Deployable mechanical metamaterials with multistep programmable transformation.

Transformations in shape are critical to actuation in engineered metamaterials. Existing engineering metamaterials are typically limited to a small number of shape transformations that must be built-in during material synthesis. Here, inspired by the multistability and programmability of kirigami-based self-folding elements, a robust framework is introduced for the construction of sequentially programmable and reprogrammable mechanical metamaterials. The materials can be locked into multiple stable deployed configurations and then, using tunable bistability enabled by temperature-responsive constituent materials, return to their original reference configurations or undergo mode bifurcation. The framework provides a platform to design metamaterials with multiple deployable and reversible configurations in response to external stimuli.

INTRODUCTION

Programmable materials that morph their internal architecture for actuation abound in nature (–) and are of interest for robotics (–) and biomedical devices (). These applications benefit from engineered materials with microstructures that can encode shape reconfigurations (–) and topological protections (, ) and that can drive high strength-to-density ratios () and unusual mechanical behaviors such as negative Poisson ratios (, ).

Single transformation materials are now widespread in engineering based on macroscale folding including deployable structures such as folding tables (), vascular stents (), flexible electronics (), and solar panels for spacecraft (, ). Many origami-inspired deployable metamaterials such as the Miura-ori and Kresling folding patterns derive complex and controllable kinematics from folds and deforming facets (–). Kirigami-based metamaterials can reconfigure from two-dimensional (2D) sheets into 3D target shapes (, , ). Mechanical metamaterials that harness mechanical instabilities (–), swelling (, ), electromagnetic actuation (, ), and thermal expansion () have been developed. Multimodal transformation materials have also been obtained using rational design of structural geometry (, ) or combinatorial design (, ). In addition, deformations with multiple transformations can be realized by imposing different actuations and excitations, as in certain cases of magnetic (, , ), pneumatic (), thermal (), and mechanical actuation (–).

However, these systems are typically limited to either a single mode of transformation that must be locked in during fabrication or, instead, require complicated actuation or control for mode switching, e.g., coupled rotation and stretching. Most material systems cannot be programmed to have multimodal shape transformation based on a single input.

We therefore sought to develop a multimodal and reprogrammable material system. We were inspired by 3D polyhedral origami with periodic space-filling tessellation, in which initial 3D architectures can be reconfigured into multiple 3D modes () and by modular kirigami metamaterials that can be transformed into versatile configurations (). However, these metamaterials lack structural stability in their deformed shapes. Instead, a bistable self-folding (BSF) element is needed.

To fill the need for a metamaterial capable of multistep transformation in a high-dimensional deformation space (, ) and of programmable, reversible, and stable reconfiguration (, , ), we developed a kirigami-based design framework based on BSF elements (, ) and applied it to build metamaterials. Initially, in the form of 1D or 2D flat structures, these metamaterials can be programmed to self-fold sequentially to generate multiple modes of transformation. Through temperature-responsive components, the metamaterials are reversible and capable of being reprogrammed with mode bifurcation. We demonstrate the framework through the construction of multistep, multimodal metamaterials.

RESULTS

Design of multistep, multimodal mechanical metamaterials

The design strategy is illustrated by an idealized model of rigid rods connected by dots that rotate in a programmed sequence when actuated by stretch, heat, magnetism, or humidity (Fig. 1A). Connections between the rods are realized by black blocks. Rotations of hinges at blue, pink, and green dots indicate relative rotations between two adjacent rods of the first, second, and third steps, with hollow and solid dots representing concave and convex rotations, respectively (see Fig. 1A and movie S1).

Fig. 1.

A multistep multimodal mechanical metamaterial enabled by bistable hinges.

