## 3 Results and Discussion
### The Phase Formation Mechanism and Mechanical Properties
However, the thermodynamic and mechanical stability change when x \(\geq\) 0.75, which indicates that the phases easily decompose or react with other substances, although the structure could form in the non-equilibrium process when the Cu content in Ti\({}_{(1-x)}\)M\({}_{x}\)N phase is high enough. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Phase** & \(B\) & \(G\) & \(E\) & \(v\) & \(G\)/B & \(C_{12}\)–\(C_{44}\) & \(H_{V}\) \\ \hline AlCu\({}_{2}\)Ti & 130 & 63 & 164 & 0.29 & 0.49 & 5 & 6.75 \\  & 135 a & 55 a & & & 0.41 a & & \\  & 131 b & 42 b & 120 b & 0.354 b & 0.32 b & & \\ \hline Al\({}_{3}\)Cu\({}_{8}\)Ti\({}_{5}\) & 124 & 54 & 141 & 0.34 & 0.44 & 25 & 4.81 \\ \hline Al\({}_{5}\)Cu\({}_{3}\)Ti\({}_{3}\) & 131 & 47 & 125 & 0.31 & 0.36 & 16 & 2.68 \\ \hline \hline \end{tabular}  
* Calculated data reported in Ref. [40]; b Calculated data reported in Ref. [41].  
\end{table} 
Table 11: The calculated bulk modulus \(B\) (GPa), shear modulus \(G\) (GPa), [PERSON]’s modulus \(E\) (GPa), [PERSON]’s ratio \(v\), [PERSON]’s ratio \(G\)/\(B\), Cauchy pressure \(C_{12}\)–\(C_{44}\), and Vickers hardness \(H_{V}\) for the Al-Cu-Ti intermetallics. Figure 12: (**a**) The change in the solution energy for N in Cu-Ti intermetallics; (**b**) the formation enthalpy of Ti\({}_{(1-x)}\)M\({}_{x}\)N.  
The volume modulus, shear modulus, Young's modulus, Poisson's ratio, \(G/B\) value, Cauchy-pressure, and Vickers hardness of Ti\({}_{\{1-x\}}\)M\({}_{x}\)N were also calculated, as shown in Table 14. It can be seen that the volume modulus, shear modulus, elastic modulus, and hardness all decreased with the increasing content of Fe and Cu. When the intermetallics contained the element Al, the volume modulus decreased while the shear modulus increased, which means that both Young's modulus and hardness grew. The shear modulus was positively correlated with phase stability, which was consistent with the changing minimum enthalpy and high stability when Ti atoms were replaced by Al atoms. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{\begin{tabular}{c} **Replacing** \\ **Atom** \\ \end{tabular} } & \multirow{2}{*}{\
\begin{tabular}{c} **Atomic** \\ **Fraction** \\ \end{tabular} } & \multicolumn{5}{c}{**Single-Crystal Elastic Constants (GPa)**} & \multirow{2}{*}{
\begin{tabular}{c} **Stability** \\ **Stability** \\ \end{tabular} } \\ \cline{3-10}  & & 0.25 & 505.64 & 145.68 & & & 179.63 & 505.64 & Yes \\  & 0.5 & 548.5 & 83.81 & 158.53 & 394.15 & 187.57 & 179.15 & Yes \\ Al & 0.75 & 409 & 165.74 & & & 307.61 & 409 & Yes \\  & 1.0 & 434.96 & 154.94 & & & 232.1 & 434.96 & Yes \\ \hline \multirow{2}{*}{Fe} & 0.25 & 525.79 & 120.65 & & & 150.52 & 525.79 & Yes \\  & 0.5 & 441.77 & 150.14 & 125.4 & 500.89 & 117.23 & 168.54 & Yes \\  & 0.75 & 404.59 & 118.68 & & & 100.84 & 404.59 & Yes \\  & 1.0 & 400.13 & 267.5 & & & \(-\)73.56 & 400.13 & No \\ \hline \multirow{2}{*}{Cu} & 0.25 & 485.49 & 153.49 & & & 144.72 & 485.49 & Yes \\  & 0.5 & 421.49 & 136.71 & 149.48 & 496.81 & 94.77 & 124.03 & Yes \\  & 0.75 & 373 & 150.51 & & & 82.45 & 373 & Yes \\  & 1.0 & 294.83 & 158.19 & & & 60.05 & 294.83 & Yes \\ \hline \hline \end{tabular}
\end{table} 
Table 13: The calculated elastic constants for Ti\({}_{\{1-x\}}\)M\({}_{x}\)N intermetallics.