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arXiv · 2603.21112

Isometric Incompatibility in Growing Elastic Sheets

Yafei Zhang·Michael Moshe·Eran Sharon
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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 smooth stretching-free configurations, 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.

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Yafei Zhang, Michael Moshe, Eran Sharon. 2026-03-22. Isometric Incompatibility in Growing Elastic Sheets. https://doi.org/10.1103/bczk-bfy5

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