: " How does Gauss' theorem "materialize": Wrinkling, crumpling, crystalline defects & delamination of films on curved topographies":
The complex morphologies of thin sheets consist of wrinkles, crumples, folds, creases, and blisters. These descriptive words may sound lucid but do they carry any quantitatively distinguishable content? Following the classical approach of pattern formation theory, we seek to impart a universal meaning to these modes of deformation as distinct types of symmetry-breaking instabilities of a flat, featureless sheet. This idea motivates us to consider the family of "radial stretching" problems. A familiar realization of this problem is the map maker's conflict: projecting a flat sheet onto a foundation of spherical shape.
I will introduce a set of morphologically-relevant dimensionless parameters: bendability, confinement, stiffness, and adhesiveness, that span a phase space for the morphology of radially stretched sheets. In this phase space, wrinkling, crumpling, folding, creasing and blistering could be identified as primary and secondary symmetry-breaking instabilities of an appropriate featureless state of the sheet. The phase diagram spanned by these parameters enables us to predict numerous types of novel transitions between distinct morphological types. A particularly interesting prediction is a "pro-lamination" effect, relevant for ultra-thin film, whereby delamination is suppressed through small-scale wrinkles without distorting the macroscopic shape of a curved substrate.