# Fundamental Solution in the Theory of Thermomicrostretch Elastic Diffusive Solids

Authors: Rajneesh Kumar; Tarun Kansal
Journal: ISRN Applied Mathematics
 (2011)

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## Abstract

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.


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## Body

## 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. 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 high-concentration 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 [11–14] 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 [20, 21] 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 [25, 26].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.


## 2. Basic Equations

Letx=(x1,x2,x3) be the point of the Euclidean three-dimensional space E3: |x|=(x12+x22+x32)1/2,  Dx=(∂/∂x1,∂/∂x2,∂/∂x3) 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(2.1)(μ+K*)Δu¯+(λ+μ)graddiv⁡u¯+K*curl⁡φ¯+χ*gradψ*¯-β1gradT¯-β2gradC¯=ρu¯̈,(f*Δ-2K*)φ¯+(α*+β*)graddiv⁡φ¯+K*curl⁡u¯=ρjφ¯̈,(b*Δ-c*)ψ*¯-χ*div⁡u¯-g*T¯-h*C¯=ρζψ*¯̈,(1+τ0∂∂t)(β1T0div⁡u¯̇-g*T0ψ*¯̇+ρCET¯̇+aT0C¯̇)=KΔT¯,Dβ2Δdiv⁡u¯++Dh*Δψ*¯+DaΔT¯-DbΔC¯+C¯̇+τ0C¯̈=0,
where β1=(3λ+2μ+K*)αt, β2=(3λ+2μ+K*)αc. Here αt,αc are the coefficients of 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 the microstretch function; ρ,CE are, respectively, the density and specific heat at constant strain; 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 temperature T0(T0≠0) and C¯ is the concentration; τ0 is diffusion relaxation time and τ0 is thermal relaxation time; Δ is the Laplacian operator. Here τ0=τ0=0 for coupled thermoelastic diffusion model.We define the dimensionless quantities:(2.2)x′=w1*xc1,u¯′=ρw1*c1u¯β1T0,φ¯′=ρc12φ¯β1T0,ψ*¯′=ρζw1*2ψ*¯β1T0,T¯′=T¯T0,C¯′=β2C¯β1T0,t′=w1*t,τ0′=w1*τ0,τ0′=w1*τ0,δ1=μ+K*λ+2μ+K*,δ2=λ+μλ+2μ+K*,δ3=K*λ+2μ+K*,δ4=χ*ρζw1*2,δ5=f*w1*2ρc14,δ6=(α*+β*)w1*2ρc14,δ7=jw1*2c12,δ8=b*ζ(λ+2μ+K*),δ9=c*ρζw1*2,δ10=χ*λ+2μ+K*,δ11=g*β1,δ12=h*β2,ζ1=aT0c12β1w1*Kβ2,ζ2=β12T0ρKw1*,ζ3=g*β1T0c12ρζKw1*3,q1*=Dw1*β22ρc14,q2*=Dw1*β2aβ1c12,q3*=Dw1*bc12,q4*=Dh*β2ρζw1*c12.
Here w1*=ρCEc12/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(2.3)δ1Δu¯+δ2graddiv⁡u¯+δ3curl⁡φ¯+δ4gradψ*¯-gradT¯-gradC¯=u¯̈,(δ5Δ-2δ3)φ¯+δ6graddiv⁡φ¯+δ3curl⁡u¯=δ7φ¯̈,(δ8Δ-δ9)ψ*¯-δ10div⁡u¯-δ11T¯-δ12C¯=ψ*¯̈,τt0(ζ2div⁡u¯̇-ζ3ψ*¯̇+T¯̇+ζ1C¯̇)=ΔT¯,q1*Δdiv⁡u¯+q4*Δψ*¯+q2*ΔT¯-q3*ΔC¯+τc0C¯̇=0,
where(2.4)τt0=1+τ0∂∂t,τc0=1+τ0∂∂t.
We assume the displacement vector, microrotation, microstretch, temperature change, and concentration functions as(2.5)(u¯(x,t),φ¯(x,t),ψ*¯(x,t),T¯(x,t),C¯(x,t))=Re[(u,φ,ψ*,T,C)e-ιωt],
where ω is oscillation frequency and ω>0.Using (2.5) into (2.3), we obtain the system of equations of steady oscillations as(2.6)(δ1Δ+ω2)u+δ2graddiv⁡u+δ3curl⁡φ+δ4gradψ*-gradT-gradC=0,(δ5Δ+μ*)φ+δ6graddiv⁡φ+δ3curl⁡u=0,-δ10div⁡u+(δ8Δ+ζ*)ψ*-δ11T-δ12C=0,-τt10[ζ2div⁡u-ζ3ψ*+ζ1C]+(Δ-τt10)T=0,q1*Δdiv⁡u+q4*Δψ*+q2*ΔT-q3*ΔC+τc10C=0,
where(2.7)τt10=-ιω(1-ιωτ0),τc10=-ιω(1-ιωτ0),μ*=δ7ω2-2δ3,ζ*=ω2-δ9.We introduce the matrix differential operator(2.8)F(Dx)=‖Fgh(Dx)‖9×9,
where(2.9)Fmn(Dx)=[δ1Δ+ω2]δmn+δ2∂2∂xm∂xn,Fm,n+3(Dx)=Fm+3,n(Dx)=δ3∑r=13ɛmrn∂∂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)=-ζ2τt10∂∂xn,F87(Dx)=ζ3τt10,F88(Dx)=Δ-τt10,F89=-ζ1τt10,F9n(Dx)=q1*Δ∂∂xn,F97(Dx)=q4*Δ,F98(Dx)=q2*Δ,F99(Dx)=-q3*Δ+τc10,m,n=1,2,3.
Here ɛmrn is alternating tensor and δmn is the Kronecker delta function.The system of equations (2.6) can be written as(2.10)F(Dx)U(x)=0,
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 fundamental matrix of operator F) is the matrix G(x)=∥Ggh(x)∥9×9 satisfying condition [25]
(2.11)F(Dx)G(x)=δ(x)I(x),
where δ is the Dirac delta, I=∥δgh∥9×9 is the unit matrix, and xϵE3.
Now we constructG(x) in terms of elementary functions.


