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

The smallest M-eigenvalue \(\tau _M ({\mathcal {A}})\) ${\tau }_M(\mathcal{A})$ of a fourth-order partial symmetric tensor \({\mathcal {A}}\) $\mathcal{A}$ plays an important role in judging the strong ellipticity condition (abbr. SE-condition) in elastic mechanics. Specifically, if \(\tau _M ({\mathcal {A}})>0\) ${\tau }_M(\mathcal{A})>0$ , then the SE-condition of \({\mathcal {A}}\) $\mathcal{A}$ holds. In this paper, we establish lower and upper bounds of \(\tau _M ({\mathcal {A}})\) ${\tau }_M(\mathcal{A})$ via extreme eigenvalues of symmetric matrices and tensors constructed by the entries of \({\mathcal {A}}\) $\mathcal{A}$ . In addition, when \({\mathcal {A}}\) $\mathcal{A}$ is an elasticity Z-tensor, we establish lower bounds for \(\tau _M ({\mathcal {A}})\) ${\tau }_M(\mathcal{A})$ 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 \({\mathcal {A}}\) $\mathcal{A}$ .