Large recoverable elastic energy in chiral metamaterials via twist buckling | Nature

Nature (2025)Cite this article

Metrics details

Mechanical metamaterials with high recoverable elastic energy density, which we refer to as high-enthalpy elastic metamaterials, can offer many enhanced properties, including efficient mechanical energy storage1,2, load-bearing capability, impact resistance and motion agility. These qualities make them ideal for lightweight, miniaturized and multi-functional structures3,4,5,6,7,8. However, achieving high enthalpy is challenging, as it requires combining conflicting properties: high stiffness, high strength and large recoverable strain9,10,11. Here, to address this challenge, we construct high-enthalpy elastic metamaterials from freely rotatable chiral metacells. Compared with existing non-chiral lattices, the non-optimized chiral metamaterials simultaneously maintain high stiffness, sustain larger recoverable strain, offer a wider buckling plateau, improve the buckling strength by 5–10 times, enhance enthalpy by 2–160 times and increase energy per mass by 2–32 times. These improvements arise from torsional buckling deformation that is triggered by chirality and is absent in conventional metamaterials. This deformation mode stores considerable additional energy while having a minimal impact on peak stresses that define material failure. Our findings identify a mechanism and provide insight into the design of metamaterials and structures with high mechanical energy storage capacity, a fundamental and general problem of broad engineering interest.

Materials with high recoverable elastic energy density (that is, enthalpy ϕ) are in demand for many applications, but this requires a combination of high modulus (Es), high strength and large recoverable strain (εlimit)12,13. Carbon nanotubes reach extremes at the nanoscale14,15, yet achieving high enthalpy at the macroscale remains challenging16,17. Metamaterials18,19,20,21 can offer extraordinary stiffness, shape control22,23 and wave manipulation24,25 by tailoring their topologies and deformation modes18,26, including elastic buckling27,28. A higher-yet-wider buckling plateau on load-deformation curves is desirable for greater energy storage (Fig. 1h). However, in existing designs (Fig. 1c–e), achieving large recoverable strain is accompanied by a reduction of plateau stress, even for tensegrity lattices29. This limitation arises from bending-dominated buckling modes, which limits macroscopic stresses while generating large local strains that drive failure. Here we propose an approach to circumvent this limitation and improve recoverable enthalpy: chiral structures that incorporate twisting, compression and bending.

a, A 1/2-order lateral bending buckling of clamped-sliding rods under compression force F1rod, observed in chiral metamaterials in f. b, A first-order buckling mode of doubly clamped vertical and oblique rods in non-chiral lattices. The colour bars in a, b and f show the distribution of Mises stress σv. c–e, Octahedron (c), prism (d) and tensegrity (e) lattices. f, Compression-twist buckling of chiral metacells. The cente schematic shows a purple straight chiral arm buckling into a red curved arm with a global twisting angle (θ). g, Multilayered and 3D chiral metamaterials consisting of decoupled metacells. h, Schematic showing the enhanced energy density (ϕ), load-bearing strength (σbk) and high stiffness (E) of chiral metamaterials.

Elastic energy in conventional rod-based lattices (Fig. 1c–e) is stored in rods that generally exhibit the first buckling mode when compressed (Fig. 1b and Methods). We use analytical modelling and nonlinear finite element analyses (FEAs) to explain the scaling of energy storage during buckling, which are then extended to include torsion30 (Methods and Supplementary Notes).

To begin, consider a doubly clamped rod of radius r, length L0 and Young’s modulus Es subjected to a vertical compression force (F1rod) or axial displacement (Δ): the rod responds initially in pure compression followed by bending buckling (Supplementary Video 1). This individually induces maximal von Mises stress σv = σcpr and σv = σcpr + σbend on the outer surface, with σcpr from compression and σbend from bending. Failure is assumed to be controlled by σv. The analytical model shows that a buckled rod stores strain energy

F1rod and U1rod are normalized by \({F}_{{\rm{s}}}={E}_{{\rm{s}}}{\rm{\pi }}{r}^{2}/4{L}_{0}^{2}\) and Us = πr2L0Es/1,000, respectively.

Straight rods undergo sudden buckling, whereas small imperfections smoothen the buckling process30 with a slightly reduced plateau load, max(F1rod) (Fig. 2a and Extended Data Fig. 2a,b). Unless otherwise stated, we use rods with moderate slenderness ratio of 2r/L0 = 0.1 and r = 1.5 mm for comparison. Properties of thick rods are provided in Extended Data Figs. 3 and 6.

a,b, Angled non-chiral rods. c–f, Chiral rods. a, Normalized force (F1rod/Fs) versus macroscopic strain (ε = s/L0 sin θ) of angled rods with oblique angle θ. Micro-bending imperfection is included here. b, Normalized energy (U1rod/Us) under different maximal von Mises stresses (σv) versus θ of perfectly straight rods. c, Theoretical force–strain curves of typical chiral models. Filled circles are points for σv = 0.1Es. None of the curves reach σv = 0.2Es at ε = 0.2. d, Proportions of energy from different deformation modes. e, Energy–stress plot of straight rods with 2r/L0 = 0.1 (U1rod/Us versus σv/Es). Discrete points are FEA results for R = 7 mm. f, Influence of initial angle (α0) and radius (R) on U1rod/Us at ε = 0.2.

Lattices consisting of angled rods (Fig. 1c–e) show improved stiffness and max(F1rod) with increased oblique angle (θ) (Fig. 2a). However, in buckled rods, σcpr ≪ σbend, resulting in nearly constant energy \({U}_{1{\rm{r}}{\rm{o}}{\rm{d}}}\approx {\rm{\pi }}{r}^{2}{L}_{0}{{\sigma }}_{{\rm{v}}}^{2}/16{E}_{{\rm{s}}}\) for a given stress σv, regardless of the angle θ (Fig. 2b). Moreover, Ubend in equation (1) is independent of buckling order (n). Thus, adjusting the oblique angle or inducing high-order buckling modes (n > 1) can change max(F1rod) but cannot effectively improve U1rod for a specified material strength (σv). This severely limits energy storage in strut-based metamaterials.

To overcome this limitation, we must introduce additional deformation modes beyond bending and compression. Chiral metamaterials31,32,33, with coupling between axial deformation and twisting, offer exciting possibilities to store more energy by torsion. The potential of these structures is defined by their post-buckling behaviour and its implications on force, stress and energy34,35,36. We propose chiral metamaterials with independent and rotatable cylindrical metacells (Fig. 1g) that are distinct from previous chiral structures37,38,39. They exhibit high-energy buckling modes that improve enthalpy (ϕ) and plateau strength (σbk) for fixed material strength (Fig. 1h).

Here we consider a mirror-symmetric metacell (Fig. 1f and Extended Data Fig. 1), in which every half-cell incorporates two coaxial tori (O1, O2) with radii R1 and R2 at a distance h0. N oblique rods, serving as chiral arms, are periodically fixed to the tori. At rest, the two tori are relatively rotated by an angle α0. When fixing torus O1 and vertically compressing torus O2 by s, a relative rotation angle θ ensues between them (Supplementary Video 2). The global strain ε = s/h0 is used throughout. Chiral buckling causes the twisted rod to bulge outwards and forms a helix under large compression (Fig. 1f). We define this mode as twist buckling.

