Buckling thin films with inhomogeneous swelling

The shapes of many leaves and flowers are determined, at least in part, by their inhomogeneous growth. Additional growth at the edges of leaves, for example, result in saddle-splay curvature and, ultimately, wrinkling. Recent experiments on thin, polymer films imprinted with a predefined pattern of inhomogeneous swelling provide a controlled, experimental playground for describing how swelling-induced buckling leads to a prescribed three-dimensional shape. For example, one expects a sufficiently thin sheet to buckle into a shape that eliminates most of its in-plane strain. Though this is always possible locally, a particular swelling pattern may be either globally frustrated, having no stress-free shapes even for vanishing thickness, or lead to a large degeneracy of stress-free shapes. In both cases, the bending energy remains important even for very thin sheets. I will describe theoretical work on disks and narrow ribbons with swelling-induced, negative Gaussian curvature. Perhaps surprisingly, when the prescribed Gaussian curvature is constant, there are families of stress-free and nearly stress-free shapes, none of which seem to appear in experiments. To understand this behavior, I will identify regimes in which the minimal energy ribbon shape can be determined and discuss the role of stretching and bending energies. We will consider both strips and closed ribbons.