Some Topics in Topological Mechanics

Abstract

We apply mechanical theories of the elasticity of thin rods and topological theories of entanglement to model experimental systems of detangling hair with a comb and the internal structure of bird’s nests. By applying these theoretical frameworks to novel systems, we open up new ways of studying classes of problems including the above, the carding and decarding of textiles, and biologically-inspired materials engineering. In the first section, we take a pedagogical approach to analyzing the topology, geometry and mechanics of a single elastic filament. We walk advanced undergraduate or early graduate student readers through Cosserat elastic rod theory. We then introduce a topological quantity called Link and connect it to the geometry and mechanics of a single filament. In the experimental sections that follow, we investigate physical entanglement in two model systems: detangling filaments in a double helix and analyzing the relationship between physical stress and internal geometry and topology of a system of randomly packed steel rods. In the first system, we use a double helix filament geometry as a simplified model for detangling hair by combing. We investigate the relationships between geometry, mechanics and topology of the double helix to analyze the key role that topology plays in the mechanics of detangling. We also use the model to design a feedback-driven hair combing robot that uses force-sensing and computer vision to find an optimal hair combing procedure. In the second system, we use a network of randomly packed steel rods as a simplified model to study the roles of geometry and topology in the mechanical properties of bird’s nests made of entangled sticks. Finally, we discuss areas for further research including: studying detangling in more varieties of human and animal hair, characterizing more complex hair interactions using topology and studying disentanglement in randomly packed steel rods.