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# Propagation of pop ups in kirigami shells

Edited by John A. Rogers, Northwestern University, Evanston, IL, and approved March 12, 2019 (received for review October 15, 2018)

## Significance

Kirigami—the Japanese art of cutting paper—has become an emergent tool to realize highly stretchable devices and morphable structures. While kirigami structures are fabricated by simply perforating an array of cuts into a thin sheet, the applied deformation and associated instabilities can be exploited to transform them into complex 3D morphologies. However, to date, such reconfiguration always happen simultaneously through the system. By borrowing ideas from phase-transforming materials, we combine cuts and curvature to realize kirigami structures in which deformation-induced shape reconfiguration initially nucleates near an imperfection and then, under specific conditions, spreads through the system. We envision that such control of the shape transformation could be used to design the next generation of responsive surfaces and smart skins.

## Abstract

Kirigami-inspired metamaterials are attracting increasing interest because of their ability to achieve extremely large strains and shape changes via out-of-plane buckling. While in flat kirigami sheets, the ligaments buckle simultaneously as Euler columns, leading to a continuous phase transition; here, we demonstrate that kirigami shells can also support discontinuous phase transitions. Specifically, we show via a combination of experiments, numerical simulations, and theoretical analysis that, in cylindrical kirigami shells, the snapping-induced curvature inversion of the initially bent ligaments results in a pop-up process that first localizes near an imperfection and then, as the deformation is increased, progressively spreads through the structure. Notably, we find that the width of the transition zone as well as the stress at which propagation of the instability is triggered can be controlled by carefully selecting the geometry of the cuts and the curvature of the shell. Our study significantly expands the ability of existing kirigami metamaterials and opens avenues for the design of the next generation of responsive surfaces as demonstrated by the design of a smart skin that significantly enhances the crawling efficiency of a simple linear actuator.

Kirigami—the Japanese art of cutting paper—has recently inspired the design of highly stretchable (1⇓⇓⇓⇓⇓⇓–8) and morphable (9⇓⇓⇓⇓⇓⇓⇓–17) mechanical metamaterials that can be easily realized by embedding an array of cuts into a thin sheet. An attractive feature of these systems is that they are manufactured flat and then, exploit elastic instabilities to transform into complex 3D configurations (2⇓⇓⇓–6, 13). Remarkably, the morphology of such buckling-induced 3D patterns can be tuned by varying the arrangement and geometry of the cuts (2⇓–4) as well as the loading direction (13). However, in all kirigami systems proposed to date, the buckling-induced pop-up process occurs concurrently through the entire system, resulting in a simultaneous shape transformation.

The coexistence of two phases has been observed at both the microscopic and macroscopic scales in a variety of systems, including phase-transforming materials (18⇓⇓⇓–22), dielectric elastomers (23, 24), and thin-walled elastic tubes (25, 26) (Movie S1). While these systems are very different in nature from each other, they all share a nonconvex free energy function that, for specific conditions, has two minima of equal height. When such a situation is reached, the homogeneous deformation becomes unstable, and a mixture of two states emerges. The new phase initially nucleates near a local imperfection and then, under prevailing conditions, propagates through the entire system (21, 24⇓–26).

Here, we demonstrate via a combination of experiments and numerical/theoretical analyses that kirigami structures can also support the coexistence of two phases, the buckled one and the unbuckled one. Specifically, we show that, in thin cylindrical kirigami shells subjected to tensile loading, the buckling-induced pop-up process initially localizes near an imperfection and then, as the deformation is increased, progressively spreads through the cylinder at constant stress. We find that the curvature of the cylinder is the essential ingredient to observe this phenomenon, as it completely changes the deformation mechanism of the hinges. In kirigami sheets, the initially flat hinges buckle out of plane, leading to a monotonic stress–strain relationship for the unit cell. By contrast, in kirigami shells, the initially bent ligaments snap to their second stable configuration, resulting in a nonmonotonic stress–strain curve typical of phase-transforming materials (18⇓⇓⇓–22).

