Sharp Bounds for the Smallest M-eigenvalue of an Elasticity Z-tensor and Its Application

The smallest M -eigenvalue τ M ( A ) of a fourth-order partial symmetric tensor A plays an important role in judging the strong ellipticity condition (abbr. SE-condition) in elastic mechanics. Specifically, if τ M ( A ) > 0 , then the SE-condition of A holds. In this paper, we establish lower and upper bounds of τ M ( A ) via extreme eigenvalues of symmetric matrices and tensors constructed by the entries of A . In addition, when A is an elasticity Z -tensor, we establish lower bounds for τ M ( A ) via the extreme C -eigenvalues of piezoelectric-type tensors. Finally, numerical examples show the efficiency of our proposed bounds in judging the SE-condition of A .