## 3. Fundamental Solution of System of Equations of Steady Oscillations

We consider the system of equations(3.1)δ1Δu+δ2graddiv⁡u+δ3curl⁡φ-δ10gradψ*-ζ2τt10gradT+q1*gradC+ω2u=H′,(3.2)(δ5Δ+μ*)φ+δ6graddiv⁡φ+δ3curl⁡u=H′′,(3.3)δ4div⁡u+(δ8Δ+ζ*)ψ*+ζ3τt10T+q4*C=Z,(3.4)-div⁡u-δ11ψ*+(Δ-τt10)T+q2*C=L,(3.5)-Δdiv⁡u-δ12Δψ*-ζ1τt10ΔT-q3*ΔC+τc10C=M,
where H′ and H′′ are three-component vector functions on E3and  Z,L, and M are scalar functions on E3.The system of equations (3.1)–(3.5) may be written in the form(3.6)Ftr⁡(Dx)U(x)=Q(x),
where Ftr is the transpose of matrix F, Q=(H′,H′′,Z,L,M), and xϵE3.Applying the operator div to (3.1) and (3.2), we obtain(3.7)(Δ+ω2)div⁡u-δ10Δψ*-ζ2τt10ΔT+q1*ΔC=div⁡H′,(υ*Δ+μ*)div⁡φ=div⁡H′′,δ4div⁡u+(δ8Δ+ζ*)ψ*+ζ3τt10T+q4*C=Z,-div⁡u-δ11ψ*+(Δ-τt10)T+q2*C=L,-Δdiv⁡u-δ12Δψ*-ζ1τt10ΔT-q3*ΔC+τc10C=M,
where υ*=δ5+δ6.Equations (3.7)1, (3.7)3, (3.7)4, and (3.7)5 may be written in the form(3.8)N(Δ)S=Q¯,
where S=(divu,ψ*,T,C), Q¯=(d1,d2,d3,d4)=(divH′,Z,L,M), and(3.9)N(Δ)=‖Nmn(Δ)‖4×4=‖Δ+ω2-δ10Δ-ζ2τt10Δq1*Δδ4δ8Δ+ζ*ζ3τt10q4*-1-δ11Δ-τt10q2*-Δ-δ12Δ-ζ1τt10Δ-q3*Δ+τc10‖4×4.
Equations (3.7)1, (3.7)3, (3.7)4, and (3.7)5 may be also written as(3.10)Γ1(Δ)S=Ψ,
where(3.11)Ψ=(Ψ1,Ψ2,Ψ3,Ψ4),Ψn=e*∑m=14Nmn*dm,Γ1(Δ)=e*det⁡N(Δ),e*=-1q3*δ9,n=1,2,3,4,
and Nmn* is the cofactor of the elements Nmn of the matrix N.From (3.9) and (3.11), we see that(3.12)Γ1(Δ)=∏m=14(Δ+λm2),
where λm2, 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(3.13)(Δ+λ72)div⁡φ=1δ*div⁡H′′,
where λ72=μ*/υ*.Applying the operatorsδ5Δ+μ* and δ3curl to (3.1) and (3.2), respectively, we obtain(3.14)(δ5Δ+μ*)[δ1Δu+δ2graddiv⁡u+ω2u]+δ3(δ5Δ+μ*)curl⁡φ=(δ5Δ+μ*)[H′+δ10gradψ*+ζ2τt10gradT-q1*gradC],(3.15)δ3(δ5Δ+μ*)curl⁡φ=-δ32curl⁡curl⁡u+δ3curl⁡H′′.
Now(3.16)curl⁡curl⁡u=graddiv⁡u-Δu.
