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Programmable gear-based mechanical metamaterials

Programmable gear-based mechanical metamaterials Articles https://doi.org/10.1038/s41563-022-01269-3 Programmable gear-based mechanical metamaterials 1,2,3 1 3 1 1  ✉  ✉ Xin Fang    , Jihong Wen    , Li Cheng    , Dianlong Yu , Hongjia Zhang 2,4  ✉ and Peter Gumbsch    Elastic properties of classical bulk materials can hardly be changed or adjusted in operando, while such tunable elasticity is highly desired for robots and smart machinery. Although possible in reconfigurable metamaterials, continuous tunability in existing designs is plagued by issues such as structural instability, weak robustness, plastic failure and slow response. Here we report a metamaterial design paradigm using gears with encoded stiffness gradients as the constituent elements and organizing gear clusters for versatile functionalities. The design enables continuously tunable elastic properties while preserving stability and robust manoeuvrability, even under a heavy load. Such gear-based metamaterials enable excellent properties such as continuous modulation of Young’s modulus by two orders of magnitude, shape morphing between ultrasoft and solid states, and fast response. This allows for metamaterial customization and brings fully programmable materials and adaptive robots within reach. 1,2 aterials featuring tunable elastic properties offer tre- Design concept mendous possibilities for smart machines, robots, air- Overcoming these challenges requires an unprecedented design 3–5 Mcraft and other systems . For example, robotic systems paradigm. First, tunability may be realized by assembling elements with variable stiffness can adapt to missions like grabbing and with built-in stiffness gradients. Second, the coupling between jumping, or maintain optimal performance in a changeable envi- elements must comply with large deformation. Achieving tunable ronment . However, elastic properties of conventional materials yet strong solids requires ensuring tunability under large force are barely tunable even if phase changes are induced. Mechanical and robust controllability while avoiding plastic deformation in 8–11 metamaterials are artificial architected materials that exhibit tuning. We find that such a mutable-yet-strong coupling can be 12–16 properties beyond those of classical materials . Most existing realized with gear clusters. Gears provide an ideal mechanism to metamaterials integrate monofunctional load-bearing elementary smoothly transmit rotation and heavy compressive loads thanks structures (such as rods, beams or plates) in specified topologies to the reliable gear engagement (meshing). Stiffness gradients can with fixed or hinged nodes (Fig. 1). Reconfigurable metamaterials be built into an individual gear body or realized with hierarchical 17–19 open possibilities for drastic changes in properties . When stim- gear assemblies. Gear clusters can be assembled into manifolds ulated by stress, heat or electromagnetic fields, reconfigurations in and can, as metacells, be periodically arranged to form metamate- these metamaterials are induced by the formation of new contacts, rials (Fig. 1c). The proposed design concept is very general since 20–22 23–25 buckling or rotating hinges . Due to node constraints, this there exist numerous architectures for gear assembly. Exotic func- 26,27 permits the reshaping among only a few stable states and often tionality and flexible tunability can emerge from the diversity of includes unstable states, which limits the tunability. Reducing the gear types, built-in variability and cluster organization. We cre- connectivity or relaxing the constraints (for example, with chiral ate several metamaterial prototypes with different gear clusters to 9 29 structures or by connecting elements with flexural traps ) can demonstrate this. enable more states to improve the shape-changing capability, but this inevitably deteriorates the robustness and structural stability Metamaterial based on Taiji gears that are essential for most applications. Moreover, reconfigura- The first prototype is created using compactly coupled periodic tion, including the shape-memory effect, usually involves large gears and two lattice frames (front and back) to arrange the gears deformation that either leads to irreversible plastic deformation or into a simple quadratic pattern (Fig. 2a). The plane gears contain adversely competes with the commonly required high stiffness . hollow sections. The outer part forms two elastic arms whose 30,31 Although the chemical-responsive materials enable some radial thickness smoothly varies with the rotation angle θ (Fig. 2b in situ tunability, the regulation process of their elastic proper- and Supplementary Figs. 1 and 2). Subject to compressive loading, ties is usually very slow, just like for thermal-responsive materi- deformation in the arms is dominated by bending (Fig. 2c). The als . Assembling rods mounted with gears into special lattices can effective stiffnesses of both an arm k and a pair of arms K depend arm p improve the stability while preserving the rotatable nodes , but on the angle θ (Supplementary Fig. 3). The homogenized Young’s engineering practical and robust metamaterials with continuously modulus of the metamaterial along the y axis E = K /B + E has con- y p f tunable elasticity, especially with fast in situ tunability in service, tributions from the gears and frame (Methods, ‘Equivalent method’ remains a major challenge. section). Here, E is the stiffness of the frame and B denotes the gear Laboratory of Science and Technology on Integrated Logistics Support, College of Intelligent Science and Technology, National University of Defense 2 3 Technology, Changsha, China. Institute for Applied Materials, Karlsruhe Institute of Technology (KIT), Karlsruhe, Germany. Department of Mechanical Engineering, Hong Kong Polytechnic University, Hong Kong, China. Fraunhofer Institute for Mechanics of Materials IWM, Freiburg, Germany. e-mail: xinfangdr@sina.com; wenjihong@vip.sina.com; peter.gumbsch@kit.edu Na Ture Ma TeriaLs | VOL 21 | AUGUST 2022 | 869–876 | www.nature.com/naturematerials 869 node Hinge NATurE MATEriAlS Articles Element: elementary Unit cell Metamaterial structure Non-tunable Buckling Discrete states Built-in Excellent tunability variability Engagement (meshing) Gears Gear clusters Control variable Fig. 1 | Design concepts of mechanical metamaterials. a,b, Classical paradigm: non-tunable metamaterial (a) and typical reconfigurable metamaterial (b). c, Gear-based metamaterials. width. E is continuously tunable by rotating the gears and domi- the anisotropy in orthogonal directions also changes with θ. For nantly depends on K since min(K /B) ≫ E = 2.06 MPa here. example, the P (3°) metamaterial can be continuously modulated p p f The tunability relies on the shape of the built-in hollow sec- from E = E to the maximum ratio (E /E ) = 24.8. The latter gives x y y x max tion. Among diverse choices, the shape inspired by the Chinese a metamaterial with negligible lateral expansion upon compression Taiji diagram (Fig. 2b), characterized by a spiral direction, can give (Supplementary Fig. 7). Compared to existing designs, the node smooth variation and polarity. The angle difference between the constraints in gear-based metamaterials are relaxed, but the connec- two local coordinates is β (Fig. 2g). The spin rotations are opposite tion stability and reconfiguration robustness are maintained at any in any two meshing gears. Also, the spiral directions of the Taiji θ even under large compressive loads (Supplementary Video 1). The patterns on the front and the back faces are reversed. Therefore, design is also robust to accommodate manufacturing inaccuracies the meshing mode of a pair of gears has two polarities. When the when regarding the angle β as an indicator of the alignment error spiral directions of patterns are opposite (Fig. 2b), the polarity is of the gears. Figure 2d shows that the programmability is preserved positive, labelled as P (β). The meshing pair in Fig. 2g features even for large β. negative polarity, P (β). The all-metallic metamaterial introduced above is manufac- We employ finite element analysis (FEA) to simulate the contact tured by assembling individual gears. For scale-up and miniatur- problem in gear-based metamaterials (Methods and Supplementary ization, it is desirable to avoid the assembly of individual parts. Figs. 4 and 5). Contact nonlinearity becomes apparent for high K Next, we demonstrate that integrated gear-based metamaterials (Supplementary Fig. 3). Young’s modulus is evaluated from the can be directly manufactured with three-dimensional (3D) print- slope of the uniaxial stress–strain curves at relatively large strain ε. ing, even on the microscale. The major challenge for such inte- An all-metallic prototype, consisting of 5 × 5 copper gears and steel grated manufacturing is to guarantee that the meshing teeth are + − frames, is manufactured and assembled with P (3°) and P (15°) not fused together but still reliably engaged. To tackle this problem, metacells, respectively (Supplementary Fig. 2). The gear has 60 a small clearance is reserved between the surfaces of the meshing teeth, with the tooth thickness t = 0.35π mm, diameter D = 42 mm teeth in the assembled digital model to overcome manufactur- to and width B = 20 mm. Measured cyclic loading–unloading curves ing errors (Methods). Here we manufacture an integrated micro show some hysteresis (Fig. 2e). This is ascribed to the sliding friction metamaterial consisting of 5 × 6 Taiji gears (Fig. 2h) by adopting between the meshed teeth (Supplementary Fig. 6). Figure 2f,g dem- the projection micro-stereolithography 3D printing technique. onstrates that experimental results of E (θ) are in excellent agree- The diameter and tooth thickness of the Taiji gear are 3.6 mm and ment with FEA. The modulation period is 180° in both the P (3°) 235 µ m, respectively; the thickest arm is 75 µ m (Supplementary − + and P (15°) cases. The smooth E (θ) curve indicates that the obtain- Fig. 8). The micro gears are arranged with P (0°), and the reserved able stable states are dense and that continuous tunability is achieved. minimal clearance between the teeth is 32 µ m. The sample is made Both polarity and β affect the tunable range and the correlation of a photosensitive resin with a Young’s modulus of 3.5 GPa. As between the tunable properties and θ (Fig. 2d). For P (3°) in Fig. 2f, experimentally demonstrated in Fig. 2i, the equivalent