We established an analytical model (Methods) that fully captures post-buckling behaviours of chiral rods (Fig. 2c,e and Extended Data Fig. 4). This model illustrates that chiral twist buckling involves four deformation modes in each rod: in-plane bending, out-of-plane bending, twisting and compression (Extended Data Fig. 1). The stored elastic energy can be written as

where 2ain, 2aout and Δcpr are the maximal amplitudes of in-plane, out-of-plane and compression deformation modes, respectively; υ is Poisson’s ratio; and Lx is the length of the twisted rod.

Both in-plane and out-of-plane bending follow the 1/2-order buckling mode with clamped-sliding ends (Fig. 1a). Out-of-plane bending couples with global twisting between the two tori, introducing two more energy sources related to aout and angle θ that are absent in non-chiral structures in equation (1). Chiral metamaterials show high stiffness at small strain ε < 0.02, at which the rods and torus are twisted equally by θ. They buckle smoothly with increasing load instead of abruptly as with purely bending buckling. Increasing α0 gives a smoother force-deformation curve (Fig. 2c). For unequal torus radii (R1 ≠ R2), both plateau load max(F1rod) and energy (U1rod) are lower than the case of equal radii (Extended Data Fig. 3a–c). We consider R = R1 = R2 in the following. For specified ε, reducing α0 or R enhances max(F1rod) and U1rod, with R having greater effects (Fig. 2f and Extended Data Fig. 5). Performance evaluation is based on the limitation of maximal von Mises stress (σv) in chiral rods, which occurs in the twisted ribbon region on the rod surface (Fig. 1f).

In prismatic or octahedron lattices, rods with angles θ > 45° may not buckle within σv < 0.1Es (Fig. 2a). Here, the same chiral rod with R = 6 mm and α0 = 5° gives max(F1rod/Fs) = 0.8 (Fig. 2c), surpassing that of a non-chiral rod with θ = 40° by 30%. The global limit strain εlimit for σv = 0.1Es also increases from 0.076 to 0.1, providing a broader safe plateau for storing more energy.

The stress–energy plot of the moderate slenderness rod (Fig. 2e) shows the superior energy storage of chiral buckling compared with bending buckling for specific strength σv. Different chiral models show convergent stress–energy curves, indicating robust performance. As non-chiral rods store energy by pure compression before buckling, they can store slightly more energy than chiral rods near the buckling point if imperfections can be avoided (Extended Data Fig. 2b). However, beyond that point, chiral rods store more energy. Larger deformation or allowable stress further enhances this advantage. For σv = 0.2Es, a chiral rod stores twice the energy of a non-chiral rod.

This advantage is greater in thinner rods because of their low buckling thresholds (Extended Data Fig. 3d). Metamaterials consisting of thick rods may favour non-chiral designs if suitable constraints ensure the ideal first buckling. Otherwise, their rods generally follow the 1/2-order lateral buckling mode, which has only 1/16 of the critical energy compared with the first mode (Methods). In this case, chiral rods still store about two to four times higher energy (Extended Data Fig. 6).

We can map the twist buckling of freely rotatable chiral metacell into the micropolar elasticity framework40,41,42 (Supplementary Fig. 8). It suggests that chirality, defined as the coupling of compression and global torsion, dominates the properties under large compression (Supplementary Note 4). This is confirmed by the energy proportions (η) of the four deformation modes predicted by the chiral twist buckling theory (Fig. 2d). Axial compression dominates for ε < 0.02, but as deformation increases, contributions from in-plane bending and in-rod twisting rise. At ε > 0.05, the in-rod twisting accounts for 40% energy, as confirmed by phased loading FEA in Extended Data Fig. 7. In-rod twisting together with out-of-plane bending—the two modes absent in non-chiral lattices—can contribute 55% energy. Therefore, the enhanced performance of chiral rods is attributed to invoking torsion.

Furthermore, chiral rods have a favourable stress distribution. Although bending causes maximal stresses at the rod ends in confined sections (Fig. 1a), the in-plane and out-of-plane bending modes are orthogonal and do not compound stress. The maximal normal stress is given by σnorm = σin + σcpr, with in-plane being the dominant contributor. Torsion induces nearly uniform shear stress (τ) perpendicular to σnorm across the entire rod surface. Thus, the maximal von Mises stress becomes

As τ ≪ σin, invoking torsion adds stored energy while only marginally raising σv (Fig. 2e and Methods). Reducing α0 or R increases F1rod, U1rod and the contribution from twisting (Extended Data Fig. 5).

Beyond a single rod, the entire chiral metamaterial shows greater improvement for specified volume (Methods). We compare the performance of chiral metamaterials to extensive non-chiral models using FEA and analytical methods (Fig. 3a,f). They consist of identical rods unless otherwise specified. In theory, half a chiral metacell can tightly pack πR/r rods around its torus. Here the internal core remains vacant and we only take half the number, N = πR/2r, ensuring ample space to avoid contact between arms. By contrast, parallel rods in the prism lattice are densely arranged with a distance of 4r. All metamaterials are characterized by macroscopic properties: load-bearing strength σbk (Fig. 1h), limit deformability εlimit at rod strength, modulus E, energy density (enthalpy) ϕ and mass density ρ. They are normalized by the modulus (Es), enthalpy (ϕs = Es/1,000) or density (ρs) of rods (Methods).

εlimit denotes the global strain at material strength (σv). Performances of octahedron and prism lattices are controlled by the oblique angle (θ) of the rod, whose εlimit increases as θ decreases. For chiral metamaterials, h0 = 30 mm and radius (R) is variable. a–e, Rod-based metamaterials. a, Legend and deformation modes for metamaterial configurations in b–e. b, Ashby map of E/Es versus ρ/ρs. c, Bearing strength σbk = max(σeq) on the buckling plateau versus εlimit for σv = 0.2Es. d, Enthalpy (ϕ/ϕs) versus εlimit for σv = 0.1Es. e, ϕ/ϕs ~ εlimit for σv = 0.2Es. f–h, Plate- and beam-based metamaterials, in which the plate and square beam have the thickness t = 1 and length L0 = 30 mm. Max(σv) occurs at the corner edge of the twisted beam. f, Legend and deformation modes for configurations in g and h. g,h, σbk (g) and ϕ/ϕs (h) versus εlimit for σv = 0.09Es. Inset in g: an E–ρ map.

Bend-dominated tetradecahedron (Kelvin) trusses possess low stiffness and enthalpy, which can be improved by tensegrity design29. However, despite withstanding higher εlimit, the tensegrity trusses—even with doubled rod thickness—exhibit 50× lower modulus and 5× lower enthalpy than the octahedron lattice of equal density (Fig. 3b,d and Extended Data Fig. 8). In octahedron and prism lattices, all oblique rods are ideally assumed to follow the first buckling mode (Fig. 3a); reducing the oblique angle (θ) of the rod increases the recoverable strain εlimit (Fig. 2a) but affects the density (ρ/ρs) non-monotonously. Their stiffness and density follow the proportional law E/Es ~ (ρ/ρs)1 (where ~ means ‘distributed as’) for θ > 60° (Fig. 3b), in which the response is stretch-dominated18,43. Plotting σbk/Es and ϕ/ϕs against εlimit shows that these non-chiral lattices follow the undesirable trade-off laws between bearing strength σbk (or enthalpy ϕ) and εlimit (Fig. 3c–e). Although prism lattices outperform octahedron lattices in σbk and ϕ owing to larger ρ/ρs, their σbk and ϕ are low and nearly constant beyond a small εlimit.

