Tuning superelasticity in high entropy alloy via a hidden strain order

Abstract

Superelasticity – exhibiting either Hookean (linear) or non-Hookean (nonlinear) recoverable strain beyond 2% – has been realized in distinct material systems such as metallic glasses, shape memory alloys, strain glass alloys and Gum metals, enabling diverse technological applications. Here we demonstrate that, through compositional tuning in a high-entropy alloy, the elastic behavior can be continuously and reversibly modulated between Hookean superelasticity, non-Hookean superelasticity with an ultrahigh recoverable strain of ~8%, and back to the Hookean regime. By combining atomic-scale strain mapping and extensive first-principles calculations, we reveal that this tunability is governed by a hidden strain order, arising from frustrated crystallization of two competing phases. As a result, local lattice distortion arises, producing a heterogeneous strain landscape that modulates phase stability, phase transformation propensity, and elastic response. Our findings establish a materials design strategy for programming Hookean and non-Hookean elasticity behavior on demand, with promising applications in microelectromechanical systems, high-precision actuators, and adaptive damping devices.

Designing superelastic materials remains challenging. Here, the authors reveal that hidden atomic-scale strain order in a high-entropy alloy enables reversible switching between linear and nonlinear superelasticity through subtle composition tuning.

Introduction

Crystalline alloys (e.g., steel, titanium, aluminum) typically exhibit elastic deformation through reversible atomic bond stretching and shearing¹, obeying Hooke’s law with a linear stress-strain relationship². In these systems, elastic strains are typically constrained to below 1%^3,4 before the activation of plasticity carriers – primarily dislocation gliding⁵ and deformation twinning⁶—initiates irreversible deformation. Remarkably, certain alloy systems transcend these classical limitations through nonlinear elastic mechanisms. Shape memory alloys (SMAs) (e.g., NiTi-based, Cu-based, Fe-based alloys)^7–10 and Gum metals^11–13 demonstrate superelasticity, achieving recoverable pseudo-elastic strains exceeding 2% via stress-induced martensitic transformations^14,15. In these materials, the crystal lattice undergoes reversible phase transitions under mechanical loading, enabling exceptional elasticity without permanent structural damage. Recent discoveries have pushed these boundaries further. Xu et al.¹⁶ reported a low modulus strain glass alloy (SGA) (e.g., Ti–50.8Ni) exhibiting elastic strains of ~8%—nearly an order of magnitude beyond conventional Hookean limits. Unlike traditional SMAs where martensitic transformations occur cooperatively across long ranges, SGAs achieve superelasticity through the activation of nanoscale martensitic domains^16,17.

The advent of high-entropy alloys (HEAs)^18,19 has revolutionized the design of superelastic alloys by exploiting multi-principal-element compositions²⁰. Unlike traditional superelastic alloys—including SMAs^7–10, Gum metals^11–13, and SGAs^16,17—HEAs contain more than 4 principal elements with significant atomic size mismatches and diverse chemical affinities^21,22. This unique composition generates pronounced chemical and elastic fluctuations at the atomic scale, inducing substantial lattice distortion and heterogenous lattice strains^23–27. Remarkably, HEAs exhibit both Hookean and non-Hookean elasticity that defy conventional elastic limits. Chen et al. observed a massive pseudo-elastic strain of ~15% in Ni₂₅Co₂₀Fe₁₈Ga₂₇ fibers²⁸, while Gou et al. reported ~4% in (TiZrHf)₄₄Ni₂₅Cu₁₅Co₁₀Nb₆ single crystals¹⁷—both attributed to stress-induced phase transformations facilitated by lattice distortion. Intriguingly, He et al. discovered a purely Hookean-type superelasticity with 2–3% recoverable strain in (TiZrHf)₅₀Ni₂₅Co₂₅ alloy, surpassing the traditional 1% limit while maintaining linear elasticity, an effect also rooted in lattice distortion²⁰. These findings raise a fundamental question: How can severe lattice distortion simultaneously enable both Hookean and non-Hookean superelasticity in HEAs? The underlying mechanisms remain unresolved, presenting a critical knowledge gap in understanding atomic-scale lattice strain heterogeneity and its role in elastic deformation. Resolving this paradox could unlock new pathways for designing alloys with tailored elastic responses beyond current limitations.

In this study, we resolve this fundamental question through systematic investigation of the (TiZrHf)₅₀Ni_50-xCo_x HEA system. By precisely tailoring Co/Ni ratio, we demonstrate direct control over lattice distortion and internal strain fields—enabling a continuous transition between non-Hookean and Hookean superelasticity within a single alloy family. Remarkably, this transition occurs through minute compositional variations (<2 at.%), unifying phenomena previously observed only in distinct alloy systems. Our discovery bridges the apparent dichotomy between conventional Hookean elasticity and transformational superelasticity, providing fundamental insights for designing alloys with tailored elastic responses.

