Three-Dimensional Steady-State General Solution for Isotropic Thermoelastic Materials with Applications I: General Solutions

Abstract

Three general solutions of the three-dimensional steady-state governing equations of isotropic thermoelastic materials are derived in this article. For this object, two displacement functions are first introduced to simplify the govering equation. Then, using the differential operator theory, three general solutions can be expressed in terms of two functions, one satisfies a harmonic equation and the other satisfies a six-order partial differential equation. By virtue of Almansi's theorem, three general solutions can be further transferred to two general solutions, which are expressed in terms of three harmonic functions. At last, one more relatively completed general solution expressed in four harmonic functions is obtained by superposing the two general solutions. The proposed general solutions are simple in form and hence they may bring more convenience to certain boundary problems. As two examples, the fundamental solutions for both a point heat source in the interior of infinite thermoelastic body and a point heat source on the surface of semi-infinite thermoelastic body are presented by virtue of the obtained general solutions.