Untangling the Mechanics and Topology in the Frictional Response of Long Overhand Elastic Knots

Abstract

We combine experiments and theory to study the mechanics of overhand knots in slender elastic rods under tension. The equilibrium shape of the knot is governed by an interplay between topology, friction, and bending. We use precision model experiments to quantify the dependence of the mechanical response of the knot as a function of the geometry of the self-contacting region, and for different topologies as measured by their crossing number. An analytical model based on the nonlinear theory of thin elastic rods is then developed to describe how the physical and topological parameters of the knot set the tensile force required for equilibrium. Excellent agreement is found between theory and experiments for overhand knots over a wide range of crossing numbers.

Second-order phase transition and universality of self-buckled elastic slender columns

“…By comparing both papers bending at three lengths: 2L, 2.3L, and 2.5L, it is clearly observed that they show precisely identical bending. Let us look back at Equation 15which can be written as L cr = (N 3 cr ξ YI/λg) 1/3 . For columns having constant λ and Y, the critical length for buckling is solely controlled by area moment, and the area moment itself depends on the column geometry.…”

“…For example, understanding the mechanism of formation of ring stains from dried liquid drops open the opportunity to develop high-quality printer inks [1]. Reis group from MIT has published many papers describing likely common phenomena around us and some of them have technological impacts [2][3][4][5][6][7][8].…”

Stable elastic knots with no self-contact

“…It is characterized by 2 isolated points of self-contact surrounding an interval of self-contact. When twist is increased by rotating one of the clamped ends, this self-contacting solution becomes unstable and the rod jumps to a nearly planar configuration with 4 points of self-contact [6,5]; see also [20]. The combination of these studies seems to suggest that such an open trefoil will always have points of self-contact in stable equilibrium.…”

The roles of impact and inertia in the failure of a shoelace knot

“…These differences are illustrated in appendix A (figure 14 a – g ) and we encourage the reader to experiment with their own shoelaces. While the inferior performance of the weak version is well known in both common knot lore and surgical knot literature [1–4], we have yet to find a proposed structural characteristic to explain the inferior behaviour. Likewise, we have not found a study explaining the failure mechanism for a shoelace knot.…”

“…Owing to the interest in simulating and animating strands of hair, ropes and sutures in computer graphics, progress in this area has been rapid in the past decade (cf. [4,9]). However, simulating the dynamics of the shoelace knot under conditions experienced during walking remains a challenging goal.…”