## 3 Results and Discussion
### Definition of Damage Variable
#### 4.1.2 Definition of the Damage Variable under Uniaxial Load  
The concept of micro-element strength is introduced to quantitatively characterize the degree of micro-damage in rock. Assuming that the micro-element strength of the rock follows a Weibull distribution [57], the damage variable \(D_{l}\) of granite under uniaxial compression is given by:  
\[D_{l}=\int_{0}^{F_{*}}P(F_{*})dF=1-\exp\left[-\left(\frac{F_{*}}{F_{0}}\right)^ {m}\right] \tag{18}\]  
where \(F_{*}\) is the statistical distribution variable of the micro-element strength, while \(m\) and \(F_{0}\) are undetermined parameters in the Weibull distribution.  
Therefore, according to the Lemaitre strain equivalence principle, the constitutive model of rocks containing only micro-defects under uniaxial compression can be derived as follows:  
\[\sigma_{1}=E_{c}\varepsilon_{1}(1-D_{l})=E_{c}\varepsilon_{1}\exp\left[- \left(\frac{F_{*}}{F_{0}}\right)^{m}\right] \tag{19}\]  
where \(E_{c}\) is the elastic modulus of rock at room temperature and \(\varepsilon_{1}\) is the axial strain of rock at room temperature.  
Based on the Drucker-Prager criterion, \(F_{*}\) it can be expressed as follows:  
\[F_{*}=\sqrt{f_{*2}}-\alpha_{0}I_{*} \tag{20}\]  
where \(\alpha_{0}\) is the test constant related to the internal friction angle and cohesion of the rock, \(\alpha_{0}=\frac{\sin\varphi}{\sqrt{\left(9+3\sin^{2}\varphi\right)}}\); \(I_{*}\) is the first invariant of the effective stress, \(I_{*}=\sigma_{*1}\); \(J_{*}\) is the second invariant of the effective stress deviator, \(J_{*2}=\frac{\sigma_{*1}^{2}}{3}\); and \(\sigma_{*1}\) is the effective stress in the axial direction, \(\sigma_{*1}=\sigma_{1}/(1-D_{l})=E_{\text{C}}\varepsilon_{1}\).  
The expression for \(F_{*}\) can be obtained as follows:  
\[F_{*}=\frac{E_{\text{C}}\varepsilon_{1}}{\sqrt{3}}-\alpha_{0}E_{\text{C}} \varepsilon_{1} \tag{21}\]  
Therefore, the constitutive model for granite with cracks under uniaxial load is given as follows:  
\[\sigma_{1}=E_{\text{C}}\varepsilon_{1}(1-D_{l})=E_{\text{C}}\varepsilon_{1} \exp\left\{-\left[\frac{\left(\frac{1}{\sqrt{3}}-\alpha_{0}\right)E_{\text{C }}\varepsilon_{1}}{F_{0}}\right]^{m}\right\} \tag{22}\]  
Using the extreme conditions at the peak value of the model curve, the theoretical expressions of the model parameters \(m\) and \(F_{0}\) of the rock with only micro-defects under uniaxial compression can be obtained. The extreme conditions that the \(\sigma_{1}-\varepsilon_{1}\) curve satisfies at the peak point are as follows:  
\(\textcircled{1}\)\(\varepsilon_{1}=\varepsilon_{P}\), \(\sigma_{1}=\sigma_{P}\); \(\textcircled{2}\)\(\varepsilon_{1}=\varepsilon_{P}\), \(\frac{\partial\sigma_{1}}{\partial\varepsilon_{1}}=0\).  
It can be obtained that:  
\[F_{p}=\frac{E_{\text{C}}\varepsilon_{p}}{\sqrt{3}}-\alpha_{0}E_{\text{C}} \varepsilon_{p} \tag{23}\]  
From the extreme value condition \(\textcircled{1}\) and Equation (19), it can be obtained that:  
\[\left(\frac{F_{p}}{F_{0}}\right)^{m}=\ln\left(\frac{\varepsilon_{p}E_{\text{C }}}{\sigma_{p}}\right) \tag{24}\]  
From the extreme value condition \(\textcircled{2}\), it can be obtained that:  
\[\frac{\partial\sigma_{1}}{\partial\varepsilon_{1}}=E_{\text{C}}\exp\left[- \left(\frac{F_{*}}{F_{0}}\right)^{m}\right]\left[1-m\varepsilon_{1}\frac{F_{* }^{m-1}}{F_{0}}\left(\frac{E_{\text{C}}}{\sqrt{3}}-\alpha_{0}E_{\text{C}} \right)\right]=E_{\text{C}}\exp\left[-\left(\frac{F_{*}}{F_{0}}\right)^{m} \right]\left(1-m\frac{F_{p}^{m}}{F_{0}^{m}}\right)=0 \tag{25}\]\[F_{0}=F_{p}m^{1/m} \tag{26}\]  
The expression for the undetermined parameters \(m\) and \(F_{0}\), obtained by combining Equations (24) and (26), is:  
\[m=\frac{1}{\ln(\varepsilon_{p}E_{C}/\sigma_{p})} \tag{27}\]  
\[F_{0}=F_{p}\Bigg{[}\frac{1}{\ln(\varepsilon_{p}E_{C}/\sigma_{p})}\Bigg{]}^{\ln (\varepsilon_{p}E_{C}/\sigma_{p})} \tag{28}\]