Crumpling-origami transition for twisting cylindrical shells.

Origami and crumpling are two processes to reduce the size of a membrane. In the shrink-expand process, the crease pattern of the former is ordered and protected by its topological mechanism, while that of the latter is disordered and generated randomly. We observe a morphological transition between origami and crumpling states in a twisted cylindrical shell. By studying the regularity of the crease pattern, acoustic emission, and energetics from experiments and simulations, we develop a model to explain this transition from frustration of geometry that causes breaking of rotational symmetry. In contrast to solving von Kármán-Donnell equations numerically, our model allows derivations of analytic formulas that successfully describe the origami state. When generalized to truncated cones and polygonal cylinders, we explain why multiple and/or reversed crumpling-origami transitions can occur.