Slender elastic objects such as a column tend to buckle under loads. While static buckling is well understood as a bifurcation problem, the evolution of shapes during dynamic buckling is much harder to study. Elastic rings under normal pressure have emerged as a theoretical and experimental paradigm for the study of dynamic buckling with controlled loads. Experimentally, an elastic ring is placed within a soap film. When the film outside the ring is removed, surface tension pulls the ring inward, mimicking an external pressurization. Here we present a theoretical analysis of this process by performing a post-bifurcation analysis of an elastic ring under pressure. This analysis allows us to understand how inertia, material parameters, and loading affect the observed shape. In particular, we combine direct numerical solutions with a post-bifurcation asymptotic analysis to show that inertia drives the system towards higher modes that cannot be selected in static buckling. Our theoretical results explain experimental observations that cannot be captured by a standard linear stability analysis.

中文翻译：

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不可伸展弹性环的动态屈曲：线性和非线性分析

细长的弹性物体（例如圆柱）往往会在负载下弯曲。尽管将静态屈曲理解为一个分叉问题，但在动态屈曲过程中形状的演变却很难研究。常压弹性环已经成为研究控制载荷下的动态屈曲的理论和实验范式。实验上，将弹性环放置在肥皂膜内。当除去环外的薄膜时，表面张力将环向内拉，模仿了外部加压。在这里，我们通过在压力下对弹性环进行分叉后分析，对这一过程进行了理论分析。通过这种分析，我们可以了解惯性，材料参数和载荷如何影响观察到的形状。尤其是，我们将直接数值解与分叉后渐近分析相结合，以表明惯性将系统推向无法在静态屈曲中选择的更高模式。我们的理论结果解释了标准线性稳定性分析无法捕获的实验观察结果。