Complete general solution for Lord–Shulman generalized thermoelastodynamics by using potential functions for transversely isotropic solids

Classical theories of thermoelasticity predict an infinite speed for thermal signals which contradicts the physical phenomena. Non-classical thermoelasticity theories have been proposed to remove this paradox. One of the non-classical theories of thermoelasticity is the Lord–Shulman theory. The main purpose of this study is to present a complete general solution for the Lord–Shulman non-classical equations of thermoelasticity for three-dimensional transversely isotropic solids. In the Lord–Shulman theory, three equations of motion accompanied with the energy equation make a set of four coupled partial differential equations. In this study, four new scalar potential functions are introduced for uncoupling the coupled displacement–temperature equations. The solution is given in terms of four scalar potential functions satisfying uncoupled partial differential equations which are of lesser complexity than the coupled displacement–temperature equations. The potential functions are governed by a wave or a repeated wave–heat equation. Completeness of the solution is proved based on retarded logarithmic and retarded Newtonian potential functions, solutions for repeated wave and heat equations, the extended Boggio’s theorem for transversely isotropic media, and the perturbation theory. It is shown that the solution presented in this paper degenerates to six special problems previously reported in the literature.