Energy scaling laws for thin elastic sheets with topological defects

Abstract: We derive energy scaling laws for thin elastic sheets with topological defects --- disclinations and dislocations --- for a fully nonlinear 3D model. For disclinations, the scaling laws are tight in the thickness parameter, and improve upon previous results by applying simultaneously to positive and negative disclinations (e-cones) and by giving an explicit dependence on the defect parameter; this latter dependence is, however, still not tight. For thin bodies with dislocations, these are, to the best of our knowledge, the first rigorous bounds for models of finite thickness, are tight when the Burgers vector is not large with respect to the thickness, and relate to a well-known conjecture from the physics literature about the scaling. A main tool is modeling these bodies in the framework of non-Euclidean elasticity, as bodies with a curl-free pre-strain; the curl-freeness allows us to obtain geometric rigidity estimates for the lower bounds.