Results

Composition-dependent phase transformation and elastic response

Figure 1a presents the X-ray diffraction (XRD) patterns of (TiZrHf)₅₀Ni_50-xCo_x (x = 15, 20, 22, 23, 25) alloys (hereafter denoted as Co15, Co20, Co22, Co23 and Co25 alloys, respectively; see “Methods”). The Co15 alloy exhibits a single-phase B19′ structure, whereas the Co20 alloy reveals the emergence of B2 characteristic peaks, indicating a dual-phase (B19′ + B2) microstructure. With further increase in Co content (Co22, Co23 and Co25 alloys), the alloys transition to a single-phase B2 structure. Figure 1b displays the differential scanning calorimeter (DSC) curves of the (TiZrHf)₅₀Ni_50-xCo_x alloys (see “Methods”). The Co15 alloy undergoes a martensitic transformation with a martensite start temperature (Ms) of ~500K (heating) and ~480K (cooling). As the Co content increases, Ms decreases sharply, and no martensitic transformation (MT) is observed for the Co23 and Co25 alloys within the DSC detection range (~100–600K). Additionally, the exothermic/endothermic MT peaks weaken progressively with increasing Co content, reflecting a continuous reduction in transformation enthalpy (Supplementary Fig. 1). To further probe the potential for MT at cryogenic temperatures, we performed electrical resistivity (ER) measurements over a broad temperature range (~5–700K) (see “Methods”). As shown in Fig. 1c, thermal hysteresis loops are evident in the Co15, Co20 and Co22 alloys during heating and cooling cycles, consistent with the DSC-observed MT signatures (Fig. 1b). Notably, no evidence of MT is observed in the Co23 and Co25 alloys even upon cooling to ~5K, confirming the complete suppression of martensitic transformation in these compositions.

Fig. 1. Structural characterization and transformation behaviors of (TiZrHf)₅₀Ni_50-xCo_x alloys.

a XRD patterns revealing the phase evolution from B19′ to B2 with increasing Co content. b Differential scanning calorimetry (DSC) curves demonstrating the suppression of martensitic transformation peaks as Co content rises. c Electrical resistivity (ER) hysteresis loops during thermal cycling, corroborating martensitic transformation (MT) behavior. d–f Storage modulus and internal friction curves (different colours represent the frequency of 0.2, 0.4, 1, 2, 4, 10 and 20 Hz, respectively) for Co15, Co22 and Co25 alloys, highlighting distinct elastic responses across compositions.

To probe the elastic behavior of these alloys, we conducted dynamic mechanical analysis (DMA) measurement on the Co15, Co22 and Co25 alloys (see “Methods”, Supplementary Fig. 1d–f). In the Co15 alloy, the storage modulus exhibits a distinct dip near 490K, accompanied by a corresponding peak in internal friction—consistent with the MT temperature identified by DSC and ER measurements. Notably, neither the position of internal frictional peak nor the storage modulus dip displays frequency dependence, a hallmark of conventional martensitic transformations^29,30. A similar DMA response was observed for the Co20 alloy (Supplementary Fig. 2a). In contrast, the Co22 alloy exhibits clear frequency-dependent behavior in both storage modulus and internal friction, with a local modulus minimum and friction maximum near 200K (Fig. 1e), which resembles a strain glass transition (STG)²⁹. For the Co23 (Supplementary Fig. 2b) and Co25 alloys (Fig. 1f), neither MT nor STG-like behavior is observed. Instead, these alloys display a pronounced Elinvar effect—near-invariant elastic modulus across a broad temperature range (130–630K)—aligning with our prior finding²⁰.

Combining these results with additional thermogravimetric analysis data (see Supplementary Fig. 3) spanning a wide compositional range (TiZrHf)₅₀Ni_50-xCo_x (0 < x < 30), we construct a comprehensive phase diagram for the (TiZrHf)₅₀Ni_50-xCo_x system (Fig. 2). At low Co concentrations (x < 15), the alloys stabilize into a high-entropy B2 phase at elevated temperatures, transforming to a high entropy B19′ phase upon cooling, with the MT temperature decreasing monotonically with increasing Co content. In the intermediate composition range (15 < x < 22), we observe a STG-like behavior, suggesting frustrated local transformations³¹. Remarkably, when the Co content exceeds that of the Co23 alloy (x > 23), the high-entropy B2 phase persists as the sole stable phase across an exceptionally wide temperature range from ~1000 K to ~5K. This complete suppression of martensitic transformation highlights the profound influence of composition tuning on phase stability in this multi-principal-element alloy system.

Fig. 2. The composition-temperature phase diagram of (TiZrHf)₅₀Ni_50-xCo_x alloys.

