On strain measures and the geodesic distance to SOn in the general linear group

Raz Kupferman, Asaf Shachar

Research output: Contribution to journalArticlepeer-review

2 Scopus citations

Abstract

We consider various notions of strains-quantitative measures for the deviation of a linear transformation from an isometry. The main approach, which is motivated by physical applications and follows the work of [12], is to select a Riemannian metric on GLn, and use its induced geodesic distance to measure the distance of a linear transformation from the set of isometries. We give a short geometric derivation of the formula for the strain measure for the case where the metric is left-GLn-invariant and right-On-invariant. We proceed to investigate alternative distance functions on GLn, and the properties of their induced strain measures. We start by analyzing Euclidean distances, both intrinsic and extrinsic. Next, we prove that there are no bi-invariant distances on GLn. Lastly, we investigate strain measures induced by inverse-invariant distances.

Original languageAmerican English
Pages (from-to)437-460
Number of pages24
JournalJournal of Geometric Mechanics
Volume8
Issue number4
DOIs
StatePublished - Dec 2016

Bibliographical note

Publisher Copyright:
©American Institute of Mathematical Sciences.

Keywords

  • General linear group
  • Strain measure
  • Symmetries

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