## 3 Uniaxial Compression Test and Particle Flow Numerical Simulation
### Construction of the Particle Flow Numerical Model
At the failure stage, the polyurea-coated coal specimen exhibits plastic characteristics due to the presence of transverse cracks within the specimen, which induces a lateral slip in the fractured coal blocks. No shear stress component exists in the radial direction under the action of axial stress; thus, the interface parallel to the radial direction remains intact. As the coal body is a sedimentary rock, the interfaces between different sedimentary layers are weak; therefore, when the direction forms an angle greater than \(0^{\circ}\) and less than \(90^{\circ}\) with the axial direction, shear stresses cause slippage and cracking along these interfaces. The specimen undergoes radial deformation along these transverse cracks and friction occurs between the rock on either side of the cracks. The frictional force and axial stress are related by \(\sigma_{\text{f}}=w\sigma_{z}\), where \(v\) is the friction coefficient, which depends on the specimen's strain and loading conditions. The axial compression of the polyurea-coated specimen closes the radial cracks, thereby increasing the friction coefficient. During the unloading process, the friction coefficient decreases. We assume the friction coefficient can be expressed as \(v=v_{0}\text{C}(\varepsilon_{\text{z}}-\varepsilon_{i})\), where C is a constant related to the loading method and material, \(v_{0}\) is the friction coefficient during the elastic stage, and \(\varepsilon_{i}\) is the strain at the state change of the sample (with subscript \(i=1,2\) representing the strain at the end of the compaction phase and during unloading, respectively). At the point of failure, the internal radial stress of the coal sample decreases considerably, whereas the confining pressure \(P_{2}{}^{\prime}\) from the coating remains. When the axial displacement remains constant, the fractured coal does not move; therefore, the residual radial stress within the coal body can be ignored. It can be assumed that the confining pressure and friction force are equal, i.e., \(P_{2}{}^{\prime}=\sigma_{\text{f}}\). At this stage, the coal body is fractured and contains numerous internal cracks. Thus, the radial and axial displacements of the polyurea-coated specimen are determined by the deformation of the coating. Polyurea (volume \(V_{0}=2\pi\sigma_{0}ml_{0}\)) is incompressible; thus, the radial and circumferential strains at the failure stage are given as follows: \(\varepsilon_{\theta}{}^{\prime}=\varepsilon_{r}{}^{\prime}=V_{0}/[2\pi\sigma_{0 }m(I_{0}+\delta)]-1\)From Equation (6), the confining pressure at the failure stage of the specimen can be obtained as follows:  
Figure 5:7  
Therefore, the stress-strain relationship of the coal body during uniaxial loading at the failure stage can be expressed as follows:  
Figure 5:8  
where C = 0.0021. During unloading, both the stress and friction coefficient vary with time, and the friction coefficient is given by \(v^{\prime}=v_{0}\text{C}^{\prime}(\epsilon_{z}-\epsilon_{i})\). The relationship between the axial stress and confining pressure can be written as \(P_{2}^{\prime}=v\sigma_{z}+\sigma_{z}\dot{v}\), where the variation of axial stress with time can be approximated as \(\dot{\sigma}_{z}=\Delta F/\left(\pi r_{0}^{2}\right)\); \(\Delta F\) is the load unloaded per unit time, and the unloading time \(t=\left[\pi r_{0}{}^{2}(\sigma_{0}-\sigma_{z})\right]/\Delta F\), where \(\sigma_{0}\) is the axial stress at the beginning of unloading. Therefore, we can derive the following:  
Figure 5:9  
By numerically solving Equation (18), we can express the axial strain as \(\epsilon_{z}=\Big{(}\delta_{0}+\dot{\delta}t\Big{)}/l_{0}\), where \(\delta_{0}\) is the axial deformation at the beginning of unloading, which is 0.004 m, and the stress during unloading is 4.19 MPa. The coefficient \(C^{\prime}\) is \(-0.0021\). Figure 8 shows the theoretical stress-strain curves of coal samples under different thicknesses of polyurea coatings during loading and unloading.