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Université Pierre et Marie Curie
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Matched asymptotic expansions for twisted elastic knots: a self-contact problem with non-trivial contact topology
N. Clauvelin, B. Audoly and S. Neukirch
Journal of the Mechanics and Physics of Solids, 57 #9 (2009) 1623--1656
Abstract : We derive solutions of the Kirchhoff equations for a knot tied on an infinitely long elastic rod subjected to combined tension and twist. We consider the case of simple (trefoil) and double (cinquefoil) knots; other knot topologies can be investigated similarly. The rod model is based on Hookean elasticity but is geometrically non-linear. The problem is formulated as a non-linear self-contact problem with unknown contact regions. It is solved by means of matched asymptotic expansions in the limit of a loose knot. Without any a priori assumption, we derive the topology of the contact set, which consists of an interval of contact flanked by two isolated points of contacts. We study the influence of the applied twist on the equilibrium and find an instability for a threshold value of the twist.
Key words : Knots, Rods and cables, Elastic material, Contact mechanics, Asymptotic analysis.
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