Influence of material stretchability on the equilibrium shape of a M\"obius band

We use a discrete, lattice-based model for two-dimensional materials to show that M\"obius bands made with stretchable materials are less likely to crease or tear. This stems a delocalization of twisting strain that occurs if stretching is allowed. The associated low-energy configurations provide strategic target shapes for the guided assembly of nanometer and micron scale M\"obius bands. 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 M\"obius bands from materials with continuum parameters that can be measured experimentally or estimated by upscale averaging. To take advantage of stretchability in the case of M\"obius bands made of graphene, DNA, and other effectively unstretchable materials, we develop and explore a novel architecture that uses the Chinese finger trap as a fundamental building block and imparts notable stretchability to otherwise unstretchable materials.