Coiling of elastic rods from a geometric perspective

We present results from a systematic numerical investigation of the pattern formation of coiling obtained when a slender elastic rod is deployed onto a moving substrate; a system known as the elastic sewing machine (ESM). The Discrete Elastic Rods method is employed to explore the parameter space, construct phase diagrams, identify their phase boundaries and characterize the morphology of the patterns. The nontrivial geometric nonlinearities are described in terms of the gravito-bending length and the deployment height. 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 of the ESM, 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.