Geometrically nonlinear configurations in rod-like structures

Abstract

Rod-like structures are ubiquitous in both nature (e.g. bacterial flagella) and engineering (ropes, cables, pipelines, and Carbon nanotubes), from nanometer to kilometer scale, and often undergo geometrically nonlinear deformation. Due to the slender geometry of the structures, the material strain usually remains in linear elastic limit even though the deformed geometry deviates significantly from the original geometry. The geometric nonlinearities that result from the deformation process pose enormous challenges to traditional analytical and numerical tools. Moreover, it is often unfeasible to perform experiments at the original length-scale of these systems. We overcome these challenges by combining model experiments with cutting-edge computational tools ported from computer graphics, and theoretical analysis towards developing predictive physical understanding of these systems. The prominence of geometry in this class of systems enables the scaling (up or down) of the problem to the desktop scale, which allows for systematic experimental exploration of parameter space. In parallel, we conduct numerical simulations using the Discrete Elastic Rods (DER) method, which was originally developed for the animation industry for special effects of the visually dramatic dynamics of hair, fur, and other rod-like structures. We port DER into engineering as a predictive computational tool and test ride it against our own model experiments. A collection of problems from three a priori unrelated scenarios, at disparate length-scales, are explored. First, as a model for laying of submarine cables onto the seabed (kilometer scale), we consider deployment of elastic rods onto a moving substrate (conveyor belt) and quantify the resulting nonlinear coiling patterns. The DER method is employed to identify the phase boundaries between different patterns and characterize the morphology. Our results are interpreted using a reduced geometric model for the evolution of the position of the contact point with the belt and the curvature of the rod in its neighborhood. This geometric model reproduces all of the coiling patterns, which allows us to establish a universal link between our elastic problem and the analogous patterns obtained when depositing a viscous thread onto a moving surface; a well-known system referred to as the fluid mechanical sewing machine. Secondly, we consider a macroscopic analog model for the locomotion of uni-flagellar bacteria (micron scale) in a viscous fluid. Our precision experiments are compared against numerical simulations that employ the Lighthill's slender body theory (a non-local description of fluid force) and DER, with excellent quantitative agreement without any fitting parameter. A novel mechanical instability is uncovered, whereby the filament buckles above a critical rotation frequency. We then augment our experiments and simulations to include the effect of a no-slip rigid boundary on the locomotion of uni-flagellar bacteria. We find that the propulsive force is strongly affected by the presence of a nearby boundary. Thirdly, we perform precision desktop experiments with overhand knots (centimeter scale) with increasing values for the crossing number (our measure of topology) and characterize their mechanical response through tension-displacement tests. The tensile force required to tighten the knot is governed by an intricate balance between topology, bending, and friction. A reduced theory is developed, which predictively rationalizes the process.