Mechanical metamaterials based on origami and kirigami

Mechanical metamaterials are artificially engineered materials with unusual mechanical properties.^1,2 In recent years, the ancient paper arts of origami (paper folding; in Japanese, “ori” means “fold” and “gami” means “paper”) and kirigami (paper cutting; in Japanese, “kiri” means “cut”) have become popular for building mechanical metamaterials and now provide valuable design guidelines.^3,4 By means of folding and cutting, simple two-dimensional thin-film materials can be transformed into complex three-dimensional structures with unique and programmable mechanical properties, such as shape morphing,⁵ flexibility,³ tunable Poisson's ratio,⁴ tunable stiffness,⁶ and multi-stability.⁷ Recently, origami and kirigami structures have been created from not only paper but also metals,^8–10 polymers,¹¹ hydrogels,^12,13 and graphene,¹⁴ with sizes ranging from macroscale to microscale¹⁵ to nanoscale.¹⁰ Origami- and kirigami-based mechanical metamaterials have been applied in many fields, including flexible electronics,^16–19 medical devices,²⁰ and robotics.^6,21–25

Regarding mechanical properties, origami and kirigami are similar because both folding and cutting are mechanical means of dividing thin materials into flexible areas (i.e., creases in origami, and linkages in kirigami) and stiff areas (i.e., thin panels in both origami and kirigami). Therefore, the mechanical behavior of origami or kirigami structures is determined to a considerable degree by the balance between flexibility and rigidity conferred by the origami or kirigami pattern. However, origami and kirigami offer different and unique behaviors. An origami structure is folded from the initial planar state into a compacted volume,^{4–9,12,15,16,18,20–24,26–140} whereas a kirigami structure is stretched from the initial state into an expanded configuration.^{10,11,14,17,19,25,141–205} Also, hybrid origami–kirigami designs are emerging that combine the two concepts to take advantage of both.^{21,22,98,112,119,150,206–266}

When designing mechanical metamaterials, one of the most important concepts is the mechanical energy landscape, which describes how the strain energy varies with different geometrical and/or deformation variables in the deformed configuration space of metamaterials. The mechanical energy landscape affects almost all the properties of a mechanical metamaterial, such as its deployability, stability, and stiffness.² For example, bistability is due inherently to the double local minima of elastic energy,⁷ and self-deployment occurs as the mechanical energy decreases along the deployment path.^7,24 Origami and kirigami both provide elegant ways to design the energy landscape through the folding and cutting patterns. As shown in Fig. 1, for some origami and kirigami patterns under specific conditions, the panels remain rigid during deformation and energy is stored only in the crease or linkage areas; known as rigid origami or kirigami, this type confers more-predictable kinetics and mechanical behavior and is ideal for applications such as shape morphing.^{5,26,144,146,147} Otherwise, the panels deform as well, and elastic energy is stored in both the crease or linkage areas and the panel areas; known as deformable origami or kirigami, the complex energy landscape of this type offers more programmability regarding forces, stiffness, and stability.^{6,7,12,24,116}

In this review, we categorize origami/kirigami-based mechanical metamaterials in three groups, as shown in Fig. 1. First, based on the folding or cutting of thin-film materials, mechanical metamaterials based on origami and/or kirigami are divided into three categories: (i) origami-based ones (with folding only), (ii) kirigami-based ones (with cutting only), and (iii) hybrid origami–kirigami-based ones (with both folding and cutting). These three groups of mechanical metamaterials are discussed in Secs. II–IV, respectively. Each group is then subdivided into rigid (energy stored increases or linkages only) and deformable (energy stored in both creases or linkages and panels) based on the elastic energy landscape. For each category of origami and kirigami, we review the deformation mechanisms, design principles, functions, and applications from a mechanical perspective. Finally, in Sec. V we discuss the future directions and challenges of origami- and kirigami-based mechanical metamaterials.

Origami-based mechanical metamaterials are designed by introducing folding creases into thin-film materials. These creases allow rotation of nearby panels and affect the elastic energy during folding and unfolding, thus origami confers programmable kinetic and mechanical properties. Depending on the landscape of elastic energy during the folding and unfolding processes, origami-based mechanical metamaterials can be divided into rigid ones and deformable ones.

Rigid origami (also known as rigid foldable origami) involves folding and unfolding without deforming the origami panels. There are three requirements for origami to be rigid: (i) the pattern must be mathematically foldable;²⁶⁷ (ii) the panels must be much stiffer than the creases; and (iii) the material should be under uniform loading along the folding/unfolding directions instead of under concentrated, bending, or twisting loads. In rigid origami, elastic energy (or strain) is stored only in the folding creases, so the energy can be expressed in terms of only the changes of the dihedral angles of the creases.

Depending on the transformation mode, rigid origami can be divided into one-DOF (i.e., degree of freedom) and multi-DOF types. The DOF of rigid origami depends on the number of creases at each vertex;²⁶⁷ for example, rigid origami with four creases per vertex has only one DOF. The best-known one-DOF rigid origami is the Miura pattern, which has an elegant geometry and unique mechanical properties.^4,61,88 Assembled Miura-based mechanical metamaterials can be formed by assembling multiple layers of Miura origami and have more-complex geometries and performances.^{50,54,66,78,88} Multi-DOF origamis have more than four creases per vertex, for example, Ron Resch and waterbomb origamis.^{4,22,49,62,64} Multi-DOF rigid origami is more flexible than one-DOF origami.

