## Introduction

The familiar metals and alloys in our daily lives are mostly crystalline materials, and they elastically deform because of temporary stretching or contracting of bonds between atoms1,2. The ideal elastic strain for a crystalline metal is the strain at which the lattice itself disintegrates and hence sets a firm upper bound of the elastic strain of the material3. Theoretically, this strain value can be in the order of 10% for most crystalline metals with absolutely no defects4,5,6. However, the bulk shape of crystalline metals, which is the state mostly used for practical engineering applications, generally suffers a very limited elastic strain of <0.5% because of the occurrence of inelastic relaxation mediated by discrete dislocations7, deformation twinning8, and stress-induced phase transformations9. The stress–strain relationship of bulk crystalline metals in the very small elastic regime is generally linear, thus obeying the well-known Hooke’s law10.

Metals with a large elastic deformation can be used as components of mechanical vibrational systems, sporting goods, and medical devices because of their large reversible strains and mechanical energy storage11,12. One approach to achieving a huge elastic deformation in crystalline metals is the downscaling method. When reducing the size of the sample to micro- or nanoscales, such as that of freestanding whiskers and nanowires, certain metals can demonstrate ultra-high yield strength and elastic strains of 4%–7%, thus approaching their theoretical values because of a dislocation starvation mechanism13,14,15,16. However, when crystalline metals are scaled up to bulk sizes, the realization of a large elastic deformation has been confirmed to be challenging. It has been only reported in certain Ti-based alloys previously, such as the so-called gum metals, which are now used in medical and dental equipment. The elastic strain at room temperature of Ti-based gum metals can be as large as 2.5% with a Young’s modulus of ~60 GPa when the material is severely cold worked, probably because of a dislocation-free deformation mode17. Recent research also demonstrates that a large compressive elastic strain of ~2% as well as a Young’s modulus of ~100 GPa can be achieved in bulk chemically complex (or high-entropy) alloys because of the existence of localized high lattice distortion18. Note that the macroscopic elastic strain should correspond to the reversible lattice strain of a single phase for a true elastic deformation. On the other hand, certain shape memory alloys may exhibit elastic-like strains of >2% with slim stress hystereses as well as reduced apparent Young’s moduli, which are, however, associated with continuous or localized martensitic phase transition; hence such a deformation is pseudoelastic rather than truly elastic19,20,21,22. For example, the single-crystalline Ni–Co–Fe–Ga alloy fibers show a non-hysteretic recoverable strain of >15% and an incipient apparent Young’s modulus of ~25 GPa due to the continuous transformation from an ordered body-centered cubic (BCC) parent phase (B2) to a tetragonal martensite phase (L10)20,23. The nanocrystalline Ni-Ti alloys may also show an elastic-like strain of >3% and an apparent Young’s modulus of ~35 GPa because of the reversible nanoscale B2 → B19’ phase transformation21. Here, to stabilize the localized martensite phase, sometimes pre-straining as a training procedure is even required21,22,24.

As seen from Hooke’s law, to achieve a large yet true elastic strain in bulk metals, both a low Young’s modulus and high strength are required. However, there is usually a trade-off between these properties for conventional metallic materials. For example, although having a low Young’s modulus of ~45 GPa, Mg alloys are not sufficiently strong to be elastically deformed to a large extent25. On the contrary, certain maraging steels can be very strong without yielding at stresses of >1 GPa, but they are extremely stiff with a Young’s modulus of >200 GPa to tolerate a larger elastic deformation26. We noticed that in a series of Cu–Al-based alloys, low Young’s moduli along the <100> orientation of a cubic-structured crystal can be identified27,28,29. A typical example is the Cu–Al–Ni alloy, which shows a very low Young’s modulus of ~25 GPa along the <100> crystal orientation, although a stress-induced martensitic phase transformation appears at relatively low elastic strains27. Cu–Al-based alloys also have ordered atomic structures in the parent phase; thus, a high yield strength can be expected because of the ordering strengthening effect30. We therefore speculate that a large elastic strain can be obtained in these alloys if the stress-induced phase transformation can be properly suppressed.

