Non-Hookean large elastic deformation in bulk crystalline metals

Crystalline metals can have large theoretical elastic strain limits. However, a macroscopic block of conventional crystalline metals practically suffers a very limited elastic deformation of <0.5% with a linear stress–strain relationship obeying Hooke’s law. Here, we report on the experimental observation of a large tensile elastic deformation with an elastic strain of >4.3% in a Cu-based single crystalline alloy at its bulk scale at room temperature. The large macroscopic elastic strain that originates from the reversible lattice strain of a single phase is demonstrated by in situ microstructure and neutron diffraction observations. Furthermore, the elastic reversible deformation, which is nonhysteretic and quasilinear, is associated with a pronounced elastic softening phenomenon. The increase in the stress gives rise to a reduced Young’s modulus, unlike the traditional Hooke’s law behaviour. The experimental discovery of a non-Hookean large elastic deformation offers the potential for the development of bulk crystalline metals as high-performance mechanical springs or for new applications via “elastic strain engineering.”


On the deviation from Hooke's law
The deviation from Hooke's law within large elastic deformation may have been experimentally observed in nanowires or whiskers due to lattice anharmonicity owing to the large deviations in interatomic separation from the equilibrium values 1 , but it is not a frequent case for bulk crystalline metals in experiments. The linear elastic response of a crystalline material is represented by a quadratic term in the interatomic potential energy; however, nonlinear elasticity requires higher order terms 2 . The nonlinear large tensile elastic response in the present bulk Cu-Al-Mn single-crystalline alloy would suggest a high-order elastic behaviour, in which the stressstrain relationship for uniaxial loading can be expressed as follows 3 : where T is the true stress, which is the applied force F divided by the actual cross-sectional area A of the tensile specimen at that load, T = ; T is the true strain, which is equal to the natural log of the quotient of current length L over the original length L0, T = ln 0 ; and E and D are the incipient Young's modulus and third-order modulus, respectively. This relationship is valid for any dimensionality and has been used in evaluating large-strain elastic responses in nanostructures such as Pd nanowires and graphene membrane 3,4 . Our data for the tensile true stress-strain curves in the present bulk Cu-Al-Mn alloy shows that a quadratic fit describes the nonlinear response until fracture ( Supplementary Fig. 8), where E and D are determined to be 23.3 and −202.8 GPa, respectively. The ratio between D and E (b = D / E), known as the strain-expanded nonlinearity parameter directly scaling the level of anharmonicity 3 , was found to be −8.70. This large magnitude of nonlinearity parameter indicates a strong elastic softening upon tension beyond the linear region.
An elastic response implies the existence of a potential energy that is a function of strain.
By considering Poisson's effect, a relationship between Equation (1) and the strain energy ( T ) related to strain of a crystalline material with Poisson's ratio v and initial volume V0 subjected to uniaxial tension can be appropriated as follows 3,5 : where the second-order term is the harmonic term, the third-order term represents the anharmonic contribution. Subsequently, a dimensionless expression for the fractional anharmonic contribution to the strain energy density as function of T can be expressed as 2 3 (1 − 2 + ) T . Using v = 0.4665 (obtained from Fig. 3), this value would be −6% and −25% for tensile true strains of 1% and 4.3%, respectively, in the present case ( Supplementary Fig. 9a), indicating that lattice anharmonicity plays a substantial role for the strong elastic softening upon uniaxial tension particularly at a large-strain deformation.
Following Milstein's work 6 , the following relation is obtained as a lattice stability criterion for a cubic crystal: The β-BCC lattice becomes unstable to the strain at which the K vanishes. Supplementary Fig. 9b plots the strain energy density 0 and force constant K as a function of tensile true strain for the present alloy upon <100> tension. The ideal elastic strain limit can be then estimated to be 5.79% at a strength of 669 MPa (true strain and true stress). The experimental results of 639 MPa (true stress) and 4.47% (true strain) greatly approach the ideal values, putting the present Cu-Al-Mn alloy into the "ultrastrength materials" catalogue. Here, the term "ultrastrength" was coined in 2009 to describe material components sustaining sample-wide stress level at a significant fraction (>1/10th) of its ideal strength or 1% elastic strain pervasively 7,8 .
By using Equation (2), the strain energy density 0 as a function of true strain in both tension and compression is plotted in Supplementary Fig. 9c. It is seen that there is an asymmetry in strain energy density between tension and compression because of the lattice anharmonicity.

