## 1 Introduction  
Coarse-grained soils exhibit good compaction performance, high strength, strong permeability, and high bearing capacity. It is generally acknowledged that the mechanical behavior of coarse-grained soils is very complex. The crushing mechanism of coarse-grained soils and their behavior under high stress has garnered the attention of scholars. It is important to thoroughly understand the phenomenon of particle breakage and comprehend its effect on engineering properties [1]. A large number of experiments and numerical studies have been carried out on the crushing and mechanical behavior of coarse-grained soil particles. However, the mechanism of particle breakage needs further study. At present, the main methods used to experimentally study particle breakage include the triaxial test [2; 3; 4; 5; 6; 7; 8; 9; 10], the shear test [11; 12; 13; 14; 15], the uniaxial compression test [8; 9; 16], the particle contact test [17; 18] and the centrifuge modeling test [19]. However, previous studies have primarily focused on the behavior of soil aggregates. The physical and mechanical properties of particle breakage have been examined from the perspective of the average stress of the aggregate, but the mechanical behavior of a single particle has rarely been analyzed. The failure processes and fracture patterns of single particle breakage have been analyzed in recent years. However, it is still difficult to observe the evolution of crushing or reveal the basic mechanism of particle breakage through a conventional experiment. Numerical simulation provides a new method for studying particle breakage from the microscopic perspective. Currently, many numerical methods have been used to simulate particle breakage, such as the discrete element method (DEM) [18; 20; 21; 22; 23; 24; 25; 26; 27], the combined finite discrete element method (CFDEM) [28; 29; 30], the extended finite element method (XFEM) [31], and the coupling of the discrete element method and the scaled boundary finite element method (DEM-SBFEM) [32]. Among them, the DEM and the CFDEM are the most widely used methods in particle breakage simulations. After many years of development, the DEM-based particle breakage simulation has been widely used and has obtained many achievements, providing an effective way to study the mechanism of particle breakage from the microscopic perspective. The CFDEM-based particle breakage simulation has the advantages of considering the complex shape of particles, simulating the process of crack propagation in the particles, and obtaining the field distributions of internal stress and deformation. However, the CFDEM requires the generation of a grid for each particle, which is computationally expensive. Moreover, the degree of particle breakage is related to grid density. The denser the grid is, the better the particle breakage simulation. Current studies using numerical methods have mainly focused on the effect of microparameters on mechanical behaviors. More attention is needed to elucidate the mechanism of particle breakage. The DEM was first introduced by [PERSON] and [PERSON] [33], and it has been widely recognized as a powerful tool to study the mechanical behavior of granular materials [34]. Using the bonded particle model (BPM), the DEM can now simulate specimens of any shape. There are two essential ways to simulate particle breakage using the DEM. The first is to form aggregates by bonding a certain number of elementary particles, which simulates the formation of cracks by breaking the bonds between elementary particles. This is called a BPM-based simulation. The other uses the fragment replacement method (FRM) to simulate particle breakage. By the time the particle contact force exceeds the preset breaking criteria, the to-break particle will be replaced by a set of unbonded elementary particles. The BPM-based simulation can simulate complex particle breakage problems without the requirement of a preset criterion, which is more suitable for the particle breakage study of small-scale aggregates.