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arXiv · 2207.09617

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

M. H. B. M. Shariff
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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.

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M. H. B. M. Shariff. 2022-07-20. On the Smallest Number of Functions Representing Isotropic Functions of Scalars, Vectors and Tensors. https://arxiv.org/abs/2207.09617

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