Elastic Knots

We study the mechanical response of elastic rods bent into open knots, focusing on the case of trefoil and cinquefoil topologies. The limit of a weak applied tensile force is studied both analytically and experimentally: the Kirchhoff equations with self-contact are solved by means of matched asymptotic expansions; predictions on both the geometrical and mechanical properties of the elastic equilibrium are compared to experiments. The extension of the theory to tight knots is discussed.

Figure 1

Geometry of an open trefoil knot ($3_1$) under tension. Inset: open cinquefoil knot ($5_1$).

Figure 2

Difference problem for the braid region. (a) Geometry. (b) Visualization of the solution for a trefoil knot, projected in the $(v,w)$ plane, and (c) in the plane $(u,v)$ perpendicular to the axis of the cylinder. Only one half of the symmetric solution is shown. Contact is denoted with thick curves and points.

Figure 3

Photograph and close-up of an experimental braid, with $ϵ\approx .074$. The two symmetric openings are predicted by the theory.

Figure 4

Braid length $\ell$ vs intermediate length $(hR{)}^{1/2}$ in log-log plot. In the inset, the same data are shown in a linear plot. Open symbols and the thin line are for trefoil knots. Filled symbols and the thick line are for cinquefoil knots. All rods are Nitinol wires, except for the single circular data point, computed from data from Ref. 3. The lines are the theoretical prediction (7), with no adjustable parameter.

Figure 5

Traction-displacement curve for a Nitinol wire with a trefoil knot. The knot is first tightened (upper curve) and then loosened (lower curve) until it becomes locked by friction, for a value $ϵ={ϵ}_0$. The curves are the theoretical prediction (8), with the Coulomb friction coefficient adjusted to $\mu =.07$ from Eq. (9). There is no other adjustable parameter in the plot. The agreement between theory and experiments is good as soon as $ϵ\lesssim 0.1$.

Figure 6

Curvature of a tight elastic trefoil knot ($ϵ=0.52$, bold curve) and comparison with the geometrical calculation of a tight trefoil knot in Ref. 11 (dotted line).