Stable Möbius Bands from Isometrically Deformed Circular Helicoids

Abstract

We consider the problem of producing a ruled Möbius band by subjecting an unstretchable, homogeneous, isotropic, elastic material surface material surface in a circular helicoidal reference configuration to a deformation that is isometric and chirality preserving. We find that such a Möbius band is completely determined by the unit binormal of the Frenet frame of its midline, which must be a geodesic and must have uniform torsion inversely proportional to the pitch of the helicoidal reference configuration. Granted that the energy density of the material surface depends quadratically on the mean curvature of its deformed configuration, we show that the total energy stored in producing a ruled Möbius band as described reduces, in closed form and without approximation, to an integral over the midline of the Möbius band. We formulate and numerically solve a constrained variational problem for finding relative minima of the dimensionally reduced bending energy and construct corresponding stable Möbius bands. The only input parameter entering our variational problem is the number \(\nu \) of turns in a helicoidal reference configuration. We only find solutions if \(\nu \) exceeds a certain threshold, which we compute to machine precision. Above that threshold, an interplay between the operative constraints leads to a multiplicity of coexisting stable solutions with \(n\ge 3\) half twists. For each \(n\ge 3\), we construct an energetically optimal Möbius band which exhibits \(n\)-fold rotational symmetry. All other energy minima yield Möbius bands which lack symmetry. To our knowledge, this study contains the first examples of stable Möbius bands produced by isometrically deforming reference configurations that are not flat.