Conditions for strong ellipticity and M-eigenvalues

doi:10.1007/s11464-009-0016-6

Front Math Chin

2009, Vol. 4

Issue (2) : 349-364 https://doi.org/10.1007/s11464-009-0016-6

RESEARCH ARTICLE

Conditions for strong ellipticity and M-eigenvalues

Liqun QI¹(

), Hui-Hui DAI², Deren HAN³

1. Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong, China; 2. Department of Mathematics, The City University of Hong Kong, Hong Kong, China; 3. School of Mathematics and Computer Sciences, Nanjing Normal University, Nanjing 210097, China

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Abstract

The strong ellipticity condition plays an important role in nonlinear elasticity and in materials. In this paper, we de?ne 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 de?nite. The elasticity tensor is rank-one positive de?nite 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 de?nite, then the strong ellipticity condition holds. The converse conclusion is not right. Computational methods for ?nding M-eigenvalues are presented.

Keywords Elasticity tensor strong ellipticity M-eigenvalue Z-eigenvalue

Corresponding Author(s): QI Liqun,Email:maqilq@polyu.edu.hk

Issue Date: 05 June 2009

Cite this article:

Liqun QI,Hui-Hui DAI,Deren HAN. Conditions for strong ellipticity and M-eigenvalues[J]. Front Math Chin, 2009, 4(2): 349-364.

URL:

https://academic.hep.com.cn/fmc/EN/10.1007/s11464-009-0016-6
https://academic.hep.com.cn/fmc/EN/Y2009/V4/I2/349

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