## 2 Numerical Simulation Method  
RFPA\({}^{\rm 3D}\) is a widely used numerical simulation software in rock mechanics and geotechnical engineering. It is capable of effectively simulating the fracture mechanism, stress-strain behavior, and failure evolution of rock materials, with its computational process shown in Figure 1. Built on the elastic damage theory and the modified Coulomb failure criterion, RFPA\({}^{\rm 3D}\) constructs a deformation and failure analysis module for rock media. With its robust computational and visualization capabilities, the software enables the dynamic simulation of the entire rock fracture process, including crack initiation, propagation, coalescence, and penetration. Moreover, RFPA\({}^{\rm 3D}\) supports various loading conditions, such as uniaxial compression, triaxial compression, tension, and shear, making it applicable to large-scale engineering challenges, including slope instability prediction, tunnel failure analysis, mining-induced ground pressure disasters, and hydraulic fracturing in complex geological environments. During the simulation, RFPA\({}^{\rm 3D}\) can also capture acoustic emission signals under stress, effectively reproducing the acoustic emission characteristics observed in laboratory experiments. This allows for the inference of crack propagation paths and rock damage evolution. Furthermore, the software provides an intuitive visualization of the spatial distribution, propagation direction, and interaction of internal cracks, offering valuable insights into the fracture mechanisms of rock materials. Rock is a highly heterogeneous material, and its inherent defects lead to a non-uniform distribution of internal strength, making its failure process somewhat random [29; 30]. Therefore, it is essential to account for the microscopic composition of the rock, the reduction in heterogeneity, and the determination of the failure criterion in numerical methods [31; 32; 33]. RFPA\({}^{\rm 3D}\) introduces the representative volume element (RVE) as the fundamental unit to address this. The size and mechanical properties of the microscopic unit RVE are spatially homogeneous [34; 35]. To capture the randomness of defects and the mineral composition, the Weibull distribution function is employed to define the distribution characteristics of RVE strength parameters, simulating the heterogeneity of rock materials [35]:  
\[\varphi(\alpha)=\frac{m}{\alpha_{0}}(\frac{\alpha}{\alpha_{0}})^{m-1}e^{-(\frac {\alpha}{\alpha_{0}})^{m}} \tag{1}\]  
where \(\alpha\) represents the strength or elastic modulus of the unit, \(\alpha_{0}\) is the average value, and \(m\) is the homogeneity coefficient, which reflects the uniformity of the material properties. The larger the value of \(m\), the less discrete the material parameters, leading to a more uniform distribution. RFPA3D treats the nonlinear deformation of rock materials as a process where the mechanical properties degrade after RVE damage until being completely destroyed. When the RVE experiences stress that satisfies the relevant failure criteria, it evolves according to tensile and shear constitutive equations. The primitive damage variable \(D\) can be defined as follows [35]:  
\[D=1-\frac{E}{E_{0}} \tag{2}\]  
where \(E_{0}\) is the initial elastic modulus, and \(E\) is the elastic modulus after damage. The damage variable \(D\) ranges from 0 to 1: when \(D\) = 1, the RVE is ituly damaged; and when \(D\) is between 0 and 1, the RVE is weakening. Figure 1: RFPA3D calculation process diagram. In RFPA[30], the damage is categorized into tensile and shear damage. When the uniaxial tensile stress reaches the RVE tensile strength (\(\sigma\leq-f_{0}\)), the maximum tensile strain criterion is applied to initiate damage evolution [35]:  
\[D=\left\{ \begin{array}{cl}0\,,&\varepsilon>\varepsilon_{t0}\\ 1-\frac{\lambda\varepsilon_{t0}}{\varepsilon}\,,&\varepsilon_{tu}\leq \varepsilon\leq\varepsilon_{t0}\\ 1\,,&\varepsilon\leq\varepsilon_{tu}\end{array}