The presence of incomplete cuts in a thin planar sheet can dramatically alter its mechanical and geometrical response to loading, as the cuts allow the sheet to deform strongly in the third dimension, most beautifully demonstrated in kirigami art-forms. We use numerical experiments to characterize the geometric mechanics of kirigamized sheets as a function of the number, size and orientation of cuts. We show that the geometry of mechanically loaded sheets can be approximated as a composition of simple developable units: flats, cylinders, cones and compressed Elasticae. This geometric construction yields scaling laws for the mechanical response of the sheet in both the weak and strongly deformed limit. In the ultimately stretched limit, this further leads to a theorem on the nature and form of geodesics in an arbitrary kirigami pattern, consistent with observations and simulations. Finally, we show that by varying the shape and size of the geodesic in a kirigamized sheet, we can control the deployment trajectory of the sheet, and thence its functional properties as an exemplar of a tunable structure that can serve as a robotic gripper, a soft light window or the basis for a physically unclonable device. Overall our study of disordered kirigami sets the stage for controlling the shape and shielding the stresses in thin sheets using cuts.

Kirigami, the art of paper cutting to create an articulating single sheet, is now increasingly viewed as a paradigm for the design of mechanical metamaterials that exhibit exceptional geometric and structural properties [1][2][3]. The basis for kirigami is the well-known observation that the mechanical response of thin sheets is characterized by the scale separation induced by slender geometries, which makes bending deformations inexpensive compared to stretching deformations. In kirigami, varying the number, size and location of cuts provides extra degrees of control via the internal localization of large bending deformations anchored at the ends of the cuts. This raises a number of questions associated with both the forward problem of understanding the mechanics of these topologically and geometrically complex materials, as well as the inverse problem of designing the cuts to obtain different types of articulated deformations for shape optimization. Recent work in the context of the forward problem has focused primarily on the mechanics of kirigami with simple distributions of periodic cuts, aimed at characterizing the response using a combination of theory, experiment and computation [4][5][6]. By contrast, the inverse problem of designing cuts that allow for articulated shape transformations has been limited primarily to geometric optimization [7,8], without much discussion of the mechanical response of the resulting structures by incorporating bending energies. To design cut patterns to control the shape and response of kirigamized sheets, we need to combine aspects of both these classes of problems by understanding the geometric mechanics of sheets with multiple aperiodic cuts. Here, we take a step in this direction by characterizing the geometry and mechanics of a single cut subject to deformation, and then generalize our study to kirigami sheets with randomly located cuts, but staying in the dilute limit where the cuts do not intersect.

To get a sense of the geometry of deformations in a kirigamized sheet, in figure 1, we show a sheet with four parallel cuts that has been strongly stretched, leading to a shape like a kabuki mask. In figure 2a, we show the shape of a thin circular sheet of radius R, thickness h and a single horizontal cut of length l. When the nominal strain γ induced by the applied horizontal force crosses a threshold, the initially planar sheet buckles out of the plane into a complex geometrical shape. Casual observations of the sheet show that the deformed sheet is constituted of simple conical and cylindrical domains connected by transition regions, as expected from thin sheet mechanics; increasing the topological complexity of the sheet increases the number, size, shape and orientational order of such domains.

We use experimental observations and numerical simulations to study the geometrical mechanics of kirigamized sheets (see electronic supplementary material, S2), starting from the nonlinear elastic energy of a triangulated plate that is minimized using a conjugate gradient method. In our simulations, cuts are defined as thin rectangular slits of length l and small width w, with a semicircular tip of diameter w added to the ends of each cut.