(A) Schematic of three-step transformation using a dot-rod model. In the left-most panel, the blue, pink, and green dots indicate rotation nodes in different sequences, connecting the gray rigid rods to rotate. The hollow and solid dots represent concave and convex rotations, respectively. In the first step, the BSF elements represented by blue dots fold, leading the 1D structure to deform to a quadrilateral cellular structure (mode 1). In the second step, those represented by pink dots subsequently fold, resulting in an octagonal cellular structure (mode 2). In the third step, BSF elements represented by green dots subsequently fold, resulting in a decagonal cellular structure (mode 3). (B) Conceptual realization of the strategy through arrangement and combination of different BSF units. A three-step transformation unit cell is illustrated. (C) The BSF element. (D) Self-folding of the element under stretching. (E) Stretching force as a function of strain for three BSF elements of different sizes. (F) Transformation snapshots of a model consisting of three elements in series when stretched to ε1, ε2, and ε3. (G) Corresponding relationship between tensile force and strain. See movie S1. Photo credits: Zhiqiang Meng, Tsinghua University.

We introduce this framework by considering a three-step, multimodal mechanical metamaterial of 2 × 2 units that emerges from an initial 1D unfolded structure (Fig. 1A). In the first step, a quadrilateral structure (mode 1) results when the blue hinges fold, while all other hinges remain unchanged. In the second step, an octagonal structure (mode 2) arises when the pink hinges fold, while the blue hinges remain folded and the green hinges remain unfolded. In the third step, a decagonal structure (mode 3) arises when the green hinges fold, while the pink and blue hinges remain folded. Using other combinations of dots and rods, an entire class of multistep, multimodal mechanical metamaterials can be constructed. Note that, in principle, the proposed framework for designing multimodal metamaterials can be used to realize metamaterials with n-step deformation (see the Supplementary Materials and figs. S16 to S22 for details).

To fabricate and test this three-step mechanical metamaterial, we built BSF elements (Fig. 1, C to E) (, ) and assembled them in series (Fig. 1, F and G). The element consisted of an elastic layer (thickness t, length 2a) bonded to a bistable unit (Fig. 1C), the latter made by laser cutting a thin plate (thickness b) into four square plates (edge length L, gray triangles) and four cantilevers (width h and length L, gray circles; Fig. 1C; fabrication details in the Supplementary Materials). Transformation proceeded first by stretching the element axially (x direction) until switched to its second stable state (Fig. 1D), with external work stored as strain energy in the elastomeric layer. Then, after releasing the axial load, the BSF element bent to a controllable, preprogrammed folding angle φ0 (see fig. S12 and movie S1 for details) owing to eccentric compression imposed by the stretched elastomeric layer. The thickness b of the plate was chosen to be large enough to prevent out-of-plane deformation during stretching. As shown by our mathematical model of this metamaterial (see fig. S11 for details), the width h and thickness t dictated these transformation responses.

To demonstrate ab initio design of a programmed transformation pattern, finite element method (FEM) simulations and experimental measurements of three different BSF elements were conducted (cases 1 to 3 in Fig. 1E), with a = 11 mm, b = 1.5 mm, and L = 2.5 mm and different h and t (fig. S7). Results show that peak forces for switching can be tuned and that bistability can be verified from negative force regions on force-strain curves, where strain is defined as normalized displacement, i.e., ε = V/(2a). By connecting these three BSF elements in series and stretching them (Fig. 1F), the resulting structure transformed sequentially to three preprogrammed shapes in a way that can be fully predicted via a theoretical model (figs. S14 and S15). The corresponding stretching force (Fig. 1G and fig. S7) showed three increasing peaks followed by decreases associated with the snap-through of the BSF elements.

These three cases represent embodiments of BSF elements (blue, pink, and green hinges in Fig. 1A) in a three-step mechanical metamaterial (Fig. 1B), with the choice of concave versus convex folding angles determined by placing the elastomeric layer above or below the bistable unit. In step 1, four BSF elements (case 1, marked with blue dots) undergo snap-through when axial strain reaches ε1, producing two mountain folds and two valley folds upon unloading to transform the 1D unfolded structure into a 2D quadrilateral structure (mode 1 in Fig. 1B). In step 2, further stretching to ε2 causes four more elements (case 2, marked with pink dots) to snap-through, and an octagonal structure (mode 2) is obtained upon unloading. In step 3, snap-through of the last four elements (case 3, marked with green dots) at a strain of ε3 produces a 2D decagonal structure (mode 3) upon unloading. This three-step mechanical metamaterial can be extended to an arbitrary number of preprogrammed transformations by a rational arrangement of the self-folding elements (figs. S16 to S18).

