# Failure Mechanism of Rock Specimens with a Notched Hole under Compression--A Numerical Study
## 3 Simulation of Failure in a Rock Specim with a Notched Hole
### Influence of the Notch Length on the Failure Mechanism  
A rectangular specimen including a hole with a radius of \(R\) = 10 mm is modeled. Two notches are created on the wall of the hole at \(\theta\) = 0\({}^{\circ}\) and 180\({}^{\circ}\) in the model. The length of the notch is varied using \(L\) = 2.5, 5, 7.5, and 10 mm, resulting in \(L\)/\(R\) ratios of 0.25, 0.5, 0.75, and 1, respectively. The model size, material properties, and boundary conditions explained in Section 3.1 are used. The failure patterns in the models with different notch lengths at 80% post-peak stress level are illustrated in Figure 11. The figure shows that, when the length of the notch is increased, the tensile failure at the bottom of the hole is longer. In the models with longer notches, the free space resulting from the notches is larger; therefore, a higher number of particles can move towards the free space. As a result, more tensile stress is induced at  
Figure 10: (**a**) The load bearing closed ring around a circular hole; (**b**) the destroyed closed ring around a notched hole. the top (\(\theta=90^{\circ}\)) and bottom of the hole (\(\theta=270^{\circ}\)). Figure 11 indicates that in the models with different notch lengths, cracks are mainly formed at the tips of the notches. In the models with longer notches, the failure zone is far away from the hole wall. When the length of the notch is decreased, the failure zone is closer to the wall of the hole. The failure pattern around the notched hole in the model with \(L=2.5\) mm (\(L/R\) ratio of 0.25) is shown in Figure 12a. The figure indicates that, in this model, the formed cracks are turned back towards the hole surface and result in the failure of the rock at the wall (\(\theta=0^{\circ}\) and \(180^{\circ}\)). The displacement vectors of the particles in this model are shown in Figure 12b. The figure indicates higher displacements of the particles in the zone of \(\theta=315^{\circ}\) to \(0^{\circ}\) compared to the other particles in the model domain. This means that, at the right wall of the hole, the particles moved toward the hole. This can be interpreted as a possible rock failure and fall in the hole in the real-world situation for circular holes with short notch lengths. The failure patterns given in Figure 12 indicate that it is possible to select \(L/R\) ratios, including a minimum value, which depends on the in-situ stresses, rock material properties, as well as the boundary and loading conditions, to prevent failure occurring at the wall of the hole. Relation between the peak strength of the specimens and the notch length is shown in Figure 13. The figure indicates that the maximum strength of the specimens decreases as the notch length increases. From the numerical results presented in this section, it can be concluded that the influence of the notches on the fracturing mechanism around a hole is a function of the notch dimensions. Figure 11: The failure patterns in the models with different notch lengths (80% post-peak stress). Figure 12: Failure pattern and displacement vector distribution around the notched hole with \(L=2.5\) mm (\(L/R\) ratio of 0.25). So far, in this study, the specimen width and hole diameter were set to 100 mm and 20 mm, respectively, resulting in a ratio of 5 for the model width to the hole diameter. The models also included notches with a varying length, which may give rise to possibilities of producing results affected by the external boundaries. Therefore, wider specimens with a width of 200 mm, which results in a ratio of 10 for the model width to the hole diameter, are modeled to make sure that the presented results in Figures 11-13 are not affected by the boundary effect. Figure 14 shows the failure patterns in the models with the width of 200 mm and a different notch length (80% post-peak stress). The figure indicates that, as the length of the notch increases, the failure zone around the hole is transferred into the rock away from the hole surface.