Analytical studies show that reducing α0 or R increases chiral metamaterial E, σbk and ϕ (Extended Data Fig. 5). We verified this through simulations with R ranging from 5 mm to 10 mm (N = 5–10) for a specified height h0 = 30 mm. A moderate dimension around R = 8 mm offers the largest εlimit. The stiffness (E/Es) is comparable with high-density non-chiral lattices (Fig. 3b). Notably, chiral metamaterial maintains 5–10 times higher σbk and 5–20 times higher enthalpy than the prism and octahedron lattices. The chiral metamaterial surpasses the tensegrity trusses with more than 100× enthalpy. Greater improvement is observed at higher material strengths (σv = 0.2Es), but even metals with limited strength show improved performance in the experiments below.

If we made chiral metacells non-rotatable under compression to show the first bending in rods (Fig. 3a–e), the properties would be similar to other stretch-dominated lattices composed of nearly vertical rods: achieving high E, σbk and ϕ but at the cost of small εlimit. Disregarding εlimit, chiral buckling entails about two to three times greater enthalpy than non-chiral mode. This directly reflects the additional contributions from in-rod twisting and out-of-plane bending.

Straight rectangle-sectioned beam fails to support large chiral buckling, as they preferentially bend along the thinnest axis (Supplementary Video 3). Although chiral units based on X-sectioned beams can display large twist buckling, they underperform the units comprising same-size round rods (Fig. 3 and Supplementary Video 2). We further compare chiral models comprising N square beams to the prism lattices comprising plates with the same thickness t (Fig. 3f–h). Here, t = 1, R ≈ 4–9 mm and N = 2πR/1.2t. The prism lattice follows identical trade-off laws in Fig. 3b–e. Despite having lower E/Es under equal density (ρ/ρs), chiral configurations provide 2.8 times higher σbk and ϕ at large strains.

All these findings indicate that chiral metamaterials outperform non-chiral lattices in maintaining high stiffness and offering superior load-bearing capacity and enthalpy at large limit strain.

For experimental demonstration, we fabricated various rod-, beam- and plate-based specimens using three-dimensional printing, using rubber and TC4 titanium alloy. Layered chiral metamaterials were assembled from separated units (Fig. 4a,b). These chiral, prism, octahedron, Kelvin and tensegrity specimens were designed following the guidance in Fig. 3e,h, ensuring their primitives exhibit approximate von Mises stress (σv) at specified global strain levels (Methods). Their compression buckling behaviours were tested (Extended Data Figs. 9 and 10 and Supplementary Videos 4–7). The chiral twist buckling behaviours closely align with analytical predictions (Fig. 4a,c). The performance indicators of all specimens are summarized in Extended Data Table 1, with the main results presented in Fig. 4g–j, showing the improvements from non-chiral to chiral metamaterials consisting of primitives with the same dimensions (plate thickness = rod diameter = 2r = 3 mm).

a–d, Three-dimensional printed samples. a, Chiral twist buckling of a 3 × 3 metamaterial under compression. b, Pictures of typical titanium-alloy samples. Top: chiral metamaterial. Bottom: the prism lattice. c, Force–strain curves and deformation of typical rubber chiral metacells. In the chiral-50° sample, the neighbouring rods contact for ε > 0.3. d, Octahedron, tensegrity tetradecahedron and prism lattices composed of rods or plates. e,f, Equivalent stress–strain curves of rubber samples (e) and rod-based titanium-alloy samples (f), with shaded areas representing the range for calculating enthalpy ϕ and energy ratio Em = energy/mass. In f, the shaded areas are distinguished into the initial and repeatable loading processes; and ‘non-chiral’ denotes the non-rotatable chiral unit that exhibits bending, reaching equal in-rod stress σv with the chiral model. g–j, ϕ (g,h) and Em (i,j) of rod-, plate- and beam-based metamaterials made of rubber (g,i) or titanium (h,j). For titanium samples, the results from the initial loading process are used. Here, octa and tens represent octahedron and tensegrity lattices, respectively; the rod radius is r = 1.5 mm for tens1 and r = 3 mm for tens2.

Rubber samples can sustain repeatable deformation until the rods make contact. For rod-based architectures, the chiral metamaterial obtains 7.5× higher buckling strength σbk over the high-stiffness prism lattice (Fig. 4e), and achieves 11×, 20×, 167× and 105× higher enthalpy compared with prism, octahedron, tensegrity and Kelvin lattices, respectively (Fig. 4g). It even exceeds the tensegrity lattice with doubled rod thickness (2r = 6 mm) by 8.7×, and exceeds the plate-based prism lattice with 1.8× density (ρ) by two times. These improvements align with the simulations in Fig. 3.

For TC4 samples (Fig. 4b,f), the initial loading follows the theoretical buckling process. On unloading, a plastic strain and augmented angle α0 persist. Subsequent cycles remain repeatable within the macroscopic strain amplitude, preserving relative performance. Despite the limited deformability, the metal chiral architecture enhances σbk by 5.7× over prism lattice and even obtains 1.8–2× higher enthalpy than the plate-based prism, which is 3.5× denser (Fig. 4h). For chiral metamaterials, the non-rotatable mode shows higher stiffness but lower limit strain (Fig. 4f), whereas twist buckling stores twice as much energy.

With a focus on mass instead of volume, we evaluate Em = energy/mrod, where mrod is the total mass of buckling rods. The plate- and rod-based prism lattices have approximately the same Em (Fig. 4i,j), indicating comparable energy ratios for same-size primitives. At ε = 0.25, the rubber chiral metamaterial obtains 2.8×, 6.3× and 6.2–32× higher Em than the prism, octahedron and tensegrity lattices, respectively (Fig. 4i). The titanium chiral metamaterial surpasses all non-chiral lattices in Em by about 2× (Fig. 4j), validating the improvements from rod twisting (Fig. 2e).

Performance decreases from the ideal case when a lattice exhibits global lateral buckling (as in the prism lattices in Supplementary Video 5) or nonuniform buckling (as in the octahedron lattice in Supplementary Video 6). Although our chiral metamaterials are assembled with decoupled units, they reliably exhibit the desired torsional buckling without external constraints (Fig. 4a and Supplementary Video 4).

Here we propose a strategy to create metamaterials with high enthalpy by invoking torsion, alongside bending and compression, within chiral metacells, forming the unique chiral twist buckling. Compared with various existing non-chiral lattices, the non-optimized chiral metamaterials maintain high stiffness and achieve larger recoverable deformation, 5–10× buckling strength, 2–160× enthalpy and 2–32× energy per mass within the limit of material strength. The performance can be further enhanced by using a denser arrangement of chiral arms. High-enthalpy metamaterials have potential applications in lightweight and miniaturized design, elastic energy storage44, impact protection45,46, torsional modulation and actuation. The twist buckling plateau also offers low-dynamic stiffness under heavy load, opening possibilities for low-frequency vibration isolators47.

