Dimension reduction for thin films prestrained by shallow curvature

TL;DR

This paper analyzes the dimension reduction of thin prestrained elastic films with shallow curvature, identifying energy limits and optimality conditions, and introduces new bounds using advanced mathematical techniques.

Contribution

It provides a comprehensive Gamma-convergence analysis for prestrained thin films, linking curvature, prestrain components, and energy scaling laws with novel mathematical constructions.

Findings

Gamma-limit of rescaled energies identified

Optimal energy scaling laws established

New energy bounds via convex integration methods

Abstract

We are concerned with the dimension reduction analysis for thin three-dimensional elastic films, prestrained via Riemannian metrics with weak curvatures. For the prestrain inducing the incompatible version of the F\"oppl-von K\'arm\'an equations, we find the $\Gamma$-limits of the rescaled energies, identify the optimal energy scaling laws, and display the equivalent conditions for optimality in terms of both the prestrain components and the curvatures of the related Riemannian metrics. When the stretching-inducing prestrain carries no in-plane modes, we discover similarities with the previously described shallow shell models. In higher prestrain regimes, we prove new energy upper bounds by constructing deformations as the Kirchhoff-Love extensions of the highly perturbative, H\"older-regular solutions to the Monge-Ampere equation obtained by means of convex integration.