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Journal Club for March 2021: Kirigami metamaterials as a robotic matter

Ahmad Rafsanjani's picture

Journal Club for March 2021: Kirigami metamaterials as a robotic matter

Ahmad Rafsanjani, Center for Soft Robotics, SDU Biorobotics, University of Southern Denmark


Kirigami the Japanese art of paper cutting has driven tremendous activities among researchers that lead to the development of materials and structures with new functionalities. The simplicity of fabrication and versatility of designs attracted researchers from different disciplines to exploit the power of cuts to create flexible devices and robots like no other. The long list of technologies that adopted kirigami design principles goes beyond stretchable electronics [1] and batteries [2], biomedical devices [3], photonics [4], art preservation [5], and of special interest here, soft robotics [6-10]. In a sense, kirigami is a robotic matter, where new functionalities are directly programmed into materials structure through carefully placed cuts. Given the multidisciplinary nature of researchers who adopted kirigami metamaterials in one way or another, it is natural that there is no consensus in the choice of terminology or defining what is meant to be a kirigami metamaterial. For the sake of this journal club, I focus mostly on systems that meet a few requirements:

A kirigami metamaterial is a flat elastic sheet, either thin or thick, perforated with an array of patterned cuts without removing any material.

Practically, we do not remove any material if we use a knife to cut the sheet, but we burn/melt a small fraction of the material if we use a laser cutter. Under uniaxial tension, the deformation localizes around the hinges separating adjacent cuts, and depending on the geometry of cuts and the ratio of the hinge width to sheet thickness, emergent behaviors arise. In the following, we may relax one of the above restrictions or add a new one to introduce mutations that can lead to novel functionality.

Planar kirigami metamaterials: Rotating squares (and their sister structure rotating triangles) fueled mechanical metamaterials research for three decades, and still interest in them shows no sign of decay [10]. To create rotating squares by cuts, we need to introduce an array of orthogonal linear cuts into a thick elastic sheet.  Under tension, rotating squares exhibit an auxetic behavior characterized by an effective Poisson’s ratio of -1. In theory, the hinges can be idealized as a zero-energy revolute mechanism and the structure deforms uniformly up to about 41% strain. In practice, finite hinges bend in-plane and may even slightly stretch and restore the undeformed configuration upon removing the load. Several approaches have been proposed to expand the design space of planar kirigami metamaterials and enhance their functionality:

-       Fractal cuts: If the cuts divide the unit cell into self-similar domains, we can keep cutting each new domain to create a hierarchical kirigami metamaterial. Cho et al. introduced hierarchical cut patterns into material based on rotating square and triangular units to generate extremely large strains and shape changes [12]. Tang and Yin extended this approach and showed using rectangular units instead of square ones can significantly increase the stretchability of the resulting structures [13].

-       Symmetry: The cut pattern in rotating squares is essentially a bunch of lines that are regularly placed over a square grid apart from each other small gaps to preserve the connectivity of the structure. We can apply this template to other grid types and symmetry groups to create new architectures for rotating units. Shan et al. used symmetry as a guide to place cuts into rubber sheets and created different planar kirigami structures with nearly isotropic negative Poisson’s ratio [14]. Recently, Liu et al. proposed a systematic approach to create kirigami metamaterials by identifying the connections between kirigami patterns and symmetry in terms of the wallpaper groups, the set of seventeen plane symmetry groups that fully characterize the space of periodic tiling of the plane [15].

-       Geometric motifs: Planar elastic kirigami metamaterials are mostly monostable and if we remove the load, they retain their undeformed configuration. Can we lock the deformation into an elastic mechanical metamaterial, and more specifically can we cut an elastic material in a way that transforms it into a bistable or multistable structure? Inspired by architectural geometric motifs, we discovered a new family of cut patterns based on a variant of rotating squares and triangles that exhibit simultaneously auxetic behavior and multistability [16]. The point of departure in this design is an energy barrier between the undeformed and the deployed configurations. Therefore, unlike the original rotating units, a rigid mechanism with zero-energy hinges cannot represent the kinematics of this structure. We recently showed if we release the elastic energy stored in the deployed bistable auxetic sheets, large-amplitude transition waves propagate through the metamaterial that can be steered using locally introduced defects [17].

