# Elasticity $\mathscr{M}$-tensors and the Strong Ellipticity Condition

## TL;DR

This paper introduces new sufficient conditions and an algorithm to verify the strong ellipticity of fourth-order elasticity tensors, focusing on elasticity $ ext{ extscr M}$-tensors and their spectral properties.

## Contribution

It proposes two novel sufficient conditions for strong ellipticity and develops an alternating projection algorithm for verification, advancing the understanding of elasticity tensor properties.

## Key findings

- Nonsingular elasticity $ ext{ extscr M}$-tensors satisfy strong ellipticity.
- An algorithm for verifying strong ellipticity based on tensor unfolding matrices.
- Equivalent definitions of nonsingular elasticity $ ext{ extscr M}$-tensors are provided.

## Abstract

In this paper, we establish two sufficient conditions for the strong ellipticity of any fourth-order elasticity tensor and investigate a class of tensors satisfying the strong ellipticity condition, the elasticity $\mathscr{M}$-tensor. The first sufficient condition is that the strong ellipticity holds if the unfolding matrix of this fourth-order elasticity tensor can be modified into a positive definite one by preserving the summations of some corresponding entries. Second, an alternating projection algorithm is proposed to verify whether an elasticity tensor satisfies the first condition or not. Besides, the elasticity $\mathscr{M}$-tensor is defined with respect to the M-eigenvalues of elasticity tensors. We prove that any nonsingular elasticity $\mathscr{M}$-tensor satisfies the strong ellipticity condition by employing a Perron-Frobenius-type theorem for M-spectral radii of nonnegative elasticity tensors. Other equivalent definitions of nonsingular elasticity $\mathscr{M}$-tensors are also established.