Abstract
The strong ellipticity condition plays an important role in nonlinear elasticity and in materials. In this paper, we define M-eigenvalues for an elasticity tensor. The strong ellipticity condition holds if and only if the smallest M-eigenvalue of the elasticity tensor is positive. If the strong ellipticity condition holds, then the elasticity tensor is rank-one positive definite. The elasticity tensor is rank-one positive definite if and only if the smallest Z-eigenvalue of the elasticity tensor is positive. A Z-eigenvalue of the elasticity tensor is an M-eigenvalue but not vice versa. If the elasticity tensor is second-order positive definite, then the strong ellipticity condition holds. The converse conclusion is not right. Computational methods for finding M-eigenvalues are presented.
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Qi, L., Dai, HH. & Han, D. Conditions for strong ellipticity and M-eigenvalues. Front. Math. China 4, 349–364 (2009). https://doi.org/10.1007/s11464-009-0016-6
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DOI: https://doi.org/10.1007/s11464-009-0016-6