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International Scholarly Research NetworkISRN Applied MathematicsVolume 2011, Article ID 764632, 15 pagesdoi:10.5402/2011/764632

Research ArticleFundamental Solution in the Theory ofThermomicrostretch Elastic Diffusive Solids

Rajneesh Kumar and Tarun Kansal

Department of Mathematics, Kurukshetra University, Kurukshetra 136 119, India

Correspondence should be addressed to Rajneesh Kumar, rajneesh [email protected]

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 theCreative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

We construct the fundamental solution of system of differential equations in the theory ofthermomicrostretch elastic diffusive solids in case of steady oscillations in terms of elementaryfunctions. Some basic properties of the fundamental solution are established. Some special casesare also discussed.

1. Introduction

Eringen [1] developed the theory of micropolar elastic solid with stretch. He derivedthe equations of motion, constitutive equations, and boundary conditions for the class ofmicropolar solid which can stretch and contract. This model introduced and explained themotion of certain class of granular and composite materials in which grains and fibres areelastic along the direction of their major axis. This theory is generalization of the theory ofmicropolar elasticity [2, 3]. Eringen [4] developed a theory of thermomicrostretch elastic solidinwhich he includedmicrostructural expansions and contractions. Microstretch continuum isa model for Bravais lattice with a basis on the atomic level and a two-phase dipolar solid witha 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 withlattice dynamical results. The asymptotic behavior of solutions and an existence result werepresented by Bofill andQuintanilla [5]. A reciprocal theorem and a representation of Galerkintype were presented by De Cicco and Nappa [6].

In classical theory of thermoelasticity, Fourier’s heat conduction theory assumes thatthe thermal disturbances propagate at infinite speed which is unrealistic from the physicalpoint of view. Lord and Shulman [7] incorporates a flux rate term into Fourier’s law of heatconduction and formulates a generalized theory admitting finite speed for thermal signals.

2 ISRN Applied Mathematics

Lord and Shulman [7] theory of generalized thermoelasticity has been further extended tohomogeneous 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 variousrepresentative theories in the range of generalized thermoelasticity has been brought out byHetnarski and Ignaczak [10].

Diffusion is defined as the spontaneous movement of the particles from a high-concentration region to the low-concentration region, and it occurs in response to aconcentration gradient expressed as the change in the concentration due to change inposition. Thermal diffusion utilizes the transfer of heat across a thin liquid or gas toaccomplish isotope separation. Today, thermal diffusion remains a practical process toseparate isotopes of noble gases (e.g., xenon) and other light isotopes (e.g., carbon) forresearch purposes. In most of the applications, the concentration is calculated using what isknown as Fick’s law. This is a simple law which does not take into consideration the mutualinteraction between the introduced substance and the medium into which it is introduced orthe effect of temperature on this interaction. However, there is a certain degree of couplingwith temperature and temperature gradients as temperature speeds up the diffusion process.The thermodiffusion in elastic solids is due to coupling of fields of temperature, massdiffusion and that of strain in addition to heat and mass exchange with the environment.

Nowacki [11–14] developed the theory of thermoelastic diffusion by using coupledthermoelastic model. Dudziak and Kowalski [15] and Olesiak and Pyryev [16], respectively,discussed the theory of thermodiffusion and coupled quasistationary problems of thermaldiffusion for an elastic layer. They studied the influence of cross-effects arising from thecoupling of the fields of temperature, mass diffusion, and strain due to which the thermalexcitation results in additional mass concentration and that generates additional fieldsof temperature. Uniqueness and reciprocity theorems for the equations of generalizedthermoelastic diffusion problem, in isotropic media, were proved by Sherief et al. [17] onthe basis of the variational principle equations, under restrictive assumptions on the elasticcoefficients. Due to the inherit complexity of the derivation of the variational principleequations, Aouadi [18] proved this theorem in the Laplace transform domain, under theassumption that the functions of the problem are continuous and the inverse Laplacetransform of each is also unique. Aouadi [19] derived the uniqueness and reciprocitytheorems for the generalized problem in anisotropic media, under the restriction that theelastic, thermal conductivity and diffusion tensors are positive definite.

