Confined elasticae and the buckling of cylindrical shells

TL;DR

This paper analyzes the minimal elastic energy of curves slightly longer than 2π embedded in a disk, deriving scaling laws, solving an obstacle problem, and applying results to buckling in cylindrical shells.

Contribution

It introduces a new scaling law for elastic energy near the critical length, solves a related obstacle problem, and applies findings to buckling analysis of cylindrical shells.

Findings

Derived the first order coefficient in energy expansion (~37) for slightly longer curves.

Solved a fourth order obstacle problem with integral constraint.

Determined a bifurcation point for buckling in cylindrical shells based on material parameters.

Abstract

For curves of prescribed length embedded into the unit disc in two dimensions, we obtain scaling results for the minimal elastic energy as the length just exceeds $2\pi$ and in the large length limit. In the small excess length case, we prove convergence to a fourth order obstacle type problem with integral constraint on the real line which we then solve. From the solution, we obtain the first order coefficient $\Theta\approx 37$ in the energy expansion $2\pi + \Theta \delta^{1/3} + o(\delta^{1/3})$ when a curve has length $2\pi + \delta$. We present an application of the scaling result to buckling in two-layer cylindrical shells where we can determine an explicit bifurcation point between compression and buckling in terms of universal constants and material parameters scaling with the thickness of the inner shell.