## 2 Mesoscopic Parameter Calibration and Basic Theory
### Strain Rate Tensor  
Based on acoustic emission technology, the moment tensor inversion analysis [41; 42; 43] has been widely used to distinguish rock fracture types. [PERSON] et al. [44] proposed a criterion for judging rock rupture through reasonably decomposing the variability tensor. The strain rate tensor can be calculated through the moment tensor inversion, but it is still inadequate to adopt strain rate tensor to judge the crack type and analyze the crack propagation law. The PFC software developed based on \begin{table}
\begin{tabular}{c c} \hline \hline
**Particle Parameters** & **Values** \\ \hline Minimum radius \(R_{min}\) (mm) & 0.18 \\ Particle radius ratio \(R_{max}/R_{min}\) & 1.66 \\ Density \(\rho\) (g/cm\({}^{3}\)) & 2.38 \\ Friction \(\mu\) & 0.55 \\ Effective modulus \(E_{c}\) (GPa) & 5.5 \\ Normal/shear stiffness ratio \(k_{n}/k_{s}\) & 2.0 \\ Tensile strength \(\overline{\sigma_{c}}\) (MPa) & 22.5 \(\pm\) 2.0 \\ Cohesion \(\overline{c}\) (MPa) & 19.5 \(\pm\) 2.0 \\ Angle of internal friction \(\overline{\varphi}\) (0) & 35 \\ Bond effective modulus \(\overline{E_{c}}\) (GPa) & 5.5 \\ Bond normal/shear stiffness ratio \(\overline{k_{n}}/\overline{k_{s}}\) & 2.0 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Microscopic parameters used in the numerical model. Figure 2: Comparison of failure modes between laboratory experiment and numerical simulation. (**a**) \(\alpha=30^{\circ}\); (**b**) \(\alpha=45^{\circ}\); (**c**) \(\alpha=60^{\circ}\); (**d**) \(\alpha=30^{\circ}\); (**e**) \(\alpha=45^{\circ}\); (**f**) \(\alpha=60^{\circ}\). (Pictures **a**, **b** and **c** are experimental results, and pictures **d**, **e** and **f** are numerical results). discrete element theory can not only simulate the propagation and evolution of micro-cracks in rock masses, but also consistently monitor the strain rate tensor changes in specific regions through the measurement circle. The schematic diagram of the main strain rate tensor when micro-cracks appear is shown in Figure 3. When the contact between the particles breaks, the velocity of the particles will instantly change, triggering the variation of the magnitude and direction of the strain rate tensor in the monitored region. Therefore, the evolution law of crack propagation can be accurately reflected by strain rate tensor. In PFC, when the absolute value of the difference between the actual velocity of the particles in the measurement circle and the calculated velocity is minimized, the strain rate tensor in the measurement circle can be obtained by establishing a system of equations [36]. Assuming that there are \(N\) particles in the measurement circle, the particle translation speed and the centroid position are \(V_{i}\) and \(x_{i}\) respectively, and the average velocity \(\overline{V}_{i}\) and average position \(\overline{x}_{i}\) can be expressed as:  
\[\overline{V}_{i}=\frac{\sum\limits_{N}V_{i}}{N} \tag{1}\]  
\[\overline{x}_{i}=\frac{\sum\limits_{N}x_{i}}{N} \tag{2}\]  
The actual relative velocity \(\overline{V}_{i}\) and relative position of the particles \(\overline{x}_{i}\) are:  
\[\overline{V}_{i}=V_{i}-\overline{V}_{i} \tag{3}\]  
\[\overline{x}_{i}=x_{i}-\overline{x}_{i} \tag{4}\]  
Assuming that the particles move from point \(x_{i}\) to point \(x_{j}\) during infinitesimal time, the average speed difference between the two points is:  
\[dv_{i}=\dot{\alpha_{ij}}dx_{j} \tag{5}\]  
If the velocity gradient tensor \(\dot{\alpha}_{ij}\) is known, the relative velocity can be calculated as:  
\[\overline{v}_{i}=\dot{\alpha_{ij}}\overline{x}_{j} \tag{6}\]  
Figure 3: Schematic diagram of the principal strain rate tensor. (The red, green and blue micro-cracks represent tensile micro-cracks, tensile-shear micro-cracks, and compressive-shear micro-cracks, respectively, and the principal strain tensor is denoted by two sets of light blue lines with arrows).