## 3 Results and Discussion
### Mechanical Properties
Fe element has the largest variation range of \(B/G\) and \(\
u\), indicating that Fe may have the most significant effect on ductility. However, the effect of the Cr element on ductility is almost the same as Fe. The Cauchy pressure (\(C_{12}\)-\(C_{44}\)) reflects the bonding characteristics of the alloy. A more positive \(C_{12}\)-\(C_{44}\) means more metallic bonding, and otherwise more covalent bonding [63]. As shown in Figure 4c, alloys have lower negative \(C_{12}\)-\(C_{44}\) as the content of Cr and Ni decreases and the content of Co and Fe increases, which indicates that the covalent nature of the metallic bond is enhanced. Covalent bonds are more stable than metal bonds, which provides an explanation for the structural stability of the Co-Cr-Fe-Ni HEAs. The stronger the alloy's metallicity, the better the ductility of the materials. The trend of \(B/G\) and \(\
u\) is consistent with that of \(C_{12}\)-\(C_{44}\). In addition, the severe lattice distortion corresponding to the higher \(\delta\) hinders the dislocation slip of HEAs in plastic deformation and increases yield strength. Therefore, our results indicate that the structural stability of the Co-Cr-Fe-Ni HEAs decreases but the ductility enhances with increasing Cr and Ni content and decreasing Co and Fe content. The bulk modulus (\(B\)) represents the ability of the alloys to resist volume deformation under external pressure, the shear modulus (\(G\)) represents the ability to resist reversible deformation under shear stress, and Young's modulus (\(E\)) represents the ability to resist compression or tension within the elastic limit and can describe the stiffness of the materials. As shown in Figure 4d, the bulk modulus increases with increasing Fe content and decreases with increasing Ni content. Ni element has the most significant effect on the bulk modulus. The variation of Co and Cr has a slight effect on the bulk modulus, and the calculated bulk modulus is 255-262 GPa, indicating that the incompressibility of all Co and Cr series alloys is similar. This is consistent with previous work [38]. The variation of the shear modulus with composition is consistent with the variation of Young's modulus, as shown in Figure 4e,f. They increase with increasing Co and Fe contents and decrease with increasing Cr and Ni contents. The \(G\) and \(E\) variation ranges of Fe and Ni elements are almost the same. With regard to Cr, the shear modulus and Young's modulus increase firstly and then decrease. This change is consistent with the change in the electronic charge gain of the atoms in FCC-Cr material [34]. Among the four elements, only the 4s orbital unsaturation of Cr may change the chemical interactions with other elements [33; 34], as well as the unique magnetic characteristics of Cr [53], which affects the structure and elastic properties. Stiffer materials generally have a larger Young's modulus. Hence, increasing Co and Fe contents or decreasing Cr and Ni contents would improve the stiffness of Co-Cr-Fe-Ni HEAs, and the Ni element has the most significant effect on the stiffness. Combining structural stability and brittle-ductile behavior, we find that the variation of stiffness with composition is consistent with that of structural stability but opposite to the brittle-ductile behavior. The relationship between stiffness and ductility is consistent with previous work on Al\({}_{x}\)FeCuCrNi HEAs [64]. This may be because the covalent nature of the metal bond improves the structural stability, and stiffness, but reduces the ductility. For FCC Co-Cr-Fe-Ni HEAs, the atomic-level pressure variation originating from the charge transfer between neighboring elements in the alloy will also improve the stiffness of the system [59]. Overall, the incompressibility, resistance to shear deformation, and stiffness of Co-Cr-Fe-NiHEAs improve with the increasing Co and Fe contents or decreasing Cr and Ni contents, corresponding to a more stable solid-solution structure and weakened ductility.