A Polynomially Irreducible Functional Basis of Elasticity Tensors

TL;DR

This paper introduces a new, smaller functional basis of 251 isotropic invariants for the elasticity tensor, simplifying tensor function representation in mechanics.

Contribution

A novel method constructs a minimal functional basis of 251 invariants using intermediate tensors, reducing the number from previous larger bases.

Findings

Achieved a smaller functional basis of 251 invariants.

Constructed 429 invariants from intermediate tensors.

Eliminated redundant invariants to optimize the basis.

Abstract

Tensor function representation theory is an essential topic in both theoretical and applied mechanics. For the elasticity tensor, Olive, Kolev and Auffray (2017) proposed a minimal integrity basis of 297 isotropic invariants, which is also a functional basis. Inspired by Smith's and Zheng's works, we use a novel method in this article to seek a functional basis of the elasticity tensor, that contains less number of isotropic invariants. We achieve this goal by constructing 22 intermediate tensors consisting of 11 second order symmetrical tensors and 11 scalars via the irreducible decomposition of the elasticity tensor. Based on such intermediate tensors, we further generate 429 isotropic invariants which form a functional basis of the elasticity tensor. After eliminating all the invariants that are zeros or polynomials in the others, we finally obtain a functional basis of 251 isotropic invariants for the elasticity tensor.