Steady-state Green's functions for thermal stresses within rectangular region under point heat source

Abstract

ABSTRACTThis article presents new steady-state Green's functions for displacements and thermal stresses for plane problem within a rectangular region. These results were derived on the basis of structural formulas for thermoelastic Green's functions which are expressed in terms of Green's functions for Poisson's equation. Structural formulas are formulated in a special theorem, which is proved using the author's developed harmonic integral representation method. Green's functions for thermal stresses within rectangle are obtained in the form of a sum of elementary functions and ordinary series. In the particular cases for a half-strip and strip, ordinary series vanish and Green's functions are presented by elementary functions. These concrete results for Green's functions and respective integration formulas for thermoelastic rectangle, half-strip and strip are presented in another theorem, which is proved on the basis of derived structural formulas. New analytical expressions for thermal stresses to a particular plane problem for a thermoelastic rectangle under a boundary constant temperature gradient also are derived. Analytical solutions were presented in the form of graphics. The fast convergence of the infinite series is demonstrated on a particular thermoelastic boundary value problem (BVP). The proposed technique of constructing thermal stresses Green's functions for a rectangle could be extended to many 3D BVPs for unbounded, semibounded, and bounded parallelepipeds.