# 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
At temperatures of 20 \({}^{\circ}\)C, 40 \({}^{\circ}\)C, 60 \({}^{\circ}\)C, 80 \({}^{\circ}\)C, and 120 \({}^{\circ}\)C, \(E\) increases by 76.51%, 78.03%, 74.74%, 71.47%, and 72.15%, respectively. When the temperature increases from 20 \({}^{\circ}\)C to 40 \({}^{\circ}\)C, \(E\) slightly increases due to the thermal expansion effect, where the crack closure effect outweighs the expansion effect. As the temperature rises to 80 \({}^{\circ}\)C, the rate of increase in \(E\) slightly declines due to the predominant crack expansion effect caused by continuous heating. When the temperature reaches 120 \({}^{\circ}\)C, the increase in \(E\) stabilizes as the thermal expansion and crack closure effects balance out. Figure 10d demonstrates a gradual rise in \(E_{t}\) as the confining pressure increases from 0 MPa to 25 MPa. At temperatures of 20 \({}^{\circ}\)C, 40 \({}^{\circ}\)C, 60 \({}^{\circ}\)C, 80 \({}^{\circ}\)C, and 120 \({}^{\circ}\)C, the increases of \(E_{t}\) are 44.18%, 35.33%, 33.15%, 35.18%, and 29.64%, respectively. As the temperature rises from 20 \({}^{\circ}\)C to 40 \({}^{\circ}\)C, the increase in \(E_{t}\) slightly declines, possibly due to most cracks and pores being compressed and closed during the crack closure phase while, in the elastic  
Figure 10: Relationship diagrams of granite deformation modulus, tangent elastic modulus with temperature, and confining pressure: (**a**) relationship between granite deformation modulus and temperature; (**b**) relationship between granite tangent elastic modulus and temperature; (**c**) relationship between granite deformation modulus and confining pressures; (**d**) relationship between granite tangent elastic modulus and confining pressures. deformation phase, new microcracks induced by temperature rise affect structural integrity and reduce the rate of increase in elastic modulus. When the temperature rises to 80 \({}^{\circ}\)C, the increase in \(E_{t}\) stabilizes due to the balanced effect of thermal expansion on cracks. At 120 \({}^{\circ}\)C, high temperature causes the expansion of cracks and pores, influencing the trend of \(E_{t}\) changes and potentially altering mineral structures, leading to a decrease in the rate of increase in \(E_{t}\). Obviously, confining pressure has a more significant impact on the deformation modulus during the pore crack compaction phase and on the tangent elastic modulus during the elastic phase of granite. The increase in deformation modulus with rising confining pressure is greater than that caused by temperature rise, indicating that confining pressure plays a major role in the changes in deformation modulus and elastic modulus. The stress-strain curves of red sandstone, gray sandstone, and shale exhibit a single slope, indicating that the elastic modulus remains constant. The nonlinear characteristics of the crack closure compaction stage are not evident and, after loading, the stress-strain curves quickly transition into the elastic deformation stage, showing linear variation characteristics. Therefore, the tangent elastic modulus \(E_{t}\) is used to represent the elastic modulus for the entire deformation stage. This value is defined as the slope of the linear segment in the stress-axial strain relationship. Figure 11 illustrates the relationship between tangent elastic modulus and temperature, confining pressure for different rocks. From Figure 11a,b, it can be observed that the tangent elastic modulus \(E_{t}\) of red sandstone exhibits significant trends with changes in temperature and confining pressure. As the temperature rises, \(E_{t}\) first increases and then decreases. Between 20 \({}^{\circ}\)C and 40 \({}^{\circ}\)C, the increase in temperature causes thermal expansion of minerals, which closes cracks and increases the sample's density, leading to an increase in \(E_{t}\).