Computation of elastic equilibria of complete Möbius bands and their stability

Abstract

Determining the equilibrium configuration of an elastic Möbius 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 Sadowsky–Wunderlich. In particular, one such strategy used does not deliver an equilibrium configuration for the complete unsupported 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 point-wise 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 singular Wunderlich model that delivers equilibria for complete strips having sufficiently narrow widths, which can then be tested for stability.