## 3 Analysis of Numerical Simulation Results
### Research on Crack Propagation Mechanism
(Here, the dip angle of the flaws is \(\alpha=30^{\circ}\). The points **a**, **b**, **c**, **d**, **e** and **f** correspond to different loading stages respectively.). Figure 7: Particle velocity fields at different loading stages. (Pictures **a**, **b**, **c**, **d**, **e** and **f** respectively show particle velocity fields corresponding to different loading stages in Figure 6 when secondary cracks propagate. ). \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Loading Stages** & **Loading Stresses (MPa)** & \(R_{1}\) & \(R_{2}\) \\ \hline a & 35.12 & 23.52 & - \\ b & 36.96 & 37.77 & - \\ c & 37.24 & 35.75 & - \\ d & 43.40 & - & 27.53 \\ e & 43.61 & - & 41.69 \\ f & 45.61 & \(-\)45.60 & 23.02 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The values of variable R in the two measurement circles at different loading stages. Figure 8: Principal strain rate tensors at different loading stages. (Pictures **a**, **b**, **c**, **d**, **e**, and **f** respectively show principal strain rate tensors in the two measurement circles corresponding to different loading stages in Figure 6 when secondary cracks propagate.). When the secondary cracks between parallel pre-existing flaws instantaneously appear in large quantities, the values of the variable \(R\) can directly discriminate the types of newly generated cracks. 'Shear' is considered to occur for \(R\) varying between \(-30\) and \(30\). 'Tensile' is considered to occur for an event with an \(R\) larger than \(30\). 'Compressive-shear' is considered to occur for an event with \(R\) smaller than \(-30\). Therefore, the failure mechanisms of events can be exactly determined according to the values of variable \(R\). Here, it should be noted that the time step adopted by PFC is not infinitely small. Even in the region where no crack initiates, the value of the strain rate tensor still exists, leading to the existence of the value of the variable \(R\) without no crack initiating. Consequently, we only need to pay more attentions to the strain rate tensor and the value of variable \(R\) at the time of secondary crack initiation. For better analyzing crack propagation, the secondary cracks in the measurement circle 1 and 2 are named as SC-1 and SC-2 respectively. It can be seen from the analysis of Figures 7 and 8 that when the axial stress is increased to 35.12 MPa, SC-1 initiates near the tip T2 of the flaw, and the velocity fields of the upper and lower half of the newly generated cracks are Types III and I, which represent composite tensile-shear crack and tensile crack, respectively. At this time, the direction of the maximum principal strain rate is close to the horizontal direction. The ratio (\(R_{1}=22.52\)) indicates that the newly generated cracks are shear cracks in essence, where the relative motion of the particles are consistent with the direction of the maximum principal strain rate. As the axial stress is continuously increased to 36.96 MPa, one end of SC-1 extends to the tip T2 of the flaw, and the other end extends toward the tip T4 of the flaw. The velocity field on both sides of the newly generated cracks is Type I, and the value of \(R_{1}\) is 37.77, indicating that the newly generated cracks are essentially tensile cracks. When the axial stress is increased to 37.24 MPa, SC-1 continues to extend to the crack tip T4, which finally generates an obvious crack zone to connect the flaw tip T2 with T4. Here the particle velocity field on both sides of the newly generated cracks near the flaw tip T4 are Type IV. Since relative shearing trend appears between the particles, the particle velocity field on both sides of the newly generated cracks near the flaw tip T2 are Type I. However, the variable (\(R_{1}=45.61\)) indicates that the newly generated crack are essentially tensile cracks. As the strain increases and the axial stress is slowly increased up to 43.40 MPa, SC-2 suddenly initiates between the crack tip T1 and T3.