## 3 Results and Discussion
### Definition of Damage Variable  
#### 4.1.1 Definition of the Macroscopic Fracture Damage Variable  
The presence of prefabricated open cracks increases the additional strain energy in the rock mass. The damage variable of fractured rock can be defined by the strain release energy criterion [55; 56]. According to the far-field stress component of prefabricated cracks (Equation (2)), there is no interaction force among the open cracks. The fracture  
Figure 12: Schematic diagram of the crack propagation mode. 1 Wing crack (WC, blue line). 2 anti-wing crack (AWC, green line). 3 Secondary crack (SC, yellow line). Figure 13: The development process of macroscopic cracks in fractured granite. (**a**) F-T 0 cycles. (**b**) F-T 40 cycles. (**c**) F-T 80 cycles. stress intensity factor under uniaxial action (\(\sigma_{2}=\sigma_{3}=0\)) can be obtained as shown in Equation (3):  
\[\left\{ \begin{array}{l}\sigma_{x}=\sigma_{1}\sin^{2}\alpha\\ \tau=\frac{1}{2}\sigma\sin(2\alpha)\\ \sigma_{y}=\sigma_{1}\cos^{2}\alpha\end{array} \right. \tag{2}\]  
\[\left\{ \begin{array}{l}K_{\rm I}=\sigma_{1}\sqrt{\pi a}\cos^{2}\alpha\\ K_{\rm II}=\frac{\sigma_{1}\sqrt{\pi a}\sin(2\alpha)}{2}\end{array} \right. \tag{3}\]  
where \(K_{\rm I}\) and \(K_{\rm II}\) are the stress intensity factors for type I and type II cracks, respectively, \(a\) is the half-length of the prefabricated fissure, \(\sigma_{1}\) is the axial stress, \(\alpha\) is a constant, \(\alpha=\frac{2\sin\varphi}{\sqrt{3}(3-\sin\varphi)}\), and \(\varphi\) is the internal friction angle of the rock. In the plane stress state, the additional elastic strain energy increment \(U_{1}\) caused by a single prefabricated crack at the center can be calculated using Equation (4):  
\[U_{1}=\int_{0}^{A}GdA=\frac{1-\mu^{2}}{E}\int_{0}^{A}\Big{(}K_{\rm I}^{2}+K_{ \rm II}^{2}\Big{)}dA \tag{4}\]  
where \(A\) is the area of the prefabricated crack, \(\mu\) is the Poisson's ratio of the granite sample, and \(E\) is the elastic modulus of the granite sample. According to damage mechanics theory, the damage variable is defined as \(D\), and the complete sample is defined as undamaged. Then, under uniaxial compression, the elastic strain energy per unit volume of fractured rock is given by:  
\[Y=-\frac{\sigma_{1}^{2}}{2E(1-D)^{2}} \tag{5}\]  
Let \(U_{E}\) denote the elastic strain energy per unit volume corresponding to the stress under uniaxial compression, which can be expressed as:  
\[U_{E}=-(1-D)Y \tag{6}\]  
Putting Equation (5) into Equation (6), we obtain:  
\[U_{E}=\frac{{\sigma_{1}}^{2}}{2E(1-D)} \tag{7}\]  
Therefore, under uniaxial compressive stress (\(\sigma_{1}\)), the strain energy per unit volume of intact rock is given by Formula (8), and that of fractured rock is given by Formula (9):  
\[U_{E_{0}}=\frac{{\sigma_{1}}^{2}}{2E} \tag{8}\]  
\[U_{E_{\rm d}}=\frac{{\sigma_{1}}^{2}}{2E(1-D)} \tag{9}\]  
where \(E_{0}\) is the elastic modulus of the intact rock sample and \(E_{\rm d}\) is the elastic modulus of the fractured rock sample. Assuming the volume of the rock sample is \(V\), the change in elastic strain energy per unit volume caused by the crack is given by Equation 10. If only the initial damage caused by the crack is considered, the change in elastic strain energy should conform to Equation (11):  
\[\Delta U_{E}=V\big{[}U_{E_{\rm d}}-U_{E_{0}}\big{]}=V\Bigg{[}\frac{\sigma_{1 }^{2}}{2E(1-D)}-\frac{\sigma_{1}^{2}}{2E}\Bigg{]} \tag{10}\]  
\[U_{1}=\Delta U_{E},\ \frac{1-\mu^{2}}{E}\int_{0}^{A}\Big{(}K_{\rm I}^{2}+K_{ \rm II}^{2}\Big{)}dA=V\Bigg{[}\frac{\sigma_{1}^{2}}{2E(1-D)}-\frac{\sigma_{1}^ {2}}{2E}\Bigg{]} \tag{11}\]Therefore, we have:  
\[D=1-\frac{1}{1+\frac{2(1-\mu^{2})\int_{0}^{A}\left(K_{1}^{2}+K_{1}^{2}\right)dA}{ VE}} \tag{12}\]  
Because of the complex fracture surface topography of real rock, the following assumptions should be made when studying the initial damage caused by prefabricated cracks:  
(1) Assume that the surface of the prefabricated crack in the rock sample is flat and simplify its shape to a planar rectangle.