Figure 2a shows the mean curvature of the simulated sheet for different values of the strain γ when the cut is initially orthogonal (θ = π/2) to the loading axis (the line connecting the loading points O 1 and O 2 , the separation of which defines the strain γ ) (figure 2a(i)). We observe that for very low strains γ 1, the sheet stretches, but remains planar. When γ exceeds a critical value γ c , the sheet buckles with bending deformations becoming localized along two conical domains centred near each end of the cut, and a large cylindrical domain of nearly uniform mean curvature κ appears to connect the cut ends and the loading points on both sides. Further stretching of the sheet causes an increase in the curvature of these cylindrical domains, and reduces the Euclidean distance between the end-points (A and B) of the cuts from its initial length (see electronic supplementary material, movie S1). This deformation localization enables the three-dimensional straightening of geodesics, the minimal-length curves connecting the loading points do not cross any cuts, shown in figure 2a,b; details of the geodesic construction and behaviour are described in §4. Simultaneously, the free edges associated with the cut deform into a shape that resembles Euler's elastica [9] as shown in figure 2c. When the applied strain becomes very large, the end points of these elasticae, corresponding to the ends of the cut, come together consistent with the geodesics straightening in three dimensions. If the sheet thickness is very small (h/R → 0), the bending energy of the sheet becomes negligible relative to the stretching energy, and any deformations must become approximately isometric. Based on our observations, we expect that in this limit, the sheet will approach a flat folded state, a configuration discussed in §4.

Moving from this geometric description of the sheet to its mechanical properties, the forcedisplacement response of the kirigami sheet described in figure 2a-c is shown in figure 2d. At low values of strain γ , the force is linearly proportional to γ , but once the sheet buckles, the force plateaus as the sheet deforms by bending out of the plane. Eventually, as the ends of the cut come together, the sheet stiffens as it cannot deform any further without significant stretching and the force increases, showing a divergent behaviour. In the buckled state, there are two alternate modes of deformation: a symmetric mode when both cylindrical domains on either side of the cut are in-phase, and an antisymmetric mode where both cylindrical domains on either side of the cut are out-of-phase. The plateau force for these two cases differs marginally, but the linear and the divergent responses away from the plateaus are indistinguishable as can be seen in figure 2d.

To understand the origin of this divergence, we use a scaling approach. The torque due to the applied force f acting over a length S corresponding to the (small) distance between the ends of the cut is balanced by the internal elastic torque Bκ c where κ c is the characteristic mean curvature in the neighbourhood of the end of the cut, with B as the flexural modulus of the sheet, so that fS ∼ Bκ c . As shown in figure 3b, at the ends of the cut of small width w, the characteristic curvature κ c ∼ θ c /w, where θ c is the angle at the cut's corner. As the sheet is pulled apart by the forces so that the cut edges are 2Rγ apart at their widest, geometry implies that θ c S/2 ∼ 2Rγ and S ≈ 2 (l/2) 2 -2R 2 γ . Substituting these geometric relations into the overall torque balance then yields the relation

where l = l/(2 √ 2R). Writing the stretch ratio of deformation as the end-to-end displacement R(1 + γ ) normalized by the length of the shortest segment connecting the force application points to the ends of the cut, i.e. the piecewise linear geodesic length l g = 2 (l/2) 2 + R 2 , equation (2.1) can be (c) The evolving shape of the cut edges is similar to an elastica [9]. Here, the cut boundary is extracted from (a), and darker colour shows the cut shapes at higher strains. (d) Numerically obtained force displacement curves for the case shown in (a). The curves show three distinct regimes corresponding to the cases shown in (a). The green curve corresponds to the configurations shown in (a), while the red curve corresponds to the case where the two cut edges buckle out-of-plane in opposite directions. Inset: a log-log plot of the scaled strain plotted against the force shows a form similar to the response of a freely jointed chain with the same divergent force-strain behaviour. The shaded region in the inset corresponds to the shaded region in the main figure . expressed in a more familiar form f ∼ 1/(1 -2R(1 + γ )/l g ) where f = f /(BwR) which we see is the same as the divergent response of a freely jointed polymer chain [10]. The inset to figure 2d shows that the mechanical response with the rescaled definition of the stretch agrees well with this simple scaling estimate. We pause to make two comments: (i) the divergent mechanical response of freely jointed chain is intimately linked to the balance between entropic effects and a finite chain length, quite unlike the divergent response of the athermal kirigami sheet, which is due to the localization of curvature of the sheet at the ends of the cut and (ii) the geometric origin of the divergent behaviour suggests that single-cut kirigami can serve as a building block for a new type of lockable mechanical response [11] that is material independent, and can instead be controlled by geometry and topology.

