*3.1. Mechanical Properties of Ni3Al*

In the elastic stage, the stress and strain relationship can be described by Hooke's law: σ*ij* = *Dijkl*ε*ij*, where *Dijkl* denotes the elastic constants. According to the symmetry of the crystal lattice, the elastic stress–strain matrix **[***Dijkl***]** of single-crystalline Ni3Al can be expressed as


where σ*ij*, ε*ij*, and γ*ij* are the stresses, normal strains, and shearing strains respectively. The stiffness matrix **[***Dijkl***]** was calculated by linearly fitting four small strains (±0.001 and ±0.003) under nine deformation conditions (see Supplementary Figure S1); the calculated elastic constants are listed in Table 1. The flexibility matrix **[***Sijkl***]** was calculated as the inverse matrix of the stiffness matrix **[***Dijkl***]**, i.e., **[***Sijkl***]** = **[***Dijkl***]** <sup>−</sup>1; the results for elastic constants *Sijkl* are given in Table 2.


**Table 1.** Elastic constants *Dijkl* for single-crystalline Ni3Al.

**Table 2.** Elastic constants *Sijkl* for single-crystalline Ni3Al.


According to the Voigt–Reuss–Hill approximation method [28], the bulk modulus (*B*) and shear modulus (*G*) of polycrystalline Ni3Al can be calculated by Equations (1) and (2), respectively:

$$B = \frac{1}{2} \left[ \frac{1}{3S\_{1111} + 6S\_{1122}} + \frac{1}{3} (D\_{1111} + 2D\_{1122}) \right] \tag{1}$$

$$G = \frac{1}{2} \left[ \frac{15}{4S\_{1111} - 4S\_{1122} + 3S\_{1212}} + \frac{1}{5} (D\_{1111} - D\_{1122} + 3D\_{1212}) \right]. \tag{2}$$

The calculated values of *B* and *G* are 186.72 GPa and 78.86 GPa respectively. Moreover, the elastic modulus (Young's modulus, *E*) and Poisson's ratio (*v*) can be evaluated by Equations (3) and (4) respectively:

$$E = \frac{\Re G}{\Im B + G'} \tag{3}$$

$$\nu = \frac{3B - E}{6B}.\tag{4}$$

The calculated values of *E* and *v* for polycrystalline Ni3Al are 207.38 GPa and 0.315 respectively; these values are similar to the experimental values *E* = 203.1 GPa and *v* = 0.305 at room temperature [29].
