Global Injectivity and Topological Constraints for Spatial Nonlinearly Elastic Rods

Abstract

In this paper we study the local and global injectivity of spatial deformations of shearable nonlinearly elastic rods. We adopt an analytical condition introduced by Ciarlet & Necas in nonlinear elasticity to ensure global injectivity in that case. In particular we verify the existence of an energy-minimizing equilibrium state without self-penetration which may also be restricted by a rigid obstacle. Furthermore we consider the special situation where the ends of the rod are glued together. In that case we can still impose topological restrictions such as, e.g., that the shape of the rod belongs to a given knot type. Again we show the existence of a globally injective energy minimizer which now in addition respects the topological constraints. Note that the investigation of super-coiled DNA molecules is an important application of the presented results.