Upper bounds estimates of the distance to cubic or orthotropic elasticity

TL;DR

This paper develops analytical methods to estimate how close a given elasticity tensor is to cubic or orthotropic symmetry, improving accuracy over covariant-based approaches.

Contribution

It introduces a new analytical technique for estimating the distance to cubic and orthotropic symmetry, considering multiple second-order tensors and extending previous methods.

Findings

Multiple second-order tensors carry the likely symmetry coordinate system.

Covariants of cubic tensors are isotropic, affecting estimation accuracy.

Numerical examples demonstrate the effectiveness of the proposed bounds.

Abstract

We address the problem, not of the determination -- which usually needs numerical methods -- but of an accurate analytical estimation of the distance of a raw elasticity tensor to cubic symmetry and to orthotropy. We point out that there are not one but several secondorder tensors that carry the likely cubic/orthotropic coordinate system of the raw tensor. Since all the second-order covariants of an (exactly) cubic elasticity tensor are isotropic, distance estimates based only on such covariants are not always accurate. We extend to cubic symmetry and to orthotropy the technique recently suggested by Klime\v{s} for transverse isotropy: solving analytically an auxiliary quadratic minimization problem whose solution is a second-order tensor that carries the likely cubic coordinate system. Numerical examples are provided, on which we evaluate the accuracy of different upper bounds estimates of the distance to cubic or orthotropic symmetry.