Two irreducible functional bases of isotropic invariants of a fourth-order three-dimensional symmetric and traceless tensor

Zhongming Chen, Yannan Chen, Liqun Qi, Wennan Zou

Research output: Journal article publicationJournal articleAcademic researchpeer-review

1 Citation (Scopus)


The elasticity tensor is one of the most important fourth-order tensors in mechanics. Fourth-order three-dimensional symmetric and traceless tensors play a crucial role in the study of the elasticity tensor. In this paper, we present two isotropic irreducible functional bases for a fourth-order three-dimensional symmetric and traceless tensor. One of them is exactly the minimal integrity basis introduced by Smith and Bao in 1997. It has nine homogeneous polynomial invariants of degrees two, three, four, five, six, seven, eight, nine and ten, respectively. We prove that it is also an irreducible functional basis. The second irreducible functional basis also has nine homogeneous polynomial invariants. It has no quartic invariant but has two sextic invariants. The other seven invariants are the same as those of the Smith–Bao basis. Hence, the second irreducible functional basis is not contained in any minimal integrity basis.

Original languageEnglish
Pages (from-to)3092-3102
Number of pages11
JournalMathematics and Mechanics of Solids
Issue number10
Publication statusPublished - 1 Oct 2019


  • fourth-order tensor
  • Invariant
  • irreducible functional basis
  • symmetric and traceless tensor

ASJC Scopus subject areas

  • Mathematics(all)
  • Materials Science(all)
  • Mechanics of Materials

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