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# Origami tubes assembled into stiff, yet reconfigurable structures and metamaterials

Edited by David A. Weitz, Harvard University, Cambridge, MA, and approved August 7, 2015 (received for review May 14, 2015)

## Significance

Origami, the ancient art of folding paper, has recently emerged as a method for creating deployable and reconfigurable engineering systems. These systems tend to be flexible because the thin sheets bend and twist easily. We introduce a new method of assembling origami into coupled tubes that can increase the origami stiffness by two orders of magnitude. The new assemblages can deploy through a single flexible motion, but they are substantially stiffer for any other type of bending or twisting movement. This versatility can be used for deployable structures in robotics, aerospace, and architecture. On a smaller scale, assembling thin sheets into these tubular assemblages can create metamaterials that can be deployed, stiffened, and tuned.

## Abstract

Thin sheets have long been known to experience an increase in stiffness when they are bent, buckled, or assembled into smaller interlocking structures. We introduce a unique orientation for coupling rigidly foldable origami tubes in a “zipper” fashion that substantially increases the system stiffness and permits only one flexible deformation mode through which the structure can deploy. The flexible deployment of the tubular structures is permitted by localized bending of the origami along prescribed fold lines. All other deformation modes, such as global bending and twisting of the structural system, are substantially stiffer because the tubular assemblages are overconstrained and the thin sheets become engaged in tension and compression. The zipper-coupled tubes yield an unusually large eigenvalue bandgap that represents the unique difference in stiffness between deformation modes. Furthermore, we couple compatible origami tubes into a variety of cellular assemblages that can enhance mechanical characteristics and geometric versatility, leading to a potential design paradigm for structures and metamaterials that can be deployed, stiffened, and tuned. The enhanced mechanical properties, versatility, and adaptivity of these thin sheet systems can provide practical solutions of varying geometric scales in science and engineering.

- stiff deployable structures
- origami tubes
- rigid origami
- thin sheet assemblages
- reconfigurable metamaterials

Introducing folds into a thin sheet can restrict its boundaries, cause self-interaction, and reduce the effective length for bending and buckling of the material (1⇓⇓–4). These phenomena make thin sheets practical for stiff and lightweight corrugated assemblies (5, 6); however, such systems tend to be static, i.e., functional in only one configuration. For creating dynamic structures, origami has emerged as a practical method in which continuous thin sheet panels (facets) are interconnected by prescribed fold lines (creases). Existing origami patterns and assemblages can easily be deployed; however, they tend to be flexible and need to be braced or locked into a fixed configuration for a high stiffness-to-weight ratio to be achieved (7⇓⇓–10). The zipper-coupled system is different because it is stiff throughout its deployment without having to be locked into a particular configuration.

Origami principles have broad and varied applications, from solar arrays (11) and building façades (12) to robotics (13), mechanisms in stent grafts (14), and DNA-sized boxes (15). The materials and methods used for fabricating, actuating, and assembling these systems can vary greatly with length scale. On the microscale, metallic and polymer films or, more often, layered composites consisting of stiff and flexible materials can be folded by inducing current, heat, or a chemical reaction (16, 17). Large-scale origami structures can be constructed from thickened panels connected by hinges and can be actuated with mechanical forces (11, 18, 19). The kinematic motion, functionality, and mechanical properties of the origami are governed largely by the folding pattern geometry. For example, rigid origami systems are defined as those having a kinematic deformation mode in which movement is concentrated along the fold lines, whereas the panels remain flat (20, 21). Among various rigid folding patterns, the Miura-ori has attracted attention for its folding characteristics (22, 23), elastic stiffness properties beyond rigid folding (24, 25), geometric versatility (26, 27), and intrinsic material-like characteristics (28, 29).

