A minimal integrity basis for the elasticity tensor

TL;DR

This paper presents the first complete minimal integrity basis of 297 polynomial invariants for the elasticity tensor, achieved through invariant theory and tensor decomposition techniques.

Contribution

It provides a definitive solution to finding a minimal integrity basis for the elasticity tensor using tensor decomposition and classical invariant theory.

Findings

Established a minimal integrity basis of 297 invariants.

Reduced the problem to joint invariants of triplet (a, b, D).

First time this basis has been explicitly obtained.

Abstract

We definitively solve the old problem of finding a minimal integrity basis of polynomial invariants of the fourth-order elasticity tensor C. Decomposing C into its SO(3)-irreducible components we reduce this problem to finding joint invariants of a triplet (a, b, D), where a and b are second-order harmonic tensors, and D is a fourth-order harmonic tensor. Combining theorems of classical invariant theory and formal computations, a minimal integrity basis of 297 polynomial invariants for the elasticity tensor is obtained for the first time.