Non-stationary Problems of Generalized Elastothermodiffusion for Inhomogeneous Media

The method of investigation of non-stationary boundary value problems of the theory of thermodiffusion using the Laplace integral transform is described. In the classical theory of elasticity this method was first used by V. Kupradze and the author. The interconnection of deformation, thermal conductivity and diffusion processes in an elastic isotropic solid body is described by a system of five scalar partial differential equations of general type. In the classical case this system is hyperbolic with respect to some part of components of an unknown vector function and parabolic with respect to the rest components. A system of equations of the conjugate (connected) theory of thermoelasticity is a particular case [1–4]. In the classical theory of elastothermodiffusion it is assumed that propagation velocity of heat and of diffusing substance is infinitely large. In particular, however, it is often necessary to take into account the fact that heat propagates not with an infinitely large but with a finite velocity. The heat flux does not occur in the body instantly but is characterized by the finite relaxation time. The consideration of these physical factors makes the main system of differential equations very complicated. There exist various generalizations of this theory. Three-dimensional non-stationary problems of non-classical (generalized) thermodiffusion are treated in [5–8]. In this paper the Green–Lindsay theory is generalized to problems of elastothermodiffusion. Initial boundary value problems are investigated for the considered physical system of differential equations in piecewisehomogeneous media with boundary and contact conditions; a substantiation of the Riesz–Fischer–Kupradze method is given and approximate solutions are considered. 1991 Mathematics Subject Classification. 73B30, 73C25. 587 1072-947X/1994/1100-0587$07.00/0 c © 1994 Plenum Publishing Corporation 588 T. BURCHULADZE Let us consider a three-dimensional homogeneous isotropic elastic medium in which a thermodiffusion process takes place. The deformed state is described by the displacement vector v(x, t) = (v1, v2, v3) = ‖vk‖3×1 (onecolumn matrix), the temperature change v4(x, t) and the ”chemical potential” of the medium v5(x, t); C(x, t) = γ2 div v(x, t)+a12v4(x, t)+a2v5(x, t), where C(x, t) is the diffusing substance concentration; x = (x1, x2, x3) is a point in the Euclidean space R3, t ≥ 0 is the time and X = (X1, X2, X3), X4, X5 are the given functions. We consider a system of partial differential equations of the generalized elastothermodiffusion theory written in the form A ( ∂ ∂x ) v − 2 ∑ k=1 γk grad v3+k + X = ρ ∂2v ∂t2 + +τ1 2 ∑ k=1 γk ∂ ∂t grad v3+k, δ1∆v4 + X4 = a1 ( 1 + τ0 ∂ ∂t )∂v4 ∂t + γ1 ∂ ∂t div v + +a12 ( 1 + τ0 ∂ ∂t )∂v5 ∂t , δ2∆v5 + X5 = a2 ( 1 + τ0 ∂ ∂t )∂v5 ∂t + γ2 ∂ ∂t div v + +a12 ( 1 + τ0 ∂ ∂t )∂v4 ∂t , (1) where A( ∂ ∂x ) ≡ ‖μδjk∆ + (λ + μ) ∂ ∂xj∂xk ‖3×3 is the statical operator of Lamé [8], δjk being the Kroneker symbol. The elastic, thermal, diffusion and relaxation constants satisfy the natural restrictions μ > 0, 3λ + 2μ > 0, ρ > 0, ak > 0, δk > 0, γk > 0, k = 1, 2, (2) a1a2 − a12 > 0, τ1 ≥ τ0 > 0. In particular, for relaxation constants τ1 = τ0 = 0 we have the classical case. Let D1 ⊂ R3 be a finite domain bounded by the closed Liapunov surface S and D2 = R\D̄1 be an infinite domain, n = (n1, n2, n3) is the unit normal on S. Elastothermodiffusion constants of the domain Dj will be denoted by the left-hand subscripts jλ, jμ, jρ, jτ, jτ, . . . , j = 1, 2. Problem At. Define in the infinite cylinder Z∞ = {(x, t) : x ∈ D1 ∪ D2, t ∈]0,∞[} the regular vetor V = (v, v4, v5) ∈ C(Z̄∞) ∩ C(Z∞) from the conditions ∀(x, t) ∈ Z∞ : jμ∆v(x, t) + (jλ + jμ) grad div v − NON-STATIONARY PROBLEMS 589