## 2 Model Establishment and Parameter Selection  
### Model Composition and Main Principles  
In PFC2D, several constitutive contact models are commonly employed, including the Linear Model (LM), the Linear Contact Bond Model (LCBM), and the Linear Parallel Bond Model (LPBM) [18]. Rock is a heterogeneous composite material, and consists of discrete mineral particles bonded by cementing agents. In contrast to the Linear Model and Linear Contact Bond Model, which are typically used for simulating soil mechanical behavior, the Linear Parallel Bond Model is more commonly adopted to accurately capture the mechanical response and micro-structural evolution of rock materials [19; 20; 21]. The Linear Parallel Bond Model (LPBM) can transmit both normal and shear contact forces, \(F_{c}\), as well as bending and twisting moments, \(M_{c}\), thereby enabling a more realistic representation of particle interactions and their resultant mechanical behavior (Equation (1)) [22; 23]. Therefore, employing the LPBM enables more accurate and reliable simulations of the mechanical response of rocks subjected to external loads.  
\[F_{c}=F^{l}+F^{d}+\overline{F},M_{c}=\overline{M} \tag{1}\]  
(where \(F^{l}\) is the linear force, \(F^{d}\) the damping force, \(\overline{F}\) is the parallel bond force, and \(\overline{M}\) is the moment generated by the bonding load).  
As shown in Figure 1, the Linear Parallel Bond Model (LPBM) comprises a frictional interface and a linear elastic bonding interface. The frictional interface mainly resists frictional forces between particles and transmits external loads. The bonding interface forms a bond between particles, capable of resisting the action of both forces and moments. When the tensile force exceeds the tensile strength or the shear stress exceeds the shear strength, a bond-break callback event is triggered, resetting both the contact force and moment to zero. This results in bond rupture and the formation of mesoscale cracks. With the accumulation of such bond failures, macroscopic cracks eventually develop [24; 25].  
As illustrated in Figure 1, the Linear Parallel Bond Model (LPBM) was adopted as the contact constitutive model to construct a two-dimensional standard core sample. The numerical model has a height of 100 mm and a diameter of 50 mm, with the minimum particle radius set to 0.2 mm. Particles were randomly generated at a porosity of 20% to simulate the actual structure of the rock. To simulate the compressive stress conditions characteristic of subsurface rock formation, the wall command was utilized to confine and compact the particles.  
After shaping and compaction, the CAMT command was employed to assign Linear Parallel Bond Model (LPBM) parameters to particle contacts, thereby achieving particle cementation and completing the rock modeling. To simulate the axial loading conditions of a rock mechanics testing apparatus, the left and right boundary walls were removed, leaving only the top and bottom boundaries. The movement of the top and bottom walls was precisely controlled using a servo mechanism, and a strain-controlled loading mode was applied with a strain rate set to 0.05%. According to these criteria, four types of uniaxial compression specimens with different particle size ratios were generated.