Key factors for achieving large tensile elastic deformation
Supplementary Fig.10 compares the strain energy landscape of the present Cu-Al-Mn alloy to that of typical bulk BCC-structured metallic elements for uniaxially stretching along the <100> direction, including Fe, W, Mo and Nb 9,10,11 . One can see that the increase in strain energy due to straining is far smaller in the present Cu-Al-Mn alloy, indicating that a smaller applied stress could induce a large elastic strain within the lattice stability region. Incidentally, this mild 4 strain energy landscape is in consistent with the results of Heusler Cu2MnAl crystals obtained by first-principles calculations 12,13 , which indicates that the crystal could distort easily due to intrinsic weak interatomic bonds. The mild characteristics in energy landscape can also reflect a low value of C′ modulus and a low Young's modulus in cubic crystals 14 .
Nevertheless, the inhibition of plastic deformation is a key issue for achieving a high yield strength. It is traditionally considered that metals showing a low Young's modulus have a low strength, such as Sn alloys, Al alloys, and Mg alloys. Typically, for example, in β-Sn or its alloys, a low C′ modulus or a low Young's modulus may be identified, indicating a mild energy landscape on straining, yet their elastic strain is smaller than 0.2% due to the very small yield strength related to the occurrence of dislocation-stimulated plastic deformation 15 . However, being different from those conventional soft metals, the present Cu-Al-Mn alloy is featured by its long-range ordered atomic structure (ordered BCC structure, L21) 16 , which should give rise to a high resistance to dislocation creation or movement for plastic deformation due to ordering strengthening effects 17 .
Ordering strengthening can be mainly considered to be related to the antiphase boundary (APB) energy. Because the creation of superlattice dislocations in ordered structures will form an APB interface, which means that the critical stress to generate and move the dislocations should be larger due to the involvement of APB energy. The high APB energy (503 mJ m −2 ) featured in Cu2MnAl system indicates a high yield strength to plastic deformation 18 Fig.12), may also contribute to the ordering strengthening effect 17 , which represent a future research topic for the present Cu-Al-Mn alloy. Hence, large elastic deformation can be obtained in bulk BCC crystals with low resistance to distortion (this can be reflected by a low shear modulus C′ or a low Young's modulus) and a long-range ordered atomic structure for high yield strength.
3. On the difference between elasticity and pseudoelasticity 5 Reversible deformations caused by mechanical stresses in crystalline metals can be due to a true elasticity or a pseudoelasticity. A true elastic behaviour in crystalline metals is a macroscopic manifestation of atomic bonding and thus a result of bonds against extension or compression and distortion. The elastic strain should then correspond to the reversible lattice strain in the single phase. Being different from a true elasticity, pseudoelasticity can be attributed to several other mechanisms, such as the stress-induced martensitic transformation (also known as superelasticity in shape memory alloys) 20,21 , the twinning-untwining behaviour 22 , or the reversible glide of dislocations (in rare cases) 23,24 . The strain in pseudoelasticity is actually inelastic; however, it is reversible. A true elasticity is noticeable for its features such as the zero stress hysteresis and the monotonicity in stress-strain relations.
Considering the more possible confuses between a true elasticity and a pseudoelasticity induced by stress-induced martensitic transformation (superelasticity), here the evidence for distinguishing them is presented.
• Stress-strain characteristics: In superelasticity, the stress-strain curve is usually highly nonlinear with a hysteresis loop (e.g., Supplementary Fig. 13a), because stress-induced martensitic transformation usually occurs in an avalanche-like manner, leading to strain burst and large energy dissipation 20 . For some alloys exhibiting confined stress-induced martensitic transformation or continuous phase transformation due to defects, the stress hysteresis can be greatly reduced; however, the inflection points in the stress-strain curve would indicate the presence of a weak first-order phase transition 25,26,27 . For a true elasticity, since there is no occurrence of first-order phase transition, the stress-strain relation is monotonic without any stress hysteresis (Fig. 1a).
• Microstructure characteristics: In superelasticity, stress-induced martensitic transformation occurs with the creation of interface between parent phases and martensite phases, which presents the surface relief in the specimen 21 . In a true elasticity, the deformation is homogeneous across the single phase; therefore, no surface relief or trace appears (Fig. 2a).
• Phase identification characteristics: In superelasticity, due to the transformation from parent phase to martensite phase with a different crystal structure, there should be additional peaks or lines that appear in the X-ray or neutron diffraction patterns. Upon the deformation processing, the intensity for the peaks or lines of martensite phases increases whereas that of parent phase decreases. In a true elasticity, the single-phase nature ensures that no additional 6 peaks or lines appear. The positions of peaks or lines shift continuously due to the lattice parameter change of the single phase during elastic deformation, whereas the intensities remain almost constant (Fig. 3).
• Thermal characteristics: In superelasticity, so-called elastocaloric effect occurs due to the entropy change between parent and martensite phases. In most cases, the temperature of the specimen greatly increases (by ~10 K in Cu-based shape memory alloys) upon adiabatic loading due to the transformation from parent phase to martensite phase, whereas it decreases upon adiabatic unloading due to reverse phase transformation 28 . In a true elasticity, a thermoelastic effect occurs due to the pure volumetric deformation of a single phase.
Generally, upon uniaxial adiabatic loading, elastic tension will cause a decrease (usually of less than 1 K) in the specimen temperature, and elastic compression will cause an increase in the specimen temperature ( Supplementary Fig. 7) 29 .