To investigate the boundary value problems of the theory of elasticity and thermoe-lasticity by potential method, it is necessary to construct a fundamental solution of systemsof partial differential equations and to establish their basic properties, respectively. Hetnarski[20, 21] was the first to study the fundamental solutions in the classical theory of coupledthermoelasticity. The fundamental solutions in the microcontinuum fields theories have beenconstructed 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 thebooks of Hormander [25, 26].

In this paper, the fundamental solution of system of equations in the case of steadyoscillations is considered in terms of elementary functions and basic properties of thefundamental solution are established. Some special cases of interest are also discussed.

ISRN Applied Mathematics 3

2. Basic Equations

Let x = (x1, x2, x3) be the point of the Euclidean three-dimensional space E3: |x| = (x21 +

x22 + x

23)

1/2, Dx = (∂/∂x1, ∂/∂x2, ∂/∂x3) and let t denote the time variable. Following Sheriefet al. [17] and Eringen [4], the basic equations for homogeneous isotropic generalizedtheromicrostretch elastic diffusive solids in the absence of body forces, body couples, bodyloads, heat and mass diffusion sources are

(μ +K∗)Δu +

(λ + μ

)grad divu +K∗ curl ϕ + χ∗ gradψ∗ − β1 grad T − β2 gradC = ρu,

(f∗Δ − 2K∗)ϕ +

(α∗ + β∗

)grad divϕ +K∗ curlu = ρjϕ,

(b∗Δ − c∗)ψ∗ − χ∗ divu − g∗T − h∗C = ρζψ∗,(1 + τ0

∂

∂t

)(β1T0 div u − g∗T0ψ∗ + ρCET + aT0C

)= KΔT,

Dβ2Δdivu + +Dh∗Δψ∗ +DaΔT −DbΔC + C + τ0C = 0,(2.1)

where β1 = (3λ + 2μ + K∗)αt, β2 = (3λ + 2μ + K∗)αc. Here αt, αc are the coefficientsof linear thermal expansion, and diffusion expansion, respectively; u = (u1, u2, u3)is the displacement vector; ϕ = (ϕ1, ϕ2, ϕ3) is the microrotation vector; ψ∗ is themicrostretch function; ρ,CE are, respectively, the density and specific heat at constantstrain; λ, μ,K,D, a, b, b∗, c∗, f∗, g∗, h∗, α∗, β∗, K∗, and χ∗ are constitutive coefficients; j and ζ

are coefficients of microintertia; T is the temperature measured from constant temperatureT0 (T0 /= 0) and C is the concentration; τ0 is diffusion relaxation time and τ0 is thermalrelaxation time; Δ is the Laplacian operator. Here τ0 = τ0 = 0 for coupled thermoelasticdiffusion model.

We define the dimensionless quantities:

x′ =w∗

1xc1

, u′ =ρw∗

1c1uβ1T0

, ϕ′ =ρc21ϕ

β1T0, ψ∗′ =

ρζw∗21 ψ

∗

β1T0, T

′=T

T0, C

′=β2C

β1T0,

t′ = w∗1t, τ ′0 = w

∗1τ0, τ0

′= w∗

1τ0, δ1 =

μ +K∗

λ + 2μ +K∗ , δ2 =λ + μ

λ + 2μ +K∗ ,

δ3 =K∗

λ + 2μ +K∗ , δ4 =χ∗

ρζw∗21

, δ5 =f∗w∗2

1

ρc41, δ6 =

(α∗ + β∗

)w∗2

1

ρc41, δ7 =

jw∗21

c21,

δ8 =b∗

ζ(λ + 2μ +K∗) , δ9 =

c∗

ρζw∗21

, δ10 =χ∗

λ + 2μ +K∗ , δ11 =g∗

β1, δ12 =

h∗

β2,

ζ1 =aT0c

21β1

w∗1Kβ2

, ζ2 =β21T0

ρKw∗1, ζ3 =

g∗β1T0c21ρζKw∗3

1

, q∗1 =Dw∗

1β22

ρc41,

q∗2 =Dw∗

1β2a

β1c21

, q∗3 =Dw∗

1b

c21, q∗4 =

Dh∗β2ρζw∗

1c21

.

