration in an alternative way by incorporating the initial conditions into the field equations and into the governing functionals which arise in the variational formulation. For this, we apply Laplace transformation to Eqs. (1), (2) and (3). Therefore, using the initial conditions (11) we get σ¯ i j, j + F¯i + ρpdi + ρvi = ρp 2 u¯ i , ¯ ρT0 p S¯ = ρT0 S0 − q¯i,i + Q,

(18)

σ¯ − pC¯ + C0 = η¯ i,i

(19)

(17)

where p is the Laplace transform parameter. Laplace transform of Eqs. (7) and (8) yields (1 + τ0 p) q¯i = −k θ¯ ,i +τ0 q0i , ¯ i +τ η0i . (1 + τ p) η¯ i = −D P,

(20) (21)

¯ C, ¯ q¯i and η¯ i and taking inverse Laplace transforms we get Solving (17), (18), (19), (20) and (21) for u¯ i , S, ρu i = g ∗ σi j, j + f i , qi,i T0 S = −l ∗ + W, ρ C = −l ∗ ηi,i + Y, qi = −M1 ∗ (kθ,i ) + L 1i , ηi = −M2 ∗ (D P,i ) + L 2i

(22) (23) (24) (25) (26)

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where we denote g (t) = t, l (t) = 1, M1 (t) = L 1i = q0i e

− τt

0

1 − τt 1 t e 0 , M2 (t) = e− τ , τ0 τ t

, L 2i = η0i e− τ , 0 ≤ t < ∞,

f i = g ∗ Fi + ρ (tvi + di ), Q + T0 S0 , W =l∗ ρ Y = C0 + l ∗ σ.

(27.1) (27.2)

(28) (29) (30)

Therefore as in [15,22,24] we obtain Theorem 1 The functions u i , qi , S, σi j , C, P, ηi satisfy the Eqs. (1), (2), (3) and the initial conditions (11) if and only if g ∗ σi j, j + f i = ρu i , qi,i T0 S = −l ∗ + W, ρ C = −l ∗ ηi,i + Y.

(31) (32) (33)

With the help of this theorem, an alternative formulation of the mixed problem is given by the following theorem:   Theorem 2 Let = u i , qi , ei j , S, σi j , θ, ηi , C, P be an admissible state. Then is a solution of the mixed problem if and only if it satisfies the Eqs. (31)–(33), (25), (26), (4)–(6), (9) and the boundary conditions (12). 4 Variational theorem In this Section, we establish a variational principle to generalized thermoelastic diffusion [7] on the basis of the alternative formulation and the theorems provided in the previous Section. This variational principle is valid in the space of functions which possess the Laplace transforms (see Gurtin [12]). The term functional identifies a real valued function whose domain is a subset of a linear space. Let X be a linear space and Y be the subset of X . We consider a functional defined on Y . Let y, y ∈ X, y + λ y ∈ Y for all real λ , and we formally define δ y {y} =

  d  .

y + λ y  dλ λ =0

(34)

(35)

The variation of {.} is said to be zero at y over Y and is written as δ {y} = 0, over Y

(36)

if and only if δ y {y} exists and is equal to zero for all y consistent with (34).   Theorem 3 Let E be the set of all admissible states. Let  = u i , qi , ei j , S, σi j , θ, ηi , C, P ∈ E, and for each t ∈ [0, ∞) we define the functional t {} on E through

λ0 1 err ∗ ess + μei j ∗ ei j +

t {} = g ∗ M1 ∗ M2 ∗ (ρ S − γ1 ekk − d P) ∗ (ρ S − γ1 ekk − d P) 2 2c V  1 + (C − γ2 ekk − dθ) ∗ (C − γ2 ekk − dθ) d x − g ∗ M1 ∗ M2 ∗ σi j ∗ ei j 2n V

