## 2 Construction of a Numerical Model of Particle Flow  
### Introduction to Particle Flow Code Simulation Methods  
Particle Flow Code 5.0 (PFC 5.0) is an essential discrete element numerical simulation software developed by ITASCA, Itasca, IL, USA [23]. It has been extensively utilized to study fundamental problems such as rock-like materials' basic mechanical properties, particulate matter's dynamic responses, and fracture development. The macroscopic mechanical behavior of materials in PFC is determined by the type of contact between particles and their variation characteristics.  
The Linear model, as implemented in the PFC, represents the fundamental contact model in rock mechanics. It characterizes the mechanical behavior of contact surfaces through parallel spring and damper elements. However, it should be noted that the Linear model solely facilitates the transfer of forces (\(F\)) and does not account for moments (\(M\)). The Linear-bond model (Linear Bond Model), which is one of the common granular models in PFC, was developed within the PFC framework to address this limitation. By incorporating bonding capabilities into the Linear model, the Linear-bond model enables the transmission of both forces (\(F\)) and moments (\(M\)), as depicted in Figure 2. In this model, the bonding elements and spring elements are parallel to each other, and the maximum normal stress (\(\sigma_{max}\)) and shear stress (\(\tau_{max}\)) around the parallel bond in the two-dimensional plane are defined as [24]:  
\[\sigma_{\max}\gets F_{n}/A+M_{b}\overline{R}/I \tag{1}\]  
\[\tau_{\max}\gets F_{s}/A \tag{2}\]  
\[F_{n}\gets F_{n}+k_{n}A\Delta\delta_{n} \tag{3}\]  
\[F_{s}\gets F_{s}+k_{s}A\Delta\delta_{s} \tag{4}\]  
\[M_{b}\gets M_{b}+k_{b}\Delta\theta_{b} \tag{5}\]  
\[\overline{R}=\overline{\lambda}\min(R_{1},R_{2}) \tag{6}\]  
\[I=\frac{2}{3}\overline{\lambda}^{3} \tag{7}\]  
where A is the area, \(F_{n}\) is the normal force, \(F_{s}\) is the shear force, \(M_{b}\) is the bending moment, \(\overline{R}\) is the bond radius, \(I\) is the moment of inertia of the parallel bond cross-section, \(k_{n}\), and \(k_{s}\) are the normal and shear stiffnesses, respectively, \(\Delta\delta_{b}\) and \(\Delta\delta_{s}\) are the displacement increments in the normal and tangential directions respectively, \(R_{1}\) and \(R_{2}\) are the particle radius, \(\Delta\theta_{b}\) is the contact angle, and \(\overline{\lambda}\) is the radius factor.  
The strength properties of the parallel bond adhere to the Moore-Coulomb criterion. When the forces (\(F\)) or moments (\(M\)) exerted on the contact surfaces attain their maximum strength, the bond undergoes failure, causing the Linearpbond model to transition from a bonded state to an unbonded state. Notably, in its unbonded state, the Linearpbond model aligns completely with the Linear model [25]. The linear parallel bond contact model represents the parallel bond as a bonding material characterized by constant stiffness (both normal and shear) and strength between two particles. This model is commonly utilized to simulate the discontinuous deformation of rock formations. Previous studies conducted on rock materials using the Particle Flow Code (PFC) have demonstrated that employing a linear parallel bond contact model between particles yields a more accurate representation of rocks' mechanical properties and particulate media's dynamic response [26; 27]. Consequently, this paper adopts the linear parallel bond contact model for the numerical simulation of particle flow.