# 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
### Failure Evolution in the Circular Hole and the Notched Hole  
In this section, models with and without notches are built to study the effect of notches on the failure behavior of rock specimens under compression. Firstly, the uniaxial compression test on a rectangular specimen with a circular hole is simulated (Figure 5a). The stress-strain curve obtained for this specimen is plotted in Figure 7. In this figure, the stress-strain curve obtained from the uniaxial compressive test on the intact rock specimen (i.e., with no hole and no notches) is also plotted. The figure shows that the specimen with a central circular hole reaches a maximum stress of 15.65 MPa before it fails. This is 17.4% less than that in the intact specimen, in which the maximum uniaxial strength is 18.95 MPa, which is similar to laboratory experimental observations [20]. The uniaxial strength of the specimen with a notched hole is 14.53 MPa, which is 23.3% and 7.1% less than that of the intact specimen and the specimen with the circular hole, respectively. Figure 5: Geometry of the models with **(a)** a circular hole and **(b)** a notched hole. Figure 6: Particle assembly in the notched hole model (\(L=10\) mm). Figure 8a shows the failure evolution in the specimen with the circular hole. In this figure, the failure patterns at the 75% peak stress, 90% peak stress, peak stress, and 80% post-peak stress are illustrated. The black and the yellow lines in Figure 8 represent the tensile and the shear cracks, respectively. When the specimen is loaded at 75% peak stress, a few cracks are nucleated at the left side of the hole (\(\theta=180^{\circ}\)) where the tangential stress is concentrated (see Equation (5)). In addition, tensile cracks are formed at the bottom of the hole (\(\theta=270^{\circ}\)). Moreover, some minor cracks appear, which are mainly distributed around the hole. At 90% of the peak stress level, more fractures are formed around the hole, and localized failure zones appear at the two sides of the hole (\(\theta=0^{\circ}\) and \(180^{\circ}\)). At the peak stress, the cracks are coalesced and formed into obvious failure planes. Figure 9a shows a close view of the failure zone at the left side of the hole at the peak stress level. In this figure, the rock at the surface of the hole is crushed because it is the location of the maximum tangential stress around the circular hole (Equation (5)). By applying more strain to the model and bringing it to 80% of the peak stress (post-peak), the number of cracks is increased, and the failure planes become wider. Heterogeneity is a characteristic of rocks and rock-like materials. Therefore, the failure in a symmetric rock specimen under uniform loads is not necessarily symmetric. PFC models are heterogeneous models in which the particle sizes and the micro-parameters are non-uniform. Thus, the rock failure in the PFC models can be asymmetric. Figure 8 indicates that, despite slight differences in failure patterns around the hole, the overall failure patterns in the specimens are symmetric. Figure 7: The stress–strain curves obtained from PFC\({}^{\text{2d}}\) models. Figure 8: Failure evolution in **(a)** circular hole, and **(b)** notched hole (black and yellow lines represent tensile and shear cracks, respectively). Figure 9: The close views of failure zones around the left side of the hole at peak-stress in the models with **(a)** a circular hole, and **(b)** a notched hole with \(L=5\) mm. In a separate model, two notches with the length of \(L\) = 5 mm, which results in \(L\)/\(R\) = 0.5, were created on the wall of the hole at \(\theta\) = 0\({}^{\circ}\) and 180\({}^{\circ}\) (see Figure 5b). This model is referenced as the notched hole. The same modeling procedure explained in Section 3.1 was used. The stress-strain curve for this model is shown in Figure 7. The figure shows a maximum strength of 14.53 MPa for the notched hole model. This is 7.1% and 23.3% less than that in the circular hole model and the intact model, respectively.