# Study on Dynamic Mechanical Properties and Failure Pattern of Thin-Layered Schist
## 3 Dynamic Mechanical Properties
### Energy Dissipation Law  
According to one-dimensional stress wave theory, the incident energy, the reflection energy and the transmission energy can be calculated by the following formula:  
\[\begin{array}{l}W_{I}(t)=E_{0}C_{0}A_{0}\int_{0}^{\tau}\varepsilon_{I}^{2}(t)dt \\ W_{R}(t)=E_{0}C_{0}A_{0}\int_{0}^{\tau}\varepsilon_{R}^{2}(t)dt\\ W_{T}(t)=E_{0}C_{0}A_{0}\int_{0}^{\tau}\varepsilon_{T}^{2}(t)dt\end{array} \tag{1}\]  
where \(W_{I}\), \(W_{I}\) and \(W_{T}\), respectively, are the incident energy, reflection energy and transmission energy. \(E_{0}\), \(A_{0}\) and \(C_{0}\) are the elastic modulus, cross-sectional area and elastic wave velocity of the impact pressure bar, respectively. \(\varepsilon_{I}\), \(\varepsilon_{R}\) and \(\varepsilon_{T}\) are, respectively, incident, reflected and transmitted strain.  
The energy loss between the pressure rod and the rock specimen is ignored during the test and, according to the principle of conservation of energy, can be calculated by the following formula:  
\[W_{\text{s}}=W_{I}-W_{R}-W_{T} \tag{2}\]  
where \(W_{\text{s}}\) is the dissipated energy.  
The dissipative energy calculated in Equation (2) includes the total energy absorbed by the sample, the kinetic energy generated by the crushing and flying of the sample, and the heat energy and sound energy in the rock deformation process, amongst others. The  
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**No.**} & \multirow{2}{*}{**Schistosity**} & \multirow{2}{*}{**Volume**} & \multicolumn{3}{c}{**Compressive Strength [MPa]**} & \multirow{2}{*}{**Incident**} & \multirow{2}{*}{**Reflection**} & \multirow{2}{*}{**Transmission**} & \multicolumn{3}{c}{**Dissipated Energy [J]**} \\ \cline{3-3} \cline{5-10}  & & & & & **Standard** & & **Energy** & & **Energy** & & **Energy** & \\ \cline{3-3} \cline{5-10}  & & **Angle** [\({}^{\circ}\)] & **[cm\({}^{3}\)]** & **Value** & & & **Mean** & & **Dexium** & & **(J]** & **(J]** & **(J]** & **(J]** & **(J]  
research results show that the latter two kinds of energy account for less than 5% of the total dissipative energy [27]. Therefore, the dissipative energy of the specimen is approximately represented by the total absorbed energy in the SHPB test, which can show how much energy the rock absorbs during the crushing process.  
Therefore, according to Equation (2), the incident energy, reflection energy and transmission energy of thin-layered schist specimens at different schistity angles in the dynamic compression test can be calculated, as shown in Table 1. And the variation law between the schistity angles and mean dissipated energy are shown in Figure 7.  
As can be seen from Figure 6, the relationship between the dissipated energy and dissipated energy density of the specimen and the schistosity angle presents an approximate \"U\"-shaped curve feature. The dynamic compressive strength of the specimens with 0\({}^{\circ}\) and 90\({}^{\circ}\) schistity angles is the best, which results in the greatest dissipation energy. Although the specimens with 30\({}^{\circ}\) schistity angles are not very strong, more energy is still consumed in the dynamic destruction process. This is because the internal structure of the specimen is broken more violently, resulting in more rock blocks and smaller lumps. Therefore, due to the existence of thin-layered schistosity, the schist failure is characterized by a mixture of splitting, stretching and shearing failure, and its dissipative energy and dynamic compressive strength change law are not completely consistent.