Dimension reduction through gamma convergence for general prestrained thin elastic sheets

Abstract

We study thin films with residual strain by analyzing the Gamma -limit of non-Euclidean elastic energy functionals as the material’s thickness tends to 0. We begin by extending prior results (Bhattacharya et al. in Arch Ration Mech Anal 228: 143–181, 2016); (Agostiniani et al. in ESAIM Control Opt Calculus Var 25: 24, 2019); (Lewicka and Lucic in Commun Pure Appl Math 73: 1880–1932, 2018); (Schmidt in J de Mathématiques Pures et Appliquées 88: 107–122, 2007) , to a wider class of films, whose prestrain depends on both the midplate and the transversal variables. The ansatz for our Gamma -convergence result uses a specific type of wrinkling, which is built on exotic solutions to the Monge-Ampere equation, constructed via convex integration (Lewicka and Pakzad in Anal PDE 10: 695–727, 2017). We show that the expression for our Gamma -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 Gamma -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.