All FEAs were performed using ANSYS, considering large nonlinear geometrical deformations. Typically, one end is fixed while a displacement boundary condition is applied to the other end, allowing us to obtain the von Mises stress, strain energy and reaction force.

We consider the compressive buckling of a doubly clamped rod. Its bending deflection is assumed to be w(x) = a(1 − cos bx), b = 2nπ/L, where L is the rod length, n denotes the order of bending buckling and x is the position along the rod (Supplementary Note 1). The maximal deflection is wmax = 2a. The rod in the chiral metacell bends in the n = 1/2 mode. A slender primitive rod in non-chiral lattices typically follows the first-order buckling mode n = 1 under suitable constraints, but without stiff constraints, some global buckling mode may lead to n = 1/2 for the oblique rods (Extended Data Fig. 9b).

We note that the assumption for deflection, w(x) = a(1 − cos bx), is valid for moderate deformation ε < 0.2. When a rod is deeply buckled, it exhibits more complex deformation patterns. In FEA, these transformations in the bending deflection are adaptively accounted for through geometrical nonlinearity. This transformation is not considered in the analytical theory, which explains the increasing deviation from FEA results in Fig. 2 for ε > 0.2.

Here we briefly describe the results of our analytical model for the compression buckling of a rod. The complete modelling process and mathematical details are provided in Supplementary Note 1. When an axial compression displacement (Δ) is applied to a doubly clamped rod of radius r, length L0 and Young’s modulus Es, the axial reaction force is F1rod. At first, the stress σcpr is induced by pure compression. Buckling occurs at a critical force

where n denotes the order of buckling mode. The critical axial stress for buckling is

When F1rod < Fc, the rod undergoes uniform strain ε = Δ/L0 and stress σcpr = Esε, storing energy

where Vs = πr2L0 denotes rod volume. In this case, von Mises stress is σv = σcpr. Moreover, the critical energy for buckling is

This equation shows that the critical energy is proportional to n4. If the buckling order changes from n = 1 to 1/2, the critical energy greatly reduces to 1/16.

When buckling occurs, the stress induced by pure compression (σcpr) remains nearly constant. In the general case with n = 1, bending induces maximal tensile or compressive stresses on the surface near the clamped ends

where 2a denotes the bending deflection. Consequently, maximal von Mises stress σv = σbend + σcpr is found there. In this case, the strain energy stored in a compressive buckling rod is

This equation indicates that σcpr gets an eight times higher increment ratio of energy than σbend when increasing stress. Thus, for a specified moderate stress, such as σv = 0.1Es, a thicker rod obtains higher U1rod/Us until buckling disappears at this stress level. This equation also indicates that combining bending and compression increases the increment ratio of U1rod/Us as strain ε = Δ/h0 increases.

We note that when analysing the angled rod and chiral rod, the vertical compression displacement is defined as s. In the above equations, Δ denotes the axial compression displacement. Δ = s for a vertical rod with angle θ = 90°.

Based on the theory above, we adopt \({F}_{{\rm{s}}}={E}_{{\rm{s}}}I/{L}_{0}^{2}\) to normalize the compressive force. Moreover, specifying ξ = σcpr/σbend, the energy inside a buckled rod can be simplified as

Here, Vs = πr2L0 is the rod volume and λ = σv/Es is the normalized strength. Then, the enthalpy of a single rod is

The factor f can be approximated as f ≈ 2.5/1,000 for ξ → 0 and λ < 0.2. For convenience, it is reasonable to use ϕs = Es/1,000 and Us = Vsϕs to normalize the enthalpy and total energy, respectively.

The process of deriving the analytical model for chiral buckling is extensive and involves rigorous mathematical calculations. In Supplementary Information section ‘Chiral twist buckling theory’, we systematically elaborate on the geometry, deformation, force, energy, and stress induced by in-plane bending, out-of-plane bending, in-rod twisting, helix and compression modes (Supplementary Notes 2 and 3). Definitions of all variables are also listed there. Here we summery the analytical theory (also in Supplementary Note 3.8), with key parameters labelled in Extended Data Fig. 1.

When a vertical compression displacement s is applied to half a chiral metacell, a variable rotation angle θ forms between the two tori. The equivalent strain is ε = s/h0. The reaction force and elastic energy contributed by a single rod are F1rod, U1rod, respectively. Point B is fixed on torus O1. Point A0 is the original point on torus O2, which moves to point A under the compression displacement s. Length is given as

The in-plane bending deflection is denoted as 2ain and the out-of-plane bending deflection is denoted as 2aout. Specifying β = ∠ABA0, 2ain = Ls sin β, Lx = Ls cos β. The term 2aout is defined by length |CD| shown in Extended Data Fig. 1b.

The moments induced by in-plane bending and in-rod twisting (Supplementary Note 3) are

Here b = π/Lx, I = πr4/4 is the second moment of the rod, G is the shear modulus, θr = θ-γ. The bulge-out angle γ arises from the helical deformation of rod, as shown in Extended Data Fig. 1c.

The components of the two moments in three-dimensional spaces are

where nin is the unit normal vector of the in-plane bending and ntwist is the unit normal vector in the twisting deformation. The concentrated force applied on a single rod denotes (Fx, Fy, Fz). We use the following equations to analyse chiral buckling deformation and calculate energy.

Force equilibrium equation (Supplementary Fig. 7)

Here α = θ + α0. Based on this equation, the compressive load applied on a chiral rod is F1rod = −Fz.

Compatibility equation of deformation

where \({{\Delta }}_{{\rm{o}}{\rm{u}}{\rm{t}}}={{\rm{\pi }}}^{4}{a}_{{\rm{o}}{\rm{u}}{\rm{t}}}^{2}/4{L}_{{\rm{s}}}\) denotes the shortening made by out-of-plane deformation; ∆cpr is the pure compression displacement along the rod, and \({{\Delta }}_{{\rm{c}}{\rm{p}}{\rm{r}}} < {\rm{\pi }}{r}^{2}/{L}_{0}\); \({L}_{{\rm{i}}{\rm{n}}}=\frac{{E}_{{\rm{k}}}({\rm{\pi }},-{a}_{{\rm{i}}{\rm{n}}}^{2}{b}^{2})}{b}\) denotes the length of the rod under in-plane deformation, and Ek(·) is the elliptical equation; η denotes a correction factor explained in Supplementary Note 3.4.

Energy equation

Here Kc = Esπr2/L0 denotes the longitudinal stiffness of the rod. Based on the energy method, the compressive load is

Equation of stresses

The maximal stresses induced by in-plane bending, in-rod compression and twisting are

Here σin and σcpr are in the same direction, σnorm = σin + σcpr. At the same point, the direction of twisting shear stress τ is perpendicular to σnorm. The stress in the radial direction of the rod is negligible. According to the theory of equivalent von Mises stress,

We adopt this equivalent stress to evaluate the strength of the rods in the chiral structure.

The shear modulus is G = Es/2(1 + υ), where Poisson’s ratio is υ = 0.3. In the chiral metacell, both the in-plane and out-of-plane bending deflection follow the function 1 − cos(πx/L), with their deflection amplitudes denoted as 2ain and 2aout, respectively. In-rod twisting generates shear stress given by τ = Esπr/2.6L0 = σbend(L0 − s)2/2.6L0πa. A proper chiral structure sets L0 ≈ 4R; under large deformation, s ≈ R and the in-plane bending deflection 2a ≈ 2R. Thus, we find that τ ≈ 0.3σbend. Under the same compressive displacement s, σnorm in the chiral rod and the lattice rod (Fig. 1a) are approximately equal.

