# Experimental Study on Damage Constitutive Model of Rock under Thermo-Confining Pressure Coupling
## 3 Results and Discussion
### Model Analysis of Elastic Modulus of Rock
#### 3.3.2 Modeling the Elastic Modulus of Rock under the Influence of Confining Pressure  
Through the mechanical properties of the rock samples, it can be observed that the tangent elastic modulus of the four types of rocks studied in this paper exhibits nonlinear growth with increasing confining pressure. Researchers have proposed many regression formulas or empirical formulas based on experimental results and most models use quadratic curves for fitting [6], which can efficiently fit the data within a limited range. However, these models fail to accurately describe the part where the growth rate of the elastic modulus decreases with increasing confining pressure. The experiments indicate that the rate of increase in the tangent elastic modulus (\(E_{t}\)) gradually slows down as the confining pressure increases, suggesting the existence of a limiting value for \(E_{t}\). Based on the micromechanical  
Figure 12: Comparison of model fitting goodness for four rock types under temperature influence. damage theory, we use a composite exponential formula to represent the relationship between \(E_{t}\) and \(\sigma_{3}\). The formula is expressed as follows:  
\[E(\sigma_{3})=E_{0}[1-(\frac{A}{E_{0}}-1)(1-e^{(-\frac{\sigma_{3}}{B})})] \tag{4}\]  
where \(E_{0}\) represents the elastic modulus of the rock under uniaxial compression; \(A\) and \(B\) are the parameters to be determined. The fitting results under different confining pressures are shown in Figure 13.  
By integrating the data from the four rock types, the overall R\({}^{2}\) value exceeded 0.90, indicating a good fit. This suggests that the model effectively describes the variation of the tangent elastic modulus of rocks under the influence of confining pressure. In particular, it accurately captures the phenomenon where the rate of change in the tangent elastic modulus decreases as the confining pressure reaches a certain value. This behavior is primarily attributed to the closure of internal cracks within the rock mass, leading to an increase in elastic modulus, which ultimately approaches its limit. The composite exponential model successfully reflects this trend. The fitting results for calcareous sandstone at 60 \({}^{\circ}\)C were less satisfactory, likely due to the higher porosity of calcareous sandstone compared to the other rock types, as discussed in Section 3.2.2. This higher porosity causes abrupt changes in the tangent elastic modulus within the 10 MPa to 20 MPa range, thereby reducing the effectiveness of the model fitting. Modeling the Elastic Modulus of Rock under the Influence of Temperature and Confining Pressure  
The thermo-mechanical coupling effect is not a simple additive relationship between temperature and confining pressure but rather a complex interaction. Therefore, drawing on the fitting models from Sections 3.3.1 and 3.3.2, both temperature and confining pressure factors are jointly considered. By referencing the development trend of the elastic modulus of rocks under thermo-mechanical coupling, the following formula is used for nonlinear fitting:  
\[E(T,\sigma_{3})=AT^{3}+BT^{2}+CT+E_{0}[1-(\frac{D}{E_{0}}-1)(1-e^{(-\frac{ \sigma_{3}}{E})})] \tag{5}\]  
where \(E_{0}\) represents the elastic modulus of the rock under uniaxial compression at room temperature (with room temperature taken as \(T=20\)\({}^{\circ}\)C), and \(A\), \(B\), \(C\), \(D\), \(E\) are the fitting parameters to be determined. Table 2 demonstrates the results of fitting the tangent elastic modulus under temperature-confining pressure coupling. Figure 14 demonstrates the fitting results of the tangent elastic  
Figure 13: Comparison of model fitting goodness for four rock types under confining pressure influence. modulus in relation to temperature and confining pressure for the four types of rocks. The results indicate that the nonlinear fitting model effectively describes the development trend of the tangent elastic modulus under thermo-mechanical coupling for all four rock types.