# Influence of Crack Geometry on Dynamic Damage of Cracked Rock: Crack Number and Filling Material
## 3 SHPB Test Results and Analysis  
### Energy Analysis  
Based on one-dimensional stress wave theory, energy can be indirectly calculated according to the incident, reflected and transmitted stress wave signals obtained from the SHPB test. The calculation formula is shown as Equation (2) [38].  
\[\begin{array}{l}W_{I}=\int_{0}^{t}A_{i}\sigma_{i}(t)ce_{i}(t)dt=\frac{A_{c}c}{ \sum_{t}\int_{0}^{t}\sigma_{t}^{2}(t)dt}\\ W_{R}=\int_{0}^{t}A_{r}\sigma_{r}(t)ce_{r}(t)dt=\frac{A_{c}c}{\sum_{t}\int_{0}^{t }\sigma_{r}^{2}(t)dt}\\ W_{T}=\int_{0}^{t}A_{t}\sigma_{t}(t)ce_{t}(t)dt=\frac{A_{c}c}{\sum_{t}\int_{0}^{t }\sigma_{t}^{2}(t)dt}\end{array} \tag{2}\]  
where \(W_{I},W_{R},W_{T}\) are the incident wave energy, reflected wave energy and transmitted wave energy, respectively; \(A_{i}=A_{r}=A_{t}=A\) represent the cross-sectional area of the bar; \(E\) is the elastic modulus of the bar; \(\sigma_{i}(t),\sigma_{r}(t),\sigma_{t}(t)\) are the incident, reflected and transmitted stress signals, respectively. The longitudinal wave velocity, \(c\), can be calculated using the density, \(\rho\), and the elastic modulus of the bar, as shown in Equation (3).  
\[c=\sqrt{E/\rho} \tag{3}\]  
The stress wave can be deduced by strain signals, so Equation (2) can be expressed by Equation (4), as follows.  
\[\begin{array}{l}W_{I}=AcE\int_{0}^{t}\varepsilon_{t}^{2}(t)dt\\ W_{R}=AcE\int_{0}^{t}\varepsilon_{r}^{2}(t)dt\\ W_{T}=AcE\int_{0}^{t}\varepsilon_{t}^{2}(t)dt\end{array} \tag{4}\]  
where \(\varepsilon_{i}(t),\varepsilon_{r}(t),\varepsilon_{t}(t)\) are the incident, reflected and transmitted strain signals, respectively, and can be obtained from the SHPB test.  
Ignoring the energy loss between the rock specimen and bars, the mathematical expression of total energy absorption is shown in Equation (5).  
\[W_{d}=W_{I}-W_{R}-W_{T} \tag{5}\]  
Using the stress wave obtained from the records of strain gauges and employing Equations (4) and (5), the total absorption energy of the specimens with different numbers of cracks was obtained. The curves of absorption energy versus time of specimens with different numbers of cracks filled with sand are shown in Figure 8. It can be observed from Figure 8 that the tendency of the absorption energy was similar when the number of cracks was different. With the increasing number of cracks, the energy absorption increased. The experimental results suggested that cracked rock with more cracks usually absorbs more energy during its failure process, indicating that cracked rock with more cracks usually has a lower strength.  
Figure 8: Curves of total absorption energy versus time of specimens with different numbers of cracks filled with sand.  
Similarly, the total absorption energy of the specimens with different filling material was calculated. The curves of absorption energy versus time of specimens with two cracks filled with different materials are shown in Figure 9. Figure 9 indicates that the specimens with different filling materials have a similar tendency of the absorption energy. The filling material that had great strength and good cohesion could contribute more strength. As a result, the energy absorption decreased accordingly, indicating that the cracked rock filled with material having great strength and good cohesion holds the higher strength.