Using (3.15) and (3.16) in (3.14), we obtain(3.17)(δ5Δ+μ*)[δ1Δu+δ2graddiv⁡u+ω2u]+δ32Δu-δ32graddiv⁡u=(δ5Δ+μ*)[H′+δ10gradψ*+ζ2τt10gradT-q1*gradC]-δ3curl⁡H′′.
The above equation can also be written as(3.18){[(δ5Δ+μ*)δ1+δ32]Δ+(δ5Δ+μ*)ω2}u=-[δ2(δ5Δ+μ*)-δ32]graddiv⁡u+(δ5Δ+μ*)[H′+δ10gradψ*+ζ2τt10gradT-q1*gradC]-δ3curl⁡H′′.
Applying the operator Γ1(Δ) to the (3.18) and using (3.10), we get(3.19)Γ1(Δ)[δ5δ1Δ2+(μ*δ1+δ5ω2+δ32)Δ+μ*ω2]u=-[δ2(δ5Δ+μ*)-δ32]gradΨ1+(δ5Δ+μ*)[Γ1(Δ)H′+δ10gradΨ2+ζ2τt10gradΨ3-q1*gradΨ4]-δ3Γ1(Δ)curl⁡H′′.
The above equation may be written in the form(3.20)Γ1(Δ)Γ2(Δ)u=Ψ′,
where(3.21)Γ2(Δ)=f*det⁡‖δ1Δ+ω2δ3Δ-δ3δ5Δ+μ*‖2×2,f*=1δ1δ5,(3.22)Ψ′=f*{-[δ2(δ5Δ+μ*)-δ32]gradΨ1+(δ5Δ+μ*)[Γ1(Δ)H′+δ10gradΨ2+ζ2τt10gradΨ3-q1*gradΨ4]-δ3Γ1(Δ)curl⁡H′′[δ2(δ5Δ+μ*)-δ32]}.
It can be seen that(3.23)Γ2(Δ)=(Δ+λ52)(Δ+λ62),
where λ52, λ62 are the roots of the equation Γ2(-κ)=0 (with respect to κ).Applying the operatorsδ3curl and δ1Δ+ω2 to (3.1) and (3.2), respectively, we obtain(3.24)δ3(δ1Δ+ω2)curl⁡u=δ3curl⁡H′-δ32curl⁡curl⁡φ,(3.25)(δ1Δ+ω2)(δ5Δ+μ*)φ+δ6(δ1Δ+ω2)graddiv⁡φ+δ3(δ1Δ+ω2)curl⁡u=(δ1Δ+ω2)H′′.
Now(3.26)curl⁡curl⁡φ=graddiv⁡φ-Δφ.
Using (3.24) and (3.26) in (3.25), we obtain(3.27)(δ1Δ+ω2)(δ5Δ+μ*)φ+δ6(δ1Δ+ω2)graddiv⁡φ+δ32Δφ-δ32graddiv⁡φ=(δ1Δ+ω2)H′′-δ3curl⁡H′.
The above equation may also be written as(3.28){[(δ5Δ+μ*)δ1+δ32]Δ+(δ5Δ+μ*)ω2}φ=-[δ6(δ1Δ+ω2)-δ32]graddiv⁡φ+(δ1Δ+ω2)H′′-δ3curl⁡H′.
Applying the operator Δ+λ72 to the (3.28) and using (3.13), we get(3.29)(Δ+λ72)[δ5δ1Δ2+(μ*δ1+δ5ω2+δ32)Δ+μ*ω2]φ=-δ3(Δ+λ72)curl⁡H′+(Δ+λ72)(δ1Δ+ω2)H′′-1υ*[δ6(δ1Δ+ω2)-δ32]graddiv⁡H′′.
The above equation may also be rewritten in the form(3.30)Γ2(Δ)(Δ+λ72)φ=Ψ′′,
where(3.31)Ψ′′=f*{1υ*-δ3(Δ+λ72)curl⁡H′+(Δ+λ72)(δ1Δ+ω2)H′′-1υ*[δ6(δ1Δ+ω2)-δ32]graddiv⁡H′′}.
From (3.10), (3.20), and (3.30), we obtain(3.32)Θ(Δ)U(x)=Ψ̂(x),