The zipper-coupled tubes introduced here are derived from the Miura-ori pattern and can undergo the same type of rigid kinematic deployment. All other deformations are restrained as they require stretching and shear of the thin sheets. Thus, the structure is light and retains a high stiffness throughout its deployment. It has only one flexible degree of freedom and can be actuated by applying a force at any point (Fig. 1 and Movie S1). To explore unique mechanical properties of the zipper tubes, we introduce concepts of eigenvalue bandgaps and cantilever analyses to the field of origami engineering. Zipper assemblages can be fabricated with a variety of materials and methods. We envision applications of these assemblages will range in size from microscale metamaterials that harness the novel mechanical properties to large-scale deployable systems in engineering and architecture (Movies S2–S4).

This paper is organized as follows. First, the Miura-ori pattern is introduced, and the geometries of three fundamental coupling orientations are discussed. Next, we demonstrate how the system stiffness changes as we assemble two sheets into a tube and then two tubes into the unique zipper-coupled tubes. The fundamental coupling orientations are then studied as deployable cantilevers that can carry perpendicular loads. Next, we discuss cellular assemblages, geometric variations, and practical applications that can be created from coupled tubes, and we conclude with some final remarks.

## Geometric Definitions

A Miura-ori cell consists of four equivalent panels, defined by a height *a*, width *c*, and vertex angle *α* (Fig. 2*A*). For our analytical investigation, we use a cell with *X* direction (Fig. 2*B*). We define the configuration of this sheet as a percentage of extension (a percentage of the maximum extended length *C* and Movie S5). Previously, Miura-ori sheets and tubes have been coupled (or stacked) to create several different folding assemblages (7, 9, 30, 31). For the zipper orientation of tube coupling, one tube is rotated and connected in a zig-zag (zipper) manner (Fig. 2*D* and Movie S1). We explore the mechanical properties of this assemblage and compare it with an aligned-coupled system where tubes are only translated (30, 31) and an internally coupled system consisting of two compatible tubes (7) (Fig. 2 *E* and *F*). Symmetry in these assemblages permits them to be rigid foldable, meaning that they can fold with no bending of the panels, but only the zipper and aligned systems can reach a flat state when deployed to a 100% extension. The range of extension for internally coupled systems is limited because the internal tube reaches a flat configuration and locks the system in place. Geometric parameters of the compatible internal tube *SI Text*, section S5.

## Eigenvalue Analyses

To model the stiffness and mass of origami, we use a bar and hinge approach that provides insight into the structural behavior of the origami assemblages. It captures three fundamental physical behaviors: bending along fold lines, bending of initially flat panels, and stretching and shearing of panels (*SI Text*, section S1 and Fig. S1) (24). The simplicity of the bar and hinge model allows for straightforward implementation and versatility where tubes can be coupled and analyzed as more complex assemblages (see *SI Text*, sections S2 and S5 and Figs. S2 and S3). The model is made scalable and incorporates thickness (*SI Text*, section S3 and Fig. S4, we show that the value of *SI Text*, section S4; Fig. 3; and Figs. S5–S7). The bar and hinge model cannot capture all localized effects, but it is significantly faster and provides a sufficiently high level of accuracy for global origami behaviors.

Stiffness (**K**) and mass matrices (**M**) are used to construct a linear dynamics system of equations *i*th eigenvalue and *A–C*), and representative deformation modes are shown for each structure at *D–F*). Eigenvalues scale linearly with mass and stiffness and therefore comparisons remain scale independent and highlight the influence of coupling on the global system behaviors. The single sheet can bend and twist globally in several directions with little energy input, resulting in low eigenvalues. The distribution of energy for the seventh and eighth deformation modes (Fig. 3*G* and Fig. S7) illustrates that bending occurs in the central panels and folds whereas the remainder of the structure remains unstressed. As expected, the bending energy in the panels is highest at the vertices, where the curvature approaches infinity (1, 37). The ninth mode of the sheet and the seventh mode of the tube represent a rigid folding motion where bending is primarily concentrated in the folds and the panels remain essentially flat throughout the deformation (Fig. 3 *D* and *E*).

