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Matched asymptotic expansions for twisted elastic knots: a self-contact
 

Summary: .
Matched asymptotic expansions
for twisted elastic knots: a self-contact
problem with non-trivial contact topology
N. Clauvelin, B. Audoly and S. Neukirch
UPMC Univ Paris 06, UMR 7190, Institut Jean Le Rond d'Alembert, F-75005
Paris, France.
CNRS, UMR 7190, Institut Jean Le Rond d'Alembert, F-75005 Paris, France.
Abstract
We derive solutions of the Kirchhoff equations for a knot tied on an infinitely long
elastic rod subjected to combined tension and twist, and held at both endpoints
at infinity. 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 nonlinear. The problem is formulated as a
nonlinear self-contact problem with unknown contact regions. It is solved by means
of matched asymptotic expansions in the limit of a loose knot. We obtain a family
of equilibrium solutions depending on a single loading parameter U (proportional
to applied twisting moment divided by square root of pulling force), which are
asymptotically valid in the limit of a loose knot, 0. Without any a priori
assumption, we derive the topology of the contact set, which consists of an interval

  

Source: Audoly, Basile - Institut Jean Le Rond D'Alembert, Université Pierre-et-Marie-Curie, Paris 6

 

Collections: Materials Science; Physics