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.

Automated summary

In this article, it is proven that for isotropic functions depending on P vectors, N symmetric tensors, and M non-symmetric tensors, the minimal number of irreducible invariants for a scalar-valued isotropic function is 3P+9M+6N-3, the minimal number of irreducible vectors for a vector-valued isotropic function is 3, and the minimal number of irreducible tensors for a tensor-valued isotropic function is at most 9. The minimal irreducible numbers obtained in this study are generally much lower than those found in previous literature. This significant reduction in the numbers of irreducible isotropic functions has the potential to substantially reduce modeling complexity.