The diagram delineates the stability regimes of B2, B19′, and STG phases, constructed from electrical resistivity measurements. The solid black squares and open blue circles are experimental data set. The black solid line denotes the martensite start temperature (M_s), while the solid and dashed blue lines indicate the strain glass transition temperature (T_g). STG strain glass transition.

Hookean versus non-Hookean superelastic behavior

Conventional compression tests were conducted to characterize the elastic response of the polycrystalline (TiZrHf)₅₀Ni_50-xCo_x bulk alloys (see “Methods”). As illustrated in Fig. 3a, a striking transition from linear to nonlinear elasticity— followed by a reversion to linear elasticity—is observed as the Co content increases from 15 to 25 at%. The single-phase B19’ Co15 alloy exhibits classical Hookean elasticity, displaying a linear elastic stress-strain relationship. In contrast, the Co20 (B19’ and B2) and Co22 (B2 single phase) alloys exhibit a nonlinear elastic behavior, as evidenced by their curved stress-strain responses. Remarkably, further Co addition restores linear elasticity in the single-phase B2 Co23 alloy and Co25 alloy. Cyclic loading-unloading tests (Fig. 3b) confirm ideal Hookean elasticity in the Co15 alloy, with perfect overlap between loading and unloading curves. However, the Co20 and Co22 alloys develop mechanical hysteresis, indicting intrinsic energy dissipation consistent with their non-Hookean behavior. Conversely, the Co23 and Co25 alloys revert to linear elasticity, demonstrating a composition-dependent recovery of Hookean behavior.

Fig. 3. Multi-scale mechanical characterization of (TiZrHf)₅₀Ni_50-xCo_x alloys at room temperature.

a Macroscopic compressive stress-strain curves, revealing a composition-dependent transition from linear to nonlinear and back to linear elasticity with increasing Co content (15–25 at%). b Cyclic loading-unloading responses, demonstrating ideal Hookean elasticity in Co15 and restored linearity in Co23/Co25 alloys, contrasted with nonlinear hysteresis in Co20/Co22 alloys. c Orientation-dependent microcompression behavior of Co22 single crystals, showing nonlinear superelasticity along [011] and [111] versus linear elasticity along [001]. d Cyclic microcompression of [011]-oriented Co22 micropillars, exhibiting ~8% recoverable strain through stress-induced martensitic transformation. The inset figure shows the morphology of a typical micropillar. e Comparative analysis of damping performance, demonstrating superior figures of merit (1.1–1.2) for Co22/Co23 alloys relative to conventional high-damping materials^{10,36,49–76}.

To investigate the intrinsic elasticity of single-crystalline alloys, we performed microcompression tests on the single-phase B2 Co22, Co23, Co25, and Co27 alloys (see “Methods”). The inverse pole figure maps in Supplementary Fig. 4 confirm the crystalline orientations of individual micropillars. As shown in Fig. 3c, the [001]-oriented Co22 micropillar exhibits Hookean (linear elastic) behavior until yielding, whereas [011]- and [111]-oriented micropillars display pronounced nonlinear superelasticity, indicative of stress-induced martensitic transformation. The strong anisotropic martensitic transformation behavior was also reported in other B2 austenite systems, such as CuZr-based SMAs^32,33. The deformation behavior of the B2 phase is strongly orientation-dependent because the B2 to B19’ martensitic transformation is activated through shear along the {011} < 100 > _B2 slip system. The resolved shear stress on these systems varies significantly with crystal orientation³³. The [001] axis has an almost zero Schmid factor for all transformation variants, suppressing the stress-induced martensitic transformation and resulting in purely linear elastic behavior. In contrast, the [011] and [111] orientations possess much higher Schmid factors, enabling activation of the transformation shear and giving rise to the observed nonlinear superelasticity. Cyclic microcompression tests reveal recoverable strains of ~8% in the [011]-oriented Co22 micropillars (Fig. 3d), with similar behavior observed for the [111] orientation (Supplementary Fig. 5). The Co23 alloy exhibits analogous orientation-dependent elasticity (Supplementary Fig. 6), with linear elasticity along [001] and nonlinear elasticity along [011] and [111]. This anisotropy can be attributed to the absence of sufficient shear strain to drive the B2–B19’ transformation along the [001] orientation¹⁰. In contrast, Co25 and Co27 micropillars exhibit purely linear elasticity across all crystallographic orientations (see ref. ³⁴ and Supplementary Figs. 7 and 8), with no evidence of martensitic transformation. These findings suggest that a critical Co concentration between Co23 and Co25 alloys fully suppresses martensitic transformation, stabilizing the B2 phase under mechanical loading. Moreover, the observed difference in elastic strain between bulk and micropillar compression is strongly influenced by microstructures. In polycrystalline bulk samples, the measured elastic strain represents an average over multiple grains and grain boundary constrains, which reduces the apparent elastic limit. In contrast, the micropillar compression would enable us to probe the intrinsic orientation-dependent and superelastic behavior within a single crystal free from grain boundary constraints. Because the transformation strain is highly anisotropic, the single-crystal micropillars naturally exhibit a larger recoverable strain than the orientation-averaged bulk material. To further investigate the temperature dependence of the superelastic behavior, we performed cyclic micro compression tests on [011] oriented Co22 micropillars at various temperatures, as shown in Supplementary Fig. 9. The results clearly demonstrate a transition from nonlinear elastic behavior to linear elastic behavior with increasing temperature, with a critical transition occurring around ~60 °C. This temperature-dependent evolution is fully consistent with the classical framework for deformation in SMAs³⁵.

