On Some Sufficient Conditions for Strong Ellipticity

TL;DR

This paper provides new sufficient conditions for the strong ellipticity of fourth-order elasticity tensors, extending positive definiteness concepts and proposing an algorithm for verification, with applications to isotropic and anisotropic materials.

Contribution

It introduces an extension of positive definite matrices as a sufficient condition and an alternating projection algorithm for verifying strong ellipticity in elasticity tensors.

Findings

The first condition ensures strong ellipticity if the unfolding matrix can be made positive definite.

An algorithm is developed to verify the first condition for given tensors.

Additional conditions are explored for isotropic and specific anisotropic materials.

Abstract

We establish several sufficient conditions for the strong ellipticity of any fourth-order elasticity tensor in this paper. The first presented sufficient condition is an extension of positive definite matrices, which states 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. An alternating projection algorithm is proposed to verify whether an elasticity tensor satisfies the first condition or not. Conditions for some special cases beyond the first sufficient condition are further investigated, which includes some important cases for the isotropic and some particular anisotropic linearly elastic materials.