## 1 Introduction
### Numerical Methods in Structural Analysis  
One of the earliest and most widely used numerical analysis techniques in engineering is the finite element method (FEM). However, its effectiveness decreases when addressing problems with cracking, such as granular materials and joint rocks. To address the limitations of FEM in handling discontinuities and to incorporate fracturing phenomena, [PERSON] et al. [37] developed the finite-discrete element method (FDEM), which addresses these issues. This approach integrates the FEM with the discrete element method (DEM).  
Furthermore, this method considers fracturing through the cohesive crack elements (CCEs) embedded within the finite element mesh to accurately model fracture processes in solids. A potential function is used to calculate the nodal forces generated by the interaction of overlapping discrete elements, which is a crucial feature of FDEM in its contact interaction algorithm [38].  
This method enables the transition from a continuous to a discontinuous domain as fractures initiate and propagate. The intact material begins with a distributed micro-crack zone at the crack tip called a fracture process zone (FPZ), representing the initial damage (Figure 1a). This process then progresses to a bridging zone, where interlocking phenomena help distribute stresses across the cracks (Figure 1b). Finally, it transitions to a traction-free macro-crack zone, where the material completely separates. This sequence is marked by the Mode I crack walls beginning to separate between elements once the bond stress threshold (\(f_{t}\)) is reached (Figure 1c). As the separation grows, the bonding stress between the elements decreases accordingly, ultimately dropping to zero once the separation surpasses a critical threshold (\(O_{r}\)). Propagation under mode II fracture initiation occurs when the tangential slip at an integration point, denoted as s, attains a critical threshold \(s_{p}\), which aligns with the element's intrinsic shear strength \(f_{\text{s}}\) (Figure 1d). Lastly, crack elements in the FDEM can undergo mixed-mode fracture behavior (Figure 1) when Mode I and Mode II displacements interact (Figure 1e). The model continuously updates the stress and displacement fields to reflect the crack opening and the relative sliding of the faces.  
We chose the FDEM model to study the capacity and fracturing behavior because it has been successfully validated, including the accurate replication of crack patterns observed in various tests [39; 40; 41]. Furthermore, FDEM effectively simulates the dynamic behavior and fracturing of CCEs by applying energy conservation principles. This principle involves the use of fracture energy (\(G_{c}\)), denoted as the energy necessary for a crack to propagate; frictional dissipation (\(E_{f}\)), representing the energy loss due to internal frictional forces; and kinetic energy (\(E_{k}\)), which accounts for the energy related to the motion of the elements. These energy components enable FDEM to model the fracturing process comprehensively,  
Figure 1: Material failure modeling in FDEM, (**a**) conceptual progression within the FPZ, (**b**) transition from a continuous to a discontinuous domain with elements representing different stages, (**c**) Mode I failure, (**d**) Mode II slip weakening approach, and (**e**) mix—mode failure interaction.  
capturing the intricate interactions between the material properties, energy dissipation, and the dynamic changes within the system [42; 43; 44].