Global curvature and self-contact of nonlinearly elastic curves and rods

Many different physical systems, e.g. super-coiled DNA molecules, have been successfully modelled as elastic curves, ribbons or rods. We will describe all such systems as framed curves, and will consider problems in which a three dimensional framed curve has an associated energy that is to be minimized subject to the constraint of there being no self-intersection. For closed curves the knot type may therefore be specified a priori. Depending on the precise form of the energy and imposed boundary conditions, local minima of both open and closed framed curves often appear to involve regions of self-contact, that is, regions in which points that are distant along the curve are close in space. While this phenomenon of self-contact is familiar through every day experience with string, rope and wire, the idea is surprisingly difficult to define in a way that is simultaneously physically reasonable, mathematically precise, and analytically tractable. Here we use the notion of global radius of curvature of a space curve in a new formulation of the self-contact constraint, and exploit our formulation to derive existence results for minimizers, in the presence of self-contact, of a range of elastic energies that define various framed curve models. As a special case we establish the existence of ideal shapes of knots.