We use the following quantities to evaluate the performance.

Here, force F, energy U, volume V, mass m, the projected area in load-bearing direction A and the vertical compression displacement s, with the subscript ‘cell’ are measured on the unit cell. The equivalent elastic modulus E denotes the slope of the ε–σeq curve at a small strain ε < 0.02. The maximal stress on the buckling plateau, σbk = max(σeq), symbolizes the load-bearing strength of the entire metamaterial. These variables are normalized for evaluation as E/Es, σbk/Es, ϕ/ϕs and ρ/ρs, where ρs is the mass density of the rod material, ϕs = Es/1,000 is defined in the section ‘Parameter generalization’.

For a chiral metamaterial, there are N rods around the circle with radius R. Here we adopt the integral N = Round(πR/2r) with enough space left to avoid contact between the deformed arms. Thus, Vcell = 4h0(R + r)2, Acell = 4(R + r)2, Fcell = NF1rod, Ucell = NU1rod, mcell = Nmrod, where mrod is the mass of a rod. Here we adopt the maximal cubic volume and square area, instead of the smaller cylinder and circular sizes, to evaluate the performance of chiral metamaterials.

For octahedron lattices, we calculate only 1/8 metacell corresponding to one rod. The parameters are defined as follows: Fcell = F1rod, Ucell = U1rod, mcell = mrod, \({V}_{{\rm{cell}}}=({L}_{0{\rm{r}}}^{3}{\cos }^{2}\theta \sin \theta )/2\), \({A}_{{\rm{cell}}}=({L}_{0{\rm{r}}}^{2}{\cos }^{2}\theta )/2\) where θ denotes the oblique angle between the rod and the horizontal. L0r denotes the distance between the neighbour connection nodes.

For prism lattices, we calculate only 1/4 metacell corresponding to one rod. Fcell = F1rod, Ucell = U1rod, mcell = mrod, \({V}_{{\rm{cell}}}={l}_{{\rm{p}}}{L}_{0{\rm{r}}}^{2}{\cos }^{2}\theta \sin \theta \), \({A}_{{\rm{cell}}}={l}_{{\rm{p}}}{L}_{0{\rm{r}}}\cos \theta \), where lp denotes the distance between neighbouring rhombuses, that is, the lattice constant along the x-axis in Fig. 4. Here we adopt lp = 4r to form a dense lattice.

For octahedron and prism lattices, the useful length of a rod is L0. In the above equations, L0r denotes the distance between two connection nodes in lattices. If L0r = L0, we will obtain ρ/ρs > 1, when the oblique angle θ > 70°, which is impractical. In reality, the connection nodes also occupy space, as shown by the samples in Fig. 4. Therefore, the practical lattices have L0r > L0. For our calculations, we adopt a reasonable value L0r = 1.1L0 for the rod with r = 1.5 mm.

The rubber chiral metacell in Fig. 4a features N = 8 rods with r = 1.5 mm, whereas the titanium version has N = 20 and r = 0.6 mm. All samples have h0 = 30 mm or L0 = 30 mm. Both the rubber and titanium chiral samples in Fig. 4a,b have R = 7.5 mm and α0 = 5°. The oblique angle in prism lattices is θ = 40°. The distance between the neighbour parallel rods is 3 mm for the TC4 lattice and 10 mm for the rubber lattice. The rods inside octahedron and tetradecahedron lattices have equal length. Based on Fig. 3, the in-rod maximal Mises stresses in chiral, prism and octahedron lattices are approximate when the same global strain ε is specified. Although tetradecahedron and tensegrity lattices can theoretically endure higher global deformation, their neighbouring rods make contact when ε > 0.4. Our performance evaluation takes these differences into account. Other parameters and results are listed in Extended Data Table 1.

In Fig. 4c, chiral-20° has R = 5.5, 2r = 1.8, α0 = 20°, h0 = 20 mm. Chiral-50° has R = 6, 2r = 1.7, α0 = 50°, h0 = 20 mm. Es ≈ 5.5 MPa for rubber samples in Fig. 4e, whereas other rubber samples have Es ≈ 15 MPa.

Cyclic compression experiments were performed. The rods in newly fabricated metal samples are initially straight. However, after the first compressive cycle, the all-metal samples exhibit some residual plastic deformation. In this case, α0 increases, and the rod becomes slightly bent. Moreover, the first cycle strengthens the material, improving its overall strength. Subsequent compression cycles are fully repeatable, that is, no further plastic deformation occurs. The enthalpy is calculated as \(\phi =\int {\sigma }_{{\rm{eq}}}{\rm{d}}\varepsilon \).

For the prism lattice, the one-layer sample shown in Extended Data Fig. 9a consists of two half metacells. We tested the deformation of two-layer and four-layer samples to confirm consistency. For multi-layer prism lattices, out-of-plane bulge-out deformation (corresponding to 1/2-order buckling) occurs when there are no lateral constraints (see Extended Data Fig. 9b and Supplementary Video 5). This deformation reduces the buckling strength σbk and enthalpy ϕ by a factor of 10 compared with the desired in-plane buckling mode shown in Fig. 1a (Extended Data Fig. 9h). Therefore, as shown in Extended Data Figs. 9c and 10e, we placed the prism lattices inside a box to constrain the lateral bulge-out deformation.

The analytical models, sample parameters, simulation and test data supporting the findings of this study are available in the paper, Extended Data figures and Extended Data table, and Supplementary Notes. Further details are available from the corresponding authors upon request.

The FEAs are performed with ANSYS software package, which is available at https://www.ansys.com. Other codes about analytical model can be programmed with equations listed in Methods and Supplementary Notes.

Koohi-Fayegh, S. & Rosen, M. A. A review of energy storage types, applications and recent developments. J. Energy Storage 27, 101047 (2020).

Article MATH Google Scholar

Amirante, R., Cassone, E., Distaso, E. & Tamburrano, P. Overview on recent developments in energy storage: mechanical, electrochemical and hydrogen technologies. Energy Convers. Manage. 132, 372–387 (2017).

Article ADS CAS Google Scholar

Alexander, R. M. & Bennet-Clark, H. C. Storage of elastic strain energy in muscle and other tissues. Nature 265, 114–117 (1977).

Article PubMed MATH ADS CAS Google Scholar

Zhan, H., Dong, B., Zhang, G., Lu, C. & Gu, Y. Nanoscale diamane spiral spring for high mechanical energy storage. Small 18, e2203887 (2022).

Article PubMed Google Scholar

Hawkes, E. W. et al. Engineered jumpers overcome biological limits via work multiplication. Nature 604, 657–661 (2022).

Article PubMed ADS CAS Google Scholar

Liu, B., Ge, W., Zhao, D., Zou, Z. & Li, B. Mechanical design and energy storage efficiency research of a variable stiffness elastic actuator. Int. J. Adv. Robot. Syst. https://doi.org/10.1177/1729881420930950 (2020).

Article MATH Google Scholar

Scarfogliero, U., Stefanini, C. & Dario, P. The use of compliant joints and elastic energy storage in bio-inspired legged robots. Mech. Mach. Theory 44, 580–590 (2009).