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 1 +ρ S ∗ θ + C ∗ P − dθ ∗ P d x + M1 ∗ M2 ∗ ρu i ∗ u i d x − M1 ∗ M2 ∗ g ∗ σi j, j + f i ∗ u i d x 2 V V ρ 1 1 + g ∗ M1 ∗ M2 ∗ W ∗ θ d x + g ∗ l ∗ M 2 ∗ qi ∗ qi d x + g ∗ l ∗ M1 ∗ ηi ∗ ηi d x T0 2kT0 2D V V V 1 1 + g ∗ l ∗ M1 ∗ M2 ∗ qi ∗ kθ,i d x + g ∗ l ∗ M1 ∗ M2 ∗ ηi ∗ D P,i d x kT0 D V V 1 1 − g ∗ l ∗ M2 ∗ L 1i ∗ qi d x − g ∗ l ∗ M1 ∗ L 2i ∗ ηi d x + g ∗ M1 ∗ M2 ∗ Y ∗ Pd x kT0 D V V V g ∗ M1 ∗ M2 ∗ σi ∗ u˜ i da + g ∗ M1 ∗ M2 ∗ (σi − σ˜ i ) ∗ u i da + 

∂ V1

∂ V2

  1 1 − g ∗ l ∗ M1 ∗ M2 ∗ θ ∗ qda ˜ − g ∗ l ∗ M1 ∗ M2 ∗ θ − θ˜ ∗ qda T0 T0 ∂ V3 ∂ V4   − g ∗ l ∗ M1 ∗ M2 ∗ P ∗ ηda ˜ − g ∗ l ∗ M1 ∗ M2 ∗ P − P˜ ∗ ηda. ∂ V5

(37)

∂ V6

Then δ t {} = 0, over E, t ∈ [0, ∞[ ,

(38)

if and only if  is the solution of the mixed boundary-initial value problem. 

Proof : Let  = u i , qi , ei j , S , σi j , θ , ηi , C , P ∈ E, which implies that  + λ  ∈ E for every real λ . Then Eq. (36), together with (13), (14), (15), (35), the properties of admissible states as mentioned earlier and the divergence theorem implies

  γ1 γ2 g ∗ M1 ∗ M2 ∗ λ0 err δi j + 2μei j − (ρ S − γ1 ekk −d P) δi j − (C −γ2 ekk −dθ ) δi j −σi j ∗ ei j c n V



 1 1 +ρ g ∗ M1 ∗ M2 ∗ (ρ S − γ1 ekk − d P) − θ ∗ S d x + g ∗ M1 ∗ M2 ∗ (C − γ2 ekk −dθ )− P ∗ C d x c n V V



 1 1 −d g ∗ M1 ∗ M2 ∗ (C − γ2 ekk − dθ ) − P ∗ θ d x − d g ∗ M1 ∗ M2 ∗ (ρ S −γ1 ekk −d P)−θ ∗ P d x n c V V     1 g ∗ M1 ∗ M2 ∗ ρW − ρT0 S − l ∗ qi,i ∗ θ d x− M1 ∗ M2 ∗ g ∗ σi j, j + f i − ρu i ∗ u i d x + T0 V V   1

g ∗ l ∗ M2 ∗ [qi + M1 ∗ (kθ,i ) − L 1i ] ∗ qi d x + g ∗ M1 ∗ M2 ∗ u (i, j) − ei j ∗ σi j d x+ kT0 V V  1

+ g ∗ M1 ∗ M2 ∗ Y − C − l ∗ ηi,i ∗ P d x+ g ∗ l ∗ M1 ∗ [ηi + M2 ∗ (D P,i ) − L 2i ] ∗ ηi d x D V V g ∗ M1 ∗ M2 ∗ (u˜ i − u i ) ∗ σi da+ g ∗ M1 ∗ M2 ∗ (σi − σ˜ i ) ∗ u i da + δ t {} =

∂ V1

∂ V2

  1 1 g ∗ l ∗ M1 ∗ M2 ∗ (q − q) ˜ ∗ θ da − g ∗ l ∗ M1 ∗ M2 ∗ θ − θ˜ ∗ q da + T0 T0 ∂ V3 ∂ V4  

+ g ∗ l ∗ M1 ∗ M2 ∗ (η − η) ˜ ∗ P da − g ∗ l ∗ M1 ∗ M2 ∗ P − P˜ ∗ η da, (0 ≤ t < ∞) ∂ V5

where q = qi n i , σi = σi j n i , η = ηi n i .

(39)

∂ V6

 

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First, we suppose that  is a solution of the mixed boundary value problem. Then according to the Theorem-2, Eq. (39) and the choice of  we find that (38) holds. The “if” part of the theorem is therefore proved. Conversely, to prove the “only if” part, let (38) hold. Then we need to prove that  is a solution of the mixed problem. Since (38) holds, we get δ t {} = 0, t ∈ [0, ∞[ , for every  ∈ E.

