Fundamental Solution in the Theory of Thermomicrostretch Elastic Diffusive Solids

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.

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

Let 𝐱=(𝑥₁,𝑥₂,𝑥₃) be the point of the Euclidean three-dimensional space E³: |𝐱|=(𝑥²¹+𝑥²²+𝑥²³)^1/2,  𝐃_𝐱=(𝜕/𝜕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𝜇+𝐾^∗Δ𝐮+(𝜆𝑓+𝜇)graddiv𝐮+𝐾^∗curl𝝋+𝜒^∗grad𝜓^∗−𝛽₁grad𝑇−𝛽₂grad𝐶=𝜌𝐮̈,_∗Δ−𝑏2𝐾^∗𝝋+𝛼^∗+𝛽^∗graddiv𝝋+𝐾^∗curl𝐮=𝜌𝑗𝝋̈,_∗1+𝜏₀Δ−𝑐^∗𝜓^∗−𝜒^∗div𝐮−𝑔^∗𝑇−ℎ^∗𝐶=𝜌𝜁𝜓^∗̈,𝜕𝜕𝑡𝛽¹𝑇₀div𝐮̇−𝑔^∗𝑇₀𝜓^∗̇‌+𝜌𝐶_𝐸̇‌𝑇+𝑎𝑇₀̇‌𝐷𝛽₂𝐶=𝐾Δ𝑇,Δdiv𝐮++𝐷ℎ^∗Δ𝜓^∗+𝐷𝑎Δ𝑇−𝐷𝑏Δ𝐶+𝐶̇+𝜏⁰̈‌𝐶=0,(2.1) where 𝛽₁=(3𝜆+2𝜇+𝐾^∗)𝛼_𝑡, 𝛽₂=(3𝜆+2𝜇+𝐾^∗)𝛼_𝑐. Here 𝛼_𝑡,𝛼_𝑐 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,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^∗, 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:𝐱^=𝑤^∗1𝐱𝑐₁,𝐮^=𝜌𝑤^∗1𝑐₁𝐮𝛽₁𝑇₀,𝝋^=𝜌𝑐²¹𝝋𝛽₁𝑇₀,𝜓^∗=𝜌𝜁𝑤^1∗2𝜓^∗𝛽₁𝑇₀,𝑇^=𝑇𝑇₀,𝐶^=𝛽²𝐶𝛽₁𝑇₀𝑡_,=𝑤^∗1𝑡,𝜏^0=𝑤^∗1𝜏₀,𝜏^0′=𝑤^∗1𝜏⁰,𝛿₁=𝜇+𝐾^∗𝜆+2𝜇+𝐾^∗,𝛿₂=𝜆+𝜇𝜆+2𝜇+𝐾^∗𝛿₃,=𝐾^∗𝜆+2𝜇+𝐾^∗,𝛿₄=𝜒^∗𝜌𝜁𝑤^1∗2,𝛿₅=𝑓^∗𝑤^1∗2𝜌𝑐⁴¹,𝛿₆=𝛼^∗+𝛽^∗𝑤^1∗2𝜌𝑐⁴¹,𝛿₇=𝑗𝑤^1∗2𝑐²¹𝛿₈,=𝑏^∗𝜁(𝜆+2𝜇+𝐾^∗),𝛿⁹=𝑐^∗𝜌𝜁𝑤^1∗2,𝛿₁₀=𝜒^∗𝜆+2𝜇+𝐾^∗,𝛿₁₁=𝑔^∗𝛽₁,𝛿₁₂=ℎ^∗𝛽₂𝜁₁,=𝑎𝑇⁰𝑐²¹𝛽₁𝑤^∗1𝐾𝛽₂,𝜁₂=𝛽²¹𝑇₀𝜌𝐾𝑤^∗1,𝜁₃=𝑔^∗𝛽₁𝑇₀𝑐²¹𝜌𝜁𝐾𝑤^1∗3,𝑞^∗1=𝐷𝑤^∗1𝛽²²𝜌𝑐⁴¹𝑞_∗2,=𝐷𝑤^∗1𝛽₂𝑎𝛽₁𝑐²¹,𝑞^∗3=𝐷𝑤^∗1𝑏𝑐²¹,𝑞^∗4=𝐷ℎ^∗𝛽₂𝜌𝜁𝑤^∗1𝑐²¹.