Elasticity \(\mathcal{M} \)-tensors and the strong ellipticity condition. (English) Zbl 1433.74028
Summary: In this paper, we establish two sufficient conditions for the strong ellipticity of any fourth-order elasticity tensor and investigate a class of tensors satisfying the strong ellipticity condition, the elasticity \(\mathcal{M}\)-tensor. The first sufficient condition is 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. Second, an alternating projection algorithm is proposed to verify whether an elasticity tensor satisfies the first condition or not. Besides, the elasticity \(\mathcal{M} \)-tensor is defined with respect to the M-eigenvalues of elasticity tensors. We prove that any nonsingular elasticity \(\mathcal{M} \)-tensor satisfies the strong ellipticity condition by employing a Perron-Frobenius-type theorem for M-spectral radii of nonnegative elasticity tensors. Other equivalent definitions of nonsingular elasticity \(\mathcal{M} \)-tensors are also established.
MSC:
| 74B20 | Nonlinear elasticity |
| 74B10 | Linear elasticity with initial stresses |
| 15A18 | Eigenvalues, singular values, and eigenvectors |
| 15A69 | Multilinear algebra, tensor calculus |
| 15B48 | Positive matrices and their generalizations; cones of matrices |
Keywords:
elasticity tensor; strong ellipticity; m-positive definite; s-positive definite; alternating projection; \(\mathcal{M}\)-tensor; nonnegative tensorReferences:
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