## 4 Damage Constitutive Model and Damage Evolution Law
### Evolution Law of Loading Damage of Single-Fissured Quasi-Sandstone after Freezing-Thawing  
In the triaxial compression test, repeatability tests of three samples were conducted under each condition. To reduce the test error, first, the suspicious data or individual abnormal values are abandoned, and then the average value of each mechanical parameter is calculated. The final summary is shown in Table 3. According to Table 3 and Equation (13), the theoretical curves of total damage evolution of quasi-sandstone with four kinds of single-fissured angles subjected to static loading after freezing-thawing can be calculated, as shown in Figure 6. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Fissure Angle @\({}^{\circ}\)**} & **Freezing** & \multirow{2}{*}{**Peak Axial Strain \(\varepsilon_{1}\)/\(\%\)**} & \multirow{2}{*}{**Peak Axial Stress \(\sigma_{1}\)/\(\%\)**} & **Elastic Modulus** & \multirow{2}{*}{**Poisson’s Ratio \(\mu\)**} \\  & **Temperature \(T\)/\(e^{\circ}\)C** & & & & \(E\)/\(GP\)** & \\ \hline \multirow{4}{*}{Intact} & Unfrozen & 1.35 & 14.30 & 2.67 & 0.220 \\  & \(-10\) & 1.98 & 13.21 & 2.55 & 0.234 \\  & \(-20\) & 1.23 & 11.67 & 2.34 & 0.259 \\  & \(-30\) & 1.02 & 10.15 & 1.99 & 0.272 \\ \hline \multirow{4}{*}{0} & Unfrozen & 0.71 & 13.17 & 2.33 & 0.247 \\  & \(-10\) & 1.31 & 12.63 & 2.16 & 0.289 \\  & \(-20\) & 0.87 & 11.49 & 1.91 & 0.302 \\  & \(-30\) & 0.88 & 10.36 & 1.78 & 0.327 \\ \hline \multirow{4}{*}{45} & Unfrozen & 1.06 & 11.34 & 2.27 & 0.319 \\  & \(-10\) & 1.95 & 10.31 & 2.08 & 0.317 \\  & \(-20\) & 1.08 & 9.49 & 1.72 & 0.338 \\  & \(-30\) & 0.98 & 8.48 & 1.53 & 0.367 \\ \hline \multirow{4}{*}{75} & Unfrozen & 1.01 & 13.19 & 2.50 & 0.260 \\  & \(-10\) & 0.92 & 12.62 & 2.21 & 0.309 \\ \cline{1-1}  & \(-20\) & 1.03 & 11.72 & 1.80 & 0.319 \\ \cline{1-1}  & \(-30\) & 0.86 & 10.68 & 1.67 & 0.334 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Main mechanical parameters of single-fissured quasi-sandstone after thawing. Figure 6 shows that the initial damage of quasi-sandstone samples gradually increases with the decrease in freezing temperature. For the freeze-thaw samples with \(T_{f}=-20\)\({}^{\circ}\)C and \(T_{f}=-30\)\({}^{\circ}\)C, with the gradual increase in axial strain, the cumulative damage \(D_{g}\) increases slowly first, then rises sharply, then increases slowly, and finally reaches 1 when the quasi-sandstone sample is unstable and fails. For the unfrozen sample and that with \(T_{f}=-10\)\({}^{\circ}\)C, the cumulative damage \(D_{g}\) increases sharply in the early stage, but the subsequent growth slows, especially for the samples with \(T_{f}=-10\)\({}^{\circ}\)C.  
Similarly, the total damage evolution curves of single-fissured quasi-sandstone subjected to static loading after freezing-thawing at four temperatures can be obtained (Figure 7). In terms of the initial damage, the order from small to large is as follows: intact quasi-sandstone \(<\theta=75^{\circ}<\theta=0^{\circ}<\theta=45^{\circ}\). However, according to the order in which the cumulative damage first reaches 1 is \(\theta=0^{\circ}\), \(75^{\circ}\), \(45^{\circ}\), and intact quasi-sandstone. In terms of the shape of the curve, the intact quasi-sandstone is similar to \(\theta=45^{\circ}\), and \(\theta=0^{\circ}\) is more similar to \(\theta=75^{\circ}\). According to a study by [PERSON] et al. [46], the S-shaped damage evolution curve corresponds to each stage of triaxial compression of quasi-sandstone. (i) The stage of fissure compaction and elastic deformation corresponds to the quasi-horizontal section at the beginning of the damage-strain curve under loading, and the damage variable is almost unchanged. (ii) In the plastic deformation stage, the damage variable increases rapidly, corresponding to the concave section of the damage-strain curve. (iii) In the strain-softening stage, corresponding to the convex section of the damage-strain curve, the slope of the curve gradually decreases. (iv) In the residual deformation stage, the quasi-sandstone damage variable remains 1.