Invariant properties for finding distance in space of elasticity tensors

TL;DR

This paper develops algorithms to compute the closest elasticity tensor with specific symmetries to a given tensor by minimizing distance measures based on eigenvalues and invariant traces, considering orientation dependence.

Contribution

It introduces a method to find the nearest symmetric elasticity tensor using orthogonal projections and eigenvalue invariants, applicable to isotropy, cubic, and transverse isotropy.

Findings

Algorithms for symmetry-specific tensor approximation

Distance minimization based on eigenvalues and invariant traces

Applicable to isotropic, cubic, and transverse isotropic materials

Abstract

Using orthogonal projections, we investigate distance of a given elasticity tensor to classes of elasticity tensors exhibiting particular material symmetries. These projections depend on the orientation of the elasticity tensor, hence the distance is obtained as the minimization of corresponding expressions with respect to the action of the orthogonal group. These expressions are stated in terms of the eigenvalues of both the given tensor and the projected one. The process of minimization is facilitated by the fact that, as we prove, the traces of the corresponding Voigt and dilatation tensors are invariant under these orthogonal projections. For isotropy, cubic symmetry and transverse isotropy, we formulate algorithms to find both the orientation and the eigenvalues of the elasticity tensor that is endowed with a particular symmetry and is closest to the given elasticity tensor.