A Variational Rod Model with a Singular Nonlocal Potential

In the classic theory of elastic rods, two non-adjacent points along the rod may upon contact occupy the same physical space. We develop an elastic rod model with a pairwise repulsive potential such that if two non-adjacent points along the rod are close in physical space, there is an energy barrier that prevents contact. For adjacent pairs, the repulsive potential is negligible and the elastic rod is described by a classical elastic rod model. The framework for this model is developed to prove the existence of minimizers within each homotopy class, where the idea of topological homotopy of a curve is generalized to a framed curve, or an elastic rod. Finally, the first-order necessary conditions are derived.