In this work, we report a large true elastic strain of >4.3%, which is experimentally determined by a tensile test at room temperature (~298 K), for a bulk and crystalline Cu–Al–Mn alloy. To date, this true tensile elastic strain of >4.3% represents the highest experimental value ever reported for bulk crystalline metals at room temperature. This bulk alloy also exhibits a nonlinear elastic response in stress–strain relationships, which does not obey the traditional Hooke’s law.

## Results

To gain further insight into the anomalous deformation behavior, in situ neutron diffraction measurements were performed on a single crystal stretched along the <100> direction at room temperature. The experimental configuration is schematically shown in Fig. 3a. Using the time-of-flight method, neutron diffraction patterns in the loading direction (LD) and transverse direction (TD) of the tensile specimen at different stress levels were simultaneously collected. Panels b and c in Fig. 3 show the plane view of neutron diffraction profiles during loading and unloading in the LD and TD, respectively. A maximum applied stress of 580 MPa was selected so as not to damage the specimen. The diffraction patterns shift continuously and reversibly on applying and releasing the stresses in both the LD and TD. For d-spacing ranging from 0.67 to 3.33 Å, no additional lines were detected during the measurement, indicating the absence of phase transformations or decomposition during the reversible deformation, in agreement with the results obtained for in situ microstructure observations.

To quantitatively evaluate the structural changes on the atomic scale, the one-dimensional diffraction patterns corresponding to the (400) reflection peak in the LD and (004) reflection peak in the TD were analyzed during loading and unloading, as shown in panels d and e in Fig. 3, respectively. The peaks shift continuously and reversibly, whereas no other new peaks can be identified at higher stresses. The direction of the peak shift for the TD is opposite to that in the LD, indicating shrinkage in the TD and an extension in the LD. The lattice strains, which refer to a relative change of the interplanar distance in comparison with its initial value before loading in the single β phase, are plotted against applied engineering stress in Fig. 3f. One can see that the engineering stress–lattice strain curve in the LD has a similar shape to that of the macroscopic engineering stress–strain curve shown in Fig. 1a. The maximum reversible lattice strain in the LD at an applied stress of 580 MPa is 3.81%, which is extremely large and in good agreement with the macroscopically measured engineering strain of 3.83% (Supplementary Fig. 6). Poisson’s ratio, defined as the negative ratio between TD and LD lattice strains, reaches a value of 0.4665. A very similar value of the Poisson’s ratio (ν = 0.4670) can be also calculated using the experimentally determined elastic constants (Table 1). This value is close to 0.5, a Poisson’s ratio for ideal rubber materials, indicating that the present crystal shows a rubber-like mechanical behavior in which the shape easily changes while volume tends to remain invariant during the tensile deformation35. The relative volumetric change at a lattice strain of 3.81% is very small, ~0.15%, as shown in Fig. 3g. Furthermore, at a high elastic strain prior to fracture, the initial cubic crystal (lattice constant a0 = 5.8955 Å for an L21-type cell) becomes tetragonal with a huge tetragonality c/a of 1.065 where c and a are the lattice parameters for the c and a axes, respectively. Such a large tetragonality on elastic deformation has been rarely observed in cubic structured crystalline metallic materials. The abovementioned results obtained by in situ neutron diffraction measurements demonstrate that the large reversible tensile deformation at room temperature in the present alloy is truly an elastic deformation of the single β phase. This conclusion is supported by the observation of thermoelastic effect, in which a typical thermal phenomenon is induced by a true elastic deformation36, in the present alloy (Supplementary Fig. 7).

