A new approach to problems of thermoelasticity in stresses

It is known that the plane problems of the theory of elasticity and thermoelasticity are usually reduced to solving the biharmonic equation with respect to the Airy stress function. Solutions to spatial problems can also be defined using the Maxwell and Morera stress functions, etc. In this work, for solving a spatial thermoelastic problem, in the framework of the Beltrami-Michell equations, Poisson-type differential equations are found for each component of the stress tensor. In this case, the right-hand sides of these equations depend on temperature and elastic constants. The boundary conditions consist of the usual surface forces and "additional" boundary conditions derived from the equilibrium equations. The differential equations and boundary conditions of the plane problem of thermoelasticity in stresses are discussed in more detail. Grid equations for plane and spatial problems of thermoelasticity were compiled by the finite-difference method and solved by the iterative method. The problems of a thermoelastic rectangle and a parallelepiped located in a given temperature field are solved numerically. The validity of the formulated boundary value problems and the reliability of the results obtained are substantiated by comparing the numerical results with the solutions of the problems of a thermoelastic plate and a parallelepiped known in the literature.