To demonstrate the generality of the design strategy and explore more transformation pathways, we fabricated centimeter-scale prototypes using BSF elements consisting of silicone layers and laser-cut, 1.5-mm-thick thermoplastic polyurethane (TPU) plates (figs. S1 and S3). These BSF elements were assembled into multistep, multimodal mechanical metamaterials (see fig. S4 for details) following the above design approach to obtain a variety of geometric shapes and mechanisms (Figs. 2 and 3, and fig. S23).

Fig. 2.

Transformation of 1D structures into stable 2D structures via multistep loading pathways.

(A to C) One-step transformation pathway. (D and E) Two-step transformation pathway. (F to K) Three-step transformation pathway with three different transformation methods. (A, D, F, H, and J) Schematic of multistep transformation structures. (B, C, E, G, I, and K) Images of different stable transformation modes that can be programmed into mechanical metamaterials. See movie S2. Scale bars, 2 cm. Photo credits: Zhiqiang Meng, Tsinghua University.

Fig. 3.

Transformation of 2D structures into stable 3D structures via multistep loading pathways.

(A, C, E, G, and I) Schematics of 2D structures. The unit cell (A) can be arranged in closely packed (C) and non–closely packed (D) configurations to transform into 3D lattices. An additional unit cell (G) can be arranged to form a 2D configuration (I). (B and H) FEM predictions and experimental characterization of the transformation of the two unit cells. (D, F, and J) Images of 3D lattices after transformation. See movie S3. Scale bars, 2 cm. Photo credits: Zhiqiang Meng, Tsinghua University.

We first constructed a metamaterial that followed a one-step transformation from a 1D flat structure into a tessellated 2D hexagonal honeycomb when released after stretching (Fig. 2, A and B). The loading process was similar to that in Fig. 1B. The metamaterial could be instead programmed to transform into a 2D pyramid structure in one step by changing the arrangement of BSF elements (Fig. 2C and movie S2). The metamaterial could be programmed to follow a two-step transformation by combining two different BSF elements (Fig. 2D and fig. S23B) so that it sequentially transformed into tessellated 2D quadrilaterals and 2D hexagons in two steps of loading and unloading (Fig. 2E and movie S2). These could all be designed and predicted by FEM simulations that showed excellent agreement with experiments (Fig. 2, B, C, and E; see the Supplementary Materials for details).

Mechanical metamaterials with three-step transformations could be programmed from unit cells (Fig. 2, F, H, and J) with prescribed arrangements of BSF elements (fig. S23C) having distinct transformation pathways (Fig. 2, G, I, and K). For sample 1, a three-step metamaterial could be generated from a sequentially stretched 1D flat structure (Fig. 2G), as in the three 2D cellular structures of Fig. 1 (A and B) (modes 1, 2-1, and 3-1). In sample 2, swapping the positions of pink and green hinges of the unit cell in sample 1 (Fig. 2H) kept the first shape transformation the same as that of sample 1 (mode 1) but changed the transformed shape to a hexagonal honeycomb (mode 2-2; Fig. 2I) for step 2. Sample 2 recovered the same third step shape as sample 1 (mode 3-1) following a third transformation. For sample 3 (Fig. 2J), changing the rotation direction of the pink hinges of sample 1 again left the first transformation unchanged (mode 1), but now, a different octagonal structure emerged for the second transformation (mode 2-3; Fig. 2K) and a decagonal structure (mode 3-2) for the third transformation, the latter forming enantiomorphs with the third transformation structure (mode 3-1) of samples 1 and 2.