(2.2) Here 𝑤^∗1=𝜌𝐶_𝐸𝑐²¹/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 obtain𝛿₁Δ𝐮+𝛿₂g𝛿raddiv𝐮+𝛿₃curl𝝋+𝛿₄grad𝜓^∗−grad𝑇−grad𝐶=𝐮̈,₅𝛿Δ−2𝛿₃𝝋+𝛿₆graddiv𝝋+𝛿₃curl𝐮=𝛿₇𝝋̈‌,₈Δ𝜏_0𝑡−𝛿₉𝜓^∗−𝛿₁₀div𝐮−𝛿₁₁𝑇−𝛿₁₂𝐶=𝜓^∗̈,𝜁²div𝐮̇−𝜁₃𝑞𝜓^∗̇‌+𝑇̇+𝜁₁𝐶̇‌=Δ𝑇,_∗1Δdiv𝐮+𝑞^∗4Δ𝜓^∗+𝑞^∗2Δ𝑇−𝑞^∗3Δ𝐶+𝜏^0𝑐̇‌𝐶=0,(2.3) where𝜏^0𝑡=1+𝜏₀𝜕𝜕𝑡,𝜏^0𝑐=1+𝜏⁰𝜕𝜕𝑡.(2.4) We assume the displacement vector, microrotation, microstretch, temperature change, and concentration 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+𝛿𝛿₂graddiv𝐮+𝛿₃curl𝝋+𝛿₄grad𝜓^∗−grad𝑇−grad𝐶=𝟎,₅−𝛿Δ₁₀+𝜇^∗𝝋+𝛿₆graddiv𝝋+𝛿₃curl𝐮=𝟎,−𝜏_𝑡10div𝐮+𝛿₈Δ+𝜁^∗𝜓^∗−𝛿₁₁𝑇−𝛿₁₂𝐶=0,𝑞𝜁₂div𝐮−𝜁₃𝜓^∗+𝜁₁𝐶+Δ−𝜏^𝑡10𝑇=0,_∗1Δdiv𝐮+𝑞^∗4Δ𝜓^∗+𝑞^∗2Δ𝑇−𝑞^∗3Δ𝐶+𝜏^𝑐10𝐶=0,(2.6) where𝜏^𝑡10=−𝜄𝜔1−𝜄𝜔𝜏₀,𝜏^𝑐10=−𝜄𝜔1−𝜄𝜔𝜏⁰,𝜇^∗=𝛿₇𝜔²−2𝛿₃,𝜁^∗=𝜔²−𝛿₉.(2.7)

We introduce the matrix differential operator𝐅𝐃_𝐱=‖‖𝐹_𝑔ℎ𝐃_𝐱‖‖_9×9,(2.8) where𝐹_𝑚𝑛𝐃_𝐱=𝛿₁Δ+𝜔²𝛿_𝑚𝑛+𝛿₂𝜕²𝜕𝑥_𝑚𝜕𝑥_𝑛,𝐹_𝑚,𝑛+3𝐃_𝐱=𝐹_𝑚+3,𝑛𝐃_𝐱=𝛿₃₃_𝑟=1𝜀_𝑚𝑟𝑛𝜕𝜕𝑥_𝑟𝐹_𝑚7,𝐃_𝐱=𝛿₄𝜕𝜕𝑥_𝑚,𝐹_𝑚8𝐃_𝐱=𝐹_𝑚9𝐃_𝐱=−𝜕𝜕𝑥_𝑚,𝐹_{𝑚+3,𝑛+3}𝐃_𝐱=𝛿