Research Article Fundamental Solution in the Theory of Thermomicrostretch Elastic Diffusive Solids

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1 International Scholarly Research Network ISRN Applied Mathematics Volume 2011 Article ID pages doi: /2011/ Research Article Fundamental Solution in the Theory of Thermomicrostretch Elastic Diffusive Solids Rajneesh Kumar and Tarun Kansal Department of Mathematics Kurukshetra University Kurukshetra India Correspondence should be addressed to Rajneesh Kumar rajneesh Received 11 March 2011; Accepted 6 April 2011 Academic Editors: F. Amirouche and F. Wang Copyright q 2011 R. Kumar and T. Kansal. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. We construct the fundamental solution of system of differential equations in the theory of thermomicrostretch elastic diffusive solids in case of steady oscillations in terms of elementary functions. Some basic properties of the fundamental solution are established. Some special cases are also discussed. 1. Introduction Eringen 1 developed the theory of micropolar elastic solid with stretch. He derived the equations of motion constitutive equations and boundary conditions for the class of micropolar solid which can stretch and contract. This model introduced and explained the motion of certain class of granular and composite materials in which grains and fibres are elastic along the direction of their major axis. This theory is generalization of the theory of micropolar elasticity 2 3. Eringen 4 developed a theory of thermomicrostretch elastic solid in which he included microstructural expansions and contractions. Microstretch continuum is a model for Bravais lattice with a basis on the atomic level and a two-phase dipolar solid with a core on the macroscopic level. In the framework of the theory of thermomicrostretch solids Eringen established a uniqueness theorem for the mixed initial boundary value problem. The theory was illustrated with the solution of one-dimensional waves and compared with lattice dynamical results. The asymptotic behavior of solutions and an existence result were presented by Bofill and Quintanilla 5. A reciprocal theorem and a representation of Galerkin type were presented by De Cicco and Nappa 6. In classical theory of thermoelasticity Fourier s heat conduction theory assumes that the thermal disturbances propagate at infinite speed which is unrealistic from the physical point of view. Lord and Shulman 7 incorporates a flux rate term into Fourier s law of heat conduction and formulates a generalized theory admitting finite speed for thermal signals.

2 2 ISRN Applied Mathematics Lord and Shulman 7 theory of generalized thermoelasticity has been further extended to homogeneous anisotropic heat conducting materials recommended by Dhaliwal and Sherief 8. All these theories predict a finite speed of heat propagation. Chanderashekhariah 9 refers to this wave-like thermal disturbance as second sound. A survey article of various representative theories in the range of generalized thermoelasticity has been brought out by Hetnarski and Ignaczak 10. Diffusion is defined as the spontaneous movement of the particles from a highconcentration region to the low-concentration region and it occurs in response to a concentration gradient expressed as the change in the concentration due to change in position. Thermal diffusion utilizes the transfer of heat across a thin liquid or gas to accomplish isotope separation. Today thermal diffusion remains a practical process to separate isotopes of noble gases e.g. xenon and other light isotopes e.g. carbon for research purposes. In most of the 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. However there is a certain degree of coupling with temperature and temperature gradients as temperature speeds up the diffusion process. The thermodiffusion in elastic solids is due to coupling of fields of temperature mass diffusion and that of strain in addition to heat and mass exchange with the environment. Nowacki developed the theory of thermoelastic diffusion by using coupled thermoelastic model. Dudziak and Kowalski 15 and Olesiak and Pyryev 16 respectively discussed the theory of thermodiffusion and coupled quasistationary problems of thermal diffusion for an elastic layer. They studied the influence of cross-effects arising from the coupling of the fields of temperature mass diffusion and strain due to which the thermal excitation results in additional mass concentration and that generates additional fields of temperature. Uniqueness and reciprocity theorems for the equations of generalized thermoelastic diffusion problem in isotropic media were proved by Sherief et al. 17 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 18 proved this theorem 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 19 derived the uniqueness and reciprocity theorems for the generalized problem in anisotropic media under the restriction that the elastic thermal conductivity and diffusion tensors are positive definite. To investigate the boundary value problems of the theory of elasticity and thermoelasticity by potential method it is necessary to construct a fundamental solution of systems of partial differential equations and to establish their basic properties respectively. Hetnarski was the first to study the fundamental solutions in the classical theory of coupled thermoelasticity. The fundamental solutions in the microcontinuum fields theories have been constructed by Svanadze 22 Svanadze and De Cicco 23 and Svanadze and Tracina 24. The information related to fundamental solutions of differential equations is contained in the books of Hörmander In this paper the fundamental solution of system of equations in the case of steady oscillations is considered in terms of elementary functions and basic properties of the fundamental solution are established. Some special cases of interest are also discussed.

