Conditions of strong ellipticity and calculations of M-eigenvalues for a partially symmetric tensor

Abstract

Strong ellipticity condition (SE-condition) of the equilibrium equations for general nonlinear elastic materials can be equivalently transformed into the SE-condition of a partially symmetric tensor. Qi et al. in 2009 proved that the SE-condition of a partially symmetric tensor holds if and only if its smallest M-eigenvalue is positive. In this paper, a criterion for the SE-condition of a partially symmetric tensor is first given. And then, an alternative method to compute all M-eigentriples of a partially symmetric tensor is presented. Finally, two numerical examples show the effectiveness of the proposed criterion and the calculation of M-eigenvalues in judging the SE-condition of the equilibrium equations.