A two-director Cosserat rod model using unconstrained quaternions

Starting from a constrained three-dimensional analysis, we propose a new model for an initially curved Cosserat rod with two independent directors, in which the cross-section can inflate and ovalize. The usage of unconstrained quaternions gets rid of the non-linear constraints and singularities in the parametrization of objective strains, related to the orthonormality of rotation matrices. Furthermore, the classic definitions of curvatures and twist have been modified and tuned for a linear constitutive law (i.e. a linear relationship between one-dimensional stresses and strains). The capabilities and the limitations of the model in describing the non-linear effects of cross-section deformation on bending and torsion, and usually handled with bi- and three-dimensional theories, are then discussed.