The same design and fabrication methods were used to create metamaterials with four-step transformations (fig. S24). When four different BSF elements were combined, a rich diversity of shapes and transformation pathways was obtained by simulations and experiments (see fig. S24, E and F for details). Controlled self-deployment of these samples required 70 ms after releasing the load (fig. S30), much faster than traditional self-assembled metamaterials (). These demonstrations show the feasibility of the design of generalized transformation pathways.

The framework was next used to transform initially 2D flat structures (Fig. 3, A, C, E, G, and I) into 3D structures (Fig. 3, B, D, F, H, and J). Note that the 2D-to-3D design framework can also be used to design metamaterials with n-step deformation (see the Supplementary Materials and figs. S19 to S22 for details). We first considered a metamaterial that followed a one-step transformation. The unit cell (Fig. 3A) consisted of eight BSF elements (blue hinges) arranged in an orthogonal cross. In this demonstration, all BSF elements were chosen to be identical (fig. S23D). The bidirectional loading along the x and z axis (Fig. 3B) was applied to the unit cell, resulting in a transformation that was well predicted by FEM simulations. Arranging four-unit cells in closely packed (Fig. 3C) and non–closely packed (Fig. 3E) 2D structures and stretching them in the x and z directions yielded two tessellated 3D structures (Fig. 3, D and F, and movie S3), and stacking of these in the y direction yielded two- and four-layer lattice structures.

Similarly, a metamaterial could be designed to follow a two-step transformation. In a 2D unit cell (Fig. 3G) composed of 16 BSF elements (8 units each in the x and z directions), with two types of BSF elements with different dimensions for the blue and pink hinges, a two-step transformation was formed by two-step deployment under dual-axis loading (Fig. 3H). Arranging these four-unit cells in a closely packed array (Fig. 3I) yielded two additional 3D structures via two-step transformation (Fig. 3J and movie S3). A rich variety of 3D deformation modes can be realized following the design framework (see the Supplementary Materials and figs. S20 to S22).

Reprogrammable, recoverable, and mode-bifurcating metamaterials

The abundant programmable, multistep metamaterials demonstrated above hold their transformed configurations because of the bistability of the BSF elements and do not shift between their encoded and independent transformation pathways (Fig. 2, F to K). To enable mode switching and recovery, we developed a reprogrammable strategy using tunable bistability enabled by temperature-responsive materials (). To do so, we partially replaced the cantilever of the BSF element with a temperature-responsive part (TRP; yellow cuboid in Fig. 4A).

Fig. 4.

Recoverable transformations of metamaterials using temperature-responsive polymers.

(A) A BSF element composed of two materials (PLA and TPU), with the inset showing an enlarged view of temperature-responsive part (TRP). (B) Temperature-dependent storage moduli of PLA and TPU, showing a loss in stiffness of PLA at ≈73°C. (C) FEM simulations and experimental measurements of the force-strain behaviors of BSF elements at temperatures of 25° and 80°C. The element changes from a bistable state to a monostable state above the glass transition temperature of PLA. (D) Images of the experimentally observed two-step transformation of the BSF element. (E) Snapshots of the transition from the structure of mode 1 back to the initial unfolded state when the BSF element is immersed in 80°C water. (F) Transformation from a 2D closely packed structure to a lattice. (G) Images of the experimentally observed recovery of 3D lattices back to the reference state. See movie S4. Scale bars, 2 cm. Photo credits: Zhiqiang Meng, Tsinghua University.

Using multimaterial 3D printing techniques (see the Supplementary Materials and fig. S2), composite BSF elements were fabricated. The TRP (width h1; Fig. 4A) was printed with polylactic acid (PLA), and the other parts were printed with TPU. As characterized by dynamic mechanical analysis (DMA) (see the Supplementary Materials and fig. S6), PLA has a relatively high modulus at room temperature ( at 25°C) that drops significantly when PLA is heated above its glass transition temperature, Tg ( at 80°C). In contrast, TPU has a relatively constant modulus from 78.2 to 40 MPa when the temperature increases from 25° to 80°C (Fig. 4B). FEM-simulated and experimentally measured force-strain curves of two BSF elements under axial stretching (Fig. 4C and fig. S9) revealed bistability (gray circles in Fig. 4C) at 25°C, which means that they will lock after unloading at 25°C, and monostability at 80°C, indicating that they will return to their initial unfolded state after unloading at 80°C. As evident from a simplified mechanical model (see the Supplementary Materials and fig. S13), this reversibility occurred because the cantilevers of the BSF element become “softer” than other parts when heated, resulting in too low of an energy barrier to prevent the square elements from returning to their original configurations.