modulus the zigzag curve of E (θ) reaches the maximum value E = 7.67 GPa E (θ) of this micro specimen can be smoothly tuned by 35 times y max y at θ = 78°, where the solid parts of the gears are in contact, and a (from 8.3 MPa to 295 MPa). Using this integrated design strategy, minimal value E = 0.102 GPa at θ = 114°, where the meshing con- gear-based metamaterials could be scaled up in size and number min nects the forearms. This experimentally obtained modulation range of gears with appropriate high-resolution large-scale 3D print- of E /E = 75 demonstrates the spectacular reconfigurability. ing facilities. Modulation of such integrated metamaterials can be max min For P (15°), E (θ) is sombrero-shaped with a tunable range of 33 achieved with distributed drives or motors (Supplementary Video from E = 0.156 GPa to E = 5.13 GPa. Since E (θ) = E (θ + 90°), 2 and Supplementary Fig. 9a). min max x y Na Ture Ma TeriaLs | VOL 21 | AUGUST 2022 | 869–876 | www.nature.com/naturematerials Rod Beam Plate node Fixed Continuous Rotation Large range log (E (θ, β)) 10 y NATurE MATEriAlS Articles a b c d θ = 0 1.0 0.5 –θ –0.5 Frame –1.0 0 60 120 180 θ (°) Stress, σ e f g P (β = 3°) 2.5 P (β = 15°) FEA θ = 78° 2.0 Experiment β θ = 56° 1.5 θ = 30° 1.0 FEA θ = 12° 0.5 –1 Experiment 0 0.05 0.10 0.15 0.20 0 30 60 90 120 150 180 0 30 60 90 120 150 180 ε (%) θ (°) θ (°) h i P (β = 0) FEA 1 Experiment 0 30 60 90 120 150 180 215 μm θ (°) Fig. 2 | Mechanical metamaterial based on Taiji gears. a, Metamaterial architecture. The blue and green colours show different orientations of gears. b, A pair of meshing gears for P (3°). The x, y and z denote the global coordinates; the origin of the local cylindrical coordinates for spin rotation θ is set at the gear centre. t , the thickness of the arm. c, Typical Mises stress contour of a meshing pair for θ = 0 and compressive strain ε = 0.18%. Deformation is enlarged 20 times. d, Numerical value of log (E (θ, β)) for P (β). e, Measured loading–unloading stress–strain curves for metamaterials with different 10 y + − rotation angles θ. f,g, Young’s modulus as a function of the gear rotation angle θ for positive P (3°) and negative P (15°) polarities, respectively. Inset: the − + meshing pair for P (15°) h, Directly printed micro metamaterial consisting of P (0°) 5 × 6 Taiji gears, with close-up images showing dimensions. i, E (θ) of the P (0°) micro specimen. The error bars and the average values in f,g and i are evaluated by choosing different intervals along the curve in e. Metamaterial based on planetary gears tunability emerges from the relative rotation of the gears inside the Obviously, this first metamaterial is tunable only under compres- metacell. The thickness of the ring t is uniform. Neighbouring rings sive loading. The tensile load is carried by the frame, and the tensile are rigidly connected in a quadratic lattice, which ensures structural modulus is E = E . One may also aim at strong metamaterials whose integrity. Planetary gears revolve along the ring when rotating the t f compressive and tensile moduli are both tunable while preserving sun gear by θ . Their position is given by the revolution angle sun structural integrity. This can be achieved by organizing a planetary θ = θ r /(R + r ), where R and r are the radii of the ring and pr sun sun in sun in sun gear system as a metacell (Fig. 3a). In this example, the metacell sun gears, respectively (Supplementary Table 1 for parameter val- contains six gears: an inner-toothed ring gear (Supplementary Fig. ues). The teeth prevent relative slippage between the two gears even 10), a central sun gear and two pairs of planetary gears A –A and under tension. The metamaterial elastic properties are given by the 1 2 B –B . Gear centres A –O–A (and B –O–B ) are colinear. Using this effective stiffness of the annulus ring supported by the planetary 1 2 1 2 1 2 gear cluster, we create a hierarchical and strong metamaterial whose gears that act as fulcrums (Fig. 3b). The position θ of the planetary pr Na Ture Ma TeriaLs | VOL 21 | AUGUST 2022 | 869–876 | www.nature.com/naturematerials 75 μm Elastic arm 3.6 mm σ (MPa) 8 mm E (GPa) E (MPa) E (GPa) y y β (°) Planetary gear Ring gear NATurE MATEriAlS Articles a b Ring Compress Compress Stretch Both 1 y F F x x O x 0° 22.5° 22.5° 45° B Sun gear Transmission gears c d e f 950 μm Planetary gear 400 μm 20 mm 20 mm 5 mm Sun gear g h i FEA: compress FEA: compress Compress FEA: stretch FEA: stretch Experiment: compress 4 Experiment: compress Experiment: stretch Experiment: stretch θ = 0° θ = 45° 73 times –0.4 0.4 –1 25 times ε (%) 10 –1 0 15 30 45 60 75 90 0 15 30 45 60 75 90 Stretch y θ (°) θ (°) pr pr Fig. 3 | Metamaterial consisting of planetary gear systems. a, Metacell, a planetary gear system. The origin (O) of the coordinates x,y is the centre of the sun gear (green). A and B denote the centres of planetary gears (grey). Their revolution angle denotes θ  = ∠YOA . The case A A  ⊥ B B is drawn here, i i pr 1 1 2 1 2 and thus θ  = ∠XOB  = ∠YOA . A transmission gear (yellow) connects to the sun gear through a shaft. b, Typical compressive and tensile deformation of pr 1 1 a metacell. c, Macro metallic metamaterial architecture. d, Macro polymer metamaterial with 6 × 6 metacells. e, Micro polymer metamaterial with 3 × 4 metacells. f, Micrograph for the metamaterial in e. Specimens in d and e are fabricated with integrated manufacturing. g, Measured strain–stress curves. h,i, Measured and simulated Young’s moduli under compression and tensile deformation, E and E , of the macro metallic (h) and micro (i) specimens. c t Results of the macro polymer sample are shown in Supplementary Fig. 12b. The error bars and the average values in h and i are evaluated by choosing different intervals near the maximum strain along the curve in g. gears determines the elastic properties. We adopt the orthogonal because four pairs of meshing teeth in a metacell bear loads. Both relation A A ⊥ B B , which gives a large tunable range and symmet- the compressive and tensile moduli E and E reach maxima at θ = 0, 1 2 1 2 c t pr max max ric behaviour with a modulation period of 90°. The tunable range but E ≫ E , and thus the static compressive-tension symme- c t can be further modified using the angle ∠A OB . For the assembled try is broken (Fig. 3g). Moreover, at θ = 45°, no stress is transmitted 1 1 pr metamaterial, all sun gears are connected to transmission gears by to the planetary gears (Fig. 3b); both moduli reach minima there, min min shafts (Fig. 3a), and those transmission gears are compactly cou- and . E = E c t pled. Thereby robust reconfiguration of all metacell patterns can be We fabricate three kinds of specimens using this strategy. An achieved by rotating transmission gears. all-steel macro metamaterial is manufactured by assembling 3 × 3 F and F denote the compressive loads in the x and y directions, metacells (Fig. 3c) with lattice constants a = a = 27 mm. The steel x y x y respectively (Fig. 3a). Under uniaxial compression (F > 0, F = 0), gears have a small tooth thickness t = 0.15π mm, and R = 12 mm, y x to in only the pair of planetary gears with an angle smaller than that of r = 6 mm and t = 1 mm. Integrated manufacturing of this proto- sun r the loading axial (min(∠YOA , ∠YOB ) < 45°) supports the load type is more challenging than that of the Taiji pattern because there 1 1 (Fig. 3b). Stress in the other pair is zero. Conversely, under uniaxial are two layers and every metacell possesses eight pairs of mesh- tension, only the other pair is load-bearing. The two pairs exchange ing teeth. The integrated prototype can also be directly manufac- roles at θ = 45°, and the material is orthogonally isotropic. This tured by 3D printing, at both macro and micro scales (Methods, pr metamaterial presents a more remarkable compressive nonlinearity Supplementary Figs. 11 and 12 and Supplementary Video 3 for Na Ture Ma TeriaLs | VOL 21 | AUGUST 2022 | 869–876 | www.nature.com/naturematerials sun pr pr σ (MPa) E (GPa) E (MPa) y NATurE MATEriAlS Articles a b c τ P (β = 3°) 0.45 0.40 0.35 τ 0.30 0.25 FEA 0.20 Experiment 0.15 0 30 60 90 120 150 180 Shear interlock θ (°) d f Free Cavity Cavity Geometrical interlock Gear ultrasoft matter 0 5 10 15 20 25 30 γ (%) Fig. 4 | strong or ultrasoft metamaterials under shear. a, Schematic shear deformation of a meshing pair. b, Four meshing gears in the state of shear interlock, θ = 60°. The circular arrows show the planetary and spin rotations, which block each other. Periodic boundary conditions are applied on this metacell (Supplementary Fig. 14). c, Theoretical and experimental shear stiffness K /B under shear interlock. The 3 × 3 architecture with P (3°) is shear studied here (Supplementary Fig. 15). The stiffness is strong (>150 MPa) and narrowly tunable (~3 times). The error bars and the average values are evaluated by choosing different intervals along the tested strain–stress curves. d,e, Real-life photographs for the deformation modes of the prototype under diagonal and surface compressions, respectively. The light blue, white and dark blue parts are the background, rubber frame and gears (d,e). f, Experimental strain–stress curves for the shear interlocked metamaterial, the ultrasoft gear matter and the rubber frame by itself. Inset: three blue disks illustrate the geometrical interlock, which occurs when three meshing gears form a closed triangle. more details). We print a 6 × 6 macro specimen (Fig. 3d) using a be synchronously controlled with distributed motors at both the polymer with a Young’s modulus of 2.5 GPa, and print a 3 × 4 micro macro and micro scales (Supplementary Video 4). specimen (Fig. 3e) using a resin with a Young’s modulus of 3.5 GPa. The size and the number of metacells are limited by the capability Mechanisms for stability of the 3D printer rather than the design strategy. The micro poly- Interestingly, the metamaterial in Fig. 2a (a discrete gear lattice with mer sample (Fig. 3e) has R = 2.4 mm, r = 0.6 mm, t = 0.3 mm a very soft frame) remains stable under compressive stresses and in sun r and a = a = 5.4 mm, with a tooth width and height of 135 µm and shows large rigidity in shear. One of the contributing factors under- x y 225 µm, respectively. pinning the observed stability stems from the non-uniform load- The experimental results are consistent with the FEA simula- ing of the meshing teeth at different points, which leads to bending tions for all specimens (Fig. 3g–i and Supplementary Fig. 11). In deformations that tightly grip the teeth together (Supplementary this strong