## 3 Numerical Model and Calculation Scheme  
### Overview of PFC Simulation Methods  
The PFC particle flow code is a discrete element method. Its basic unit is granular particles, which generate displacement and force through the extrusion contact between particles and the movement of particles. Particle flow was first mainly used in the analysis of granular materials; with the deepening of people's research, granular materials can not meet people's research on solid materials, where the introduction of bonding properties between the particles (including parallel bonding and contact bonding) to achieve similar mechanical properties of solid materials. At the same time, particles can generate a variety of shapes to reach the basic unit of rock mass, similar to the actual engineering geology through bonding. Because the characteristics of particles can be bonded and formed at will and can move freely without being limited by displacement, they have obvious advantages in simulating large deformation problems, especially failure and fracture problems. At the same time, they can simulate and analyze the progressive failure and fracture process of  
Figure 13: Three-dimensional contour map of internal force distribution of structural arch. (**a**) Structural arch bending moment diagram; (**b**) Structural arch stress contour diagram. materials, which is very suitable for analyzing the mechanical characteristics of rock failure, the PFC calculation flow is shown in Figure 14.  
The PFC particle flow numerical method is based on [PERSON]'s second law and force-displacement law to calculate the particles of the model. The motion of the particles follows [PERSON]'s second law, and the interaction between the particles obeys the force-displacement law. [PERSON]'s second law microscopically reflects the movement of granular matter, and the macroscopic performance is the displacement of the object. The law of force and displacement reflects the change of the internal contact force chain of granular matter, and the macroscopic performance is the change of the stress of the object. Both the force-displacement law and [PERSON]'s second law act simultaneously during the operation until the system reaches equilibrium or the set calculation time step is stopped, as shown in Figure 15.  
In the PFC model, in addition to the presence of spherical particles representing the material, a 'wall' representing the boundary is also included. The contact force generated by the mutual contact between the particles and the wall acts, and the motion equation is satisfied for each particle, but the motion equation is not satisfied for the wall; that is, the contact force acting on the wall will not affect the movement of the wall. The movement  
Figure 14: PFC calculation process. Figure 15: Cycle calculation process. of the wall is artificially given speed and is not affected by the contact force acting on it. Similarly, there is no contact force between the two walls, so PFC only has a particle-particle contact model and particle-wall contact model, as shown in Figure 16.  
In this paper, a linear parallel bonding model is established. The bonding component is parallel to the linear element, and the elastic interaction is established between the contacts. Parallel bonding can transfer force and torque between different entities. The contact bond can be regarded as a set of springs. The normal and tangential stiffnesses are kept as constants, which are evenly distributed on the contact surface and the central contact point. These springs are parallel to the linear element springs. After the parallel key is generated, the relative motion occurs at the contact, which causes the force and moment inside the bonding material. This force and moment act on two contact blocks and are related to the maximum normal stress and shear stress of the bond material around the bond.