Conditions for strong ellipticity and M-eigenvalues

Liqun Qi, Hui Hui Dai, Deren Han

Research output: Journal article publicationJournal articleAcademic researchpeer-review

51 Citations (Scopus)

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.
Original languageEnglish
Pages (from-to)349-364
Number of pages16
JournalFrontiers of Mathematics in China
Volume4
Issue number2
DOIs
Publication statusPublished - 1 Jun 2009

Keywords

  • Elasticity tensor
  • M-eigenvalue
  • Strong ellipticity
  • Z-eigenvalue

ASJC Scopus subject areas

  • Mathematics (miscellaneous)

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