## 6 Stress Analysis Around the Hole
### Stress Monitoring  
In accordance with the numerical model, the mechanical behavior of arbitrary rock particles during loading can be directly monitored by employing the FISH language tool. As illustrated in Figure 4c, the pore stress concentration area is designated as the stress detection area at the inception of the modeling process. As demonstrated in Figure 11, the stress-strain curves of the average stress of the marble particles inside the monitoring area of the intact, cement paste-filled, and hole-containing specimens are shown. The monitoring area is capable of producing an output of the stress state in each direction: \(xx\), \(yy\), \(xy\), and \(yx\). However, since the tangential stress \(\sigma_{yy}\) is more prominent for the stress concentration phenomenon at the tip [41], this paper only studies the tangential stress \(\sigma_{yy}\) at the tip of the hole. The stress concentration factor k is then as follows:  
\[k=\frac{\sigma_{yy}}{q} \tag{4}\]  
The directional stress in the hole monitoring area, denoted by \(\sigma_{yy}\), and the uniaxial compressive strength, denoted by \(q\), are the primary variables of interest.  
As demonstrated in Figure 12, the maximum stress value recorded in the monitoring region of the intact specimen is 109.51 MPa, whereas the peak stress achieved in the numerical test is 130.1 MPa. This indicates that the stress concentration phenomenon does not occur in the monitoring region and is not present in the peak region of uniaxial compression damage. The stress concentration factor is approximately 0.84.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \(\theta\) & \(\mathbf{0}^{\circ}\) & \(\mathbf{30}^{\circ}\) & \(\mathbf{45}^{\circ}\) & \(\mathbf{60}^{\circ}\) & \(\mathbf{90}^{\circ}\) \\ \hline \(\sigma_{\theta}\) & \(q\) & \(0\) & \(-q\) & \(-2q\) & \(-3q\) \\ \hline \end{tabular}
\end{table}
Table 4: The stress level around the hole.  
The maximum stress value recorded in the monitoring area of the hole-containing specimens is 175.39 MPa, which is a mere 3% of the error value of the analytical solution in Section 5.1 and highly restores the stress concentration state in the direction of the principal stress of the hole rock sample. The peak strength of the rock sample in question is 59.32 MPa, and the stress concentration factor is approximately 2.96, which is consistent with the theoretical solution of the planar elliptic model in elastic-plastic mechanics.  
The maximum stress value recorded in the monitoring area of the cement paste-filled specimen is 164.41 MPa, whereas the corresponding peak strength of the rock sample is 72.91 MPa, and the stress concentration factor is approximately 2.3 times the initial value. This finding suggests that the stress concentration persists in the hole edge region of the rock samples following hole filling, albeit at a significantly lower level than that observed in the unfilled rock samples. Under the condition of unfilled holes, there is a critical surface at the hole of the rock samples, which is in a two-way stress state during the stressing process. The presence of cement paste filler results in a transition from a two-way to a three-way force state within the monitoring area. The damage process, as observed in both the indoor test and numerical test, indicates that cement paste detachment from the rock particles occurs in the late stage of the test. Furthermore, stress concentration is evident at this location during the middle and late stages of the test. However, the mechanical properties of the filled rock samples were found to be enhanced to a certain extent by the reduction in the stress concentration time and the weakening of the degree of stress concentration.