# Study on Dynamic Loading Characteristics of Rock Containing Holes
## 2 Numerical Model  
### Experimental Equipment and Data-Processing Methods  
As depicted in Figure 1, dynamic mechanical experiments were conducted on specimens using the Split Hopkinson Pressure Bar (SHPB) apparatus. The terminal surfaces of the rock specimens were polished to ensure uniform contact before positioning them between the incident and transmission bars of the SHPB system. A high-speed camera documented the deformation and fracture dynamics of the specimens during the testing period.  
After the dynamic impact test was concluded, the stress (\(\sigma_{\rm s}\)(t)), axial strain (\(\varepsilon_{\rm s}\)(t)), and strain rate \(\big{(}\dot{\varepsilon}_{\rm s}(\)t\()\()\(\)\(\)\(\)\(\)\) for the specimens were determined using the stress uniformity theory and the three-wave method, as detailed in Equations (1)-(3). Here, A\({}_{0}\), C\({}_{0}\), and E\({}_{0}\) symbolize the cross-sectional area of the pressure bar (m), the velocity of the elastic longitudinal wave (m/s), and the elastic modulus of the waveguide bar (GPa), respectively. The variables \(\varepsilon_{\rm I}\)(t), \(\varepsilon_{\rm R}\)(t), and \(\varepsilon_{\rm T}\)(t) represent the three types of electrical signals generated. Furthermore, A\({}_{1}\), I\({}_{0}\), and t are used to denote the specimen's cross-sectional area (m\({}^{2}\)), length (m), and the time span of the dynamic loading, respectively.  
\[\sigma_{\rm s}(\rm t)=E_{0}A_{0}/2A_{1}-\varepsilon_{\rm R}(\rm t)-\varepsilon_ {\rm T}(\rm t) \tag{1}\]  
\[\varepsilon_{\rm s}=C_{0}/l_{0}\int_{0}^{t_{0}}(\varepsilon_{\rm I}(\rm t)- \varepsilon_{\rm R}(\rm t)-\varepsilon_{\rm T}(\rm t))\rm dt \tag{2}\]  
\[\dot{\varepsilon}=C_{0}/l_{0}(\varepsilon_{\rm I}-\varepsilon_{\rm R}- \varepsilon_{\rm T}) \tag{3}\]  
Sandstone, widely recognized as the prevalent type of material in underground engineering, was selected and shaped into cylindrical specimens measuring 50 mm in diameter and 50 mm in height using professional cutting tools. The deviation in parallelism and perpendicularity of the end surfaces of each specimen was maintained below 0.02 mm, meeting the International Society for Rock Mechanics' (ISRM) criteria for surface flatness. The key mechanical properties measured were a density of 2570 kg/m\({}^{3}\), an elastic modulus of 8.7 GPa, and a Poisson's ratio of 0.23.  
To ensure the experimental results' precision, it is critical that the specimen's ends reach dynamic stress equilibrium, as demonstrated by the close alignment of stress history data at both ends during the impact sequence [43]. Figure 2 shows a schematic diagram of the action of a dynamic stress wave. Figure 3 depicts the stress-time curve for a specimen featuring a single 8 mm hole exposed to a strain rate of 100 s\({}^{-1}\) in an indoor dynamic loading test. The diagram shows that the stress numerical curves at the incident (I\({}_{\rm n}\) + R\({}_{\rm e}\)) and transmitted (T\({}_{\rm r}\)) ends converge, virtually superimposing on each other. This overlap suggests that the specimen achieved a dynamic equilibrium state, thus confirming the accuracy of the numerical findings derived from the experiment.  
Figure 1: Schematic diagram of the SHPB system.