Smooth wrinkling patterns in strongly compressed shells

Changing the gaussian curvature of a plate or of a thin shell generally leads to compressive stresses and induces folding and crumpling instabilities. However, the localised folds first observed as a shallow shell is compressed between parallel plates progressiveley evolve into a smooth wrinkling pattern at high compaction. While azimutal wrinkles are observed in the case of a plain spherical cap, cutting a wide hole at center results in a different wrinkling pattern: radial wrinkles appear around the hole and are followed by circular wrinkles. The radius of the transition if found to be the geometric mean of the radius of the hole and the radius of the base of the shell. We show how the wavelength and, more generally, the global geometry of the wrinkles can be inferred from the stress distribution corresponding to the full compression of the shell, as previously demonstrated in the case of embossed plates. This description is not limited to axisymmetric shells and can be applied to complex topographies.