## 5 Model Validation and Variation in Damage Variable  
### Model Validation  
To validate the damage constitutive model of argillaceous siltstone under cyclic hydrothermal conditions proposed in this study, experimental data were substituted into Equations (22) and (23). The void strain coefficient a was fitted and solved, resulting in all the parameters of the constitutive equation, as shown in Table 2. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Water Temperature** & \(\varepsilon_{\rm lg}\) **(\%)** & \(\varepsilon_{\rm Id}\) **(\%)** & \(a\) & \(m\) & \(\lambda\) &
\begin{tabular}{c} **Soaking Duration** \\ **(d)** \\ \end{tabular} \\ \hline
5 & 0.0600 & 0.48 & 8.26 & 3.24 & 0.26 & \\
15 & 0.0460 & 0.67 & 7.12 & 2.79 & 0.27 & \\
25 & 0.0370 & 0.76 & 9.72 & 2.95 & 0.25 & \\
35 & 0.0018 & 0.76 & 6.34 & 3.32 & 0.23 & \\
5 & 0.1200 & 0.80 & 5.83 & 2.72 & 0.29 & \\
15 & 0.0035 & 0.48 & 7.29 & 2.34 & 0.31 & \\
25 & 0.0400 & 0.61 & 5.34 & 2.52 & 0.33 & \\
35 & 0.0027 & 0.67 & 6.56 & 3.03 & 0.28 & \\
5 & 0.0110 & 0.50 & 5.98 & 2.36 & 0.34 & \\
15 & 0.2600 & 0.84 & 2.03 & 2.04 & 0.44 & \\
25 & 0.1100 & 0.62 & 3.16 & 1.84 & 0.41 & 7 \\
35 & 0.0180 & 0.55 & 6.57 & 2.48 & 0.31 & \\
5 & 0.0430 & 0.68 & 10.53 & 2.02 & 0.36 & \\
15 & 0.1500 & 0.72 & 6.99 & 1.44 & 0.52 & \\
25 & 0.1300 & 0.81 & 4.65 & 1.64 & 0.48 & \\
35 & 0.0047 & 0.34 & 10.31 & 2.23 & 0.40 & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Constitutive model parameters. By substituting the constitutive model parameters obtained into Equations (19) and (20), the theoretical stress-strain curve for argillaceous siltstone under different water temperature effects can be derived, as shown in Figure 9. It can be observed that the theoretical curves of the constitutive model align closely with the experimental curves, accurately reflecting the deformation characteristics of argillaceous siltstone. Evolution Law of Parameters \(m\) and \(\lambda\) in Damage Variable Under the Action of Different Water Temperatures  
The parameter \(\lambda\) lambda represents the scale parameter in the Weibull distribution function, which determines the scale or range of the distribution. In a physical sense, it can, to some extent, reflect the plastic characteristics of the rock. The shape parameter mm indicates the degree of material homogeneity, where a higher \(m\) value suggests greater homogeneity in the mechanical properties of the rock, while a lower \(m\) value indicates lesser homogeneity [30]. As shown in Figure 10, When the water temperature is between 5 \({}^{\circ}\)C and 15 \({}^{\circ}\)C, the shape parameter \(m\) decreases with rising water temperature, leading to a decrease in strength. Conversely, between 15 \({}^{\circ}\)C and 35 \({}^{\circ}\)C, the shape parameter \(m\) increases with rising water temperature, resulting in an increase in strength. The trend of the scale parameter \(\lambda\) with increasing water temperature is opposite to that of the shape parameter \(m\). When the water temperature is near room temperature (15-25 \({}^{\circ}\)C), the scale parameter \(\lambda\) increases, indicating that the plastic characteristics of argillaceous siltstone  
Figure 9: Comparison of experimental and theoretical curves. Soaked for (**a**) 1 day, (**b**) 3 days, (**c**) 7 days, and (**d**) 14 days. become more pronounced. Between 5 \({}^{\circ}\)C and 15 \({}^{\circ}\)C, as the water temperature rises, the scale parameter \(\lambda\) increases. However, between 15 \({}^{\circ}\)C and 35 \({}^{\circ}\)C, as the water temperature rises, the scale parameter \(\lambda\) decreases, indicating a reduction in the plastic characteristics of argillaceous siltstone.