On the Smallest Number of Functions Representing Isotropic Functions of Scalars, Vectors and Tensors

TL;DR

This paper determines the minimal number of irreducible functions needed to represent isotropic functions of scalars, vectors, and tensors, significantly reducing previous counts and simplifying modeling complexity.

Contribution

It proves new lower bounds on the number of irreducible invariants, vectors, and tensors for isotropic functions, improving upon existing literature.

Findings

Number of scalar irreducible invariants: 3P+9M+6N-3

Number of vector irreducible functions: 3

Number of tensor irreducible functions: at most 9

Abstract

In this paper, we address the open problem (stated in Pennisi and Trovato, 1987. Int. J. Engng Sci., 25(8), 1059-1065) associated with the irreducibility of representations for isotropic functions. In particular, we prove that for isotropic functions that depend on $P$ vectors, $N$ symmetric tensors and $M$ non-symmetric tensors (a) the number of irreducible invariants for a scalar-valued isotropic function is $3P+9M+6N-3$ (b) the number of irreducible vectors for a vector-valued isotropic function is $3$ and (c) the number of irreducible tensors for a tensor-valued isotropic function is at most $9$. The irreducible numbers in given (a), (b) and (c) are much lower than those obtained in the literature. This significant reduction in the number of irreducible scalar/vector/tensor-valued functions have the potential to substantially simplify modelling complexity.