Rational design of reconfigurable prismatic architected materials

Advances in fabrication technologies are enabling the production of architected materials with unprecedented properties. Most such materials are characterized by a fixed geometry, but in the design of some materials it is possible to incorporate internal mechanisms capable of reconfiguring their spatial architecture, and in this way to enable tunable functionality. Inspired by the structural diversity and foldability of the prismatic geometries that can be constructed using the snapology origami technique, here we introduce a robust design strategy based on space-filling tessellations of polyhedra to create three-dimensional reconfigurable materials comprising a periodic assembly of rigid plates and elastic hinges. Guided by numerical analysis and physical prototypes, we systematically explore the mobility of the designed structures and identify a wide range of qualitatively different deformations and internal rearrangements. Given that the underlying principles are scale-independent, our strategy can be applied to the design of the next generation of reconfigurable structures and materials, ranging from metre-scale transformable architectures to nanometre-scale tunable photonic systems.

of the systems. We find that qualitatively different responses can be achieved, including shear, uniform expansion along one or two principal directions, and internal reconfigurations that do not alter the macroscopic shape of the materials. Therefore, this research paves the way for a new class of structures that can tune their shape and function to adapt and even influence their surroundings, bringing origami-inspired metamaterials closer to application.

Design strategy
To design 3D reconfigurable architected materials, we start by selecting a space-filling and periodic assembly of convex polyhedra (Fig. 1). We then perform two operations on the tessellation. (i) We separate adjacent polyhedra while ensuring that the normals of the overlapping faces remain aligned. This can be achieved by imposing that for each overlapping face pair where dp j denotes the displacements applied to the polyhedra to separate the jth pair of faces, and the subscripts a and b indicate to which polyhedron the two overlapping faces belong. Moreover, L j is the distance between the jth pair of faces in the separated state, and n j is the unit normal to the faces pointing outward from the polyhedron indicated by the subscript a. (ii) We extrude the edges of the polyhedra in the direction normal to their faces to form a connected thin-walled structure ( Fig. 1), which we refer to as a prismatic architected material (Supplementary Video 1). Importantly, for the periodic space-filling tessellations considered here, it is sufficient to focus on a unit cell that consists of only a few polyhedra and covers the entire assembly when translated by the three lattice vectors l i 0 (i = 1, 2, 3). While equation (1) can be directly imposed on all internal face pairs in the unit cell, for overlapping faces that are periodically located (that is, lie on the external boundary of the unit cell) the constraint needs to be updated as ,a 0 Advances in fabrication technologies are enabling the production of architected materials with unprecedented properties. Most such materials are characterized by a fixed geometry, but in the design of some materials it is possible to incorporate internal mechanisms capable of reconfiguring their spatial architecture, and in this way to enable tunable functionality. Inspired by the structural diversity and foldability of the prismatic geometries that can be constructed using the snapology origami technique, here we introduce a robust design strategy based on space-filling tessellations of polyhedra to create three-dimensional reconfigurable materials comprising a periodic assembly of rigid plates and elastic hinges. Guided by numerical analysis and physical prototypes, we systematically explore the mobility of the designed structures and identify a wide range of qualitatively different deformations and internal rearrangements. Given that the underlying principles are scale-independent, our strategy can be applied to the design of the next generation of reconfigurable structures and materials, ranging from metre-scale transformable architectures to nanometre-scale tunable photonic systems.