For practical purposes, we prefer that *A*). The single and zipper-coupled tubes display a continuous bandgap and do not experience switching of the rigid folding mode (seventh). The eighth mode of the single tube constitutes a “squeezing” deformation, where one end of the structure is folded and the other end is unfolded (Fig. 3*E* and Movie S5). This mode requires bending of folds and panels at the ends of the structure; however, the panels do not stretch or shear, and thus *SI Text*, section S5 and Fig. S8).

However, when coupling the tubes into a zipper assemblage we observe that the structure has a substantially increased bandgap, where *F* and *I*). When actuated at one end, a single, aligned, or internally coupled tube squeezes (Movie S5 and Fig. S8), whereas zipper-coupled tubes expand and contract in a consistent rigid folding motion (Fig. 1 and Movie S1). The zipper-coupled tubes are not prone to the squeezing-type deformation, as it would require differential movement between the tubes and stretching the thin sheet (*SI Text*, section S2 and Fig. S3).

## Structural Cantilever Analyses

The zipper, aligned, and internally coupled tube systems can be applicable as deployable cantilever structures when restrained on one end (Fig. 4). At the supported end all nodes are fully fixed (*X*, *Y*, and *Z* displacements), whereas a total load of 1 is distributed on the other end of the structure (Fig. S9 *D**–**F*). When used as cantilevers, the tubes exhibit behavior similar to that of an I-beam, wherein the second moment of area (or area moment of inertia) is increased by distributing material away from the centroid. The aligned and internally coupled tube systems often experience squeezing-type deformations when loaded on one end, whereas the zipper-coupled tubes experience more uniform bending deformations (Fig. 4*A*). The stiffness of the structures is calculated for the three Cartesian directions for different extensions (Fig. 4 *B–D*). The loads are applied for each configuration, and the resultant displacements (**F** is a force vector. A quantitative stiffness of the system is then calculated as *δ*). The zipper-coupled tubes tend to be stiffer than the other two systems and have a greater stiffness when closer to full extension. The internally coupled tubes tend to be stiff for *Z*-direction loading and when locked into a fixed configuration (i.e., 80% extension). The results for stiffness presented here show the general system behavior and are in consistent units of force per length (e.g., newtons per millimeter). A realistic length scale and elastic modulus can be substituted to find quantitative results for the cantilevers.

The stiffness for different directions of loading, orthogonal to the *X* axis, is investigated, by rotating the load in the *E–G*). The structure is analyzed with the same constraint and load distributions, and only the direction of the load vector (equating to 1) is rotated in the *E–G*).

The fixed end of the cantilevers may be constrained in a different fashion, so that a mechanism is installed to fold and deploy the entire structure (Fig. S9 *G–I*). The mechanism would control the rigid folding mode of the system, thus permitting easy deployment (especially for zipper-coupled tubes—see actuation in Movie S1). When the mechanism is contracted, the structure will deploy, and when it is extended, the structure will fold. When the length of the mechanism is fixed, the cantilever will behave much as if the support is fully fixed.

## Cellular Assemblages

Cellular origami can permit self-assembly of engineered hierarchical materials (5, 38), whose mechanical properties depend on the microstructure geometry. The structural stiffness and energy absorption properties of cellular origami can be optimized to complement and improve naturally occurring materials (39). Zipper-coupled tubes can be integrated with aligned or internal coupling to create layered foldable assemblages (Fig. 5 and Movie S3). Structures that incorporate zipper coupling inherit the large bandgap, while also retaining properties from the other coupling techniques (e.g., space filling from aligned or locking from internal coupling). The zipper/aligned metamaterial studied in Fig. 5 *B–E* has the same parameters as before, except the fold to panel stiffness ratio is set to *F* and *G*). Origami metamaterials created with 3D printing do not fold like traditional origami, but possess novel characteristics such as the single flexible mode of zipper coupling.

