Collaborative Research: Mechanics of Knots and Tangles of Elastic Rods

University Of Maine, Orono ME

Investigators

Abstract

In settings from shipping to sailing to surgery, thousands of different types of knots are used every day, each requiring a specific amount of force to tighten. This force depends on the material properties, friction, and the topology of the knot. A knot typically also has a load-bearing capacity; force beyond this level causes the knot to be undone, and excessive force may result in material failure in the knot. Moreover, some knots hold tight without any external force while others easily get untangled. In other words, the knots can store energy in the material. This concept is called a topological battery with implications in nanometer-sized knots in DNA to macroscopic knots in structural engineering. This award supports research to understand the fundamental science of knots. The work will develop modeling and computational methods for the analysis of the mechanics of knots and tangles. In parallel, it will formulate experimental techniques to systematically study this mechanics. The research will be complemented by developing teaching tools (videos, notes, and demonstrations) for undergraduate and graduate courses. The computational software will also be made publicly available. The research objective of this project is to quantify the mechanical response of knots tied in elastic rods. The project will employ (1) fast numerical simulations inspired by computer graphics, (2) innovative materials with customizable friction, and (3) autonomous robotic experiments to untangle the mechanics of knots. Even in the case of the most basic type of knots (overhand knots), the force required to tie the knot depends on an intricate interplay of (1) elasticity, (2) friction, and (3) topology. Interestingly, the overhand knot may undergo a snap-through buckling instability beyond a critical amount of pull. Such instability in a basic knot points to the richness of the mechanical behavior of knots. After developing simulation and experimental tools, the mechanical response and instabilities of a few common knots, e.g. overhand and shoelace knots, will be investigated. Exploiting the computational speed of the simulation tool and autonomy of robotic experiments, the mechanical response of several types of knots will be quantified to build a library of their mechanics. This data will be used to rationalize the variation of a knot’s mechanical response as a function of the topological, material, and frictional parameters. Similar to the periodic table of elements, a mechanics-based classification scheme of knots will be formulated, where the knots will be grouped into various classes, such as, friction-dominated knots, bending-dominated knots, and others. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.