-       Inverse design: Intuition has been the main drive in designing mechanical metamaterials but soon we may realize that expanding their design space requires to involve computers.  Choi et al. developed an inverse design algorithm that can generate a kirigami pattern from regular rotating square cuts (and other cut patterns including our bistable auxetic structure) to enable shape-shifting from an initial geometry into a target shape [18]. They explored the deployment of resulting kirigami patterns by tracking the energetics of the system with a mechanical model by having linear springs along the edges and diagonals of the quads, and simple torsional springs at the nodal hinges. A remarkable outcome of this work is that although the periodic rotating squares are monostable, the inversely designed shape-shifting structure exhibits bistability for sufficiently thin hinges.

Figure 1: Planar kirigami metamaterials (images adopted from [12, 13, 14, 15, 17])



Buckling-induced kirigami metamaterials: Let’s revisit the rotating squares and investigate the mechanical response of thin sheets perforated with a square array of mutually orthogonal cuts, which leaves a network of squares connected by small ligaments. Unlike conventional rotating squares, we found under uniaxial tension the ligaments buckle out of the plane, inducing the formation of 3D patterns whose morphology is controlled by the load direction [18]. This class of materials that are not limited to mutually orthogonal cuts provides a myriad of opportunities to create kirigami metamaterials that are not only highly stretchable, also their self-assembled 3D surface can enable new functionalities. To distinguish them from planar kirigami metamaterials that primarily deform in-plane, we call them “buckling-induced kirigami”. The out-of-plane component of their deformation significantly diversifies their behavior.  Theoretically, there is no preferential direction for the out-of-the-plane pop-ups due to symmetry and in the presence of imperfections, this may lead to a non-uniform distribution of pop-ups. Also, for specific cut patterns, different deformation modes can be dominant depending on the spacing and size of the cuts. As a result, in recent years, a large body of research has been devoted to buckling-induced kirigami metamaterials from different perspectives ranging from the fundamental understanding of their rich nonlinear mechanics to the development of multifunctional materials and innovative devices for biomedical applications. In the following, I briefly introduce several designs for buckling-induced kirigami metamaterials and show different deformation regimes that are programmed into their material structure.

Mechanics of buckling-induced kirigami metamaterials: The simplest buckling-induced kirigami consists of linear parallel cuts that are hexagonally arranged perpendicular to the loading direction. By simply varying the spacing of the cuts, the structure can adopt multiple shapes when stretched. Isobe and Okumura showed the tensile response of kirigami metamaterials with linear cuts consists of three deformation regimes [20]. First, the initial response is linear, and hinges bend in-plane under small, applied deformation. Second, the response shows a sudden departure from linearity to a large plateau region caused by out-of-plane deformation. Finally, for large enough applied deformation, the force rises again and the deformation mechanism of hinges changes from bending to stretching. This deformation behavior resembles the softening and consequent stress-induced crystallinity in lightly crosslinked elastomers. Therefore, buckling-induced kirigami has the potential to transform almost any rigid material into a soft matter. Remarkably, almost all buckling-induced kirigami metamaterials share the same deformation behavior which makes them a robust platform to design programmable matter with predictable response irrespective of the specific shape of cuts. The simplicity of linear cuts has attracted mechanicians to further investigate their rather rich local mechanics and relate that to the global behavior of kirigami metamaterial. Dias et al. studied the connections between this kirigami pattern and the mechanics of a single, non-propagating crack in a sheet [21]. Yang et al. proposed e-cone or excess angle cone as the most fundamental geometric building block of buckling-induced kirigami metamaterials with linear cut and investigated the dependence of the variation in buckling configurations on geometric parameters of the cuts [22]. Recently, Sadik and Dias used a reduced two-dimensional plate model of a circular thin disk with a radial slit and unraveled its deformation map following the opening of the slit and the rotation of its lips [23].

Programming buckling-induced kirigami metamaterials: A variety of novel functionalities have been programmed into the material structure by harnessing the behavior of buckling-induced kirigami metamaterials:

-       Kirigami display: An et al. showed introducing hierarchical cut patterns into thin elastic sheets triggers a variety of different buckling‐induced 3D deformation patterns, and the hierarchy can be used to program the stress-strain response of the surface [24]. They assembled distinct unit cells that exhibit contrasting 3D deformation but similar force-displacement to encode visual information in kirigami sheets that can be revealed under tension.