where Ψ̂=(Ψ′,Ψ′′,Ψ2,Ψ3,Ψ4)(3.33)Θ(Δ)=‖Θgh(Δ)‖9×9,Θmm(Δ)=Γ1(Δ)Γ2(Δ)=∏q=16(Δ+λq2),Θm+3,n+3(Δ)=Γ2(Δ)(Δ+λ72)=∏q=57(Δ+λq2),Θgh(Δ)=0,Θ77(Δ)=Θ88(Δ)=Θ99(Δ)=Γ1(Δ),m=1,2,3,g,h=1,2,…,9,g≠h.Equations (3.11), (3.22), and (3.31) can be rewritten in the form(3.34)Ψ′=[f*(δ5Δ+μ*)Γ1(Δ)J+q11(Δ)graddiv⁡]H′+q21(Δ)curl⁡H′′+q31(Δ)gradZ+q41(Δ)gradL+q51(Δ)gradM,Ψ′′=q12(Δ)curl⁡H′+{f*(Δ+λ72)(δ1Δ+ω2)J+q22(Δ)graddiv⁡}H′′,Ψ2=q13(Δ)div⁡H′+q33(Δ)Z+q43(Δ)L+q53(Δ)M,Ψ3=q14(Δ)div⁡H′+q34(Δ)Z+q44(Δ)L+q54(Δ)M,Ψ4=q15(Δ)div⁡H′+q35(Δ)Z+q45(Δ)L+q55(Δ)M,
where J=∥δgh∥3×3 is the unit matrix.In (3.34), we have used the following notations: (3.35)qm1(Δ)=f*e*{(δ5Δ+μ*)[δ10Nm2*+ζ2τt10Nm3*-q1*Nm4*]-(δ2(δ5Δ+μ*)-δ32)Nm1*},q21(Δ)=-f*δ3Γ1(Δ),q12(Δ)=-f*δ3(Δ+λ72),q22(Δ)=-f*υ*[δ6(δ1Δ+ω2)-δ32],qmn(Δ)=e*Nmn*,m=1,3,4,5,n=3,4,5.
Now from (3.34), we have that(3.36)Ψ̂(x)=Rtr⁡(Dx)Q(x),
where(3.37)R=‖Rgh‖9×9,Rmn(Dx)=f*(δ5Δ+μ*)Γ1(Δ)δmn+q11(Δ)∂2∂xm∂xn,Rm,n+3(Dx)=q12(Δ)∑r=13ɛmrn∂∂xr,Rmp(Dx)=q1,p-4(Δ)∂∂xm,Rm+3,n(Dx)=q21(Δ)∑r=13ɛmrn∂∂xr,Rm+3,n+3(Dx)=f*(Δ+λ72)(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.
From (3.6), (3.32), and (3.36), we obtain(3.38)ΘU=Rtr⁡Ftr⁡U.
It implies that(3.39)Rtr⁡Ftr⁡=Θ,F(Dx)R(Dx)=Θ(Δ).
We assume that(3.40)λm2≠λn2≠0,m,n=1,2,3,4,5,6,7m≠n.
Let(3.41)Y(x)=‖Yrs(x)‖9×9,Ymm(x)=∑n=16r1nςn(x),Ym+3,m+3(x)=∑n=57r2nςn(x),Y77(x)=Y88(x)=Y99(x)=∑n=14r3nςn(x),Yvw(x)=0,m=1,2,3,v,w=1,2,…,9,v≠w,
where(3.42)ςn(x)=-14π|x|exp⁡(ιλn|x|),n=1,2,…,7,r1l=∏m=1,m≠l6(λm2-λl2)-1,l=1,2,3,4,5,6,r2v=∏m=5,m≠v7(λm2-λv2)-1,v=5,6,7,r3w=∏m=1,m≠w4(λm2-λw2)-1,w=1,2,3,4.
We will prove the following lemma.Lemma 3.1.
The matrixY defined above is the fundamental matrix of operator Θ(Δ), that is
(3.43)Θ(Δ)Y(x)=δ(x)I(x).Proof.
To prove the lemma, it is sufficient to prove that(3.44)Γ1(Δ)Γ2(Δ)Y11(x)=δ(x),Γ2(Δ)(Δ+λ72)Y44(x)=δ(x),Γ1(Δ)Y77(x)=δ(x).
We find that
