# Experimental Study on the Mechanical Properties of Deep Granite under Gradient-Confining Pressure
## 3 Results and Discussion
### Stress-Strain Curves and Characteristic Stresses
In practical engineering, it is necessary to conduct a detailed assessment and analysis of the post rock fracture stage to take appropriate measures to ensure the safety and stability of the project, such as using reinforcement measures or conducting monitoring. Figure 7 represents the stress-strain relationship curve of deep granite under a confining pressure gradient, which shows the interrelationship between axial strain, radial strain, and volumetric strain. The volumetric strain of the rock can be considered to be composed of the crack volumetric strain and the elastic volumetric strain. The indirect method is ordinarily employed to compute the crack volumetric strain, which is derived by deducting the elastic volumetric strain from the total volumetric strain of the rock. Under the conventional triaxial stress state where \(\sigma_{2}=\sigma_{3}\), the volumetric strain of the rock mass can be expressed via Equation (1):  
\[\varepsilon_{V}=\varepsilon_{1}+2\varepsilon_{3} \tag{1}\]  
In the equation, \(\varepsilon_{V}\) is the volumetric strain, \(\varepsilon_{1}\) is the axial strain, and \(\varepsilon_{3}\) is the radial strain. According to the generalized Hooke's law, combined with the conventional triaxial stress state, the elastic strain of the rock is [28]:  
\[ \begin{cases}\varepsilon_{1}^{\epsilon}=\frac{1}{E}(\sigma_{1}-2\mu\sigma_{3} )\\ \varepsilon_{3}^{\epsilon}=\frac{1}{E}[\sigma_{3}-\mu(\sigma_{1}+\sigma_{3})] \end{cases}  \tag{2}\]  
In the equation, \(\sigma_{1}\) is the maximum principal stress, \(\sigma_{3}\) is the minimum principal stress, \(\varepsilon_{1}^{\epsilon}\) is the axial elastic strain of the rock, \(\varepsilon_{3}^{\epsilon}\) is the radial elastic strain of the rock, \(\mu\) is Poisson's ratio, and E is the elastic modulus. By combining Equations (1) and (2), the elastic volumetric strain of the rock can be derived as:  
\[\varepsilon_{V}^{\epsilon}=\frac{1-2\mu}{E}(\sigma_{1}+2\sigma_{3}) \tag{3}\]  
where \(\varepsilon_{V}^{\epsilon}\) is the elastic volumetric strain of the rock. Then, the volumetric strain of the crack can be expressed as:  
\[\varepsilon_{V}^{\zeta}=\varepsilon_{V}-\frac{1-2\mu}{E}(\sigma_{1}+2\sigma_{ 3}) \tag{4}\]  
Substituting the mechanical properties in Table 1 into Equation (4) enables f the crack volumetric strain of the rock under different stress states at each confining pressure to be derived. Via the stress-strain curve, four characteristic stress values of the rock can be acquired: the closure stress \(\sigma_{\rm cc}\), the initiation stress \(\sigma_{\rm ci}\), the dilatancy stress \(\sigma_{\rm cd}\), and the yield stress \(\sigma_{\rm f}\). The starting and ending points of the horizontal section of the crack volumetric strain-axial strain curve correspond to \(\sigma_{\rm cc}\) and \(\sigma_{\rm ci}\), respectively, the turning point of the rock volumetric strain pertains to \(\sigma_{\rm cd}\), and the point where the rock volumetric strain is zero, relates to \(\sigma_{\rm f}\). Figure 7: _Cont._  
Figure 8, and an exponential function is utilized to fit the relationship between the confining pressure and the characteristic stress to obtain a fitting function such as Equation (5). \[\left\{\begin{aligned} \sigma_{cc}=20.514\lx@arcdegree 0.0417\sigma_{3}, \lx@par R^{2}=0.6077\\ \sigma_{ci}=50.232\lx@par\lx@par\lx@par\lx@par\lx@par\lx@par \lx@par\lx@par\lx@par\lx@par\lx@par\lx@par\lx@par\lx@par\lx@par\@par \lx@par\lx@par\lx@par\lx@par\lx@par\lx@par\lx@par\lx@par\@par\lx@par \lx@par\lx@par\lx@par\lx@par\lx@par\lx@par\@par\lx@par\lx@par\@par\lx@par \lx@par\lx@par\@par\lx@par\lx@par\@par\lx@par\@par\lx@par\@par\lx@par\@par \lx@par\@par\lx@par  
function. With increasing confining pressure, the values of each characteristic stress also increase accordingly, and the rate of increase becomes increasingly faster. Nevertheless, dispersion still occurs, which is attributed to the heterogeneity of the deep granite.