Following the methodology in ref. ³⁶, we evaluated the damping performance of both Co22 and Co23 alloy single crystals using the loss factor $\eta =\frac{\mathrm{\Delta }W}{\pi \cdot W_{\max }}$, where ∆W represents the energy dissipated per hysteresis cycle, and W_max denotes the maximum stored energy. For structural applications involving elastic loading, the figure of merit for optimal damping is defined as $E^{1/2}\cdot \eta$^36,37, where E is the Young’s modulus. As shown in Fig. 3e and Supplementary Data 1, the Co22 and Co23 alloys exhibit exceptional damping characteristics, with figures of merit reaching 1.1–1.2. This performance surpasses that of conventional high-damping materials, including Cu-Al-Ni alloys (maximum figure of merit ~ 0.9). The superior damping capacity of these alloys, combined with their excellent mechanical properties, makes them particularly promising for microelectromechanical systems. Such systems require passive damping solutions to control oscillations and mitigate shocks in micro-resonators, sensors and actuators³⁸, suggesting significant potential for practical applications of these alloys in advanced microdevice technologies.

Atomic-scale structural variation and lattice strain

To elucidate the atomic-scale mechanisms governing the transition from Hookean to non-Hookean superelasticity, we conducted a detailed investigation of the crystalline and local atomic structures in the representative Co15, Co22, and Co25 alloys. As shown in Supplementary Fig. 10, the Co15 alloy adopts the martensitic B19′ structure. In contrast, the Co22 alloy maintains a single-phase B2 structure (Supplementary Fig. 11a). Notably, the Co25 alloy maintains a single-phase B2 structure (Supplementary Fig. 11b–d) without any detectable martensitic variants even at 90K³⁹, indicating exceptional stability of the B2 phase across the entire temperature range probed. Atomic-scale structural and chemical evolution across these compositions is revealed by aberration-corrected scanning transmission electron microscopy high-angle annular dark-field (STEM-HAADF) imaging (Fig. 4a–c). The images show distinct contrast variations, indicative of chemical heterogeneity inherent to these multi-principal-element alloys. The corresponding atomic intensity line profiles (Fig. 4d–f), extracted from the boxed regions in the STEM-HAADF images, quantitatively corroborate these compositional fluctuations. The Co22 and Co25 alloys exhibit more pronounced intensity variations than Co15 alloy, signifying enhanced chemical short-range ordering and local elemental partitioning with increasing Co content. This chemical evolution is accompanied by a clear structural progression, as confirmed by selected-area electron diffraction patterns (Fig. 4g–i).

Fig. 4. Atomic-scale chemical fluctuation and lattice strain evaluations.

a–c The STEM-HAADF images of Co15, Co22 and Co25 alloys, respectively. d–f The atomic intensity line profiles corresponding to the square regions marked in a–c, respectively. g–i The selective area diffraction pattern of Co15, Co22 and Co25 alloys, respectively. j–l The atomic von mises strain distributions of Co15, Co22 and Co25 samples, respectively. The insets show frequency distributions of von Mises strain.

Critical insights into lattice strain were revealed through atomic-scale strain mapping based on geometric phase analysis (GPA) of high-resolution STEM images. The von Mises strain distributions (Fig. 4j–l) exhibit striking variations across the three alloys. The Co15 alloy exhibits uniform, low-magnitude strain fields, with a narrow distribution (inset histogram) peaked near zero—a signature of its homogeneous, low-strain martensitic (B19′) structure. In stark contrast, the Co22 alloy displays a substantially higher and heterogeneous strain distribution, with a broad peak extending to elevated values. This signifies pronounced lattice distortion, which we attribute to strong chemical fluctuations and local elastic misfit within the high-entropy B2 phase. Notably, the Co25 alloy exhibits a strain distribution that is reduced in magnitude and heterogeneity compared to Co22 alloy, indicating that further Co addition beyond a critical concentration alleviates, rather than enhances, the lattice strain. This non-monotonic evolution of strain— peaking at an intermediate composition—is a key predictor of the anomalous superelastic behavior.

