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. 2017 Aug 21;7:8949. doi: 10.1038/s41598-017-09218-w

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© The Author(s) 2017

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.

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Schematics of the theoretical model of the 3D planar auxetic metamaterial subject to a uniform tensile stress along horizontal stretching: (a) the lattice materials with 5 × 5 unit cells under horizontal stretching is taken as an example to illustrate the model; (b) a representative unit cell is under horizontal stretching and due to the anti-symmetric one corner structure is analyzed; (c) free body diagrams of the corner structure; (d) a curved beam subject to axial forces, shear forces and a moment at the right end.