Having understood the geometry and mechanical response of a sheet with a single symmetrically placed cut, we ask what would happen when the cut's length l and/or orientation θ is varied. Figure 3a inset shows the mechanical response for various cases where l is varied keeping the cut orientation orthogonal to the clamped axis (θ = π/2) in red curves. The mechanical response in all cases is qualitatively similar to figure 3a, with a shift in the applied strain γ at the onset of plateau response, the magnitude of the plateau force, and the strain at the onset of divergent response. Larger l results in a lower value of the threshold in γ and a lower force plateau persisting for longer, before the force diverges. Similarly, changing the orientation θ of the cut changes the mechanical response by increasing the plateau force value, as the initial cut direction is more aligned with the direction of the clamping axis. The deformed geometric configurations for all these cases are shown in electronic supplementary material, figure S3.

To quantify these observations, we note that the linear response at very small strain corresponds to the planar stretching of sheet. We note similarities of our scaling analysis to previous studies [4,12,13], and hence keep the discussion on the linear regime and the onset of the nonlinear response very brief here. The presence of the cut leads to a stress intensification near the ends of the cut. As first described in Inglis' seminal work [14], the stress near the tip of a cut of length 2l and a very small radius of curvature w can be approximated as σ 0 l/w, where σ 0 is the far field applied stress on a large plate. Assuming that the planar elastic deformation is confined near the cut tip, we get σ 0 l/w ∼ Eγ . Hence we expect σ 0 ∼ f ∼ 1/ √ l (see electronic supplementary material, figure S2b). Following the initial linear increase in the force with applied strain, the sheet buckles when the compressive load on a plate of size corresponding to the length of a cut reaches a threshold given by the buckling stress σ b ∼ B/hl 2 , with B being the flexural rigidity of the sheet, so that the buckling force f b ∼ (B/h 2 R)(1/l 2 ) (see electronic supplementary material, figure S2d). Then the strain at which the sheet buckles (γ b ) can be estimated from a balance of the in-plane stress (∼ Eγ b / l/w) with the buckling stress, so that γ b ∼ 1/(l/R) 3/2 (see also electronic supplementary material, figure S2c).

Following the onset of buckling, for small out-of-plane displacements δ l of the cut boundary, the curvature of the deformed cylindrical core of the sheet scales as κ ∼ δ/l 2 . Thus the total bending energy of the sheet scales as 2a. Further, the out-of-plane displacement can be expressed as δ ≈ Rγ 1/2 , and hence U ∼ Eh 3 R 3 γ /l 3 . Thus the force scales as f = ∂U/∂(Rγ ) ∼ Eh 3 R 2 /l 3 (see electronic supplementary material, §S3 and figure S1). For a cut oriented at an arbitrary angle θ to the clamped axis, we replace l by its orthogonal projection to the loading axis l sin θ . Figure 3a shows the collapsed data from figure 3a inset, indicating that the bending energy localization in the elasticae corresponding to the two edges of cut determines the magnitude of plateau response in the force-displacement curves. We note that we have ignored the contribution of energy from the conical domains since the size of conical domains is much smaller than the cylinder-like domains just above the onset of out-of-plane deformation, and the mean curvature decays away from the cone tip as κ ∼ 1/r. At large applied strain, when the geodesics are relatively better aligned with the loading axis, application of further strain induces a strong bending deformation at the cut corners. Sheets with a finite tearing threshold stress generally tear as a result of this deformation, and this raises a class of different questions about the nature and shape of the curve of tearing [15], which we do not discuss here. Figure 3c depicts the rescaled forcedisplacement curves showing the divergence response of sheets with varying cut length, which agree well with the scaling arguments given by equation (2.1).

We now turn to understand how the geometric mechanics of single-cut kirigami translates into our understanding of sheets with multiple cuts. In figure 4a, the mean curvature maps of the deformed sheet show the similarity in the localized deformation for the different cases with the number of cuts N, all of which are perpendicular to the loading axis, varying from 2 to 12 (see electronic supplementary material, movie S2 and figure S6). We see that the deformed geometry in all cases consists of four conical domains, and N + 1 connected elasticae N. Indeed, regardless of the number of cuts, the conical domains localize to the ends of cuts that are nearest to the loading points while the sheet remains flat in between. Figure 4b shows the mechanical response of sheets kirigamized with varying number of cuts and shows that increasing the number of cuts softens the system, reducing the plateau force and delaying the transition to the ultimately divergent force-displacement response.

