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

Abstract

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 Analysis of Steel Structure Bent Frame Column Bearing Transverse Concentrated Force at the Top in Factory Buildings

“…A straight compression rod with equal sections was taken out from the elastic structure, and any constraints were imposed at both ends, so that there were the axial force, shear force, and bending moment on the end section, as shown in Figure 2. In the figure, the original vertical position without the load was presented by the straight line, while the new balanced position under the load was shown by the curve [8,9]. An endpoint before deformation was taken out as the original point of X-axis, another end as the positive direction of X-axis.…”

Viscoelastic ribbons

“…However, any viscoelastic deformation of a thin sheet can accomplish the same task, as we illustrate here, and is familiar from the practicalities of curling hair (Barnes & Roberts 2000;Zuidema et al 2003). Other curling problems in which viscoelasticity likely plays a prominent role have been considered recently by Audoly, Callan-Jones & Brun (2015), Tadrist, Brochard-Wyart & Cuvelier (2012) and Arriagada, Massiera & Abkarian (2014).…”

Dynamic adhesion behavior of natural benders

“…Villey 等人 [10] 通 过剥离实验测定了不同压敏胶的能量释放率，结果表明，黏附能剥离角度与剥离速度依赖性是可 分离的，当剥离速度在 0.1~10 m/s 时，能量释放率随速度的增加而减小. Dalbe 等人 [11,13] 2 卷曲运动理论研究 本文基于Callan-Jones等人 [4,5] 量平衡方程和弯矩平衡方程 [4,24] ： ,, ,…”

“…将这些解代入式(11)和(12)，进而求出黏附基底上自然弯曲体运动 的前沿速度v a 和剥离强度γ、自然曲率κ 0 之间的关系, 即 表 1 自然弯曲体的尺寸参数和实验数据 . 上述结论与 Callan-Jones 等人[4,5] 工作类似，只是无量纲系数有细微差异，原…”