3 ISRN Applied Mathematics 3 2. Basic Equations Let x x 1 x 2 x 3 be the point of the Euclidean three-dimensional space E 3 : x x 2 1 x 2 2 x2 3 1/2 D x / x 1 / x 2 / x 3 and let t denote the time variable. Following Sherief et al. 17 and Eringen 4 the basic equations for homogeneous isotropic generalized theromicrostretch elastic diffusive solids in the absence of body forces body couples body loads heat and mass diffusion sources are μ K ) Δu λ μ ) grad div u K curl ϕ χ grad ψ β 1 grad T β 2 grad C ρü f Δ 2K ) ϕ α β ) grad div ϕ K curl u ρj ϕ b Δ c ψ χ div u g T h C ρζψ ) 1 τ 0 β 1 T 0 div t u ) g T 0 ψ ρc E Ṫ at 0 Ċ KΔT Dβ 2 Δ div u Dh Δψ DaΔT DbΔC Ċ τ 0 C where β 1 3λ 2μ K α t β 2 3λ 2μ K α c. Here α t α c are the coefficients of linear thermal expansion and diffusion expansion respectively; u u 1 u 2 u 3 is the displacement vector; ϕ ϕ 1 ϕ 2 ϕ 3 is the microrotation vector; ψ is the microstretch function; ρ C E are respectively the density and specific heat at constant strain; λ μ K D a b b c f g h α β K andχ are constitutive coefficients; j and ζ are coefficients of microintertia; T is the temperature measured from constant temperature T 0 T 0 / 0 and C is the concentration; τ 0 is diffusion relaxation time and τ 0 is thermal relaxation time; Δ is the Laplacian operator. Here τ 0 τ 0 0 for coupled thermoelastic diffusion model. We define the dimensionless quantities: x w 1 x c 1 u ρw 1 c 1u β 1 T 0 ϕ ρc2 1 ϕ β 1 T 0 δ 3 δ 8 t w 1 t τ 0 w 1 τ 0 τ 0 w 1 τ 0 δ 1 K λ 2μ K δ 4 χ b ζ λ 2μ K ) ρζw 2 δ 9 1 ψ ρζw 2 1 ψ β 1 T 0 T T T 0 C β 2C β 1 T 0 μ K λ 2μ K δ 5 f w 2 1 δ ρc c ρζw 2 1 δ 10 χ λ 2μ K δ 2 α β ) w 2 1 ρc 4 1 δ 11 g β 1 λ μ λ 2μ K δ 7 jw 2 1 c 2 1 δ 12 h β 2 ζ 1 at 0c 2 1 β 1 w 1 Kβ ζ 2 β2 1 T 0 2 ρkw 1 q 2 Dw 1 β 2a β 1 c 2 1 ζ 3 g β 1 T 0 c 2 1 ρζkw 3 1 q 1 Dw 1 β2 2 ρc 4 1 q 3 Dw 1 b q c 2 4 Dh β 2. 1 ρζw 1 c