To characterize the mechanical response in the case with multiple cuts, we approximate the bending deformations as localized in N + 1 elasticae close to the onset of out-of-plane deformation so that the elastic energy U ∼ Eh 3 R 2 dγ /l 3 , and hence the force f ∼ ∂U/∂(Rγ ) ∼ Eh 3 Rd/l 3 , where d is the distance between the loading point and the nearest cut (see electronic supplementary material, §S3 and figure S4). Consistent with this, rescaling the force with d collapses the data over the scale of intermediate deformations as shown in the inset of figure 4b.

At large strains, the strong bending deformations near the cone tips appear similar to the case with a single cut discussed earlier. Predicting the potential sites of structural failure in practical applications requires knowledge of the location of conical domains. Our observations suggest that the high-stress conical domains will appear at the ends of a particular cut, if the presence of that cut increases the geodesic length l g . To formalize this, we define a binary participation ratio (PR) for each cut as

where l 0 g is the geodesic length evaluated for a given arrangement that includes cut i, and l -i g is the geodesic length with the ith cut removed from the arrangement. For the cases shown in figure 4a, the two cuts nearest to the loading points have a PR = 1 and all other cuts have PR = 0. Similarly, for the cases with two cuts of varying projected lengths (see electronic supplementary material, figure S5), conical domains disappear at the corners of cuts with PR = 0. In figure 4a, we see that an increase in the number of cuts increases the geodesic length l g . Since the divergent forcedisplacement response emerges as 2R(1 + γ ) → l g , the plateau response in the force-displacement curves is observed at larger γ for larger N. In cases with N > 2, fixing the location of the cuts closest to the points of force application sets the trajectory of geodesics as well as the geodesic length l g , regardless of the presence of the inner cuts. In figure 4c, we see that the mean curvature field of the kirigamized sheets with different cut configurations results in similar profiles. Each case has two conical domains that appear at the corners of the cuts closest to the points of force application, a cylindrical core extending between the two loading points whose extent is defined by the cut length. Since the geodesic length is the same for all the cases in figure 4c, the divergent response occurs at the same strain regardless of the presence of the inner cuts; in fact, When cuts are distributed in a disordered manner on a loaded sheet, the geometry of the geodesics continues to control the mechanical response of the overall system. In figure 5a, we show an example with 21 cuts of the same length (totaling 8R), with randomly chosen locations of the cut midpoint and its orientation, but with a minimum separation between the cuts and between the cut and the loading points. The localization of deformation in participating cuts' elasticae and conical domains is evident, while the sheet remains flat and undeformed near cuts with PR = 0 (also see electronic supplementary material, figure S7).

To obtain the average response for the case of disordered kirigami, we repeat the simulations keeping the sum l constant. Figure 5d shows the mechanical response for these disordered kirigami structures averaged over 10 samples per case. The overall nature of the forcedisplacement curve is similar to that of a single cut, with the plateau force decreasing for longer cuts. When the sum of the cut lengths l is small, we see a very weak deviation from the initial linear response, but with increasing l a clear plateau is observed. The divergent forcedisplacement response is observed when the geodesic connecting the two loading points nearly straightens out under applied strain, and we observe relatively large variance in force beyond the initial linear response due to the strong dependence on the cut length and location. Increasing the number of cuts while keeping l constant results in similar observations (see electronic supplementary material, figure S8) with a lower number of longer length cuts resulting in a lower plateau force response and an increased variance.

Since the cuts with PR = 0 do not significantly affect the geometric mechanics of the sheet, simply removing them from the given cut arrangement results in a sheet with a nearly identical geometric and mechanical response to a sheet with a much smaller number of cuts. In figure 5b, we show a kirigami pattern derived from that shown in figure 5a where the cuts with PR = 0 are removed. When such a structure is loaded, we see that the mean curvature field is quantitatively similar to that in figure 5a; indeed figure 5c shows that the force-strain curves are nearly the same.