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0 denote the distance between the two periodically located faces in the expanded and initial configuration, respectively, l i being the lattice vectors of the expanded unit cell and α j,i ∈ {− 1, 0, 1}. As shown by equations (1) and (2), for a unit cell with F face pairs the expanded configuration is fully described by F extrusion lengths L j (j = 1,…, F) ( Fig. 1). However, for most unit cells the extrusion lengths cannot all be specified independently owing to the constraints introduced by equations (1) and (2). As a result, each unit cell is characterized by F indep ≤ F independent extrusion lengths as illustrated in Supplementary Fig. 6. For the sake of convenience we chose the F indep independent extrusion lengths to be as close as possible to an average extrusion length L, by solving while ensuring that the constraints imposed by equations (1) and (2) are not violated. Finally, we note that all periodic and space-filling assemblies of convex polyhedra tested in this study were successfully extruded following the proposed design strategy (that is, we always found F indep ≥ 1). As an example, in Fig. 1 we show three prismatic architected materials based on unit cells containing two triangular and one hexagonal prism (Fig. 1a), an octahedron and cuboctahedron (Fig. 1b), and four triangular prisms (Fig. 1c).

Characterizing reconfigurability
Although the aforementioned design strategy represents a robust and efficient approach to construct prismatic architected materials, it does not provide any indication of their reconfigurability. To determine whether, and to what extent, the meso-structure of the designed architected materials can be reshaped, we started by fabricating centimetre-scale prototypes from cardboard and double-sided tape ( Fig. 2a-c), using a stepwise layering and laser-cutting technique (see the 'Methods' subsection of Supplementary Information) 26,27 .
Focusing on the three architected materials shown in Fig. 1, we find that the structure based on triangular prisms and the one based on a combination of triangular and hexagonal prisms can be reconfigured by bending the edges and without deforming the faces, and are respectively characterized by one and two deformation modes (Fig. 2d, e and Supplementary Video 2). In contrast, the material based on a combination of octahedra and cuboctahedra is completely rigid ( Fig. 2b and Supplementary Video 2). Furthermore, our experiments reveal that these architected materials have fewer degrees of freedom than their constituent individual extruded polyhedra ( Supplementary Fig. 7), indicating that the additional constraints introduced by the connections between the polyhedra effectively reduce their reconfigurability.

Numerical algorithm
While the examples of Fig. 2a-e illustrate the potential of our strategy to design reconfigurable architected materials, they also show that the design of systems with specific behaviour is not straightforward. To improve our understanding of the reconfigurability of the proposed architected materials, we implemented a numerical algorithm that predicts their mobility and corresponding deformation modes. In our numerical analysis, each extruded unit cell is modelled as a set of rigid faces connected by linear torsional springs, with periodic boundary conditions applied to the vertices located on the boundaries. To characterize the mobility of the structure we solved the following eigenproblem ω = Figure 1 | Design strategy to construct 3D prismatic architected materials. Space-filling and periodic assemblies of convex polyhedra are used as templates to construct prismatic architected materials (Supplementary Video 1). After selecting a space-filling tessellation, we focus on a unit cell spanned by the three lattice vectors l i 0 (i = 1, 2, 3) and identify all pairs of overlapping faces. We then separate the polyhedra while ensuring that the normals of all face pairs remain aligned. Finally, we extrude the edges of the polyhedra in the direction normal to their faces to construct the extruded unit cell. Note that the architected material can be constructed by tessellating the extruded unit cell along the three new lattice vectors l i . Using this approach, we designed three architected materials that are based on space-filling tessellations comprising triangular prisms and hexagonal prisms (a), octahedra and cuboctahedra (b) and triangular prisms (c).

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3D prismatic architected materials with rigid faces' subsection of Supplementary Information). Figure 2f, g shows the simulated eigenmodes for the two reconfigurable architected materials considered in Fig. 1a, c. Although the simulations predict only the deformation for small rotations, the modes are strikingly similar to the deformation observed in the experiments (Fig. 2d, e). Solving the aforementioned eigenproblem therefore provides a convenient approach to determine the mobility of the structures and gives insight into their deformation without the need for specific boundary conditions.

