Exact solutions for wrinkle patterns from geometrically incompatible confinement

Thin elastic shells readily wrinkle in complex and potentially controllable ways. A reliable way of making wrinkles appear is to impose an overall shape on the shell that is different from its natural one. For instance, a shell cut out of a sphere and put onto a planar water bath tends to wrinkle in a mixed “ordered-disordered” fashion, wherein one part exhibits a robust response and a second part behaves statistically instead. In contrast, a shell cut out from a saddle tends to exhibit a totally ordered response with a well-defined wrinkle pattern throughout. We present a simple yet complete set of geometric rules for determining the direction of wrinkling that emerges in a general spherical or saddle-shaped shell. We show how the patterned response is set in any case by the medial axis, also known as the skeleton, of the shell. This distinguished, one-dimensional set can be the target of control. Underlying this result is a heretofore unnoticed reciprocity between positively and negatively curved wrinkle patterns, as well as a general method based on Lagrange multipliers for finding what we call the shell's "locking stress".