Confined developable elastic surfaces:: cylinders, cones and the Elastica

We consider two of the simplest problems associated with the packing of a naturally flat thin elastic sheet. Both problems involve packing the sheet into a hollow cylinder; the first considers the partial contact of a cylindrically curved sheet with a cylindrical surface, while the second considers the partial contact of a conically curved sheet with the edge of a cylindrical surface. In each case, we solve the free-boundary problems to determine the shape, response and stability of the confined surfaces. In particular, we show that an exact description of both the cylindrical and conical structures is given by solutions of the Elastica equation, allowing us to present a unified description of a large class of elastic developable surfaces. This includes what is possibly the simplest example of strain localization, occurring at a point and forming one of the constituent elements of a crumpled elastic sheet.