₅Δ+𝜇^∗𝛿_𝑚𝑛+𝛿₆𝜕²𝜕𝑥_𝑚𝜕𝑥_𝑛𝐹_𝑚+3,7,𝐹𝐃_𝐱=𝐹_7,𝑛+3𝐃_𝐱=𝐹_𝑚+3,8𝐃_𝐱=𝐹_8,𝑛+3𝐃_𝐱=𝐹_𝑚+3,9𝐃_𝐱=𝐹_9,𝑛+3𝐃_𝐱=0,_7𝑛𝐃_𝐱=−𝛿₁₀𝜕𝜕𝑥_𝑛𝐹_8𝑛,𝐹₇₇𝐃_𝐱=𝛿₈Δ+𝜁^∗,𝐹₇₈𝐃_𝐱=−𝛿₁₁,𝐹₇₉𝐃_𝐱=−𝛿₁₂,𝐃_𝐱=−𝜁₂𝜏^𝑡10𝜕𝜕𝑥_𝑛𝐹_9𝑛,𝐹₈₇𝐃_𝐱=𝜁₃𝜏^𝑡10,𝐹₈₈𝐃_𝐱=Δ−𝜏^𝑡10,𝐹₈₉=−𝜁₁𝜏^𝑡10,𝐃_𝐱𝐹=𝑞^∗1Δ𝜕𝜕𝑥_𝑛,𝐹₉₇𝐃_𝐱=𝑞^∗4Δ,𝐹₉₈𝐃_𝐱=𝑞^∗2Δ,₉₉𝐃_x=−𝑞^∗3Δ+𝜏^𝑐10,𝑚,𝑛=1,2,3.(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 as𝐅𝐃_𝐱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 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 𝐱𝜖E³.
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 equations𝛿₁Δ𝐮+𝛿₂graddiv𝛿𝐮+𝛿₃curl𝝋−𝛿₁₀grad𝜓^∗−𝜁₂𝜏^𝑡10grad𝑇+𝑞^∗1grad𝐶+𝜔²𝐮=𝐇^,(3.1)₅𝛿Δ+𝜇^∗𝝋+𝛿₆graddiv𝝋+𝛿₃curl𝐮=𝐇^,(3.2)₄−didviv𝐮𝐮+−𝛿𝛿₈Δ₁₁+𝜁^∗𝜓^∗+𝜁₃𝜏^𝑡10𝑇+𝑞^∗4𝐶=𝑍,(3.3)−Δdiv𝐮−𝛿₁₂𝜓^∗+Δ−𝜏^𝑡10𝑇+𝑞^∗2𝐶=𝐿,(3.4)Δ𝜓^∗−𝜁₁𝜏^𝑡10Δ𝑇−𝑞^∗3Δ𝐶+𝜏^𝑐10𝐶=𝑀,(3.5) 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 form𝐅^tr𝐃_𝐱𝐔(𝐱)=𝐐(𝐱),(3.6) where 𝐅^tr 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Δ+𝜔²div𝐮𝜐−_∗𝛿₁₀Δ𝜓^∗−𝜁₂𝜏^𝑡10Δ𝑇+𝑞^∗1Δ𝐶=div𝐇^,𝛿₄Δ+𝜇^∗div𝝋=div𝐇^,−didvi𝐮v𝐮+−𝛿𝛿₈Δ₁₁+𝜁^∗𝜓^∗+𝜁₃𝜏^𝑡10𝑇+𝑞^∗4𝐶=𝑍,−Δdiv𝐮−𝛿₁₂𝜓^∗+Δ−𝜏^𝑡10𝑇+𝑞^∗2𝐶=𝐿,Δ𝜓^∗−𝜁₁𝜏^𝑡10Δ𝑇−𝑞^∗3Δ𝐶+𝜏^𝑐10𝐶=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 