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.
Subscribe to Journal
Get full journal access for 1 year
only $4.92 per issue
All prices are NET prices.
VAT will be added later in the checkout.
Tax calculation will be finalised during checkout.
Rent or Buy article
Get time limited or full article access on ReadCube.
All prices are NET prices.
The data that support the findings of this study are available from the corresponding author on reasonable request.
Computer codes used in this study are available from the corresponding author on request.
Grima, J. N. & Evans, K. E. Auxetic behavior from rotating squares. J. Mater. Sci. Lett. 19, 1563–1565 (2000).
Grima, J. N., Alderson, A. & Evans, K. E. Negative Poisson’s ratios from rotating rectangles. Comp. Methods Sci. Technol. 10, 137–145 (2004).
Grima, J. N., Alderson, A. & Evans, K. E. Auxetic behaviour from rotating rigid units. Phys. Status Solidi B 242, 561–575 (2005).
Rafsanjani, A. & Pasini, D. Bistable auxetic mechanical metamaterials inspired by ancient geometric motifs. Extreme Mech. Lett. 9, 291–296 (2016).
Sussman, D. M. et al. Algorithmic lattice kirigami: a route to pluripotent materials. Proc. Natl Acad. Sci. USA 112, 7449–7453 (2013).
Blees, M. K. et al. Graphene kirigami. Nature 524, 204–207 (2015).
Zhang, Y. et al. A mechanically driven form of Kirigami as a route to 3D mesostructures in micro/nanomembranes. Proc. Natl Acad. Sci. USA 112, 11757–11764 (2015).
Shyu, T. C. et al. A kirigami approach to engineering elasticity in nanocomposites through patterned defects. Nat. Mater. 14, 785 (2015).
Rafsanjani, A. & Bertoldi, K. Buckling-induced kirigami. Phys. Rev. Lett. 118, 084301 (2017).
Konaković, M. et al. Beyond developable: computational design and fabrication with auxetic materials. ACM Trans. Graph. 35, 89 (2016).
Tang, Y. & Yin, J. Design of cut unit geometry in hierarchical kirigami-based auxetic metamaterials for high stretchability and compressibility. Extreme Mech. Lett. 12, 77–85 (2017).
Gatt, R. et al. Hierarchical auxetic mechanical metamaterials. Sci. Rep. 5, 8395 (2015).
Kolken, H. M. & Zadpoor, A. A. Auxetic mechanical metamaterials. RSC Adv. 7, 5111–5129 (2017).
Mitschke, H., Robins, V., Mecke, K. & Schröder-Turk, G. E. Finite auxetic deformations of plane tessellations. Proc. R. Soc. Lond. A 469, 20120465 (2013).
Shan, S., Kang, S. H., Zhao, Z., Fang, L. & Bertoldi, K. Design of planar isotropic negative Poissons ratio structures. Extreme Mech. Lett. 4, 96–102 (2015).
Isobe, M. & Okumura, K. Initial rigid response and softening transition of highly stretchable kirigami sheet materials. Sci. Rep. 6, 24758 (2016).
Neville, R. M., Scarpa, F. & Pirrera, A. Shape morphing kirigami mechanical metamaterials. Sci. Rep. 6, 31067 (2016).
Celli, P. et al. Shape-morphing architected sheets with non-periodic cut patterns. Soft Matter 14, 9744–9749 (2018).
Grünbaum, B. & Shephard, G. C. Tilings and Patterns (Freeman, 1987).
Choi, G. P.-T. & Lui, L. M. A linear formulation for disk conformal parameterization of simply-connected open surfaces. Adv. Comput. Math. 44, 87–114 (2018).
Meng, T. W., Choi, G. P.-T. & Lui, L. M. TEMPO: feature-endowed Teichmüller extremal mappings of point clouds. SIAM J. Imaging Sci. 9, 1922–1962 (2016).
Dudte, L. H., Vouga, E., Tachi, T. & Mahadevan, L. Programming curvature using origami tessellations. Nat. Mater. 15, 583–588 (2016).
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.
We have filed a patent on our algorithms for kirigami design.
Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
About this article
Cite this article
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
Extreme Mechanics Letters (2021)
A predictive deep-learning approach for homogenization of auxetic kirigami metamaterials with randomly oriented cuts
Modern Physics Letters B (2021)
International Journal of Mechanical Sciences (2021)
Rapidly deployable and morphable 3D mesostructures with applications in multimodal biomedical devices
Proceedings of the National Academy of Sciences (2021)
Science Robotics (2021)