Using origami design principles to fold reprogrammable mechanical metamaterials

Metamaterials are rapidly emerging at the frontier of scientific and technological innovation due to their exotic and tunable material properties, which arise from arrangements of smaller units within the bulk system (1–5). Once fabricated, structural modifications are generally difficult, making it rare to find metamaterials that can be reconfigured beyond their original design. Origami-inspired mechanical metamaterials offer enhanced flexibility in performance because their properties are coupled to a dynamically alterable folding pattern (6–16). Hence, multiple stable configurations can manifest from a single structure yielding programmable metamaterials.

We studied the Miura-ori tessellated folding pattern, which has recently been proposed as an origami metamaterial (Fig. 1, A to C, figs. S1 to S3, and movie S1) (14, 15). Historically, this design was invented to efficiently pack solar panels for space missions (17), but the morphology also naturally occurs in leaves (18) and embryonic intestine (19, 20) and generally arises when thin sheets tethered to a surface undergo biaxial compression. The geometry of a Miura-ori is a herringbone pattern that emerges from a series of convex mountain and concave valley creases (Fig. 1C). Vertices are formed when four creases intersect, and four adjacent vertices bound equal-area facets arranged with inversion symmetry (Fig. 1B). This fold pattern defines a lattice characterized by two static crease lengths , and one static plane angle α. To quantify folding, a vertex angle ε is required (Fig. 1C); when ε = 180° – 2α, the structure is maximally contracted into a folded state, and when ε = 180°, the structure is an unfolded flat sheet.

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If folded from an ideal material with infinite stretching modulus, the Miura-ori would have only one degree of freedom described by ε (14). However, generating samples from laser-cut sheets of paper and mylar (Fig. 1C) (see supplementary materials for details), we find additional degrees of freedom, as evidenced by soft bending modes (14, 15). As an extreme example, the Miura-ori supports highly localized heterogeneity introduced after folding by applying force to a vertex in the normal direction and popping it into a categorically different mechanically stable state (Fig. 2A, fig. S4A, and movie S2). This pop-through defect (PTD) changes lattice topology by suppressing one fold and bending adjacent facets through an angle ϕ (Fig. 2, B and C). Quantifying lattice distortion by the surface’s Monge patch mean curvature calculated from three-dimensional (3D) digitized scans, we find that displacements of vertices in the xy plane are negligible within a distance of one unit cell; this compares favorably with theoretical predictions that estimate a distortion decay length ≈1/2 the unit cell width (Fig. 2D) (see supplementary materials for details) (21, 22). Although highly localized, PTDs are unlike conventional lattice defects because they involve only elastic facet bending and hence are reversibly removable.

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To determine the robustness of PTDs to mechanical perturbations, we theoretically model a single unit cell with one mountain and three valley folds in mechanical equilibrium (similar to the paper model in Fig. 2, B and C). To account for experimentally observed facet bending, two additional symmetrically placed creases are included whose folding angles ϕ are equal and have equilibrium values of 180° (14, 15). We assume a Hookean energy expression for all creases and calculate the dimensionless energy u as a function of the folding configuration (Fig. 2E and fig. S5) (see supplementary materials for details). We observe that the energy landscape has two distinct regions, each with its own local minimum (Fig. 2E, points I and II), that are connected at a single point corresponding to a fully unfolded unit cell where ϕ = ε = 180°. For materials with a finite stretching modulus, this connecting point widens along the ϕ axis and allows for otherwise forbidden stretching-enabled transitions. Because the stretching modulus is proportional to sheet thickness (23), the energy required to transition between states is a tunable design feature. Thus, while PTDs are introduced reversibly, their stability to mechanical perturbations is controlled by the competition between the material’s bending and stretching energy scales.

The mechanical consequences of a PTD were determined by folding laser-cut structures from 24-lb paper with s² unit cells, where s was varied from 3 to 8 and α was fixed at 60°. A centrally located defect was introduced by hand, and the compressive modulus was measured as a function of defect density n = 1/s² (see supplementary materials for details). Normalizing these data by the modulus of a defect-free lattice, , revealed a divergence in stiffness as a function of compression limited only by tearing of the sheet at the highest strains (Fig. 2F and fig. S4B). This divergence arises from a violation of Maekawa’s theorem, which requires the number of mountain folds minus the number of valley folds at every vertex to be ±2 for a structure to fold flat (24). By linking enhanced mechanical stiffness to violations of a canonical paper-folding theorem, these data experimentally demonstrate how geometric constraints in origami-based design can dominate bulk material properties. Nevertheless, such constraints are still malleable; the value of ε where increases is shifted simply by varying n. Thus, PTDs make attractive elements for metamaterial design because the onset and strength of are rationally tunable, and moreover, these “mechanical pixels” can be activated ondemand to vary the bulk compressive properties. Although PTDs were manually introduced here, soft robotics-based technology (25), microelectromechanical systems actuator arrays (26), or even thermal fluctuations (27), provide alternative avenues to introduce and remove PTDs.

