Classifications of ideal 3D elastica shapes at equilibrium

We investigate the equilibrium configurations of the ideal 3D elastica, i.e., inextensible, unshearable, isotropic, uniform, and naturally straight and prismatic rods, with linear elastic constitutive relations. Infinite solution trajectories are expressed analytically and classified in terms of only three parameters related to physical quantities. Orientation of sections and mechanical loading are also well described analytically with these parameters. Detailed analysis of solution trajectories yields two main results. First, all particular trajectories are completely characterized and located in the space of these parameters. Second, a general geometric structure is exhibited for every ideal 3D elastic rod, where the trajectory winds around a core helix in a tube-shaped envelope. This remarkable structure leads to a classification of the general case according to three properties called chirality components. In addition, the geometry of the envelope provides another characterization of the ideal 3D elas...