hierarchical metamaterial, we can smoothly tune the Fig. 5). The relatively large shear modulus of the metamaterial, compressive modulus E of the macro metallic specimen by 46 G = G + G , is composed of the shear moduli generated by gears c g f times (5.2–0.11 GPa), the macro polymer specimen by 55 times (G ) and by frames (G = 1.04 MPa). Shear force induces both spin g f (69–1.25 MPa; Supplementary Fig. 12b) and the micro specimen and the planetary rotation of gears. For a pair of gears, the relative by 25 times (100–4 MPa). Meanwhile, their tensile modulus E can planetary rotation leads to zero shear resistance, G = 0, giving a t g be tuned by 5 times (0.52–0.11 GPa), 5.6 times (7–1.25 MPa) and 5 highly unstable state (Fig. 4a). However, in a group of four gears times (20–4 MPa), respectively. In Fig. 3h, some differences between (shown in Fig. 4b), shear stress τ induces mutual locking of the plan- experiment and FEA near θ = 45° arise from the boundary condi- etary rotation by the opposite spin of the neighbouring gears, which pr tions (Supplementary Fig. 13). The in situ tunability combined with is referred to as shear interlock. We calculate the shear stiffness the reasonably large moduli in tension and compression as well as of the metacell with periodic boundary conditions and the finite large shear rigidity makes this metamaterial design particularly n × n gear lattice (Supplementary Figs. 14–16). Owing to the shear robust and strong, yet tunable. Furthermore, the metamaterials can interlock, the shear modulus is large but only marginally tunable Na Ture Ma TeriaLs | VOL 21 | AUGUST 2022 | 869–876 | www.nature.com/naturematerials Rubber frame Frame Shear interlock τ (kPa) K /B (GPa) shear NATurE MATEriAlS Articles a b 4 4 10 10 Gear-based metamaterials 3 3 10 10 Thermal- (35) responsive Thermal-responsive metamaterials 2 metamaterials 2 10 10 1 1 10 10 (36) (36) (37) (30) 0 0 (31) 10 10 (30) (31) (34) (37) Chemical- responsive MMs –1 –1 10 10 Shape-morphing (38) metamaterials (38) Chemical-responsive MR Magneto- Shape-morphing –2 –2 (18) metamaterials: 10 10 fluids responsive MMs metamaterials hydrogels (18) In situ –3 –3 10 10 ε = 0 10 50 100 150 200 1 10 100 1,000 10,000 Required strain, ε (%) Shortest response time (s) Fig. 5 | Properties of active mechanical metamaterials. a,b, Tunable effective modulus versus the required strain ε (a) and shortest response time (b) for tunability. The shaded colour regions (rectangles and ovals) represent the possible tunable ranges of different tunable materials or metamaterials. Pale green, gear-based metamaterials; yellow, chemical-responsive metamaterials; blue, magneto-responsive metamaterials, brown: shape-morphing metamaterials; grey, thermal-responsive metamaterials. These regions partially overlap. The straight lines indicate the tunable ranges. The solid (dashed) lines represent the continuous (non-continuous) tunability. Arrows represent uncertainties. Numbers in parentheses denote the references. The green shaded region at the bottom in a signifies that the required strain is zero, that is, the in situ tunability. MMs, metamaterials; MR, magnetorheological. Properties of our gear-based metamaterials are shown by red lines and are also labelled by figure numbers. (Supplementary Fig. 16), which is demonstrated by the measured formance and efficiency in variable environments. Programmable generalized shear stiffness K /B of the finite 3 × 3 architecture materials featuring tunable elastic properties, including active shear in Fig. 4c. mechanical metamaterials, are much anticipated in intelligent 1,2 machines and systems . Here we offer a comparison of typical 2,19 Gear metamaterial for shape morphing material designs from the literature . The programmability of gear-based metamaterials is not limited The response time, stability, force and energy required for to elastic constants. Removing every second gear in every second property changes are all critical attributes for variable-stiffness row of the metamaterial in Fig. 2 can release the shear interlock structures . We take the strain and response time required to (inset in Fig. 4f ) to generate a state with G = 0. The effective shear accomplish a tunable period of material stiffness as metrics to posi- modulus is then determined solely by the low stiffness frame, and tion our gear-based metamaterials among the existing active mate- 18,34 the metamaterial can be considered as ultrasoft matter. The van- rials (Fig. 5). Shape-morphing metamaterials enable tunability ishing shear modulus enables complex deformation modes (Fig. between or among two or a few stable states. The achieved tunabil- 4d,e and Supplementary Video 5), conducive to shape morphing. ity is non-continuous and requires a large deformation (ε ≈ 30%). To verify this, an ultrasoft prototype consisting of 4 × 4 metacells Thermal-responsive composites made of shape-memory alloys or with rubber frames is manufactured and tested (Fig. 4f ). The gears polymers may give a continuously tunable modulus. However, they are made of aluminium alloy (Supplementary Fig. 17). Shear tests require a long response time, and some suffer from nearly 100% 36,37 30,31 on the prototype give a tiny modulus of G = 21.52 kPa. Independent strain . Chemical-responsive materials containing hydrogels measurement of the frames gives G = 21.11 kPa, so that G = G – G can offer continuous and in situ tunability, but also require hours of f g f is indeed negligibly small. Moreover, the modulus G remains tiny response time. Conventional magneto- or electro-responsive meta- until the shear strain γ reaches 25%, at which point the semi-free materials based on elastomers or magnetorheological fluids can gear interacts with two other gears, which builds a new meshing give fast, continuous, but narrow tunability, which usually requires a connection among the three gears. The new connection supports high active voltage (~5 kV) and complex facilities . Our gear-based high shear stresses and leads to a sharp rise in G, switching the soft metamaterials in Figs. 2 and 3 can offer a fast response and the matter to a stiff solid. The resulting solid represents a geometrically desired broad-range, continuous and in situ tunability of stiffness. interlocked state (Methods). Oscillations of the shear stress in Fig. We propose several scenarios to showcase the broad appli- 4c arise from the critical meshing state among the three gears before cation potential of the proposed gear-based metamaterials in they interlock (Supplementary Fig. 18). The shear strain at which Supplementary Figs. 19–21 and Supplementary Table 2. For robots, the geometrical interlock occurs can be adjusted by the size of the a tunable-stiffness leg/actuator can offer high stiffness to stably sup- neighbouring gears, which in turn determines the limiting (strong) port a heavy load while walking and a low stiffness for shock pro- 4,39 states of a shape-morphing structure. Previous metamaterial tection while jumping or running (Supplementary Figs. 19 and designs with vanishing shear modulus, like pentamode metamateri- 20). A similar tunable-stiffness isolator is desired in the aero-engine als , show vanishing moduli only at small strains and in extremely pylon system to maintain the best performance and efficiency fragile structures. at different flight stages (Supplementary Fig. 21). Moreover, the fast-response gear-based metamaterial may give rise to a sensitive Potential applications variable-stiffness skin, which has been attracting wide attention . Conventional machines generally rely on materials with constant Furthermore, resonators with tunable stiffness are critical compo- 41,42 stiffness and therefore show constant stiffness themselves. 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The images or other 1373–1377 (2014). third party material in this article are included in the article’s Creative Commons license, Na Ture Ma TeriaLs | VOL 21 | AUGUST 2022 | 869–876 | www.nature.com/naturematerials 875 NATurE MATEriAlS Articles unless indicated otherwise in a credit line to the material. If material is not included in from the copyright holder. To view a copy of this license, visit http://creativecommons. the article’s Creative Commons license and your intended use is not permitted by statu- org/licenses/by/4.0/. tory regulation or exceeds the permitted use, you will need to obtain permission directly © The Author(s) 2022 Na Ture Ma TeriaLs | VOL 21 | AUGUST 2022 | 869–876 | www.nature.com/naturematerials 876 NATurE MATEriAlS Articles show the shear state of a finite n × n gear lattice, we fix the lower row/column of Methods the gears and apply a displacement field to the upper row/column of the gears. Integrated manufacturing. The printer used for the projection As a second method, periodic boundary conditions are applied on a metacell to micro-stereolithography micro metamaterial fabrication is a BMF NanoArch S130, calculate the shear modulus. These periodic boundary conditions present the with a precision of about 5 µm. The material used in microscale 3D printing is a shearing state ε = (0, 0, γ). In both cases, the strain energy density W = Gγ /2 is photosensitive resin with a Young’s modulus of about 3.5 GPa. The manufacturing extracted to evaluate the shear modulus G. For the n × n finite structure without process for the integrated micro metamaterial sample consisting of 5 × 6 Taiji gears periodic boundary conditions, although the equation of the generalized shear follows three steps. First, the assembled gears are printed on a baseplate; those stiffness G′ = K /B is the same as the formula for shear modulus G = τ/γ, gears are adhered to the plate. Second, the sample is wrapped in a box to constrain shear the value of G′ may not equal the real shear modulus G (Supplementary Fig. the motion of the gears (Supplementary Fig. 9a). Last, everything including the box 15) due to the free edge effects in finite structures (Supplementary Figs. 13a is removed from the plate. The box with a frame helps maintain the relative angle and 14a). of the assembly in the removal process. For the metamaterial based on a planetary gear system, the load is applied For the metamaterial based on planetary gears, the layer of the planetary gears on the four blocks of the ring. For a metacell in FEA, we specify the uniaxial is printed first (Supplementary Fig. 12a and Supplementary Video 3). Except for deformation v = ε a and make ε free for solving E . the preserved clearance, the connection shaft between the transmission gear and y y x y the sun gear is