The closest elastic tensor of arbitrary symmetry to an elasticity tensor of lower symmetry

Maher Moakher, Andrew N. Norris

Research output: Contribution to journalArticlepeer-review

128 Scopus citations


The closest tensors of higher symmetry classes are derived in explicit form for a given elasticity tensor of arbitrary symmetry. The mathematical problem is to minimize the elastic length or distance between the given tensor and the closest elasticity tensor of the specified symmetry. Solutions are presented for three distance functions, with particular attention to the Riemannian and log-Euclidean distances. These yield solutions that are invariant under inversion, i.e., the same whether elastic stiffness or compliance are considered. The Frobenius distance function, which corresponds to common notions of Euclidean length, is not invariant although it is simple to apply using projection operators. A complete description of the Euclidean projection method is presented. The three metrics are considered at a level of detail far greater than heretofore, as we develop the general framework to best fit a given set of moduli onto higher elastic symmetries. The procedures for finding the closest elasticity tensor are illustrated by application to a set of 21 moduli with no underlying symmetry.

Original languageEnglish (US)
Pages (from-to)215-263
Number of pages49
JournalJournal of Elasticity
Issue number3
StatePublished - Dec 2006

All Science Journal Classification (ASJC) codes

  • Materials Science(all)
  • Mechanics of Materials
  • Mechanical Engineering


  • Anisotropy
  • Closest moduli
  • Elastic symmetry
  • Log-Euclidean
  • Reimannian metric

Fingerprint Dive into the research topics of 'The closest elastic tensor of arbitrary symmetry to an elasticity tensor of lower symmetry'. Together they form a unique fingerprint.

Cite this