On Some Sufficient Conditions for Strong Ellipticity

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.