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The Nonlinear Dynamics of Filaments

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Abstract

The Kirchhoff equations provide a well-established framework tostudy the statics and dynamics of thin elastic filaments. The study ofstatic solutions to these equations has a long history and provides thebasis for many investigations, both past and present, of theconfigurations taken by filaments subject to various external forces andboundary conditions. Here we review recently developed techniquesinvolving linear and nonlinear analyses that enable one to study, insome detail, the actual dynamics of filament instabilities and thelocalized structures that can ensue. By introducing a novel arc-lengthpreserving perturbation scheme a linear stability analysis can beperformed which, in turn, leads to dispersion relations that provide theselection mechanism for the shape of an unstable filament. Thesedispersion relations provide the starting point for nonlinear analysisand the derivation of new amplitude equations which describe thefilament dynamics above the instability threshold. Here we will mainlybe concerned with the analysis of rods of circular cross-sections andsurvey the behavior of rings, rods, helices and show how these resultslead to a complete dynamical description of filament buckling.

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Authors and Affiliations

  1. Department of Mathematics and Program in Applied Mathematics, University of Arizona, Building #89, Tucson, AZ, 85721, U.S.A.

    Alain Goriely

  2. Department of Mathematics and Program in Applied Mathematics, University of Arizona, Building #89, Tucson, AZ, 85721, U.S.A.

    Michael Tabor

Authors
  1. Alain Goriely
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  2. Michael Tabor
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Goriely, A., Tabor, M. The Nonlinear Dynamics of Filaments. Nonlinear Dynamics 21, 101–133 (2000). https://doi.org/10.1023/A:1008366526875

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