Gamma-limit of a model for the elastic energy of an inextensible ribbon

Abstract

A \Gamma-convergence result involving the elastic energy of a narrow inextensible ribbon is established. A non-dimensional form of the elastic energy is reduced to a one-dimensional integral over the centerline of the ribbon with the aspect ratio of the ribbon being a small parameter. That integral is observed to increase monotonically with the aspect ratio. The \Gamma-limit of the family of non-dimensional elastic energies is taken in a Sobolev space of centerlines with non-vanishing curvature. In that space, it is shown that the \Gamma-limit is a functional first proposed by Sadowsky in the context of narrow ribbons that form M\"{o}bius bands. The results obtained here do not apply to such ribbons, since the centerline of a M\"{o}bius band must have at least one inflection point. As a first step toward dealing with such inflection points, a result is presented on the lower semicontinuity of the Sadowsky functional with inflection points comprising a set of measure zero within the domain of an arclength parameterization.