TY - CHAP
T1 - Dynamic curling of an Elastica
T2 - a nonlinear problem in elastodynamics solved by matched asymptotic expansions
AU - Audoly, B.
AU - Callan-Jones, A.
AU - Brun, P. T.
PY - 2015/1/1
Y1 - 2015/1/1
N2 - 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).
AB - 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).
KW - Constant Curvature
KW - Master Curve
KW - Matched Asymptotic Expansion
KW - Nonlinear Problem
KW - Physical Review Letter
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U2 - 10.1007/978-3-7091-1877-1_3
DO - 10.1007/978-3-7091-1877-1_3
M3 - Chapter
AN - SCOPUS:85052454788
T3 - CISM International Centre for Mechanical Sciences, Courses and Lectures
SP - 137
EP - 155
BT - CISM International Centre for Mechanical Sciences, Courses and Lectures
PB - Springer International Publishing
ER -