As a periodic parallelogram pattern, the Miura pattern was designed by Koryo Miura in the 1970s.⁹⁰ As shown in Fig. 2(a),⁴ the geometry of Miura origami can be determined by the lengths a and b and the plane angle β, and the folding state of Miura origami can be determined by the folding angle ϕ.^4,61,88 All the other geometric variables during folding (e.g., the dihedral angles) can be expressed in terms of these four parameters.^4,88

The Miura pattern has many unique yet intrinsic mechanical properties during folding and unfolding, such as tunable Poisson's ratio, panel directions, and stiffness.^4,88 These behaviors of the Miura pattern have been harnessed to make flexible lithium-ion batteries,¹⁶ artificial muscles,²² heat-dissipation enhancement,⁷³ and solar cells.²⁶⁸ The Poisson's ratio of Miura origami as a function of geometry and folding state is plotted in Fig. 2(a). Depending on the geometry, the Poisson's ratio can be either negative, positive, or in transition between negative and positive during folding. More interestingly, Miura origami exhibits bistability when the geometrical and mechanical parameters are in a specific range, thereby conferring efficient deployment.⁴⁶ By changing the folding direction of Miura origami between upward and downward, different chirality can be tuned for electromagnetic behaviors.⁴⁴ Similar functions of Miura origami are used to control electromagnetic waves,^32,85 deflect light,⁷⁰ and reduce radar cross section.⁸⁶ Miura origami also has interesting dynamic properties. By analyzing and designing the transformation dynamics of Miura origami, a branching Miura origami structure can have 17 distinct configurations activated by different dynamic inputs of a single actuator.⁴² 

The mechanical properties of Miura origami can be programmed by modifying the creases. In Fig. 2(b), by changing the material distribution at the creases and panels, a metamaterial based on Miura origami can have a tunable coefficient of thermal expansion (CTE), from negative to zero to positive, with potential applications in aerospace and optics.⁸⁰ Stretchable materials are also used at the creases of Miura origami to realize dual stiffness, which has been applied in robotics for rapid and robust gripping.²¹ Moreover, photoactive materials and four-dimensional printing methods are applied to control the deformation of creases and realize self-folding Miura origami.⁴³ Thin creases design are realized via three-dimensional (3D) printing, thereby enabling Miura origami to confer either loadbearing or flexibility properties.⁵⁷ Meanwhile, for enhanced panel rigidity for better application in engineering structures, origami with thick panels has been designed and the related theories have been developed.^27,91

By modifying the geometry of Miura origami facets from periodic parallelograms to aperiodic general quadrilaterals, complex surfaces with designed curvature can be created. The planar and folded states of modified Miura origami are shown in Fig. 2(c).^5,45 The modified Miura origami tessellations can be folded into complex curvatures, such as cylinders or spheres.⁷⁷ Optimization algorithms have also been developed for designing the 3D shapes of folded Miura origami.^26,39 Modified Miura origami strings are used in robot hands for programmed motions.⁵¹ 

Although Miura origami has many interesting mechanical properties, the original single-layer Miura origami can only be folded as a plane or a shell, which somehow limits its applications. To develop metamaterials that can deploy in 3D, the original Miura origamis are assembled in different ways to form 3D metamaterials, while keeping their original mechanical properties, for example, negative Poisson's ratio. The most straightforward assembly method is to connect identical Miura origami sheets layer by layer and form cellular materials.⁵³ Layer-stacked Miura origami-based mechanical metamaterials are flexible in the in-plane directions but stiff along the stacking direction. These simply assembled Miura origamis can be used as sandwich cores for anti-blast structures and as frequency-selective surfaces.^34,83

Miura origami sheets with different geometries and layers can be assembled and have more interesting and nonlinear mechanical properties. Assembled Miura-origami-based metamaterials with “ABAB” stacking mode have been designed that enable self-locking behavior due to their mathematical nonflat foldability.⁸⁸ A more general case is shown in Fig. 2(d), where the unit cell of the Miura pattern is modified into a collinear pattern consisting of four different parallelograms.⁶⁶ The assembled metamaterials have even more programmability with multi-stage nonlinear force responses. The stacking of Miura origami has been widely used for graded stiffness.^65,68,82 The assembled origami has nested-in and bulged-out design with bistability²⁸ and can be used as a mechanical dianode,⁶⁹ as well as metamaterials with tunable bandgaps.⁸⁴ The bistability can also be realized by coupling the elasticity of origami with magnetics.⁵² Inspired by the fluid mechanism of plants, Miura origami tubes can be assembled with pressurized air pouches inside for tunable stability,⁴⁰ tunable stiffness,^78,269 and vibration isolation.^33,41

Using nonidentical parallelogram patterns instead of the original Miura pattern, tubular origami structures can be designed with reconfigurable polygonal cross sections.^31,67 Using polygonal cross-sectioned origami tubes as building blocks, cellular origami metamaterials with either foldable or self-locking properties can be achieved.^35,37 Another family of Miura-derived polygonal tubes is based on the Tachi–Miura polyhedron (TMP), fabricated by connecting two identical papers with different patterns.⁹² The TMP structures have negative Poisson's ratio and bistability between concave and convex configurations.⁵⁴ Metamaterials assembled by TMPs provide a platform for forming rarefaction waves because of their controllable strain-softening behavior.^55,81 Moreover, TMP structures can be tuned between a collapsible state and a load-bearing state.⁹³ 

Other than changing the Miura origami pattern in each layer, the manners of assembly play an even more important role in tuning the stiffness of Miura origami. As shown in Fig. 2(e), by assembling two stripes of Miura origami together, a Miura tube can be made.^9,72 A “zipper” Miura origami tube made of two perpendicular tubes has much higher bending stiffness than that of the original Miura origami.⁵⁰ This zipper tube design provides Miura origami tubes and tessellations with much higher stiffness and so is excellent for load bearing and rapid deployment. Moreover, the metamaterials have tunable stiffness in about four orders of magnitude by changing the loading direction and folding ratio. Zipper-coupled Miura origami tubes are also fabricated using 3D direct laser writing fabrication, thereby achieving tunable stiffness and deployability at the microscale.¹⁵ 