Mechanistic understanding through first-principles calculation

To elucidate the atomic-scale mechanisms governing composition-dependent elastic transitions in (TiZrHf)₅₀Ni_50-xCo_x alloys, we conducted first-principles density functional theory (DFT) calculations (see “Methods”). Structural models of B2 (cubic) and B19′ (monoclinic) phases—key to understanding divergent superelastic behaviors—were constructed and relaxed. Figure 5a reveals distinct atomic packing geometries in these phases at the equiatomic Ni:Co ratio (25:25), visualized along major crystallographic orientations. DFT-derived formation enthalpies (ΔH) (Fig. 5b) exhibit a critical composition-dependent crossover. Below 25% Co, the B19′ structure is energetically favored (ΔH_B19′ < ΔH_B2); while beyond this threshold, the B2 structure becomes thermodynamically preferred. This trend, corroborated by the TOTEN energy calculations (Supplementary Fig. 12), persists even without configurational entropy contributions. The enthalpy reversal at x ≈ 25 aligns with experimental phase stability observations and implies a fundamental link between Co content and the B2↔B19’ structural energetics.

Fig. 5. The atomic structure evaluations revealed by DFT calculations.

a, b Structural snapshots and formation enthalpies of the relaxed B2 and B19’ structure with Ni:Co = 25:25. Box plots of c Von Mises strain and d Volumetric strain for B2 and B19 structures across varying Ni:Co ratios are presented. The box is determined by the 25th and 75th percentiles, the whiskers are determined by the 5th and 95th percentiles, the purple and orange lines indicate the mean and median values, respectively. The normal distribution of the strain data is illustrated using green curve. e Electron localization function on the (100) plane of binary B2-ZrNi and B2-ZrCo alloys and (001) plane of B19’-ZrNi and B19’-ZrCo alloys. The unit is electron/Bohr³. Atom coordinates are provided in Supplementary Data 2.

Figure 5c, d presents the calculated von Mises and volumetric strain distributions as a function of Ni:Co ratio (see “Methods”), resulting from lattice distortions in the B2 and B19′ phases. In the B2 phase, both the von Mises strain and atomic displacements (Supplementary Fig. 13) decrease monotonically with increasing Co content. In contrast, the B19′ phase exhibits the opposite trend, with von Mises strain increasing as Co content rises (Fig. 5c). Given that lattice strain correlates with system potential energy and enthalpy²³, these strain variations align with the computed enthalpies of both phases across different Co concentrations. Notably, these results uncover a hidden strain order imposed by Co that influences the B2–B19′ transformation and superelastic behavior. At low Co content, the B19′ phase is less distorted than B2, thermodynamically favoring the transformation from B2 to B19’. This preference reverses at high Co content, where B19’ becomes more strained. The crossover point—where both phases experience comparable lattice distortion—is particularly significant, as it suggests a geometrically “frustrated” state that may inhibit coordinated phase transition. Crucially, this transition can be precisely modulated by adjusting the Ni:Co ratio, as demonstrated experimentally in Fig. 2. In comparison, volumetric strain remains largely insensitive to Co content in both phases, indicating a secondary role in strain-governed phase transformation.

To further elucidate the influence of the Ni:Co ratio, we performed detailed electronic structure calculations (see “Methods”). Given that Ti, Zr and Hf share the same valence electron configuration, Zr was selected as a representative model system to simplify the presentation of electronic structure results. The electron localization function (ELF) on the (100) plane of the B2 phase (Fig. 5e) reveals enhanced electron localization between Zr atoms in Zr-Co pairs relative to Zr-Ni pairs. This suggests that Co forms stronger covalent-like bonds with Zr, thereby stabilizing the Zr-occupied B2 sublattice and reducing lattice displacement and lattice distortion. Consistent with this, the displacements of Ti and Hf atoms also decrease systematically with increasing Co content (Supplementary Fig. 13). In contrast, the ELF distribution on the (001) plane of the B19′ phase exhibits the opposite behavior (Fig. 5e): substitution of Ni with Co weakens electron localization, resulting in a more compliant Zr sublattice with larger atomic displacements. To assess the influence of Ti and Hf elements, we further performed DFT calculations for TM-X systems (TM = Ti and Hf, and X = Ni and Co). Our calculations show that Co can induce a BCC-like atomic order not only in ZrCo but also in TiCo and HfCo. Ni, however, does not produce this effect in any of the three systems (see Supplementary Fig. 14). Notably, the Ti-Co system undergoes the most sensitive transformation, fully converting to a standard B2 phase. At higher Co concentrations, this promotes atomic slip along the [100] direction and facilitates the formation of B2-like clusters within the B19′ matrix (Supplementary Fig. 15a). These electronic and structural changes align with the calculated enthalpies (Fig. 5b), which indicate that the B2 phase is more stable than B19′ at a Ni:Co ratio of 0:50. No such B2-like clustering is observed in B19′ at a Ni:Co ratio of 50:0 after structural relaxation (Supplementary Fig. 15b). Both the magnitude of atomic displacements (Supplementary Fig. 15c) and the extent of B2-like clustering (Supplementary Fig. 15d) increase with Co content in B19′, indicating a growing tendency for B19’-to-B2 reversion at higher Co concentrations.

