Isometric Incompatibility in Growing Elastic Sheets

TL;DR

This paper introduces a new type of geometric incompatibility in growing elastic sheets that prevents stretching-free configurations, leading to the formation of d-cone-like dimples due to topological frustration.

Contribution

It identifies a novel incompatibility mechanism in elastic sheets that extends beyond classical theories, with experimental, simulation, and theoretical validation.

Findings

A new incompatibility prevents stretching-free states in growing sheets.

Surface growth with positive Gaussian curvature leads to geometric horizons and frustration.

Sheets develop periodic d-cone-like dimples as a response to incompatibility.

Abstract

Geometric incompatibility, the inability of a material's rest state to be realized in Euclidean space, underlies shape formation in natural and synthetic thin sheets. Classical Gauss and Mainardi-Codazzi-Peterson (MCP) incompatibilities explain many patterns in nature, but they do not exhaust the mechanisms that frustrate thin elastic sheets. We identify a new incompatibility that forbids any stretching-free configuration, even when the rest state of the elastic sheet locally satisfies the Gauss and MCP compatibility conditions. We demonstrate this principle in a model of surface growth with positive Gaussian curvature, where a geometric horizon forms, leading to the onset of frustration. Experiments, simulations, and theory show that the sheet responds by nucleating periodic d-cone-like dimples. We show that this obstruction to stretching-free configurations is topological, and we point to open questions concerning the origin of frustration.