Two-dimensional piezoelectricity. Part I: eigensolutions of nondegenerate and degenerate materials

General solutions of two-dimensional piezoelectricity, which yield all solutions of 2-D boundary values problems, are obtained by combining four complex conjugate pairs of independent eigensolutions, each containing an arbitrary analytic function. The forms of representation are fundamentally different for 14 different classes of nondegenerate and degenerate piezoelectric materials, as determined by the multiplicity and types of eigenvalues. Degenerate materials possess high-order eigensolutions, in which the eigenvectors of equal and lower orders are intrinsically coupled. Such coupling is nonexistent in nondegenerate cases including the well-known and analytically simple case with no multiple eigenvalues. The present analysis is drastically simplified by using the compliance-based formalism, instead of the stiffness- based, extended Eshelby–Stroh formalism. Explicit expressions are obtained for the eigensolutions, the pseudometrics, and the intrinsic tensors characterizing piezoelectric materials of every type.