# Influence of Crack Geometry on Dynamic Damage of Cracked Rock: Crack Number and Filling Material
## 3 SHPB Test Results and Analysis
### Dynamic Damage  
According to the method proposed previously, the damage variable can be calculated using Equation (6) [39,40]. \[D=1-\frac{\sigma_{j}}{\sigma_{r}} \tag{6}\]  
where \(\sigma_{j}\) is the dynamic peak stress of cracked rock and \(\sigma_{r}\) is the dynamic peak stress of intact rock mass. The histories of stress of the specimen with different geometric characteristics can be obtained by analyzing the stress wave obtained from the records of strain gauges and can be derived as  
\[\sigma(t)=\frac{A}{2A_{0}}E[\varepsilon_{\mathrm{t}}(t)+\varepsilon_{\mathrm{ r}}(t)+\varepsilon_{\mathrm{t}}(t)] \tag{7}\]  
where \(A_{0}\) is the initial cross-sectional area of the specimen. Assuming that the stress equilibrium condition or uniform deformation prevails during dynamic loading (i.e., \(\varepsilon_{\mathrm{t}}+\varepsilon_{\mathrm{r}}=\varepsilon_{\mathrm{t}}\)) in Equation (7), Equation (7) can be rewritten as  
\[\sigma(t)=\frac{A}{A_{0}}E\varepsilon_{\mathrm{t}}(t) \tag{8}\]  
The pulse shaping technique was used in this study to ensure the stress equilibrium. The force balance can be found in Figure 7.  
Figure 10 shows the curves of stress versus time with different numbers of cracks. It can be seen that the stress form of specimens was very similar when the number of cracks was different. In addition, the peak stresses decreased with an increasing number of cracks. The largest stresses were 129.6, 125.8, 110.1 and 101.9 MPa when the number of cracks was zero, one, two and three, respectively. Figure 9: Curves of total absorption energy versus time of specimens with two cracks filled with different materials. According to Equation (6) and Figure 10, the damage variable of cracked rock with different numbers of cracks can be calculated (see Figure 11). The results showed that dynamic damage of cracked rock is obviously influenced by the number of cracks. The damage variable increased with an increasing number of cracks, which suggested that the cracked rock with more cracks was damaged more seriously under dynamic loading. Further, it can be seen in Figure 12 that the stress form of specimens was very similar when the filling material was different. However, the peak stress was various. The extreme stress was 34.72, 57.33, 70.29, 58.63 and 46.98 MPa for the filling material being soil, 2:1 (the weight ratio of soil to sand), 1:1, 1:2 and sand, respectively. The corresponding stress was 77.52 MPa for the intact specimen. As shown in Figure 12, the peak stress of intact rock mass was bigger than that of the jointed rock mass. For the specimens with different filling materials, the order of the peak stress was 1:1 > 1:2 > 2:1 > sand > soil. Figure 11: Curves of damage variable versus different number of cracks. Figure 10: Curves of stress versus time of specimens with different numbers of cracks filled with sand. The damage variable of cracked rock with different filling materials can be calculated by Equation (6) and Figure 12, as is shown in Figure 13. The results suggested that the damage variable shown in Figure 13 was obviously influenced by the filling material. With the difference of the strength or cohesion of the filling material, the strength of cracked rock was various. The Liaoning soil and the ISO sand were chosen as initial materials in this study. Five types of filling materials were prepared: Liaoning soil, ISO sand, and the weight ratios of soil/sand = 2, 1, and 0.5. The cohesion of sand can be regarded as null. As a result, with the increasing ratio of sand, the cohesion of the filling material decreased. However, many studies have shown that the strength of sand is greater than that of soil. There is no doubt that the filling material with a higher ratio of sand had greater strength. As the strength of sand is greater than soil, the jointed rock filled with sand was harder than that filled with soil. At the same time, the cohesion of the filling material should be considered. The filling material that has great strength and good cohesion can contribute more strength.