## Experiments

We start by testing under uniaxial tension a kirigami flat sheet and a corresponding cylindrical shell. Both structures are fabricated by laser cutting triangular cuts arranged on a triangular lattice with lattice constants *A*). The flat kirigami sheet comprises an array of *B*) and gluing the two overlapping edges with a thin adhesive layer (*SI Appendix*, section 1 and Movie S2 have fabrication details).

In Fig. 1 *C* and *D*, we show snapshots of the kirigami sheet and kirigami shell at different levels of applied deformation. We find that the responses of the two structures are remarkably different (Movie S3). In the kirigami sheet at a critical strain, all triangular features simultaneously pop up, forming a uniform 3D textured surface that becomes more accentuated for increasing deformation (Fig. 1*C*). By contrast, in the cylindrical kirigami shell, the pop-up process initiates at the top end of the sample and then, spreads toward the other end as the applied deformation is increased (Fig. 1*D*). Note that the pop-up process in our shells typically starts at one of the ends of the shell, since these act as imperfection. As a matter of fact, a local reduction in the size of the hinges has to be introduced to cause the propagation to start from a different location (*SI Appendix*, Fig. S7 and Movie S9).

Next, to better characterize the response of our structures, during the tests, we monitor black circular markers located at the base of the triangular cuts (Fig. 1 *C* and *D*) and use their position to determine both the applied strain,

In Fig. 2 *A* and *B*, we report the evolution of the local strain *A*). Differently, the contour map for the kirigami cylindrical shell shows a nonvertical boundary between popped/open (yellow in Fig. 2*B*) and unpopped/closed (blue in Fig. 2*B*) regions (Fig. 2*B*)—a clear signature of sequential opening. Furthermore, the constant slope of such boundary indicates that the pop-up process propagates at constant rate of applied deformation (*SI Appendix*, section 2).

To gain more insight into the physics behind the different behavior observed in the kirigami sheet and kirigami shell, we then investigate the deformation mechanism of their hinges. By inspecting their 3D-scanned profiles (Fig. 2 *C* and *D*), we find that they deform in a very different way. In the kirigami sheet, the hinges are initially flat and act as straight beams (5, 13); for a critical level of applied deformation, they buckle and subsequently bend out of plane. By contrast, in the kirigami shell, the initially bent hinges behave as bistable arches (27) and snap to their second stable configurations, which are characterized by curvature inversion. This observation is fully consistent with the results of Figs. 1 and 2, since snapping is always accompanied by a highly nonlinear stress–strain response, which is typical of phase-transforming materials (18⇓⇓⇓–22). As a matter of fact, while elastic structures comprising arrays of beams that buckle under the applied load have been shown to display homogeneous pattern transformations (13, 28), sequential events are typically observed in systems based on snapping units (29⇓–31).

Finally, in Fig. 2*E*, we compare the stress–strain curves of the kirigami sheet and the kirigami shell. We find that the response of the kirigami sheet is typical of buckling-based structures (32) and that it is characterized by an initial linear regime (during which all hinges bend in plane) followed by a plateau stress (caused by the homogeneous buckling-induced pop-up process). The cylindrical kirigami shell also exhibits these two regimes, but the transition between them is more abrupt and characterized by a sharp load drop. At the peak, a small portion of the kirigami shell near the top end pops up, causing the unloading of the rest of the structure and a drop in stress. Subsequently, when the ligaments of the buckled region start to be stretched and become resistant to additional deformation, the pop-up process spreads sequentially through the entire structure, and the stress reaches a steady-state value

## Modeling

Having understood how the imposed curvature affects the deformation mechanism as well as the response of our kirigami structures, we now use a combination of numerical and analytical tools to quantify this effect. To begin with, we conduct nonlinear finite element (FE) within Abaqus/Standard to investigate the effect of both the ligament width δ and the curvature *SI Appendix*, section 4). We find that, for *A* and Movie S7). However, the stress–strain response is found to be significantly affected by both δ and n (Fig. 3 *B* and *C*). For large values of n (i.e., for small curvatures), all unit cells are characterized by monotonic stress–strain curves (Fig. 3*B*) irrespective of *B*) or increasing *C*), the peak becomes more accentuated and is eventually followed by a sharp drop.