More DOFs are desired in some applications, for example, multi-state shape morphing. When there are more than four creases at one vertex, rigid origami has multiple DOFs. A well-known multi-DOF origami is Ron Resch origami, which can be folded in different modes. As shown in Fig. 2(f), the planar folding pattern of Ron Resch consists of periodic triangle cells. The fully folded state of Ron Resch origami is a flat plate that has remarkable load-bearing capacity because of its sixfold structure.⁴ When forming a tubular structure, Ron Resch origami still has good load-bearing ability for axial loads because of its negative Poisson's ratio and self-locking phenomenon. Based on these properties, Ron Resch-inspired structures have been designed for impact protection.^58,270

Another well-known multi-DOF origami is waterbomb origami (also known as magic-ball origami). The unit cell of the waterbomb pattern is shown in Fig. 2(g), with six creases on each vertex. The unit cell of waterbomb origami has bistability and can represent binary states 0 and 1 by upward and downward folding, respectively.^63,64,76 Therefore, the tessellation of waterbomb origami can be designed for mechanical logistic calculation⁶⁴ in which humid sensitive materials are used for the creases to enable humidity-responsive computation. The binary states 0 and 1 are two locally stable states, since the strain energy of the origami unit cell that is only stored in origami creases due to the properties of the rigid origami has local minimum at both states. The periodic folding pattern of waterbomb origami at planar state is shown in Fig. 2(h). Waterbomb origami has also been used for tunable robotic wheels to adapt to different environments;⁴⁷ wheels made using waterbomb origami can change radius because of the rich DOFs of the latter. Integrated with a pneumatic system, waterbomb origami has also been designed into a soft gripper that can grip different objects by shape transformation.²² The radius-changing property of waterbomb origami has also been harnessed to design crawling robots.³⁸ Under symmetric folding, the number of DOFs of waterbomb origami reduces to one, and it can be used in deployable structures with predictable kinetics.⁸⁹ Waterbomb origami has been used as a building block to design metamaterials with programmable stiffness and deformation,^49,56,59,62 and topology optimization has been performed on waterbomb-like origami to design actuators with specific movement.³⁰ By adding a single quadrilateral face to the unit cell, multi-DOF origami can become one-DOF origami and is easier to control.⁹⁴ 

Deformable origami involves storing energy in both creases and panels during folding, so it confers a complex energy landscape and mechanical performances. Patterns of deformable origami include rigid-foldable patterns, Kresling/Yoshimura patterns, and curved patterns, among others. In rigid-foldable origami, although the panels remain rigid during folding in theory, they may still deform in reality. Therefore, these patterns are still categorized as deformable origami and specifically deformable rigid origami. For the other patterns that are not mathematically foldable, panels must undergo deformation to be folded.

In practice, during folding and unfolding, theoretically rigid origami will introduce deformation in the panels, especially when they are flexible. Consequently, these deformable rigid origami patterns have more DOFs and unusual behaviors. For example, Miura origami made of paper can be easily bent or twisted, although these deformations are not theoretically allowed by rigid-body kinetics.^61,101 Consequently, the deformation energy of the panels is as important as the energy stored in the creases when analyzing the mechanical responses of deformable rigid origami structures. As shown in Fig. 3(a), to analyze the stiffness of Miura tubes under bending, eigenvalue analysis is performed by considering the deformation of both creases and panels. Zipper-coupled Miura tubes have much higher eigenvalues of bending and so are stiffer under bending.^50,130 A bar-and-hinge model can efficiently capture the elastic energy of both creases and panels to simulate the mechanics of deformable Miura patterns.^102,110,130

Deformable rigid origami patterns enable many applications. Considering the energy absorption of both crease folding and panel bending, deformable rigid origami has been used for impact energy absorption.^8,23,36,128 Deformable Miura origami also allows flexible and nonuniform deformation that is effective for protecting robots from rotary collisions.²³ The elastic behavior of panels also enables rigid origami tessellations to work as acoustic metamaterials.¹²⁷ The nonrigid deformation of Miura origami has been harnessed for complex shape morphing by using distributed actuators.^129,135 With some panels being floppy while others are rigid, the designed metamaterials can have a certain number of DOFs and so can be used to store information.⁹⁵ 

Imperfection can be introduced artificially and locally into Miura origami panels to reprogram their mechanical properties. As shown in Fig. 3(b), pop-up defects have been introduced at the vertices of Miura origami to change the latter into defected stable states.¹¹⁶ Defects on vertices can improve the in-plane stiffness of Miura-origami-based metamaterials. By tuning the number and locations of the defects, the stiffness response of the metamaterial can be reprogrammed in situ. It has been discovered that small imperfections and modifications in rigid origami can influence the mechanical properties considerably.^105,109 Topological principles have also been used to understand and design deformable-origami-based mechanical metamaterials.¹⁰³ 

Kresling origami is folded to a cylindrical shape with triangulated unit cells. The folding pattern is shown in Fig. 3(c), where the parameters n, α, and β determine the geometry as well as the mechanical performance of the Kresling origami. To satisfy the flat-foldable condition, the smallest angle of the triangular unit cell is π/n, where n is the number of edges per circle.²⁷¹ By modifying the unit-cell geometry (α and β), Kresling origami can become either stiff or flexible. As shown in Fig. 3(c), flexible Kresling origami (left) can be fully collapsed from its deployed state and exhibits bistability, whereas diamond origami (or Yoshimura origami, which can be modified from the Kresling pattern) is stiff and not foldable.⁷ The differences between Kresling and diamond origamis can be analyzed using a truss-based model.^7,111,117 From the perspective of energy, Kresling origami has double-well energy and an in-plane strain of less than 2% [as shown in Fig. 3(c)]. The vanishing strain energy at initial and deployed states means that Kresling origami has zero strain at both states. For the other configuration of Kresling origami (α = β = 50°), the elastic energy under compression increases monotonically and the in-plane strain is much higher. Compared with diamond origami, Kresling origami has more unusual mechanical properties, including bistability, tunable stiffness, and coupled compressing-twisting deformation.

