A sheet on a drop reveals wrinkling and crumpling as distinct symmetry-breaking instabilities

Smooth wrinkles and sharply crumpled regions are familiar motifs in biological or synthetic sheets, such as rapidly growing plant leaves and crushed foils. Previous studies have addressed both morphological types, but the generic route whereby a featureless sheet develops a complex shape remains elusive. Here we show that this route proceeds through an unusual sequence of distinct symmetry-breaking instabilities. The object of our study is an ultrathin circular sheet stretched over a liquid drop. As the curvature is gradually increased, the surface tension stretching the sheet over the curved drop causes compression along circles of latitude. The compression is relieved first by a transition into a wrinkling pattern, and then into a crumpled state via a geometry-driven continuous transition. Classical buckling theory describes wrinkling as a perturbation of the unbuckled state. However, our measurements prove that this textbook analysis fails for very thin sheets, and highlight the importance of a new, mechanical notion of “thinness” that is distinct from the geometric aspect ratio of the sheet. Our data provide the first conclusive evidence that the wrinkled state in ultra-thin sheets is described by a theory based on a compression-free, symmetric stress field. This insight leads to the surprising recognition that the wrinkle-to-crumple transition is not merely a further elaboration of the shape, but rather signals the symmetry-breaking of the stress field. Furthermore, our observations indicate that crumpling emerges through a continuous transition, where the imposed geometry frustrates the growth of wrinkles.