4 4 ISRN Applied Mathematics Here w 1 ρc E c 2 1 /K and c 1 λ 2μ K /ρ are the characteristic frequency and longitudinal wave velocity in the medium respectively. Upon introducing the quantities 2.2 in the basic equations 2.1 after suppressing the primes we obtain δ 1 Δu δ 2 grad div u δ 3 curl ϕ δ 4 grad ψ grad T grad C ü δ 5 Δ 2δ 3 ϕ δ 6 grad div ϕ δ 3 curl u δ 7 ϕ δ 8 Δ δ 9 ψ δ 10 div u δ 11 T δ 12 C ψ ζ 2 div u ) ζ 3 ψ Ṫ ζ 1Ċ ΔT τ 0 t 2.3 q 1 Δ div u q 4 Δψ q 2 ΔT q 3 ΔC τ c 0 C 0 where τ 0 t 1 τ 0 t τ0 c 1 τ 0 t. 2.4 We assume the displacement vector microrotation microstretch temperature change and concentration functions as ) u xt ϕ xt ψ xt T xt C xt Re [ u ϕψ TC ) e ιωt] 2.5 where ω is oscillation frequency and ω>0. Using 2.5 into 2.3 we obtain the system of equations of steady oscillations as δ 1 Δ ω 2) u δ 2 grad div u δ 3 curl ϕ δ 4 grad ψ grad T grad C 0 δ5 Δ μ ) ϕ δ 6 grad div ϕ δ 3 curl u 0 δ 10 div u δ 8 Δ ζ ψ δ 11 T δ 12 C 0 [ ζ2 div u ζ 3 ψ ζ 1 C ] ) Δ τ 10 T 0 τ 10 t t 2.6 q 1 Δ div u q 4 Δψ q 2 ΔT q 10 3ΔC τc C 0 where τt 10 ιω 1 ιωτ 0 τc 10 ιω 1 ιωτ 0) μ δ 7 ω 2 2δ 3 ζ ω 2 δ We introduce the matrix differential operator F D x Fgh D x

5 ISRN Applied Mathematics 5 where F mn D x [δ 1 Δ ω 2] 2 3 δ mn δ 2 F mn 3 D x F m 3n D x δ 3 x m x n ε mrn r 1 x r F m7 D x δ 4 F m8 D x F m9 D x F m 3n 3 D x δ 5 Δ μ ) 2 δ mn δ 6 x m x m x m x n F m 37 D x F 7n 3 D x F m 38 D x F 8n 3 D x F m 39 D x F 9n 3 D x 0 F 7n D x δ 10 F 77 D x δ 8 Δ ζ F 78 D x δ 11 F 79 D x δ 12 x n F 8n D x ζ 2 τ 10 t F 87 D x ζ 3 τt 10 F 88 D x Δ τt 10 F 89 ζ 1 τ 10 x n F 9n D x q 1 Δ F 97 D x q 4 x Δ F 98 D x q 2 Δ n F 99 D x q 10 3Δ τc mn t 2.9 Here ε mrn is alternating tensor and δ mn is the Kronecker delta function. The system of equations 2.6 can be written as F D x U x where U u ϕψ TC is a nine-component vector function on E 3. Definition 2.1. The fundamental solution of the system of equations 2.6 the fundamental matrix of operator F is the matrix G x G gh x 9 9 satisfying condition 25 F D x G x δ x I x 2.11 where δ is the Dirac delta I δ gh 9 9 is the unit matrix and x ɛ E 3. Now we construct G x in terms of elementary functions. 3. Fundamental Solution of System of Equations of Steady Oscillations We consider the system of equations δ 1 Δu δ 2 grad div u δ 3 curl ϕ δ 10 grad ψ ζ 2 τt 10 grad T q 1 grad C ω2 u H 3.1 δ5 Δ μ ) ϕ δ 6 grad div ϕ δ 3 curl u H 3.2 δ 4 div u δ 8 Δ ζ ψ ζ 3 τt 10 T q 4C Z 3.3 ) div u δ 11 ψ Δ τt 10 T q 2C L 3.4 Δ div u δ 12 Δψ ζ 1 τt 10 ΔT q 10 3ΔC τc C M 3.5

