Generalized Solutions in Isotropic and Anisotropic Elastostatics

Abstract

Linear elasticity comprises the fundamental branch of continuum mechanics that is extensively used in modern structural analysis and engineering design. In view of this concept, the displacement field provides a measure of how solid materials deform and become internally stressed due to prescribed loading conditions, a fact which is associated with linear relationships between the components of strain and stress, respectively. The mathematical characteristics of these dyadic fields are combined within the Hooke’s law via the stiffness tetratic tensor, which embodies either the isotropic or the anisotropic behavior, exhibited by materials with linear properties. In fact, Hooke’s law is incorporated into the general law of Newton that actually defines the principal spatial and temporal second-order non-homogeneous partial differential equation for the displacement. In this study, we construct handy closed-form solutions for Newton’s law in the Cartesian regime, implying time-independence and considering the case of absence of body forces. Towards this direction, our aim is twofold, in the sense that an efficient analytical technique is introduced that generates homogeneous polynomial solutions of the displacement field for both the typical isotropic and the cubic-type anisotropic structure in the invariant Cartesian geometry. The reliability of the presented methodology is verified by reducing the results for each polynomial degree from the anisotropic to the isotropic eigenspace, in terms of a simple transformation, while we demonstrate our theory with an important application, wherein the effect of a prescribed force on an isotropic half-space to the neighboring half-space of cubic anisotropy is examined.