A Reformulation of the Browaeys and Chevrot Decomposition of Elastic Maps

Abstract

An elastic map \(\mathbf{T}\) associates stress with strain in some material. A symmetry of \(\mathbf{T}\) is a rotation of the material that leaves \(\mathbf{T}\) unchanged, and the symmetry group of \(\mathbf{T}\) consists of all such rotations. The symmetry class of \(\mathbf{T}\) describes the symmetry group but without the orientation information. With an eye toward geophysical applications, Browaeys & Chevrot developed a theory which, for any elastic map \(\mathbf{T}\) and for each of six symmetry classes \(\Sigma \), computes the “\(\Sigma \)-percentage” of \(\mathbf{T}\). The theory also finds a “hexagonal approximation”—an approximation to \(\mathbf{T}\) whose symmetry class is at least transverse isotropic. We reexamine their theory and recommend that the \(\Sigma \)-percentages be abandoned. We also recommend that the hexagonal approximations to \(\mathbf{T}\) be replaced with the closest transverse isotropic maps to \(\mathbf{T}\).