Designs based on uniform tessellations
To further explore the potential of prismatic architected materials, and to establish relations between their reconfigurability and the initial space-filling polyhedral assembly, we next focus on extruded materials based on the 28 uniform tessellations of the 3D space, which comprise regular polyhedra, semiregular polyhedra and semiregular prisms [28][29][30] . Owing to their relative simplicity, these uniform templates provide a convenient starting point to explore the design space.
Using the numerical algorithm, we first determined the number of degrees of freedom, n dof , of the resulting 28 architected materials ( Supplementary Fig. 9). We find that the mobility of the unit cells is affected by two parameters: the average connectivity of the unit cell, , and the average number of modes of the individual polyhedra, = ∑ = n n P p P p 1 1 , where P is the number of polyhedra in the unit cell and z p and n p are the number of extruded faces and modes of the pth polyhedron, respectively ( Supplementary Fig. 8). The results for the 28 architected materials reported in Fig. 3 show three key features. First, higher values for z lead to rigid materials (that is, n dof = 0 for z > 8). Second, if all the constituent extruded polyhedra are rigid (that is, = n 0), the resulting architected material is rigid as well. Third, only 13 of the 28 designs are reconfigurable (that is, n dof > 0).
Interestingly, we find that all of the 13 reconfigurable structures are based on unit cells comprising only prisms, such that they recover the relation previously demonstrated for extruded individual prisms, = − n z 3 (ref. 31). Moreover, our results indicate that most of the reconfigurable structures are characterized by fewer degrees of freedom than the constituent individual polyhedra (that is, Having determined the number of modes for the 28 architected materials, we next characterize the macroscopic deformation associated to each of them. More specifically, we determine the macroscopic volumetric strain δ = ∑ = ε j j 1 3 for each mode, where ε j are the macroscopic principal strains (see the 'Mode analysis for 3D prismatic architected materials with rigid faces' subsection of Supplementary  Information). Interestingly, we find that for the 13 reconfigurable architectures all modes are characterized by δ = 0, which indicates pure macroscopic shearing deformation, as also confirmed by visual inspection of the modes (Supplementary Fig. 9).
To characterize the reconfigurability of prismatic architected materials, so far we had assumed the faces to be completely rigid and   Supplementary Fig. 9.

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the hinges to act as linear torsional springs. However, fabrication will always result in deformable faces, raising the question of whether prismatic architected materials can be reconfigured when their faces are deformable. To explore this direction, we updated our numerical algorithm by introducing a set of springs to account for the deformability of the faces 12,13,21 (see the 'Stiffness of 3D prismatic architected materials with deformable faces' subsection of Supplementary Information). We then deformed the extruded unit cells uniaxially and investigated their macroscopic stiffness for different loading directions (identified by the two angles γ and θ as shown in Fig. 4).
In Fig. 4 we report the normalized stiffness K/E as a function of γ and θ for four prismatic architected materials characterized by t/L = 0.01, where E is the Young's modulus of the material and t is the thickness of the faces. We find that the response of the architected material based on template number 28, which was previously qualified as rigid (that is, n dof = 0), is fairly isotropic because its stiffness does not vary much as a function of the loading direction (that is, 3.1 × 10 −3 ≤ K/E ≤ 4.0 × 10 −3 ). In contrast, the stiffness of architected materials for which n dof > 0 drops noticeably for specific directions (that is, K min /K max = O(10 −3 )). Interestingly, these are the loading directions for which the reconfiguring modes are activated, as indicated by the deformed structures shown in Fig. 4. Therefore, these results indicate that the deformation modes we found in the limit of rigid faces still persist even when the faces are deformable. We used the same stiffness for bending of the faces and bending of the hinges, and from the results we can therefore conclude that the architecture of these systems makes bending of the faces energetically costly (because it is typically accompanied by stretching and shearing of the faces). Finally, materials characterized by higher n dof are characterized by more 'soft' deformation modes. As such, materials with n dof = 1 seem most promising for the design of reconfigurable architected materials, since they can be reconfigured along a specific direction, while still being able to carry loads in all other directions ( Fig. 4 and Supplementary Fig. 10).

