Invariant Tensor-to-Matrix Mappings for Evaluation of Tensorial Expressions

J. C. Nadeau, M. Ferrari

Research output: Contribution to journalArticle

25 Scopus citations

Abstract

A class of mappings is presented, parameterized by a variable η, that operates on tensorial expressions to yield equivalent matrical expressions which are then easily evaluated, either numerically or symbolically, using standard matrix operations. The tensorial expressions considered involve scalar, second- and fourth-order tensors, double contractions, inversion and transposition. Also addressed is coordinate transformation and eigenanalysis of fourth-order tensors. The class of mappings considered is invariant, meaning that for a given η the corresponding mapping depends only on the order of the tensor upon which it acts and not, for example, on its physical interpretation (e.g., stress vs. strain, or stiffness vs. compliance). As a result the proposed mappings avoid ad hoc definitions like that of engineering shear strain (i.e., γij := 2εij for i ≠ j) which is inconsistent with an invariant mapping. Two convenient choices for the parameter η are presented. Appendix B presents a convenient summary for instructional purposes.

Original languageEnglish (US)
Pages (from-to)43-61
Number of pages19
JournalJournal of Elasticity
Volume52
Issue number1
DOIs
StatePublished - 1998

Keywords

  • Condensed/ engineering/ matrix notation
  • Tensor-to-matrix mappings
  • Tensorial expression evaluation

ASJC Scopus subject areas

  • Engineering (miscellaneous)
  • Computational Mechanics
  • Mechanics of Materials
  • Materials Science(all)

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