Existence of energy minimums for thin elastic rods in static helical configurations

We characterize families of solutions of the static Kirchhoff model of a thin elastic rod physically. These families, which are proved to exist, depend on the behavior of the so-called register and also on the radius and pitch. We describe the energy densities for each of the solutions in terms of the elastic properties and geometric shape of the unstrained rod, which allows determining the selection mechanism for the preferred helical configurations. This analysis promises to be a fundamental tool for understanding the close connection between the study of elastic deformations in thin rods and coarse-grained models with widespread applications in the natural sciences.