It is evident from the observed geometry of disordered kirigami that the dominant modes of deformation become localized near a few cuts. Disordered cuts behave similarly to structured cuts, where the mechanical response depends on the distance of the cut from the loading point(s), its projected length, and its PR. At the onset of the buckling transition from the initial planar stretching response at very low strains, each cut with PR = 1 introduces a soft bending deformation mode in the sheet with a characteristic bending force f * , given by the smallest buckling load, i.e.

where d i is the minimum distance of the mid-point of the cut of length l i from the loading points, and 2Rd i is the distance of the cut midpoint from the farthest loading point, and θ d i and θ 2R-d i are the angles that the cut makes with the line joining the midpoint of the cut to the points of force application. The above result follows from the assumption that at the onset of the plateau regime, the characteristic mean curvature for bending localization is set by the cut that corresponds to f * , so that the two terms in equation (3.2) follow from the energy of two elasticae that exist on both sides of the cut. Here, we note that the cuts with PR = 0 do not alter the plateau response near its onset as seen with the cases in figures 4b and 5c. These observations allow us to determine the rescaled mechanical response shown in the inset of figure 5d, providing a reasonable collapse in the plateau region of force-displacement data. The spread in the scaled data is likely due to the simplification that the area of the sheet where energy is localized is assumed to span the sheet (hence the factor 2R in equation (3.2)), and that additional cuts which buckle following the onset of first buckling also contribute to localizing the bending deformation. All together, this allows us to reduce a given disordered kirigamized sheet to a simpler 'mechanical equivalent' [16].) The path-shortening algorithm yields a polygonal competing to be a geodesic. (b-d) Strongly stretched inextensible sheets of negligible bending rigidity, h → 0, can be flat-folded to two-dimensional sheets while simultaneously rectifying all geodesics to lie along a single straight segment in space.

creases as the sheet folds on itself, as shown in figure 6b-d. These observations of the geometry of strongly deformed kirigamized sheets show that the polygonal geodesic in the plane become an approximately straight geodesic in R 3 connecting the points of force application. The sheet can be then flat-folded, leading to a configuration that is a piecewise affine isometric immersion of the plane. We leave precise theorems and proofs of these statements for a separate study [16], but provide a brief summary of the results here.

We represent the given set of cuts L, contained within an open, bounded, convex domain Ω ⊂ R 2 , as the union of the edges of a graph G. Without loss of generality, G may be taken as planar, i.e. each pair of its edges intersects at most at a single common vertex. The polygonal structure of geodesics follows from the idea of a path-shortening algorithm (figure 6a

one successively replaces its portions by segments, as follows. Firstly, for t > 0 sufficiently small, the segment O 1 τ (t) does not intersect L. Let t 1 ∈ (0, 1] be the first time that the segment

) must contain some of the vertices of the cuts. Call p 1 the closest one of these vertices to τ (t 1 ) and perform the concatenation of the segment p 1 τ (t 1 ) with the curve τ restricted to [t 1 , 1]. The process is now repeated from p 1 . After finitely many such steps one obtains a polygonal connecting O 1 and O 2 , with a shorter length than τ .

The construction of the desired isometric immersion follows from the folding algorithm below that consists of three steps. Let O 1 , O 2 ∈ ∂Ω and denote by l g the length of the geodesics between O 1 , O 2 . Then, we start with two preparatory steps. Step 1. Sealing portions of inessential cuts that do not affect l g . To this end, label cuts (the edges of G) by l 1 , . . . , l n . Move the first endpoint vertex of l 1 toward its second vertex, and start "sealing" the portion of the cut l 1 left behind. The length of the geodesics connecting O 1 and O 2 may drop initially, in which case the configuration is left unchanged. Otherwise, the geodesic distance is continuously non-increasing, although it may initially remain constant. The sealing process is stopped when the aforementioned distance becomes strictly less than the original one, and the new position point is labeled as the new vertex endpoint of l 1 . In the next step, the remaining endpoint is moved along l 1 toward the (new) first endpoint and the process is repeated, thus possibly sealing the cut l 1 further. The same procedure is carried out for each l i in the given order i = 1, . . . , n. It follows that upon repeating the same process for the newly created configuration, labeled the minimal configuration will not be further altered. While different ordering of cuts and vertices may yield different minimal configurations and new geodesics may be created in the cutsealing process, all original geodesics are preserved. Also, since the newly created set L is a subset of the original one, finding the isometric immersion relative to the new L will yield the desired isometric immersion.