form𝐍(Δ)𝐒=𝐐,(3.8) where 𝐒=(div𝐮,𝜓^∗,𝑇,𝐶), 𝐐=(𝑑₁,𝑑₂,𝑑₃,𝑑₄)=(div𝐇^,𝑍,𝐿,𝑀), and𝐍(Δ)=‖‖𝑁_𝑚𝑛(Δ)‖‖_4×4=‖‖‖‖‖‖‖Δ+𝜔²𝛿₄−𝛿₁₀Δ−𝜁₂𝜏^𝑡10Δ𝑞^∗1Δ𝛿₈Δ+𝜁^∗𝜁₃𝜏^𝑡10𝑞^∗4−1−𝛿₁₁Δ−𝜏^𝑡10𝑞^∗2−Δ−𝛿₁₂Δ−𝜁₁𝜏^𝑡10Δ−𝑞^∗3Δ+𝜏^𝑐10‖‖‖‖‖‖‖_4×4.(3.9) Equations (3.7)₁, (3.7)₃, (3.7)₄, and (3.7)₅ may be also written asΓ₁(Δ)𝐒=𝚿,(3.10) where𝚿=Ψ₁,Ψ₂,Ψ₃,Ψ₄,Ψ_𝑛=𝑒^∗4_𝑚₌₁Γ₁𝑁^∗𝑚𝑛𝑑_𝑚,(Δ)=𝑒^∗det𝐍(Δ),𝑒^∗=−1𝑞^∗3𝛿₉,𝑛=1,2,3,4,(3.11) and 𝑁^∗𝑚𝑛 is the cofactor of the elements 𝑁_𝑚𝑛 of the matrix 𝐍.

From (3.9) and (3.11), we see thatΓ₁(Δ)=⁴_𝑚=1Δ+𝜆^2𝑚,(3.12) where 𝜆^2𝑚, 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 𝛿₃curl to (3.1) and (3.2), respectively, we obtain𝛿=₅Δ+𝛿𝜇^∗𝛿₁Δ𝐮+𝛿₂graddiv𝐮+𝜔²𝐮+𝛿₃𝛿₅Δ+𝜇^∗curl𝝋₅𝛿Δ+𝜇^∗𝐇^+𝛿₁₀grad𝜓^∗+𝜁₂𝜏^𝑡10grad𝑇−𝑞^∗1grad𝐶,(3.14)₃𝛿₅Δ+𝜇^∗curl𝝋=−𝛿²³curlcurl𝐮+𝛿₃curl𝐇^.(3.15) Nowcurlcurl𝐮=graddiv𝐮−Δ𝐮.(3.16) Using (3.15) and (3.16) in (3.14), we obtain𝛿₅Δ=+𝛿𝜇^∗𝛿₁Δ𝐮+𝛿₂graddiv𝐮+𝜔²𝐮+𝛿²³Δ𝐮−𝛿²³graddiv𝐮₅Δ+𝜇^∗𝐇^+𝛿₁₀grad𝜓^∗+𝜁₂𝜏^𝑡10grad𝑇−𝑞^∗1grad𝐶−𝛿₃curl𝐇^.(3.17) The above equation can also be written as𝛿=₅Δ−+𝛿𝜇^∗₂𝛿₁+𝛿²³Δ+𝛿₅Δ+𝜇^∗𝜔²𝐮+𝛿𝛿₅Δ+𝜇^∗−𝛿²³graddiv𝐮₅Δ+𝜇^∗𝐇^+𝛿₁₀grad𝜓^∗+𝜁₂𝜏^𝑡10grad𝑇−𝑞^∗1grad𝐶−𝛿₃curl𝐇^.