A unified theory of representations for scalar-, vector- and second order tensor-valued anisotropic functions of vectors and second order tensors

Abstract

A substantial generalization of Lokhin-Sedov-Boehler-Liu's isotropic extension method for representations of anisotropic tensor functions is suggested. It is shown that every scalar-, vector- and second order tensor-valued anisotropic tensor function with vector and second order tensor variables can be extended as an isotropic tensor function merely with augmented vector and second order tensor variables through some simple polynomial vector-valued and second order tensor-valued invariant tensor functions characterizing the anisotropy group. This result circumvents the difficulty involved in the usual direct generalization of the aforementioned LSBL method due to the introduction of structural tensor variables of order higher than two, and enables us to derive complete representations for various types of anisotropic tensor functions of vectors and second order tensors directly from the well-known results for isotropic tensor functions of vectors and second order tensors. All anisotropy groups describing symmetries of solid materials, including the thirty-two crystal classes and all infinitely many noncrystal classes, are considered.