Computation of Unconstrained Elastic Equilibria of Complete M\"{o}bius Bands and their Stability

TL;DR

This paper develops a systematic numerical approach to compute and analyze the stability of unconstrained elastic M"{o}bius band configurations using Kirchhoff and Wunderlich models, introducing a novel regularization technique.

Contribution

It presents the first method to obtain unconstrained equilibrium configurations of complete M"{o}bius bands and assesses their stability, addressing previous limitations in modeling.

Findings

Successfully computed unconstrained equilibria for the complete strip.

Demonstrated that certain equilibria are local minima of the total potential energy.

Introduced a new regularization method for the Wunderlich model.

Abstract

Determining the equilibrium configuration of an elastic M\"{o}bius band is a challenging problem. In recent years numerical results have been obtained by other investigators, employing first the Kirchhoff theory of rods and later the developable, ruled-surface model of Wunderlich. In particular, the strategy employed previously for the latter does not deliver an unconstrained equilibrium configuration for the complete strip. Here we present our own systematic approach to the same problem for each of these models, with the ultimate goal of assessing the stability of flip-symmetric configurations. The presence of pointwise constraints considerably complicates the latter step. We obtain the first stability results for the problem, numerically demonstrating that such equilibria render the total potential energy a local minimum. Along the way we introduce a novel regularization for the for the singular Wunderlich model that delivers unconstrained equilibria for the complete strip, which can then be tested for stability.