Enhancing the reconfigurability
Although we have shown that by extruding the edges of expanded assemblies of polyhedra we can construct reconfigurable architected materials, our results indicate that the mobility of the resulting structures is strongly reduced by their connectivity. Furthermore, the modes of all reconfigurable designs show a qualitatively similar shearing deformation. To overcome these limitations, we next introduce an additional step in the design strategy and reduce the connectivity of the materials by extruding some of the faces of the unit cell, while making the remaining faces rigid.
As an example, in Fig. 5 we consider the architected material based on a tessellation of truncated octahedra (number 28). When all faces are extruded, = z 14, leaving the structure rigid (that is, n dof = 0). However, by making 8 of the 14 faces rigid instead of extruding them ( Fig. 5a and Supplementary Video 3) we can reduce the connectivity to = z 6 and the resulting architected material is no longer rigid, because n dof = 1. As shown in Fig. 5b and Supplementary Video 3, this response was also confirmed experimentally. Finally, we note that by varying the face pairs in the unit cell that are made rigid instead of extruded, a total of 2 F = 128 different architected materials can be designed using the truncated octahedra as a template. However, only 82 combinations are possible (as all the other cases will result in structures with disconnected parts) and of those designs only four are reconfigurable. Owing to symmetries in the truncated octahedron, these four configurations are identical to the one shown in Fig. 5.
Next, to determine the range of deformations that can be achieved in the proposed structures, we apply the same brute force strategy to the other 27 uniform space-filling tessellations depicted in Fig. 3. For this study we considered a maximum of 2 16 designs per tessellation, randomly selected from the 2 F possibilities, so that for 11 of the tessellations (numbers 4, 5, 9, 10, 16, 17, 20, 21, 23, 25 and 27) the results are not complete, but rather indicate a trend. We expanded the number of possible designs by removing the polyhedra for which all faces have been made rigid from the extruded unit cell, because those would have resulted in rigid parts completely disconnected from the architected materials.
Of the approximately 0.6 × 10 6 connected designs investigated here (Supplementary Table 1), 90% are rigid (that is, n dof = 0) while the other 10% are reconfigurable (that is, n dof > 0). Supplementary Fig. 11a, b shows that to achieve reconfigurability we still need ≤ z 8, with the exception of six designs based on number 5 for which = z 9 (see Supplementary Fig. 12). Moreover, fully extruded architected materials characterized by = n 0 always remain rigid, independent of the reduced number of connections. Finally, and perhaps more importantly, we also find that using this design approach the mobility of the architected materials can be greatly enhanced, as 0 ≤ n dof ≤ 16 and for many of the structures > n n dof (Supplementary Table 1). Inspection of the modes also reveals that a variety of qualitatively different types of deformation can be achieved. To characterize them better, in Fig. 6 and Supplementary Fig. 11c-f we report the magnitude To determine the stiffness in all loading directions, the architected materials are rotated by angles γ and θ before loading. In each contour plot we indicate the minimum and maximum stiffness with white and black squares, respectively. We also show the deformed architected materials for the minimum and maximum stiffness direction. Note that the deformation is magnified to facilitate visualization.

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of the principal strains, = ∑ = ε ε i i 1 3 2 , versus the volumetric strain, δ, for each deformation mode observed in the reconfigurable architected materials investigated here. Interestingly, we find that for many modes δ = = ε 0. These modes do not alter the global shape of the structure, but result only in internal rearrangements. The design labelled a, shown in Fig. 6, is an example of a structure undergoing such a local deformation. Here, most of the structure is rigid except for one-dimensional tubes that can deform independently. In contrast, the design labelled b is an example in which the whole internal structure is deforming, while maintaining the same macroscopic shape (Supplementary Video 4). Besides these local modes, Fig. 6 also indicates that there are designs capable of achieving types of macroscopic deformation that differ from pure shear (for which δ = 0 and > ε 0). For example, we find that some of the structures are characterized by an effective vanishing strain in two directions (labelled c in Fig. 6). The deformation of such architected materials is characterized by ε 1 ≠ 0 and ε 2 = ε 3 ≈ 0, resulting in δ = ε . Moreover, the results also reveal that there are a variety of structures capable of uniform biaxial expansion (or contraction), for which ε 2 = ε 3 ≠ 0 and ε 1 = 0 and δ = / ε 2 . This deformation mode is exemplified by the design labelled d shown in Fig. 6 (Supplementary  Video 4). Finally, we note that δ = ε 3 corresponds to uniform expansion (or contraction) characterized by ε 1 = ε 2 = ε 3 , and defines a boundary for possible combinations of δ and ε . In fact, none of the designs considered here exhibits this type of deformation.

