## 4 Energy Evolution of Bituminous Sandstone under Cyclic Loading and Unloading  
Under cyclic loading and unloading, internal cracks started and then connected in rocks, accompanied by energy transformation. According to the first law of thermodynamics [22; 23], the power made by the tester to rocks was transformed completely into elastic energy and dissipated energy before thermal energy exchange with the external environment. Elastic energy is the energy absorbed by rocks to recover from deformation during loading, whereas dissipated energy is the energy consumed by rocks to generate damage and plastic strains under cyclic loads. Hence, the area enclosed by the loading curve and strain (\(\epsilon\)) in the cyclic loading and unloading test is the total power made by the tester to test samples; this power \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Sample** & \(E_{0}\) **(GPa)** & \(E\) **(GPa)** & **Reduction in** & \(D_{E}\) \\
**Number** & & & **Elastic Modulus** & \(D_{E}\) \\ \hline \hline \multirow{2}{*}{1} & & 7.81 & 15.2\% & 0.133 \\  & & 7.62 & 18.1\% & 0.153 \\  & & 7.51 & 19.8\% & 0.166 \\  & & 7.31 & 23.1\% & 0.189 \\  & & 7.15 & 25.9\% & 0.206 \\  & & 7.05 & 27.7\% & 0.217 \\  & & 6.51 & 38.2\% & 0.277 \\  & & 6.32 & 42.4\% & 0.298 \\  & & 6.22 & 43.9\% & 0.309 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The elastic modulus and damage of samples. refers to the input energy density (\(U\)) of bituminous sandstone. The area enclosed by the unloading curve and strain (\(\epsilon\)) is the recoverable elastic energy density (\(U_{\epsilon}\)), which is kept in bituminous sandstone. The area between the loading curve and unloading curve, known as the area enclosed by AOBA, is the unrecoverable dissipation energy density (\(U_{d}\)), as shown in Figure 3.  
The expressions of input energy, dissipated energy, and elastic energy are [23; 24; 25; 26; 27; 28]  
\[U=U_{\epsilon}+U_{d}, \tag{3}\]  
\[U=\int_{0}^{\epsilon_{2}}\sigma_{i}d\epsilon_{i}, \tag{4}\]  
\[U_{\epsilon}=\int_{\epsilon_{1}}^{\epsilon_{2}}\sigma_{i}d\epsilon_{i}, \tag{5}\]  
\[U_{d}=\int_{0}^{\epsilon_{2}}\sigma_{i}d\epsilon_{i}-\int_{\epsilon_{1}}^{ \epsilon_{2}}\sigma_{i}d\epsilon_{i}, \tag{6}\]  
where \(\epsilon_{1}\) is the strain value at the loading stage \(\sigma_{1}\), \(\epsilon_{2}\) is the strain value at the loading stage \(\sigma_{2}\), \(\sigma_{i}\) and \(\epsilon_{i}\) are stress and strains in the ith cyclic loading and unloading, respectively. According to Equations (3)-(6), the relationship of input energy density, elastic energy density, and dissipation energy density of bituminous sandstones with the number of loading-unloading cycles could be calculated, as shown in Figure 4.  
As shown in Figure 4a, when the amplitude of cyclic loading and unloading was 20% \(\sigma_{c}\), the initial input energy density was 3.63 \(\times\) 10\({}^{-2}\) MJ/m\({}^{-3}\), and it stabilized at 2 \(\times\) 10\({}^{-2}\) MJ/m\({}^{-3}\) with the increase in the number of cycles, which was decreased by nearly 45% compared with the initial value. When the amplitude of cyclic loading and unloading was 30% \(\sigma_{c}\), the initial input energy density was 5.22 \(\times\) 10\({}^{-2}\) MJ/m\({}^{-3}\), and it stabilized at 3.1 \(\times\) 10\({}^{-2}\) MJ/m\({}^{-3}\) with the increase in the number of cycles, which was decreased by nearly 41% compared with the initial value. When the amplitude of cyclic loading and unloading was 40% \(\sigma_{c}\), the initial input energy density was 6.93 \(\times\) 10\({}^{-2}\) MJ/m\({}^{-3}\), and it stabilized at 4.4 \(\times\) 10\({}^{-2}\) MJ/m\({}^{-3}\) with the increase in the number of cycles, which was decreased by nearly 37% compared with the initial value. This result indicates that the input energy density increases with the increase in the amplitude of cyclic loading and unloading.