At this point, we want to emphasize that the nonmonotonic up–down–up behavior observed for most of our rolled unit cells is typical of elastic structures supporting propagative instabilities (25, 26). Remarkably, it has been shown that the Maxwell construction (33) can be applied to such stress–strain curves to determine several key parameters that characterize the behavior of our curved kirigami shell (25, 26). Specifically, by equating the area of the two lobes formed by the *D*), we can identify (i) the propagation stress *D*). For

While Maxwell construction enables us to easily determine several parameters, it does not provide any information on the width and the shape of the transition zone. This motivates the derivation of a more detailed model based on a 1D array of N nonlinear springs (Fig. 4*A*) in which the response of the ith element is described as*A*). We then write the strain energy of the system when subjected to a constant force *SI Appendix*, section 4). It follows that the equilibrium equations are given by**1**.

Next, we take the continuum limit of Eq. **4**, retain the nonlinear terms up to the third order, and integrate it with respect to Z to obtain**5** is the continuum-governing equation for our kirigami structures, and given a stress–strain curve of the unit cell *B*, we focus on the cylindrical kirigami shell of Fig. 1*B* and compare the evolution of the strain along its axes as predicted by our model and measured in experiments at **5** with *SI Appendix*, section 4). Moreover, since the solution of Eq. **5** is translational invariant with respect to Z, the position of the propagation front **1**). We find that our model accurately captures the shape, width, and amplitude of the transition zone as well as its position as a function of the applied strain, confirming the validity of our approach.

## Effect of Geometry

While in Fig. 4*B*, we focus on a specific geometry, it is important to point out that our model can be used to efficiently characterize the propagation front as a function of the curvature of the shell, the hinge size, and the arrangement of the cuts. In Fig. 4*C*, we focus on kirigami structures with triangular cuts and report the evolution of the normalized width of the propagation front, W [which is defined as the width of region in which the strain changes by *B*), as a function of *C*). Second, we find that, by increasing *C*). It is also interesting to note that the width of the transition zone is inversely proportional to the energy barrier *D*). The largest values of *SI Appendix*, Fig. S15*D*), each corresponding to the opening of one row of cuts (Movie S4).

Notably, using our model in combination with FE analysis conducted on the unit cells, phase diagrams similar to those shown in Fig. 4 *C* and *D* can be constructed for any cut pattern (*SI Appendix*, Fig. S16). Such diagrams can then be used to identify regions in the parameter space where propagation of instability is triggered. As examples, in Fig. 5, we report snapshots of cylindrical kirigami shells with a staggered array of linear cuts (3⇓–5) and an array of mutually orthogonal cuts (1, 13). Both images clearly show the coexistence of the popped and unpopped phases (*SI Appendix*, Figs. S5 and S6 and Movies S5–S7) and further indicate that the characteristics of the phase transition can be controlled by carefully selecting the geometry of the cuts as well as the curvature of the shell. The kirigami shell with the linear pattern is characterized by a sharp propagation front spanning across about one unit cell and a propagation stress *SI Appendix*, Fig. S18), whereas the orthogonal cuts lead to a wider front spreading across about four unit cells and *SI Appendix*, Fig. S19).

