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We seek to quantify both this effect and accompanying shape variations that bands of different aspect ratios exhibit with increasing stretchability. Even a slight degree of stretchability should alleviate such concentrations. Their results point to the emergence of singularities indicative of the onset of failure. For bands with sufficiently large width-to-length ratios, Starostin and van der Heijden 22,23 observed localized zones of concentrated bending-energy density. We depart from established tradition and explore the influence of material stretchability on the shape of an equilibrated Möbius band. Using Wunderlich's energy, Starostin and van der Heijden 22,23 computed equilibria that meet these requirements and also found evidence to suggest that Sadowsky's energy is a singular limit that produces midlines with discontinuous curvatures. The problem of constructing developable equilibrium configurations was first considered by Mahadevan and Keller, 20 whose numerically determined solutions led to a tighter upper bound on the bending energy but are inconsistent with results of Randrup and Røgen, 21 who showed that the midline of a Möbius band must have an odd number of switching points at which its curvature and torsion both vanish. Wunderlich 18,19 later sharpened Sadowsky's bound and generalized Sadowsky's bending energy to incorporate the effect of finite width. Aside from proving that it is possible to construct a developable band from a rectangular strip of width sufficiently small relative to its length, Sadowsky established an upper bound for the bending energy of a developable band and derived a dimensionally reduced expression for the bending energy of a band made from an infinitesimally thin rectangular strip. This direction was initiated by Sadowsky, 12–17 who considered materials like paper which are easy to bend but essentially unstretchable and, thus, must adopt shapes that are very closely approximated by developable surfaces. This article focuses on the problem of computing energetically preferred equilibrium shapes of Möbius bands. 8 Due to their topologically derived structural stability, these proteins have the potential to serve as drug scaffolds and pharmaceutical templates. 6,7 Möbius topology is also exhibited by cyclotides, macrocyclic plant proteins involved in plant defense.
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5 The ability of a recently synthesized expanded porphyrinoid to switch between Hückel and Möbius topologies presents the possibility of novel memory devices. 2,3 Micron-scale Möbius crystals, which were first created over a decade ago by spooling niobium triselenide ribbons onto selenium droplets, 4 can be viewed as global disclinations. In chemical topology, for example, mechanically interlinked molecules, or catenanes, have been created using Möbius molecules as intermediaries, setting the stage for the synthesis of programmable topological nanostructures. 1 With this realization and breakthroughs in the ability to fabricate objects with molecular-scale precision, research into using the one-sided topology of the Möbius band in scientific applications is burgeoning. 1 Introduction Recent technical advances have made it increasingly clear that the properties of a material are determined not only by its composition but also by geometrical and topological factors. To predict macroscopic band shapes for a given material, we establish a connection between stretchability and relevant continuum moduli, leading to insight regarding the practical feasibility of synthesizing Möbius bands from materials with continuum parameters that can be measured experimentally or estimated by upscale averaging. The associated low-energy configurations provide strategic target shapes for the guided assembly of nanometer and micron scale Möbius bands.
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This stems from a delocalization of twisting strain that occurs if stretching is allowed. We use a two-dimensional discrete, lattice-based model to show that Möbius bands made with stretchable materials are less likely to crease or tear.