The Shape of a M\"obius Strip via Elastic Rod Theory Revisited

TL;DR

This paper investigates the shape of a M"obius strip using a Cosserat rod model, capturing both smooth and localized configurations, and demonstrates the stability of these shapes as the strip becomes thinner.

Contribution

It introduces a middle-ground modeling approach that captures both smooth and localized M"obius strip shapes, differing from previous models by adjusting torsional stiffness without relying solely on classical formulas.

Findings

Successfully models both smooth and localized M"obius strip shapes

Demonstrates stability of the configurations under small perturbations

Shows the effect of decreasing thickness on shape localization

Abstract

In 1993 Mahadevan and Keller used the Kirchhoff rod theory to predict the shape of a M\"obius band. Starting from the solution for a square cross-section (isotropic), they employ numerical continuation in the cross-sectional aspect ratio in order to approach the solution for a thin strip. Certain smoothly varying configurations are obtained. More recently in 2007, Starostin and van der Heijden pointed out that an actual M\"obius band "localizes" into a nearly flat triangular configuration as the ratio of the strip width to center-line circumference is no longer small. Accordingly they return to the developable, thin-plate model of Sadowsky and Wunderlich, obtaining such localized shapes. In this work we strike a middle ground between these two approaches. We employ the standard two-director (special) Cosserat model, and we also use the cross-sectional aspect ratio as a numerical continuation parameter. Nonetheless, we are able to capture both the smoothly varying shapes and the localized, nearly triangular configurations. Our point of departure is that we do not always employ the strength-of-materials formula for the torsional stiffness - a standard feature of the Kirchhoff theory. Our key observation comes from the usual Cosserat ansatz in light of developability, suggesting an increasing torsional stiffness ratio (as well as one increasing bending stiffness ratio) as the thickness becomes small. We demonstrate that the closed-loop configurations we obtain are stable with respect to small perturbations.