6 6 ISRN Applied Mathematics where H and H are three-component vector functions on E 3 and Z L andm are scalar functions on E 3. The system of equations may be written in the form F tr D x U x Q x 3.6 where F tr is the transpose of matrix F Q H H ZLM andx ɛ E 3. Applying the operator div to 3.1 and 3.2 weobtain Δ ω 2) div u δ 10 Δψ ζ 2 τt 10 ΔT q 1 ΔC div H υ Δ μ ) div ϕ div H δ 4 div u δ 8 Δ ζ ψ ζ 3 τt 10 T q 4C Z ) div u δ 11 ψ Δ τt 10 T q 2C L 3.7 Δ div u δ 12 Δψ ζ 1 τt 10 ΔT q 10 3ΔC τc C M where υ δ 5 δ 6. Equations and may be written in the form N Δ S Q 3.8 where S div uψ TC Q d 1 d 2 d 3 d 4 div H ZLM and Δ ω 2 δ 10 Δ ζ 2 τt 10 Δ q 1 Δ δ 4 δ 8 Δ ζ ζ 3 τt 10 q 4 N Δ N mn Δ δ 11 Δ τt 10 q 2 Δ δ 12 Δ ζ 1 τt 10 Δ q 3 Equations and may be also written as Δ τ 10 c Γ 1 Δ S Ψ 3.10 where 4 Ψ Ψ 1 Ψ 2 Ψ 3 Ψ 4 Ψ n e Nmnd m m 1 Γ 1 Δ e det N Δ e 1 q 3 δ n and N mn is the cofactor of the elements N mn of the matrix N.

7 ISRN Applied Mathematics 7 From 3.9 and 3.11 weseethat 4 ) Γ 1 Δ Δ λ 2 m 3.12 m 1 where λ 2 m m are the roots of the equation Γ 1 κ 0 with respect to κ. From it follows that ) Δ λ 2 7 div ϕ 1 δ div H 3.13 where λ 2 7 μ /υ. Applying the operators δ 5 Δ μ and δ 3 curl to 3.1 and 3.2 respectively we obtain δ5 Δ μ )[ δ 1 Δu δ 2 grad div u ω 2 u] δ 3 δ5 Δ μ ) curl ϕ δ 5 Δ μ )[ H δ 10 grad ψ ζ 2 τ 10 t grad T q 1 grad C ] δ 3 δ5 Δ μ ) curl ϕ δ 2 3 curl curl u δ 3 curl H Now curl curl u grad div u Δu Using 3.15 and 3.16 in 3.14 weobtain δ5 Δ μ )[ δ 1 Δu δ 2 grad div u ω u] 2 δ 2 3 Δu δ2 3 grad div u δ 5 Δ μ )[ H δ 10 grad ψ ζ 2 τ 10 t grad T q 1 grad C ] δ 3 curl H The above equation can also be written as {[ δ5 Δ μ ) δ 1 δ3] 2 Δ δ 5 Δ μ ) ω 2} u [ δ 2 δ5 Δ μ ) ] δ 2 3 grad div u 3.18 δ 5 Δ μ )[ H δ 10 grad ψ ζ 2 τ 10 t grad T q 1 grad C ] δ 3 curl H.

