# Mechanical and Energy Evolution Characteristics of Sandstone under True Triaxial Cyclic Loading
## 3 Test Results and Analysis
### Analysis of Deformation and Failure Characteristics of Specimens
#### 3.2.1 Definition of Rock Energy Parameters in the True Triaxial State  
The deformation and damage of rocks under external force are essentially processes of internal energy release and dissipation. In a rock mechanics test, the total input energy of the external load on the rock is partly stored in the rock mass in the form of elastic energy and partly dissipated in various forms. Assuming that the rock unit has no heat exchange with the outside world during the action of external force, the deformation of the rock unit under the action of external force conforms to the first law of thermodynamics [29; 30; 31; 32; 33; 34], that is,  
\[U=U_{e}+U_{d}, \tag{1}\]  
where \(U\) is the total input energy generated by the external force of the unit, \(U_{e}\) is the total elastic energy that the unit can release, and \(U_{d}\) is the total dissipated energy of the unit. In a true triaxial axial graded cyclic loading test, the energy input of rock mass can be divided into two parts: axial input energy and circumferential input energy [25; 35]. Therefore, the total energy input per unit volume of external force can be defined as  
\[U=U_{1}+U_{2}+U_{3}, \tag{2}\]  
where \(U_{1}\), \(U_{2}\), and \(U_{3}\) are the energy inputted by the maximum, medium, and minimum principal stresses, respectively, to deform the specimen. In a graded cyclic loading test, each level of stress has corresponding upper and lower limits for carrying out regular cyclic loading and unloading. The hysteresis curve during the cycle is often not closed due to the existence of rock damage [36; 37]. In this study, the relationship between axial input energy and dissipated energy was illustrated by the \(i\)-cyclic stress-strain hysteresis curve during cyclic loading and unloading. As shown in Figure 9a, the area surrounded by ABB\({}^{\prime}\)A\({}^{\prime}\) is the energy \(u_{1,i}\) inputted by the axial stress to the sandstone during the \(i\)-cycle loading process, the area surrounded by CBB\({}^{\prime}\)C\({}^{\prime}\) is the energy \(u_{1,i}^{\epsilon}\) released by the sandstone during the \(i\)-cycle unloading process, and the area surrounded by ABCC\({}^{\prime}\)A\({}^{\prime}\) is the energy \(u_{1,i}^{d}\) dissipated by the sandstone during the \(i\)-cycle unloading process. Following the definition of calculus, the maximum principal stress input total energy \(U_{1}\) of the loading part during the whole cycle, the total energy released by the unloading part, and the total energy \(U_{1u}\) dissipated during the cycle can be obtained as follows:  
\[U_{1}=\sum_{i=1}^{z}\int_{\varepsilon_{1,i}}^{\varepsilon_{1,i}^{\prime}}\sigma_{ 1}d_{\varepsilon_{1}}, \tag{3}\]  
\[U_{1e}=\sum_{i=1}^{z}\int_{\varepsilon_{1,i}^{\prime}}^{\varepsilon_{1,i}^{ \prime}}\sigma_{1}d_{\varepsilon_{1}}, \tag{4}\]  
\[U_{1d}=U_{1}-U_{1e}, \tag{5}\]  
where \(\sigma_{1}\) and \(\varepsilon_{1}\) are axial stress and strain, respectively; \(\varepsilon_{1,i}\) is the axial strain corresponding to the starting point of \(i\)-cyclic loading; \(\varepsilon_{1,i}^{\prime}\) is the axial strain corresponding to the upper limit of \(i\)-cyclic loading; \(\varepsilon_{1,i}^{\prime\prime}\) is the axial strain corresponding to the lower limit of \(i\)-cyclic unloading; and \(z\) (\(z=1,2,\cdots\)) is the maximum number of cycles. As shown in Figure 9b, the intermediate principal stress and the minimum principal stress remain constant throughout the cycle. The area surrounded by ABB\({}^{\prime}\)A\({}^{\prime}\) is the energy \(u_{j,i}\) inputted by the radial stress to the sandstone during the \(i\)-cycle loading process. The area surrounded by CBB\({}^{\prime}\)C\({}^{\prime}\) is the energy (\(u_{j,i}^{e}\)) released by the sandstone in the radial direction during the \(i\)-cycle unloading process. The area surrounded by ACC\({}^{\prime}\)A\({}^{\prime}\) is the radial dissipated energy \(u_{j,i}^{d}\) of sandstone during the \(i\)-cyclic loading and unloading process.