(2.2)


Here w∗1 = ρCEc

21/K and c1 =

√(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

δ1Δu + δ2 grad div u + δ3 curl ϕ + δ4 gradψ∗ − grad T − gradC = u,

(δ5Δ − 2δ3)ϕ + δ6 grad divϕ + δ3 curlu = δ7ϕ,

(δ8Δ − δ9)ψ∗ − δ10 divu − δ11T − δ12C = ψ∗,

τ0t

(ζ2 div u − ζ3ψ∗ + T + ζ1C

)= ΔT,

q∗1Δdivu + q∗4Δψ∗ + q∗2ΔT − q∗3ΔC + τ0c C = 0,

(2.3)

where

τ0t = 1 + τ0∂

∂t, τ0c = 1 + τ0

∂

∂t. (2.4)

We assume the displacement vector, microrotation, microstretch, temperature change, andconcentration functions as

(u(x, t),ϕ(x, t), ψ∗(x, t), T(x, t), C(x, t)

)= Re

[(u,ϕ, ψ∗, T, C

)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

(δ1Δ +ω2

)u + δ2 grad divu + δ3 curl ϕ + δ4 gradψ∗ − grad T − gradC = 0,

(δ5Δ + μ∗)ϕ + δ6 grad divϕ + δ3 curl u = 0,

−δ10 divu + (δ8Δ + ζ∗)ψ∗ − δ11T − δ12C = 0,

−τ10t[ζ2 divu − ζ3ψ∗ + ζ1C

]+(Δ − τ10t

)T = 0,

q∗1Δ divu + q∗4Δψ∗ + q∗2ΔT − q∗3ΔC + τ10c C = 0,

(2.6)

where

τ10t = −ιω(1 − ιωτ0), τ10c = −ιω(1 − ιωτ0

), μ∗ = δ7ω2 − 2δ3, ζ∗ = ω2 − δ9.

(2.7)

We introduce the matrix differential operator

F(Dx) =∥∥Fgh(Dx)

∥∥9×9, (2.8)


where

Fmn(Dx) =[δ1Δ +ω2

]δmn + δ2

∂2

∂xm∂xn, Fm,n+3(Dx) = Fm+3,n(Dx) = δ3

3∑

r=1

εmrn∂

∂xr,

Fm7(Dx)=δ4∂

∂xm, Fm8(Dx)=Fm9(Dx)=− ∂

∂xm, Fm+3,n+3(Dx)=

(δ5Δ + μ∗)δmn + δ6

∂2

∂xm∂xn,

Fm+3,7(Dx) = F7,n+3(Dx) = Fm+3,8(Dx) = F8,n+3(Dx) = Fm+3,9(Dx) = F9,n+3(Dx) = 0,

F7n(Dx) = −δ10 ∂

∂xn, F77(Dx) = δ8Δ + ζ∗, F78(Dx) = −δ11, F79(Dx) = −δ12,

F8n(Dx) = −ζ2τ10t∂

∂xn, F87(Dx) = ζ3τ10t , F88(Dx) = Δ − τ10t , F89 = −ζ1τ10t ,

F9n(Dx) = q∗1Δ∂

∂xn, F97(Dx) = q∗4Δ, F98(Dx) = q∗2Δ,

F99(Dx) = −q∗3Δ + τ10c , m, n = 1, 2, 3.(2.9)

Here εmrn is alternating tensor and δmn is the Kronecker delta function.The system of equations (2.6) can be written as

F(Dx)U(x) = 0, (2.10)

where U = (u,ϕ, ψ∗, T, C) is a nine-component vector function on E3.

Definition 2.1. The fundamental solution of the system of equations (2.6) (the fundamentalmatrix of operator F) is the matrix G(x) = ‖Ggh(x)‖9×9 satisfying condition [25]

F(Dx)G(x) = δ(x)I(x), (2.11)

where δ is the Dirac delta, I = ‖δgh‖9×9 is the unit matrix, and x ε E3.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

δ1Δu + δ2 grad div u + δ3 curlϕ − δ10 gradψ∗ − ζ2τ10t grad T + q∗1 gradC +ω2u = H′, (3.1)(δ5Δ + μ∗)ϕ + δ6 grad div ϕ + δ3 curl u = H′′, (3.2)

δ4 div u + (δ8Δ + ζ∗)ψ∗ + ζ3τ10t T + q∗4C = Z, (3.3)

−div u − δ11ψ∗ +(Δ − τ10t

)T + q∗2C = L, (3.4)

−Δdiv u − δ12Δψ∗ − ζ1τ10t ΔT − q∗3ΔC + τ10c C =M, (3.5)


where H′ and H′′ are three-component vector functions on E3and Z,L, and M are scalarfunctions on E3.

