On closest isotropic tensors and their norms

TL;DR

This paper compares four methods for approximating anisotropic elasticity tensors with the closest isotropic tensors, analyzing their similarities and differences within measurement error ranges.

Contribution

It introduces and compares four different criteria for determining the closest isotropic tensor to a given anisotropic elasticity tensor.

Findings

The four approaches yield similar isotropic tensors within measurement error ranges.

Different criteria can lead to slightly different isotropic approximations.

The methods provide a basis for choosing appropriate approximation criteria in practice.

Abstract

An anisotropic elasticity tensor can be approximated by the closest tensor belonging to a higher symmetry class. The closeness of tensors depends on the choice of a criterion. We compare the closest isotropic tensors obtained using four approaches: the Frobenius 36-component norm, the Frobenius 21-component norm, the operator norm and the L2 slowness-curve fit. We find that the isotropic tensors are similar to each other within the range of expected measurement errors.