## 2 Uniaxial Compression Test of Hard Coal
### Analysis Method  
The test result analyses of SHPB are based on homogenization assumption. The DPO 3024 digital oscilloscope records the waveform of the specimen and the voltage signal corresponding to each time process during the test. The incident wave strain \(\epsilon_{\mathrm{i}}(t)\), the reflected wave strain \(\epsilon_{\mathrm{r}}(t)\) and the transmitted wave strain \(\epsilon_{\mathrm{t}}(t)\) of the specimen can be obtained via analysis, and the stress \(\sigma(t)\), strain rate \(\dot{\epsilon}(t)\), and strain \(\epsilon(t)\) of the hard coal test piece are obtained according to the three-wave method [49; 50; 51] as follows:  
\[\left\{\begin{array}{l}\sigma(t)=\frac{A_{\mathrm{e}}E_{\mathrm{e}}}{2A_{ \mathrm{s}}}[\epsilon_{\mathrm{t}}(t)+\epsilon_{\mathrm{r}}(t)+\epsilon_{ \mathrm{i}}(t)]\\ \dot{\epsilon}(t)=\frac{C_{\mathrm{e}}}{\Gamma_{\mathrm{s}}}[\epsilon_{ \mathrm{i}}(t)-\epsilon_{\mathrm{r}}(t)-\epsilon_{\mathrm{t}}(t)]\\ \epsilon(t)=\frac{C_{\mathrm{e}}}{\Gamma_{\mathrm{s}}}\int_{0}^{t}[\epsilon_ {\mathrm{i}}(t)-\epsilon_{\mathrm{r}}(t)-\epsilon_{\mathrm{t}}(t)]\mathrm{d}t \end{array}\right. \tag{2}\]  
where \(E_{\mathrm{e}}\) is the elastic modulus of the pressure bar, GPa; \(A_{\mathrm{e}}\) is the cross-section area of the pressure bar in the test device, mm\({}^{2}\); \(A_{\mathrm{s}}\) is the sectional area of the specimen, mm\({}^{2}\); \(C_{\mathrm{e}}\) is the propagation velocity of the stress wave, m/s; and \(L_{\mathrm{s}}\) is the initial height of the test piece, mm.  
The energy is calculated as follows [52]:  
\[W=\frac{A_{\mathrm{e}}C_{\mathrm{e}}}{E_{\mathrm{e}}}\int_{0}^{t}\sigma^{2}(t) \mathrm{d}t \tag{3}\]  
where \(W\) is the energy carried by the stress wave, J, and the wave velocity \(C_{\mathrm{e}}\) can be expressed by the elastic modulus \(E_{\mathrm{e}}\) and the density \(\rho_{\mathrm{e}}\) of the pressure bar:  
\[C_{\mathrm{e}}=\sqrt{E_{\mathrm{e}}/\rho_{\mathrm{e}}} \tag{4}\]  
Figure 3: Stress balance curve.  
Figure 2: The schematic diagram of SHPB.  
Therefore, the stress wave energy equation can be simplified by the following Equation (5):  
\[W=\frac{A_{\rm e}}{\rho_{\rm e}C_{\rm e}}\int_{0}^{\tau}\sigma^{2}(t)\mathrm{d}t \tag{5}\]  
where \(\rho_{\rm e}C_{\rm e}\) is the wave impedance of the pressure bar, \(\mathrm{g/cm^{3}\cdot m/s}\).  
The incident energy \(W_{\rm I}\), the reflected energy \(W_{\rm R}\), and the transmitted energy \(W_{\rm T}\) on the pressure bar are, respectively, expressed in Equations (6)-(8).  
\[W_{\rm I}=\frac{A_{\rm e}}{\rho_{\rm e}C_{\rm e}}\int_{0}^{\tau}\sigma_{\rm I} ^{2}(t)\mathrm{d}t \tag{6}\]  
\[W_{\rm R}=\frac{A_{\rm e}}{\rho_{\rm e}C_{\rm e}}\int_{0}^{\tau}\sigma_{\rm R} ^{2}(t)\mathrm{d}t \tag{7}\]  
\[W_{\rm T}=\frac{A_{\rm e}}{\rho_{\rm e}C_{\rm e}}\int_{0}^{\tau}\sigma_{\rm T} ^{2}(t)\mathrm{d}t \tag{8}\]  
where \(\sigma_{\rm I}(t)\), \(\sigma_{\rm R}(t)\), and \(\sigma_{\rm T}(t)\) are the incident stress, the reflected stress, and the transmitted stress at a certain time \(t\).  
According to Equations (8)-(10), the destructive energy \(W_{\rm S}\) consumption of the hard coal specimen under a SHPB dynamic impact can be expressed as:  
\[W_{\rm S}=W_{\rm I}-W_{\rm R}-W_{\rm T} \tag{9}\]  
The energy dissipation rate \(N\) is usually used to represent the strength of coal rock energy dissipation under SHPB dynamic impact. \(N\) is calculated as follows [46]:  
\[N=\frac{W_{\rm S}}{W_{\rm I}}\times 100\% \tag{10}\]