The system of equations (3.1)–(3.5)may be written in the form

Ftr(Dx)U(x) = Q(x), (3.6)

where Ftr is the transpose of matrix F, Q = (H′,H′′, Z, L,M), and x ε E3.Applying the operator div to (3.1) and (3.2), we obtain

(Δ +ω2

)divu − δ10Δψ∗ − ζ2τ10t ΔT + q∗1ΔC = divH′,

(υ∗Δ + μ∗)divϕ = divH′′,

δ4 divu + (δ8Δ + ζ∗)ψ∗ + ζ3τ10t T + q∗4C = Z,

−divu − δ11ψ∗ +(Δ − τ10t

)T + q∗2C = L,

−Δdivu − δ12Δψ∗ − ζ1τ10t ΔT − q∗3ΔC + τ10c C =M,

(3.7)

where υ∗ = δ5 + δ6.Equations (3.7)1, (3.7)3, (3.7)4, and (3.7)5 may be written in the form

N(Δ)S = Q, (3.8)

where S = (divu, ψ∗, T, C),Q = (d1, d2, d3, d4) = (divH′, Z, L,M), and

N(Δ) = ‖Nmn(Δ)‖4×4 =

∥∥∥∥∥∥∥∥∥∥∥

Δ +ω2 −δ10Δ −ζ2τ10t Δ q∗1Δ

δ4 δ8Δ + ζ∗ ζ3τ10t q∗4

−1 −δ11 Δ − τ10t q∗2−Δ −δ12Δ −ζ1τ10t Δ −q∗3Δ + τ10c

∥∥∥∥∥∥∥∥∥∥∥4×4

. (3.9)

Equations (3.7)1, (3.7)3, (3.7)4, and (3.7)5 may be also written as

Γ1(Δ)S = Ψ, (3.10)

where

Ψ = (Ψ1,Ψ2,Ψ3,Ψ4), Ψn = e∗4∑

m=1

N∗mndm,

Γ1(Δ) = e∗ detN(Δ), e∗ = − 1q∗3δ9

, n = 1, 2, 3, 4,

(3.11)

andN∗mn is the cofactor of the elementsNmn of the matrix N.


From (3.9) and (3.11), we see that

Γ1(Δ) =4∏

m=1

(Δ + λ2m

), (3.12)

where λ2m,m = 1, 2, 3, 4 are the roots of the equation Γ1(−κ) = 0 (with respect to κ).From (3.7)2, it follows that

(Δ + λ27

)divϕ =

1δ∗

divH′′, (3.13)

where λ27 = μ∗/υ∗.

Applying the operators δ5Δ+ μ∗ and δ3 curl to (3.1) and (3.2), respectively, we obtain

(δ5Δ + μ∗)

[δ1Δu + δ2 grad div u +ω2u

]+ δ3

(δ5Δ + μ∗) curlϕ

=(δ5Δ + μ∗)

[H′ + δ10 gradψ∗ + ζ2τ10t grad T − q∗1 gradC

],

(3.14)

δ3(δ5Δ + μ∗) curl ϕ = −δ23 curl curl u + δ3 curl H′′. (3.15)

Now

curl curl u = grad div u −Δu. (3.16)

Using (3.15) and (3.16) in (3.14), we obtain

(δ5Δ + μ∗)

[δ1Δu + δ2 grad divu +ω2u

]+ δ23Δu − δ23 grad divu

=(δ5Δ + μ∗)


]− δ3 curl H′′.

(3.17)

The above equation can also be written as

{[(δ5Δ + μ∗)δ1 + δ23

]Δ +

(δ5Δ + μ∗)ω2

}u

= −[δ2(δ5Δ + μ∗) − δ23

]grad div u

+(δ5Δ + μ∗)


]− δ3 curlH′′.

(3.18)


Applying the operator Γ1(Δ) to the (3.18) and using (3.10), we get

Γ1(Δ)[δ5δ1Δ2 +

(μ∗δ1 + δ5ω2 + δ23

)Δ + μ∗ω2

]u

= −[δ2(δ5Δ + μ∗) − δ23

]gradΨ1

+(δ5Δ + μ∗)

[Γ1(Δ)H′ + δ10 gradΨ2 + ζ2τ10t gradΨ3 − q∗1 gradΨ4

]− δ3Γ1(Δ) curlH′′.