We analyze the assemblage in Fig. 5 by applying symmetric uniform forces (summing to 1) on opposing faces of the system and calculate the compression stiffness as *δ* is the mean total displacement in the direction of loading. Because of the zipper geometry, the system is primarily flexible in the *X* direction at lower extensions (0–70%) and in the *Y* direction at higher extensions (70–100%). The peak in the *Y*-direction stiffness (at 96.4% extension) corresponds to a bistable state, where the tube cross section is square, and can transition to a different rhombus, depending on the direction of folding (Movie S3). In addition to the stiffness, the deformation characteristics of the material are also anisotropic. The perceived Poisson’s ratio is negative in the *Y* direction when compressed in *X*, whereas it is positive in the *Z* direction when compressed in *Y* (Fig. 5 *C* and *D*). The structure is substantially stiffer in the *Z* direction, and deformations do not follow the kinematic folding mode.

## Geometric Variations

There are numerous other ways in which rigid foldable tubes can be defined, combined, and coupled. For example, a different vertex angle *α* and panel height *a* may be used with rigid and flat foldability of the system preserved. Zipper coupling can be continued in one direction to create an easily deployable slab structure resistant to out-of-plane loads; potential applications are architectural canopies (Fig. 6*A*), bridges (Fig. 6*B*), solar arrays, or synthetic materials. When the vertex angle is varied for the coupled zipper tubes, we can create arbitrarily curved systems with high out-of-plane stiffness (Fig. 6*A* and Movie S4). The bridge deck uses tubes with a vertex angle of *B* and Movie S2). If the zipper coupling is repeated in more than one direction, the structure will self-interlock during the deployment, creating a stiff array of coupled thin sheets (Fig. 6*C* and Movie S2). The structure does not necessarily need to have a square final cross section (four segments of zipper-coupled tubes), but can be any radially symmetric shape (*SI Text*, section S6).

The cross section of the tubes is not restricted to a quadrilateral shape (30, 31); any polygon with translational symmetry can be used to extrude a rigid, foldable origami tube. The tubes can have any cross section where each segment of the cross section has an equal-length counterpart (or counterparts) that is (are) translationally symmetric. The internal panels of the six-sided polygonal tube (blue in Fig. 7*A* and Movie S2) are defined to conform with the flat and rigid foldability of the system. The direction of the folds in these segments can be reversed (e.g., mountain to valley), thus changing the geometry. Polygonal tubes with more sides can be reconfigured in the same way, with many more variable cross sections. In Fig. 7*B* we show that finite thickness may also be incorporated into the construction of the coupled origami tubes. Using available techniques for thick origami (18, 19), we can create structures of thick panels adjoined with physical hinge elements. With these techniques cost-effective materials (e.g., thin wood panels with metal hinges) can be used to create large structures that can be easily deployed, but possess large global stiffness from the zipper-coupling framework. Finally, the characteristics of origami assemblages can be modified by coupling only specific portions of a tube. The long tube in Fig. 7*C* has zipper coupling only in the middle portion to restrict the global squeezing and bending of the system. The ends remain uncoupled, allowing for a rigid connection to the outside edge while still permitting the system to fold and unfold.

## Concluding Remarks

This paper introduces an approach for coupling origami tubes in a zipper fashion and explores their unique mechanical characteristics. These assemblages engage the thin sheets in tension, compression, and shear for any deformation mode that does not follow the kinematic deployment sequence. The origami tubes can be stacked and coupled into a variety of cellular assemblages that can further enhance the mechanical characteristics and versatility of the systems. Extensions and refinements of this work may explore geometric arrangements that improve the stiffness-to-weight ratio, and impact energy dissipation and other mechanical properties. Further study of the hierarchical system properties with respect to fabrication, scale, and materials will be needed to inform potential applications. As origami becomes more widely used in science and engineering, the coupled tube assemblages will serve as an important component that allows flexible deployment while simultaneously retaining a high global stiffness.