(3.45)r11+r12+r13+r14+r15+r16=0,r12(λ12-λ22)+r13(λ12-λ32)+r14(λ12-λ42)+r15(λ12-λ52)+r16(λ12-λ62)=0,r13(λ12-λ32)(λ22-λ32)+r14(λ12-λ42)(λ22-λ42)+r15(λ12-λ52)(λ22-λ52)+r16(λ12-λ62)(λ22-λ62)=0,r14(λ12-λ42)(λ22-λ42)(λ32-λ42)+r15(λ12-λ52)(λ22-λ52)(λ32-λ52)+r16(λ12-λ62)(λ22-λ62)(λ32-λ62)=0,r15(λ12-λ52)(λ22-λ52)(λ32-λ52)(λ42-λ52)+r16(λ12-λ62)(λ22-λ62)(λ32-λ62)(λ42-λ62)=0,r16(λ12-λ62)(λ22-λ62)(λ32-λ62)(λ42-λ52)(λ52-λ62)=1,(Δ+λm2)ςn(x)=δ(x)+(λm2-λn2)ςn(x),m,n=1,2,3,4,5,6.
Now consider
(3.46)Γ1(Δ)Γ2(Δ)Y11(x)=(Δ+λ22)(Δ+λ32)(Δ+λ42)(Δ+λ52)(Δ+λ62)∑n=16r1n[δ+(λ12-λn2)ςn]=(Δ+λ22)(Δ+λ32)(Δ+λ42)(Δ+λ52)(Δ+λ62)∑n=26r1n(λ12-λn2)ςn=(Δ+λ32)(Δ+λ42)(Δ+λ52)(Δ+λ62)∑n=26r1n(λ12-λn2)[δ+(λ22-λn2)ςn]=(Δ+λ32)(Δ+λ42)(Δ+λ52)(Δ+λ62)∑n=36r1n(λ12-λn2)(λ22-λn2)ςn=(Δ+λ42)(Δ+λ52)(Δ+λ62)∑n=36r1n(λ12-λn2)(λ22-λn2)[δ+(λ32-λn2)ςn]=(Δ+λ42)(Δ+λ52)(Δ+λ62)∑n=46r1n(λ12-λn2)(λ22-λn2)(λ32-λn2)ςn=(Δ+λ52)(Δ+λ62)∑n=46r1n(λ12-λn2)(λ22-λn2)(λ32-λn2)[δ+(λ42-λn2)ςn]=(Δ+λ62)∑n=56r1n(λ12-λn2)(λ22-λn2)(λ32-λn2)(λ42-λn2)[δ+(λ52-λn2)ςn]=(Δ+λ62)ς6=δ.
Similarly, (3.44)2 and (3.44)3 can be proved.
We introduce the matrix(3.47)G(x)=R(Dx)Y(x).
From (3.39), (3.43), and (3.47), we obtain
(3.48)F(Dx)G(x)=F(Dx)R(Dx)Y(x)=Θ(Δ)Y(x)=δ(x)I(x).
Hence, G(x) is a solution to (2.11).Therefore we have proved the following theorem.Theorem 3.2.
The matrixG(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 matrixG(x) is the solution of the system of equations (2.6) at every point xϵE3 except the origin.Property 2.
The matrixG(x) can be written in the form
(4.1)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.


## 5. 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 solutionG(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].



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*Source: 764632-2011-05-17.xml*