On Some Sufficient Conditions for Strong Ellipticity
Provides theoretical conditions for strong ellipticity in elasticity, relevant to materials science and continuum mechanics, but the results are theoretical and incremental.
The paper establishes sufficient conditions for strong ellipticity of fourth-order elasticity tensors, including an extension of positive definiteness to an unfolding matrix, and proposes an alternating projection algorithm to verify the condition.
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.