# Dynamic buckling of an inextensible elastic ring: Linear and nonlinear analyses

## TL;DR

This paper investigates the dynamic buckling behavior of elastic rings under pressure, combining experiments, numerical simulations, and asymptotic analysis to reveal how inertia influences shape evolution beyond static buckling predictions.

## Contribution

It provides a comprehensive theoretical framework for understanding dynamic buckling of elastic rings, incorporating inertia effects through post-bifurcation analysis and numerical methods.

## Key findings

- Inertia causes higher mode shapes during dynamic buckling.
- Standard linear stability analysis fails to predict observed behaviors.
- Theoretical results align with experimental observations.

## Abstract

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 properties, 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.