conical at both the macro and micro scales (Supplementary Fig. Equivalent method. For the metamaterial based on Taiji gears, the deformation 10), which ensures that every printing part, especially the teeth of the transmission mode for meshing gears can be represented by the overall stiffness of a pair of gears, is tightly attached on the formed structure. Otherwise, the teeth could move meshing elastic arms K = 1/(1/k + 1/k + 1/k ) (Supplementary Fig. 3 for and then fuse together during the printing. At the macro scale, the integrated p arm1 arm2 tooth their definitions). The stiffnesses of the two arms k and k are independent of arm1 arm2 model is printed with two photosensitive resins using polymer injection with the compressive deformation. As shown in Supplementary Fig. 5, the meshing of a the printer Stratasys Objet260, with a precision of about 50 µm. The stiff model pair of teeth features a line of contact on their surfaces. With compression, a small material is wrapped in the soft, soluble support material. The metamaterial contact area is generated near the line where sliding occurs during the process. acquires the targeted tunability after removing the support material. At the Therefore, the contact stiffness of the teeth k depends on the contact pressure microscale, the material is immersed in the fluid resin during the printing. No tooth on the involute teeth. A high pressure leads to significant contact nonlinearity and support material/structure is required for this model owing to the conical shaft and results in a dependence of K on the displacement/load. By contrast, deformation high precision of projection micro-stereolithography. The sample shown in Fig. 3d mainly occurs in the elastic arm rather than the teeth if k ≪ k , and K is is printed with a resin (polymer) with a Young’s modulus of 2.5 GPa. arm tooth p constant in this case. The homogenized Young’s modulus in the y direction of In integrated manufacturing, the clearance reserved between the surfaces of the metamaterial is E = K /B + E . The equivalent methods for shear modulus are meshing teeth in the assembled digital model depends on the precision of the y p f explained with Supplementary Fig. 14. printer, the structure and the materials. The minimal clearance Δ should be higher For the metamaterial consisting of a periodic planetary gear cluster, the Young’s than the printer’s precision p (manufacturing errors) but much smaller than the modulus depends on the deformation of the ring. The influences of contact tooth height (h = 2.25m for standard gears), where the gear module m denotes nonlinearity between teeth on E are the same as described above. the ratio between the gear diameter D and the number of teeth z, m = D/z (see Supplementary Text). Here, the minimal clearance between meshing teeth in all Geometrical interlock. In a meshing pair, the rotation directions of the driving macro specimens printed with Objet260 is 86 µm. The minimal clearance for the and driven gears are opposite. In a group of gears, if every meshing is viewed as a micro metamaterial consisting of Taiji gears is set to 32 µm, and that for the micro connection line, n gears form a closed polygon as shown in Fig. 4f. If n is odd, spin planetary gear-based metamaterial is 21 µm. These clearances are sufficient to rotation is incompatible, leading to the locking among the gears. This meshing alleviate the manufacturing uncertainties to keep the meshing teeth separated state is referred to as geometrical interlock. but reliably engaged. Based on our 3D printers and tests, we suggest Δ > 1.5p and Δ ≤ h /10 = 0.225m. This requires p < 0.15m, which helps us determine the required Mechanical tests for Young’s modulus. When measuring the Young’s modulus precision scale with a specified gear size. E in the metamaterial based on Taiji gears, a compressive load F is applied and y y released from the top of the prototype in Fig. 2a. We control the strain ε for Actuation. As shown in Supplementary Video 2, we prepare a microscale sample different θ to overcome clearance nonlinearity while avoiding plastic deformation. consisting of 5 × 5 Taiji gears to show its actuation process. They are embedded The rotation angle θ is manually controlled. Similar cyclic loading–unloading tests into a box, and those gears connect to the frames through micro shafts. The are performed for the measurement of the shear modulus. sample is synchronously driven by four d.c. brushless motors (8 mm diameter) The experimental setting for the test on the metamaterial based on planetary connected to the 1 × 1st, 1 × 4th, 4 × 1st and 4 × 4th gears. Here n × m denotes the gear systems is shown in Supplementary Fig. 9. When measuring the compressive position at the nth row and mth column in the array. As shown in Supplementary modulus, a compressive load is applied on the top and the bottom blocks on the Video 4, the macro metamaterial in Fig. 3d is synchronously actuated by four-step rings; when measuring the tensile modulus, we fix the tails on the sample to a pair motors whose diameter is 20 mm. These motors are synchronously controlled by of clamps and apply tensile loads through the tails. an electronic controller. The revolving speed of the step motor depends on the As shown in Figs. 2e and 3g, the cyclic loading–unloading process features high impulse frequency generated by the controller. Similarly, the micro sample in Fig. repeatability, thus testifying to the experimental accuracy. Moduli E and G are 3e is put in a box and actuated by five micro step motors whose diameter is 5 mm. both calculated as the slope around the maximum ε. The initial cycle is excluded The controller is identical to the one used for the macro sample. when fitting E and G. The choice of the strain interval for the slope calculation affects the final modulus value. The error bars and the average values are evaluated FEA. FEA simulations are carried out with the commercial software ANSYS. We by choosing different intervals along the curve. compare the accuracy of different finite element models, including two-dimensional (2D), 3D, linear and nonlinear models. The plane stress state is considered in the Mechanical tests for shear stiffness. For the metamaterial based on Taiji 2D model. In the linear models, the meshing points of gears are bonded by fixing gears, a sample consisting of 3 × 3 gears and steel frames is manufactured for together the two surfaces in contact, resulting in a linear stress–strain relationship. the measurement of the shear stiffness in the shear interlock state, as shown in In the nonlinear models, the size of the contact area on the tooth surface at the Supplementary Fig. 15. A fixture apparatus is fabricated to obtain the shearing meshing points depends on the load, and there is a relative sliding between the state. For the shape-morphing metamaterial, the sample is put in two right-angle contact surfaces. The sliding induces frictional damping if the coefficient of friction grooves, and the load from the testing machine directly transfers to the sample. is non-zero. We also use a simplified model by removing all teeth, where the contact between two gears becomes that between two cylinders. In principle, the 3D nonlinear model should be the most realistic representation of the experimental Data availability set-up. Supplementary Fig. 4 demonstrates that the 2D nonlinear model is in The main data and models supporting the findings of this study are available within excellent agreement with the 3D nonlinear model. The two linear models produce a the paper and Supplementary Information. Further information is available from large discrepancy with the nonlinear ones, although they still can capture the general the corresponding authors upon reasonable request. variation trend. The simplified model approximately presents the standard results. To enhance the simulation efficiency, we use the 2D nonlinear models in most cases. a cknowledgements The 3D model is adopted only when considering the frictional contact. This research was funded by the National Natural Science Foundation of China (projects Our metamaterials embrace a periodic architecture. To evaluate the no. 12002371 and no. 11991032), the Hong Kong Scholars Program, the Fraunhofer homogenized elastic and shear moduli, ideal periodic boundary conditions Cluster of Excellence ‘Programmable Materials’ and the Excellence Cluster EXC 2082 ‘3D are applied on the unit cell in the FEA. Boundary conditions depend on the Matter Made to Order’ (3DMM2O) in Germany. deformation mode of the unit cell. The homogenized strain vector is ε = (ε , ε , γ). x y These strains are realized by enforcing the displacement fields (u, v) in the plane stress state. a uthor contributions As explained in Supplementary Fig. 14, two types of boundary condition are X.F. and P.G. designed the study. X.F. conceived the idea and performed the experiments. considered when calculating the shear modulus in the shear interlock state. To X.F. and P.G. carried out the numerical simulations. L.C., D.Y. and H.Z. analysed the Na Ture Ma TeriaLs | www.nature.com/naturematerials NATurE MATEriAlS Articles data. All authors interpreted the results. X.F., L.C., J.W. and P.G. wrote the manuscript a dditional information with input from all authors. P.G. supervised the study. Supplementary information The online version contains supplementary material available at https://doi.org/10.1038/s41563-022-01269-3. Funding Correspondence and requests for materials should be addressed to Open access funding provided by Fraunhofer-Gesellschaft zur Förderung der Xin Fang, Jihong Wen or Peter Gumbsch. angewandten Forschung e.V. Peer review information Nature Materials thanks Amir A. Zadpoor and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Competing interests Reprints and permissions information is available at www.nature.com/reprints. The authors declare no competing interests. Na Ture Ma TeriaLs | www.nature.com/naturematerials http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Nature Materials Springer Journals

Programmable gear-based mechanical metamaterials

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Abstract