To understand both the density dependence and compression dependence of the Miura-ori with a PTD, we analytically calculated the intrinsic PTD compressive modulus, (see supplementary materials for details). This was found by noting that the stress a PTD exerts is proportional to its extension relative to the neighboring undefected unit cells. At low defect densities for a lattice of arbitrary size, the compressive modulus of a lattice with a PTD can be expressed in a dipole expansion, = + , where is a function of ε averaged over the nondefected unit cells. This expression is valid when n << 1 and holds for a lattice with multiple defects as long as defect interactions can be neglected. Moreover, it suggests a data collapse with a one-parameter fit to the theory that sets the scale of . Indeed, we find such a collapse (Fig. 2F, inset) and excellent agreement with the analytic expression (Fig. 2F, inset, black line), demonstrating that the contribution of PTDs to the modulus at low densities is linearly additive. This mechanical response is useful for design purposes because it offers great simplicity in predicting the compressive modulus of a Miura-ori with PTDs.

With increasing density, PTD interactions become important, particularly when pairs of defects are on adjacent vertices. There are four unique defect pairs: 1–2, 1–3, 1–4, and 2–4, with the other combinations degenerate by symmetry (Fig. 1B and Fig. 3, A to D). Folding 4 by 4 lattices with α = 60°, we measured in the presence of the four unique defect pair configurations. Normalizing by the defect-free modulus, we found that the 1–2, 1–3, and 1–4 defect pair moduli diverge as the compression increases (Fig. 3E and movie S3).

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Surprisingly, the 2–4 configuration has qualitatively different compressive behavior: The modulus first increases during compression similar to other defect-pair configurations, but then decreases, and finally reduces to a near defect-free value at the highest compressions (Fig. 3E). Effectively, the respective mechanical signatures of the 2–4 defect pair are annihilated during compression. To illustrate the folding sequence, we mark the centers of a 3 by 3 sublattice (Fig. 3F, green dots), as well as the location of each defect (Fig. 3F, red and blue dots). During compression, facets near the defect sites are increasingly bent as ε decreases, leading to the enhanced stiffness. This enhancement progresses until defect site 4 folds under defect site 2 and all four creases on the facet flip. The defect sites then collapse onto, and overlap with, nearby vertices (Fig. 3F) so that the center lattice site is tucked underneath the adjacent unit cell (movie S4). Visually inspecting the 2–4 defect pair reveals a notable consequence: This defect configuration generates a lattice vacancy analogous to those seen in crystallographic systems (28).

To understand why the 2–4 defect pair is special, we compute the mean curvature from 3D scans of laser-cut sheets (Fig. 3, A to D). Comparing the four cases, we see that the 2–4 pair is distinguished by overlapping bending facets of equal magnitude but opposite sign, as quantified by the zero mean curvature on the central facet. This cancellation is possible due to the symmetry of the 2–4 placement and the reversibility of a PTD’s deformation of a Miura-ori lattice. When fully compressed, the defect-modified crease pattern is compatible with Maekawa’s theorem for flat-foldability (Fig. 3G and fig. S6), and hence the compressive modulus is restored to a nearly defect-free value.

Recognizing the 2–4 defect pair as a lattice vacancy immediately suggests that the Miura-ori supports other types of crystallographic defects (23). For example, a column of vacancies forms an edge dislocation (Fig. 4A), and several adjacent edge dislocations form a grain boundary (Fig. 4B). In both cases, these reconfigurations of the lattice are flat-foldable; however, non–flat-foldable configurations are possible as well. For example, a column of PTDs on alternating vertices generates a hinge-like structure that allows the Miura-ori to easily bend out of plane (Fig. 4C). Conversely, a column of PTDs on consecutive vertices forms a corrugated structure that is highly resistive to out-of-plane bending (Fig. 4D). In fact, rationally introducing patterns of defects as design elements swiftly generates a vast library of origami-inspired mechanical metamaterials. To demonstrate in experiments the ability to place these features without permanently altering the crease pattern, we programmed eight configurations into a single sheet with 13 consecutive reprogramming events (Fig. 4E) (29). Measuring for a lattice compressed to 68% of its initial width, we found that once calibrated (dashed lines), we are able to predict and realize target modulus values when defect concentration and interactions are altered (solid black lines). This capability to dynamically reprogram elastic properties illustrates the power of our approach.

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Extending programmable metamaterial design principles to self-folding robotic (25, 30–33) and polymer (8–10, 13, 16) systems opens the door to engineering devices that can alter their mechanical functionality on demand. Such enhanced capabilities would move these transforming systems whose function depends on configuration from what was merely science fiction in past decades to real-world applications.