## Materials and Methods

Prototypes of the origami were created for demonstrating the mechanical characteristics and for showing the kinematic folding. The origami assemblages in Movies S1, S2, and S5 and Figs. 1, 6, and 7 are created from perforated paper that is folded and adhered together. The Miura-ori sheets were created from 160-g/m^{2} paper by perforating along the fold lines with cuts of length 0.5 mm spaced evenly at 1-mm intervals. Because each tube cannot be developed from a single flat piece of paper, it is assembled by connecting two Miura-ori sheets (Fig. 2*C* and Movie S5). One of the sheets is constructed with perforated tabs at the edge, which can be folded and attached with standard paper adhesive to a mirror-image Miura-ori sheet. When connecting two tubes into either the zipper or the aligned assemblage, the adjacent facets are adhered together (Fig. 2 *D* and *E* and Movie S1). For internally coupled tubes, the structure is assembled in sequence (Fig. 2*F*) by connecting individual Miura-ori sheets with the joining tabs. The numerous variations of the coupling configurations (Figs. 6 and 7) were also assembled using the same techniques. In Fig. 5 *F* and *G*, we show how additive manufacturing can be used to create cellular metamaterials with characteristics inherited from the zipper tubes. The model in Fig. 5*F* was created with a Keyence AGILISTA 3000 3D printer that uses digital light processing (DLP) technology. The model material is an AR-M2 transparent resin and the support material is an AR-S1 water-soluble resin. The cube has dimensions of 68 *G* was created with an EOS Formiga P 100 selective laser sintering (SLS) system. The model uses a PA 2200 (Polyamide 12) material that is sintered together and does not require support material. This assemblage has dimensions of 102

## SI Text

## S1) Stiffness Calculations

For our analyses and discussion in the main text, we examine the thin sheet origami in a consistent fashion to highlight the behaviors, while staying independent of scale (eigenvalue and static analyses). The eigenvalues (*λ*) have units of

For our analyses we use the following parameters unless otherwise noted. The Miura-ori sheet (Fig. 2*B*) with unit dimensions *SI Text*, section S5 for additional details on internally coupled tubes). A Miura-ori sheet with

Here, we describe the simple bar and hinge method for the numerical modeling of thin sheets in origami systems. A previously established model (24) is used as a basis, and several improvements are incorporated to make the model scalable and isotropic and to incorporate material characteristics (32). We make the stiffness dependent on the material’s thickness (*t*), Young’s modulus (*E*), and Poisson’s ratio (*ν*). The stiffness matrix (**K**) for the origami structure incorporates stiffness parameters for (*i*) panels stretching and shearing (*ii*) panels bending (*iii*) bending along prescribed fold lines (**C**) and Jacobian matrices (*X*, *Y*, and *Z* displacement) and the stiffness matrix is of size (*ρ*. A mass matrix **M** for the entire structure is constructed by distributing one-quarter of the mass of each panel to each of its connecting nodes.

### S1.1) Panel Stretching and Shearing.

In-plane axial and shear stiffness is simulated using the indeterminate bar frame (Fig. S1*A*). A general formulation for bar elements is used with an equilibrium matrix (**A**) relating internal bar forces (**t**) to nodal forces (**f**) as **C**) relating bar nodal displacements (**d**) to bar extensions (**e**) as **e**) to local forces (**t**) as **S1**. The crossed bar frame (Fig. S1*A*) has six bars connected at the four corner nodes of the origami panel. This crossed bar geometry results in the frame behaving as an isotropic panel. The bar stiffness parameters (i.e., components of *X*), vertical (*Y*), and diagonal (*D*) bars, respectively. The height (*H*) and width (*W*) of the panel are taken as an average for skewed panels. To verify the frame model we define a panel with *B*) and a shear patch test (Fig. S1*C*). For tensile loading the system always satisfies the patch test, but for shear loading, the behavior of the model is highly dependent on the chosen Poisson’s ratio. From Eq. **S4**, when a low *ν* is used, the diagonal bars have a low area, and the frame demonstrates a low shear stiffness. The converse is also true, countering the behavior expected in a truly isotropic material. Alternatively, when *ν* is set to *X* direction, and the top of the frame laterally displaces in the direction of loading. When *G* is the shear modulus, defined as