Finally, we find that the coexistence of the buckled and unbuckled phases observed in our kirigami cylindrical shells provides opportunities to realize surfaces with complex behavior that can be programmed to achieve a desired functionality. To demonstrate this, we consider a kirigami surface with 20 rows of triangular cuts separated by hinges with two different sizes. Specifically, we choose *A*). If such a surface is planar, no clear signature of the two different *B* and *SI Appendix*, Fig. S9*B*). By contrast, if we use the heterogeneous kirigami sheet to form a cylinder with *C*, *SI Appendix*, Fig. S9*C*, and Movie S8). Remarkably, this sequencing achieved by simply patterning the sheet with regions characterized by different ligament widths can be exploited to design a smart skin that significantly enhances the crawling efficiency of a linear actuator (Fig. 6 *D*–*F*). While all three kirigami-skinned crawlers advance on elongation and contraction of the actuator because of the anisotropic friction induced by the pop ups (7) (*SI Appendix*, section 3), the programmed pop up achieved in our patterned shell enhances the anchorage of the crawler to the substrate at two ends and significantly reduces the backslide (*SI Appendix*, Fig. S11*B*). As a result, the patterned crawler (Fig. 6*F*), which benefits from coexistence of popped and unpopped regions at desired locations, proceeds about twice as fast as the crawlers with a homogeneous array of triangular cuts with either *D*) or *E*).

## Discussion and Conclusions

To summarize, we have shown that, in cylindrical kirigami shells, the buckled and unbuckled phases can coexist, with the pop-up process initially starting near an end and then, propagating along the cylinder at constant stress. In contrast to flat kirigami sheets, which can only support continuous phase transitions, by introducing curvature, the buckling-induced transformation exhibits discontinuity in the first derivative of the free energy, resulting in the coexistence of two phases (37). This remarkable difference in behavior arises, because the curvature transforms the ligaments from straight columns that buckle to bistable arches that snap. It should be also noted that such response is completely different from that of porous cylindrical shells, which under compression, exhibit uniform buckling-induced shape transformation (38⇓–40), whereas it shares similarities with structures consisting of an array of beams resting on flexible supports, which have recently been shown to exhibit a very rich response (41, 42). The behavior of our system can be further understood by looking at its behavior surface (Fig. 7 shows a triangular pattern with *SI Appendix*, Fig. S14). Finally, we have shown that the characteristics of discontinuous phase transition can be tuned by carefully selecting the geometry of the kirigami structure. With such control on the phase transition in kirigami structures, we envision that these mechanical metamaterials could be used to design the next generation of responsive surfaces as shown by the design of a smart skin that enhances the crawling efficiency of a linear actuator.

## Materials and Methods

Details of fabrication of kirigami shells are described in *SI Appendix*, section 1. The protocol for experiments and additional experimental data for kirigami shells with triangular, linear, and orthogonal cut patterns are provided in *SI Appendix*, section 2. Principles of kirigami-skinned crawlers are presented in *SI Appendix*, section 3. Details of FE simulations and theoretical models are presented in *SI Appendix*, section 4.

## Acknowledgments

We thank J. W. Hutchinson for fruitful discussions and Lisa Lee, Omer Gottesman, and Shmuel M. Rubinstein for technical support and access to their custom laser profilometer. A.R. acknowledges support from Swiss National Science Foundation Grant P300P2-164648. L.J. acknowledges support from National Natural Science Foundation of China Grants 11672202, 11602163, and 61727810. K.B. acknowledges support from NSF Grant DMR-1420570 and Army Research Office Grant W911NF-17-1-0147. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the Army Research Laboratory or the US Government. The US Government is authorized to reproduce and distribute reprints for government purposes notwithstanding any copyright notation herein.

## Footnotes

↵

^{1}A.R. and L.J. contributed equally to this work.- ↵
^{2}To whom correspondence should be addressed. Email: bertoldi{at}seas.harvard.edu.

Author contributions: A.R., L.J., and K.B. designed research; A.R., L.J., and B.D. performed research; A.R., L.J., and B.D. analyzed data; B.D. developed the theoretical model; and A.R., L.J., B.D., and K.B. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1817763116/-/DCSupplemental.

Published under the PNAS license.

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