One of the most interesting properties of Kresling origami is tunable stiffness. As shown in Fig. 3(c), by combining the flexible Kresling pattern and the rigid diamond pattern, truss-based mechanical metamaterials have been designed for on-demand deployability and tunable stiffness.⁷ In the designed origami-inspired metamaterials, two paths can be triggered selectively by direct compressing or twisting-compression, corresponding to a collapsible state with low stiffness or a noncollapsible state with high stiffness. The two different collapsing paths can be visualized by an energy landscape varying with deploying state and rotating state. The metamaterial deploys along the energy valley, where the normalized energy has local minimum around 0.1. There are two possible paths for collapsing. Without twisting, the metamaterial experiences a load-bearing path where the normalized energy is around 1000. If twist is applied before compressing, the metamaterial enters an easy-collapsing state, which is the same as the deploying path, and the energy is around 0.1. Four orders of magnitude differences of the strain energy during deploying and collapsing explain the mechanical responses of the metamaterial. In the load-bearing state, the prototyped metamaterials can hold 1600 times their own weight. As shown in Fig. 3(d), a series of Kresling-origami-inspired structures has been assembled with decreasing stiffness under compression loading. When an impact load is imposed on the left side, the strain field exhibits a rarefaction wave (i.e., tensile wave) traveling from left to right. This unique behavior of the designed Kresling structures can be used in reusable impact-mitigating systems. By tuning the compressive stiffness, Kresling origami can also be used for the transmission of rarefaction solitary waves,⁹⁷ tunable frequency bands,¹⁰⁴ tunable dynamic behaviors,¹⁰⁷ and vibration isolation.¹²⁰ 

The bistability and compression-twist coupling of Kresling origami have been explored for other applications, such as mechanical memory operation,^98,117 binary switches,⁹⁶ haptic feedback mechanisms,¹³³ and crawling robots.^123,139 As shown in Fig. 3(d), Kresling origami has been coupled with magnetism to change the mechanical information remotely.⁹⁸ The coupling between rotation and compression in Kresling origami allows the magnetic field to control the origami structure remotely. A Kresling origami unit cell can be triggered into binary states 0 and 1 by rotating the magnetic field. By connecting multiple magnetic Kresling origamis in series, the designed metamaterial can store binary information and perform digital computation via its multi-stable states.

Curved origami is origami with curved creases and/or panels. Curved origami has a more elegant geometry compared to the corresponding straight-crease origami. As shown in Fig. 4(a), curved-crease origami has one crease and two panels, compared with similar Miura origami with four creases and four panels.⁶ Moreover, a simple circular strip with a curved crease can be folded into unusual buckled and symmetric shapes.¹³⁴ Because of its elegant geometry, curved origami has been used for face shields and generating curved surfaces.^20,136

Unlike in the aforementioned deformable origami patterns, the bending energy in the curved panels makes an important contribution to the mechanical properties of curved origami. Recently, curved origami has been designed for the in situ manipulation of mechanical stiffness. As shown in Fig. 4(a), by changing the curvature of the curved creases, the stiffness of curved origami can be tuned from negative to zero to positive.⁶ The crease folding in the designed curved origami provides negative stiffness, while the panel bending provides positive stiffness. Therefore, via the competing folding and bending deformations, the stiffness of curved origami can be elegantly tuned. The energy and force variation of the curved origami with negative stiffness is shown on the right side of Fig. 4(a). The energy of crease folding and panel bending (and stretching) both contributes to the mechanical response of the curved origami. However, the strain energy of crease folding contributes to the negative stiffness, and the strain energy of panel bending leads to the positive stiffness. At small deformation, crease folding and panel bending have similar contribution to the total energy, while at larger deformation the panel bending contributes more than 90% to the total energy. The balance between crease folding and panel bending results in the different stiffness of the curved origami, which can be realized by tuning the normalized curvature and folding modulus of the curved creases. Because the deformation of panel bending is totally elastic, it also resolves the issue of plastic hysteresis on origami creases.⁶ The designed curved origami can be applied to robotics and many other fields. In deployable and reconfigurable structures, the coupling relationship between crease folding and panel bending has been harnessed to design self-folding curved origami, where the bending of panels also folds the curved creases.¹²⁶ A pre-cut curved crease pattern on a tube can be used to control the buckling mode and so thus improve the load-bearing capacity.¹³⁷ The bending stiffness of curved origami can be strengthened by corrugation,^138,140 with application for reinforcing lightweight structures.¹¹⁵ 

