The closest elastic tensor of arbitrary symmetry to an elasticity tensor of lower symmetry

TL;DR

This paper develops explicit methods to find the closest higher-symmetry elastic tensor to a given arbitrary tensor using various distance metrics, enhancing the analysis of material symmetries in elasticity.

Contribution

It introduces explicit solutions for the closest elastic tensors of higher symmetry classes using Riemannian, log-Euclidean, and Frobenius distances, with detailed projection methods.

Findings

Solutions invariant under inversion for Riemannian and log-Euclidean distances.

Complete description of Euclidean projection method.

Application to 21 moduli with no symmetry.

Abstract

The closest tensors of higher symmetry classes are derived in explicit form for a given elasticity tensor of arbitrary symmetry. The mathematical problem is to minimize the elastic length or distance between the given tensor and the closest elasticity tensor of the specified symmetry. Solutions are presented for three distance functions, with particular attention to the Riemannian and log-Euclidean distances. These yield solutions that are invariant under inversion, i.e., the same whether elastic stiffness or compliance are considered. The Frobenius distance function, which corresponds to common notions of Euclidean length, is not invariant although it is simple to apply using projection operators. A complete description of the Euclidean projection method is presented. The three metrics are considered at a level of detail far greater than heretofore, as we develop the general framework to best fit a given set of moduli onto higher elastic symmetries. The procedures for finding the closest elasticity tensor are illustrated by application to a set of 21 moduli with no underlying symmetry.