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Edited* by Harry L. Swinney, The University of Texas at Austin, Austin, TX, and approved September 2, 2014 (received for review May 16, 2014)
Significance
The deployment of a rodlike structure onto a moving substrate is commonly found in a variety engineering applications, from the fabrication of nanotube serpentines to the laying of submarine cables and pipelines. Predictively understanding the resulting coiling patterns is challenging given the nonlinear geometry of deposition. In this paper, we combine precision model experiments with computer simulations of a rescaled analogue system and explore the mechanics of coiling. In particular, the natural curvature of the rod is found to dramatically affect the coiling process. We have introduced a computational framework that is widely used in computer animation into engineering, as a predictive tool for the mechanics of filamentary structures.
Abstract
We investigate the deployment of a thin elastic rod onto a rigid substrate and study the resulting coiling patterns. In our approach, we combine precision model experiments, scaling analyses, and computer simulations toward developing predictive understanding of the coiling process. Both cases of deposition onto static and moving substrates are considered. We construct phase diagrams for the possible coiling patterns and characterize them as a function of the geometric and material properties of the rod, as well as the height and relative speeds of deployment. The modes selected and their characteristic length scales are found to arise from a complex interplay between gravitational, bending, and twisting energies of the rod, coupled to the geometric nonlinearities intrinsic to the large deformations. We give particular emphasis to the first sinusoidal mode of instability, which we find to be consistent with a Hopf bifurcation, and analyze the meandering wavelength and amplitude. Throughout, we systematically vary natural curvature of the rod as a control parameter, which has a qualitative and quantitative effect on the pattern formation, above a critical value that we determine. The universality conferred by the prominent role of geometry in the deformation modes of the rod suggests using the gained understanding as design guidelines, in the original applications that motivated the study.
Footnotes
↵1M.K.J. and F.D. contributed equally to this work.
- ↵2To whom correspondence may be addressed. Email: preis{at}mit.edu or eitan{at}cs.columbia.edu.
Author contributions: E.G. and P.M.R. designed research; M.K.J., F.D., J.J., E.G., and P.M.R. performed research; M.K.J., F.D., J.J., E.G., and P.M.R. contributed new reagents/analytic tools; M.K.J. and F.D. analyzed data; and M.K.J., F.D., E.G., and P.M.R. wrote the paper.
The authors declare no conflict of interest.
↵*This Direct Submission article had a prearranged editor.
This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1409118111/-/DCSupplemental.
Coiling of elastic rods on rigid substrates
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