Formation of small scales in nonlinear PDEs

Dimension reduction through Gamma-convergence in thin elastic sheets with thermal strain, with consequences for the design of controllable sheets

David Padilla Garza

New York University

In this work, we analyze thin elastic sheets with a wide class of spatially varying prestrains. Using techniques from (Friesecke, G., James, R. D., & Müller, S. (2002). A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity), we derive a rigorous Gamma-convergence result for the limiting energy. We borrow from geometric generalizations of the Friesecke-James-Müller theory, work by Bernd Schmidt and later by Marta Lewicka and collaborators, and generalize their results to a wider class of geometric strains and elastic laws. Our main result involves convex integration type techniques found in [Lewicka, M., & Pakzad, M. R. (2017). Convex integration for the Monge–Ampère equation in two dimensions]. Our ansatz for the upper bound is qualitatively different from that associated with any classical plate theory; it suggests that in a region where the limiting configuration is locally planar, the presence of prestrain could induce wrinkling (we are grateful to Marta Lewicka for suggesting the use of an ansatz involving wrinkling). Our results provide a systematic framework for modelling and analyzing the design of controllable sheets.