(3.18) Applying the operator Γ₁(Δ) to the (3.18) and using (3.10), we getΓ₁(Δ=)−𝛿₅𝛿𝛿₁₂Δ²+𝜇^∗𝛿₁+𝛿₅𝜔²+𝛿²³Δ+𝜇^∗𝜔²𝐮𝛿₅Δ+𝜇^∗−𝛿²³gradΨ₁+𝛿₅Δ+𝜇^∗Γ₁(Δ)𝐇^+𝛿₁₀gradΨ₂+𝜁₂𝜏^𝑡10gradΨ₃−𝑞^∗1gradΨ₄−𝛿₃Γ₁(Δ)curl𝐇^.(3.19) The above equation may be written in the formΓ₁(Δ)Γ₂(Δ)𝐮=𝚿^,(3.20) whereΓ₂(Δ)=𝑓^∗det‖‖‖𝛿¹Δ−𝛿+₃𝜔²𝛿₃Δ𝛿₅Δ+𝜇^∗‖‖‖_2×2,𝑓^∗=1𝛿₁𝛿₅,(3.21)𝚿^=𝑓^∗−𝛿₂𝛿₅Δ+𝜇^∗−𝛿²³gradΨ₁+−𝛿₃𝛿₅Δ+𝜇^∗Γ₁(Δ)𝐇^+𝛿₁₀gradΨ₂+𝜁₂𝜏^𝑡10gradΨ₃−𝑞^∗1gradΨ₄Γ₁(Δ)curl𝐇^.(3.22) It can be seen thatΓ₂(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 𝛿₃curl and 𝛿₁Δ+𝜔² to (3.1) and (3.2), respectively, we obtain𝛿₃𝛿₁Δ+𝜔²curl𝐮=𝛿₃curl𝐇^−𝛿²³curlcurl𝝋,(3.24)𝛿₁Δ+𝜔²𝛿₅Δ+𝜇^∗𝝋+𝛿₆𝛿₁Δ+𝜔²graddiv𝝋+𝛿₃𝛿₁Δ+𝜔²curl𝐮=𝛿₁Δ+𝜔²𝐇^.(3.25) Nowcurlcurl𝝋=graddiv𝝋−Δ𝝋.(3.26) Using (3.24) and (3.26) in (3.25), we obtain𝛿₁Δ+=𝜔²𝛿𝛿₅Δ+𝜇^∗𝝋+𝛿₆𝛿₁Δ+𝜔²graddiv𝝋+𝛿²³Δ𝝋−𝛿²³graddiv𝝋₁Δ+𝜔²𝐇^−𝛿₃curl𝐇^.(3.27) The above equation may also be written as𝛿=₅Δ−+𝛿𝜇^∗₆𝛿₁+𝛿²³Δ+𝛿₅Δ+𝜇^∗𝜔²𝝋𝛿₁Δ+𝜔²−𝛿²³graddiv𝝋+𝛿₁Δ+𝜔²𝐇^−𝛿₃curl𝐇^.(3.28) Applying the operator Δ+𝜆²⁷ to the (3.28) and using (3.13), we getΔ+=𝜆−²⁷𝛿₃𝛿₅𝛿₁Δ²+𝜇^∗𝛿₁+𝛿₅𝜔²+𝛿²³Δ+𝜇^∗𝜔²𝝋Δ+𝜆²⁷curl𝐇^+Δ+𝜆²⁷𝛿₁Δ+𝜔²𝐇^−𝜐1_∗𝛿₆𝛿₁Δ+𝜔²−𝛿²³graddiv𝐇^.