# 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
The peak stress \(\sigma_{p}\) and peak strain \(\epsilon_{p}\) satisfy the following two geometric conditions:  
\[\varepsilon=\epsilon_{p},\ \ \sigma_{1}=\sigma_{p} \tag{28}\]  
\[\varepsilon=\epsilon_{p},\ \ \frac{\text{d}\sigma_{1}}{\text{d}\epsilon_{1}}=0 \tag{29}\]  
By substituting Equations (28) and (29) into Equation (27), two relationships regarding \(m\) and \(F_{0}\) can be obtained, as shown below:  
\[m=\frac{1}{\ln\frac{E_{T}}{E_{0}}[\frac{E_{T}\epsilon_{p}+(1-2v)\sigma_{3}}{ \sigma_{p}+(1-2v)\sigma_{3}}]} \tag{30}\]  
\[F_{0}=\frac{[E_{T}\epsilon_{p}+(1-2v)\sigma_{3}][(\sigma_{1}+3\sigma_{3})\sin \varphi+\sigma_{p}\sqrt{3+\sin^{2}\varphi}]}{\sqrt{9+3\sin^{2}\varphi}[\sigma _{1}+(1-2v)\sigma_{3}]}m\frac{1}{\pi} \tag{31}\]  
By adjusting the parameters \(m\) and \(F_{0}\) based on experimental data and substituting them into Equation (27), a damage constitutive model can be constructed that accurately reflects the stress-strain state of rock under thermo-mechanical coupling. Experimental Validation of the Damage Constitutive Model of Rock under Thermo-Confining Pressure Coupling  
To validate the rationality and effectiveness of the method proposed in this paper, we evaluated the statistical damage Drucker-Prager constitutive model for granite, red sandstone, gray sandstone, and shale under conditions of 20 \({}^{\circ}\)C and 0 MPa, as well as 120 \({}^{\circ}\)C and 25 MPa. Based on the experimental results, we extracted mechanical parameters such as peak stress, peak strain, confining pressure, elastic modulus, and Poisson's ratio. These parameters were substituted into Equations (30) and (31) to calculate the Weibulldistribution parameters \(m\) and \(F_{0}\) for the four rock types under the conditions of 20 \({}^{\circ}\)C, 0 MPa, and 120 \({}^{\circ}\)C, 25 MPa. The specific results are shown in Table 3.  
By substituting the calculated Weibull distribution parameters \(m\) and \(F_{0}\) along with the values of the elastic modulus, internal friction angle, and other parameters into Equation (27), the statistical damage Drucker-Prager constitutive model for various types of rocks under the conditions of 20 \({}^{\circ}\)C, 0 MPa and 120 \({}^{\circ}\)C, 25 MPa can be obtained. The general form of the Drucker-Prager constitutive model under these conditions can be expressed in Table 4.  
Based on the damage constitutive models in Table 4, the theoretical stress-strain curves for granite, red sandstone, gray sandstone, and shale under conditions of 20 \({}^{\circ}\)C, 0 MPa and 120 \({}^{\circ}\)C, 25 MPa are compared with the experimental stress-strain curves obtained from tests. Figure 15 compares the theoretical and experimental stress-strain curves, where it can be observed that the theoretical results obtained from the established constitutive model coincide well with the experimental results, indicating that the Drucker-Prager constitutive model for the four types of rocks under conditions of 20 \({}^{\circ}\)C, 0 MPa, and 120 \({}^{\circ}\)C, 25 MPa is reliable. By comparing the tangent elastic modulus \(E_{t^{\prime}}\) derived from the theoretical stress-strain curves with the tangent elastic modulus \(E_{t}\) obtained from the experimental stress-strain curves, the accuracy and applicability of the proposed models are further validated, as shown in Table 5.