### S1.2) Panel Bending.

Out-of-plane bending of the panel is modeled as an angular constraint between two triangular segments of the panel (1–2–3 and 1–3–4 in Fig. S1*D*). For small displacements, the choice of the diagonal does not affect the kinematics of the system (7). Previous findings (1, 4) show that the bending energy is lower along the shorter length, so we formulate our model assuming that bending occurs along the shorter diagonal (i.e., 1–3 in Fig. S1*D*). An angular constraint, *F*, is formulated based on the dihedral bending angle, *θ*, which can be calculated by using cross and inner products of the vectors **a**, **b**, and **c** from the nodal coordinates of the panel **p**. This constraint is defined as**d** are the displacements of the panel nodes. The second row of Eq. **S1** incorporates panel bending stiffness where each element in the diagonal matrix

We assume that the in-plane stiffness of the thin sheet (e.g., paper) is high enough to prevent bending and buckling at the edge connecting two panels (i.e., at the fold line of a thin sheet). The bending energy of thin sheets increases when the edges of the sheet are restrained. In this case, tensile forces develop over the sheet’s surface, and flexural deformations become restricted to a small area focused at the bending ridge (i.e., the diagonal 1–3 in Fig. S1 *D–F*) (1, 2, 4). This phenomenon occurs with large displacements and the elastic energy of the panel bending scales approximately as *k* is the bending modulus of the sheet, defined as *F* at the fourth corner (Fig. S1*E*). When a small force is applied (*F*, *Top*). For a larger force (*F*, *Bottom*). As discussed in previous literature (1, 4) the large displacement case has a higher stiffness and the rotational hinge formulation in Eq. **S7** would give a realistic result with a factor of *θ* into Eq. **S7** that would scale stiffness between small and large displacement cases. For example, ref. 1 suggests that the total bending energy (and thus stiffness) of the restricted sheet should scale with

### S1.3) Folds Bending.

Folds are modeled in a similar fashion to the bending of panels. Realistic origami behavior does not allow for out-of-plane displacements along fold lines due to the restrictive nature of the perpendicularly oriented sheets. Thus, it is sufficient to use a simplified approach: modeling the origami fold as a rotational hinge along an edge. A schematic of the fold model contains a fold spanning nodes 2 and 3 connecting two panels (1–2–3–4 and 2–5–6–3) (Fig. S1*G*). The length of the fold is *i*) **a**, **b**, and **c** and (*ii*) **−a**, **d**, and **e**. This approach distributes the stiffness of the fold to all relevant nodes on the two adjacent panel elements. The initial fold angle, *θ* represents a rotation away from the initial folded configuration.

We introduce a parameter *SI Text*, we assume that the folds are less stiff than the panels, and we use an arbitrary choice of *SI Text*, section S3 we show that **a**, **b**, **c** and **−a**, **d**, **e**) is calculated similarly to Eq. **S7**, as*θ*, the angle

## S2) Coupling of Zipper Tube Structures

In the computer model, the coupling of the zipper structures is performed by inserting coupling elements that restrict relative movement between the adjacent panels of the two tubes. These coupling elements can be thought of as an adhesive joint between the adjacent faces of the tube. The relative local coordinates (*E* is Young’s modulus, *t* is the thickness of the thin sheet, and the coupling coefficients (*t*-thick piece of panel material. Finally, *C*. A rotational hinge element is constructed that restricts out-of-plane movement between each of the overlapping nodes and the three corresponding nodes on the opposing tube. For clarity, only two sample corresponding node sets are illustrated in Fig. S2*C*. The vectors groups (**a**, **b**, and **c**) and (**d**, **e**, and **f**) can be used to define the rotational hinge for each set. The stiffness for each of these rotational hinges is defined as**S7**, and the coupling coefficient