Curved-panel origami is origami with curved panels but straight creases. As shown in Fig. 4(b), curved-panel origami has been designed based on inspiration from the ladybird beetle for fast actuation.²⁴ The undeformed configuration of the curved-panel is shown on the left side, where the origami consists of a curved compliant facet and a folding crease. At a small folding angle, the curved-panel origami has high positive bending stiffness and high bending moment, leading to the self-locking property; when the folding angle is larger than the critical angle (around 10°), it has negative stiffness and low bending moment, leading to the flexible and snap-through collapsing behavior. The prototyped curve-panel origami has a self-locking moment of about 0.02 N m, which is about 20 times higher than its folding moment. When released in a folded state, the stored bending energy can quickly become dynamic energy, resulting in rapid deployment and locking functions. Curved-panel origami has also been used in designs such as those for solar deployment systems, where smooth panels are preferred rather than discrete creases.¹³² 

Other deformable origami patterns include square-twist origami and hypar bistable origami. The folding crease pattern of the square-twist origami is shown on the left top side of Fig. 4(c), and folded and deployed configurations are shown at the left bottom. It has zero DOFs when considered as rigid origami, but it has a twisting DOF when panel bending is allowed.¹² Square-twist origami exhibits bistable and bifurcated behaviors and can be used to build mechanical metamaterials with multi-stability. The reaction force with varying folding ratio is shown at the right upper side Fig. 4(c) for two different α angles. With α = 45°, the negative stiffness of square-twist origami leads to a snap-though phenomenon, while with α = 10°, the square-twist origami has only positive stiffness and no snap-through is observed. The energy of crease folding and panel bending with various folding ratio are both shown at the right bottom side of Fig. 4(c) for different α angles from 10° to 45°. The bending of panel provides negative stiffness, while the folding of creases provides positive stiffness. By changing the angle α, the energy landscape of both the crease bending and the panel folding alters. Larger angle α will result in higher bending energy but lower folding energy. At α  = 45°, the square-twist origami has the most pronounced bistability and negative stiffness. By using thermally reactive materials, metamaterials based on square-twist origami have been designed with five stable states and self-deploying properties.¹²² Another deformable origami is hypar origami with a pattern of concentrically pleated squares. As shown in Fig. 4(d), hypar origami is folded between two saddle-shaped stable states.^100,114 The energy landscape and force distribution are plotted at the bottom of Fig. 4(d). During deformation, the crease folding energy, panel bending energy, and panel stretching energy vary. Among the three energies, the panel stretching energy has the least contribution to the total strain energy (i.e., less than 5%). At the two stable states, crease folding and panel bending have a similar contribution to the strain energy. When the hypar origami is closing to the planar state, folding has larger contribution; when the hypar origami is more bent, panel bending has more contribution. The bistability and negative stiffness of hypar origami is mainly resulted from the crease folding deformation, since the crease folding provides a double-well energy landscape while the panel bending provides much lower energy gaps. Based on those distinct mechanical behaviors, the hypar origami can be used to from multi-stable metasurfaces with non-Euclidean geometries.

Kirigami involves thin-film materials with cuts and linkages between panels, and the mechanical properties of the linkages play an important role in the overall behavior of kirigami. When the linkages are thin and weak, the kirigami panels are more likely to remain rigid and rotate about the linkages; when the linkages are thick and strong, rotation of the kirigami panels is constrained and strain is distributed more evenly over the linkage and panel areas, thereby making the kirigami panels deformable. However, unlike origami, kirigami is not usually categorized by patterns. Therefore, we categorize kirigami structures based on their deformation mechanisms. For example, rigid kirigami is divided into two-dimensional (2D) and 3D types, and deformation kirigami is divided into tension-induced, compression-induced, and self-deforming types based on the cause of deformation.

When the linkages are much smaller than the panels and so are less stiff than the panels, the linkages are treated as mathematical points and the kirigami is considered as rigid kirigami with rigid-body kinetics. As with rigid-origami metamaterials, the main applications of rigid-kirigami ones are shape morphing, and based on this application, rigid kirigami can be divided into 2D and 3D rigid kirigami, which transforms into 2D and 3D shapes, respectively.

Two-dimensional rigid kirigami transforms in a 2D plane, and its basic design requirements are (i) the compact pattern in the initial state and (ii) zero Gaussian curvature in the deployed state.¹⁴⁴ The reverse design method of reconfigurable 2D rigid kirigami is shown in Fig. 5(a). First, a guessed pattern is created in the deployed state, then constraint equations must be solved. The reverse design method can be used to create 2D kirigami with complex 2D configurations. Theoretically, any 2D shape can be formed by deploying a designed regular kirigami tessellation. Other principles are involved in developing 2D rigid kirigami, including kirigami with double compact states¹⁵² and kirigami designs with stochastic cuts and linkages.¹⁴¹ As shown in Fig. 5(b), hierarchical rigid kirigami is more flexible compared to classical kirigami structures.¹⁴⁵ Metamaterials based on two-level hierarchical kirigami have 156% stretchability and 81% compressibility, thereby greatly expanding the application of kirigami for situations in which extreme flexibility is needed, for example, stretchable electronics and soft robotics. By using active materials, 2D rigid kirigami can be controlled thermally and magnetically.¹⁴³ As shown in Fig. 5(c), 2D rigid kirigami can also be designed with multi-stability.²⁷² Using an elastic material (e.g., latex rubber), kirigami linkages can be made flexible and so can store elastic energy, and kinetic analysis shows that the linkages deform elastically during transformation while the initial and deployed configurations remain undeformed, which explains the bistability. The strain energy landscape is plotted for 2D kirigami with different geometric designs. The double-well energy landscape indicates the bistability and negative stiffness behaviors. By modifying the pattern, kirigami-based metamaterials can also be mono-stable with thinner and more-flexible linkages. Through rational optimization of the geometry and materials, kirigami-based multi-stable metamaterials are made more durable and can survive under 10 000 loading cycles.¹⁵³ 

The deployed configuration of rigid kirigami depends on the facet geometry and the linkage thickness. As shown in Fig. 5(d), kirigami can also be designed to deploy in 3D space, for which a reverse design method similar to that for 2D rigid kirigami is used.¹⁴⁴ The deformation mode of kirigami depends on the unit-cell geometry and the kirigami thickness and linkage width. Figure 5(e) shows a 3D rigid kirigami design combined with inflatables to make 3D shapes with pneumatic actuation.¹⁴⁶ The shape can be programmed by axial and circumferential strains, which are determined by the geometric ratio of the unit cells.

