Cylinder Buckling: The Mountain Pass as an Organizing Center

TL;DR

This paper investigates the buckling behavior of cylindrical shells, emphasizing the mountain pass as a critical energy barrier, and combines theoretical proofs with numerical simulations to understand imperfection sensitivity and validate findings with experiments.

Contribution

It introduces the mountain pass as an organizing principle for understanding cylinder buckling, providing existence proofs and numerical examples for the first time.

Findings

The lowest energy mountain pass solution resembles a single dimple.

The energy landscape determines the shell's sensitivity to imperfections.

Numerical results align with experimental observations.

Abstract

We revisit the classical problem of the buckling of a long thin axially compressed cylindrical shell. By examining the energy landscape of the perfect cylinder we deduce an estimate of the sensitivity of the shell to imperfections. Key to obtaining this is the existence of a mountain pass point for the system. We prove the existence on bounded domains of such solutions for all most all loads and then numerically compute example mountain pass solutions. Numerically the mountain pass solution with lowest energy has the form of a single dimple. We interpret these results and validate the lower bound against some experimental results available in the literature.