8 8 ISRN Applied Mathematics Applying the operator Γ 1 Δ to the 3.18 and using 3.10 weget [ ) Γ 1 Δ δ 5 δ 1 Δ 2 μ δ 1 δ 5 ω 2 δ 2 3 Δ μ ω 2] u [ δ 2 δ5 Δ μ ) ] δ 2 3 grad Ψ 1 δ 5 Δ μ )[ ] Γ 1 Δ H δ 10 grad Ψ 2 ζ 2 τt 10 grad Ψ 3 q 1 grad Ψ 4 δ 3 Γ 1 Δ curl H The above equation may be written in the form Γ 1 Δ Γ 2 Δ u Ψ 3.20 where Γ 2 Δ f δ 1 Δ ω 2 det δ 3 δ 3 Δ f 1 δ 5 Δ μ δ 1 δ Ψ f { [ δ 2 δ5 Δ μ ) ] δ 2 3 grad Ψ 1 δ 5 Δ μ )[ ] Γ 1 Δ H δ 10 grad Ψ 2 ζ 2 τt 10 grad Ψ 3 q 1 grad Ψ δ 3 Γ 1 Δ curl H }. It can be seen that Γ 2 Δ ) ) Δ λ where λ 2 5 λ2 6 are the roots of the equation Γ 2 κ 0 with respect to κ. Applying the operators δ 3 curl and δ 1 Δ ω 2 to 3.1 and 3.2 respectively we obtain δ 3 δ 1 Δ ω 2) curl u δ 3 curl H δ 2 3 curl curl ϕ 3.24 δ 1 Δ ω 2) δ5 Δ μ ) ϕ δ 6 δ 1 Δ ω 2) grad div ϕ δ 3 δ 1 Δ ω 2) curl u δ 1 Δ ω 2) H Now curl curl ϕ grad div ϕ Δϕ. 3.26

9 ISRN Applied Mathematics 9 Using 3.24 and 3.26 in 3.25 weobtain δ 1 Δ ω 2) δ5 Δ μ ) ϕ δ 6 δ 1 Δ ω 2) grad div ϕ δ 2 3 Δϕ δ2 3 grad div ϕ δ 1 Δ ω 2) H δ 3 curl H The above equation may also be written as {[ δ5 Δ μ ) δ 1 δ3] 2 Δ δ 5 Δ μ ) ω 2} ϕ [δ 6 δ 1 Δ ω 2) ] δ 2 3 grad div ϕ δ 1 Δ ω 2) H δ 3 curl H Applying the operator Δ λ 2 7 to the 3.28 and using 3.13 weget )[ ) Δ λ 2 7 δ 5 δ 1 Δ 2 μ δ 1 δ 5 ω 2 δ 2 3 Δ μ ω 2] ϕ ) δ 3 Δ λ 2 7 curl H Δ λ 2 7 )δ 1 Δ ω 2) H 1 [δ υ 6 δ 1 Δ ω 2) ] δ 2 3 grad div H The above equation may also be rewritten in the form ) Γ 2 Δ Δ λ 2 7 ϕ Ψ 3.30 where { ) ) Ψ f δ 3 Δ λ 2 7 curl H Δ λ 2 7 δ 1 Δ ω 2) H 1 [ υ δ 6 δ 1 Δ ω 2) ] } δ 2 3 grad div H From and 3.30 weobtain Θ Δ U x Ψ x 3.32

10 10 ISRN Applied Mathematics where Ψ Ψ Ψ Ψ 2 Ψ 3 Ψ 4 Θ Δ Θ gh Δ 9 9 Θ mm Δ Γ 1 Δ Γ 2 Δ 6 ) Δ λ 2 q q 1 ) Θ m 3n 3 Δ Γ 2 Δ Δ λ ) Δ λ 2 q q Θ gh Δ 0 Θ 77 Δ Θ 88 Δ Θ 99 Δ Γ 1 Δ m gh g/ h. Equations and 3.31 can be rewritten in the form Ψ [ f δ 5 Δ μ ) Γ 1 Δ J q 11 Δ grad div ] H q 21 Δ curl H q 31 Δ grad Z q 41 Δ grad L q 51 Δ grad M { Ψ q 12 Δ curl H f ) Δ λ 2 7 δ 1 Δ ω 2) } J q 22 Δ grad div H Ψ 2 q 13 Δ div H q 33 Δ Z q 43 Δ L q 53 Δ M 3.34 Ψ 3 q 14 Δ div H q 34 Δ Z q 44 Δ L q 54 Δ M Ψ 4 q 15 Δ div H q 35 Δ Z q 45 Δ L q 55 Δ M where J δ gh 3 3 is the unit matrix. In 3.34 we have used the following notations: q m1 Δ f e { δ5 Δ μ )[ ] δ 10 N m2 ζ 2τt 10 N m3 q 1 N m4 δ 2 δ5 Δ μ ) ) } δ 2 3 N m1 ) q 21 Δ f δ 3 Γ 1 Δ q 12 Δ f δ 3 Δ λ 2 7 q22 Δ f [ δ6 δ1 υ Δ ω 2) δ 2 ] 3 q mn Δ e Nmn m n Now from 3.34 we have that Ψ x R tr D x Q x 3.36