(3.19)

The above equation may be written in the form

Γ1(Δ)Γ2(Δ)u = Ψ′, (3.20)

where

Γ2(Δ) = f∗ det

∥∥∥∥∥

δ1Δ +ω2 δ3Δ

−δ3 δ5Δ + μ∗

∥∥∥∥∥2×2, f∗ =

1δ1δ5

, (3.21)

Ψ′ = f∗{−[δ2(δ5Δ + μ∗) − δ23

]gradΨ1

+(δ5Δ + μ∗)

[Γ1(Δ)H′ + δ10 grad Ψ2 + ζ2τ10t gradΨ3 − q∗1 gradΨ4

]

−δ3Γ1(Δ) curlH′′}.

(3.22)

It can be seen that

Γ2(Δ) =(Δ + λ25

)(Δ + λ26

), (3.23)

where λ25, λ26 are the roots of the equation Γ2(−κ) = 0 (with respect to κ).

Applying the operators δ3 curl and δ1Δ+ω2 to (3.1) and (3.2), respectively, we obtain

δ3(δ1Δ +ω2

)curlu = δ3 curlH′ − δ23 curl curlϕ, (3.24)

(δ1Δ +ω2

)(δ5Δ + μ∗)ϕ + δ6

(δ1Δ +ω2

)grad div ϕ + δ3

(δ1Δ +ω2

)curl u =

(δ1Δ +ω2

)H′′.

(3.25)

Now

curl curl ϕ = grad div ϕ −Δϕ. (3.26)


Using (3.24) and (3.26) in (3.25), we obtain

(δ1Δ +ω2

)(δ5Δ + μ∗)ϕ + δ6

(δ1Δ +ω2

)grad divϕ + δ23Δϕ − δ23 grad div ϕ

=(δ1Δ +ω2)H′′ − δ3 curl H′ .

(3.27)

The above equation may also be written as

{[(δ5Δ + μ∗)δ1 + δ23

]Δ +

(δ5Δ + μ∗)ω2

}ϕ

= −[δ6(δ1Δ +ω2

)− δ23

]grad div ϕ +

(δ1Δ +ω2

)H′′ − δ3 curlH′.

(3.28)

Applying the operator Δ + λ27 to the (3.28) and using (3.13), we get

(Δ + λ27

)[δ5δ1Δ2 +

(μ∗δ1 + δ5ω2 + δ23

)Δ + μ∗ω2

]ϕ

= −δ3(Δ + λ27

)curl H′ +

(Δ + λ27

)(δ1Δ +ω2

)H′′ − 1

υ∗[δ6(δ1Δ +ω2

)− δ23

]grad div H′′.

(3.29)

The above equation may also be rewritten in the form

Γ2(Δ)(Δ + λ27

)ϕ = Ψ′′, (3.30)

where

Ψ′′ = f∗{− δ3

(Δ + λ27

)curl H′ +

(Δ + λ27

)(δ1Δ +ω2

)H′′

− 1υ∗

[δ6(δ1Δ +ω2

)− δ23

]grad div H′′

}.

(3.31)

From (3.10), (3.20), and (3.30), we obtain

Θ(Δ)U(x) = Ψ(x), (3.32)


where Ψ = (Ψ′,Ψ′′,Ψ2,Ψ3,Ψ4)

Θ(Δ) =∥∥Θgh(Δ)

∥∥9×9,

Θmm(Δ) = Γ1(Δ)Γ2(Δ) =6∏

q=1

(Δ + λ2q

),

Θm+3,n+3(Δ) = Γ2(Δ)(Δ + λ27

)=

7∏

q=5

(Δ + λ2q

),

Θgh(Δ) = 0, Θ77(Δ) = Θ88(Δ) = Θ99(Δ) = Γ1(Δ),

m = 1, 2, 3, g, h = 1, 2, . . . , 9, g /=h.

