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.