Self-contact in closed and open Kirchhoff rods

We consider the problem of an impenetrable Kirchhoff rod in self-contact. The contact is assumed to be frictionless and of point-contact type. The self-contacted rod is modeled as a set of contact-free segments each having unknown length and connected in series with each other through frictionless hard-contact condition. The Landau-Lifshitz approach is used to express analytically the centerline of each contact-free segment in terms of unknown parameters which are obtained using the hard-contact condition at each contact point and boundary condition on the full rod. The presented formulation is applicable to both closed and open rods. For the open rod case, its two end cross-sections need not be parallel either. Numerical solution of following three example problems are presented both before and after self-contact (i) twisting of a closed rod (H) compression and twisting of a straight rod (Hi) compression and twisting of a rod with opening angle. For the case of twisting of a closed rod in the pre-self-contact regime, we also derive a minimal set of three integral equations to obtain the ring's buckled spatial configuration. We also point to a difficulty in numerical integration of the rod's centerline torsion over the rod length whenever the torsion value becomes large locally: a finer discretization of the rod's length is necessitated in such regions to obtain an accurate integral value. An analytical approximation of the torsion integral is derived in such regions to make the full numerical integration computationally efficient.