Rigid kirigami can also buckle into a 3D configuration without panel deformation. As shown in Fig. 5(f), when the linkage thickness is less than a critical value, the kirigami buckles in the out-of-plane direction, while the kirigami panels remain rigid.¹⁴⁸ Kirigami-based hierarchical mechanical metamaterials have been designed with two levels of kirigami with different buckling loads and so have multi-stage stiffness and programmable buckling patterns.¹⁴⁷ Three-dimensional rigid kirigami has been designed into actuators for complex output from simple input.¹⁴² Another application of 3D rigid kirigami is a kirigami-based stretchable Li-ion battery,¹⁷ which can undergo stretching, bending, and twisting deformations, while the kirigami panels remain undeformed and thus protected.

In some conditions, kirigami structures cannot deform with rigid-body motion, and bending strain exists in the kirigami panels. For example, the kirigami pattern in Fig. 6(a) does not allow rigid-body motion and the linkage areas are not sufficiently flexible. Based on the loading and triggering methods, deformable kirigami can be divided into stretch-buckled, compression-buckled, and self-deforming types.

Stretch-buckled kirigami has been used widely for stretchable structures, such as flexible electronics,^{19,160,171,190–192,201,273} mechanical logic,¹⁹⁶ adhesion,^176,198 and graphene kirigami.^14,185 The best-known stretch-buckled kirigami is linear-cut kirigami, which responds interestingly under tensile loading.^{14,19,25,191,201,203,205,274} As shown in Fig. 6(a), the mechanical responses of linear-cut kirigami can be divided into three regions:²⁰⁰ (i) the first is the rigid region with high stretching stiffness in which the kirigami undergoes only in-plane deformation; (ii) when the tensile force reaches a critical value, the kirigami enters the softening region and its panels start to bend and rotate out of plane; and (iii) the third region is another rigid region in which the panels are mainly stretched. Similar multi-region buckling modes exist in other stretch-buckled kirigami patterns, for example, circular, triangular, and trapezoidal patterns.^{25,164,275,276} This unique buckling phenomenon in kirigami relieves stress and protects kirigami-based structures from tensile fracture.²⁰⁴ 

The out-of-plane deformation and rotation of kirigami panels during buckling have been harnessed in many applications. As shown in Fig. 6(b), stretch-buckled kirigami has been designed as a smart skin for crawling soft robots.²⁵ The kirigami panels buckle and rotate under stretching loads, thereby forming 3D anchor structures and increasing the friction of the skin. The stretch-buckling design can also be used to design locomotion robotics requiring minimum actuation. Based on the same mechanism, shoes with pop-up grippers have been designed for efficient and safe walking.²⁷⁶ Other applications using the buckling and rotational behaviors of stretch-buckled kirigami include actuators,^164,197 solar tracking,¹⁶⁶ microwave resonators,¹⁶⁵ and aerodynamic control.¹⁶⁹ 

Stretch-buckled kirigami can be further programmed with more distinct and multi-stage mechanical properties by tuning the buckling process. As shown in Fig. 6(c), the propagation of pop-up buckling can be realized in cylindrical kirigami shells with linear and orthogonal cuts.²⁷⁵ The triggering force, sequence, and locations of the pop-up propagation can be tuned by changing the geometry of the kirigami linkages, and the design can be used to improve the programmability of kirigami-based metamaterials and robotics. As shown in Fig. 6(d), a so-called “kiri-kirigami” has been designed by engraving notches on a classical kirigami pattern to control the buckling direction of the kirigami panels.²⁰³ Without the engraved notches, the buckling direction of the kirigami panels is random and bistable, but with them the kirigami panels always buckle in the same direction as the minor cuts and are no longer bistable. The designed kiri-kirigami metamaterials have on-demand tilting direction and distribution of kirigami panels and so can be used as energy-efficient building skins for reflecting or transmitting sunlight. In other kirigami designs, minor cuts are also used to change the local mechanical responses of stretch-buckled kirigami.¹⁷⁵ 

When a planar kirigami pattern is under compressive load, it buckles in the out-of-plane direction to release the stress.²⁰⁴ The most interesting aspect of compression-buckled kirigami is the transformation from 2D patterns to complex 3D configurations, which is also called mechanically guided 3D assembly.¹⁹⁴ This transformation mechanism has been used to form 3D structures from the nanoscale to the macroscale.²⁰² 

Figure 6(e) shows 2D kirigami membranes being compressed and buckled to determined complex 3D configurations. Before the buckling, the outer squares of the kirigami patterns are attached to a pre-stretched elastomer substrate. By releasing the substrate, compressive stress generates in the kirigami structure and the kirigami buckles out of plane. The deployed shape of the buckled kirigami structure can be predicted by finite-element simulation, and the planar kirigami pattern can be reverse designed.^167,199 Different materials, such as silicone-epoxy bilayers,^162,174,195 shape-memory polymers,^158,159,174 and magnetic materials,¹⁹⁴ are used to make compression-buckled kirigami for applications in microelectromechanical systems (MEMS), robotics, and bio-integrated electronics.^162,181