-       Pop-up propagation: We combined cut and curvature to create kirigami shells that to our surprise showed a propagating pop-up behavior [25]. We found that the interaction of cuts and curvature results in local inversion of curved hinges generating a nonmonotonic response that shares similarities with phase transforming materials in which two phases coexist. In a sense, for specific geometric parameters, kirigami shells behave the way bulges propagate at constant pressure in a party balloon [26, 27]. The propagation of pop-ups in kirigami shells is universal among different cut patterns and we showed it can be achieved for triangular, linear, and orthogonal cut patterns.

-       Programming stiffness: Yang et al. exploited the post-buckling of kirigami metamaterials based on linear cuts to program the stiffness of kirigami sheets in-situ [23]. They found if the post-buckling configuration of a unit cell is symmetric, it can reversibly switch between symmetric and antisymmetric configurations. They demonstrated the precise control of material stiffness by locally and reversibly switching individual unit cells. Kirigami is also used for the mounting of single leaf parchment and vellum objects for display and storage in a museum [5]. Because of the nature of animal skin, parchment and vellum objects have acutely hygroscopic properties and are liable, even over a very short period, to unevenly shape change in all dimensions. The flexibility of kirigami allows for the change in size while the applied force is almost kept constant over a wide range in the plateau region.

-       Kirigami for friction control and locomotion: The morphology of 3D textured surface emerging from synchronized buckling of hinges can be tuned by stretching kirigami sheet enabling applications in which the desired function relies on surface properties such as active friction control. Inspired by the rectilinear locomotion of snakes, we coupled the deformation of a kirigami skin with asymmetric cut patterns to the deformation of a simple soft extending actuator to generate propulsive force for locomotion of a soft crawling robot. The asymmetry in the geometry of cuts results in a tunable friction anisotropy that when it is activated in a cyclic manner enables the crawler to inch forward. Liu et al. extended this approach to soft earthworm a robot for anchoring and locomotion under cohesive soil [28], and Branyan et al. applied it snake-inspired robot for lateral undulation [29]. Recently, we applied a similar concept to develop a shoe grip by attaching a metallic kirigami surface to the outsole of a shoe [30]. The bending of the foot of the wearer activates the pop-ups and increases the friction with the ground potentially reducing the risk of falling on slippery surfaces such as ice, especially among the elderly.


Figure 2: Buckling-induced kirigami metamaterials (images adopted from [5, 7, 19, 23, 24, 25, 30])


Concluding remarks: Kirigami metamaterials as a robotic matter

A robot is a machine capable of sensing, planning, and action [31], and for a soft robot being compliant is an additional requirement. Kirigami metamaterials imbue both the body and skin of soft robots with a wide range of properties. While the presented examples in this short article mostly highlight the programming mechanical deformation in kirigami metamaterials, there is ongoing progress in integrating kirigami in multimodal soft sensors, stimuli-responsive materials, novel soft actuators, and embedded logical operations [not all cited here]. While unleashing the full potentials of kirigami metamaterials in soft robotics requires system-level developments and cross-interdisciplinary approaches, mechanics plays a central role. In the end, I invite all researchers who are active in this field or have a general interest in mechanical metamaterials and soft robotics to share their perspectives on the mechanics of kirigami metamaterials or highlight their recent progress on this subject in the comment section. I am looking very much forward to having a fruitful discussion.




1.     Shyu, T.C., Damasceno, P.F., Dodd, P.M., Lamoureux, A., Xu, L., Shlian, M., Shtein, M., Glotzer, S.C. and Kotov, N.A., 2015. A kirigami approach to engineering elasticity in nanocomposites through patterned defects. Nature materials14(8), pp.785-789.

2.     Song, Z., Wang, X., Lv, C., An, Y., Liang, M., Ma, T., He, D., Zheng, Y.J., Huang, S.Q., Yu, H. and Jiang, H., 2015. Kirigami-based stretchable lithium-ion batteries. Scientific reports, 5(1), pp.1-9.

