## 2 Deformation Rate Threshold-Based Solution for Slope Safety Factor  
### Fundamentals of the Discrete Element-Intensity Discounting Method  
Methods such as the finite element method and the finite difference method are based on the assumption of a continuous medium. Therefore, they can only deal with a small number of discontinuous surfaces, while the engineering rock mass is rich in discontinuous surfaces, such as joints and fissures, and its deformation and damage are subject to these discontinuous surfaces, which have significant discontinuity characteristics. For these reasons, its mechanical behavior is more suitable for simulation by discontinuous medium methods, such as discrete element and DDA [21; 22]. Therefore, this paper investigatesthe solution method for the safety factor of rocky slopes based on the three-dimensional discrete element method 3 DEC. The discrete unit method [23] is a numerical calculation method proposed by [PERSON] in 1971, which is a powerful tool for analyzing the deformation and motion problems of discrete media, and is also widely used in the field of geotechnical engineering. The method regards the jointed rock body as a discrete medium system composed of rock blocks and jointed surfaces, which allows the rock blocks to translate, rotate and deform, while the jointed surfaces can slip, separate or flip, so that the discontinuous large deformation and damage behavior of the jointed rock body can be simulated. In the discrete cell method, the motion of a single block follows [PERSON]'s second law of motion, and the equations of motion of the block can be obtained based on the combined force transmitted to the block by neighboring blocks through the nodal plane, combined with its deformation intrinsic relationship:  
\[m\frac{\mathrm{d}^{2}u(t)}{\mathrm{d}t^{2}}+c\frac{\mathrm{d}u(t)}{\mathrm{d }t}+ku(t)=f(t) \tag{1}\]  
Within this formula: \(m\) is the mass of the block, \(u\) is displacement, \(t\) is time, \(c\) is viscous damping, \(k\) is the stiffness coefficient and \(f\) is the external force applied to the block. The interaction between blocks is described by the force-displacement relationship, which is usually assumed to be proportional, i.e.,  
\[F_{\mathrm{n}}=K_{\mathrm{n}}\times U_{\mathrm{n}} \tag{2}\]  
\[F_{\mathrm{s}}=K_{\mathrm{s}}\times U_{\mathrm{s}} \tag{3}\]  
Within this formula: \(F_{\mathrm{n}}\) and \(F_{\mathrm{s}}\) are the normal and tangential components of the contact force, \(K_{\mathrm{n}}\) and \(K_{\mathrm{s}}\) are the normal and tangential stiffnesses, and \(U_{\mathrm{n}}\) and \(U_{\mathrm{s}}\) are the normal and tangential components of the contact displacement, respectively. For engineered rock masses, the force-displacement relationship between blocks is characterized by nonlinearity, which is usually described by the Coulomb slip model, i.e., the normal direction is not subject to tension, and the tangential direction can generate slip:  
\[F_{\mathrm{n}}= \begin{cases}K_{\mathrm{n}}\times U_{\mathrm{n}},&U_{\mathrm{n }}\leq 0\\ 0,&U_{\mathrm{n}}>0\end{cases}  \tag{4}\]  
\[F_{\mathrm{s}}= \begin{cases}K_{\mathrm{s}}*U_{\mathrm{s}},\ |F_{\mathrm{s}}| \leq|F_{\mathrm{n}}|\tan\varphi+cA\\ sign(U_{\mathrm{s}})(|F_{\mathrm{n}}|\tan\varphi+cA),|F_{\mathrm{s}}|>|F_{ \mathrm{n}}|\tan\varphi+cA\end{cases}  \tag{5}\]  
Within this formula: \(c\) is the cohesive force on the contact surface, \(\varphi\) is the friction angle, and \(A\) is the contact area. The discrete elements are calculated using the strength discount method to obtain the factor of safety of the slope. The basic principle of the strength discount method is to divide the strength parameters of the slope's geotechnical body (the rock mass and joint surface) by a discount factor at the same time to obtain a new set of strength parameters, which are then input as the new material parameters and then calculated. When the slope is in a critical state, the corresponding discount factor is defined as the slope's safety factor.