The sensitivity of the model eigenvalues vs. the magnitude of each of the coupling coefficients is explored for the zipper-coupled tubes deployed to 70% extension (Fig. S2). The rigid folding mode (*Y* direction. The

When changing the value of the vertex angle *α* or the extended configuration of the zipper-coupled tubes, there is little change in the sensitivity of the different coupling elements, and the general trends remain. The tubes are effectively coupled when the coupling coefficients are about equal to 1. The

To better understand why the zipper coupling results in the large bandgap increase, we investigate how the squeezing deformation mode is restrained by the new geometry. We perform an eigenvalue analysis on two tubes arranged in a zipper-coupling fashion, when the stiffness of all coupling elements is substantially reduced (*A–C*). However, this motion is incompatible for an effectively coupled zipper system. On the left side, the first vertex of the bottom tube moves downward (point I in Fig. S3*A*), whereas the first vertex on the top tube moves upward (point II in Fig. S3*B*) and vice versa on the right side. This transverse motion between the two tubes can be quantified by tracking the distance between adjacent panel-edge center points on the two tubes (Fig. S3*D*). In an undeformed (or effectively coupled) system the distance between adjacent edge center points is uniform at 0.7 units. The squeezing of the loosely coupled system results in separation on the left side (distance increases up to 0.9 units) and a closing on the right side (distance decrease down to 0.5 units). In an effectively coupled zipper system these in-plane motions are restrained, and it would be necessary to stretch and shear the thin sheet to achieve a squeezing-type deformation.

## S3) Sensitivity of Model and Analysis

In Fig. S4 we show differences in scaling of eigenvalues, for the tube and zipper-coupled tubes, with respect to different model parameters. The seventh eigenvalue for both the single tube and the zipper-coupled tubes corresponds to the rigid folding mode, in which deformation primarily occurs as bending of the prescribed folds (Fig. 3 and Fig. S7). The eighth mode for the single tube corresponds to squeezing, in which bending occurs in the fold and panel elements. However, the eighth mode for the zipper-coupled tubes requires stretching and shearing of the thin sheet, which require much more energy than bending and result in drastic differences for the scaling of eigenvalues.

All eigenvalues (*E* (i.e., doubling *E* doubles the eigenvalue) and inversely proportionally with the material density *ρ* (Fig. S4 *A*, *B*, *E*, and *F*). This is expected because *E* scales the stiffness proportionally, and *ρ* scales the mass proportionally in equation *t*, its stretching/shearing stiffness and the system’s mass scale proportionally. On the other hand, the bending stiffness for both the panels and the folds scales as **S7** and **S8**. Therefore, when scaling the thickness, the eigenvalues scale as *C* and *G*). The eighth eigenvalue for the zipper-coupled tubes does not change because both mass and stretching/shear stiffness scale proportionally (both are defined by *t*). When scaling the fold stiffness ratio, *D* and *H*). The system behaviors that lead to these scaling relations are complemented by the energy distributions shown in Fig. S7.

In summary, Young’s Modulus *E* and material density *ρ* directly scale all system eigenvalues, whereas the material thickness *t* and fold stiffness ratio

## S4) Model Verification with Finite-Element Analysis

In this section, we verify and explore the accuracy of the bar and hinge model by comparing the results from the bar and hinge model to those of a shell model created using the commercial finite-element (FE) analysis software ABAQUS (41). Details of the FE implementation are shown in Fig. S5*A* for a single Miura-ori cell. For the FE model, we discretize each panel into *D* segments in each direction, such that each panel will now be modeled using *B* and for all FE analyses herein). General purpose, four-node shell elements with finite membrane strains are used (S4 elements) and are connected with one node at each corner. The fully 3D FE model contains 6 DOFs at each node (3 displacements and 3 rotations). Mass in the model is distributed based on the volume and the density, *ρ*, of the shell elements. All model parameters are defined to be the same as those for the bar and hinge model.