3.     Vachicouras, N., Tarabichi, O., Kanumuri, V.V., Tringides, C.M., Macron, J., Fallegger, F., Thenaisie, Y., Epprecht, L., McInturff, S., Qureshi, A.A. and Paggi, V., 2019. Microstructured thin-film electrode technology enables proof of concept of scalable, soft auditory brainstem implants. Science translational medicine, 11(514).

4.     Liu, Z., Du, H., Li, J., Lu, L., Li, Z.Y. and Fang, N.X., 2018. Nano-kirigami with giant optical chirality. Science advances4(7), p.eaat4436.

5.     D. Norman. 1993. The mounting of single leaf parchment & vellum objects for display and storage, V&A Conservation Journal, Issue 9.

6.     Rossiter, J. and Sareh, S., 2014, March. Kirigami design and fabrication for biomimetic robotics. In Bioinspiration, Biomimetics, and Bioreplication 2014 (Vol. 9055, p. 90550G).

7.     Rafsanjani, A., Zhang, Y., Liu, B., Rubinstein, S.M. and Bertoldi, K., 2018. Kirigami skins make a simple soft actuator crawl. Science Robotics, 3(15).

8.     Jin, L., Forte, A.E., Deng, B., Rafsanjani, A. and Bertoldi, K., 2020. Kirigami‐Inspired Inflatables with Programmable Shapes. Advanced Materials, 32(33), p.2001863.

9.     Truby, R.L., Della Santina, C. and Rus, D., 2020. Distributed proprioception of 3d configuration in soft, sensorized robots via deep learning. IEEE Robotics and Automation Letters, 5(2), pp.3299-3306.

10.  Rafsanjani, A., Bertoldi, K. and Studart, A.R., 2019. Programming soft robots with flexible mechanical metamaterials. Science Robotics, 4(29).

11.  Grima, J.N. and Evans, K.E., 2000. Auxetic behavior from rotating squares. Journal of Materials Science Letters19(17), pp.1563-1565.

12.  Cho, Y., Shin, J.H., Costa, A., Kim, T.A., Kunin, V., Li, J., Lee, S.Y., Yang, S., Han, H.N., Choi, I.S. and Srolovitz, D.J., 2014. Engineering the shape and structure of materials by fractal cut. Proceedings of the National Academy of Sciences, 111(49), pp.17390-17395.

13.  Tang, Y. and Yin, J., 2017. Design of cut unit geometry in hierarchical kirigami-based auxetic metamaterials for high stretchability and compressibility. Extreme Mechanics Letters, 12, pp.77-85.

14.  Shan, S., Kang, S.H., Zhao, Z., Fang, L. and Bertoldi, K., 2015. Design of planar isotropic negative Poisson’s ratio structures. Extreme Mechanics Letters, 4, pp.96-102.

15.  Rafsanjani, A. and Pasini, D., 2016. Bistable auxetic mechanical metamaterials inspired by ancient geometric motifs. Extreme Mechanics Letters9, pp.291-296.

16.  Jin, L., Khajehtourian, R., Mueller, J., Rafsanjani, A., Tournat, V., Bertoldi, K. and Kochmann, D.M., 2020. Guided transition waves in multistable mechanical metamaterials. Proceedings of the National Academy of Sciences, 117(5), pp.2319-2325.

17.  Choi, G. P., Dudte, L. H., & Mahadevan, L. (2019). Programming shape using kirigami tessellations. Nature materials, 18(9), 999-1004.

18.  Liu, L., Choi, G. and Mahadevan, L., 2021. Wallpaper group kirigami. arXiv preprint arXiv:2102.10754.

19.  Rafsanjani, A. and Bertoldi, K., 2017. Buckling-induced kirigami. Physical review letters, 118(8), p.084301.

20.  Isobe, M. and Okumura, K., 2016. Initial rigid response and softening transition of highly stretchable kirigami sheet materials. Scientific reports, 6(1), pp.1-6.

21.  Dias, M.A., McCarron, M.P., Rayneau-Kirkhope, D., Hanakata, P.Z., Campbell, D.K., Park, H.S. and Holmes, D.P., 2017. Kirigami actuators. Soft matter, 13(48), pp.9087-9092.

22.  Sadik, S. and Dias, M.A., 2021. On local kirigami mechanics I: Isometric conical solutions. Journal of the Mechanics and Physics of Solids, p.104370.

