I finally received a copy from Pedro Reis of his paper; let me know if anyone else needs a copy. Untangling the Mechanics and Topology in the Frictional Response of Long Overhand Elastic Knots M.K. Jawed, 1 P. Dieleman, 2 B. Audoly, 3,* and P.M. Reis 1,2, 1 Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA 2 Department of Civil and Environmental Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA 3 Sorbonne Universités, UPMC Univ Paris 06, CNRS, UMR 7190 Institut Jean Le Rond dÂAlembert, F-75005 Paris, France (Received 23 April 2015; revised manuscript received 7 August 2015; published 11 September 2015) 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 onthe 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. DOI: 10.1103/PhysRevLett.115.118302 PACS numbers: 46.25.-y, 02.10.Kn, 46.70.Hg Shoelaces are commonly tied using the reef knot, which comprises two trefoil knots: the first is left handed and the other right handed. Mistakenly tying two consecutive left-handed trefoil knots leads to the mechanically inferior granny knot [1], whose lower performance illustrates the important interplay between topology and mechanics.
From polymer chains [2] to the shipping industry, knots are ubiquitous across length scales [3]. Whereas they can appear spontaneously [4] and are sometimes regarded as a nuisance (e.g., in hair and during knitting), knots as fasteners of filamentary structures have applications in biophysics [5], surgery [6,7], fishing [8], sailing [9], and climbing [10]. Frictional knots have also been added to fibers for increased toughness [11].
Even if the quantitative study of knots has remained primarily in the realm of pure mathematics [12], there have been empirical attempts to characterize their mechanical properties according to strength or robustness [13,14]. However, these metrics rely strongly on material-specific properties and are therefore of limited applicability across different systems and length scales [3]. Recent studies have addressed the mechanics of knots from a more fundamental perspective [15,16]. For example, existing theories on flexible strings (with zero bending stiffness) [17,18] treat friction using the capstan equation [19]. Finite element simulations of knots have also been performed in instances where bending cannot be neglected [20] and friction has been treated perturbatively for trefoil knots tied in elastic rods [21,22]. Still, predictively understanding the mechan- ics of knots remains a challenging endeavor, even for the simplest types of elastic knots, due to the complex coupling of the various physical ingredients at play. Here, we perform a systematic investigation of elastic knots under tension and explore how their mechanical response is influenced by topology. We perform precision model experiments and rationalize the observed behavior through an analysis based on KirchhoffÂs geometrically nonlinear model for slender elastic rods. Our theory takes into account regions of self-contact, where friction dominates. Focus is given to open overhand knots [Figs. 1(a)Â1(d)]. These knots comprise a braid with arc length l, a loop with arc length ?, and two tails onto which a tensile load is applied. The topology of the braid is quantified by the unknotting number n ¼ ð? - 1Ã=2 (number of times the knot must be passed through itself to untie it), where ? is the crossing number (number of apparent crossing nodes). In Fig. 1(e), we plot the traction force F as a function of the end-to-end shortening, e (e ¼ 0 corresponds to a straight configuration, without a knot) for a variety of knots in the range 1 = n = 10. We find that F depends nonlinearly on e and varies significantly with n. We shall provide an analytical solution for the relation between the knot topology (defined by n) and the braid geometry. We then extend our analysis to identify the underlying physical ingredients and predictively capture the experimental mechanical response.