At the fold lines, collocated (overlapping) nodes are placed with one node connected to the shell elements of each adjoining panel. Each set of collocated nodes, shown in Fig. S5*A*, *Inset*, is connected with an element that restricts the nodes to the same **a** vector shown in Fig. S5*A*, *Inset*. The stiffness of the fold is based on the dimensions of connected shell elements and is defined as**S8**) based on the fold length and the same material properties as before. Eq. **S12** distributes the stiffness of the fold (as calculated for the bar and hinge model) based on the tributary length of the shell elements used. Collocated nodes that are at the end of a fold (i.e., at a vertex) will have only one adjacent shell element and thus the stiffness will be based only on

Discretized FE models of the sheet, the single tube, and zipper-coupled tubes are created with *D–F* and Fig. S6 *D–F*), which we expect because these deformation modes correspond to global behaviors of the structure such as bending and rigid folding. The magnitudes of the eigenvalues follow similar trends between the two models but do not always match exactly. For example, with the single tube, **S8** and **S12**), and in the FE model, panel bending occurs globally on the diagonal (as predicted for Eq. **S7**). However, the magnitude of the eigenvalues

We also use the FE model to perform energy analyses for the Miura-ori sheet, the single tube, and the zipper-coupled tubes (Fig. S7). The energies are calculated based on the structural deformation for the normalized mode shapes (Fig. S6 *D–F*). The percentage of distribution between the different sets of element deformations (i.e., fold bending, panel bending, and panel stretching/shear) and the total energy for each mode are shown. The total energy for eigenvalues of the single sheet is relatively low, because deformations consist of localized bending in panels and folds. On the other hand, the eighth and ninth modes of the zipper-coupled tubes require much more energy, because the thin sheets are engaged in stretching and shearing. The rigid folding modes, ninth for the sheet and seventh for the tube structures, primarily engage the fold lines in bending as previously expected.

## S5) Aligned and Internally Coupled Tubes

The eigenvalue vs. configuration plots and representative eigenmodes at *E*). The tube could be a copy of the original geometry or could have different parameters *a* and/or *α*. For internally coupled tubes, the geometries of the internal and external tubes have to conform to preserve flat and rigid foldability as discussed in ref. 7. Here, we define the internal tube geometry, using a parameter *a*, *c*, and *α* define the external tube geometry. The internally coupled tubes discussed in this paper (Figs. 2*F* and 4 and Figs. S8 and S9) use

The eigenvalues *B*), because coupling tubes in this fashion doubles the system’s mass, and simultaneously adding an identical set of elements to the system doubles the stiffness. The seventh and eighth mode shapes are identical to the single tube and are not restricted by this form of coupling (Fig. 3*E* and Fig. S8D). The magnitude of the ninth eigenvalue increases slightly, which can be attributed to the aligned coupling.

Coupling tubes internally (Fig. 2*F*) has more of an effect on their behavior; however, *B* and *E*). When approaching an extension of *A–C*). However, multiple coupling orientations can be integrated into a single system to take advantage of the different behaviors. For example, a combined zipper-coupled and internally coupled system would not experience squeezing and it would lock at a specified configuration.

## S6) Self-Interlocking Structure

Each side of the self-interlocking structure (Fig. 6*C*) can be composed of any number of zipper-coupled tubes together (of same *α*). The structure can be any radially symmetric shape with *n* sides, as long as there is no self-intersection. The structure will interlock when an angle (*γ*) between the two faces on the

## Acknowledgments

This work was partially funded by the National Science Foundation (NSF) through Grant CMMI 1538830. E.T.F. is grateful for support from the NSF Graduate Research Fellowship and the Japan Society for the Promotion of Science Fellowship. The authors also acknowledge support from the Japan Science and Technology Agency Presto program and the Raymond Allen Jones Chair at the Georgia Institute of Technology.

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. Email: paulino{at}gatech.edu.

Author contributions: E.T.F., T.T., and G.H.P. designed research, performed research, contributed new reagents/analytic tools, analyzed data, and wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

See Commentary on page 12234.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1509465112/-/DCSupplemental.

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