Step 2: Ordering of the geodesics and connected components of Ω \ L. There are two important properties of a minimal configuration: the graph G has no loops (i.e. it is a collection of its connected components that are trees), and each vertex that is a leaf in some tree is a vertex of some geodesic. With these properties, one proceeds to label all geodesics in a consecutive order, with τ 1 . . . τ N . Here, τ r τ r+1 means that the concatenated polygonal curve from O 1 to O 2 via τ r and then back to O 1 via τ r+1 encloses a region D r and it is oriented counterclockwise with respect to D r . Next, one labels and orders the trees {T m } s m=1 in Dr so that D r \ L is partitioned into subregions {P m } s m=0 and {Q m } s m=1 in the following way: each P m is a polygon bounded by the "right most" path from the tree T m , the "left most" path from T m+1 , and the intermediate portions of τ r and τ r+1 which are concave with respect to P m . Each Q m is a finite union of polygons enclosed within the single tree T m , again bounded by the intermediate portions of geodesics τ r and τ r+1 . Note that τ r and τ r+1 may have nontrivial overlaps and some of {Q m } s m=1 may be absent.

Step 3: Constructing a desired isometric immersion. Finally, we fix the segment I = [0, l g ] along the x 1 -axis in R 3 . To construct an isometric immersion u of Ω \ L such that u(O 1 ) = 0, u(O 2 ) = l g e 1 , u(τ r ) = I for r = 1, . . . , N, and where each segment on τ r is mapped onto a designated segment portion of I, we first note that u consists exclusively of planar folds and returns the image that is a subset of R 2 . By Step 2, D r = s m=0 P m ∪ s m=1 Q m , for r = 1, • • • , N -1, so that it is possible to construct u on P 0 , Q 1 , Q 2 , . . . , Q m , P m , even though the step to construct u on P 1 , . . . , P m-1 can be highly technical [16]. Since the exterior region D 0 = Ω \ N-1 r=0 D r does not contain trees, the two outermost geodesics σ 1 and σ N are convex, and so the definition of u on D 0 consists of several simple folds. We note that the condition O 1 , O 2 ∈ ∂Ω is essential here: indeed there exist minimal configurations for O 1 , O 2 / ∈ ∂Ω, that do not admit any isometry u with the property that the Euclidean distance from u(O 1 ) to u(O 2 ) equals the geodesic distance from O 1 to O 2 in Ω \ L (for further details, we refer to [16]).

The geometric mechanics of ordered and disordered kirigami leads us naturally to questions of design for function. We apply the principles discussed so far to three examples: a soft gripper with multiple modes of grasping, a family of kirigami designs displaying optical transmittance and shielding properties, and a physically unclonable function based on the inherent disorder in the deformation of kirigami sheets.

The opening of a kirigami slit upon loading suggests its potential as a soft grasper, its shape modulated by an applied force or strain at the boundary [12,18]. To be effective, a grasper We define a passive grasper as one that only requires external work to be done in order to grasp or release the object, with energy-free relocation while an active grasper requires continuous application of an external force (external energy) to grasp and relocate, as shown in the schematics in figure 7b,c. Kirigami enables both designs; while the active grasper accommodates the target object in the curved features of the deformed sheet, the passive grasper uses the cuts to accommodate objects.

The simplest passive grasper design, a single slit perpendicular to the loading axis, requires a prestretch to deform the cut into a hole that can accommodate the target object. On releasing the prestretch, the cut boundary relaxes to contact the object, grasping it with no further energy input. After relocation, the arrested object can be released by applying an extensional strain to the sheet.

However, a potential drawback of this solution is that the object is liable to rotate in order to accommodate the two point-like forces from the cut edge on either side. Our disordered kirigami observations suggest adding more cuts softens the grasper's loading curve, enabling symmetric deformation and reducing the severity of sudden reorientations at a minor cost to the maximum load bearing capacity. Hence, an effective design of a kirigamized grasper is one that has multiple smaller cuts in addition to a larger cut in the middle as shown in figure 7a. The additional cuts are prescribed in a way that all cuts have a PR = 1, and buckle under the applied strain. Further, the largest cut is split in the middle by a small cut, oriented along the pulling axis. Together these features enable symmetric deformation of the flat kirigami sheet, and improve the stability of contact with the grasped object. The additional contact points may not help stabilize the grasped object if the size of the object is much smaller than that cut length, in which case the grasped object may slip during handling. We note that the simplest kirigamized grasper with a single cut (figure 2a) deforms asymmetrically, and hence cannot be effectively used as a grasper in both active and passive modes. In practical scenarios however symmetric deformation can be realized if the sheet is stretched significantly [19].