23.  Yang, Y., Dias, M.A. and Holmes, D.P., 2018. Multistable kirigami for tunable architected materials. Physical Review Materials, 2(11), p.110601.

24.  An, N., Domel, A.G., Zhou, J., Rafsanjani, A. and Bertoldi, K., 2020. Programmable hierarchical kirigami. Advanced Functional Materials, 30(6), p.1906711.

25.  Rafsanjani, A., Jin, L., Deng, B. and Bertoldi, K., 2019. Propagation of pop ups in kirigami shells. Proceedings of the National Academy of Sciences, 116(17), pp.8200-8205.

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27.  Chater, E. and Hutchinson, J.W., 1984. On the propagation of bulges and buckles.

28.  Liu, B., Ozkan-Aydin, Y., Goldman, D.I. and Hammond, F.L., 2019, April. Kirigami skin improves soft earthworm robot anchoring and locomotion under cohesive soil. In 2019 2nd IEEE International Conference on Soft Robotics (RoboSoft) (pp. 828-833).

29.  Branyan, C., Hatton, R.L. and Mengüç, Y., 2020. Snake-Inspired Kirigami Skin for Lateral Undulation of a Soft Snake Robot. IEEE Robotics and Automation Letters, 5(2), pp.1728-1733.

30.  Babaee, S., Pajovic, S., Rafsanjani, A., Shi, Y., Bertoldi, K. and Traverso, G., 2020. Bioinspired kirigami metasurfaces as assistive shoe grips. Nature Biomedical Engineering, 4(8), pp.778-786.

31.  Corke, P., 2017. Robotics, vision and control: fundamental algorithms in MATLAB® second, completely revised (Vol. 118). Springer.


Jie. Yin's picture

Dear Ahmad,

Many thanks for the excellent and inspiring introduction and summary of kirigami matter. 

I would like to add one more point on shape morphing feature of kirigami approach. Shape plays an important role in determining its properties and functionalities. Introducing prescribed cut patterns leads to different 3D structures on different scales, e.g., the work by Dr. Zhang and Dr. Rogers’s groups, Zhang et al., A mechanically driven form of Kirigami as a route to 3D mesostructures in micro/nanomembranes. Proceedings of the National Academy of Sciences 112, 11757- 11764 (2015).

Non-periodic cut patterns could generate non-developable curved surfaces through buckling, e.g., the work by Dr. Daraio’s group, P. Celli et al., Shape-morphing architected sheets with non-periodic cut patterns. Soft Matter, 14, 9744-9749 (2018)

Very recently, my group posted a manuscript on arXiv on utilizing the control of cut boundary’s curvature to program the deployed 3D curved shapes through both forward and inverse design (Hong, et al., Boundary cuvature guided shape-programming kirigami sheets, arXiv preprint, arXiv:2103.11076 (2021)

Moving forward, can we extend the kirigami approach beyond 2D sheets to other forms or a generic inverse design of cuts for target mechanical properties and shapes?


Ahmad Rafsanjani's picture

Dear Jie,

Thank you very much for your comments. 

1) You are right and the pioneer works from Prof. Yonggang Huang, and Prof. John A. Rogers groups have shown the potential of out-of-plane kirigami structure in compression. The main difference of their works with examples of kirigami metamaterials presented here is the requirement for pre-stretched substrates to enable local buckling of cuts and restrict the global buckling of the whole structure (given the small thickness of the kirigami sheet) while in tension, this is not the case.

2) The recent works of Prof. Daraio's group are very interesting for programming 3D shapes. I am wondering if we can achieve such 3D shapes solely by cuts without removing the material?

3) Thank you for sharing the fresh work from your group. This is very interesting because it opens the possibility of actuating kirigami from the boundaries for soft robotics applications. Do you think can you extend this approach to other kirigami patterns with higher stiffness or composite kirigami structures (e.g. kirigami embedded in a soft layer)?


Jie. Yin's picture

Dear Ahmad,

Thanks for your inspiring thoughts. 

For 2), I think it is possible if one can carefully design the inhomoenenous cut patterns without cuting holes. The in-plane mismatch deformation could lead out of plane buckling to form 3D shapes. 

For 3), it is a great idea to have krigami patterns embedded in soft layers. In that case, it could be doable by acutating the boundary of kirigami patterns to form target shapes.    

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