## 3 Multi-Scale Numerical Modeling of Rock Failure Process under Compression
### Macroscale Simulation
#### 3.3.2 Macroscale Numerical Simulation Results  
The three-dimensional CJRM reconstructed by digital image processing can actually reflect the spatial distribution of the joints on the engineering site (Figure 12). The spatial distribution of the joints greatly influences the fracture process and the final failure pattern of CJRM. Due to the low strength of the joints, they generate a tensile stress area around the joints in the initial stage of loading (Figure 13), causing the joints to fracture under a relatively small external load. The failure characteristics of the three-dimensional numerical simulation and the in-situ tests are very similar. Most of the failure occurs along the joint planes due to their lower tensile strength. Most longitudinal joints suffer from tensile failure, and the low-angle horizontal joints suffer from compression shear failure (Figure 14). The method proposed provides a useful reference for research into irregular CJRM. \begin{table}
\begin{tabular}{l l l l l l l} \hline \hline
**Materials** & \(\mathbf{E_{L}}\)**(MPa)** & \(\mathbf{P_{L}}\)**(MPa)** & \(\mathbf{m}\) & \(\mathbf{\
u}\) & \(\mathbf{\phi}\) (\({}^{\circ}\))** & \(\mathbf{C/T}\) \\ \hline Rock matrix & 65,100 & 280 & 2 & 0.21 & 38 & 10 \\ Joint & 1160 & 10 & 3 & 0.31 & 30 & 12 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Material properties of the CJRM FE model [58]. Figure 12: Configurations of columnar jointed basalt. (**a,b**) Structural descriptions of columnar jointed basalt in transverse sections [35]. (**c**) A numerical model for the columnar jointed basalt. (**d**) The spatial shape of the joints is characterized. Figure 14: Numerical uniaxial compression test of columnar jointed basalt at the macroscale. (**a**–**i**) Vertical displacement contours at different loading stages, (**a**–**i**) correspond to points A–F in Figure 15. The unit of displacement in (**a**–**i**) is the millimeter. Figure 13: The maximum principal stress distribution; the unit is MPa. The stress-strain curves of CJRM are different from those of intact rock samples, and the brittle characteristics of intact rocks are significantly higher than those of CJRM. The stress-strain curve and the AE count for CJRM are shown in Figure 15. There is a stepwise increase when the curve rises to the DE stage before the peak strength. The strain increases from 0.36% to 0.46% at this stage, an increase of 27.8%. The stress increases from 131.2 MPa to 137.5 MPa, a rise of 4.8%. This may be due to the crushed low-angle horizontal joints and the longitudinal blocks being squeezed. After the peak strength, the stress-strain curve of the complete rock sample under uniaxial compression generally decreases sharply. The stress-strain curve of CJRM decreases gradually, without prominent brittle failure characteristics. It can be seen from the changing trend of AE events that AE events occur at low stress levels, and AE events maintain a relatively stable level before peak point E. Due to the low strength of the joints, the joints fracture at low stress levels, and AE events occur. AE events in the A-D interval are mainly caused by longitudinal joint failure, while AE events in the D-E interval are mainly caused by low-angle horizontal joint failure. After the peak point E, the acoustic emission events are mainly concentrated in the E-F interval, and the fracture of the basalt column leads to a large number of AE events. At this stage, the AE events account for about 50% of the total AE events in the fracture process. The mechanical responses of joints and intact rock vary with the same external displacement (e.g., mismatch stress). Figure 13 shows the maximum principal stress distribution at 0.1 mm displacement. Even under compression, tensile stresses (in red) can be observed in the surroundings of joints. Figure 14 illustrates the vertical displacement field during fracture. Due to the existence of joints when the strain level is 0.40 \(\epsilon_{\rm P}\) (Figure 14b), there are obvious cracks in the longitudinal joints in the middle of the CJRM.