The boundary layer phenomena in two-dimensional transversely isotropic piezoelectric media by exact symplectic expansion

Abstract

A separable variable method is introduced to find the exact homogeneous solutions of a two-dimensional transversely isotropic piezoelectric media to handle general boundary conditions. The usual method of separable variables for partial differentiation equations cannot be readily applicable due to the tangling of the unknowns and their derivatives. Introducing dual variables of stresses, we obtain a set of first-order Hamiltonian equations whose eigensolutions are symplectic spanning over the solution space to cover all possible boundary conditions. The solutions consist of two parts. The first part is the derogative zero-eigenvalue solutions of the Saint Venant type together with all their Jordan chains. The second part is the decaying non-zero-eigenvalue solutions describing the boundary layer effects. The classical solutions are actually the zero-eigenvalue solutions representing the simple extension, bending, equipotential field, and the uniform electric displacement. On the other hand, the non-zero-eigenvalue solutions represent the localized solutions, which are sensitive to the boundary conditions and are decaying rapidly with respect to the distance from the boundaries. Some rate-of-decay curves of the newly found non-zero-eigenvalue solutions are shown by numerical examples. Finally, the complete boundary layer effects are quantified for the first time.