(3.33)

Equations (3.11), (3.22), and (3.31) can be rewritten in the form

Ψ′ =[f∗(δ5Δ + μ∗)Γ1(Δ)J + q11(Δ) grad div

]H′ + q21(Δ) curl H′′

+ q31(Δ) gradZ + q41(Δ) gradL + q51(Δ)gradM,

Ψ′′ = q12(Δ) curl H′ +{f∗

(Δ + λ27

)(δ1Δ +ω2

)J + q22(Δ)grad div

}H′′,

Ψ2 = q13(Δ) div H′ + q33(Δ)Z + q43(Δ)L + q53(Δ)M,



(3.34)

where J = ‖δgh‖3×3 is the unit matrix.In (3.34), we have used the following notations:

qm1(Δ) = f∗e∗{(δ5Δ + μ∗)

[δ10N

∗m2 + ζ2τ

10t N

∗m3 − q∗1N∗

m4

]−(δ2(δ5Δ + μ∗) − δ23

)N∗

m1

},

q21(Δ) = −f∗δ3Γ1(Δ), q12(Δ) = −f∗δ3(Δ + λ27

), q22(Δ) = −f

∗

υ∗[δ6(δ1Δ +ω2) − δ23

],

qmn(Δ) = e∗N∗mn, m = 1, 3, 4, 5, n = 3, 4, 5.

(3.35)

Now from (3.34), we have that

Ψ(x) = Rtr(Dx)Q(x), (3.36)


where

R =∥∥Rgh

∥∥9×9,

Rmn(Dx) = f∗(δ5Δ + μ∗)Γ1(Δ)δmn + q11(Δ)∂2

∂xm∂xn,

Rm,n+3(Dx) = q12(Δ)3∑

r=1

εmrn∂

∂xr, Rmp(Dx) = q1,p−4(Δ)

∂

∂xm,

Rm+3,n(Dx) = q21(Δ)3∑

r=1

εmrn∂

∂xr,

Rm+3,n+3(Dx) = f∗(Δ + λ27

)(δ1Δ +ω2

)δmn + q22(Δ)

∂2

∂xm∂xn,

Rm+3,p(Dx) = Rp,m+3(Dx) = 0,

Rpm(Dx) = qp−4,1(Δ)∂

∂xn, Rps(Dx) = qp−4,s−4(Δ), m = 1, 2, 3, p, s = 7, 8, 9.

(3.37)


ΘU = RtrFtrU. (3.38)

It implies that

RtrFtr = Θ,

F(Dx)R(Dx) = Θ(Δ).(3.39)

We assume that

λ2m /=λ2n /= 0, m, n = 1, 2, 3, 4, 5, 6, 7 m/=n. (3.40)

Let

Y(x) = ‖Yrs(x)‖9×9, Ymm(x) =6∑

n=1

r1nςn(x), Ym+3,m+3(x) =7∑

n=5

r2nςn(x),

Y77(x) = Y88(x) = Y99(x) =4∑

n=1

r3nςn(x),

Yvw(x) = 0, m = 1, 2, 3, v,w = 1, 2, . . . , 9, v /=w,

(3.41)


where

ςn(x) = − 14π |x| exp(ιλn|x|), n = 1, 2, . . . , 7,

r1l =6∏

m=1,m/= l

(λ2m − λ2l

)−1, l = 1, 2, 3, 4, 5, 6,

r2v =7∏

m=5,m/=v

(λ2m − λ2v

)−1, v = 5, 6, 7,

r3w =4∏

m=1,m/=w

(λ2m − λ2w

)−1, w = 1, 2, 3, 4.

(3.42)

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). (3.43)

Proof. To prove the lemma, it is sufficient to prove that

Γ1(Δ)Γ2(Δ)Y11(x) = δ(x), Γ2(Δ)(Δ + λ27

)Y44(x) = δ(x), Γ1(Δ)Y77(x) = δ(x).

(3.44)

We find that

r11 + r12 + r13 + r14 + r15 + r16 = 0,

r12(λ21 − λ22

)+ r13

(λ21 − λ23

)+ r14

(λ21 − λ24

)+ r15

(λ21 − λ25

)+ r16

(λ21 − λ26

)= 0,

r13(λ21 − λ23

)(λ22 − λ23

)+ r14

(λ21 − λ24

)(λ22 − λ24

)+ r15

(λ21 − λ25

)(λ22 − λ25

)

+ r16(λ21 − λ26

)(λ22 − λ26

)= 0,

r14(λ21 − λ24

)(λ22 − λ24

)(λ23 − λ24

)+ r15

(λ21 − λ25

)(λ22 − λ25

)(λ23 − λ25

)

+ r16(λ21 − λ26

)(λ22 − λ26

)(λ23 − λ26

)= 0,

r15(λ21 − λ25

)(λ22 − λ25

)(λ23 − λ25

)(λ24 − λ25

)+ r16

(λ21 − λ26

)(λ22 − λ26

)(λ23 − λ26

)(λ24 − λ26

)= 0,

r16(λ21 − λ26

)(λ22 − λ26

)(λ23 − λ26

)(λ24 − λ25

)(λ25 − λ26

)= 1,

(Δ + λ2m

)ςn(x) = δ(x) +

(λ2m − λ2n

)ςn(x), m, n = 1, 2, 3, 4, 5, 6.