Articles https://doi.org/10.1038/s41563-022-01269-3 Programmable gear-based mechanical metamaterials 1,2,3 1 3 1 1  ✉  ✉ Xin Fang    , Jihong Wen    , Li Cheng    , Dianlong Yu , Hongjia Zhang 2,4  ✉ and Peter Gumbsch    Elastic properties of classical bulk materials can hardly be changed or adjusted in operando, while such tunable elasticity is highly desired for robots and smart machinery. Although possible in reconfigurable metamaterials, continuous tunability in existing designs is plagued by issues such as structural instability, weak robustness, plastic failure and slow response. Here we report a metamaterial design paradigm using gears with encoded stiffness gradients as the constituent elements and organizing gear clusters for versatile functionalities. The design enables continuously tunable elastic properties while preserving stability and robust manoeuvrability, even under a heavy load. Such gear-based metamaterials enable excellent properties such as continuous modulation of Young’s modulus by two orders of magnitude, shape morphing between ultrasoft and solid states, and fast response. This allows for metamaterial customization and brings fully programmable materials and adaptive robots within reach. 1,2 aterials featuring tunable elastic properties offer tre- Design concept mendous possibilities for smart machines, robots, air- Overcoming these challenges requires an unprecedented design 3–5 Mcraft and other systems . For example, robotic systems paradigm. First, tunability may be realized by assembling elements with variable stiffness can adapt to missions like grabbing and with built-in stiffness gradients. Second, the coupling between jumping, or maintain optimal performance in a changeable envi- elements must comply with large deformation. Achieving tunable ronment . However, elastic properties of conventional materials yet strong solids requires ensuring tunability under large force are barely tunable even if phase changes are induced. Mechanical and robust controllability while avoiding plastic deformation in 8–11 metamaterials are artificial architected materials that exhibit tuning. We find that such a mutable-yet-strong coupling can be 12–16 properties beyond those of classical materials . Most existing realized with gear clusters. Gears provide an ideal mechanism to metamaterials integrate monofunctional load-bearing elementary smoothly transmit rotation and heavy compressive loads thanks structures (such as rods, beams or plates) in specified topologies to the reliable gear engagement (meshing). Stiffness gradients can with fixed or hinged nodes (Fig. 1). Reconfigurable metamaterials be built into an individual gear body or realized with hierarchical 17–19 open possibilities for drastic changes in properties . When stim- gear assemblies. Gear clusters can be assembled into manifolds ulated by stress, heat or electromagnetic fields, reconfigurations in and can, as metacells, be periodically arranged to form metamate- these metamaterials are induced by the formation of new contacts, rials (Fig. 1c). The proposed design concept is very general since 20–22 23–25 buckling or rotating hinges . Due to node constraints, this there exist numerous architectures for gear assembly. Exotic func- 26,27 permits the reshaping among only a few stable states and often tionality and flexible tunability can emerge from the diversity of includes unstable states, which limits the tunability. Reducing the gear types, built-in variability and cluster organization. We cre- connectivity or relaxing the constraints (for example, with chiral ate several metamaterial prototypes with different gear clusters to 9 29 structures or by connecting elements with flexural traps ) can demonstrate this. enable more states to improve the shape-changing capability, but this inevitably deteriorates the robustness and structural stability Metamaterial based on Taiji gears that are essential for most applications. Moreover, reconfigura- The first prototype is created using compactly coupled periodic tion, including the shape-memory effect, usually involves large gears and two lattice frames (front and back) to arrange the gears deformation that either leads to irreversible plastic deformation or into a simple quadratic pattern (Fig. 2a). The plane gears contain adversely competes with the commonly required high stiffness . hollow sections. The outer part forms two elastic arms whose 30,31 Although the chemical-responsive materials enable some radial thickness smoothly varies with the rotation angle θ (Fig. 2b in situ tunability, the regulation process of their elastic proper- and Supplementary Figs. 1 and 2). Subject to compressive loading, ties is usually very slow, just like for thermal-responsive materi- deformation in the arms is dominated by bending (Fig. 2c). The als . Assembling rods mounted with gears into special lattices can effective stiffnesses of both an arm k and a pair of arms K depend arm p improve the stability while preserving the rotatable nodes , but on the angle θ (Supplementary Fig. 3). The homogenized Young’s engineering practical and robust metamaterials with continuously modulus of the metamaterial along the y axis E = K /B + E has con- y p f tunable elasticity, especially with fast in situ tunability in service, tributions from the gears and frame (Methods, ‘Equivalent method’ remains a major challenge. section). Here, E is the stiffness of the frame and B denotes the gear Laboratory of Science and Technology on Integrated Logistics Support, College of Intelligent Science and Technology, National University of Defense 2 3 Technology, Changsha, China. Institute for Applied Materials, Karlsruhe Institute of Technology (KIT), Karlsruhe, Germany. Department of Mechanical Engineering, Hong Kong Polytechnic University, Hong Kong, China. Fraunhofer Institute for Mechanics of Materials IWM, Freiburg, Germany. e-mail: xinfangdr@sina.com; wenjihong@vip.sina.com; peter.gumbsch@kit.edu Na Ture Ma TeriaLs | VOL 21 | AUGUST 2022 | 869–876 | www.nature.com/naturematerials 869 node Hinge NATurE MATEriAlS Articles Element: elementary Unit cell Metamaterial structure Non-tunable Buckling Discrete states Built-in Excellent tunability variability Engagement (meshing) Gears Gear clusters Control variable Fig. 1 | Design concepts of mechanical metamaterials. a,b, Classical paradigm: non-tunable metamaterial (a) and typical reconfigurable metamaterial (b). c, Gear-based metamaterials. width. E is continuously tunable by rotating the gears and domi- the anisotropy in orthogonal directions also changes with θ. For nantly depends on K since min(K /B) ≫ E = 2.06 MPa here. example, the P (3°) metamaterial can be continuously modulated p p f The tunability relies on the shape of the built-in hollow sec- from E = E to the maximum ratio (E /E ) = 24.8. The latter gives x y y x max tion. Among diverse choices, the shape inspired by the Chinese a metamaterial with negligible lateral expansion upon compression Taiji diagram (Fig. 2b), characterized by a spiral direction, can give (Supplementary Fig. 7). Compared to existing designs, the node smooth variation and polarity. The angle difference between the constraints in gear-based metamaterials are relaxed, but the connec- two local coordinates is β (Fig. 2g). The spin rotations are opposite tion stability and reconfiguration robustness are maintained at any in any two meshing gears. Also, the spiral directions of the Taiji θ even under large compressive loads (Supplementary Video 1). The patterns on the front and the back faces are reversed. Therefore, design is also robust to accommodate manufacturing inaccuracies the meshing mode of a pair of gears has two polarities. When the when regarding the angle β as an indicator of the alignment error spiral directions of patterns are opposite (Fig. 2b), the polarity is of the gears. Figure 2d shows that the programmability is preserved positive, labelled as P (β). The meshing pair in Fig. 2g features even for large β. negative polarity, P (β). The all-metallic metamaterial introduced above is manufac- We employ finite element analysis (FEA) to simulate the contact tured by assembling individual gears. For scale-up and miniatur- problem in gear-based metamaterials (Methods and Supplementary ization, it is desirable to avoid the assembly of individual parts. Figs. 4 and 5). Contact nonlinearity becomes apparent for high K Next, we demonstrate that integrated gear-based metamaterials (Supplementary Fig. 3). Young’s modulus is evaluated from the can be directly manufactured with three-dimensional (3D) print- slope of the uniaxial stress–strain curves at relatively large strain ε. ing, even on the microscale. The major challenge for such inte- An all-metallic prototype, consisting of 5 × 5 copper gears and steel grated manufacturing is to guarantee that the meshing teeth are + − frames, is manufactured and assembled with P (3°) and P (15°) not fused together but still reliably engaged. To tackle this problem, metacells, respectively (Supplementary Fig. 2). The gear has 60 a small clearance is reserved between the surfaces of the meshing teeth, with the tooth thickness t = 0.35π mm, diameter D = 42 mm teeth in the assembled digital model to overcome manufactur- to and width B = 20 mm. Measured cyclic loading–unloading curves ing errors (Methods). Here we manufacture an integrated micro show some hysteresis (Fig. 2e). This is ascribed to the sliding friction metamaterial consisting of 5 × 6 Taiji gears (Fig. 2h) by adopting between the meshed teeth (Supplementary Fig. 6). Figure 2f,g dem- the projection micro-stereolithography 3D printing technique. onstrates that experimental results of E (θ) are in excellent agree- The diameter and tooth thickness of the Taiji gear are 3.6 mm and ment with FEA. The modulation period is 180° in both the P (3°) 235 µ m, respectively; the thickest arm is 75 µ m (Supplementary − + and P (15°) cases. The smooth E (θ) curve indicates that the obtain- Fig. 8). The micro gears are arranged with P (0°), and the reserved able stable states are dense and that continuous tunability is achieved. minimal clearance between the teeth is 32 µ m. The sample is made Both polarity and β affect the tunable range and the correlation of a photosensitive resin with a Young’s modulus of 3.5 GPa. As between the tunable properties and θ (Fig. 2d). For P (3°) in Fig. 2f, experimentally demonstrated in Fig. 2i, the equivalent modulus the zigzag curve of E (θ) reaches the maximum value E = 7.67 GPa E (θ) of this micro specimen can be smoothly tuned by 35 times y max y at θ = 78°, where the solid parts of the gears are in contact, and a (from 8.3 MPa to 295 MPa). Using this integrated design strategy, minimal value E = 0.102 GPa at θ = 114°, where the meshing con- gear-based metamaterials could be scaled up in size and number min nects the forearms. This experimentally obtained modulation range of gears with appropriate high-resolution large-scale 3D print- of E /E = 