Article MATH Google Scholar

Gopalakrishnan, S. & Rajapakse, Y. (eds) Blast Mitigation Strategies in Marine Composite and Sandwich Structures (Springer, 2018).

Begley, M. R., Gianola, D. S. & Ray, T. R. Bridging functional nanocomposites to robust macroscale devices. Science 364, eaav4299 (2019).

Article PubMed ADS CAS Google Scholar

Raabe, D., Tasan, C. C. & Olivetti, E. A. Strategies for improving the sustainability of structural metals. Nature 575, 64–74 (2019).

Article PubMed MATH ADS CAS Google Scholar

Liu, L. et al. Making ultrastrong steel tough by grain-boundary delamination. Science 368, 1347–1352 (2020).

Article PubMed MATH ADS CAS Google Scholar

Lu, K. The future of metals. Science 328, 319–320 (2010).

Article PubMed MATH ADS CAS Google Scholar

Li, H. et al. Uniting tensile ductility with ultrahigh strength via composition undulation. Nature 604, 273–279 (2022).

Article PubMed MATH ADS CAS Google Scholar

Dusoe, K. J. et al. Ultrahigh elastic strain energy storage in metal-oxide-infiltrated patterned hybrid polymer nanocomposites. Nano Lett. 17, 7416–7423 (2017).

Article PubMed MATH ADS CAS Google Scholar

Bai, Y. et al. Storage of mechanical energy based on carbon nanotubes with high energy density and power density. Adv. Mater. 31, e1800680 (2019).

Article PubMed Google Scholar

Ali, A., Rahimian Koloor, S. S., Alshehri, A. H. & Arockiarajan, A. Carbon nanotube characteristics and enhancement effects on the mechanical features of polymer-based materials and structures—a review. J. Mater. Res. Technol. 24, 6495–6521 (2023).

Article MATH CAS Google Scholar

Ramezani, M., Dehghani, A. & Sherif, M. M. Carbon nanotube reinforced cementitious composites: a comprehensive review. Constr. Build. Mater. 315, 125100 (2022).

Article MATH CAS Google Scholar

Zheng, X. et al. Ultralight, ultrastiff mechanical metamaterials. Science 344, 1373–1377 (2014).

Article PubMed MATH ADS CAS Google Scholar

Fang, X. et al. Programmable gear-based mechanical metamaterials. Nat. Mater. 21, 869–876 (2022).

Article PubMed PubMed Central MATH ADS CAS Google Scholar

Zhang, X., Zangeneh-Nejad, F., Chen, Z., Lu, M. & Christensen, J. A second wave of topological phenomena in photonics and acoustics. Nature 618, 687–697 (2023).

Article PubMed MATH ADS CAS Google Scholar

Fang, X., Wen, J., Bonello, B., Yin, J. & Yu, D. Ultra-low and ultra-broad-band nonlinear acoustic metamaterials. Nat. Commun. 8, 1288 (2017).

Article PubMed PubMed Central MATH ADS Google Scholar

Kim, M. et al. Harnessing a paper-folding mechanism for reconfigurable DNA origami. Nature 619, 78–86 (2023).

Article PubMed MATH ADS CAS Google Scholar

Chen, T., Pauly, M. & Reis, P. M. A reprogrammable mechanical metamaterial with stable memory. Nature 589, 386–390 (2021).

Article PubMed MATH ADS CAS Google Scholar

Qi, J. et al. Recent progress in active mechanical metamaterials and construction principles. Adv. Sci. 9, 2102662 (2022).

Article Google Scholar

Wenz, F. et al. Designing shape morphing behavior through local programming of mechanical metamaterials. Adv. Mater. 33, 2008617 (2021).

Article PubMed PubMed Central MATH CAS Google Scholar

Zheng, X. et al. Multiscale metallic metamaterials. Nat. Mater. 15, 1100–1106 (2016).

Article PubMed MATH ADS CAS Google Scholar

Frenzel, T., Findeisen, C., Kadic, M., Gumbsch, P. & Wegener, M. Tailored buckling microlattices as reusable light‐weight shock absorbers. Adv. Mater. 28, 5865–5870 (2016).

Article PubMed CAS Google Scholar

Findeisen, C., Hohe, J., Kadic, M. & Gumbsch, P. Characteristics of mechanical metamaterials based on buckling elements. J. Mech. Phys. Solids 102, 151–164 (2017).

Article MathSciNet MATH ADS Google Scholar

Bauer, J., Kraus, J. A., Crook, C., Rimoli, J. J. & Valdevit, L. Tensegrity metamaterials: toward failure‐resistant engineering systems through delocalized deformation. Adv. Mater. 33, 2005647 (2021).

Article CAS Google Scholar

Simitses, G. J. & Hodges, D. H. Fundamentals of Structural Stability (Elsevier, 2006).

Frenzel, T., Kadic, M. & Wegener, M. Three-dimensional mechanical metamaterials with a twist. Science 358, 1072–1074 (2017).

Article PubMed MATH ADS CAS Google Scholar

Chen, Y., Kadic, M., Guenneau, S. & Wegener, M. Isotropic chiral acoustic phonons in 3D quasicrystalline metamaterials. Phys. Rev. Lett. 124, 235502 (2020).

Article PubMed MATH ADS CAS Google Scholar

Kadic, M., Milton, G. W., van Hecke, M. & Wegener, M. 3D metamaterials. Nat. Rev. Phys. 1, 198–210 (2019).

Article Google Scholar

Patil, V. P., Sandt, J. D., Kolle, M. & Dunkel, J. Topological mechanics of knots and tangles. Science 367, 71–75 (2020).

Article MathSciNet PubMed MATH ADS CAS Google Scholar

Lin, G. et al. Buckling of lattice columns made from three-dimensional chiral mechanical metamaterials. Int. J. Mech. Sci. 194, 106208 (2021).

Article MATH ADS Google Scholar

Majumdar, A. & Raisch, A. Stability of twisted rods, helices and buckling solutions in three dimensions. Nonlinearity 27, 2841–2867 (2014).

Article MathSciNet MATH ADS Google Scholar

Frenzel, T., Köpfler, J., Jung, E., Kadic, M. & Wegener, M. Ultrasound experiments on acoustical activity in chiral mechanical metamaterials. Nat. Commun. 10, 3384 (2019).

Article PubMed PubMed Central ADS Google Scholar

Zhou, S. et al. Chiral assemblies of pinwheel superlattices on substrates. Nature 612, 259–265 (2022).

Article PubMed MATH ADS CAS Google Scholar

Fernandez-Corbaton, I. et al. New twists of 3D chiral metamaterials. Adv. Mater 31, 1807742 (2019).

Article MATH Google Scholar

Eringen, A. C. Microcontinuum Field Theories. I. Foundations and Solids (Springer, 1999).

Liu, X. N., Huang, G. L. & Hu, G. K. Chiral effect in plane isotropic micropolar elasticity and its application to chiral lattices. J. Mech. Phys. Solids 60, 1907–1921 (2012).

Article MathSciNet MATH ADS Google Scholar

Chen, Y., Frenzel, T., Guenneau, S., Kadic, M. & Wegener, M. Mapping acoustical activity in 3D chiral mechanical metamaterials onto micropolar continuum elasticity. J. Mech. Phys. Solids 137, 103877 (2020).