(40)

In what follows we use three Lemmas (1)–(3) given in Gurtin [12,15] and Nickell and Sackman [22]  and the Corollary 2.4 provided in Nickell and Sackman [22]. We choose  = u i , 0, 0, 0, 0, 0, 0, 0, 0 and let u i

together with all of its space derivatives vanish on ∂ V × [0, ∞[. Therefore from Eqs. (39) and (40), we obtain   M1 ∗ M2 ∗ g ∗ σi j, j + f i − ρu i ∗ u i d x = 0, for t ∈ [0, ∞[ . V

Hence, Lemma-1 [15,22], Eqs. (16) and (27.1) imply that Eq. (31) holds.  Now let  = u i , 0, 0, 0, 0, 0, 0, 0, 0 , but requiring this time u i vanish on ∂ V1 × [0, ∞[. Then (39), (40), (31) and Lemma-2 [15,22] yield g ∗ (σi − σ˜ i ) = 0 on ∂ V2 × [0, ∞[ ,

  and this result, because of (16) and (27.1) implies (12.2). Next, we select  = 0, qi , 0, 0, 0, 0, 0, 0, 0 and let qi together with all of its space derivatives vanish on ∂ V × [0, ∞[. Then (39), (40) and Lemma-1 [15,22] imply g ∗ l ∗ M2 ∗ [qi + M1 ∗ (kθ,i ) − L 1i ] = 0, and this result together with (16) and (27.1) imply that (25) holds. Considering the same choice for  , but requiring now qi vanishing on ∂ V3 × [0, ∞[ we conclude that (39), (40), (25), Corollary-2.4 [22], (16) and (27.1) imply the validity of  the boundary condition  (12.4). Next, we select  = 0, 0, 0, 0, 0, 0, ηi , 0, 0 and let ηi together with all of its space derivatives vanish on ∂ V × [0, ∞[. Then (39), (40) and Lemma-1[15,22] yield g ∗ l ∗ M1 ∗ [ηi + M2 ∗ (D P,i ) − L 2i ] = 0. The above result together with (16) and (27.1) verifies that (26) holds. Considering the same choice for  , but requiring now ηi vanish on ∂ V5 × [0, ∞[,  (39), (40), (26), Corollary-2.4 [22] and (16) imply (12.6). Next, let  = 0, 0, 0, S , 0, 0, 0, 0, 0 , where S together with all its space derivatives vanishes on ∂ V × [0, ∞[ .Then (39), (40), Lemma-1[15,22] and (16) imply the validity of Eq. (5).  Now let  = 0, 0, 0, 0, 0, 0, 0, C , 0 , where C together with all its space derivatives vanishes on ∂ V × [0, ∞[ . Then (39), (40), Lemma-1[15,22] and (16) imply that (6) holds. 



Now, let  = 0, 0, ei j , 0, 0, 0, 0, 0, 0 where ei j and all its space derivatives vanish on ∂ V × and (16) therefore imply (4). [0, ∞[ . Equations (39), (40), (5), (6), Lemma-1[15,22]  Now, select  = 0, 0, 0, 0, σi j , 0, 0, 0, 0 , where σi j and all its space derivatives vanish on ∂ V ×[0, ∞[ .

Then (39), (40), Lemma-1 [15,22] and (16) imply (9). Considering the same choice of  but supposing now that σi j and its space derivatives vanish on ∂ V2 × [0, ∞[, Eqs. (39), (40), (9) and Lemma-3 [15,22] verify that   (12.1) holds. Next, let  = 0, 0, 0, 0, 0, θ , 0, 0, 0 where θ together with all its space derivatives vanishes on ∂ V × [0, ∞[ . Then (39), (40), (6) and Lemma-1[15,22] imply (32). Considering the same choice of  but requiring now that θ vanishes on ∂ V4 × [0, ∞[, we obtain from (39), (40), (32), (6) and Lemma-2 [15,22] that the boundary condition (12.3) is satisfied.   Now let  = 0, 0, 0, 0, 0, 0, 0, 0, P where P together with all its space derivatives vanishes on ∂ V × [0, ∞[ .Then (39), (40), (5) and Lemma-1 [15,22] imply (33). Considering the same choice of  but requiring now that P vanishes on ∂ V6 ×[0, ∞[, we obtain from Eqs. (39), (40), (33), (5) and Lemma-2 [15,22] that the boundary condition (12.5) is satisfied. Therefore,  satisfies (4), (5), (6), (9), (25), (26), (31)–(33) and boundary conditions (12.1–6), and hence we conclude from our Theorem-2 that  is the solution of the mixed problem. This completes the proof of our theorem.

