## 4 Damage Characterization of Sandstone Under Different Soaking Times and Temperatures
### Sandstone Damage Characterization  
Under external forces, damage occurs within the rock, and the development and propagation of cracks consume a significant amount of energy. Therefore, the degree of energy dissipation can be used to quantitatively reflect the damage state of the rock. [PERSON] et al. [28] defined the damage variable of rock using normalized dissipated energy. \[D=\frac{U_{d}}{U_{dm}} \tag{6}\]  
Figure 9: Plot of energy and rate of change in sandstone for different temperatures and immersion times. (**a**) Plot of energy changes in sandstone at different temperatures and immersion times; (**b**) plot of the rate of change in energy in sandstone at different temperatures and immersion times. where \(U_{d}\) is the cumulative dissipated energy at any stress state, \(U_{dm}\) is the cumulative dissipated energy for rock, and the units of each of the above quantities are \(\text{MJ}\cdot\text{m}^{-3}\). D = 1 means that the rock is completely destroyed, and D = 0 means that the rock is intact and undamaged. However, the sandstone retains its load-bearing capacity in the post-peak stage. To reflect the existence of the rock's residual strength, a correction factor is introduced. By modifying the relationship between dissipated energy and the damage variable, the damage variable D does not completely reach 1 in the post-peak stage, thereby providing a more precise representation of the damage evolution characteristics of sandstone. Consequently, modifications are made to Equation (6) [29]  
\[D=\left(1-\frac{\sigma_{p}}{\sigma_{r}}\right)\frac{U_{d}}{U_{dm}} \tag{7}\]  
where \(\sigma_{r}\) is the peak strength, \(\sigma_{p}\) is the residual value stress; the unit of each of the above quantities are MPa. The \(\sigma_{p},\sigma_{r}\), \(U_{dm}\) of sandstone at different temperatures and immersion times are shown in Table 2.  
By substituting the values from Table 2 into Equation (6), the damage evolution of sandstone under varying temperatures and immersion times can be determined, as illustrated in Figure 10.  
From Figure 10, it is evident that the damage variable accurately represents the stress-strain behavior of the sandstone. During the compaction stage, under external force, the initial cracks inside the rock close, and the dissipated energy shows little change, with the damage variable remaining nearly almost constant. In the elastic stage, a significant portion of the energy input into the rock is stored as elastic energy, while a small fraction is allocated for crack development. During this stage, the damage variable increases slightly, but the increase is minimal. In the plastic stage, the rock begins to undergo plastic deformation, and the initiation and development of new cracks require significant energy, leading to a gradual increase in the damage variable with fluctuating changes, which indicates that the damage to the rock is intensifying. In the post-peak stage, when the stress exceeds the peak strength of the rock, the internal cracks continue to develop, resulting in macroscopic damage, with through cracks forming, a large release of elastic energy, and a sharp increase in dissipated energy. The damage variable shows a sudden increase but does not reach 1, indicating that the rock has not been completely destroyed after the peak and still retains some bearing capacity. \begin{table}
\begin{tabular}{c c c c c} \hline
**Temperature/\({}^{\circ}\)C** & **Soaking Time/h** & \(\sigma_{p}\)/MPa & \(\sigma_{r}\)/MPa & \(\text{U}_{dm}\)/(MJ\cdot\text{m}^{-3}\))** \\ \hline
25 \({}^{\circ}\)C & 0.5 & 12.06 & 63.45 & 0.06634 \\
55 \({}^{\circ}\)C & 0.5 & 13.34 &  
Figure 10: _Cont._  
From Figure 10, it can be observed that under 0.5 h soaking, the maximum damage variable values at each temperature are 0.81, 0.72, 0.76, and 0.78, respectively. Under 1 h soaking, the maximum damage variable values at each temperature are 0.63, 0.52, 0.55, and 0.54, respectively. Under 3 h soaking, the maximum damage variable values at each temperature are 0.56, 0.33, 0.47, and 0.48.