  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.