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5 Reciprocity theorem In this Section, we follow the method given in Iesan [24] to obtain the reciprocity theorem without using the Laplace transform. We consider the body subjected to two different systems of thermoelastic loadings   (β) (β) (β) (β) (β) (β) (β) (β) (β) L (β) = Fi , Q (β) , C (β) , u˜ i , q˜ (β) , σ˜ i , θ˜ (β) , P˜ (β) , η˜ (β) , di , vi , S0 , C0 , q0i , η0i , β = 1, 2 (41) and the two corresponding thermoelastic configurations   (β) I (β) = u i , θ (β) , P (β) , β = 1, 2.

(42)

The following reciprocity theorem states the relation between these two sets of thermoelastic loading and thermoelastic configurations: Theorem 4 If a thermoelastic solid is subjected to two different systems of thermoelastic loadings L (β) , (β = 1, 2), then there is the following reciprocity relation between the corresponding thermoelastic configurations I (β) , (β = 1, 2):



ρ 1 (1) (2) (1) (2) f i ∗ u i − g ∗ W (1) ∗ θ (2) − g ∗ Y (1) ∗ P (2) d x + g ∗ σ˜ i ∗ u i + l ∗ q˜ (1) ∗ θ (2) T0 T0 V ∂V

  1 (1) (2) (1) (2) +l ∗ η˜ (1) ∗ P (2) da − g ∗ l ∗ L 1i ∗ θ,i +L 2i ∗ P,i dx T0 V



ρ 1 = f i(2) ∗ u i(1) − g ∗ W (2) ∗ θ (1) − g ∗ Y (2) ∗ P (1) d x + g ∗ σ˜ i(2) ∗ u i(1) + l ∗ q˜ (2) ∗ θ (1) T0 T0 V ∂V

  1 (2) (1) (2) (1) +l ∗ η˜ (2) ∗ P (1) da − g ∗ l ∗ L 1i ∗ θ,i +L 2i ∗ P,i dx (43) T0 V

(α)

(α)

(α)

where L 1i , L 2i , f i

, W (α) , Y (α) are given by Eqs. (27.2), (28)–(30) and (41).

Proof : From Eq. (4), we have   (1) (1) (1) σi j = 2μei j + λ0 ekk − γ1 θ (1) − γ2 P (1) δi j ,   (2) (2) (2) σi j = 2μei j + λ0 ekk − γ1 θ (2) − γ2 P (2) δi j . (2)

(44) (45)

(1)

Taking convolutions of Eq. (44) with ei j and of Eq. (45) with ei j and subtracting the results we get     (1) (2) (2) (1) σi j + γ1 θ (1) δi j + γ2 P (1) δi j ∗ ei j = σi j + γ1 θ (2) δi j + γ2 P (2) δi j ∗ ei j .

(46)

From Eq. (5), we have γ1 (1) e − ρ rr γ1 (2) − err − ρ

S (1) − S (2)

d (1) c (1) θ , P = ρ ρT0 d (2) c (2) θ . P = ρ ρT0

Taking convolution of Eq. (47) with θ (2) and of (48) with θ (1) and subtracting the results, we get

 

γ1 (1) d (1) γ1 (2) d (2) − P − P ∗ θ (2) = S (2) − err ∗ θ (1) . S (1) − err ρ ρ ρ ρ

(47) (48)

(49)

Some theorems on generalized thermoelastic diffusion

295

From Eq. (6), we have (1)

C (1) − γ2 ekk − dθ (1) = n P (1) , C

(2)

−

(2) γ2 ekk

− dθ

(2)

= nP

(2)

.

Taking convolution of (50) with P (2) and of (51) with P (1) and subtracting, we get     (1) (2) C (1) − γ2 ekk − dθ (1) ∗ P (2) = C (2) − γ2 ekk − dθ (2) ∗ P (1) .

