二维弹性曲线的θ公式——屈曲与边界层理论
The θ-formulation of the 2D elastica -- Buckling and boundary layer theory
AI总结：
本研究提出二维弹性曲线的θ-公式，通过局部取向角参数化方程，在弱弯曲刚度下开展边界层分析，推导出无需椭圆函数的复合解，还发现弹性环的snapping失稳并经数值和实验验证。
摘要：
通过以局部取向角θ而非弧长为参数对变量进行参数化，受压平面弹性曲线的方程可被改写为一种实用形式。这种“θ-公式”特别适用于弯曲刚度较弱极限下的边界层分析。在该参数化框架下，边界层位于θ取极值的拐点处，连接低曲率与高曲率区域。我们无需借助椭圆函数和椭圆积分，便推导出了一种简单的复合解。该近似解可作为基本单元来描述复杂形状。将该理论应用于研究受均匀压力且承受一组集中力的弹性环时，我们发现了一种 snapping 失稳现象，该失稳已通过数值模拟得到验证。最后，我们开展了实验，发现理论结果与变形后弹性曲线的实验形状吻合良好。
英文摘要：
The equations of a planar elastica under pressure can be rewritten in a useful form by parametrising the variables in terms of the local orientation angle, $θ$, instead of the arc length. This ``$θ$-formulation'' lends itself to a particularly easy boundary layer analysis in the limit of weak bending stiffness. Within this parameterization, boundary layers are located at inflexion points, where $θ$ is extremum, and they connect regions of low and large curvature. A simple composite solution is derived without resorting to elliptic functions and integrals. This approximation can be used as an elementary building block to describe complex shapes. Applying this theory to the study of an elastic ring under uniform pressure and subject to a set of point forces, we discover a snapping instability. This instability is confirmed by numerical simulations. Finally, we carry out experiments and find good agreement of the theory with the experimental shape of the deformed elastica.
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