Sharp bounds for the smallest \(M\)-eigenvalue of an elasticity \(Z\)-tensor and its application. (English) Zbl 1543.15006
Summary: The smallest \(M\)-eigenvalue \(\tau_M (\mathcal{A})\) of a fourth-order partial symmetric tensor \(\mathcal{A}\) plays an important role in judging the strong ellipticity condition (abbr. SE-condition) in elastic mechanics. Specifically, if \(\tau_M (\mathcal{A})>0\), then the SE-condition of \(\mathcal{A}\) holds. In this paper, we establish lower and upper bounds of \(\tau_M (\mathcal{A})\) via extreme eigenvalues of symmetric matrices and tensors constructed by the entries of \(\mathcal{A}\). In addition, when \(\mathcal{A}\) is an elasticity \(Z\)-tensor, we establish lower bounds for \(\tau_M (\mathcal{A})\) 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 \(\mathcal{A}\).
MSC:
| 15A18 | Eigenvalues, singular values, and eigenvectors |
| 15A42 | Inequalities involving eigenvalues and eigenvectors |
| 15A69 | Multilinear algebra, tensor calculus |
Keywords:
partial symmetric tensors; elasticity \(Z\)-tensors; \(M\)-eigenvalues; \(C\)-eigenvalues; strong ellipticity conditionReferences:
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