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Programming shape using kirigami tessellations

Abstract

Kirigami tessellations, regular planar patterns formed by partially cutting flat, thin sheets, allow compact shapes to morph into open structures with rich geometries and unusual material properties. However, geometric and topological constraints make the design of such structures challenging. Here we pose and solve the inverse problem of determining the number, size and orientation of cuts that enables the deployment of a closed, compact regular kirigami tessellation to conform approximately to any prescribed target shape in two or three dimensions. We first identify the constraints on the lengths and angles of generalized kirigami tessellations that guarantee that their reconfigured face geometries can be contracted from a non-trivial deployed shape to a compact, non-overlapping planar cut pattern. We then encode these conditions into a flexible constrained optimization framework to obtain generalized kirigami patterns derived from various periodic tesselations of the plane that can be deployed into a wide variety of prescribed shapes. A simple mechanical analysis of the resulting structure allows us to determine and control the stability of the deployed state and control the deployment path. Finally, we fabricate physical models that deploy in two and three dimensions to validate this inverse design approach. Altogether, our approach, combining geometry, topology and optimization, highlights the potential for generalized kirigami tessellations as building blocks for shape-morphing mechanical metamaterials.

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Fig. 1: Inverse design framework.
Fig. 2: Generalized planar kirigami patterns.
Fig. 3: Planar deployment of generalized kirigami tessellation.
Fig. 4: Generalized kirigami patterns for three-dimensional surface fitting.
Fig. 5: Three-dimensional deployment of generalized kirigami tessellation.

Data availability

The data that support the findings of this study are available from the corresponding author on reasonable request.

Code availability

Computer codes used in this study are available from the corresponding author on request.

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Acknowledgements

This work was supported in part by the Croucher Foundation (G.P.T.C.), National Science Foundation grant no. DMR 14-20570 (L.M.), DMREF grant no. 15-33985 (L.M.) and EFRI grant no. 18-30901 (L.M.). We thank M. Goldberg for contributing to preliminary numerical work, M. Gazzola for helpful initial discussions, and A. Nagarkar and the Whitesides Group for help with fabrication of some of the models using PDMS.

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L.H.D. and L.M. conceived the research. G.P.T.C., L.H.D. and L.M designed the research. G.P.T.C. and L.H.D. conducted the simulations and built the models. G.P.T.C., L.H.D. and L.M. analysed the results and wrote the manuscript.

Corresponding author

Correspondence to L. Mahadevan.

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We have filed a patent on our algorithms for kirigami design.

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Supplementary Information

Supplementary Sections 1–7, Supplementary Figs. 1–13 and supplementary references.

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Choi, G.P.T., Dudte, L.H. & Mahadevan, L. Programming shape using kirigami tessellations. Nat. Mater. 18, 999–1004 (2019). https://doi.org/10.1038/s41563-019-0452-y

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