Axial compression of a thin elastic cylinder: bounds on the minimum energy scaling law

TL;DR

This paper derives bounds on the minimum elastic energy of a thin cylinder under axial compression around a cylindrical core, revealing how energy scales with thickness, compression, and core size, and identifying different wrinkling regimes.

Contribution

It provides the first rigorous bounds on energy scaling laws for compressed cylinders with a core, considering both linear and nonlinear elasticity models.

Findings

Bounds match in the large mandrel case, revealing energy scaling laws.

Three wrinkling patterns identified, each dominating in different regimes.

In the neutral mandrel case, energy scales as unbuckled configuration under small compression.

Abstract

We consider the axial compression of a thin elastic cylinder placed about a hard cylindrical core. Treating the core as an obstacle, we prove upper and lower bounds on the minimum energy of the cylinder that depend on its relative thickness and the magnitude of axial compression. We focus exclusively on the setting where the radius of the core is greater than or equal to the natural radius of the cylinder. We consider two cases: the "large mandrel" case, where the radius of the core exceeds that of the cylinder, and the "neutral mandrel" case, where the radii of the core and cylinder are the same. In the large mandrel case, our upper and lower bounds match in their scaling with respect to thickness, compression, and the magnitude of pre-strain induced by the core. We construct three types of axisymmetric wrinkling patterns whose energy scales as the minimum in different parameter regimes, corresponding to the presence of many wrinkles, few wrinkles, or no wrinkles at all. In the neutral mandrel case, our upper and lower bounds match in a certain regime in which the compression is small as compared to the thickness; in this regime, the minimum energy scales as that of the unbuckled configuration. We achieve these results for both the von K\'arm\'an-Donnell model and a geometrically nonlinear model of elasticity.