Dimension reduction through Gamma convergence for general prestrained thin elastic sheets
Abstract
We study thin films with residual strain by analyzing the -limit of non-Euclidean elastic energy functionals as the material's thickness tends to 0. We begin by extending prior results bhattacharya2016plates agostiniani2018heterogeneous lewicka2018dimension schmidt2007plate, to a wider class of films, whose prestrain depends on both the midplate and the transversal variables. The ansatz for our -convergence result uses a specific type of wrinkling, which is built on exotic solutions to the Monge-Ampere equation, constructed via convex integration lewicka2017convex. We show that the expression for our -limit has a natural interpretation in terms of the orthogonal projection of the residual strain onto a suitable subspace. We also show that some type of wrinkling phenomenon is necessary to match the lower bound of the -limit in certain circumstances. These results all assume a prestrain of the same order as the thickness; we also discuss why it is natural to focus on that regime by considering what can happen when the prestrain is larger.