Thermoelastodynamics with Scalar Potential Functions

Abstract

A linear thermoelastic isotropic material is considered. A complete solution in terms of three scalar potential functions for the coupled displacement-temperature equations of motion and heat equation is presented, where the governing equations for the potential functions are the wave, heat, or a repeated wave-heat equation. The completeness theorem is proven based on a retarded Newtonian potential function, existence of solutions for the repeated wave equation and heat equation, and perturbation theory. A connection is also made between a complete solution already existing in the literature and the solution introduced in this paper, which in itself is another method to prove the completeness. If no heat source exists, the number of potential functions is reduced to two. In some circumstances, the number of potential functions is reduced to only one, and the required conditions for this case are discussed. As a special case, the torsionless and rotationally symmetric configuration with respect to an arbitrary axis is carefully assessed. The degenerated case to elastodynamics is obtained, and it is shown that the solutions given here are identical to a complete solution, which already exists in the literature for the equations of motion. The potential functions may be used for elastostatics, if the time dependence of the displacement vector and the temperature, and also the potential functions, are suppressed. In analogy with the deformation theory of a fluid saturated poroelastic material, the results given here may be used in the theory related to steady-state water flow.