The transformation of compression-buckled kirigami is also influenced by other factors such as loading sequence,¹⁶⁸ substrate,¹⁸³ and defects.¹⁵⁷ As shown in Fig. 6(f), ring-shaped kirigami can buckle into two different shapes upon different triggering.¹⁶⁸ The kirigami of shape I was formed by releasing in the x and y directions simultaneously, while that of shape II was formed by releasing in the y direction first and then in the x direction. Using this strategy, microstructures can reconfigure between multiple stable states by using different loading sequences and can be applied in reconfigurable microelectronic devices. The substrate design is also important in making compression-buckled kirigami structures. A heterogeneous substrate with soft, medium, and hard materials can be used to form buckled kirigami structures with nonuniform shapes.¹⁸³ 

Similar to self-folding origami, kirigami can also self-deform under specific triggering methods. The basic means by which kirigami self-deforms is a nonuniform stress distribution that can be generated by different methods, such as bilayer materials with different properties, focused ion beams (FIBs),^10,161,172 temperature-responsive materials,^{11,156,180,182} solvent-activated materials,¹⁵⁶ electroactive polymers,^179,188 magnetically activated materials,¹⁷⁸ shape-memory polymers,¹⁷⁷ photoactive materials,¹⁶³ and electrothermal materials.¹⁷³ The applications of self-deforming kirigami include shape morphing,^{11,156,163,180,182} stents,¹⁷⁷ actuators for robotics,^{161,163,178,179,188} MEMS,¹⁷³ tunable optical properties,^10,172 and tunable thermal expansion.¹¹ 

Figure 6(g) shows propeller-shaped and pinwheel-shaped kirigamis made of gold film buckling under the triggering of an FIB.¹⁰ Because of the tensile and compressive stresses introduced by the FIB, the kirigami pattern can be cut by a high-dose FIB and bent by a low-dose FIB. Self-deforming FIB-triggered kirigami can be used to design complex nanoscale structures with precise buckling and twisting transformations, and the fabricated metamaterials have giant optical charity and can be applied in many fields.

As shown in Fig. 6(h), self-deforming-kirigami-inspired metamaterials with different structures can either expand or shrink when heated, showing positive and negative CTEs, respectively.¹¹ In this design, bilayer materials with different CTEs are used for each beam of the kirigami-inspired metamaterial and result in it bending when heated. The bending of the kirigami beams is then converted into linear deformation through the kirigami-inspired structure. With hierarchical design, kirigami-inspired metamaterials can realize a huge range of positive and negative CTEs.

Because origami and kirigami both pertain to thin sheets, they are easily combined. To maximizing the advantages of origami and kirigami, hybrid metamaterials have been created with the coexistence of folding and cutting. Depending on the energy distribution, hybrid origami–kirigami can also be divided into rigid and deformable.

Similar to original rigid origami or kirigami, rigid hybrid origami–kirigami is also for shape morphing, but with higher programmability. Depending on the assembly requirement, hybrid origami–kirigami can be divided into cut-and-fold and assembled.

Cut-and-fold hybrid origami–kirigami metamaterials are made simply by cutting and folding thin materials, but nevertheless this simple method has solved many issues in classical origami and kirigami. The linkage areas in classical rigid kirigami are always made of thin and narrow materials, which are usually fragile and limit the DOFs of rigid kirigami. To solve this problem, kirigami linkages are replaced by origami creases, as shown in Fig. 7(a).²⁴⁵ The designed hybrid metamaterials transform similarly to the original kirigami structures but have more-robust mechanical performances and more DOFs. This combination also introduces the self-folding techniques of origami to kirigami; thus, the hybrid metamaterials are easier to actuate and are used to make self-foldable 3D metastructures and easy-tuning soft robots.^{150,228,237,245} Similar principles are applied to make hybrid metamaterials in many applications, such as deployable structures,^206,221,233 actuators,^236,241 wave manipulation,^225,230 tunable electromagnetic responses,^217,218 and haptic devices.²¹⁰ 

As well as introducing creases into kirigami, kirigami cuts can also be introduced into classical origami structures, for example, Miura origami. As shown in Fig. 7(b), zigzag-shaped hybrid metamaterials have been designed by cutting off some of the creases in Miura origami.²⁴⁸ The hybrid metamaterials have similar kinetics to those of the original Miura origami but have more geometric parameters to change and so have a broader range of applications.²¹³ Kirigami designs are also introduced into origami patterns as stops to change the kinetics, and they can be used in deployable structures.²⁰⁶ 

Hybrid origami–kirigami metamaterials have great potential in shape morphing. Through cutting and folding, thin materials can form honeycomb structures and have applications in wingboxes,²²⁹ sandwich structures,²¹⁵ shape morphing,^223,244 tunable Poisson's ratio,^212,224 blast resistance,²¹⁹ among others. Figure 7(c) shows a thin sheet with both cuts and creases along a honeycomb lattice being transformed into several different complex 3D surfaces following the rules of “lattice kirigami.”²⁰⁹ These rules allow simple 2D sheets with lattice patterns to form 3D surfaces, and algorithms have been developed to solve the reverse problem.²⁵⁰ Allowing different types of lattice and the overlapping of facets, “additive lattice kirigami” is then designed for more-complex geometries.^231,249 The hybrid lattice structures have self-locking properties and so can bear high loads.²⁴⁶ The cut-and-fold strategy can also be used to transform 2D sheets into 3D polyhedrons and curved 3D surfaces.^{207,216,220,234}

In some conditions, the desired structure cannot be transformed from a developable surface, even though both folding and cutting are used, for example, eggbox origami.¹²⁷ To make structures that are not developable, origami and kirigami can be fabricated modularly by cutting, folding, and assembling, and such mechanical metamaterials are known as assembled origami–kirigami-based mechanical metamaterials.

