## 4 Numerical Examples
### Uniaxial Compression Test on Rock with a Prefabricated Crack  
To simulate and study the crack propagation law of rock under uniaxial compression, a 2D numerical model of a rock square plate with a single prefabricated crack is constructed, as shown in Figure 7. The simulation adopts the plane strain assumption. The model has dimensions of 75 mm \(\times\) 100 mm, a thickness of 1 mm, and a prefabricated crack at the  
Figure 5: Single-edge notched tensile test phase field simulation results: (**a**) 200 steps; (**b**) 400 steps; (**c**) 600 steps; (**d**) 800 steps; (**e**) 1000 steps; (**f**) 1200 steps. Figure 6: Crack propagation behavior under different mesh sizes. center with an angle of 45 degrees to the horizontal line, a crack length of \(h\) = 20.00 mm, and a width of \(w\) = 2.00 mm. Material parameters are set according to reference [24], and the specific values are shown in Table 1.  
The model's bottom boundary is constrained in the y-direction, while quasi-static displacement loading is prescribed vertically downward at the top boundary with a constant displacement rate of 2 \(\times\) 10\({}^{-4}\) mm/step. Simulations were performed using custom FE code developed on the COMSOL Multiphysics platform, employing a hybrid meshing strategy that combines quadrilateral elements with triangular elements for local refinement. Mesh size transitions from 0.25 mm near the crack to 0.5 mm at the specimen boundaries. Boundary conditions include normal constraint on the bottom edge; horizontal constraint at the bottom-left node; and displacement-controlled loading at the top. Figure 8 presents the simulation results. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Symbol of Material** & & & \\
**Parameter** & & & \\ \hline \(E\) & elastic modulus & 32.15 & GPa \\ \(v\) & [PERSON]’s ratio & 0.3 & \\ \(\varphi\) & angle of internal friction & 38.8\({}^{\circ}\) & \\ \(c\) & cohesion & 18.5 & \\ \(G_{\mathrm{cl}}^{+}\) & critical energy release rate for tensile fracture & 11.0 & J/m\({}^{2}\) \\ \(G_{\mathrm{cl}}^{+}\) & critical energy release rate for tension–shear fracture & 50 \(G_{\mathrm{cl}}^{+}\) & \\ \(G_{\mathrm{c}}^{-}\) & critical energy release rate for & 27 \(G_{\mathrm{cl}}^{+}\) & \\ \(G_{\mathrm{pD}}\) & critical energy release rate for plastic damage & 10 \(G_{\mathrm{cl}}^{+}\) & \\ \(K_{p}\) & plastic bulk modulus & 10.0 & GPa \\ \(\mu_{p}\) & plastic shear modulus & 5.0 & GPa \\ \(\alpha\) & material parameter & 0.65 & \\ \(k\) & hardening parameter & 0.5 & \\ \(\ell_{0}\) & scale parameter & 0.5 & mm \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Material parameters and their values used in the simulation. Figure 7: Uniaxial compression test model for rock with a prefabricated crack. Figure 8 illustrates the phase field simulation results of the uniaxial compression test on prefabricated crack rock, capturing the progressive failure process from crack initiation to macroscopic fracture. During initial loading, the material remains in the elastic stage with no crack propagation. As loading progresses, stress concentration at the crack tip induces localized damage, manifesting as wing cracks propagating perpendicular to the prefabricated crack surface under tensile dominance. With further loading, shear stress accumulation triggers tensile-shear, resulting in through-going fracture surfaces and approaching macroscopic failure. These results align closely with numerical findings from [PERSON] et al. [12] and [PERSON] et al. [14], confirming the model's capability to replicate realistic fracture mechanics. As shown in Figure 9, the load-displacement curve of the uniaxial compression test on rock with prefabricated cracks is presented, where the load represents the vertical load on the model. As displacement gradually increases, the load grows linearly, indicating the elastic deformation stage. When displacement reaches a certain value, cracks initiate, and the load reaches the ultimate bearing capacity, followed by a sharp drop until complete fracture.