Generalized strain space formulations and eigenprojection tensors

The relations between the symmetric second-order tensors of deformation rate and of {q, r}-generalized strain rate are given by transformations with fourth-order tensors, which are determined by the eigenprojection algorithm and summed up from functions of the distinct eigenvalues multiplied with dyadic products of the corresponding eigenprojections of the stretch tensors. These fourth-order transformation tensors also define the tensors of {q, r}-generalized stress which are work-conjugate to the corresponding {q, r}-generalized strain (rate). For finite deformations, every tensor pair of {q, r}-generalized stress and strain constitutes a distinct material model and forms a unique element of the generalized strain space formulations. For $q=r=0$, the tensors of logarithmic strain and stress emerge from the {q, r}-generalized strains and stresses together with the corresponding fourth-order logarithmic transformation tensors. The resulting logarithmic strain space formulation has proven as most accurate, stable, and efficient by its implementation into the special-purpose finite element simulation tools AutoForm, Pafix, and Urmel.