International Scholarly Research Network ISRN Applied Mathematics Volume 2011, Article ID 764632, 15 pages doi:10.5402/2011/764632 Research Article Fundamental Solution in the Theory of Thermomicrostretch 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 kuk@rediffmail.com 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 1developed 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 4developed 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 7incorporates a flux rate term into Fourier’s law of heat conduction and formulates a generalized theory admitting finite speed for thermal signals.
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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,
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)
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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,
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)
6 ISRN Applied Mathematics
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
Theorem 3.2. The matrix G(x) defined by (3.47) is the fundamental solution of system of equations(2.6).
14 ISRN Applied Mathematics
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
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ISRN Applied Mathematics 15
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