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

In this article, we prove that for isotropic functions that depend on $P$ vectors, $N$ symmetric tensors and $M$ non-symmetric tensors (a) the minimal number of irreducible invariants for a scalar-valued isotropic function is $3P+9M+6N-3,$ (b) the minimal number of irreducible vectors for a vector-valued isotropic function is $3$ and (c) the minimal number of irreducible tensors for a tensor-valued isotropic function is at most $9$. The minimal irreducible numbers given in (a), (b) and (c) are, in general, much lower than the irreducible numbers obtained in the literature. This significant reduction in the numbers of irreducible isotropic functions has the potential to substantially reduce modelling complexity.