Wrapping liquids, solids, and gases in thin sheets

Many objects in nature and industry are wrapped in a thin sheet to enhance their chemical, mechanical, or optical properties. There are similarly a variety of methods for wrapping, from pressing a film onto a hard substrate, to using capillary forces to spontaneously wrap droplets, to inflating a closed membrane. Each of these settings raises challenging nonlinear problems involving the geometry and mechanics of a thin sheet, often in the context of resolving a geometric incompatibility between two surfaces. Here we review recent progress in this area, focusing on highly bendable films that are nonetheless hard to stretch, a class of materials that includes polymer films, metal foils, textiles, graphene, as well as some biological materials. Significant attention is paid to two recent advances: (i) a novel isometry that arises in the doubly-asymptotic limit of high flexibility and weak tensile forcing, and (ii) a simple geometric model for predicting the overall shape of an interfacial film while ignoring small-scale wrinkles, crumples, and folds.