## 2 General and Theoretical Aspects Regarding Slope Stability  
A slope represents a flat surface inclined at a certain angle \(\alpha\) relative to the horizontal, ensuring the connection between two different elevations and bordering an earth mass. The design of slopes involves determining their optimal inclination to ensure the stability of the earth mass. Too steep slope angles may lead to instability, while overly gentle slopes represent uneconomical solutions (requiring large volumes of excavation and occupying significant land areas). The classical methods used in geotechnical practice for analyzing the general stability of sloped earth masses evaluate the static equilibrium of an earth mass, which is limited at the bottom by a slip surface and at the top by the ground surface. This earth mass tends to move under the influence of gravitational forces. Plane sections are analyzed, and the conditions of a plane strain state are considered. More recently, the method of strength reduction combined with the finite element method has been increasingly applied to solve geotechnical slope stability problems. This method proves to be a useful tool for determining the factor of safety (FoS), which can be applied to evaluating the stability of slopes, natural hillsides, specific embankment slopes, open-pit steps, and waste dumps. Besides the slope's geometry, stability is influenced by the physical and mechanical characteristics of the material (soil or rock) making up the earth mass. In the strength reduction method, the material's characteristic properties are progressively reduced until failure occurs. The FoS is considered to be an indicator of stability, which, in the context of geotechnical problems, is defined as the ratio between the limit (failure) value of the material's strength parameter (typically shear stress) and its actual value under given geometric and loading conditions. The strength reduction method is applicable primarily to linear failure criteria such as the Mohr-Coulomb criterion. When using the Mohr-Coulomb criterion with this method, the values of cohesion, the internal friction angle, and the dilatation angle are reduced simultaneously until mechanical equilibrium is lost. As the material parameters decrease, the soil's shear strength reduces, eventually leading to slope instability. This phenomenon causes the slope to slide under a specific combination of loads, material properties, and boundary conditions. The ratio between the initial cohesion and the cohesion at failure determines the FoS. The COMSOL software package is suitable for solving slope stability problems as it includes modules for implementing models in terms of geometry, material properties, failure (yielding) assumptions, and the necessary computational algorithms for applying the strength reduction method. The material properties for applying the Mohr-Coulomb model are expressed in relation to the FoS. Thus, in a parametric study the soil's strength characteristics are reduced by gradually increasing the FoS with each iteration step. The actual FoS is reached when the model no longer converges, signifying the point at which slope failure (sliding) occurs. The yield function \(F\) and plastic potential Q for the Mohr-Coulomb failure hypothesis are given by the relations:  
\[F=m\sqrt{J_{2}}+\frac{\sin\Phi}{3}I_{1}-C\cos\Phi \tag{1}\]  
\[Q=m_{q}\sqrt{J_{2}}+\frac{\sin\Psi}{3}I_{1}-C\cos\Psi \tag{2}\]  
where \(I_{1}\) and \(J_{2}\) are the first and second invariants of stress. The parametrized material properties (\(\Phi\)--parametrized internal friction angle; C--parametrized cohesion; and \(\Psi\)--parameterized angle of dilatation) are expressed as:  
\[C=\frac{c}{\text{FoS}},\ \Phi=\arctan\left(\frac{\tan\phi}{\text{FoS}}\right),\ \Psi=\left(\frac{\tan\Psi}{\text{FoS}}\right) \tag{3}\]  
where \(c\) is the cohesion, \(\phi\) is the internal friction angle, and \(\psi\) is the angle of dilatation; these are the unreduced initial parameters of the material.