(50) (51)

(52)

Now from Eqs. (46), (49) and (52), we obtain (1)

(2)

(2)

(1)

σi j ∗ ei j − ρ S (1) ∗ θ (2) − C (1) ∗ P (2) = σi j ∗ ei j − ρ S (2) ∗ θ (1) − C (2) ∗ P (1) . We introduce the notation   (β) (α) L αβ = g ∗ σi j ∗ ei j − ρ S (α) ∗ θ (β) − C (α) ∗ P (β) d x, α, β = 1, 2.

(53)

(54)

V

Using relations (9), (25), (26) and (31)–(33), we find   (β) (α) g ∗ σi j ∗ ei j − ρ S (α) ∗ θ (β) − C (α) ∗ P (β)   (α)   qi,i ρ (β) (α) (α) (α) = g ∗ σi j ∗ u i, j − g ∗ −l ∗ ∗ θ (β) − g ∗ −l ∗ ηi,i + Y (α) ∗ P (β) +W T0 ρ 1 ρ (α) (α) g ∗ l ∗ qi,i ∗ θ (β) − g ∗ W (α) ∗ θ (β) + g ∗ l ∗ ηi,i ∗ P (β) − g ∗ Y (α) ∗ P (β) T0 T0     1 (β) (β) (β) (α) (α) (α) (α) = g ∗ σi j ∗ u i , j −ρu i ∗ u i + f i ∗ u i + g ∗ l ∗ qi ∗ θ (β) ,i T0     1 ρ (β) (α) (α) (α) − g ∗ l ∗ −M1 ∗ kθ,i +L 1i ∗ θ,i − g ∗ W (α) ∗ θ (β) + g ∗ l ∗ ηi ∗ P (β) ,i T0 T0   (β) (α) (α) − g ∗ l ∗ −M2 ∗ D P,i +L 2i ∗ P,i −g ∗ Y (α) ∗ P (β) . (55) (β)

= g ∗ σi(α) j ∗ u i, j +

Therefore from (54) we finally obtain 

ρ (β) (α) (α) (β) (α) (β) L αβ = fi ∗ u i − g ∗ W ∗ θ − g ∗ Y ∗ P dx T0 V

 1 (β) (α) (α) (β) (α) (β) + g ∗ σ˜ i ∗ u i + l ∗ q˜ ∗ θ + l ∗ η˜ ∗ P da T0 ∂V 

k (β) (β) (β) (α) (α) (α) − ρu i ∗ u i − g ∗ l ∗ M1 ∗ θ,i ∗θ,i −Dg ∗ l ∗ M2 ∗ P,i ∗P,i dx T0 V

 1 (α) β β (α) L 1i ∗ θ,i + L 2i ∗ P,i d x. − g∗l ∗ T0

(56)

V

Now, from Eqs. (53) and (54) we obtain L 12 = L 21 .

(57)

Therefore, we find that Eqs. (56) and (57) yield the relation (43), which completes the proof of our Theorem-4.   Acknowledgments The authors are grateful to the reviewers for their valuable and constructive comments and suggestions which have helped to improve the quality of our paper. The authors thankfully acknowledge the financial support from PURSE PROGRAM, IT-BHU under Department of Science and Technology, India. One of the authors (RK) thankfully acknowledges the support and encouragement of BIT, Mesra (Deoghar), India.

