The number of independent invariants for m unit vectors and n symmetric second order tensors is 2m+ 6n-3

TL;DR

This paper establishes the maximum number of independent invariants for a set of m unit vectors and n symmetric second order tensors as 2m+6n-3, aiding in continuum mechanics modeling.

Contribution

It provides a rigorous proof for the maximum count of independent invariants and explores relations among classical invariants in minimal integrity bases.

Findings

Maximum number of invariants is 2m+6n-3.

Relations between classical invariants are characterized.

Supports more efficient constitutive modeling in continuum mechanics.

Abstract

Anisotropic invariants play an important role in continuum mechanics. Knowing the number of independent invariants is crucial in modelling and in a rigorous construction of a constitutive equation for a particular material, where it is determined by doing tests that hold all, except one, of the independent invariants constant so that the dependence in the one invariant can be identified. Hence, the aim of this paper is to prove that the number of independent invariants for a set of $n$ symmetric tensors and $m$ unit vectors is at most 2m+ 6n-3. We also give relations between classical invariants in the corresponding minimal integrity basis.