Article MathSciNet MATH Google Scholar

Cheung, K. C. & Gershenfeld, N. Reversibly assembled cellular composite materials. Science 341, 1219–1221 (2013).

Article PubMed MATH ADS CAS Google Scholar

Cheng, H. et al. Mechanical metamaterials made of freestanding quasi-BCC nanolattices of gold and copper with ultra-high energy absorption capacity. Nat. Commun. 14, 1243–1243 (2023).

Article PubMed PubMed Central MATH ADS CAS Google Scholar

Siddique, S. H., Hazell, P. J., Wang, H., Escobedo, J. P. & Ameri, A. A. H. Lessons from nature: 3D printed bio-inspired porous structures for impact energy absorption—a review. Addit. Manuf. 58, 103051 (2022).

Google Scholar

Zhang, T. et al. Compressive mechanical behaviors of PPR and NPR chiral compression–twist coupling lattice structures under quasi-static and dynamic loads. Thin-Walled Struct. 182, 110234 (2023).

Article Google Scholar

Dalela, S., Balaji, P. S. & Jena, D. P. Design of a metastructure for vibration isolation with quasi-zero-stiffness characteristics using bistable curved beam. Nonlinear Dyn. 108, 1931–1971 (2022).

Article MATH Google Scholar

Download references

This research was funded by the National Natural Science Foundation of China (projects numbers 52322505, 52241103 and 11991032), the Natural Science Foundation of Hunan Province (projects number 2023JJ10055). P.G. acknowledges support from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) through the Excellence Cluster 3D Matter Made to Order (EXC-2082/1–390761711). We thank X. Liu and S. Forest for their discussions on mapping to micropolar elasticity.

National Key Laboratory of Equipment State Sensing and Smart Support, College of Intelligent Science and Technology, National University of Defense Technology, Changsha, China

Xin Fang, Dianlong Yu, Jihong Wen & Yifan Dai

Department of Mechanical Engineering, University of California, Santa Barbara, Santa Barbara, CA, USA

Matthew R. Begley

Mechanics-X Institute, Tsinghua University, Beijing, China

Huajian Gao

Institute for Applied Materials, Karlsruhe Institute of Technology, Karlsruhe, Germany

Peter Gumbsch

Fraunhofer Institute for Mechanics of Materials IWM, Freiburg, Germany

Peter Gumbsch

You can also search for this author in PubMed Google Scholar

X.F. conceived the idea, established the analytical models and performed the experiments. X.F. and P.G. designed the study. X.F., P.G., H.G. and D.Y. conducted the numerical simulations and designed the experiments. J.W. designed experimental setups. Y.D. fabricated samples. Y.D., D.Y. and J.W. conceived application cases. M.R.B. and H.G. analysed the properties. All authors interpreted the results. X.F., H.G., M.R.B. and P.G. wrote the paper with input from all authors.

Correspondence to Xin Fang or Peter Gumbsch.

The authors declare no competing interests.

Nature thanks Muamer Kadic and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Peer reviewer reports are available.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

(a) Compressive buckling of an oblique rod in a nonchiral lattice. (b) Model of half a chiral metacell (Supplementary Note 2), showing the in-plane and out-of-plane bending deformations (Supplementary Note 3.1 and 3.2). XYZ denotes the global coordinates. (c) A twisted rod becomes helical, indicating the bulge-out angle γ (Supplementary Notes 3.3 and 3.4). Taking arm A0B as an example, point B is fixed to axis O1X, and point A0 is on torus O2. The relative rotation angle at rest between the two tori is α0. When torus O1 is fixed, torus O2 generates a dynamic rotation angle θ under the compressive displacement s, resulting in chiral deformation. Point A0 moves to A, while the rod is twisted by angle θ. The subpoints of A and A0 on plane XOY are C and A1, respectively.

Unless otherwise labeled, r = 1.5, L0 = 30 mm. For metamaterial consisting of oblique rods, applying a vertical compressive displacement (s) gets an equivalent strain ε = s/h0. In (b, c, d), the rods are vertical. (a) Normalized compressive force of the perfectly straight rod with different oblique angle. The dashed range represents σv > 0.1Es. Bending buckling happens abruptly. (b) Normalized strain energy of a micro-bending rod under different stress. The rod center is bended by 2°. This small imperfection greatly reduces the normalized energy under σv = 0.05Es, σv = 0.1Es and σv = 0.2Es by about 2/3, compared to the perfectly straight rod shown in Fig. 2 in the main text, due to stress concentration at the bent position. Fortunately, comparing Fig. 2a in the main text, the maximum force of this imperfect rod nearly unchanged. (c, d) Analytical solution of energy and von Mises stress as a function of equivalent strain for a rod. (e) Energy-stress curves combined from panels (c, d). The original rod is perfectly straight. Panel (d,e) indicates the two-stage deformation process of a nonchiral straight rod. Panel (c) indicates that: (1) When buckling occurs, the normalized energy U1rod/Us increases at a higher rate as strain increases; (2) When r < 2.5 mm, thicker rod achieves higher U1rod/Us within moderate strain (ε = 0.12) because the contribution from pure compression increases. However, further increasing r cannot increase U1rod/Us because buckling no longer occurs. (3) In the pure compression stage, U1rod/Us is independent of rod thickness. As indicated by equation (1), σcpr gets 8 times higher increment ratio than σbend when increasing stress. Thus, for a specified moderate stress, like σv = 0.1Es, a thicker rod obtains higher U1rod/Us until buckling disappears at this stress level. If the material strength limit is relaxed, hybrid bending and axial shortening will always result in higher energy storage within the specified material volume.

(a) Deformation of a chiral metacell with R1 > R2. (b, c) Influence of radius R2 on the maximum force max(F1rod) and energy of the chiral rod for R1 = 8 mm and α0 = 10°. (d) Influences of rod radius r on the stress and energy of a chiral rod at a specified strain ε = 0.2. (e) Energy-stress plot of a thinner rod with r = 0.6 mm. For all plots, L0 ≈ h0 = 30 mm, and r = 1.5 mm for (a-d).

The four panels show the generalized force F1rod/Fs, generalized energy U1rod/Us, twisting rotation angle of the chiral unit θ, and generalized stress in rod σv/Es versus global equivalent strain ε = s/h0. All panels share the same legends as panel 1. In these models, R1 = R2, r = 1.5 mm.

(a-c) Maximum normalized force F1rod/Fs, normalized energy U1rod/Us, and proportion of twisting energy ηtwist under specified strain ε = 0.2. (d-f) Normalized elastic modulus E/Es, buckling strength σbk/Es, (c) enthalpy ϕ/ϕs of chiral metamaterials under specified equivalent strain ε = 0.2.

The rod has 2r/L0 = 0.27, L0 = 30 mm. Thick rods may favor nonchiral designs in moderate stress ranges, assuming the 1st bending mode remains dominant. However, in practice, thick rods without the assistance of rigid constraints often follow a 1/2-order lateral buckling mode because its critical buckling energy is only 1/16 of the 1st mode (Methods). In this case, chiral rod still offers 2–4 times higher energy storage. Moreover, reducing the initial angle α0 to 0.1° can improve chiral rod’s performance near the critical point. The influences of α0, R, and the rod thickness-length ratio 2r/L0 highlight the importance of design in chiral metamaterials.

