The natural -conformation tensor constitutive laws

In this work, we revisit the seminal work of Renardy [M. Renardy, J. Non-Newtonian Fluid Mech. 52(1), 91–95 (1994)] on the reformulation of the stress tensor in its “natural” basis and present a generic framework for the natural-conformation tensor for a large class of differential constitutive models. We show that the proposed dyadic transformation can be equated as an orthogonal transformation of the conformation tensor into a streamlined orthonormal basis given by a rotation tensor expressed in terms of the unit velocity vectors. We also show that the natural-conformation tensor formulation is a particular sub-case of the kernel-conformation tensor transformation [A. M. Afonso, F. T. Pinho, M. A. Alves, J. Non-Newtonian Fluid Mech. 167–168, 30–37 (2012)] with the kernel function acting on the rotation of the eigenvectors rather than on the magnitude of the extension of the conformation tensor.