Anisotropic Symmetries of Linear Elasticity

TLDR

Traditional treatments of anisotropic elastic symmetries rely on a full set of crystallographic symmetry elements, including centers, rotation axes, inversion axes, and mirror planes. This paper aims to develop anisotropic symmetry theory for linear elasticity using only the plane of mirror symmetry as the fundamental element. The authors catalog the thirteen distinct mirror planes and generate crystal system symmetry groups via orthogonal transformations associated with these planes. They show that crystal system symmetry groups of elastic crystallographic symmetries arise from combinations of orthogonal transformations of the thirteen mirror planes, and that the resulting restrictions on Hooke’s law coefficients follow directly from a small subset of these planes.

Abstract

The objective of this paper is to present a development of the anisotropic symmetries of linear elasticity theory based on the use of a single symmetry element, the plane of mirror symmetry. In this presentation the thirteen distinct planes of mirror symmetry are catalogued. Traditional presentations of the anisotropic elastic symmetries involve all the crystallographic symmetry elements which include the center of symmetry, the n-fold rotation axis and the n-fold inversion axis as well as the plane of mirror symmetry. It is shown that the crystal system symmetry groups, as opposed to the crystal class symmetry groups, of the elastic crystallographic symmetries can be generated by the appropriate combinations of the orthogonal transformations corresponding to each of the thirteen distinct planes of mirror symmetry. It is also shown that the restrictions on the elastic coefficients appearing in Hooke’s law follow in a simple and straightforward fashion from orthogonal transformations based on a small subset of the small catalogue of planes of mirror symmetry.