To demonstrate the temperature-induced recoverability of BSF elements in a multistep mechanical metamaterial, we introduced TRP elements into the two-step transformation design of Fig. 2D (Fig. 4D). At a temperature of 25°C, the unit cell had two stable configurations (modes 1 and 2, Fig. 4D) following the two-step loading. When the unit cell of the hexagonal mode 2 structure was heated to 80°C in a water bath, the metamaterial disassembled back to its flat 1D state with elements releasing their folds over 44 s (Fig. 4E and movie S4). The 2D unit cells in a closely packed array that transformed into a 3D lattice when stretched at 25°C (Fig. 4F) likewise returned to the 2D unfolded structure when heated (Fig. 4G and movie S4). Analogous to this heating-based recovery, stimulus-responsive materials such as humidity-sensitive hydrogels () and light-sensitive liquid crystal elastomers () can be introduced into BSF elements to achieve deployable metamaterials with autonomous recoverability.

TRP components also enabled mode bifurcation in mechanical metamaterials. As a demonstration, three BSF elements were constructed with cantilevers of the same width (h = 6.2 mm) but with TRP components of different thicknesses (h1 decreasing from BSF elements ① to ③). In Fig. 5B, FEM-simulated and experimentally measured force-strain curves showed all three BSF elements to be bistable at both 25° and 80°C, with the transformation forces being F1 > F2 > F3 at 25°C but switching to F2 > F1 at 80°C. When BSF elements ① and ② were connected in series and stretched at 25°C, BSF element ② transformed first, followed by BSF element ① (②→①; Fig. 5C). At 80°C, the order switched (①→②; Fig. 5D), demonstrating mode switching by altering the temperature (movie S5).

Fig. 5.

Temperature-induced modal bifurcation.

(A) Three different BSF elements with h = 6.2 mm (① h1 = 2.5 mm, ② h1 = 1.0 mm, and ③ h1 = 0). (B) FEM-predicted and experimentally measured force-strain curves for the three BSF elements when stretched at 25° or 80°C. At 25°C, the transformation forces scale as F1 < F2; at 80°C, F1 > F2. (C and D) Shape transformation in a metamaterials unit cell with ① and ② in series at 25°C (C) and 80°C (D), showing modal bifurcation (from ②→① to ①→②). (E) Temperature-induced bifurcation of the transformation pathway. At 25°C, the transformation progresses from the initial state to modes 1, 2-1, and then 3. Following heating in mode 1, the same material transforms to mode 2-2 upon subsequent stretching and then to mode 3. See movie S5. Scale bars, 2 cm. Photo credits: Zhiqiang Meng, Tsinghua University.

When combining all three BSF elements into a three-step mechanical metamaterial analogous to that of Fig. 2F, three configurations (modes 1, 2-1, and 3) of the unit cell could again be produced by a three-step loading at 25°C (Fig. 5E). At 80°C, the positions of the green and pink BSF elements became interchanged; that is, the transformation order changed. Accordingly, in step 2, the metamaterial transformed instead into a hexagonal structure (mode 2-2). The metamaterial remained stabilized in mode 2-2 even when returned to room temperature. Last, the structure of mode 3 was obtained after the third step. These two experiments demonstrate temperature-controlled bifurcation of transformation pathways (movie S5). FEM simulations of four-step transformation unit cells showed additional and more complicated mode bifurcation of transformation pathways (see figs. S27 and S28 and movie S5).

