Imposing curvature on crystalline and non-crystalline sheets: shape deformations, grain boundaries, and asymptotic isometry

Imposing a shape with Gaussian curvature on a solid sheet, generates in it elastic stress. This coupling between geometry (curvature) and mechanics (stress) is a basic demonstration of Gauss’ theorema Egregium, and underlies the morphological richness observed in solid sheets, and their nontrivial response to exerted forces. In this talk I will attempt to provide a unifying framework for morphological transitions in elastic sheets with imposed curvature, by using asymptotic analysis around ``tension field theory”. This singular limit assumes a sheet with zero bending modulus under finite tensile load. Considering crystalline sheets with small, finite bending modulus, we predict a transition between patterns of wrinkles (shape deformation) and grain boundaries. Considering a vanishing tensile load, we predict a transition between two types of “asymptotic isometry”: a developable type (which repels Gaussian curvature) and a non-developable on (which accommodates the imposed Gaussian curvature). The predicted transitions will be demonstrated through examples from a few realistic systems.