Fundamental Solution in the Theory of Thermomicrostretch Elastic Diffusive Solids

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.


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.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 H örmander 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.

Basic Equations
where 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 * , and χ * 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:

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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 where We assume the displacement vector, microrotation, microstretch, temperature change, and concentration functions as where ω is oscillation frequency and ω > 0. Using 2.5 into 2.3 , we obtain the system of equations of steady oscillations as where

2.7
We introduce the matrix differential operator where Here ε mrn is alternating tensor and δ mn is the Kronecker delta function.
The system of equations 2.6 can be written as where U u, ϕ, ψ * , T, C 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 where δ is the Dirac delta, I δ gh 9×9 is the unit matrix, and x E 3 .Now we construct G x in terms of elementary functions.

Fundamental Solution of System of Equations of Steady Oscillations
We consider the system of equations where H and H are three-component vector functions on E 3 and Z, L, and M are scalar functions on E 3 .The system of equations 3.1 -3.5 may be written in the form where F tr is the transpose of matrix F, Q H , H , Z, L, M , and x E 3 .Applying the operator div to 3.1 and 3.2 , we obtain where υ * δ 5 δ 6 .Equations 3.7 1 , 3. 3.9 Equations 3.7 1 , 3.7 3 , 3.7 4 , and 3.7 5 may be also written as where , n 1, 2, 3, 4,

3.11
and N * mn is the cofactor of the elements N mn of the matrix N.

3.16
Using 3.15 and 3.16 in 3.14 , we obtain

3.17
The above equation can also be written as

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Applying the operator Γ 1 Δ to the 3.18 and using 3.10 , we get The above equation may be written in the form where

3.26
Using 3.24 and 3.26 in 3.25 , we obtain

3.27
The above equation may also be written as

3.28
Applying the operator Δ λ 2 7 to the 3.28 and using 3.13 , we get

3.29
The above equation may also be rewritten in the form where

3.33
Equations 3.11 , 3.22 , and 3.31 can be rewritten in the form

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

3.43
Proof.To prove the lemma, it is sufficient to prove that

3.45
Now consider

3.46
Similarly, 3.44 2 and 3.44 3 can be proved.We introduce the matrix G x R D x Y 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 .

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 x E 3 except the origin.4.1

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.

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 .

Property 2 .
The matrix G x can be written in the formG G gh 9×9 , G mn x R mn D x Y 11 x , G m,n 3 x R m,n 3 D x Y 44 x , G mp x R mp D x Y 77 x ,m 1, 2, . . ., 9, n 1, 2, 3, p 7, 8, 9.