# Research on Slope Stability Based on Bayesian Gaussian Mixture Model and Random Reduction Method
## 2 Construction of Slope Instability Criteria Using Bayesian Gaussian Mixture Model  
### Energy Dissipation Mutation Criterion  
During the progressive failure of slopes, the energy field within the slope system and its surrounding environment remains in a state of dynamic equilibrium [15]. Studying slope failure from the perspective of energy evolution allows us to bypass complex stress-strainanalyses and avoid tracking detailed damage states of geomaterials at each intermediate step. Instead, the focus shifts to the pathways of energy input, storage, transformation, and dissipation. In slope systems, part of the externally applied energy is stored as elastic strain energy within the material, whilst the remaining energy is gradually dissipated as damage-induced energy through deformation and damage processes [16; 17]. According to the first law of thermodynamics, this process can be expressed as follows:  
\[U=U^{d}+U^{e} \tag{1}\]  
Under static equilibrium, the energy input is primarily driven by the self-weight of the slope. Thus, Equation (1) can be rewritten as follows:  
\[U^{g}=U^{d}+U^{e} \tag{2}\]  
where \(U\) represents the total energy input; \(U^{d}\) denotes the dissipated energy; \(U^{e}\) is the elastic strain energy; and \(U^{g}\) stands for the gravitational potential energy. Both the storage of elastic strain energy and the dissipation of energy through deformation in a slope have inherent capacity limits. When the total energy input from external sources exceeds the slope's energy-bearing threshold, the surplus energy that cannot be stored or dissipated rapidly transforms into kinetic energy. At this point, the slope reaches a state of limit equilibrium, triggering sliding and leading to failure. By simplifying slope stability analysis to the judgment of energy evolution patterns, the critical conditions for slope instability can be identified more efficiently and accurately. The energy evolution process of the slope can then be expressed by the following equation:  
\[U^{g}=U^{d}+U^{e}+U^{k} \tag{3}\]  
where \(U^{k}\) is the kinetic energy. Taking any volumetric element within the slope system as the research object, the gravitational potential energy \(U^{g}\), dissipated energy \(U^{d}\) elastic strain energy, \(U^{e}\) and kinetic energy \(U^{k}\) can be calculated using the following formulas:  
\[U^{g}=\int_{V}\rho ghdV \tag{4}\]  
\[U^{d}=\int_{v}\sigma:de^{p} \tag{5}\]  
\[U^{e}=\int_{V}\frac{1}{2}\big{(}\sigma_{x}\varepsilon_{x}+\sigma_{y}\varepsilon _{y}+\sigma_{z}\varepsilon_{z}\big{)}dV= \tag{6}\]  
\[U^{k}=\int_{V}\frac{1}{2}\rho v^{2}dV \tag{7}\]  
where \(\rho\) represents the density; \(g\) is the gravitational acceleration; \(h\) is the height of the element's centroid; \(V\) is the volume of the element; \(\sigma\) is the stress tensor; \"\(\cdot\)\" is the inner product operation; \(\varepsilon^{p}\) is the plastic strain tensor; \(\sigma_{x}\), \(\sigma_{y}\), and \(\sigma_{z}\) are the principal stresses in the x, y, and z directions, respectively; \(\varepsilon_{x}\), \(\varepsilon_{y}\), and \(\varepsilon_{z}\) are the principal strains in the x, y, and z directions, respectively; \(E\) is the elastic modulus of the material; \(v\) is the Poisson's ratio of the material; and \(v\) is the velocity at the element's centroid. The energy dissipation process of slopes exhibits significant differences between unstable and stable states. During the stable phase, the dissipated energy typically follows a relatively smooth trend, with fluctuations within a certain range when influenced by external factors. However, as the slope approaches the critical state of instability, the internal structure of the rock and soil undergoes drastic changes, such as rapid crack propagation and imbalances in particle interactions, causing the energy dissipation to deviate from its original smooth pattern and exhibit abrupt changes [18].