11 ISRN Applied Mathematics 11 where R Rgh 9 9 R mn D x f δ 5 Δ μ ) Γ 1 Δ δ mn q 11 Δ x m x n 3 R mn 3 D x q 12 Δ ε mrn r 1 x r 3 R m 3n D x q 21 Δ 2 R mp D x q 1p 4 Δ x m ε mrn r 1 x r R m 3n 3 D x f ) Δ λ 2 7 δ 1 Δ ω 2) 2 δ mn q 22 Δ x m x n 3.37 R m 3p D x R pm 3 D x 0 R pm D x q p 41 Δ x n R ps D x q p 4s 4 Δ m ps From and 3.36 weobtain ΘU R tr F tr U It implies that R tr F tr Θ F D x R D x Θ Δ We assume that λ 2 m / λ 2 n / 0 mn m / n Let 6 7 Y x Y rs x 9 9 Y mm x r 1n ς n x Y m 3m 3 x r 2n ς n x n 1 n 5 4 Y 77 x Y 88 x Y 99 x r 3n ς n x n Y vw x 0 m vw v/ w

12 12 ISRN Applied Mathematics where ς n x 1 4π x exp ιλ n x n r 1l r 2v r 3w 6 m 1m / l λ 2 m λ 2 l ) 1 l m 5m / v 4 m 1m / w 1 λ 2 m λv) 2 v λ 2 m λw) 2 w We will prove the following lemma. Lemma 3.1. The matrix Y defined above is the fundamental matrix of operator Θ Δ that is Θ Δ Y x δ x I x Proof. To prove the lemma it is sufficient to prove that Γ 1 Δ Γ 2 Δ Y 11 x δ x ) Γ 2 Δ Δ λ 2 7 Y 44 x δ x Γ 1 Δ Y 77 x δ x We find that r 11 r 12 r 13 r 14 r 15 r 16 0 ) r 13 λ 2 1 λ2 3 r 12 λ 2 1 λ2 2 ) r 14 λ 2 1 λ2 4 ) ) ) r 15 λ 2 1 λ2 5 r 16 λ 2 1 λ2 6 0 ) ) ) ) ) ) r 13 λ 2 1 λ2 3 λ 2 2 λ2 3 r 14 λ 2 1 λ2 4 λ 2 2 λ2 4 r 15 λ 2 1 λ2 5 λ 2 2 λ2 5 ) ) r 16 λ 2 1 λ2 6 λ 2 2 λ2 6 0 ) ) ) ) ) ) r 14 λ 2 1 λ2 4 λ 2 2 λ2 4 λ 2 3 λ2 4 r 15 λ 2 1 λ2 5 λ 2 2 λ2 5 λ 2 3 λ2 5 ) ) ) r 16 λ 2 1 λ2 6 λ 2 2 λ2 6 λ 2 3 λ2 6 0 ) ) ) ) ) ) ) ) r 15 λ 2 1 λ2 5 λ 2 2 λ2 5 λ 2 3 λ2 5 λ 2 4 λ2 5 r 16 λ 2 1 λ2 6 λ 2 2 λ2 6 λ 2 3 λ2 6 λ 2 4 λ2 6 0 ) ) ) ) ) r 16 λ 2 1 λ2 6 λ 2 2 λ2 6 λ 2 3 λ2 6 λ 2 4 λ2 5 λ 2 5 λ2 6 1 ) ) Δ λ 2 m ς n x δ x λ 2 m λ 2 n ς n x mn

