Equations in stresses for the three-, two-, and one- dimensional dynamic problems of thermoelasticity

Abstract

Source systems of differential equations for the six components of the tensor of dynamic stresses are presented in Cartesian and cylindrical coordinates by using the equations of motion, Hooke’s law, Cauchy formulas, and the Saint-Venant compatibility conditions for strains. Without introducing auxiliary potential functions, these systems of interrelated equations are reduced to systems of hierarchically connected wave equations for the key function, i.e., the first invariant and unknown components of the stress tensor. We also present the systems of key equations for the solution of two-dimensional dynamic problems in stresses (the plane problem in Cartesian coordinates and the plane and axially symmetric problems in cylindrical coordinates) and the wave equations for the one-dimensional problems of evaluation of the normal and tangential components of stresses in the same coordinate systems. This enables us to use standard methods of mathematical physics for the solution of the equations obtained as a result. If, in the quasistatic problem, the influence of temperature and bulk forces is absent, then the proposed equations coincide with the equations known from the literature.