75 demonstrates the spectacular reconfigurability. ing facilities. Modulation of such integrated metamaterials can be max min For P (15°), E (θ) is sombrero-shaped with a tunable range of 33 achieved with distributed drives or motors (Supplementary Video from E = 0.156 GPa to E = 5.13 GPa. Since E (θ) = E (θ + 90°), 2 and Supplementary Fig. 9a). min max x y Na Ture Ma TeriaLs | VOL 21 | AUGUST 2022 | 869–876 | www.nature.com/naturematerials Rod Beam Plate node Fixed Continuous Rotation Large range log (E (θ, β)) 10 y NATurE MATEriAlS Articles a b c d θ = 0 1.0 0.5 –θ –0.5 Frame –1.0 0 60 120 180 θ (°) Stress, σ e f g P (β = 3°) 2.5 P (β = 15°) FEA θ = 78° 2.0 Experiment β θ = 56° 1.5 θ = 30° 1.0 FEA θ = 12° 0.5 –1 Experiment 0 0.05 0.10 0.15 0.20 0 30 60 90 120 150 180 0 30 60 90 120 150 180 ε (%) θ (°) θ (°) h i P (β = 0) FEA 1 Experiment 0 30 60 90 120 150 180 215 μm θ (°) Fig. 2 | Mechanical metamaterial based on Taiji gears. a, Metamaterial architecture. The blue and green colours show different orientations of gears. b, A pair of meshing gears for P (3°). The x, y and z denote the global coordinates; the origin of the local cylindrical coordinates for spin rotation θ is set at the gear centre. t , the thickness of the arm. c, Typical Mises stress contour of a meshing pair for θ = 0 and compressive strain ε = 0.18%. Deformation is enlarged 20 times. d, Numerical value of log (E (θ, β)) for P (β). e, Measured loading–unloading stress–strain curves for metamaterials with different 10 y + − rotation angles θ. f,g, Young’s modulus as a function of the gear rotation angle θ for positive P (3°) and negative P (15°) polarities, respectively. Inset: the − + meshing pair for P (15°) h, Directly printed micro metamaterial consisting of P (0°) 5 × 6 Taiji gears, with close-up images showing dimensions. i, E (θ) of the P (0°) micro specimen. The error bars and the average values in f,g and i are evaluated by choosing different intervals along the curve in e. Metamaterial based on planetary gears tunability emerges from the relative rotation of the gears inside the Obviously, this first metamaterial is tunable only under compres- metacell. The thickness of the ring t is uniform. Neighbouring rings sive loading. The tensile load is carried by the frame, and the tensile are rigidly connected in a quadratic lattice, which ensures structural modulus is E = E . One may also aim at strong metamaterials whose integrity. Planetary gears revolve along the ring when rotating the t f compressive and tensile moduli are both tunable while preserving sun gear by θ . Their position is given by the revolution angle sun structural integrity. This can be achieved by organizing a planetary θ = θ r /(R + r ), where R and r are the radii of the ring and pr sun sun in sun in sun gear system as a metacell (Fig. 3a). In this example, the metacell sun gears, respectively (Supplementary Table 1 for parameter val- contains six gears: an inner-toothed ring gear (Supplementary Fig. ues). The teeth prevent relative slippage between the two gears even 10), a central sun gear and two pairs of planetary gears A –A and under tension. The metamaterial elastic properties are given by the 1 2 B –B . Gear centres A –O–A (and B –O–B ) are colinear. Using this effective stiffness of the annulus ring supported by the planetary 1 2 1 2 1 2 gear cluster, we create a hierarchical and strong metamaterial whose gears that act as fulcrums (Fig. 3b). The position θ of the planetary pr Na Ture Ma TeriaLs | VOL 21 | AUGUST 2022 | 869–876 | www.nature.com/naturematerials 75 μm Elastic arm 3.6 mm σ (MPa) 8 mm E (GPa) E (MPa) E (GPa) y y β (°) Planetary gear Ring gear NATurE MATEriAlS Articles a b Ring Compress Compress Stretch Both 1 y F F x x O x 0° 22.5° 22.5° 45° B Sun gear Transmission gears c d e f 950 μm Planetary gear 400 μm 20 mm 20 mm 5 mm Sun gear g h i FEA: compress FEA: compress Compress FEA: stretch FEA: stretch Experiment: compress 4 Experiment: compress Experiment: stretch Experiment: stretch θ = 0° θ = 45° 73 times –0.4 0.4 –1 25 times ε (%) 10 –1 0 15 30 45 60 75 90 0 15 30 45 60 75 90 Stretch y θ (°) θ (°) pr pr Fig. 3 | Metamaterial consisting of planetary gear systems. a, Metacell, a planetary gear system. The origin (O) of the coordinates x,y is the centre of the sun gear (green). A and B denote the centres of planetary gears (grey). Their revolution angle denotes θ  = ∠YOA . The case A A  ⊥ B B is drawn here, i i pr 1 1 2 1 2 and thus θ  = ∠XOB  = ∠YOA . A transmission gear (yellow) connects to the sun gear through a shaft. b, Typical compressive and tensile deformation of pr 1 1 a metacell. c, Macro metallic metamaterial architecture. d, Macro polymer metamaterial with 6 × 6 metacells. e, Micro polymer metamaterial with 3 × 4 metacells. f, Micrograph for the metamaterial in e. Specimens in d and e are fabricated with integrated manufacturing. g, Measured strain–stress curves. h,i, Measured and simulated Young’s moduli under compression and tensile deformation, E and E , of the macro metallic (h) and micro (i) specimens. c t Results of the macro polymer sample are shown in Supplementary Fig. 12b. The error bars and the average values in h and i are evaluated by choosing different intervals near the maximum strain along the curve in g. gears determines the elastic properties. We adopt the orthogonal because four pairs of meshing teeth in a metacell bear loads. Both relation A A ⊥ B B , which gives a large tunable range and symmet- the compressive and tensile moduli E and E reach maxima at θ = 0, 1 2 1 2 c t pr max max ric behaviour with a modulation period of 90°. The tunable range but E ≫ E , and thus the static compressive-tension symme- c t can be further modified using the angle ∠A OB . For the assembled try is broken (Fig. 3g). Moreover, at θ = 45°, no stress is transmitted 1 1 pr metamaterial, all sun gears are connected to transmission gears by to the planetary gears (Fig. 3b); both moduli reach minima there, min min shafts (Fig. 3a), and those transmission gears are compactly cou- and . E = E c t pled. Thereby robust reconfiguration of all metacell patterns can be We fabricate three kinds of specimens using this strategy. An achieved by rotating transmission gears. all-steel macro metamaterial is manufactured by assembling 3 × 3 F and F denote the compressive loads in the x and y directions, metacells (Fig. 3c) with lattice constants a = a = 27 mm. The steel x y x y respectively (Fig. 3a). Under uniaxial compression (F > 0, F = 0), gears have a small tooth thickness t = 0.15π mm, and R = 12 mm, y x to in only the pair of planetary gears with an angle smaller than that of r = 6 mm and t = 1 mm. Integrated manufacturing of this proto- sun r the loading axial (min(∠YOA , ∠YOB ) < 45°) supports the load type is more challenging than that of the Taiji pattern because there 1 1 (Fig. 3b). Stress in the other pair is zero. Conversely, under uniaxial are two layers and every metacell possesses eight pairs of mesh- tension, only the other pair is load-bearing. The two pairs exchange ing teeth. The integrated prototype can also be directly manufac- roles at θ = 45°, and the material is orthogonally isotropic. This tured by 3D printing, at both macro and micro scales (Methods, pr metamaterial presents a more remarkable compressive nonlinearity Supplementary Figs. 11 and 12 and Supplementary Video 3 for Na Ture Ma TeriaLs | VOL 21 | AUGUST 2022 | 869–876 | www.nature.com/naturematerials sun pr pr σ (MPa) E (GPa) E (MPa) y NATurE MATEriAlS Articles a b c τ P (β = 3°) 0.45 0.40 0.35 τ 0.30 0.25 FEA 0.20 Experiment 0.15 0 30 60 90 120 150 180 Shear interlock θ (°) d f Free Cavity Cavity Geometrical interlock Gear ultrasoft matter 0 5 10 15 20 25 30 γ (%) Fig. 4 | strong or ultrasoft metamaterials under shear. a, Schematic shear deformation of a meshing pair. b, Four meshing gears in the state of shear interlock, θ = 60°. The circular arrows show the planetary and spin rotations, which block each other. Periodic boundary conditions are applied on this metacell (Supplementary Fig. 14). c, Theoretical and experimental shear stiffness K /B under shear interlock. The 3 × 3 architecture with P (3°) is shear studied here (Supplementary Fig. 15). The stiffness is strong (>150 MPa) and narrowly tunable (~3 times). The error bars and the average values are evaluated by choosing different intervals along the tested strain–stress curves. d,e, Real-life photographs for the deformation modes of the prototype under diagonal and surface compressions, respectively. The light blue, white and dark blue parts are the background, rubber frame and gears (d,e). f, Experimental strain–stress curves for the shear interlocked metamaterial, the ultrasoft gear matter and the rubber frame by itself. Inset: three blue disks illustrate the geometrical interlock, which occurs when three meshing gears form a closed triangle. more details). We print a 6 × 6 macro specimen (Fig. 3d) using a be synchronously controlled with distributed motors at both the polymer with a Young’s modulus of 2.5 GPa, and print a 3 × 4 micro macro and micro scales (Supplementary Video 4). specimen (Fig. 3e) using a resin with a Young’s modulus of 3.5 GPa. The size and the number of metacells are limited by the capability Mechanisms for stability of the 3D printer rather than the design strategy. The micro poly- Interestingly, the metamaterial in Fig. 2a (a discrete gear lattice with mer sample (Fig. 3e) has R = 2.4 mm, r = 0.6 mm, t = 0.3 mm a very soft frame) remains stable under compressive stresses and in sun r and a = a = 5.4 mm, with a tooth width and height of 135 µm and shows large rigidity in shear. One of the contributing factors under- x y 225 µm, respectively. pinning the observed stability stems from the non-uniform load- The experimental results are consistent with the FEA simula- ing of the meshing teeth at different points, which leads to bending tions for all specimens (Fig. 3g–i and Supplementary Fig. 11). In deformations that tightly grip the teeth together (Supplementary this strong hierarchical metamaterial, we can smoothly tune the Fig. 5). The relatively large shear modulus of the metamaterial, compressive modulus E of the macro metallic specimen by 46 G = G + G , is composed of the shear moduli generated by gears c g f times (5.2–0.11 GPa), the macro polymer specimen by 55 times (G ) and by frames (G = 1.04 MPa). Shear force induces both spin g f (69–1.25 MPa; Supplementary Fig. 12b) and the micro specimen and the planetary rotation of gears. For a pair of gears, the relative by 25 times (100–4 MPa). Meanwhile, their