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

This article establishes minimal irreducible bases for isotropic functions depending on vectors, symmetric tensors, and non-symmetric tensors, significantly reducing the number of invariants and basis elements needed compared to existing literature. It proves that scalar-valued isotropic functions depending on \(P\) vectors, \(N\) symmetric tensors, and \(M\) non-symmetric tensors require only \(3P + 9M + 6N - 3\) irreducible invariants; vector-valued isotropic functions need only 3 irreducible vectors; and tensor-valued isotropic functions require at most 9 irreducible tensors. These minimal sets are constructed using a spectral approach based on eigenvalues and eigenvectors of symmetric tensors, which simplifies the representation of constitutive equations in continuum mechanics and reduces modeling complexity. The article also clarifies that previously established irreducible bases by Boehler and Smith, while complete, are generally not minimal, and demonstrates how potential vectors and tensors derived from scalar isotropic functions fit within these minimal bases. This work has implications for more efficient and physically interpretable modeling of complex materials, including anisotropic and viscoelastic solids.