# Experimental Study on Damage Constitutive Model of Rock under Thermo-Confining Pressure Coupling
## 3 Results and Discussion
### Analysis of Mechanical Properties of Rocks
#### 3.2.2 Analysis of Elastic Modulus for Different Rocks  
Granite exhibits distinct stages of pore crack compaction and elastic deformation during loading. The transition from a concave upward shape to an approximately linear shape in the stress-strain curve serves as the boundary, allowing a separate discussion of the deformation characteristics of these two stages. The ratio of stress to strain at this boundary point is used as the deformation modulus \(E\) of the pore crack compaction stage, determined by Equation (1). The tangent elastic modulus \(E_{t}\) of the elastic deformation stage is represented by the tangent slope of the linear segment of the stress-strain curve, determined by Equation (2). The determination of \(E\) and \(E_{t}\) is shown in Figure 9.  
\[E=\frac{\sigma_{01}}{\varepsilon_{01}} \tag{1}\]  
\[E_{t}=\frac{\sigma_{02}-\sigma_{01}}{\varepsilon_{02}-\varepsilon_{01}} \tag{2}\]  
Figure 8: Stress–strain curves of shale under different temperature and confining pressure conditions: (**a**) shale \(\sigma_{3}\) = 0 MPa; (**b**) shale \(\sigma_{3}\) = 25 MPa; (**c**) shale \(T\) = 20 \({}^{\circ}\)C; (**d**) shale \(T\) = 120 \({}^{\circ}\)C.  
Figure 10a indicates that, as the temperature increases, the deformation modulus \(E\) increases. In the temperature range of 40 \({}^{\circ}\)C to 80 \({}^{\circ}\)C, the slopes of the curves remain relatively low, particularly under lower stress conditions (e.g., 0 MPa and 5 MPa). This suggests that, at lower temperatures, the impact of temperature on the elastic modulus is minimal, and the stiffness of the rock does not increase significantly. Conversely, in the temperature range of 100 \({}^{\circ}\)C to 120 \({}^{\circ}\)C, the slopes of the curves increase markedly, especially under higher stress conditions (e.g., 25 MPa). During this phase, the elastic modulus exhibits a rapid upward trend as temperature rises. This behavior is primarily attributed to the enhanced interactions between mineral particles, the repair or rearrangement of crystal lattice defects, and the resulting increase in rock density and hardness under high-temperature and high-pressure conditions, leading to a significant increase in the elastic modulus. At confining pressures of 0 MPa, 5 MPa, 10 MPa, 15 MPa, 20 MPa, and 25 MPa, \(E\) increases by 38.17%, 59.31%, 42.46%, 41.53%, 33.73%, and 34.76%, respectively, and the increment in rock deformation modulus initially rises and then levels off. At lower confining pressures, the increase in pressure compresses and closes microcracks and pores, enhancing the overall stiffness and load-bearing capacity of the rock sample. However, as confining pressure further increases, the effect of microcrack closure saturates, and the increase in deformation modulus stabilizes. Figure 10b illustrates that the tangent elastic modulus (\(E_{t}\)) increases as the temperature rises. At confining pressures of 0 MPa, 5 MPa, 10 MPa, 15 MPa, 20 MPa, and 25 MPa, \(E_{t}\) increases by 39.09%, 40.17%, 28.82%, 21.17%, 26.92%, and 25.06%, respectively. The significant rise in \(E_{t}\) when the confining pressure increases from 0 MPa to 5 MPa is attributed to the compaction of microcracks and pores during the crack closure phase. As the confining pressure reaches 15 MPa, the rate of increase in \(E_{t}\) declines due to the formation of new microcracks or the expansion of existing ones. However, when the confining pressure increases to 25 MPa, the rate of increase in \(E_{t}\) stabilizes, indicating that new cracks are rapidly compressed under high confining pressure, with the rock potentially nearing its density limit. Figure 9: Schematic diagram for determining the \(E\) and the \(E_{t}\). Figure 10c shows a gradual increase in the deformation modulus \(E\) as the confining pressure rises from 0 MPa to 25 MPa. The confining pressure causes internal pores and microcracks to gradually close, enhancing the load-bearing capacity of the rock sample and leading to an increase in deformation modulus.