FEA simulations are performed using the same chiral model under different boundary conditions. R = 7, r = 1.5, h0 = 30 mm; α0 = 5°. (a) Compressive-twisting buckling: rotation θ is free for a specified compression ε. This is the boundary condition used in the main text to generate compressive chiral buckling. (b) Purely twisting condition without external compression: θ is specified, with free ε. Results from this condition indicate the contribution from in-rod twisting. In this case, all energy is contributed by global torsion because the global compressive force is zero for freely compressive boundary. Comparing U1rod in (a) and (b), we obtain the energy ratio of in-rod twisting is ηtwist = 1.6/4 = 40% at θ = 120°, matching the analytical result in Fig. 2d. (c) Results for step simulation used to separate global torsion and global compression. From n to n + 1 step (εn+1 = εn, θn+1 = θn + ∆θ); from n + 1 to n + 2 step (εn+2 = εn+1 + ∆ε, θn+2 = θn+1). Separating the total energy accumulated by global compression or torsion steps shows that the contribution from global compression can reach 95 ~ 100%. This is correct because the freely rotatable boundary condition under specified compression induces zero global torque, and thus the work done by global torque is zero. The nonlinear micropolar model in the Supplementary Note 4 indicates that all energy is contributed by chirality, as “chirality” is defined as the coupled axial force (global compression force) generated by global torsion in the micropolar model.

(a-g) Tetradecahedron (Kelvin) lattice. (h-m) Tensegrity tetradecahedron lattice. (a, h) A metacell. (c,d) Rods in 1/8 model and corresponding compression bending deformation. (e, f, g) Changes in F1rod, U1rod and generalized stress σv/Es with equivalent strain ε. In Kelvin lattice, every rod is same with r = 1.5, L0 = 30 mm. The size of the round connection nodes is considered. The Kelvin lattice has: ρ/ρs = 7.78 × 10−3, E/Es = 1.7 × 10−3, σbk/Es = 5.7 × 10−5, ϕ/ϕs = 3.07 × 10−3 at σv = 0.2Es. Its buckling strength σbk/Es is 1/5 and its enthalpy is 1/19 of the octahedron lattice with rod’s oblique angle near 40°. Thus, its enthalpy is lower than 1/100 of the chiral metamaterial. In (h-m), we calculate the nonlinear compressive responses of a tensegrity metacell. All tensile and compressive rods in metacell have same radius, and ρ/ρs is varied by changing the rod radius. The stress σv is picked near the rod’s end instead the rods’ crossing positions where stress concentration occur. At σv = 0.1Es. Enthalpy ϕ/ϕs for r = 3 mm is 5 times of the case with r = 1.5 mm. However, even for the tensegrity metacell with r = 3 mm, its enthalpy is only 1/4 of the same-density octahedron lattice consisting of rod with r = 1.5 mm.

(a) Buckling of a single-layer rubber prism lattice. (b) Bulge-out buckling of a two-layer rubber lattice without rigid constrains, with the center bugle-out along the x-axis. The view in (a) is different from (b,c). (c) In-plane buckling of a four-layer rubber lattice placed inside a box. (d) Compression buckling of the octahedron lattice. (e) Deformation of the plate-based prism lattice. (f) Deformation of the tetradecahedron (Kelvin) lattice. (g) Deformation of the tensegrity tetradecahedron lattice. (h) In-plane and bulge-out buckling curves of the two-layer rubber prism lattice. (i) Equivalent strain-stress curves of rubber prism lattices with one, two or four layers, where only in-plane buckling occurs. The result for the two-layer sample is shown in Fig. 4 of the main text. This panel confirms the consistency of the performance of prism lattices across different layers. (j) Equivalent strain-stress curves of octahedron lattices consisting of rods with radii r = 1.5 and 2 mm. (k) Equivalent strain-stress curves of the Kelvin lattices and tensegrity lattices consisting of rods with different radii. Experimental processes are shown in the Supplementary Videos. Notes: In performance evaluations and comparisons, every rod in these nonchiral lattices is assumed to have ideal 1st-order bending deformation. For the prism lattice, this is achieved by constraining lateral deformation. However, in experiments, different layers in octahedron lattices do not buckle simultaneously due to intrinsic instability [Programmable Materials 1, 1–18 (2023)], and it is difficult to induce the desired bending mode in the tetradecahedron rods. As a result, the experimental performance of multi-layered octahedron and tetradecahedron lattices may be lower than theoretical predictions.

(a-f) Deformation images under compression for different metamaterials. (a) 3D model of the 3 × 3 rubber chiral metamaterial. Samples in (b-f) are made of TC4 titanium-alloy. (b,c) Deformation of rod- and beam-based chiral units. (d, e) Plate- and rod-based prism lattices. (f) Identical chiral unit under non-rotatable boundary conditions, presenting nonchiral bending rods. (g) Equivalent strain-stress curves of titanium alloy chiral metacells. There are four samples shown, with “chiral 2” used in Fig. 4 of the main text. (h) Equivalent strain-stress curves of different titanium alloy metamaterials. (i, j) FEA results for the identical TC4 chiral unit. “Chiral” and “nonchiral” refer to rotatable and non-rotatable boundary boundary, respectively. At the equivalent strain ε = 0.037, the real maximum Mises stress in the rod max(σv) is about 5 GPa for the chiral unit, but around 10 GPa for the nonchiral rod. For nonchiral deformation, ε = 0.012 for max(σv) ≈ 5 GPa. For a specified stress of max(σv) ≈ 5 GPa, the energy stored by chiral deformation is 1.84 times that of nonchiral deformation. This property is demonstrated by the experimental green curves inserted in (j), where “nonchiral1” and “nonchiral2” denote the two cases with maximal stresses of σv ≈ 5 GPa and 10 GPa, respectively.

This file contains Supplementary Notes 1–4 and Supplementary Figs. 1–8.

Non-chiral buckling deformation. In this video, the rotation of a chiral model is fixed. Applying a compression force to the model results in non-chiral bending buckling in the rods.

Chiral buckling deformation. This video shows the chiral buckling deformation for both rod-based and crossed-beam-based chiral units.

Unrobust chiral buckling based on flat wide beam. In this model, rotation occurs under moderate compression but is halted or reversed when bending buckling of the flat beam occurs.

Experimental testing process of chiral metamaterial.

Experimental testing process of prism lattices. This video demonstrates both the global bulge-out buckling and the desired high-energy buckling.

Experimental testing process of octahedron lattice.

Experimental testing process of tensegrity tetradecahedron lattice.

Open Access This article is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, which permits any non-commercial use, sharing, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if you modified the licensed material. You do not have permission under this licence to share adapted material derived from this article or parts of it. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by-nc-nd/4.0/.

Reprints and permissions

Fang, X., Yu, D., Wen, J. et al. Large recoverable elastic energy in chiral metamaterials via twist buckling. Nature (2025). https://doi.org/10.1038/s41586-025-08658-z

Download citation

Received: 07 March 2024

Accepted: 16 January 2025

Published: 12 March 2025

DOI: https://doi.org/10.1038/s41586-025-08658-z

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative