Frontiers of Mathematics in China >
Conditions for strong ellipticity and M-eigenvalues
Received date: 13 Jan 2009
Accepted date: 24 Jan 2009
Published date: 05 Jun 2009
Copyright
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.
Key words： Elasticity tensor; strong ellipticity; M-eigenvalue; Z-eigenvalue
Liqun QI , Hui-Hui DAI , Deren HAN . Conditions for strong ellipticity and M-eigenvalues[J]. Frontiers of Mathematics in China, 2009 , 4(2) : 349 -364 . DOI: 10.1007/s11464-009-0016-6
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