## Discussion

The low Young’s modulus and moderately high strength, which give rise to the large elastic deformation, are remarkable. The present alloy has an incipient Young’s modulus smaller than even that of Mg alloys (~45 GPa) but with apparently higher strength. We consider the low $$[110](1\bar{1}0)$$ shear modulus C′ of the cubic crystal to be important for that very low Young’s modulus. The value of C′ = 7.85 GPa in the present alloy is extremely small and has a direct contribution to the low <100> Young’s modulus when the bulk modulus representing the resistance to hydrostatic deformation is large (Table 1)37. This small C′ may be associated with the lattice instability of the β phase, which implies a low energy barrier for structural change in the tetragonal path to another structure during deformation38,39,40. Once the premature occurrence of such a structural change, usually a first-order martensitic phase transformation, is suppressed (this is exactly what we did for our alloy by tailoring the composition), the lattice may be highly distorted to yield a huge elastic deformation as well as a significant lattice anharmonicity38,39,41. The strong lattice anharmonicity induces the elastic softening upon tension, which results in a non-Hookean elastic response (see details in Supplementary Discussion, Supplementary Figs. 810). Note that the large elastic deformation and elastic softening phenomenon become weak or absent for stretching along the <110> or <111> direction because that C′ is less involved in those cases (Supplementary Fig. 11). The lattice anharmonicity of the β phase has a weak temperature dependence (Supplementary Fig. 13), and its existence is further demonstrated by the observation of an elastic stiffening phenomenon on uniaxial compression (see details in Supplementary Discussion, Supplementary Figs. 1415). On the other hand, the moderately high strength is considered to result from the atomic ordering of the L21 structure in the single β phase where dislocations related to plastic deformation are difficult to generate and migrate because of the involvement of antiphase–phase boundary (APB) energies30,42. Typically, the average APB energy for a Heusler Cu2MnAl alloy is reported to be extremely high (503 mJ m−2), indicating a high resistance to plastic deformation43. Although the yield strength of 612 MPa in the <100> deformation in the present alloy is slightly smaller than that of 679 MPa for a full Heusler structure at room temperature42, which is presumably caused by the lower degree of atomic ordering in an off-stoichiometric composition (Supplementary Fig. 12). The large tensile elastic deformation may be discovered in multiple other bulk BCC crystalline metals with similar features to the present alloy such as the low tetragonal shear resistance for a weak resistance to crystal distortion and the long-range atomic ordered single phase for high resistance to dislocation gliding (see details in Supplementary Discussion).

The large elastic deformation with an extremely low Young’s modulus in bulk metals at room temperature is useful for multiple potential applications. Figure 4 gives a comparison of elastic strain limit and Young’s modulus for various bulk metals, including amorphous and ultraelastic high-entropy alloys, and human bones. The metals showing pseudoelasticity are excluded because pseudoelasticity is a phase-transformation-induced shape change rather than a true elasticity. The huge elastic strain experimentally obtained in this study is the highest among bulk metals. Because of the large elastic strain and the moderately high strength, the storage of a large amount of mechanical energy is enabled. The mechanical energy stored with applied stress of 600 MPa is 13.5 MJ m−3, which is comparable to that of bulk amorphous alloys (~15 MJ m−3) and much higher than that of commercialized spring steels (~1 MJ m−3)11,44. This feature is fascinating for multiple engineering applications such as in high-performance springs, self-locking joints, and pressure gauges. The low Young’s modulus very close to that of human bones (10–30 GPa) has potential applications in medical devices12. Furthermore, the point-to-point correspondence between stress (or strain) and apparent Young’s modulus because of the elastic softening phenomenon make the alloy available for applications requiring different material stiffness at certain stress (or strain) ranges. The ease of large-scale production of bulk single crystals with a controllable orientation by cyclic heat treatment and the low cost make the present alloy particularly advantageous for these applications31,32,45.

