A Unified Formalism of Two-Dimensional Anisotropic Elasticity, Piezoelectricity and Unsymmetric Laminated Plates

Abstract

A unified formalism is presented for theoretical analysis of plane anisotropic elasticity and piezoelectricity, unsymmetric anisotropic plates, and other two-dimensional problems of continua with linear constitutive relations. Complex variables are used to reduce the governing differential equations to algebraic equations. The constitutive relation then yields an eigenrelation, which is easily solved explicitly for the material eigenvalues and eigenvectors. The latter have polynomial expressions in terms of the eigenvalues. When the eigenvectors are combined after multiplication by arbitrary analytic functions containing the corresponding eigenvalues, one obtains the two-dimensional general solution. Important results, including the orthogonality of the eigenvectors, the expressions of the pseudometrics and the intrinsic tensors, are established here for nondegenerate materials, including the case of all distinct eigenvalues. Green’s functions of the infinite domain, and of the semi-infinite domain with interior or edge singularities, are determined explicitly for the most general types of point loads and discontinuities (dislocations).