(3.29) The above equation may also be rewritten in the formΓ₂(Δ)Δ+𝜆²⁷𝝋=𝚿^,(3.30) where𝚿^=𝑓^∗−𝛿³Δ+𝜆²⁷curl𝐇^+Δ+𝜆²⁷𝛿₁Δ+𝜔²𝐇^−1𝜐^∗𝛿₆𝛿₁Δ+𝜔²−𝛿²³graddiv𝐇^.(3.31) From (3.10), (3.20), and (3.30), we obtain𝚯(Δ)𝐔(𝐱)=𝚿(𝐱),(3.32) where Ψ=(Ψ^,Ψ^,Ψ₂,Ψ₃,Ψ₄)Θ_𝑚𝑚𝚯(Δ)=‖‖Θ_𝑔ℎ(Δ)‖‖_9×9,(Δ)=Γ₁(Δ)Γ₂(Δ)=⁶Θ_{𝑚+3,𝑛+3}_𝑞=1Δ+𝜆^2𝑞,(Δ)=Γ₂(Δ)Δ+𝜆²⁷=⁷Θ_𝑔ℎ_𝑞=5Δ+𝜆^2𝑞,(Δ)=𝑚0=,1,2,Θ3,₇₇(𝑔Δ,)ℎ==Θ1,₈₈2(,Δ…)=,9Θ,₉₉𝑔(≠Δℎ).=Γ₁(Δ),(3.33)

Equations (3.11), (3.22), and (3.31) can be rewritten in the form𝚿^=𝑓^∗𝛿₅Δ+𝜇^∗Γ₁(Δ)𝐉+𝑞₁₁(Δ)graddiv𝐇^+𝑞₂₁(Δ)curl𝐇^𝚿+𝑞₃₁(Δ)grad𝑍+𝑞₄₁(Δ)grad𝐿+𝑞₅₁(Δ)grad𝑀,_Ψ₂=𝑞₁₂(Δ)curl𝐇^+𝑓^∗Δ+𝜆²⁷𝛿₁Δ+𝜔²𝐉+𝑞₂₂(Δ)graddiv𝐇^,Ψ=𝑞₁₃(Δ)div𝐇^+𝑞₃₃(Δ)𝑍+𝑞₄₃(Δ)𝐿+𝑞₅₃(Δ)𝑀,₃Ψ=𝑞₁₄(Δ)div𝐇^+𝑞₃₄(Δ)𝑍+𝑞₄₄(Δ)𝐿+𝑞₅₄(Δ)𝑀,₄=𝑞₁₅(Δ)div𝐇^+𝑞₃₅(Δ)𝑍+𝑞₄₅(Δ)𝐿+𝑞₅₅(Δ)𝑀,(3.34) where 𝐉=‖𝛿_𝑔ℎ‖_3×3 is the unit matrix.

In (3.34), we have used the following notations: 𝑞₂₁𝑞_𝑚1(Δ)=𝑓^∗𝑒^∗𝛿₅Δ+𝜇^∗𝛿₁₀𝑁^∗𝑚2+𝜁₂𝜏^𝑡10𝑁^∗𝑚3−𝑞^∗1𝑁^∗𝑚4−𝛿₂𝛿₅Δ+𝜇^∗−𝛿²³𝑁^∗𝑚1,(Δ)=−𝑓^∗𝛿₃Γ₁(Δ),𝑞₁₂(Δ)=−𝑓^∗𝛿₃Δ+𝜆²⁷,𝑞₂₂(Δ)=−𝑓^∗𝜐^∗𝛿₆𝛿₁𝑞_𝑚𝑛Δ+𝜔²−𝛿²³,(Δ)=𝑒^∗𝑁^∗𝑚𝑛,𝑚=1,3,4,5,𝑛=3,4,5.