Representation of Vector-Valued Hemitropic Functions of a Symmetric Tensor and a Vector

Abstract

We revisit the question of finding the most general representation of vector-valued functions such that \({\mathbf{f}}( {\mathbf{Q}}{\mathbf{A}}{\mathbf{Q}}^{T},{\mathbf{Q}}{\mathbf{u}}) = {\mathbf{Q}} {\mathbf{f}}({ \mathbf{A}},{\mathbf{u}}) \) for all rotation tensors \({\mathbf{Q}}\), all symmetric tensors \({\mathbf{A}}\), and all vectors \({\mathbf{u}}\). Seven scalar invariants and three vectors are identified that form the basis for the functional dependence in its simplest form. Six of the seven scalar invariants are required in the case of isotropy with respect to the full orthogonal group. The results are relevant to hemitropic vector-valued functions in continuum mechanics, such as the thermoelastic heat flux.