## 4 Analysis of Energy Evolution Characteristics of Triaxial Unloading Confining Sandstone  
### Energy Calculation Method  
Under the action of the mechanical testing machine, the energy inside the rock is in different states; in the initial loading stage, the mechanical energy and heat energy are input into the rock, the rock undergoes elastic deformation, and the energy can be stored in the rock. The whole process is dynamically balanced. As the loading goes on, the deformation capacity of rock becomes worse, and the elastic energy that can be stored becomes less. During this period, external input energy causes new micro-cracks in the rock. Although the energy at this stage is in dynamic equilibrium, it is extremely unstable. When the external load is too large, the input energy causes the internal microscopic cracks of the rock to be connected, and the macroscopic cracks begin to appear. When the energy accumulated in the rock exceeds its energy storage limit, the energy is released instantaneously, resulting in rock damage, and the dynamic image of rock block flying may be caused. When the energy is released, the rock returns to a new equilibrium state [20; 21; 22].  
In the process of rock specimens being loaded, energy storage and dissipation are always present. The whole energy evolution process can be expressed in terms of elastic energy and dissipative energy. In laboratory experiments, these variables are not easily obtained directly and can be calculated from real-time recorded stress-strain curves [23; 24; 25; 26]. Figure 5 shows the calculation principle of energy. Taking the peak stress point as an example, Eu is the unloading modulus, elastic energy \(U^{e}\) is the shadow area, and dissipated energy \(U^{d}\) is the area enclosed by the stress-strain curve, unloading modulus, and transverse axis.  
According to the first law of thermodynamics, the energy \(U\) generated by the external forces is as follows [20; 23; 24; 25; 26]:  
\[U=U^{\rm e}+U^{\rm d}, \tag{6}\]  
where \(U^{e}\) is elastic energy, and \(U^{d}\) is dissipated energy.  
It is assumed that the system is a closed and the heat transfer during the test is not taken into account, whereby all the mechanical energy generated by the testing machine enters the rock sample, i.e., \(U=U_{0}\). In the triaxial compression test, the axial absorption energy \(U_{1}\) is positive, and the circumferential consumption energy \(U_{3}\) is negative. The rock absorption energy \(U_{0}\) is equal to the sum of axial absorption energy \(U_{1}\) and circumferential consumption energy \(U_{3}\), i.e.,  
\[U_{0}=U_{1}+U_{3}. \tag{7}\]  
Figure 5: Calculation principle of energy.  
The axial absorbed energy \(U_{1}\) and circumferential consumption energy \(U_{3}\) of rock can be represented as follows [18,20]:  
\[U_{1}=\int\sigma_{1}\mathrm{d}\varepsilon_{1}=\sum_{i=0}^{n}\frac{1}{2}( \varepsilon_{1i+1}-\varepsilon_{1i})(\sigma_{1i}+\sigma_{1i+1}) \tag{8}\]  
\[U_{3}=2\int\sigma_{3}\mathrm{d}\frac{\varepsilon_{3}}{2}=\sum_{i=0}^{n}\frac{1 }{2}(\varepsilon_{3i+1}-\varepsilon_{3i})(\sigma_{3i}+\sigma_{3i+1}) \tag{9}\]  
where \(\sigma_{1i}\), \(\varepsilon_{1i}\), \(\sigma_{3i}\), and \(\varepsilon_{3i}\) are the axial stress, axial strain, circumferential stress, and circumferential strain, respectively.  
The elastic energy of triaxial compression can be expressed as follows [18,19]:  
\[U^{\mathrm{e}}=\frac{1}{2E^{\mathrm{u}}}[\sigma_{1}{}^{2}+2\sigma_{3}{}^{2}-2 \mu(2\sigma_{1}\sigma_{3}+\sigma_{2}\sigma_{3})] \tag{10}\]  
where \(E^{\mathrm{u}}\) is the elastic modulus when unloading, and \(\mu\) is 50-60% of the peak strength [19,20,25].