(3.35) Now from (3.34), we have that𝚿(𝐱)=𝐑^tr𝐃_𝐱𝐐(𝐱),(3.36) where𝑅_𝑚𝑛𝐑=‖‖𝑅_𝑔ℎ‖‖_9×9,𝑅_𝑚,𝑛+3𝐃_𝐱=𝑓^∗𝛿₅Δ+𝜇^∗Γ₁(Δ)𝛿_𝑚𝑛+𝑞₁₁(Δ)𝜕𝑥_𝑚𝜕²𝜕𝑥_𝑛,𝐃_𝐱=𝑞₁₂(Δ)³_𝑟=1𝜀_𝑚𝑟𝑛𝜕𝜕𝑥_𝑟,𝑅_𝑚𝑝𝐃_𝐱=𝑞_1,𝑝−4(Δ)𝜕𝜕𝑥_𝑚𝑅_𝑚+3,𝑛,𝐃_𝐱=𝑞₂₁(Δ)³_𝑟=1𝜀_𝑚𝑟𝑛𝜕𝜕𝑥_𝑟𝑅_{𝑚+3,𝑛+3},𝐃_𝐱=𝑅𝑓^∗Δ_𝑚+3,𝑝+𝜆²⁷𝛿₁Δ+𝜔²𝛿_𝑚𝑛+𝑞₂₂(Δ)𝜕𝑥_𝑚𝜕²𝜕𝑥_𝑛,𝑅𝐃_𝐱=𝑅_𝑝,𝑚+3𝐃_𝐱=0,_𝑝𝑚𝐃_𝐱=𝑞_𝑝−4,1(Δ)𝜕𝜕𝑥_𝑛,𝑅_𝑝𝑠𝐃_𝐱=𝑞_{𝑝−4,𝑠−4}(Δ),𝑚=1,2,3,𝑝,𝑠=7,8,9.(3.37) From (3.6), (3.32), and (3.36), we obtain𝚯𝐔=𝐑^tr𝐅^tr𝐔.(3.38) It implies that𝐅𝐃𝐑^tr𝐅^tr=𝚯,_𝐱𝐑𝐃_𝐱=𝚯(Δ).(3.39) We assume that𝜆^2𝑚≠𝜆^2𝑛≠0,𝑚,𝑛=1,2,3,4,5,6,7𝑚≠𝑛.(3.40) Let𝐘(𝐱)=‖‖𝑌_𝑟𝑠(𝐱)‖‖_9×9,𝑌_𝑚𝑚(𝐱)=⁶_𝑛=1𝑟_1𝑛𝜍_𝑛(𝐱),𝑌_{𝑚+3,𝑚+3}(𝐱)=⁷_𝑛=5𝑌𝑟_2𝑛𝜍_𝑛(𝐱),₇₇(𝐱)=𝑌₈₈(𝐱)=𝑌₉₉(𝐱)=⁴_𝑛=1𝑌𝑟_3𝑛𝜍_𝑛(𝐱),_𝑣𝑤(𝐱)=0,𝑚=1,2,3,𝑣,𝑤=1,2,…,9,𝑣≠𝑤,(3.41) where𝜍_𝑛(𝐱)=−14𝜋|𝐱|exp𝜄𝜆^𝑛𝑟|𝐱|,𝑛=1,2,…,7,_1𝑙=⁶_𝑚=_{1,𝑚≠𝑙}𝜆^2𝑚𝑟−𝜆^2𝑙⁻¹,𝑙=1,2,3,4,5,6,_2𝑣=⁷_𝑚=_{5,𝑚≠𝑣}𝜆^2𝑚𝑟−𝜆^2𝑣⁻¹,𝑣=5,6,7,_3𝑤=⁴_𝑚=_{1,𝑚≠𝑤}𝜆^2𝑚−𝜆^2𝑤⁻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 𝑟𝑟₁₁+𝑟₁₂+𝑟₁₃+𝑟₁₄+𝑟₁₅+𝑟₁₆=0,₁₂𝑟𝜆²¹−𝜆²²+𝑟₁₃𝜆²¹−𝜆²³+𝑟₁₄𝜆²¹−𝜆²⁴+𝑟₁₅𝜆²¹−𝜆²⁵+𝑟₁₆𝜆²¹−𝜆²⁶=0,₁₃𝜆+²¹𝑟−₁₆𝜆²³𝜆²²−𝜆²³+𝑟₁₄𝜆²¹−𝜆²⁴𝜆²²−𝜆²⁴+𝑟₁₅𝜆²¹−𝜆²⁵𝜆²²−𝜆²⁵𝑟𝜆²¹−𝜆²⁶𝜆²²−𝜆²⁶=0,₁₄𝜆+²¹𝑟−₁₆𝜆²⁴𝜆²²−𝜆²⁴𝜆²³−𝜆²⁴+𝑟₁₅𝜆²¹−𝜆²⁵𝜆²²−𝜆²⁵𝜆²³−𝜆²⁵𝑟𝜆²¹−𝜆²⁶𝜆²²−𝜆²⁶𝜆²³−𝜆²⁶=0,₁₅𝑟𝜆