The isotropic material closest to a given anisotropic material

TL;DR

This paper introduces methods to determine the isotropic elastic moduli closest to a given anisotropic material using various distance measures, providing explicit formulas and algorithms for different material symmetries.

Contribution

It defines and compares three approaches to find the closest isotropic moduli to anisotropic materials, including explicit formulas and a general algorithm for arbitrary anisotropy.

Findings

Closest moduli are unique under Riemannian and log-Euclidean norms.

Explicit formulas for cubic materials' bulk and shear moduli.

Algorithm for arbitrary anisotropic materials demonstrated.

Abstract

The isotropic elastic moduli closest to a given anisotropic elasticity tensor are defined using three definitions of elastic distance, the standard Frobenius (Euclidean) norm, the Riemannian distance for tensors, and the log-Euclidean norm. The closest moduli are unique for the Riemannian and the log-Euclidean norms, independent of whether the difference in stiffness or compliance is considered. Explicit expressions for the closest bulk and shear moduli are presented for cubic materials, and an algorithm is described for finding them for materials with arbitrary anisotropy. The method is illustrated by application to a variety of materials, which are ranked according to their distance from isotropy.