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References 1. Nowacki, W.: Dynamical problems of thermoelastic diffusion in solids I. Bull. Acad. Pol. Sci. Ser. Sci. Tech. 22, 55–64 (1974) 2. Nowacki, W.: Dynamical problems of thermoelastic diffusion in solids II. Bull. Acad. Pol. Sci. Ser. Sci. Tech. 22, 129– 135 (1974) 3. Nowacki, W.: Dynamical problems of thermoelastic diffusion in solids III. Bull. Acad. Pol. Sci. Ser. Sci. Tech. 22, 257– 266 (1974) 4. Nowacki, W.: Dynamical problems of thermoelastic diffusion in elastic solids. Proc. Vib. Prob. 15, 105–128 (1974) 5. Gawinecki, J., Kacprzyk, P., Bar-Yoseph, P.: Initial boundary value problem for some coupled nonlinear parabolic system of partial differential equations appearing in thermoelastic diffusion in solid body. J. Anal. Appl. 19, 121–130 (2000) 6. Gawinecki, J.A., Szymaniec, A.: Global solution of the Cauchy problem in nonlinear thermoelastic diffusion in solid body. PAMM Proc. Appl. Math. Mech. 1, 446–447 (2002) 7. Sherief, H.H., Hamza, F.A., Saleh, H.A.: The theory of generalized thermoelastic diffusion. Int. J. Eng. Sci. 42, 591– 608 (2004) 8. Dhaliwal, R.S., Singh, A.: Dynamic Coupled Thermoelasticity. Hindustan Pub. Corp., Delhi (1980) 9. Parkus, H.: Variational Principles in Thermo- and Magneto-Elasticity. Springer-Verlag, Vienna (1972) 10. Nowacki, W.: Dynamic Problems of Thermoelasticity. PWN-Polish Sci. Pub. Warszawa, Poland (1975) 11. Carlson, D.E.: Linear thermoelasticity, of Flügge’s Handbuch der Physik, vol. VI a/2 (Edited by C. Truesdell), pp. 297–345. Springer-Verlag, Berlin-Heidelberg-New York (1972) 12. Gurtin, M.E.: The linear theory of elasticity, of Flügge’s Handbuch der Physik, vol. VI a/2 (Edited by C. Truesdell), pp. 1–295. Springer-Verlag, Berlin-Heidelberg-New York (1972) 13. Lebon, G.: Variational Principles in Thermomechanics. Springer, Wien (1980) 14. Ignaczak, J.: A completeness problem for stress equations of motion in the linear elasticity theory. Arch. Mech. Stosow. 15, 225–235 (1963) 15. Gurtin, M.E.: Variational principles for linear elastodynamics. Arch. Rat. Mech. Anal. 16, 34–50 (1964) 16. Gurtin, M.E.: Variational principles for linear initial-value problems. Q. Appl. Math. 22, 252 (1964) 17. Biot, M.A.: Thermoelasticity and irreversible thermodynamics. J. Appl. Phys. 27, 240–253 (1956) 18. Biot, M.A.: New thermoelastical reciprocity relations with application to thermal stresses. J. Aeronaut Space Sci. 7, 26 (1959) 19. Hermann, G.: On variational principles in thermoelasticity and heat conduction. Q. Appl. Math. 21, 151 (1963) 20. Ben-Amoz, M.: On variational theorem in coupled thermoelasticity. J. Appl. Mech. 32, 943–945 (1965) 21. Iesan, D.: Principes variationnels dans la theorie de la thermoelasticite couplee, Ann, Sci. Univ. “Al. I. Cuza” Iasi. Mathematica 12, 439–456 (1966) 22. Nickell, R.E., Sackman, J.L.: Variational principles for linear coupled thermoelasticity. Q. Appl. Math. 26, 11–26 (1968) 23. Weiner, J.: A uniqueness theorem for the coupled thermoelastic problem. Q. Appl. Math. 15, 102–105 (1957) 24. Iesan, D.: On thermodynamics of non-simple elastic material with two temperatures. J. Appl. Math. Phys (ZAMP) 21, 583– 591 (1970) 25. Ignaczak, J.: Uniqueness in generalized thermoelasticity. J. Therm. Stress. 2, 171–179 (1979) 26. Sherief, H.H., Dhaliwal, R.S.: A uniqueness theorem and variational principle for generalized thermoelasticity. J. Therm. Stress. 3, 223–230 (1980) 27. Chandrasekharaiah, D.S., Srikantaiah, K.R.: Some theorem in temperature- rate-dependent theory of thermoelasticity. Proc. Indian Acad. Sci. (Math. Sci.) 92, 135–141 (1983) 28. Wang, J.R., Dhaliwal, S., Majumdar, S.R.: Some theorems in the generalized theory of thermoelasticity for pre-stressed bodies. Indian J. Pure Appl. Math. 28, 267–275 (1997) 29. Chandrasekharaiah, D.S.: Variational and reciprocal principles in thermoelasticity without energy dissipation. Proc. Indian Acad. Sci. (Math. Sci.) 108, 209–215 (1998) 30. Kumar, R., Prasad, R., Mukhopadhyay, S.: Variational and reciprocal principles in two-temperature generalized thermoelasticity. J. Therm. Stress. 33, 161–171 (2010) 31. Aouadi, M.: Uniqueness and reciprocity theorems in the theory of generalized thermoelastic diffusion. J. Therm. Stress. 30, 665–678 (2007) 32. Aouadi, M.: Generalized theory of thermoelastic diffusion for anisotropic media. J. Therm. Stress. 31, 1–16 (2008)