The final design sequence is demonstrated in figure 7b,c and visualized in the electronic supplementary material, movie S3. In this passive mode, the conical tips and the cylindrical core of the kirigami sheet enables confinement of the target object as shown in figure 7b and electronic supplementary material, movie S3. The same geometry also functions as an active grasper, as the two ends of the middle cut are forced toward each other as loading is applied. For the symmetric deformations, this may be used to encase an object with compressive force and relocate it, as shown in figure 7c.

The range of applicability of a general kirigami grasper can be understood from a balance of the forces due to the bending of kirigami sheet, and the weight of the grasped object. For a cut of length l, the characteristic grasping force of the deformed sheet can be written as f g ∼ μ(B/l)c(γ ), with μ being the coefficient of friction and c(γ ) is a strain-dependent geometric factor. For an object with effective density ρ and size l, force balance yields μ(B/l)c(γ ) ∼ ρgl 3 . This provides a non-dimensional parameter ρgl 4 /(μBc(γ )), which has values ranging between 1 and 100 for the successful grasping demonstrations we attempted using a plastic grasper. For a kirigami gripper with an arbitrary configuration of cuts, the geometric and mechanical response of kirigamized sheets in various ordered and disordered cases can help estimate the non-dimensional kirigami grasper parameter. This is because the nature of force-strain curve for ordered/disordered kirigami is universal (schematics in figure 7b,c) and knowing the force in the extensible regime, i.e. force corresponding to points 1, 3 in figure 7b and point 2 in figure 7c, will help determine the operating regime of the gripper.

Although our discussion so far has been restricted to piecewise linear cuts, the ideas established are applicable to cuts with finite planar curvature. Inspired by the recent efforts towards design of kirigami-based mechanically deployable structures [1], we demonstrate a family of deployable kirigami designs whose force-dependent transmittance properties can be tuned by varying a single geometric parameter. The application follows directly from our observations (in figures 2-5) and analysis in figure 6 that with the increasing strain the geodesic in kirigamized structures straightens out and the cuts with PR = 1 localize the bending deformation. Thus, a careful choice of cuts (and hence the geodesic) can be used to embed force/strain-dependent functional features in kirigamized sheets. Figure 8a shows three cases of physical kirigamized sheets perforated with five concentric circular arcs, differing only by a parameter dφ that defines the arcuate cuts that extend between ±dφ and ±(πdφ) with respect to the vertical axis, while the radius is linearly increased between the five arcs. We see that the planar kirigami sheets (with marginally different dφ) show very different geometric mechanics under a small applied strain. When dφ = π/60, none of the cuts intersect the vertical diameter, leaving a straight geodesic connecting O 1 and O 2 ; hence, tensile loading of this structure results in simple in-plane stretching along this diameter.

However, a small change in dφ to -π/180 results in each of the circular cuts now intersecting the vertical diameter, resulting in a geodesic connecting O 1 and O 2 that must meander around the corners of all cuts. Under an applied load, this structure shows a large relative out-of-plane displacement of different domains of the sheet (see electronic supplementary material, movie S4). Further increasing the angular extent of the cuts by taking dφ = -π/30 makes the geodesic qualitatively change, as it now skirts the two outermost cuts without intersecting any of the inner cuts. Since the inner cuts have PR = 0, and hence do not introduce any soft deformation modes, under an applied deformation, the outer frame localizes the bending while the inner structure stays nearly planar without any deformation, and is thus shielded mechanically and geometrically.

