Isometric Incompatibility in Growing Elastic Sheets
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866917356580634624 |
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| author | Zhang, Yafei Moshe, Michael Sharon, Eran |
| author_facet | Zhang, Yafei Moshe, Michael Sharon, Eran |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_21112 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Isometric Incompatibility in Growing Elastic Sheets Zhang, Yafei Moshe, Michael Sharon, Eran Soft Condensed Matter Materials Science Mathematical Physics Applied Physics 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. |
| title | Isometric Incompatibility in Growing Elastic Sheets |
| topic | Soft Condensed Matter Materials Science Mathematical Physics Applied Physics |
| url | https://arxiv.org/abs/2603.21112 |