# Experimental Study on Damage Constitutive Model of Rock under Thermo-Confining Pressure Coupling
## 4 Damage Constitutive Model of Rock
### Rock Damage Theory Based on the Drucker-Prager Criterion
#### 4.1.3 Total Damage of Rock under Thermo-Confining Pressure Coupling
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Rock Type** & \(T^{C}\)**C** & \(\sigma_{3}\)**/MPa** & \(\sigma_{p}\)**/MPa** & \(\epsilon_{p}\)**/\(\psi_{0}\)** & \(\varphi l^{c}\) & \(E_{T}\)**/GPa** & \(\upsilon\) & \(m\) & \(F_{0}\) \\ \hline G & 20 & 0 & 119.02 & 0.31 & 45 & 44.93 & 0.15 & 6.56 & 44.42 \\  & 120 & 25 & 344.60 & 0.42 & 50 & 81.02 & 0.21 & 35.59 & 93.30 \\ \hline RS & 20 & 0 & 22.06 & 0.29 & 45 & 8.27 & 0.23 & 14.24 & 6.92 \\  & 120 & 25 & 17.04 & 0.13 & 53 & 11.93 & 0.22 & 1.00 & 32.90 \\ \hline GS & 20 & 0 & 150.22 & 0.45 & 45 & 48.16 & 0.07 & 2.80 & 68.49 \\  & 120 & 25 & 289.53 & 0.43 & 45 & 113.34 & 0.35 & 0.94 & 124.62 \\ \hline SH & 20 & 0 & 59.09 & 0.48 & 45 & 13.99 & 0.30 & 8.07 & 20.98 \\  & 120 & 25 & 102.63 & 0.63 & 45 & 27.43 & 0.28 & 1.24 & 80.31 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Mechanical parameters of rocks under different thermo-mechanical coupling conditions. Figure 15: _Cont._  
Figure 16 illustrates the comparison between \(E_{t^{\prime}}\) and \(E_{t}\). The data points primarily cluster around the 45\({}^{\circ}\) line, indicating a high degree of correlation between the theoretical and actual values. This suggests that the theoretical model accurately predicts the experimental outcomes. Data analysis reveals that the average ratio of the theoretical to experimental tangent elastic modulus is 0.95, with a standard deviation of 0.003. This indicates that the constitutive model exhibits high precision and reliability, effectively predicting the actual values of the tangent elastic modulus. The theoretical peak stress \(\sigma_{p}\) and peak strain \(\sigma^{/p}\) obtained from the stress-strain curves of different rock types are compared with the experimental peak stress \(\varepsilon_{p}\) and peak strain \(\varepsilon_{p^{\prime}}\) as shown in Table 6. Figure 17 shows the theoretical calculated values of peak stress and peak strain compared to the test values. Data analysis indicates that the average ratio of the theoretical peak strain to the experimental peak strain is 1.088, with a standard deviation of 0.0134. Similarly, the average ratio of the theoretical peak stress to the experimental peak stress is 1.03, with a standard deviation of 0.00083. These results demonstrate that the constitutive model exhibits high accuracy and reliability, effectively predicting the actual values of peak stress and peak strain. Figure 15: Comparison of theoretical and experimental stress–strain curves: (**a**) granite 20 \({}^{\circ}\)C, 0 MPa; (**b**) granite 120 \({}^{\circ}\)C, 25 MPa; (**c**) red sandstone 20 \({}^{\circ}\)C, 0 MPa; (**d**) red sandstone 120 \({}^{\circ}\)C, 25 MPa; (**e**) gray sandstone 20 \({}^{\circ}\)C, 0 MPa; (**f**) gray sandstone 120 \({}^{\circ}\)C, 25 MPa; (**g**) shale 20 \({}^{\circ}\)C, 0 MPa; (**h**) shale 120 \({}^{\circ}\)C, 25 MPa. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Rock Type** & **T/\({}^{c}\)C** & \(\sigma_{3}\)/MPa & \(E_{t}\)/GPa & \(E_{t}\)/GPa & **Specific Value** \\ \hline G & 20 & 0 & 44.93 & 43.92 & 0.98 \\  & 120 & 25 & 81.02 & 80.68 & 1.00 \\ \hline RS & 20 & 0 & 8.27 & 7.08 & 0.86 \\  & 120 & 25 & 20.81 & 20.71 & 0.99 \\ \hline GS & 20 & 0 & 48.16 & 47.58 & 0.99 \\  & 120 & 25 & 65.71 & 63.29 & 0.96 \\ \hline SH & 20 & 0 & 13.99 & 12.25 & 0.88 \\  & 120 & 25 & 27.43 & 26.71 & 0.98 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Theoretical and experimental tangent elastic modulus for different rock types. Figure 16: Comparison of theoretical and experimental tangent elastic modulus.