Three-dimensional Green’s Functions For Transversely Isotropic Thermoelastic Materials

Green’s functions or fundamental solutions play an important role in both applied and theoretical studies on the physics of solids. They are foundations of a great many further works for theoretical analysis and numerical calculation. For example, fundamental solutions can be used to construct many analytical solutions of practical problems when boundary conditions are imposed. They are essential in the boundary element method as well as the study of cracks, defects and inclusions.Concerning the Green’s functions or fundamental solutions studies of elastic materials, many references did not research thermal coupling equations. This is mainly because adding thermal coupling equations breaks the fine properties of the pristine equations, which increases the troubles. However, for their practical work environment is not universally isothermal. This makes the thermal coupling effects of elastic materials inevitable. Against this background, as the study object of the familiar transversely isotropic thermoelastic materials, the three-dimensional Green’s functions are presented respectively for an infinite, a semi-infinite and two-phase transversely isotropic thermoelastic material with a steady point heat source or point force acting.Firstly, we introduced the consititutive equations and equilibrium of transversely isotropic thermoelastic materials considreing the thermal coupling effect. Three dimensional steady-state general solutions can be obtained by using their fundamental equations. Secondly, for various kinds of point loads in an infinite or on the surface and in the interior of a semi-infinite or in two-phase infinite transversely isotropic thermoelastic materials, their fundamental solutions can be obtained by constructing new harmonic functions. The constants of this harmonic functions are determined by using the compact general solutions of transversely isotropic thermoelastic materials and considering the corresponding boundary, equilibrium and continuity conditions. Finally, we got all dimensionless components of the coupling fields, numerical results of corresponding coupled fields are given graphically by contours. Meanwhile the contours are analyzed, some valuable engineering conclusions are obtained.