## 2 Slope Excavation Energy Discontinuity Criterion  
### Principle of Conservation of Energy for Slope Destabilization  
The slope system undergoes deformation and eventually becomes unstable due to various external environmental factors such as forces, displacement, and temperature [35]. During this process, the slope system remains in a dynamic equilibrium energy field with the surrounding environment. External factors, including force, displacement, and temperature, constrain the slope body, involving the input, transfer, and transformation of energy [36; 37]. The energy conservation equation for the slope system is given in Equation (1). \[E=\Delta E^{\mathrm{e}}+\Delta E^{\mathrm{d}}+\Delta E^{\mathrm{k}} \tag{1}\]  
In the equation, \(E\) represents the total work done by the slope system, while \(\Delta E^{\mathrm{e}}\), \(\Delta E^{\mathrm{d}}\), and \(\Delta E^{\mathrm{k}}\) represent the elastic-plastic strain energy, dissipated energy, and kinetic energy of the slope system, respectively. The dissipated energy includes various forms such as apparent energy, thermal energy, and radiative energy. When performing calculations using the finite difference method, the influence of environmental temperature and radiative energy on the slope is often ignored. Therefore, the total energy of the system is equal to the work done by the self-weight of the slope, i.e., \(E=\Delta E^{\mathrm{g}}\). The energy conservation equation for the slope system is as shown in Equation (2). \[\Delta E^{\mathrm{g}}=\Delta E^{\mathrm{e}}+\Delta E^{\mathrm{d}}+\Delta E^{ \mathrm{k}} \tag{2}\]  
In the equation, \(\Delta E^{\mathrm{g}}\) represents the change in gravitational potential energy, which is the difference in gravitational potential energy of the slope after sliding compared to its initial state. \(\Delta E^{\mathrm{e}}\), \(\Delta E^{\mathrm{d}}\), and \(\Delta E^{\mathrm{k}}\) represent the \"initial\" and \"final\" differences from the start of self-equilibrium to the stabilization end. At the beginning, there is no kinetic energy and dissipated energy (i.e., initial values are 0). This can be expressed as shown in Equation (3). \[ \begin{array}{l}\Delta E^{\mathrm{e}}=\int_{x_{0}}e^{\mathrm{d}}dV-\int_{y_{ \mathrm{t}}}e^{\mathrm{e}}dV\\ \Delta E^{\mathrm{g}}=\int_{y_{\mathrm{t}}}e^{\mathrm{g}}dV-\int_{y_{\mathrm{t} }}e^{\mathrm{g}}dV\\ \Delta E^{\mathrm{k}}=\int_{y_{\mathrm{t}}}e^{\mathrm{k}}dV-\int_{y_{\mathrm{t} }}e^{\mathrm{g}}dV\end{array}  \tag{3}\]  
By substituting Equation (3) into Equation (2), the total change in dissipated energy \(\Delta E^{\mathrm{d}}\) of the slope system at any moment \(t\) during sliding deformation can be obtained as shown in Equation (4). \[\Delta E^{\mathrm{d}}=\left(\int_{y_{0}}e^{\mathrm{g}}dV-\int_{y_{\mathrm{t} }}e^{\mathrm{g}}dV\right)-\left(\int_{y_{\mathrm{t}}}e^{\mathrm{e}}dV-\int_{y _{0}}e^{\mathrm{e}}dV\right)-\left(\int_{y_{\mathrm{t}}}e^{\mathrm{e}}dV-\int_ {y_{0}}e^{\mathrm{e}}dV\right) \tag{4}\]In the equation, \(v_{0}\) and \(v_{t}\) represent the total volume of the slope system at the initial state and at any moment during the sliding deformation process, respectively. \(e^{\prime}\), \(e^{\prime}\), \(e^{k}\) represent the elastic strain energy density, gravitational potential energy density, and kinetic energy density of any unit volume within the slope body, respectively. \[ \begin{array}{l}E^{\rm e}=\frac{1}{2}\delta_{1}\xi_{1}^{\rm e}+\frac{1}{2} \delta_{2}\xi_{2}^{\rm e}+\frac{1}{2}\delta_{3}\xi_{3}^{\rm e}\\ \xi_{1}^{\rm e}=\frac{1}{V_{1}}[\sigma_{i}-{\rm p}_{i}(\sigma_{i})+\sigma_{k} )](i,j,k=1,2,3)\\ \par e^{\rm g}=pghV_{i}\\ e^{k}=\frac{1}{2}pv^{2}V_{i}\end{array}  \tag{5}\]  
In the equation, \(\sigma_{1}\), \(\sigma_{2}\), \(\sigma_{3}\) represent the magnitudes of the principal stresses in the three directions of the unit volume. \(\xi_{1}^{\rm e}\) (i = 1, 2, 3) represents the elastic strain in the different principal stress directions.