13 ISRN Applied Mathematics 13 Now consider Γ 1 Δ Γ 2 Δ Y 11 x ) ) ) ) ) 6 Δ λ 2 2 Δ λ 2 3 Δ λ 2 4 Δ λ 2 5 n 1 ) ) ) ) ) 6 Δ λ 2 2 Δ λ 2 3 Δ λ 2 4 Δ λ 2 5 n 2 n 2 r 1n [δ r 1n λ 2 1 λ2 n ) ) ) ) 6 )[ Δ λ 2 3 Δ λ 2 4 Δ λ 2 5 r 1n λ 2 1 λ2 n δ ) ) ) ) 6 ) ) Δ λ 2 3 Δ λ 2 4 Δ λ 2 5 r 1n λ 2 1 λ2 n λ 2 2 λ2 n ) ) ) 6 ) )[ Δ λ 2 4 Δ λ 2 5 r 1n λ 2 1 λ2 n λ 2 2 λ2 n δ n 3 n 4 n 3 ] λ 2 1 λ2 n )ς n ) ς n ] λ 2 2 λ2 n )ς n ς n ] λ 2 3 λ2 n )ς n ) ) ) 6 ) ) ) Δ λ 2 4 Δ λ 2 5 r 1n λ 2 1 λ2 n λ 2 2 λ2 n λ 2 3 λ2 n ς n ) ) 6 Δ λ 2 5 ) 6 n 5 ) ς 6 δ. n 4 r 1n λ 2 1 λ2 n r 1n λ 2 1 λ2 n ) ) )[ ] λ 2 2 λ2 n λ 2 3 λ2 n δ λ 2 4 λ2 n )ς n ) ) ) )[ ] λ 2 2 λ2 n λ 2 3 λ2 n λ 2 4 λ2 n δ λ 2 5 λ2 n )ς n 3.46 Similarly and can be proved. We introduce the matrix G x R D x Y x From and 3.47 weobtain F D x G x F D x R D x Y x Θ Δ Y x δ x I x Hence G x is a solution to Therefore we have proved the following theorem. Theorem 3.2. The matrix G x defined by 3.47 is the fundamental solution of system of equations 2.6.

14 14 ISRN Applied Mathematics 4. Basic Properties of the Matrix Gx) Property 1. Each column of the matrix G x is the solution of the system of equations 2.6 at every point x ɛ E 3 except the origin. Property 2. The matrix G x can be written in the form G Ggh 9 9 G mn x R mn D x Y 11 x G mn 3 x R mn 3 D x Y 44 x 4.1 G mp x R mp D x Y 77 x m n p Special Cases i If we neglect the diffusion effect we obtain the same results for fundamental solution as discussed by Svanadze and De Cicco 23 by changing the dimensionless quantities into physical quantities in case of coupled theory of thermoelasticity. ii If we neglect the thermal and diffusion effects we obtain the same results for fundamental solution as discussed by Svanadze 22 by changing the dimensionless quantities into physical quantities. iii If we neglect both micropolar and microstretch effects the same results for fundamental solution can be obtained as discussed by Kumar and Kansal 27 in case of the Lord-Shulman theory of thermoelastic diffusion. 6. Conclusions The fundamental solution G x of the system of equations 2.6 makes it possible to investigate three-dimensional boundary value problems of generalized theory of thermomicrostretch elastic diffusive solids by potential method 28. Acknowledgment Mr. T. Kansal is thankful to the Council of Scientific and Industrial Research CSIR for the financial support. References 1 A. C. Eringen Micropolar elastic solids with stretch M. I. Anisina Ed. pp Ari Kitabevi Matbassi Istanbul Turkey A. C. Eringen Linear theory of micropolar elasticity Indiana University Mathematics Journal vol. 15 pp A. C. Eringen Theory of micropolar elasticity in Fracture H. Liebowitz Ed. vol. 2 pp Academic Press New York NY USA A. C. Eringen Theory of thermomicrostretch elastic solids International Journal of Engineering Science vol. 28 no. 12 pp F. Bofill and R. Quintanilla Some qualitative results for the linear theory of thermo-microstretch elastic solids International Journal of Engineering Science vol. 33 no. 14 pp

