A method to derive thermoelastic Green’s functions for bounded domains (on examples of two-dimensional problems for parallelepipeds)

Abstract

This article presents new constructive formulas to steady-state thermoelastic Green’s functions for a plane generalized boundary value problem of thermoelasticity for a generalized rectangle. The constructive formulas are expressed in terms of Green’s functions for Poisson’s equation. These results are formulated in a special theorem, which is proved using the author’s developed harmonic integral representations method. On the base of derived constructive formulas, it is possible to obtain many analytical expressions for Green’s functions for thermoelastic displacements and stresses to 28 concrete boundary value problems for: rectangle-16, half-strip-8, strip-4. An example of such kind is presented for a concrete plane boundary value problem for a rectangle, Green’s functions of which are presented in the form of a sum of elementary functions and ordinary series. These results are presented in another theorem, which is proved on the base of derived general constructive formulas. In the particular case for a half-strip and strip, ordinary series vanish and Green’s functions are presented in elementary functions. New analytical expressions for thermal stresses to a particular plane problem for a thermoelastic rectangle subjected to a constant boundary temperature gradient are also derived. Numerical investigation has shown that the infinite series are convergent. All solutions obtained for thermal stresses, caused by a constant temperature gradient and by a unit heat source are validated by checking the respective equilibrium equation and continuity of deformation equations (Beltrami–Michel equations), written in the terms of thermal stresses. The graphics for thermal stresses and their infinite series also are presented.