(3.45)


Now consider

Γ1(Δ)Γ2(Δ)Y11(x) =(Δ + λ22

)(Δ + λ23

)(Δ + λ24

)(Δ + λ25

)(Δ + λ26

) 6∑

n=1

r1n[δ +

(λ21 − λ2n

)ςn]

=(Δ + λ22

)(Δ + λ23

)(Δ + λ24

)(Δ + λ25

)(Δ + λ26

) 6∑

n=2

r1n(λ21 − λ2n

)ςn

=(Δ + λ23

)(Δ + λ24

)(Δ + λ25

)(Δ + λ26

) 6∑

n=2

r1n(λ21 − λ2n

)[δ +

(λ22 − λ2n

)ςn]

=(Δ + λ23

)(Δ + λ24

)(Δ + λ25

)(Δ + λ26

) 6∑

n=3

r1n(λ21 − λ2n

)(λ22 − λ2n

)ςn

=(Δ + λ24

)(Δ + λ25

)(Δ + λ26

) 6∑

n=3

r1n(λ21 − λ2n

)(λ22 − λ2n

)[δ +

(λ23 − λ2n

)ςn]

=(Δ + λ24

)(Δ + λ25

)(Δ + λ26

) 6∑

n=4

r1n(λ21 − λ2n

)(λ22 − λ2n

)(λ23 − λ2n

)ςn

=(Δ + λ25

)(Δ + λ26

) 6∑

n=4

r1n(λ21 − λ2n

)(λ22 − λ2n

)(λ23 − λ2n

)[δ +

(λ24 − λ2n

)ςn]

=(Δ + λ26

) 6∑

n=5

r1n(λ21 − λ2n

)(λ22 − λ2n

)(λ23 − λ2n

)(λ24 − λ2n

)[δ +

(λ25 − λ2n

)ςn]

=(Δ + λ26

)ς6 = δ.

(3.46)

Similarly, (3.44)2 and (3.44)3 can be proved.We introduce the matrix

G(x) = R(Dx)Y(x). (3.47)


F(Dx)G(x) = F(Dx)R(Dx)Y(x) = Θ(Δ)Y(x) = δ(x)I(x). (3.48)

Hence, G(x) is a solution to (2.11).

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).


4. Basic Properties of the Matrix G(x)

Property 1. Each column of the matrix G(x) is the solution of the system of equations (2.6) atevery point x ε E3 except the origin.

Property 2. The matrix G(x) can be written in the form

G =∥∥Ggh

∥∥9×9,

Gmn(x) = Rmn(Dx)Y11(x),

Gm,n+3(x) = Rm,n+3(Dx)Y44(x),

Gmp(x) = Rmp(Dx)Y77(x), m = 1, 2, . . . , 9, n = 1, 2, 3, p = 7, 8, 9.

(4.1)

5. Special Cases

(i) If we neglect the diffusion effect, we obtain the same results for fundamental solutionas discussed by Svanadze and De Cicco [23] by changing the dimensionless quantities intophysical quantities in case of coupled theory of thermoelasticity.

(ii) If we neglect the thermal and diffusion effects, we obtain the same resultsfor fundamental solution as discussed by Svanadze [22] by changing the dimensionlessquantities into physical quantities.

(iii) If we neglect both micropolar and microstretch effects, the same results forfundamental solution can be obtained as discussed by Kumar and Kansal [27] in case ofthe Lord-Shulman theory of thermoelastic diffusion.

6. Conclusions

The fundamental solution G(x) of the system of equations (2.6) makes it possible toinvestigate three-dimensional boundary value problems of generalized theory of thermomi-crostretch elastic diffusive solids by potential method [28].

Acknowledgment

Mr. T. Kansal is thankful to the Council of Scientific and Industrial Research (CSIR) for thefinancial support.