To further probe the electronic origins of phase stability, we compared the total density of states (TDOS) for alloys with Ni:Co ratios of 50:0, 25:25, and 0:50 before and after structural relaxation, with a focus on orbital hybridization. In the B2 structure, the TDOS of the Ni:Co=50:0 alloy undergoes a substantial reshaping after relaxation (Supplementary Fig. 16). This is attributed to significant lattice distortion, which disrupts the crystal field and promotes diverse orbital hybridization. In contrast, the TDOS profile of the Ni:Co=0:50 alloy remains largely unchanged, consistent with its minimal lattice distortion. Notably, an additional hybridization peak emerges in the bonding states near the Fermi level in the Co-rich alloy, corresponding to Co d-eg—d-eg interactions (Supplementary Fig. 17), which further enhances structural stability. For the B19′ structure, the Ni:Co=50:0 alloy exhibits a stable TDOS curve after relaxation, with the Fermi level positioned at the bottom of a pseudogap—a signature of electronic stability and low distortion (Supplementary Fig. 18). This contrasts sharply with the Co-rich B19’ alloy (Ni:Co=0:50), which shows greater electronic disorder, reflecting its structural instability and propensity for transformation.

Discussion

Based on our experimental and theoretical findings, we propose a physical mechanism for the composition-dependent superelasticity in HEAs, governed by a hidden strain order that modulates the transition between Hookean and non-Hookean elasticity. This order is revealed by a systematic dichotomy in lattice strain: with increasing Co content, the strain in the B2 phase decreases while it increases in the B19′ phase. This strain hierarchy directly dictates the thermodynamic stability of the phases and the resultant elastic behavior. As illustrated in Fig. 6, at low Co concentrations, the B19′ phase exhibits lower lattice strain than the B2 phase, thermodynamically stabilizing B19′ and resulting in conventional linear elasticity. Conversely, at high Co concentrations, the lower strain of the B2 phase stabilizes it, favoring Hookean linear behavior and resisting transformation. The most striking behavior emerges at intermediate compositions (near the equiatomic Ni/Co composition), where both phases exhibit comparable and maximized lattice distortion. This creates a frustrated thermodynamic state where neither phase is distinctly favored. While configurational entropy favors high-symmetry B2 phases at high temperatures, stabilizing it metastably, the inherent strain frustration at lower temperatures suppresses long-range martensitic transformation. Instead, it promotes a strain-glass-like state, characterized by non-Hookean superelasticity, significant mechanical hysteresis, and large recoverable strain. Our experimental results (Fig. 4j–l) directly corroborate this strain-order mechanism. The Co15 alloy, with lowest atomic lattice strain, exhibit a stable B19’ structure. The Co25 alloy, with lower lattice strain, forms a stable B2 phase. Crucially, the Co22 alloy resides in the metastable transition regime; though it retains a B2 phase, its lattice strain is significantly higher than that of the stable Co25 B2 phase, confirming the role of strain—not just structure—in governing stability and properties.

Fig. 6. The schematic illustration of lattice strain variation and hidden strain order governing hookean versus non-hookean superelasticity.

The dash blue and red curves represent the lattice strain variations in the B19’ and B2 phase respectively. The solid line represents the trend of feasible lattice strain variations in real alloys.

Here, we emphasize that the superelasticity governed by this hidden strain order is fundamentally distinct from the mechanisms observed in conventional SMAs^4,40 and strain glasses⁴¹. In traditional systems, such as Ti-Ni based alloys, transformation characteristics are primarily dictated by valence electron concentration and specific chemical substitutions that alter the local thermodynamic driving force for phase transformation (i.e., through local pinning effect)^42,43. By contrast, in the present multi-principal element HEA, the hidden strain order arises from the complex interplay of atomic-size mismatch and chemical ordering among constituent elements distributed throughout the distorted crystalline structure. The connection between chemical fluctuation, chemical short-range order, and lattice distortion is well established in chemically complex alloys^22,25,26. Atomic-size mismatches among constituent elements generate local lattice strains, while the specific arrangement of atoms, i.e., chemical short-range ordering, modulates both the magnitude and distribution of these strains²⁰. Regions with stronger chemical ordering can locally accommodate size differences more efficiently, reducing lattice distortion, whereas regions with weaker or frustrated ordering amplify local strains. This reciprocal interplay governs the heterogeneity of the atomic-scale strain field, providing a fundamental basis for the tunable Hookean and non-Hookean superelasticity observed in the (TiZrHf)₅₀Ni_50-xCo_x alloys. By adjusting the Ni/Co ratio, the balance between chemical ordering and lattice distortion can be finely controlled, resulting in a continuous strain-field landscape that can be precisely tuned through minor compositional variations (<2 at.%), unlike binary or ternary systems where similar adjustments often induce discrete phase transitions. The chemical complexity of the high-entropy system facilitates smooth evolution between strain states, enabling continuous modulation of the elastic response— from Hookean to non-Hookean and back to Hookean—within a single alloy system. Such compositional tuning of elasticity has not been achieved in conventional alloys.