Recently, a growing interest has emerged in “elastic strain engineering,” in which lattice strains are engineered as a dynamic continuous variable for manipulating multiple physical or chemical properties of materials46,47. However, the elastic strain engineering applications at present are limited in nanostructures because bulk crystalline metals rarely sustain a sufficiently large elastic deformation48,49. The large elastic strains achieved in bulk metals, such as the present Cu–Al–Mn alloy, have the potential to use elastic strain engineering for emerging technologies beyond nanoscales such as in large-volume strain-mediated sensors50,51,52.

## Methods

### Specimen preparation

Ingots of Cu69Al17Mn14 were prepared by induction melting in an argon atmosphere. Then, the ingots were remelted and subjected to unidirectional solidification so that columnar polycrystalline structures with strong <100> textures along the solidification direction could be obtained. Sheet specimens with dimensions of approximately 60 mm × 10 mm × 1 mm were cut out using electrodischarging machining (EDM) so that the length will be parallel to the direction of solidification. Then, the sheet specimens were sandwiched between Mo sheets and subjected to a cyclic heat treatment. In this manner, large <100> single crystal sheet specimens could be obtained45. Additionally, sheet specimens cut from the noncolumnar area of the ingots were subjected to cyclic heat treatment to obtain large single crystals with other orientations along the length direction.

### Mechanical tests

To determine the limit of the large elastic deformation, a dog-bone-shaped specimen (15 mm × 4 mm × 1 mm in gauge dimension) with its tensile direction close to the <100> orientation was stretched to fracture by incremental cyclic loading–unloading tests using a tensile tester (Shimadzu AG-X 10 kN). At room temperature, the specimen was initially loaded to a stress of 550 MPa and then unloaded. This was followed by reloading to a stress of 600 MPa in the second cycle, and so forth. The strains were measured using a noncontact video extensometer (Shimadzu TRViewX240S) with high accuracy. The strain rate was 5 × 10−4 s−1.

To directly evaluate the anisotropic elastic properties, quasistatic tensile tests at room temperature were also conducted for <110> and <111> bulk single crystals. Dog-bone-shaped specimens with a gauge dimension of 15 mm × 4 mm × 1 mm were cut out from the single-crystal sheets using EDM. The crystal orientations of the specimens were determined using the EBSD technique. The strains were carefully measured using strain gauges (Kyowa KFGS-2-120-C1-11) attached onto the surface of the specimens. A loading–unloading cycle with a strain rate of 5 × 10−4 s−1 was conducted for all specimens. The maximum applied stress was set to 550 MPa so as not to damage the specimens.

In the engineering stress–strain curves, engineering stress σ is defined as the ratio of the applied force F to the initial cross-sectional area A0 of the tensile specimen, $$\sigma=\frac{F}{{A}_{0}}.$$ Engineering strain ε is defined as the ratio of extension ΔL = L − L0 to the initial gauge length L0 of the tensile specimen, $$\varepsilon=\frac{\triangle L}{{L}_{0}}$$.

The uniaxial compression test at room temperature was conducted on a bulk <100> Cu–Al–Mn single crystal. A rectangular parallelepiped specimen with a dimension of 7.41 mm × 3.00 mm × 2.95 mm was used. The specimen was loaded to a stress of 600 MPa and then unloaded. The strain was measured by strain gauges (Kyowa KFGS-2-120-C1-11) attached onto the surface of the specimen. The strain rate was 5 × 10−4 s−1.

### In situ microstructure observation

In situ microstructure observation during tensile loading and unloading was conducted using a field-emission scanning electron microscope (FE-SEM, Philips XL-3000) equipped with an EBSD detector. A tensile stage tilted to 70° was used at a working distance of 18.2 mm. The specimen was in a dog-bone shape and had a gauge dimension of 10 mm × 2 mm × 1 mm. The <100> orientation of the single crystal was close to the long axis of the specimen. At room temperature, the specimen was firstly prestretched using a 14 MPa stress, then further loaded to 500 MPa in steps of 100 MPa. It was then loaded to 550 MPa, which did not lead to the fracture of the specimen, and then finally unloaded. The crosshead speed was set at 0.4 mm min−1. The EBSD data were collected at each stress point and were analyzed using a TSL-OIM software.

