Two Irreducible Functional Bases of Isotropic Invariants of A Fourth Order Three-Dimensional Symmetric and Traceless Tensor

TL;DR

This paper identifies two distinct irreducible functional bases for fourth order three-dimensional symmetric and traceless tensors, expanding understanding of their invariant structures relevant to elasticity tensor analysis.

Contribution

It introduces a second irreducible functional basis with nine invariants, distinct from the known minimal integrity basis, highlighting new invariant structures.

Findings

Proves the Smith-Bao basis is also an irreducible functional basis.

Introduces a second basis with no quartic invariant and two sextic invariants.

Shows the second basis is not contained in any minimal integrity basis.

Abstract

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 tensors. In this paper, we present two isotropic irreducible functional bases of a fourth order three-dimensional symmetric and traceless tensor. One of them is 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.