In summary, our work establishes a lattice-strain-mediated mechanism that governs the superelastic behavior of HEAs. Using the (TiZrHf)₅₀Ni_50-xCo_x model system, we demonstrate that precise tuning of the lattice strain in the B2 and B19’ phases enable reversible switching between Hookean and non-Hookean superelasticity through minimal compositional variation (e.g., <2 at.%). This strain-centric control, enabled by the intrinsic chemical complexity of multi-principal-element systems, provides a design paradigm for tailoring elastic properties beyond the limitations of conventional alloys. The resulting capability to precisely regulate energy dissipation and storage opens avenues for advanced applications in damping systems, resilient structural components, and adaptive mechanical metamaterials.

Methods

Sample preparation

The present (TiZrHf)₅₀Ni_50-xCo_x alloys were fabricated by an arc-melting furnace in Ti-gettered high-purity argon atmosphere. The master alloys were papered from commercially pure elements with purities at least 99.9 wt.%. The ingots were turned and remelted at least four times to ensure the chemical homogeneity, and then drop cast into a water-cooled copper plate mold with dimension of 5 × 10 × 50 mm³. The as-cast ingots were encapsulated into vacuum quartz tubes and thermal annealed at 1000 °C for 6 h with water quenching to eliminate possible chemical inhomogeneities.

Structural characterization

The crystalline structures of the present alloys were determined through an XRD equipment (D8 ADVANCE Da Vinci, Bruker) using Cu-Kα radiation with wavelength of 1.5418 Å. The microstructure characterization was conducted in a scanning electron microscope (SEM, LYRA3 GMU, TESCAN) equipped with electron backscatter diffraction (EBSD, Nordlys Max3, Oxford Instruments) detector for crystallographic analysis. The EBSD samples were papered by mechanical grinding using SiC sandpapers and then polished step by step using diamond and alumina oxidation suspension. The detailed structure analysis was carried out in a TEM equipment (FEI Talos F200X G2, Thermofisher). Whereas the atomic scale STEM-HAADF analysis was conducted on the spherical aberration corrected TEM (FEI Spectra 300, Thermofisher). The GPA was performed using the open-source software Strain++⁴⁴, which enables the determination of various strain components in the atomic strain field from high-resolution TEM images. The cryogenic structure analysis was carried out on a TEM equipment (JEOL 2100F) equipped with a cryogenic TEM sample holder. The TEM samples were prepared following two methods, i.e., fabrication using ion milling and focused-ion beam (FIB). For these ion milling samples, the sheet TEM foils were mechanically grinded to a thickness about 40 μm and then punched into discs with diameter of 3 mm. The TEM foils were further dimpled and thinned using ion milling in a Precision Ion-beam Polish System (PIPS, PIPS II 695, Gatan). For the FIB-fabricated TEM samples, a 2 μm-thick protective layer of Pt was first deposited over the area of interest. The lamellas with dimension about 10 × 7 × 1.5 μm³ were then extracted and attached to a copper TEM grid. Finally, the lamella was thinned to below 100 nm by alternately milling each side at tilt angles of ±1.2°.

Mechanical characterization

The macroscopic compression tests were conducted on a universal test machine (MTEST 5000 W Tensile Stage, GATAN) with quasi-static strain rate of 1 × 10⁻³s⁻¹. The cylindrical compression samples (3.5 mm in diameter and 7 mm in height) were prepared by wire cutting, followed by mechanically grinding and polishing to a mirror surface. A digital image correlated (DIC) system was used for the in-situ strain analysis (Vic-2D, Correlated Solution). The micropillar samples were prepared using the SEM/FIB system using the annual milling methods with a diameter about 1 μm and height about 2 μm. The micropillar compression tests were carried out on a Hysitron TI 950 TriboIndenter system. The high-temperature micropillar compression tests were carried out on the Hysitron TI 950 TriboIndenter system equipped with a heating stage.

Phase transformation characterization

Latent heat and characteristic temperatures associated with martensitic transformations were determined by a DSC (TA-Q200) at a cooling/heating rate of 10 K/min. Electrical resistivity (ER) curves were detected by a four-probe method under a constant current of 100 mA and at a cooling/heating rate of 3 K/min. Storage modulus and internal friction curves were measured by a dynamic mechanical analyzer (TA-Q800) in a single cantilever mode with a cooling rate of 2 K/min and a frequency range from 0.2 to 20 Hz.

