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Thermo-mechanical stresses developed in a disk, inner boundary of which is subject to the nonstationary thermo-mechanical loadings due to variation in temperature and pressure inside the disk is considered in the presented paper. Problems of theoretical analysis are divided and described as follows: The first problem implies analytical solution of the heat conduction equation with nonhomogeneous boundary conditions of Robin type, and the second problem deals with obtaining analytical solution of dynamic equation of displacement for nonisothermal elastic body (Navier equation) with traction boundary conditions in the plane strain case of uncoupled thermo-elasticity. Finite Hankel transform is used to solve the problems. Two sets of orthogonal basis functions necessary for integral transform are obtained. The first one corresponds to the boundary conditions written for the heat conduction equation, and the second fits the boundary conditions for Navier equation. Integrals defining elements of transformation matrix between the two sets of basis functions are evaluated analytically. In general, convenient method of obtaining analytical solution of linear hyperbolic equation with nonhomogeneous term, which itself is solution of nonhomogeneous parabolic equation, is elaborated within the presented work. Using finite difference method, discrete counterparts of the described problems are constructed and, via the built-in ordinary differential equation (ODE) solvers in MATLAB, numerical solutions are obtained, which exhibit excellent accordance with analytical ones.

Developing the scientific basis for making new materials with pre-planned, physical and mechanical properties by SHS (self-propagating high temperature synthesis)-compaction is a fundamental problem of modern metallurgy, which can be solved by imaginative modeling of the structural-phase state of materials. Among the many methods of obtaining metal-ceramic tiles, the ease of process management, cheapness, ecological cleanliness and the possibility of obtaining high-quality products are significantly distinguished by the innovative SHS electrrical rolling process developed at the F. Tavadze Institute of Metallurgy and Materials Science, which, by maintaining the equilibrium velocities of the rolling and combustion fronts, ensures hot deformation of the hot viscous plastic mass in the deformation core under conditions of continuous compensation of heat losses Namicheishvili, 2016 [1] and Aslamazashvili, 2017 [2].