Exemplary applications

As an example of the application of these materials, we demonstrated their potential use in tunable force transmission. This demonstration used multimodal deformation to tune force transmission from a higher stiffness mode, desirable in the responsive suspension of a sports car, to a lower stiffness mode, desirable for the smooth suspension of a sedan (). We demonstrated the application of our multistep mechanical metamaterial design scheme in a two-step mechanical metamaterial that could shift between these two modes for low-frequency (<30 Hz) excitation (see Fig. 6, A to D; fig. S29; and movie S6). Figure 6A shows two stable configurations (i.e., modes 1 and 2), which are sequentially generated by a two-step deformation metamaterial. The compressive force-strain responses of both modes were given in Fig. 6B, showing that mode 2 exhibits a higher stiffness than mode 1 in a certain strain range. We performed experimental tests (Fig. 6C) to measure the capability of vibration isolation of the structures at a load of 0.92 N. Figure 6D shows the transmissibility for the frequency in the range between 1 and 27 Hz for modes 1 and 2 under the weight of 92 g. The measured results showed that tunable force transmission could be realized by switching the deformation modes.

Fig. 6.

Applications of multistep transforming metamaterials.

(A to D) Design and experimental characterization of a multimodal metamaterial for tunable force transmission. (A) Photographs of the deployable metamaterial in modes 1 and 2. (B) Force-strain relationships of modes 1 and 2. (C) Experimental setup used to measure the acceleration transmissibility. (D) Acceleration transmissibility under various vibration frequencies from 1 to 25 Hz. (E and F) Design of an autonomous, untethered robotic material that deploys upon sensing a critical velocity. (E) The components of the robotic material include a microcontroller, a pair of drivers, and a deployable metamaterial. (F) The time sequence of sequential self-folding during free fall. After velocity is sensed to exceed a critical threshold, the microcontroller drives the metamaterial to adopt a preprogrammed configuration. See movie S6. Scale bars, 2 cm. Photo credits: Zhiqiang Meng, Tsinghua University.

In a second application, the potential use in protective equipment was demonstrated by building an autonomous, untethered robotic material (Fig. 6, E and F). The system was built by integrating an onboard microcontroller, two drivers, and a programmable metamaterial (Fig. 6E, fig. S31, and movie S6). The deployable metamaterial was first prestretched by two pistons driven by two pressure canisters. When the metamaterial robot was released from rest, the microcontroller sensed velocity through a gyro sensor and then activated the drivers to release the prestretched metamaterials when velocity exceeded a prescribed set point (Fig. 6F). The robotic material then self-deployed into a cellular structure over 37 ms. Impact tests demonstrated that deployment reduced the magnitude of high acceleration peaks on impact (see fig. S32 for details). This autonomous untethered metamaterial robot demonstrates the possibility for intelligent, active metamaterial protection in a helmet or padding.

We envision a range of additional applications for these multistep, multimodal mechanical metamaterials across different length scales. For example, millimeter-level vascular stents are limited to a single deployed shape that is determined before surgery but no longer need to be limited in this way. Metamaterials with some actuation encoded via self-folding can enable complex motions with fewer actuators and simplifies control. The methods here can enable robots and deployable structures to adopt a plurality of shapes and configurations, possibly extending to large, stable deployable structures.

DISCUSSION

In summary, we demonstrated a framework for the rational design, fabrication, and programming of a foldable unit cell that enables a mechanical metamaterial to undergo reversible, deployable, and multistep transformation. Through uniaxial or biaxial stretch and release, these metamaterials enable a variety of morphable modes that are stable without external constraints and that can be generated over multiple loading steps. Different from previous designs that only have a single transformation pathway to a preprogrammed structure (, ), reprogramming of these metamaterials is possible through bifurcations of transformation pathways that can be realized by arranging BSF elements and regulating temperature. The design methodology presented in this work could be extended to smaller and larger scales, enabling a broad range of potential applications across aerospace structures, implantable biomedical devices, and intelligent robotic systems.

