Matched asymptotic expansions for twisted elastic knots: A self-contact problem with non-trivial contact topology
Matched asymptotic expansions for twisted elastic knots: A self-contact problem with non-trivial contact topology
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扭曲弹性结的匹配渐近展开：具有非平凡接触拓扑的自接触问题
DOI：
10.1016/j.jmps.2009.05.004
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发表时间：
2008
影响因子：
5.3
通讯作者：
S. Neukirch
中科院分区：
文献类型：
--
作者：
N. Clauvelin;B. Audoly;S. Neukirch
我们推导出的基尔霍夫方程的解的结系在一个无限长的弹性杆受到联合拉伸和扭转，并在两端无穷大。我们考虑的情况下，简单的（三叶草）和双（梅花）结;其他结拓扑结构可以类似地进行研究。杆模型是基于胡克弹性，但几何非线性。该问题是一个非线性的自接触问题，未知的接触区域。它是通过匹配的渐近展开在一个松散的结的极限。我们得到了一族依赖于单个载荷参数U <$（与施加的扭矩除以拉力的平方根成比例）的平衡解，它们在松弛纽结的极限ε→0下渐近有效。在没有任何先验假设的情况下，我们推导出了接触集的拓扑结构，该接触集由两侧为两个孤立接触点的接触区间组成。我们研究了外加扭转对平衡的影响。
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, ε→0. 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.