Dimension reduction through Gamma convergence for general prestrained thin elastic sheets

Abstract

We study thin films with residual strain by analyzing the Γ−limit of non-Euclidean elastic energy functionals as the material's thickness tends to 0. We begin by extending prior results (Bhattacharya, Lewicka, & Schäffner, 2016) (Agostiniani, Lucantonio, & Lucic, 2019) (Lewicka & Lucic, 2018) (Schmidt, 2007, to a wider class of films, whose prestrain depends on both the midplate and the transversal variables. The ansatz for our Γ−convergence result uses a specific type of wrinkling, which is built on exotic solutions to the Monge-Ampere equation, constructed via convex integration (Lewicka & Pakzad, 2017). We show that the expression for our Γ−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 Γ−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.