²¹−𝜆²⁵𝜆²²−𝜆²⁵𝜆²³−𝜆²⁵𝜆²⁴−𝜆²⁵+𝑟₁₆𝜆²¹−𝜆²⁶𝜆²²−𝜆²⁶𝜆²³−𝜆²⁶𝜆²⁴−𝜆²⁶=0,₁₆Δ+𝜆²¹𝜆−_2𝑚𝜆²⁶𝜆²²−𝜆²⁶𝜆²³−𝜆²⁶𝜆²⁴−𝜆²⁵𝜆²⁵−𝜆²⁶=1,𝜍_𝑛(𝐱)=𝛿(𝐱)+𝜆^2𝑚−𝜆^2𝑛𝜍_𝑛(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=Δ+𝜆₂₂𝑟_1𝑛𝛿+𝜆²¹−𝜆^2𝑛𝜍_𝑛Δ+𝜆²³Δ+𝜆²⁴Δ+𝜆²⁵Δ+𝜆²⁶⁶_𝑛=2𝑟_1𝑛𝜆²¹−𝜆^2𝑛𝜍_𝑛=Δ+𝜆²³Δ+𝜆²⁴Δ+𝜆²⁵Δ+𝜆²⁶⁶_𝑛=2=Δ+𝜆₂₃𝑟_1𝑛𝜆²¹−𝜆^2𝑛𝛿+𝜆²²−𝜆^2𝑛𝜍_𝑛Δ+𝜆²⁴Δ+𝜆²⁵Δ+𝜆²⁶⁶_𝑛=3𝑟_1𝑛𝜆²¹−𝜆^2𝑛𝜆²²−𝜆^2𝑛𝜍_𝑛=Δ+𝜆²⁴Δ+𝜆²⁵Δ+𝜆²⁶⁶_𝑛=3=Δ+𝜆₂₄𝑟_1𝑛𝜆²¹−𝜆^2𝑛𝜆²²−𝜆^2𝑛𝛿+𝜆²³−𝜆^2𝑛𝜍_𝑛Δ+𝜆²⁵Δ+𝜆²⁶⁶_𝑛=4𝑟_1𝑛𝜆²¹−𝜆^2𝑛𝜆²²−𝜆^2𝑛𝜆²³−𝜆^2𝑛𝜍_𝑛=Δ+𝜆²⁵Δ+𝜆²⁶⁶_𝑛=4=Δ+𝜆₂₆𝑟_1𝑛𝜆²¹−𝜆^2𝑛𝜆²²−𝜆^2𝑛𝜆²³−𝜆^2𝑛𝛿+𝜆²⁴−𝜆^2𝑛𝜍_𝑛=Δ+𝜆₂₆⁶_𝑛=5𝑟_1𝑛𝜆²¹−𝜆^2𝑛𝜆²²−𝜆^2𝑛𝜆²³−𝜆^2𝑛𝜆²⁴−𝜆^2𝑛𝛿+𝜆²⁵−𝜆^2𝑛𝜍_𝑛𝜍₆=𝛿.(3.46) Similarly, (3.44)₂ and (3.44)₃ can be proved.
We introduce the matrix𝐆(𝐱)=𝐑𝐃_𝐱𝐘(𝐱).(3.47) From (3.39), (3.43), and (3.47), we obtain 𝐅𝐃_𝐱𝐆(𝐱)=𝐅𝐃_𝐱𝐑𝐃_𝐱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) at every point 𝐱𝜖E³ except the origin.

Property 2. The matrix 𝐆(𝐱) can be written in the form 𝐆_𝑚𝑛𝐆=‖‖𝐺_𝑔ℎ‖‖_9×9,𝐆(𝐱)=𝐑_𝑚𝑛𝐃_𝐱𝑌₁₁(𝐱),_𝑚,𝑛+3𝐆(𝐱)=𝐑_𝑚,𝑛+3𝐃_𝐱𝑌₄₄(𝐱),_𝑚𝑝(𝐱)=𝐑_𝑚𝑝𝐃_𝐱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 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.