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

Abstract

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 selfintersection. 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. (2000): 49J99, 53A04, 57M25, 74B20, 92C40 Mathematics Subject Classification

Self-Contact for Rods on Cylinders

“…In three dimensions a variety of different descriptions of self-contact exists for rods of finite thickness, each with subtle advantages and disadvantages (see e.g. the introduction of [10]). For a rod on a cylinder the situation is simpler, since the freedom of movement is essentially two-dimensional-similar to that of a curve in a plane.…”

“…The study of self-contact in elastic rods has seen some remarkable progress over the last ten years, with highlights such as the numerical work of Tobias, Coleman, and Swigon [22,6,5], the introduction of global curvature by Gonzalez and co-workers [10], and the derivation of the Euler-Lagrange equations for energy minimization by Schuricht and Von der Mosel [19]. Parallel advances have been made on the highly related ideal knots and Gehring links, where ropelength is minimized instead of elastic energy [4,18,3].…”

Dynamics of Elastic Rods in Perfect Friction Contact

“…Another avenue of studies of stationary states of elastic rods with self-contact [6][7][8][9][10] explicitly computes the contact forces from the existence of constraints. In our case, the rolling contact comes from friction, which does not admit any potential description.…”

Visualizing the tightening of knots

“…It is known that such minimizing curves exist for every kind of knot and link, but their exact shapes are currently the subject of active mathematical research (cf. [8,15,16]). Interestingly, the curves of minimimum ropelength in a given knot type are not always congruent: some links are known to have a family of tight realizations with different shapes.…”

“…An exciting recent development in this field of mathematics has been the formulation of a kind of Euler-Lagrange equation describing length-critical knots in terms of the set of self-contacts of their tubes [7,16,38]. This theory has allowed us to make some conjectures about the tightening process, but these conjectures have * e-mail: cantarel@math.uga.edu, supported by NSF DMS #0204862 † e-mail: piatek@cs.washington.edu, supported by NSF DMS #0311010 ‡ e-mail: rawdon@mathcs.duq.edu, supported by NSF DMS #0311010 Figure 1: A tight 7.2 knot as computed using our method.…”