MATERIALS AND METHODS

Fabrication and mechanical testing of metamaterials

Bistable units were fabricated in two ways, laser cutting and multimaterial 3D printing. For the method of laser cutting, TPU plates (1.5 mm thick) printed by 3D printer were cut with a laser cutting machine (VLS 2.30, Universal, USA) to fabricate units. A low-power multiple cutting method was used to ensure high precision of specimens. For the method of multimaterial 3D printing, two-material bistable units were manufactured using a multimaterial 3D printer (Ultimaker 3). Because of the size limitations of 3D printing, we printed subparts in batches and assembled them together using an electric soldering iron. Elastomeric layers were made of silicone rubber (Dragon Skin 30, Smooth-On) by casting. The prepared silicone was degassed in a vacuum chamber to remove bubbles and poured into 3D-printed molds to cure. Then, we took them out of the molds and trimmed off the excess. We then used silicone adhesive (RTV silicone adhesive, MingSheng) to connect the prepared bistable units and elastomeric layers to fabricate multistep mechanical metamaterials. Uniaxial tensile tests of bistable units were conducted using a uniaxial testing machine (Zwick Z005) equipped with a 50-N load cell. Specimens were fully clamped at both ends and stretched at a strain rate of 0.1 min−1. As for the tensile tests at high temperature, we built a water bath tensile test platform consisting of a metal collect, a digital water bath, and the testing machine (Zwick Z005). The specimens were immersed in water and stretched uniaxially. As for the application of tunable force transmission, vibration tests of two-step deformation metamaterials were conducted. A signal generator (AFG2021 from Tektronix Inc.) generated signals of different frequencies, which were amplified by the power amplifier (HEAS-50 from Nanjing Foneng Inc.). Two acceleration sensors (LC0101 from Lance Technologies Inc.) were attached to the bottom and top surfaces to record input and output accelerations. The data were collected through a dynamic data collection device (National Instruments NI 6341 DAQ card and LabVIEW program). The details of the above can be found in the Supplementary Materials.

Material characterization

To measure the mechanical properties of silicone elastomer, TPU, and PLA, uniaxial tensile tests were performed on dogbone-shaped specimens mounted on a uniaxial testing machine (Zwick Z005) that was equipped with a 5-kN load cell. Experimentally measured stress-strain curves (fig. S5) were fit with a Mooney-Rivlin hyperelastic constitutive model for TPU and silicone elastomer (C10 = −10.85 MPa and C01 = 23.98 MPa for TPU and C10 = 0.157 MPa and C01 = −0.037 MPa for silicone elastomer). PLA had a Young’s modulus of 2.3 GPa and a yield strength of 55.6 MPa. Dynamic thermomechanical properties of materials were obtained by DMA. Samples with dimensions of 25 mm by 8 mm by 2 mm were oscillated at 1 Hz to 0.02% strain using a TA Instruments RSA3 dynamic mechanical analyzer. The temperature-dependent results in terms of storage modulus and tanδ are shown in fig. S6. The storage modulus of PLA is 2.3 GPa at 25°C, decreasing to 2.7 MPa at 80°C as PLA transitioned from the glassy phase to the rubbery phase. TPU had a relatively constant modulus ranging from 78.2 to 40 MPa over this temperature range. See the Supplementary Materials for details.

Numerical simulations

FEM results were obtained using commercially available software (Abaqus, Dassault Systèmes, Vélizy-Villacoublay, France). We adopted an implicit solver to calculate the deformations of the deployable mechanical metamaterials. Geometric nonlinearity was considered in the calculations. The model was meshed using hybrid eight-node linear brick elements (i.e., C3D8H elements in Abaqus). Mesh sensitivity analysis was conducted to ensure numerical convergence. The Mooney-Rivlin hyperelastic model was used as described above. For simulations with varying temperatures, mechanical properties varied as in fig. S6A. Quasi-static compressive displacement was applied, and an artificial internal dissipation factor of 2 × 10−4 was specified to stabilize the simulations.