As shown in Fig. 7(d), transformable multi-DOF metamaterials have been made by assembling hybrid origami–kirigami sheets.²²⁶ The designed hybrid metamaterials have multiple DOFs and tunable stiffness, and their shape and connectivity are easily changed because of the modular design. Polyhedral tessellations with more DOFs can be rationally designed following a similar strategy.²²⁷ As shown in Fig. 7(e), the modular metamaterials can decouple the local mechanical properties from the global properties.^238,252 The designed unit cells are assembled to complex 3D geometries that can collapse and deploy. Modular metamaterials with hybrid origami–kirigami can be used for mechanical, acoustic, and electromagnetic wave manipulations,^{208,214,232,243,247} reconfigurable structures,^{211,222,235,239,240} water weirs,²⁴² and tunable thermal expansion.²⁵¹ 

Compared with rigid origami and kirigami, deformable origami and kirigami have more-complex mechanical responses because of the complex energy landscape, and so theoretically, deformable hybrid origami–kirigami should have even-more-interesting mechanical properties. However, there have been only a few studies of this category, and many possibilities for deformable hybrid metamaterials remain to be discovered.

One way to develop deformable hybrid metamaterials is to introduce cuts into deformable origami structures. Figure 8(a) shows Kresling origami modified by cutting off some of the creases.⁹⁸ The modified structure has similar mechanical properties to those of the original Kresling origami, such as bistability and coupled compressing and twisting. However, a difference is that the cuts on Kresling origami offer stress relief and allow durable folding without breaking the panels.^119,263,265

On the other hand, deformable kirigami patterns can be introduced into origami structures. As shown in Fig. 8(b), stretch-buckled kirigami has been used as origami creases for easy self-folding.²⁶⁴ The kirigami design changes the local mechanical behavior; thus, self-folding can be realized without using special materials. In other designs, the kirigami-as-crease strategy has also been harnessed to make mechanical logic gates,²⁵³ and stretch-buckled kirigami has been assembled with foldable origami structures and used as deployable reflectors and metaimplants.^254–256 

We have reviewed the categorizations, transformation mechanisms, functions, and applications of mechanical metamaterials based on origami and kirigami. Here, we provide several directions and challenges for future research work.

First, for novel mechanical properties, new origami and kirigami designs are highly desired, but most origami and kirigami patterns have been well studied, and discovering new patterns requires tremendous work both mathematically and mechanically. The existing library of origami and kirigami provides useful information for designing metamaterials with different applications. Rigid kirigami and origami are relatively straightforward to design, though with plain mechanical responses. Deformable kirigami and origami has complex mechanical responses such as multi-stability and tunable stiffness; however, the patterns are limited. An alternative solution is to combine existing patterns and knowledge from either origami or kirigami to create new hybrid patterns. There are several methods for discovering mechanical properties without discovering new patterns, such as curved origami design, hybrid origami–kirigami design, modular design, and hierarchical design.

Second, the selection of materials for origami- and kirigami-based mechanical metamaterials has been overlooked. Most origami and kirigami metamaterials are prototyped with paper, and their mechanical properties are influenced and limited by the plasticity and fragility of paper. To design origami and kirigami for real-world applications, materials with different properties should be considered, such as thin or thick, soft or hard, elastic, or plastic. The creases of origami and the linkages of kirigami, where stress concentrates, should be designed specifically to prevent fatigue and improve flexibility. A possible solution is hybrid origami–kirigami design in which origami and kirigami can be used to relieve the stress for each other. Another solution involves designs inspired by origami and kirigami, such as truss-based designs,^7,117,256 where origami and kirigami structures are redesigned and replaced by mechanical components including trusses and springs, thereby preventing unnecessary issues.

Third, energy landscape and energy distribution are two powerful tools to analyze mechanical performances of origami and kirigami and should be leveraged in future investigations. Energy landscape is an effective way to visualize and evaluate the mechanical behavior of metamaterials at different deformation configurations. For example, the valleys on energy landscape indicate the stable states, high energy barrier indicates high load-bearing ability, and the slope and curvature of energy indicate the force and stiffness responses, respectively. On the other hand, the energy distribution of the origami and kirigami deformation modes essentially determines the mechanical performances and provides important guidelines to design mechanical metamaterials with different properties. For example, the contributions of crease folding energy and panel bending energy affect the mechanical stiffness and stability in curved origami. However, the deformation modes (i.e., folding, bending, and stretching) have different contributions and functions depending on the origami and kirigami patterns. For example, folding provides negative stiffness in curved origami while contributes positive stiffness in square-twist origami. Therefore, there is no universal guideline to design the deformation modes of origami and kirigami yet. In future works, there are still a lot to discover from the perspective of energy, for example, programmable energy landscape, multi-path energy landscape, and controllable energy distribution.

Finally, to use origami and kirigami in applications such as robotics, medical devices, and deployable structures, the actuation method must be designed carefully. Currently, origami and kirigami structures are actuated by cable-driven, pneumatic, magnetic, photonic, thermal, and chemical methods, but these are not yet perfect and have their own limitations (e.g., low speed, high cost, sensitive to environment, hard to control) and may only work for specific patterns in specific environments. A universal actuation method remains highly desired to actuate different patterns efficiently and robustly.

We acknowledge support from the National Science Foundation (Grant No. CMMI-1762792). The manuscript was submitted when all authors were working at Arizona State University. The revisions were made while L.W. and H.J. worked in China.

The authors have no conflicts to disclose.

Data sharing is not applicable to this article as no new data were created or analyzed in this study.