The coarsening of folds in hanging drapes

We consider the elastic energy of a hanging drape -- a thin elastic sheet, pulled down by the force of gravity, with fine-scale folding at the top that achieves approximately uniform confinement. This example of energy-driven pattern formation in a thin elastic sheet is of particular interest because the length scale of folding varies with height. We focus on how the minimum elastic energy depends on the physical parameters. As the sheet thickness vanishes, the limiting energy is due to the gravitational force and is relatively easy to understand. Our main accomplishment is to identify the "scaling law" of the correction due to positive thickness. We do this by (i) proving an upper bound, by considering the energies of several constructions and taking the best; (ii) proving an ansatz-free lower bound, which agrees with the upper bound up to a parameter-independent prefactor. The coarsening of folds in hanging drapes has also been considered in the recent physics literature, using a self-similar construction whose basic cell has been called a "wrinklon." Our results complement and extend that work, by showing that self-similar coarsening achieves the optimal scaling law in a certain parameter regime, and by showing that other constructions (involving lateral spreading of the sheet) do better in other regions of parameter space. Our analysis uses a geometrically linear F\"{o}ppl-von K\'{a}rm\'{a}n model for the elastic energy, and is restricted to the case when Poisson's ratio is zero.Comment: 34 page