tensile modulus E can planetary rotation leads to zero shear resistance, G = 0, giving a t g be tuned by 5 times (0.52–0.11 GPa), 5.6 times (7–1.25 MPa) and 5 highly unstable state (Fig. 4a). However, in a group of four gears times (20–4 MPa), respectively. In Fig. 3h, some differences between (shown in Fig. 4b), shear stress τ induces mutual locking of the plan- experiment and FEA near θ = 45° arise from the boundary condi- etary rotation by the opposite spin of the neighbouring gears, which pr tions (Supplementary Fig. 13). The in situ tunability combined with is referred to as shear interlock. We calculate the shear stiffness the reasonably large moduli in tension and compression as well as of the metacell with periodic boundary conditions and the finite large shear rigidity makes this metamaterial design particularly n × n gear lattice (Supplementary Figs. 14–16). Owing to the shear robust and strong, yet tunable. Furthermore, the metamaterials can interlock, the shear modulus is large but only marginally tunable Na Ture Ma TeriaLs | VOL 21 | AUGUST 2022 | 869–876 | www.nature.com/naturematerials Rubber frame Frame Shear interlock τ (kPa) K /B (GPa) shear NATurE MATEriAlS Articles a b 4 4 10 10 Gear-based metamaterials 3 3 10 10 Thermal- (35) responsive Thermal-responsive metamaterials 2 metamaterials 2 10 10 1 1 10 10 (36) (36) (37) (30) 0 0 (31) 10 10 (30) (31) (34) (37) Chemical- responsive MMs –1 –1 10 10 Shape-morphing (38) metamaterials (38) Chemical-responsive MR Magneto- Shape-morphing –2 –2 (18) metamaterials: 10 10 fluids responsive MMs metamaterials hydrogels (18) In situ –3 –3 10 10 ε = 0 10 50 100 150 200 1 10 100 1,000 10,000 Required strain, ε (%) Shortest response time (s) Fig. 5 | Properties of active mechanical metamaterials. a,b, Tunable effective modulus versus the required strain ε (a) and shortest response time (b) for tunability. The shaded colour regions (rectangles and ovals) represent the possible tunable ranges of different tunable materials or metamaterials. Pale green, gear-based metamaterials; yellow, chemical-responsive metamaterials; blue, magneto-responsive metamaterials, brown: shape-morphing metamaterials; grey, thermal-responsive metamaterials. These regions partially overlap. The straight lines indicate the tunable ranges. The solid (dashed) lines represent the continuous (non-continuous) tunability. Arrows represent uncertainties. Numbers in parentheses denote the references. The green shaded region at the bottom in a signifies that the required strain is zero, that is, the in situ tunability. MMs, metamaterials; MR, magnetorheological. Properties of our gear-based metamaterials are shown by red lines and are also labelled by figure numbers. (Supplementary Fig. 16), which is demonstrated by the measured formance and efficiency in variable environments. Programmable generalized shear stiffness K /B of the finite 3 × 3 architecture materials featuring tunable elastic properties, including active shear in Fig. 4c. mechanical metamaterials, are much anticipated in intelligent 1,2 machines and systems . Here we offer a comparison of typical 2,19 Gear metamaterial for shape morphing material designs from the literature . The programmability of gear-based metamaterials is not limited The response time, stability, force and energy required for to elastic constants. Removing every second gear in every second property changes are all critical attributes for variable-stiffness row of the metamaterial in Fig. 2 can release the shear interlock structures . We take the strain and response time required to (inset in Fig. 4f ) to generate a state with G = 0. The effective shear accomplish a tunable period of material stiffness as metrics to posi- modulus is then determined solely by the low stiffness frame, and tion our gear-based metamaterials among the existing active mate- 18,34 the metamaterial can be considered as ultrasoft matter. The van- rials (Fig. 5). Shape-morphing metamaterials enable tunability ishing shear modulus enables complex deformation modes (Fig. between or among two or a few stable states. The achieved tunabil- 4d,e and Supplementary Video 5), conducive to shape morphing. ity is non-continuous and requires a large deformation (ε ≈ 30%). To verify this, an ultrasoft prototype consisting of 4 × 4 metacells Thermal-responsive composites made of shape-memory alloys or with rubber frames is manufactured and tested (Fig. 4f ). The gears polymers may give a continuously tunable modulus. However, they are made of aluminium alloy (Supplementary Fig. 17). Shear tests require a long response time, and some suffer from nearly 100% 36,37 30,31 on the prototype give a tiny modulus of G = 21.52 kPa. Independent strain . Chemical-responsive materials containing hydrogels measurement of the frames gives G = 21.11 kPa, so that G = G – G can offer continuous and in situ tunability, but also require hours of f g f is indeed negligibly small. Moreover, the modulus G remains tiny response time. Conventional magneto- or electro-responsive meta- until the shear strain γ reaches 25%, at which point the semi-free materials based on elastomers or magnetorheological fluids can gear interacts with two other gears, which builds a new meshing give fast, continuous, but narrow tunability, which usually requires a connection among the three gears. The new connection supports high active voltage (~5 kV) and complex facilities . Our gear-based high shear stresses and leads to a sharp rise in G, switching the soft metamaterials in Figs. 2 and 3 can offer a fast response and the matter to a stiff solid. The resulting solid represents a geometrically desired broad-range, continuous and in situ tunability of stiffness. interlocked state (Methods). Oscillations of the shear stress in Fig. We propose several scenarios to showcase the broad appli- 4c arise from the critical meshing state among the three gears before cation potential of the proposed gear-based metamaterials in they interlock (Supplementary Fig. 18). The shear strain at which Supplementary Figs. 19–21 and Supplementary Table 2. For robots, the geometrical interlock occurs can be adjusted by the size of the a tunable-stiffness leg/actuator can offer high stiffness to stably sup- neighbouring gears, which in turn determines the limiting (strong) port a heavy load while walking and a low stiffness for shock pro- 4,39 states of a shape-morphing structure. Previous metamaterial tection while jumping or running (Supplementary Figs. 19 and designs with vanishing shear modulus, like pentamode metamateri- 20). A similar tunable-stiffness isolator is desired in the aero-engine als , show vanishing moduli only at small strains and in extremely pylon system to maintain the best performance and efficiency fragile structures. at different flight stages (Supplementary Fig. 21). Moreover, the fast-response gear-based metamaterial may give rise to a sensitive Potential applications variable-stiffness skin, which has been attracting wide attention . Conventional machines generally rely on materials with constant Furthermore, resonators with tunable stiffness are critical compo- 41,42 stiffness and therefore show constant stiffness themselves. The nents in programmable metamaterials for wave manipulation . designed stiffness is then a compromise among stability, safety, effi- Therefore, gear-based programmable metamaterials can aid in the ciency and performance, thus hindering the pursuit of the best per- realization of extensive intelligent machines. In contrast with con- Na Ture Ma TeriaLs | VOL 21 | AUGUST 2022 | 869–876 | www.nature.com/naturematerials Effective modulus (MPa) Fig. 2f Fig. 2i Fig. 2g Fig. 3i Fig. 3h Fig. 3d Fig. 4f Effective modulus (MPa) Fig. 2i Fig. 3i Fig. 3d Electro-thermal composite NATurE MATEriAlS Articles 13. Berger, J. B., Wadley, H. N. G. & McMeeking, R. M. Mechanical ventional methods, the programmability enabled by a gear-based metamaterials at the theoretical limit of isotropic elastic stiffness. Nature 543, metamaterial does not require large deformation and heavy control- 533–537 (2017). ling systems, such as hydraulic/pneumatic or magnetic systems, and 14. Fernandez-Corbaton, I. et al. 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The images or other 1373–1377 (2014). third party material in this article are included in the article’s Creative Commons license, Na Ture Ma TeriaLs | VOL 21 | AUGUST 2022 | 869–876 | www.nature.com/naturematerials 875 NATurE MATEriAlS Articles unless indicated otherwise in a credit line to the material. If material is not included in from the copyright holder. To view a copy of this license, visit http://creativecommons. the article’s Creative Commons license and your intended use is not permitted by statu- org/licenses/by/4.0/. tory regulation or exceeds the permitted use, you will need to obtain permission directly © The Author(s) 2022 Na Ture Ma TeriaLs | VOL 21 | AUGUST 2022 | 869–876 | www.nature.com/naturematerials 876 NATurE MATEriAlS Articles show the shear state of a finite n × n gear lattice, we fix the lower row/column of Methods the gears and apply a displacement field to the upper row/column of the gears. Integrated manufacturing. The printer used for the projection As a second method, periodic boundary conditions are applied on a metacell to micro-stereolithography micro metamaterial fabrication is a BMF NanoArch S130, calculate the shear modulus. These periodic boundary conditions present the with a precision of about 5 µm. The material used in microscale 3D printing is a shearing state ε = (0, 0, γ). In both cases, the strain energy density W = Gγ /2 is photosensitive resin with a Young’s modulus of about 3.5 GPa. The manufacturing extracted to evaluate the shear modulus G. For the n × n finite structure without process for the integrated micro metamaterial sample consisting of 5 × 6 Taiji gears periodic boundary conditions, although the equation of the generalized shear follows three steps. First, the assembled gears are printed on a baseplate; those stiffness G′ = K /B is the same as the formula for shear modulus G = τ/γ, gears are adhered to the plate. Second, the sample is wrapped in a box to constrain shear the value of G′ may not equal the real shear modulus G (Supplementary Fig. the motion of the gears (Supplementary Fig. 9a). Last, everything including the box 15) due to the free edge effects in finite structures (Supplementary Figs. 13a is removed from the plate. The box with a frame helps maintain the relative angle and 14a). of the assembly in the removal process. For the metamaterial based