| Kathleen A. Hoffman
Elastic Rods with Self-ContactClassical theory of elastic rods neglects contact forces along the rod, and specifically allows two distinct points along the rod to occupy the same physical space. My goal was to develop and analyze models of elastic rods that specifically prohibit this behavior. There are essentially two classes of models that prevent self-contact of elastic rods: soft contact, in which a singular, nonlocal repulsive potential (such as electrostatic repulsion along the rod) is added to the classical energy formulation; and hard contact in which an impenetrable tube surrounds the centerline of the elastic rod.
Stability theory for elastic rods models with soft-contact%Stability theory for elastic rods with soft contact is technically %challenging and computationally difficult. The challenge in working with this model of self-contact is in the integral term of the differential-integral second variation operator, which is both non-local and singular. My paper, An extended conjugate point theory with application to the stability of planar buckling of an elastic rod subject to a repulsive self-potential, detailing the analysis and demonstrating a two-dimensional example with computational results, appeared in SIAM Journal of Mathematical Analysis.
Existence and Optimality for Elastic rods with soft-contactUnlike my previous work which focused on stability theory, the focus of this work is existence of minimizers. The proof requires a fundamentally different formulation of the elastic rod theory than that of the stability theory and also sets forth a new idea of rod homotopy, which extends the classical ideas of homotopy of a curve to that of a framed curved. This work appeared in the Archive for Rational Mechanics and Analysis.
Existence and Optimality for Elastic rods with hard-contactUnlike the previous two problems which focused on elastic rods with soft-contact, the challenge of this problem lie in modeling the impenetrable tube as an inequality constraint. Dealing with this constraint required tools from non-smooth analysis. This paper appeared in the Journal of Nonlinear Analysis Series A: Theory, Methods & Applications.