## 4 Response Mechanisms of Load Damage During Freeze-Thaw Cycles
\[D=1-\frac{E_{i}}{E}\exp\biggl{[}-(\frac{\epsilon_{1}}{\lambda})^{k}\biggr{]} \tag{12}\]  
By substituting Equation (12) into Equation (3), Equation (13) is derived, which characterizes the comprehensive stress-strain curve for uniaxial loading of the rock, depicted in \[N=N_{i}+N_{s}+N_{m}+N_{0} \tag{4}\]0  
The theoretical values exhibit a strong correlation with the experimental values at the peak point, effectively capturing their stress-strain relationship. Nevertheless, a significant divergence is observed in the fitting during the initial loading phase and the elastic deformation phase. \[N=N_{i}+N_{s}+N_{m}+N_{0} \tag{4}\]1  
In Equation (12), the compaction phase of the rock loading process is excluded, and the initial freeze-thaw damage is treated as the total damage factor for this phase. By substituting Equation (12) into Equation (3) and conducting a segmented analysis, the stress-strain relationship is established as shown in Equation (14). The comprehensive representation of the total damage variable is elaborated in Equation (15), where \(\varepsilon_{p}\) and \(\sigma_{p}\) represent the axial strain and stress at the compaction point, respectively. The efficacy of the modified constitutive model is illustrated in \[N=N_{i}+N_{s}+N_{m}+N_{0} \tag{4}\]2  
\[\sigma=\left\{ \begin{array}{l}E_{i}\varepsilon_{1}\Bigl{\{}1-\exp\biggl{[}- (\frac{\varepsilon_{1}}{\lambda_{1}})^{k}\biggr{]}\biggr{\}},\epsilon_{1} \leq\varepsilon_{p}\\ E_{i}(\epsilon_{1}-\varepsilon_{p})\exp\biggl{[}-(\frac{\epsilon_{1}- \varepsilon_{p}}{\lambda_{2}})^{k_{2}}\biggr{]}+\sigma_{p},\epsilon_{1}> \varepsilon_{p}\end{array} \right. \tag{14}\]  
\[D=\left\{ \begin{array}{l}1-\frac{E_{i}}{L},\epsilon_{1}\leq\varepsilon_{p}\\ 1-\frac{E_{i}}{L}\exp\biggl{[}-(\frac{\epsilon_{1}-\varepsilon_{p}}{\lambda_{ 2}})^{k_{2}}\biggr{]},\epsilon_{1}>\varepsilon_{p}\end{array} \right. \tag{15}\]  
Figure 17: Validation of the uniaxial loading constitutive model (uncorrected). The parameter values are obtained through data fitting, as detailed in Table 4. Substituting these values into Equation (15) yields the evolution law for the total damage variable of sandstone, which is depicted in Figure 19.  
The constitutive model presented in Equation (13) was calibrated using experimental data to ascertain the characteristic and proportional parameters. The results of this calibration are illustrated in Figure 18. Additionally, Figure 19 indicates that the total damage variable prior to the compaction point is entirely attributable to initial freeze-thaw damage. As the number of freeze-thaw cycles increases, there is a discernible upward trend in the level of initial freeze-thaw damage, although the magnitude of this damage gradually diminishes. Concurrently, the axial strain associated with the compaction point exhibits a progressive increase. This phenomenon can be attributed to freeze-thaw erosion, which modifies the original skeletal structure of the sample, resulting in the expansion and interconnection of internal cracks. Consequently, the structural integrity is increasingly compromised, requiring greater axial deformation to reach a compacted state. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Number of Freeze–Thaw Cycles** & \(\mathbf{\lambda_{1}}\) & \(\mathbf{k_{1}}\) & \(\mathbf{R^{2}}\) & \(\mathbf{\lambda_{2}}\) & \(\mathbf{k_{2}}\) & \(\mathbf{R^{2}}\) \\ \hline
0 & 0.02 & 0.16 & 0.95 & 0.01 & 5.11 & 0.99 \\
3 & 0.02 & 0.71 & 0.99 & 0.01 & 9.59 & 0.95 \\
6 & 0.02 & 0.47 & 0.99 & 0.01 & 11.00 & 0.94 \\
9 & 198.2 & 0.09 & 0.98 & 0.01 & 10.03 & 0.93 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Parameters for the sandstone probability distribution function. Figure 19: Trend of total damage variable evolution during uniaxial loading. Figure 18: Verification of the uniaxial loading constitutive model (updated). The total damage variable starts to increase from the densification point, with a relatively slow growth rate in the early stage, until it experiences a sudden increase at a certain axial strain.