Analysis of cone-like singularities in twisted elastic ribbons

Abstract

Twisting a thin elastic ribbon is known to produce a localised deformation pattern resembling a cone whose tip is located on the edge of the ribbon. Using the theory of inextensional ribbons, we present a matched asymptotic analysis of these singularities for ribbons whose width-to-length ratio w/ℓ≪1 is small. An inner layer solution is derived from the finite-w Wunderlich model and captures the fast, local variations of the bending and twisting strains in the neighbourhood of the cone-like region; it is universal up to a load intensity factor. The outer solution is given by the zero-w Sadowsky model. Based on this analysis, we propose a new standalone ribbon model that combines the Sadowsky equations with jump conditions providing a coarse-grained description of cone-like singularities, and give a self-contained variational derivation of this model. Applications to the Möbius band and to an end-loaded open ribbon are presented. Overall, the new model delivers highly accurate approximations to the solutions of the Wunderlich model in the limit w≪ℓ while avoiding the numerical difficulties associated with cone-like singularities.