Geometry, topology and mechanics of soft filaments

Abstract

Systems that can be modeled as collections of interacting filaments occur throughout nature and engineering, from supercoiled DNA and carbon nanotube fibers to solar coronae magnetic flux lines and steel construction beams. Increasing interest in biomimetics, soft biological objects (e.g. plant tendrils, neurons and DNA) and soft robotics demands a thorough and precise understanding of filament systems that are soft—that is, in which one must account for not only bend and twist, but also shear and stretch deformations. In this thesis, we develop and demonstrate a physical, computational, and mathematical framework to model and design complex systems of soft elastic filaments, interacting actively with each other and environmental factors. To this end, we employ the Cosserat model of elastic filaments that accounts for all four aforementioned types of filament cross-section deformations; we solve the model using a recently developed numerical scheme, expanding the model and simulator to incorporate additional phenomena, including internal energy dissipation, interfilament contact, and dynamical evolution of filament material properties, the last of which allows for more realistic and wide-ranging methods of modeling filament activity and indirectly modeling environmental effects such as changing temperature; we characterize filament systems in terms of knot-theoretic link, twist, writhe and the Calugareanu-Fuller-White theorem; we suggest local versions of link, twist and writhe that can be accurately computed even for simulated filaments for which time and space resolution is limited, and deploy these local topological quantities in conjunction with their global counterparts; we connect filament topology with the systems’ geometry and mechanics, demonstrating how complex nonlinear dynamics and mechanics can be understood compactly and at a fundamental level when one relies not only on a physical model of filament interactions, but also couples this model with a local and global topological characterization of the system. We demonstrate our framework on four case studies: the deformations of a stretched and twisted filament, applying our results to recent experiments on twisted and coiled artificial muscle fibers; combing a double helix, drawing conclusions about an optimal strategy for combing a tangled curl of hair; a newly-observed transient snake gait (the “S-start”), extending our model to sidewinding and uncovering potential biological connections between sidewinding and the S-start; and a newly developed soft robotic gripper, characterizing the design space and demonstrating the central role of topological entanglement in successful soft robotic grasping strategies. Through these case studies, we highlight power of topology to dictate complex observed physical phenomena and uncover a theme likely common to many filament systems: that topological constraints can be predictably harnessed to perform mechanical work.