Dynamic curling of an Elastica: a nonlinear problem in elastodynamics solved by matched asymptotic expansions

  • B. Audoly
  • A. Callan-Jones
  • P.-T. Brun
Part of the CISM International Centre for Mechanical Sciences book series (CISM, volume 562)


We consider the motion of an infinitely long, naturally curved, planar Elastica. The Elastica is flattened onto a rigid impenetrable substrate and held by its endpoints. When one of its endpoints is released, it is set off in a curling motion, which we seek to describe mathematically based on the non-linear equations of motions for planar elastic rods undergoing finite rotations. This problem is used to introduce the technique of matched asymptotic expansions. We derive a non-linear solution capturing the late dynamics, when a roll comprising many turns has formed: in this regime, the roll advances at an asymptotically constant velocity, whose selection we explain. This contribution presents an expanded version of the results published in Callan-Jones et al. (Phys. Rev. Lett. 2012).


Nonlinear Problem Constant Curvature Master Curve Physical Review Letter Matched Asymptotic Expansion 
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Copyright information

© CISM, Udine 2015

Authors and Affiliations

  • B. Audoly
    • 1
  • A. Callan-Jones
    • 2
  • P.-T. Brun
    • 3
  1. 1.UPMC Univ Paris 06, CNRS, UMR 7190 Institut Jean Le Rond d’AlembertSorbonne UniversitésParisFrance
  2. 2.CNRS, UMR 7057, Matière et Systèmes ComplexesUniv. Paris 7 Denis DiderotParisFrance
  3. 3.Department of MathematicsMassachusetts Institute of TechnologyCambridgeUSA

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