Anisotropic Elasticity and Harmonic Functions in Cartesian Geometry

Abstract

Linear elasticity in an isotropic space is a well-developed area of continuum mechanics. However, the situation is exactly opposite if the fundamental space exhibits anisotropic behavior. In fact, the area of linear anisotropic elasticity is not well developed at the quantitative level, where actual closed-form solutions are needed to be calculated. The present work aims to provide a little progress in this interesting branch of continuum mechanics. We provide a short review of isotropic elasticity in order to demonstrate in the sequel how the anisotropy modifies the final equations, via Hooke’s and Newton’s laws. The eight standard anisotropic structures are also reviewed for completeness. A simple technique is introduced that generates homogeneous polynomial solutions of the anisotropic equations in Cartesian form. In order to demonstrate how this technique is applied, we work out the case of cubic anisotropy, which is the simplest anisotropic structure, having three independent elasticities. This choice is dictated by the restricted number of calculations it requires, but it carries all the basic steps of the method. Isotropic elasticity accepts the differential representation of Papkovich, which expresses the displacement field in terms of a vector and a scalar harmonic function. Unfortunately, though, no such representation is known for the anisotropic elasticity, which can represent the anisotropic displacement field in terms of solutions of the anisotropic Laplacian, as also discussed in this work.