DFT calculations

DFT calculations were conducted using the Vienna Ab-initio Simulation Package⁴⁵ to investigate the influence of Co content on lattice strains in the (TiZrHf)₅₀Ni_50-xCo_x alloy system. To accurately model these compositions, which are highly sensitive to the Ni/Co ratio, we constrained the special quasi-random structures (SQS) supercell to 100 atoms. For the B2 alloys, SQS were generated within 5 × 5 × 2 supercells, while the B19′ alloys were modeled using 4 × 1 × 5 SQS supercells. Taking the Ni:Co = 1:1 composition as a representative case, we performed volume-energy fitting using more isotropic supercells, a 2 × 2 × 2 supercell for B2 and a 3 × 2 × 2 supercell for B19’. As shown in the Table S1, the resulting lattice constants, when normalized to their conventional unit cells, deviated by less than 0.3%. This excellent agreement confirms that the systematic bias from the anisotropic 100-atom supercells is negligible. To determine the equilibrium lattice parameters, the uniform scaling factor of each structure was systematically varied from 0.94 to 1.05 in increments of 0.01, and single-point total energies were computed for each of the resulting 12 configurations. The energy–volume data were subsequently fitted using the Birch–Murnaghan equation of state, yielding optimized lattice constants for each alloy composition. Following this, full structural relaxations were performed, allowing atomic positions to relax freely while constraining lattice shape deformations. For the lower-symmetry B19’ supercells, using the Ni:Co = 1:1 composition as a representative example, we first applied independent −1% to +1% perturbations to the three lattice parameters of the initial structure. This process confirmed that using a uniform scaling factor provides the pathway of steepest energy descent (Supplementary Fig. 19a). With the optimized lattice, we further applied another set of independent −2% to +2% perturbations to the three lattice constants. This step confirmed that the optimized structure is located at the bottom of the potential energy surface, thereby validating the uniform scaling approach (Supplementary Fig. 19b). Periodic boundary conditions were applied throughout, and the projector augmented-wave method⁴⁶ was employed in conjunction with the Perdew–Burke–Ernzerhof generalized gradient approximation for exchange-correlation interactions⁴⁷. Geometry optimizations were carried out using the conjugate gradient algorithm, with convergence thresholds set to 1 × 10⁻⁵eV/atom for total energy and 0.01 eV/Å for residual atomic forces. A plane-wave energy cutoff of 400 eV was adopted for both B2 and B19′ phases. Brillouin zone sampling was performed using Monkhorst–Pack k-point grids of 3 × 3 × 3 for B2 and 2 × 4 × 2 for B19′ structures. Formation energy is defined as the energy difference between the compound and its constituent elements in their standard states. These computational settings were also applied to the electronic structure calculations of binary B19′ ZrCo and ZrNi compounds, wherein both lattice shape and atomic positions were fully relaxed. The B2-like ordering in the relaxed B19′ structural (TiZrHf)₅₀Ni_50-xCo_x alloy is analyzed by Common Neighbor Analysis (CNA) module in OVITO software. The volumetric strain tensor η^Volume and Von Mises shear strain tensor η^Misess in this work were calculated according to the following formulations⁴⁸:

$${\eta }_i^{Volume}=\frac{{\eta }_{xx}+{\eta }_{yy}+{\eta }_{zz}}{3}$$ 1

$${\eta }_i^{Mises}=\sqrt{{\eta }_{yz}^2+{\eta }_{xz}^2+{\eta }_{xy}^2+\frac{{\left({\eta }_{yy}-{\eta }_{zz}\right)}^2+{\left({\eta }_{xx}-{\eta }_{zz}\right)}^2+{\left({\eta }_{xx}-{\eta }_{yy}\right)}^2}{6}}$$ 2

Author contributions

Y.Y., S.J.Z., and J.F.G. supervised the project. Y.Y., Q.F.H., and S.R. conceived the idea. Q.F.H. and S.R. fabricated the samples and characterized their structures and mechanical properties. X.L.G. and S.J.Z. carried out the atomistic simulations. H.G., X.F.W., Z.Q.C., R.H., and Q.W. contributed to the data analysis. Y.Y. and Q.F.H. wrote the manuscript. All authors participated in the discussion of the results.

Peer review

Peer review information

Nature Communications thanks Huilong Hou and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. A peer review file is available.

Data availability

The data generated in this study are provided in the Supplementary Information/Source Data file. Source data are provided with this paper. All the raw data relevant to the study are available from the corresponding author upon request. Source data are provided with this paper.

Competing interests

The authors declare no competing interests.

Supplementary information

The online version contains supplementary material available at 10.1038/s41467-026-69108-6.