Noncoaxiality between Two Tensors with Application to Stress Rate Decomposition and Fabric Anisotropy Variable

Abstract

The coaxial and totally noncoaxial parts of a tensor in regard to another reference tensor are derived in closed analytical form based on representation theorems of tensor-valued isotropic functions. In the process a new interpretation is obtained for a singular case of representation theorems. As a first application, the coaxial and noncoaxial parts of a stress rate tensor in regard to the stress tensor are analytically expressed, and the findings applied to the following two cases: analytically express the part of a stress rate tensor that induces change of stress principal axes at fixed principal stress values, and change of stress principal values at fixed stress principal axes such that the stress orbit on the stress π-plane is circular. A second application refers to enhancing the definition of the fabric anisotropy variable A, a quantity of cardinal importance for anisotropic critical state theory in granular mechanics, so that the orthogonal coaxial and noncoaxial parts of the fabric tensor in regard to the plastic strain rate direction participate in the definition of A.