On the relationship to pretransitional phenomena in martensitic transformation
In shape memory crystalline alloys, there are some studies reporting a drastic reduction of C′ modulus upon cooling of presumably homogeneous parent phase towards the martensitic transformation 30 , known as pretransitional phenomena. The significant softening of certain acoustic phonons has also been reported in the cooling process of a parent phase towards martensitic transformation 30 . We performed tensile test on the present Cu-Al-Mn alloy at lower temperatures by using another near-<100> single crystal, and stress-induced martensitic transformation occurs due to the reduced relative lattice stability upon cooling ( Supplementary Fig.   13a). It is seen in Supplementary Fig. 13b that the incipient Young's modulus of the parent phase is almost constant at various temperatures and that drastic change due to pretransitional softening behaviours is not detected even before stress-induced martensitic transformation. This demonstrates that the low Young's modulus and the huge elastic deformation in the present alloy originate intrinsically in the parent phase, but they are not directly related to the pretransitional phenomena.

Elastic responses on compression and bending
7 Supplementary Fig. 14a shows the engineering stress-strain curves in compression of the present <100> Cu-Al-Mn single crystal. It is seen that the Young's modulus increases gradually with the increase in stress (elastic stiffening), yet a large elastic strain of over 1.8% can still be reached at an applied engineering stress of 600 MPa. The instantaneous Young's modulus increases from ~24 GPa at a stress of 20 MPa to ~40 GPa at a stress of 600 MPa, as shown in Supplementary Fig. 14b. The asymmetric elastic behaviour, i.e., the elastic stiffening in compression and elastic softening in tension, is a result of the lattice anharmonicity described in Supplementary Fig. 9c.
Supplementary Fig. 15a shows the force-deflection curves in bending of the present <100> Cu-Al-Mn single crystal. It is the seen that the crystal could be bended to a large deflection and then spring back to its original shape after unloading, being different from the bending response of Mo sheets undergoing plastic deformation, as also indicated by Supplementary Fig. 15  The inset shows the crystal orientation of the specimen, which is parallel to the uniaxial loading direction. The dashed lines indicate the Young's modulus calculated using elastic constants in the <001>, <101>, and <111> directions.