We now quantify some of the functional responses of these kirigamized structures using dφ as the only tunable parameter. A straightforward observation is that the force required to deform such a class of structures increases monotonically with dφ (see electronic supplementary material, figure S11a). Very different geometric consequences due to changing geodesic paths can be quantified in terms of the transmittance of the kirigami structure. In practice, transmittance may correspond to the light transmitted though an optical window when illuminated with light rays perpendicular to the rest plane. In experiments, we restrict the deformation to small strains approximately 1%, a practically relevant regime, and note that geometric mechanics are generally strain-dependent. Figure 8b shows a phase diagram of transmittance (left axis) measured at 1% strain as a function of dφ, along with the computed geodesic length normalized by diameter (l g /2R) for reference. The transmittance curves clearly display non-monotonic trends. Transmittance is achievable in the cases when l g → 2R, and the strain sensitivity of transmittance is enhanced with increasing l g /2R (see electronic supplementary material, figure S11b). For cases with dφ > 0, we note l g /2R = 1 and we see only weak variation of transmittance, since these structures must deform primarily by planar stretching.

We close with an unusual application of kirigami for encoding information, inspired by the fact that for a given configuration of cuts in a sheet, every choice of a loading axis 'activates' a distinct set of deformations anchored at different cuts (i.e. cuts with PR = 0), each leading to a distinct geometrical response. Such a system in principle can be employed to encrypt information geometrically in a flat kirigamized sheet. Encoding information in such a system involves a choice of the number of cuts (N), parameters associated with each cut (x i , y i , l i and θ i ), a loading axis, and an extensional strain γ . The resulting high-dimensional design space leads to a rich nonlinear strain-dependent and possibly discontinuous shape response.

This observation allows us to use kirigami sheets that are easy to realize practically in creating physically unclonable functions (PUFs), building on the notion that the inherent disorder in many physical systems is a promising hardware route towards roots-of-trust cryptographic keys [20,21]. Specifically, a kirigami PUF is generated by activating/stretching a kirigamized sheet along a loading axis up to a fixed nominal strain. The deformed sheet is then imaged with a camera held at a fixed distance and orthogonal to the initial plane of the sheet, and a z-displacement map is created for the deformed structure. We store this z-displacement map, i.e. |z| as a PUF key. For future authentication, the user will need to know the cut configuration, loading axis and the extensional strain to duplicate such a key. Figure 9a shows a planar kirigamized sheet with disordered cuts, and figure 9b shows the greyscale z-displacement maps of the kirigami structure for different strain values and a different choice of loading axis. Since we use a twodimensional z-displacement map as a PUF key, we rely on a measure of image similarity index to match keys between different experiments. Here, we use the multiscale structural similarity index function in MATLAB [22] to compare different keys. The index quantifies the luminance, contrast and structure of several versions of the image at various scales, with its values ranging between 0 and 1 (1 being the highest possible match).

To demonstrate the effectiveness of kirigami as a PUF in a minimal setting, we keep the cut configuration of the kirigamized sheet fixed and only vary the loading axis and the applied strain. The versatility of the kirigami PUF is directly linked to the nonlinear geometric mechanics of disordered kirigami, and our observations make it evident that kirigamized sheets will yield a deterministic geometric mechanical response. Thus given complete information about all the variables involved, the PUF key will be deterministic. In practice, having a large number of cuts will make it very difficult to be reverse engineered by an adversary. A robust PUF system will return significantly different (false) PUF keys if there exists a mismatch between the loading axis and the applied strain that corresponds to the stored (true) PUF key, because even a small variation in the loading axis by dθ results in a sharp decrease in the key similarity as shown in figure 9c. As an example, setting a threshold value on the similarity index to 0.9 will enable a robust response from this PUF system, and yet allows for a margin of error in ±dθ . In figure 9d, we show that there is a similar robustness in the PUF keys when the applied extensional strain is varied. Indeed, we see that kirigami-based PUF are only effective when the strain is large enough for the structure to undergo substantial out-ofplane deformation and corresponds to the plateau regime in the force-displacement curve. In the vicinity of the strain corresponding to the stored key, a slight error in the applied strain does not impact the PUF performance, while a significant deviation results in a poor match.

Our study of ordered and disordered kirigami has shown how elementary geometric and energetic concepts allow us to understand the three-dimensional structure and mechanical response of kirigamized sheets. The general principles relating the mechanical response of such sheets to their geometry, along with scaling arguments for all regimes of deformation, reduces a complex nonlinear problem to a combination of simple geometrical constructions, potentially easing the search for novel kirigami-based engineering solutions in instances such as grasping, windowing and encoding information. We believe that these examples are just the beginning of a different way of thinking about functional aspects of topological and geometrical mechanical metamaterials.