## 3 Methodology
### Mathematical Model and Variance Analysis
\begin{table}
\begin{tabular}{c c c c c c c} \hline  & **Sum of Squares** & **Degrees of Freedom** & **Mean Variance** & **F-Value** & _p-Value_ & **Significance** \\ \hline \(E(T,R,V)\) & \(1.94\times 10^{5}\) & 19 & 10,197.12 & 19,467.81 & \textless{}0.0001 & Significant \\ \hline \(T\) & 16,112.46 & 1 & 16,112.46 & 30,761.08 & \textless{}0.0001 & Significant \\ \hline \(R\) & 35.92 & 1 & 35.92 & 68.58 & \textless{}0.0001 & Significant \\ \hline \(V\) & 265.85 & 1 & 265.85 & 507.54 & \textless{}0.0001 & Significant \\ \hline \(TR\) & 132.36 & 1 & 132.36 & 252.69 & \textless{}0.0001 & Significant \\ \hline \(TV\) & 914.05 & 1 & 914.05 & 1745.06 & \textless{}0.0001 & Significant \\ \hline \(RV\) & 3.66 & 1 & 3.66 & 6.99 & 0.0119 & \\ \hline \(T^{2}\) & 258.85 & 1 & 258.85 & 494.18 & \textless{}0.0001 & Significant \\ \hline \(R^{2}\) & 1.75 & 1 & 1.75 & 3.34 & 0.0755 & \\ \hline \(V^{2}\) & 21.88 & 1 & 21.88 & 41.77 & \textless{}0.0001 & Significant \\ \hline \(TRV\) & 0.2016 & 1 & 0.2016 & 0.3848 & 0.5388 & \\ \hline \(T^{2}R\) & 30.15 & 1 & 30.15 & 57.56 & \textless{}0.0001 & Significant \\ \hline \(T^{2}V\) & 87.22 & 1 & 87.22 & 166.51 & \textless{}0.0001 & Significant \\ \hline \(TR^{2}\) & 0.013 & 1 & 0.013 & 0.0248 & 0.8757 & \\ \hline \(TV^{2}\) & 27.9 & 1 & 27.9 & 53.27 & \textless{}0.0001 & Significant \\ \hline \(R^{2}V\) & 0.2625 & 1 & 0.2625 & 0.5012 & 0.4834 & \\ \hline \(RV^{2}\) & 1.11 & 1 & 1.11 & 2.12 & 0.1538 & \\ \hline \(T^{3}\) & 595.62 & 1 & 595.62 & 1137.13 & \textless{}0.0001 & Significant \\ \hline \(R^{3}\) & 0.9142 & 1 & 0.9142 & 1.75 & 0.1946 & \\ \hline \(V^{3}\) & 0.2686 & 1 & 0.2686 & 0.5128 & 0.4784 & \\ \hline Residual & 19.38 & 37 & 0.5238 & & & \\ \hline Total Variation & \(1.938\times 10^{5}\) & 56 & & & \\ Value & & & & & \\ \hline \end{tabular}
\end{table} 
Table 4: [PERSON]’s modulus of compressive strength regression model. \begin{table}
\begin{tabular}{c c c c c c} \hline  & **Sum of Squares** & **Degrees of Freedom** & **Mean Variance** & **F-Value** & _p-Value_ & **Significance** \\ \hline \(Q(T,R,V)\) & 79.88 & 19 & 4.2 & 372.32 & \textless{}0.0001 & Significant \\ \hline \(T\) & 3.32 & 1 & 3.32 & 293.64 & \textless{}0.0001 & Significant \\ \hline \(R\) & 0.0108 & 1 & 0.0108 & 0.9526 & 0.3354 & \\ \hline \(V\) & 0.0037 & 1 & 0.0037 & 0.327 & 0.5709 & \\ \hline \(TR\) & 0.0111 & 1 & 0.0111 & 0.9809 & 0.3284 & \\ \hline \(TV\) & 0.8068 & 1 & 0.8068 & 71.45 & \textless{}0.0001 & Significant \\ \hline \(RV\) & 0.0625 & 1 & 0.0625 & 5.53 & 0.0241 & Significant \\ \hline \(T^{2}\) & 3.51 & 1 & 3.51 & 310.68 & \textless{}0.0001 & Significant \\ \hline \(R^{2}\) & 0.0108 & 1 & 0.0108 & 0.9544 & 0.335 & \\ \hline \(V^{2}\) & 0.0135 & 1 & 0.0135 & 1.2 & 0.2807 & \\ \hline \(TRV\) & 0.1094 & 1 & 0.1094 & 9.68 & 0.0036 & Significant \\ \hline \(T^{2}R\) & 0.259 & 1 & 0.259 & 22.94 & \textless{}0.0001 & Significant \\ \hline \end{tabular}
\end{table} 
Table 5: Yield limit of compressive strength regression model. Equations (7) and (8) suggest that, when \(R\) and \(T\) are constant, as \(V\) increases, Young's modulus gradually increases due to an increase in the number of dislocations caused by vacancy inside the grains. However, the yield strength shows a decreasing trend as the defects increase. When \(V\) and \(T\) are constant, as \(R\) increases, the stiffness of low-carbon steel gradually increases, and the stage of yield strength enhancement in comparison to Figure 3 becomes shorter, while the toughness decreases. When \(R\) and \(V\) are constant, as the temperature increases, the atomic activity intensifies, and the tendency of atoms to move away from each other becomes more apparent, resulting in a decrease in both the Young's modulus and yield strength of low-carbon steel. In order to verify the reliability of the equation and its statistical significance, a variance and confidence interval analysis can be performed.