on a planetary gear system, the load is applied For the metamaterial based on planetary gears, the layer of the planetary gears on the four blocks of the ring. For a metacell in FEA, we specify the uniaxial is printed first (Supplementary Fig. 12a and Supplementary Video 3). Except for deformation v = ε a and make ε free for solving E . the preserved clearance, the connection shaft between the transmission gear and y y x y the sun gear is conical at both the macro and micro scales (Supplementary Fig. Equivalent method. For the metamaterial based on Taiji gears, the deformation 10), which ensures that every printing part, especially the teeth of the transmission mode for meshing gears can be represented by the overall stiffness of a pair of gears, is tightly attached on the formed structure. Otherwise, the teeth could move meshing elastic arms K = 1/(1/k + 1/k + 1/k ) (Supplementary Fig. 3 for and then fuse together during the printing. At the macro scale, the integrated p arm1 arm2 tooth their definitions). The stiffnesses of the two arms k and k are independent of arm1 arm2 model is printed with two photosensitive resins using polymer injection with the compressive deformation. As shown in Supplementary Fig. 5, the meshing of a the printer Stratasys Objet260, with a precision of about 50 µm. The stiff model pair of teeth features a line of contact on their surfaces. With compression, a small material is wrapped in the soft, soluble support material. The metamaterial contact area is generated near the line where sliding occurs during the process. acquires the targeted tunability after removing the support material. At the Therefore, the contact stiffness of the teeth k depends on the contact pressure microscale, the material is immersed in the fluid resin during the printing. No tooth on the involute teeth. A high pressure leads to significant contact nonlinearity and support material/structure is required for this model owing to the conical shaft and results in a dependence of K on the displacement/load. By contrast, deformation high precision of projection micro-stereolithography. The sample shown in Fig. 3d mainly occurs in the elastic arm rather than the teeth if k ≪ k , and K is is printed with a resin (polymer) with a Young’s modulus of 2.5 GPa. arm tooth p constant in this case. The homogenized Young’s modulus in the y direction of In integrated manufacturing, the clearance reserved between the surfaces of the metamaterial is E = K /B + E . The equivalent methods for shear modulus are meshing teeth in the assembled digital model depends on the precision of the y p f explained with Supplementary Fig. 14. printer, the structure and the materials. The minimal clearance Δ should be higher For the metamaterial consisting of a periodic planetary gear cluster, the Young’s than the printer’s precision p (manufacturing errors) but much smaller than the modulus depends on the deformation of the ring. The influences of contact tooth height (h = 2.25m for standard gears), where the gear module m denotes nonlinearity between teeth on E are the same as described above. the ratio between the gear diameter D and the number of teeth z, m = D/z (see Supplementary Text). Here, the minimal clearance between meshing teeth in all Geometrical interlock. In a meshing pair, the rotation directions of the driving macro specimens printed with Objet260 is 86 µm. The minimal clearance for the and driven gears are opposite. In a group of gears, if every meshing is viewed as a micro metamaterial consisting of Taiji gears is set to 32 µm, and that for the micro connection line, n gears form a closed polygon as shown in Fig. 4f. If n is odd, spin planetary gear-based metamaterial is 21 µm. These clearances are sufficient to rotation is incompatible, leading to the locking among the gears. This meshing alleviate the manufacturing uncertainties to keep the meshing teeth separated state is referred to as geometrical interlock. but reliably engaged. Based on our 3D printers and tests, we suggest Δ > 1.5p and Δ ≤ h /10 = 0.225m. This requires p < 0.15m, which helps us determine the required Mechanical tests for Young’s modulus. When measuring the Young’s modulus precision scale with a specified gear size. E in the metamaterial based on Taiji gears, a compressive load F is applied and y y released from the top of the prototype in Fig. 2a. We control the strain ε for Actuation. As shown in Supplementary Video 2, we prepare a microscale sample different θ to overcome clearance nonlinearity while avoiding plastic deformation. consisting of 5 × 5 Taiji gears to show its actuation process. They are embedded The rotation angle θ is manually controlled. Similar cyclic loading–unloading tests into a box, and those gears connect to the frames through micro shafts. The are performed for the measurement of the shear modulus. sample is synchronously driven by four d.c. brushless motors (8 mm diameter) The experimental setting for the test on the metamaterial based on planetary connected to the 1 × 1st, 1 × 4th, 4 × 1st and 4 × 4th gears. Here n × m denotes the gear systems is shown in Supplementary Fig. 9. When measuring the compressive position at the nth row and mth column in the array. As shown in Supplementary modulus, a compressive load is applied on the top and the bottom blocks on the Video 4, the macro metamaterial in Fig. 3d is synchronously actuated by four-step rings; when measuring the tensile modulus, we fix the tails on the sample to a pair motors whose diameter is 20 mm. These motors are synchronously controlled by of clamps and apply tensile loads through the tails. an electronic controller. The revolving speed of the step motor depends on the As shown in Figs. 2e and 3g, the cyclic loading–unloading process features high impulse frequency generated by the controller. Similarly, the micro sample in Fig. repeatability, thus testifying to the experimental accuracy. Moduli E and G are 3e is put in a box and actuated by five micro step motors whose diameter is 5 mm. both calculated as the slope around the maximum ε. The initial cycle is excluded The controller is identical to the one used for the macro sample. when fitting E and G. The choice of the strain interval for the slope calculation affects the final modulus value. The error bars and the average values are evaluated FEA. FEA simulations are carried out with the commercial software ANSYS. We by choosing different intervals along the curve. compare the accuracy of different finite element models, including two-dimensional (2D), 3D, linear and nonlinear models. The plane stress state is considered in the Mechanical tests for shear stiffness. For the metamaterial based on Taiji 2D model. In the linear models, the meshing points of gears are bonded by fixing gears, a sample consisting of 3 × 3 gears and steel frames is manufactured for together the two surfaces in contact, resulting in a linear stress–strain relationship. the measurement of the shear stiffness in the shear interlock state, as shown in In the nonlinear models, the size of the contact area on the tooth surface at the Supplementary Fig. 15. A fixture apparatus is fabricated to obtain the shearing meshing points depends on the load, and there is a relative sliding between the state. For the shape-morphing metamaterial, the sample is put in two right-angle contact surfaces. The sliding induces frictional damping if the coefficient of friction grooves, and the load from the testing machine directly transfers to the sample. is non-zero. We also use a simplified model by removing all teeth, where the contact between two gears becomes that between two cylinders. In principle, the 3D nonlinear model should be the most realistic representation of the experimental Data availability set-up. Supplementary Fig. 4 demonstrates that the 2D nonlinear model is in The main data and models supporting the findings of this study are available within excellent agreement with the 3D nonlinear model. The two linear models produce a the paper and Supplementary Information. Further information is available from large discrepancy with the nonlinear ones, although they still can capture the general the corresponding authors upon reasonable request. variation trend. The simplified model approximately presents the standard results. To enhance the simulation efficiency, we use the 2D nonlinear models in most cases. a cknowledgements The 3D model is adopted only when considering the frictional contact. This research was funded by the National Natural Science Foundation of China (projects Our metamaterials embrace a periodic architecture. To evaluate the no. 12002371 and no. 11991032), the Hong Kong Scholars Program, the Fraunhofer homogenized elastic and shear moduli, ideal periodic boundary conditions Cluster of Excellence ‘Programmable Materials’ and the Excellence Cluster EXC 2082 ‘3D are applied on the unit cell in the FEA. Boundary conditions depend on the Matter Made to Order’ (3DMM2O) in Germany. deformation mode of the unit cell. The homogenized strain vector is ε = (ε , ε , γ). x y These strains are realized by enforcing the displacement fields (u, v) in the plane stress state. a uthor contributions As explained in Supplementary Fig. 14, two types of boundary condition are X.F. and P.G. designed the study. X.F. conceived the idea and performed the experiments. considered when calculating the shear modulus in the shear interlock state. To X.F. and P.G. carried out the numerical simulations. L.C., D.Y. and H.Z. analysed the Na Ture Ma TeriaLs | www.nature.com/naturematerials NATurE MATEriAlS Articles data. All authors interpreted the results. X.F., L.C., J.W. and P.G. wrote the manuscript a dditional information with input from all authors. P.G. supervised the study. Supplementary information The online version contains supplementary material available at https://doi.org/10.1038/s41563-022-01269-3. Funding Correspondence and requests for materials should be addressed to Open access funding provided by Fraunhofer-Gesellschaft zur Förderung der Xin Fang, Jihong Wen or Peter Gumbsch. angewandten Forschung e.V. Peer review information Nature Materials thanks Amir A. Zadpoor and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Competing interests Reprints and permissions information is available at www.nature.com/reprints. The authors declare no competing interests. Na Ture Ma TeriaLs | www.nature.com/naturematerials

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Nature MaterialsSpringer Journals

Published: Aug 1, 2022

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