General solutions for coupled equations for piezoelectric media

Abstract

Three general solutions are obtained for the coupled dynamic equations for a transversely isotropic piezoelectric medium. These solutions are expressed in terms of the two functions ψ and F, where ψ satisfies a second-degree partial differential equation and F a sixth-degree partial differential equation, respectively. If the terms concerning the derivatives of time are removed, the results become three general solutions for the corresponding equilibrium equations, in which the function F can be represented by functions Fi (i = 1, 2, 3), each of which satisfies a second-degree partial differential equation by utilizing a generalized Almansi theorem; and the solution Wang and Zheng [Int. J. Solids Structures32, 105–115 (1995)] obtained is proved to be consistent with one case of one of the three general solutions. When the constants e15 = e31 = e33 = 0 the piezo-electric coupling is absent; then, two of the solutions reduce to the elasticity general solutions for a transversely isotropie medium, one of which is the result Hu [Acta Scientia Sin.2(2), 145–151 (1953)] obtained; the other one has not been published. Last, the solution in the limiting explicit form for the problem for a half-space with concentrated loads at the boundary is obtained by utilizing the general solutions.