Defects and boundary layers in non-Euclidean plates

TL;DR

This paper analyzes the elastic behavior of non-Euclidean plates with negative curvature, revealing the existence of boundary layers and classifying energy-minimizing deformations, with implications for understanding thin elastic sheets.

Contribution

It provides rigorous bounds on elastic energy, classifies global minimizers, and introduces boundary layers as a new phenomenon in non-Euclidean plates.

Findings

Elastic energy scales with thickness squared.

Only flat and saddle-shaped minimizers exist globally.

Boundary layers are necessary to regularize curvature jumps.

Abstract

We investigate the behavior of non-Euclidean plates with constant negative Gaussian curvature using the F\"oppl-von K\'arm\'an reduced theory of elasticity. Motivated by recent experimental results, we focus on annuli with a periodic profile. We prove rigorous upper and lower bounds for the elastic energy that scales like the thickness squared. In particular we show that are only two types of global minimizers -- deformations that remain flat and saddle shaped deformations with isolated regions of stretching near the edge of the annulus. We also show that there exist local minimizers with a periodic profile that have additional boundary layers near their lines of inflection. These additional boundary layers are a new phenomenon in thin elastic sheets and are necessary to regularize jump discontinuities in the azimuthal curvature across lines of inflection. We rigorously derive scaling laws for the width of these boundary layers as a function of the thickness of the sheet.