Differential swelling of boundary-prescribed patterns

Non-Euclidean plates, thin elastic sheets that grow or shrink inhomogeneously, can be thought as geometrically frustrated surfaces that are unable to completely eliminate in-plane stress and are forced to adopt interesting 3D rest configurations as means of reducing their elastic energy cost. We use conformal flattening methods as a natural framework to prescribe isotropic, nonuniform growth patterns on elastic sheets as a way of making them buckle into a given target shape. We tune the ratio of maximal to minimal area distortion required by modifying the planar domain shape and produce 3D patterns that range from ellipsoids and Gaussian bumps to undulating spheres and corroborate the results using finite thickness simulations of growing sheets. Though surfaces with both positive and negative Gaussian curvature behave differently from those with only one sign of Gaussian curvature, we discuss what seems to be a more general composition property of optimal swelling patterns.