A new efficient unified method to derive new constructive formulas and explicit expressions for plane and spatial thermoelastic Green’s functions

Abstract

This study is devoted to elaboration of a new efficient unified method for deriving main thermoelastic Green’s functions (MTGF) based on their new integral representations via Green’s functions for Poisson’s equation (GFPE). In comparison with the author’s previous works on the method proposed here, it is possible to prove certain new theorems about constructive formulas for MTGFs expressed via respective GFPE. For the first time, in this method, the unknown on the surfaces thermoelastic dilatation is derived using not only some equilibrium equations on the boundary, but also the integral representations for MTGFs. So, the efficiency of the proposed method is explained by its possibility to derive new constructive formulas for MTGFs for whole classes of thermoelastic BVPs. In this case, the concrete analytical expressions for MTGFs can be obtained as particular cases of the established constructive formulas by changing the respective well-known GFPEs. The unification of this method is provided by the general integral representations for MTGFs via GFPEs. So, if we know the respective GFPEs, then the MTGFs for any thermoelastic BVP can by derived in the same unified way using the mentioned-above general integral representations. As example, here is proved a theorem about constructive formulas for MTGFs for a whole class of two- and three-dimensional BVPs for a generalized semi-infinite thermoelastic body. In this case, the MTGFs are generated by a unitary point heat source, applied inside of a generalized thermoelastic octant which is subjected to different homogeneous mechanical and thermal boundary conditions. Many MTGFs for a thermoelastic BVPs for plane, half-plane, quadrant, space, quarter-space and octant may be obtained as particular cases of the established constructive formulas. As example, new explicit elementary MTGFs for a thermoelastic octant are derived. Their analytical checking and graphical evaluation are also included. The proposed method can be applied to any domain of Cartesian coordinate system.