Discussion and conclusion
In this work we introduced a convenient and robust strategy for the design of reconfigurable architected materials, and explored the design space by considering structures based on the 28 uniform space-filling tessellations of polyhedra. Our study uncovered architected materials with a wide range of qualitatively different responses and degrees of freedom, but many more designs are made possible by using different assemblies of convex polyhedra as templates (including assemblies based on Johnson solids and irregular polyhedra, and assemblies that do not fill space), by considering different extrusion lengths, or by removing faces (instead of making them rigid before the extrusion step). Given these additional possibilities in the design of reconfigurable architected materials, we have made our numerical algorithm, implemented in Matlab, available for download as Supplementary Information, to be used and expanded upon by the community. Finally, we believe that, building on the results presented in this work, prismatic  Fig. 3), we extrude only six of its faces and make the remaining eight faces rigid. Using this approach, the average connectivity is reduced from = z 14 to = z 6 and the resulting structure is no longer rigid, because n dof = 1. b, Experimental validation of the numerical predictions (scale bar, 10 cm). (based on tessellations 24 and 9, respectively) are characterized by δ = = ε 0 and experience internal rearrangements that do not alter their macroscopic shape. The structure labelled c (based on tessellation 16) deforms only in one direction (that is, δ = 4.21, = . ε 4 76), while the structure labelled d (based on tessellation 14) experiences uniform biaxial extension (or contraction) (that is, δ = 2.45, = . ε 1 73). The grey shaded region corresponds to combinations of strains that do not permit deformation.

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architected materials with specific properties may be efficiently identified by combining our numerical algorithm with stochastic optimization algorithms such as genetic algorithms. Such optimization algorithms could prove essential in the design of reconfigurable architected materials capable of handling changing environments or multiple tasks (that will probably lead to pareto optimal solutions).
To realize prismatic architected materials, in this study we used cardboard for the rigid faces and double-sided tape for the hinges. This fabrication process enables the realization of centimetre-scale prototypes (for our models we used L = 35 mm) that closely match the conceptual origami-inspired mechanisms, but real-world applications depend on the ability to efficiently manufacture assemblies comprising a large number of unit cells at different length scales using different fabrication techniques. Taking advantage of recent developments in multi-material additive manufacturing, we also built the proposed architected materials using a stiff material (with Young's modulus E ≈ 1 GPa) for the faces and a soft material (E ≈ 1 MPa) for the hinges (see the 'Methods' section of Supplementary Information). Supplementary Video 5 shows 3D printed models for two designs based on assemblies of truncated octahedra (for both models we used L = 6 mm). Although additional local deformation arises from the finite size of the flexible hinges, the 3D printed structures exhibit the same deformation modes predicted by our numerical analysis and observed in the cardboard prototypes. As such, recent advances in fabrication, including projection micro-stereolithography 7 , two-photon lithography 8,32,33 and 'pop-up' strategies [34][35][36][37][38][39][40] , open up exciting opportunities for miniaturization of the proposed architectures. Our strategy thus enables the design of a new class of reconfigurable systems across a wide range of length scales.
Data availability The Matlab model used to determine the mobility and deformation modes of the prismatic architected materials is provided in Supplementary Information. Other models and datasets generated during and/or analysed during the current study are available from the corresponding author on request.