A Reformulation of the Browaeys and Chevrot Decomposition of Elastic Maps

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2024年3月8日

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Walter Tape,
Carl Tape

摘要

An elastic map $T$ $\mathbf{T}$ associates stress with strain in some material. A symmetry of $T$ $\mathbf{T}$ is a rotation of the material that leaves $T$ $\mathbf{T}$ unchanged, and the symmetry group of $T$ $\mathbf{T}$ consists of all such rotations. The symmetry class of $T$ $\mathbf{T}$ 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 $T$ $\mathbf{T}$ and for each of six symmetry classes $Σ$ $\Sigma $ , computes the “ $Σ$ $\Sigma $ -percentage” of $T$ $\mathbf{T}$ . The theory also finds a “hexagonal approximation”—an approximation to $T$ $\mathbf{T}$ whose symmetry class is at least transverse isotropic. We reexamine their theory and recommend that the $Σ$ $\Sigma $ -percentages be abandoned. We also recommend that the hexagonal approximations to $T$ $\mathbf{T}$ be replaced with the closest transverse isotropic maps to $T$ $\mathbf{T}$ .