References

[1] A. C. Eringen, “Micropolar elastic solids with stretch,” M. I. Anisina, Ed., pp. 1–18, Ari KitabeviMatbassi, Istanbul, Turkey, 1971.

[2] A. C. Eringen, “Linear theory of micropolar elasticity,” Indiana University Mathematics Journal, vol. 15,pp. 909–923, 1966.

[3] A. C. Eringen, “Theory of micropolar elasticity,” in Fracture, H. Liebowitz, Ed., vol. 2, pp. 621–729,Academic Press, New York, NY, USA, 1968.

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[8] R. S. Dhaliwal and H. H. Sherief, “Generalized thermoelasticity for anisotropic media,” Quarterly ofApplied Mathematics, vol. 38, no. 1, pp. 1–8, 1980.

[9] D. S. Chanderashekhariah, “Thermoelasticity with second: a review,” Applied Mechanics Review, vol.39, pp. 355–376, 1986.

[10] R. B. Hetnarski and J. Ignaczak, “Generalized thermoelasticity,” Journal of Thermal Stresses, vol. 22, no.4-5, pp. 451–476, 1999.

[11] W.Nowacki, “Dynamical problem of thermodiffusion in solids—I,” Bulletin de l’Academie Polonaise desSciences. Serie des Sciences Techniques, vol. 22, pp. 55–64, 1974.

[12] W. Nowacki, “Dynamical problem of thermodiffusion in solids—II,” Bulletin de l’Academie Polonaisedes Sciences. Serie des Sciences Techniques, vol. 22, pp. 129–135, 1974.

[13] W. Nowacki, “Dynamical problem of thermodiffusion in solids—III,” Bulletin de l’Academie Polonaisedes Sciences. Serie des Sciences Techniques, vol. 22, pp. 275–276, 1974.

[14] W. Nowacki, “Dynamical problems of thermodiffusion in solids,” Engineering Fracture Mechanics, vol.8, pp. 261–266, 1976.

[15] W. Dudziak and S. J. Kowalski, “Theory of thermodiffusion for solids,” International Journal of Heatand Mass Transfer, vol. 32, no. 11, pp. 2005–2013, 1989.

[16] Z. S. Olesiak and Y. A. Pyryev, “A coupled quasi-stationary problem of thermodiffusion for an elasticcylinder,” International Journal of Engineering Science, vol. 33, no. 6, pp. 773–780, 1995.

[17] 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. 591–608, 2004.

[18] M. Aouadi, “Uniqueness and reciprocity theorems in the theory of generalized thermoelasticdiffusion,” Journal of Thermal Stresses, vol. 30, no. 7, pp. 665–678, 2007.

[19] M. Aouadi, “Generalized theory of thermoelastic diffusion for anisotropic media,” Journal of ThermalStresses, vol. 31, no. 3, pp. 270–285, 2008.

[20] R. B. Hetnarski, “The fundamental solution of the coupled thermoelastic problem for small times,”Archiwwn Mechaniki Stosowanej, vol. 16, pp. 23–31, 1964.

[21] R. B. Hetnarski, “Solution of the coupled problem of thermoelasicity in the form of series offunctions,” Archiwwn Mechaniki Stosowanej, vol. 16, pp. 919–941, 1964.

[22] M. Svanadze, “Fundamental solution of the system of equations of steady oscillations in the theoryof microstretch elastic solids,” International Journal of Engineering Science, vol. 42, no. 17-18, pp. 1897–1910, 2004.

[23] M. Svanadze and S. De Cicco, “Fundamental solution in the theory of thermomicrostretch elasticsolids,” International Journal of Engineering Science, vol. 43, no. 5-6, pp. 417–431, 2005.

[24] M. Svanadze and R. Tracina, “Representations of solutions in the theory of thermoelasticity withmicrotemperatures for microstretch solids,” Journal of Thermal Stresses, vol. 34, pp. 161–178, 2011.

[25] L. Hormander, Linear Partial Differential Operators, Springer, Berlin, Germany, 1963.[26] L. Hormander, The Analysis of Linear Partial Differential Operators—II, vol. 257, Springer, Berlin,

Germany, 1983.[27] 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. Kupradze, T. G. Gegelia, M. O. Basheleıshvili, and T. V. Burchuladze, Three-Dimensional Problems

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