Deconstructing plane anisotropic elasticity Part I: The latent structure of Lekhnitskii's formalism

摘要

General solutions of the stress and displacements in two-dimensional anisotropic elasticity may be represented by eigenvectors and analytic functions of the complex variables x + mu(i)y, but the representation takes different forms for five distinct types of materials as determined by the elastic compliance matrix [beta]. In this paper, explicit expressions of the general solutions are derived for each type of anisotropic materials in terms of the eigenvalues mu(i) and the elements of [beta]. It is shown that, for degenerate and extra-degenerate materials, the generalized eigenvectors and associated eigensolutions may be obtained by the derivative rule. The Barnett-Lothe tensors are defined in terms of unnormalized eigenvectors by the same set of relations regardless of material degeneracy. Explicit expressions of these tensors are given in concise forms depending only on the multiplicity of the eigenvalues. The six-dimensional matrix formalism and normalization of the eigenvectors are found to be neither essential nor expedient for the analysis except as a device for abridged expressions of matrix identities. (C) 2000 Elsevier Science Ltd. All rights reserved. [References: 9]