Conditions for strong ellipticity and M-eigenvalues

被引：0
作者：
Liqun Qi
Hui-Hui Dai
Deren Han
机构：
[1] The Hong Kong Polytechnic University,Department of Applied Mathematics
[2] The City University of Hong Kong,Department of Mathematics
[3] Nanjing Normal University,School of Mathematics and Computer Sciences
来源：
Frontiers of Mathematics in China | 2009年 / 4卷
关键词：
Elasticity tensor; strong ellipticity; M-eigenvalue; Z-eigenvalue; 74B99; 15A18; 15A69;
D O I：
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中图分类号：
学科分类号：
摘要：
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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