Construction of static solutions of the equations of elasticity and thermoelasticity theory

Abstract

New solutions to the theories of thermoelasticity and elasticity in the Cartesian coordinate system are found in this paper. New explicit partial solutions of thermoelasticity equations, when the temperature field is defined by 3D or 2D harmonic functions, are constructed. Displacements, deformations, and stresses determined by these partial solutions are called temperature functions. A simple formula for the expression of normal temperature stresses is obtained and it is shown that their sum is zero. Separate cases when the temperature depends on the product of harmonic functions of two variables on the degree of coordinate z are also considered. Partial and general solutions are derived for them. General solutions of thermoelasticity equations (Navier’s equations) through four harmonic functions, when the temperature field is given three-dimensional or two-dimensional harmonic functions, are constructed. The thermoelastic state of the body is divided into symmetric and asymmetric stress states. It is proposed to present the solutions of the theory of elasticity, which are expressed by the product of the harmonic function of two variables to the degree of the coordinate. Polynomial solutions that depend on three coordinate variables are recorded. An example of the application of the proposed solution is given.