In situ microstructure observation under tensile stresses was also conducted using a transmission electron microscope (TEM, JEOL JEM-3200FSK) and a single-tilt straining holder (Gatan Model-671). A single-crystal specimen with a near <100> orientation was used. The area of interest for observation was polished down to a thickness of about 100 μm, of which the electron transparency area was prepared by twin-jet polishing with a solution of 45% distilled water, 25% phosphoric acid, 25% ethanol, and 5% propanol at room temperature. Inside the electron microscope, the specimen was stretched to a maximum nominal strain of 5% with a step interval of 1%. The nominal strain is calculated as the ratio between the travel distance of the crosshead and the length of the specimen.

### Determination of elastic constants and elastic anisotropy

The elastic constants of the Cu69Al17Mn14 alloy were precisely evaluated at room temperature using the laser–ultrasonic transient grating spectroscopy method53. The measurements were performed on an approximately (110)-oriented single crystal. The alloy exhibits a very small tetragonal shear modulus ($${C}^{{\prime} }=\frac{{C}_{11}-{C}_{12}}{2}$$) of 7.85 GPa and therefore a large elastic anisotropy ($$A=\frac{{C}_{44}}{{C}^{{\prime} }}$$) of 11.1. In an anisotropic cubic crystal, the Young’s modulus E along a certain direction n (unit vector) is given by the following equation54:

$$\frac{1}{E({{{{{\bf{n}}}}}})}=\frac{1}{3({C}_{11}+2{C}_{12})}-\frac{1-3P\left({{{{{\bf{n}}}}}}\right)}{3\left({C}_{11}-{C}_{12}\right)}+\frac{1-P\left({{{{{\bf{n}}}}}}\right)}{2{C}_{44}},$$
(1)

where

$$P\left({{{{{\bf{n}}}}}}\right)={n}_{1}^{4}+{n}_{2}^{4}+{n}_{3}^{4}.$$
(2)

By using the experimentally determined elastic constants, the orientation dependence of Young’s modulus was calculated. The calculated incipient Young’s modulus along the <100> direction is 23.2 GPa, which agrees well with the value obtained from quasi-static tensile measurements in Fig. 1. The <100> Young’s modulus can also be written as a function of C′ and the bulk modulus ($$B\,=\frac{{C}_{11}+2{C}_{12}}{3}$$) as follows:

$${E}_{ < 100 > }=\frac{9{C}^{{\prime} }}{{C}^{{\prime} }/B+3}.$$
(3)

It can be seen that the low Young’s modulus along the <100> direction is mainly caused by the small C′ when B is far larger than C′. The Young’s modulus for the <110> and <111> directions were calculated using the same approaches. The results are E<110> = 69.2 GPa and E<111> = 209.1 GPa, respectively.

The Poisson’s ratio ν of a cubic crystal subjected to <100> uniaxial elastic deformation can be explicitly expressed in terms of elastic constants by54:

$${\nu }_{ < 100 > }=\frac{{C}_{12}}{{C}_{11}+{C}_{12}}=\frac{1}{2+\frac{{C}_{11}-{C}_{12}}{{C}_{12}}}=\frac{1}{2+\frac{2C^{\prime}}{{C}_{12}}}.$$
(4)

The calculated Poisson’s ratio is 0.4670. It is also seen that when the value of C′ is small, i.e., the value of $$({C}_{11}-{C}_{12})$$ approaching 0, the Poisson’s ratio in the <100> elastic deformation approaches 0.5.

### Evaluation of the thermoelastic effect

$$\triangle T=-\frac{{\alpha }_{{{{{{\rm{L}}}}}}}{T}_{0}}{{C}_{{{{{{\rm{p}}}}}}}\rho }\triangle \sigma,$$