Stability of elastic rods with self-contact

In this paper we study the numerical stability of equilibrium configurations of an elastic rod with unilateral frictionless self-contact. We use a finite-element method with penalty and augmented Lagrangian approaches to account for the self-penetration constraint. The numerical solution is carried out using the arclength continuation method. In the case where the energy expression satisfies some hypotheses, we have proved that any isolated minimum of the continuous self-contact problem is the strong limit of a sequence of local minima of the discrete problem. Our stability analysis is based on detecting the zeros of the determinant of a symmetrized version of the stiffness matrix and whether there is a sign change of the determinant or not.