15 ISRN Applied Mathematics 15 6 S. De Cicco and L. Nappa Some results in the linear theory of thermomicrostretch elastic solids Mathematics and Mechanics of Solids vol. 5 no. 4 pp H. W. Lord and Y. Shulman A generalized dynamical theory of thermoelasticity Journal of Mechanics and Physics of Solids vol. 15 pp R. S. Dhaliwal and H. H. Sherief Generalized thermoelasticity for anisotropic media Quarterly of Applied Mathematics vol. 38 no. 1 pp D. S. Chanderashekhariah Thermoelasticity with second: a review Applied Mechanics Review vol. 39 pp R. B. Hetnarski and J. Ignaczak Generalized thermoelasticity Journal of Thermal Stresses vol. 22 no. 4-5 pp W. Nowacki Dynamical problem of thermodiffusion in solids I Bulletin de l Académie Polonaise des Sciences. Série des Sciences Techniques vol. 22 pp W. Nowacki Dynamical problem of thermodiffusion in solids II Bulletin de l Académie Polonaise des Sciences. Série des Sciences Techniques vol. 22 pp W. Nowacki Dynamical problem of thermodiffusion in solids III Bulletin de l Académie Polonaise des Sciences. Série des Sciences Techniques vol. 22 pp W. Nowacki Dynamical problems of thermodiffusion in solids Engineering Fracture Mechanics vol. 8 pp W. Dudziak and S. J. Kowalski Theory of thermodiffusion for solids International Journal of Heat and Mass Transfer vol. 32 no. 11 pp Z. S. Olesiak and Y. A. Pyryev A coupled quasi-stationary problem of thermodiffusion for an elastic cylinder International Journal of Engineering Science vol. 33 no. 6 pp H. H. Sherief F. A. Hamza and H. A. Saleh The theory of generalized thermoelastic diffusion International Journal of Engineering Science vol. 42 no. 5-6 pp M. Aouadi Uniqueness and reciprocity theorems in the theory of generalized thermoelastic diffusion Journal of Thermal Stresses vol. 30 no. 7 pp M. Aouadi Generalized theory of thermoelastic diffusion for anisotropic media Journal of Thermal Stresses vol. 31 no. 3 pp R. B. Hetnarski The fundamental solution of the coupled thermoelastic problem for small times Archiwwn Mechaniki Stosowanej vol. 16 pp R. B. Hetnarski Solution of the coupled problem of thermoelasicity in the form of series of functions Archiwwn Mechaniki Stosowanej vol. 16 pp M. Svanadze Fundamental solution of the system of equations of steady oscillations in the theory of microstretch elastic solids International Journal of Engineering Science vol. 42 no pp M. Svanadze and S. De Cicco Fundamental solution in the theory of thermomicrostretch elastic solids International Journal of Engineering Science vol. 43 no. 5-6 pp M. Svanadze and R. Tracinà Representations of solutions in the theory of thermoelasticity with microtemperatures for microstretch solids Journal of Thermal Stresses vol. 34 pp L. Hörmander Linear Partial Differential Operators Springer Berlin Germany L. Hörmander The Analysis of Linear Partial Differential Operators II vol. 257 Springer Berlin Germany R. Kumar and T. Kansal Plane waves and fundamental solution in the generalized theories ofthermoelastic diffusion International Journal of Applied Mathematics and Mechanics. In press. 28 V.D.KupradzeT.G.GegeliaM.O.Basheleĭshvili and T. V. Burchuladze Three-Dimensional Problems of the Mathematical Theory of Elasticity and Thermoelasticity vol. 25 of North-Holland Series in Applied Mathematics and Mechanics North-Holland Publishing Amsterdam The Netherlands 1979.

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