## 3 Uniaxial Compression Test and Particle Flow Numerical Simulation
### Construction of the Particle Flow Numerical Model  
A cylindrical coal specimen model with dimensions matching the laboratory samples (50 mm in diameter and 100 mm in height) was constructed using PFC3D 5.00 software, as shown in Figure 5. In the model, the coal specimen was represented in blue, the polyurea coating applied to the surface was represented in yellow, and the transparent upper and lower walls served as the loading plates. The boundary conditions were defined by constructing the walls, within which particles were randomly generated according to a specific grading distribution and contact conditions. The walls were set as the top and bottom loading plates, with the upper and lower walls slowly moving toward each other to simulate the loading process. The numerical model consisted of three main components: the coal specimen, coating material, and uniaxial compression walls. The minimum and maximum particle sizes for the coal specimen were set to 0.5 mm and 1.2 mm, respectively, to balance computational accuracy and efficiency. The particle size for the polyurea coating was set to 0.15 mm. The linear model in PFC3D (Figure 6a shows the Linear Model) can only transfer inter-particle forces, the linear parallel bond model (Figure 6b shows the Linear Parallel Bond Model) can transfer both inter-particle forces and moments, and the linear contact bond model (Figure 6c shows the Linear Contact Bond Model) eliminates the possibility of slip due to the existence of contact bonds. Hence, we used the linear parallel bond model to represent coal samples, the linear model to simulate the mechanical behavior of the interface between the sprayed material and the coal sample, and the linear contact bond model to simulate the sprayed polyurea. The mechanical properties of rock mineral components and cementing materials were characterized by PFC particle strength and contact parameters. A trial-and-error method was used to adjust the parameters gradually to ensure that the obtained stress-strain curve closely matched the experimental stress-strain curve. When the macroscopic failure modes of both curves were similar, then the constructed numerical model was considered valid. Figure 5: PFC3D numerical simulation model. Figure 6: Schematic diagram of model components in PFC3D. (**a**) Linear Model; (**b**) Linear Parallel Bond Model; (**c**) Linear Contact Bond Model. Theoretical Analysis of the Load-Bearing Mechanical Characteristics of Coal Under Polyurea Spray Coating Conditions  
We established a theoretical model based on the above experimental results, as shown in Figure 7. The model consists of an internal cylindrical coal specimen and an external polyurea thin shell. During the axial loading process, the specimen undergoes radial expansion and displacement continuity is maintained at the interface between the polyurea thin shell and the coal pillar. As the axial deformation of the polyurea-coated coal specimen progresses, the upper and lower end faces of the coal pillar are subjected to a uniform load \(P_{\mathrm{C1}}\), while the wall surfaces experience radial compressive pressure \(P_{2}\) from the thin shell (as shown in Figure 7c). On the thin shell, the pressure \(P_{2}\) on the inner wall and the internal tension \(F_{\mathrm{P}}\) in the thin shell remain in equilibrium (as shown in Figure 7d). Because the cross-sectional area and elastic modulus (34.5 MPa) of the thin shell are much smaller than those of the coal specimen, we can neglect the contribution of the axial stress within the coating to the overall average stress of the specimen. Therefore, the axial stress measured during the experiment can be approximated as the axial stress \(\sigma_{\mathrm{z}}\) inside the coal body.