## 3 UDEC-Based Numerical Simulation on Discrete Element
### Rheological Model Based on Strength Attenuation of Contact Face
\tag{9}\]  
The strength attenuation of blocks can be transformed into the strength attenuation of the contact surface through the conversion coefficient \(K\). The conversion coefficient of the GBM-TtoF creep model is the defined damage coefficient, but the model in this study defines the conversion coefficient \(K\) based on the crack static fatigue failure mechanism [27] and software characteristics. The mechanism of the strength attenuation of the contact surface is illustrated in Figure 5. The contact surface between blocks 1 and 2 is called contact surface 3. Since the strength attenuation of contact surface 3 is jointly controlled by the strength attenuation of blocks 1 and 2, the strengths of the blocks can be applied to the strength of the contact surface through strength conversion. In fact, many deformable bodies exist in a block, and they also follow the law of strength attenuation. The stress states of deformable bodies in the block differed as a result of stress concentration. Therefore, we could monitor the stress states of deformable bodies inside the block using the FISH function and record the number of deformable bodies exceeding the rheological damage threshold; afterwards, through dividing this number by the total number of deformable bodies inside the block, the conversion coefficient \(K\) was obtained:  
\[K=\frac{N_{d1}+N_{d2}}{N_{1}+N_{2}} \tag{10}\]  
where \(N_{d1}\) and \(N_{d2}\) are the numbers of deformable bodies exceeding the rheological damage threshold in blocks 1 and 2, respectively; \(N_{1}\) and \(N_{2}\) are the total numbers of deformable bodies in blocks 1 and 2, respectively. Since the stress states of deformable bodies change constantly, the \(K\) value varied with time. The empirical constant F in the attenuation mechanism of the contact surface (Equation (9)) was to be determined using laboratory tests. According to the laboratory tests, the long-term strength and short-term strengths of sandy mudstone were 10.8 MPa and 12.1 MPa, respectively; the time of accelerated rheology was 28,500 s; and the time of the uniaxial compression test was 120 s. The data are substituted into Equation (5):  
\[F=\frac{1-(\sigma_{1}/\sigma_{ucS})}{\ln(t/t_{0})}=\frac{1-0.893}{5.47}=0.0196 \tag{11}\]  
Figure 5: Mechanism of contact strength attenuation. are more accurate in capturing the fracturing damage at the upper left and lower right corners of the sample, and they well describe the three stages of the rheology of sandy mudstone. The simulation results of accelerated rheology time and final deformation are basically consistent with the test results. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Meaning of Parameter** & **Value** \\ \hline \hline Block parameter & & \\ \(K_{V}\) & Bulk modulus (GPa) & 1.2 \\ \(c\) & Initial cohesion (MPa) & 3.5 \\ \(\rho\) & Density (kg/m\({}^{3}\)) & 1800 \\ \(\varphi\) & Friction (\({}^{\circ}\)) & 30 \\ \(\sigma_{I}\) & Tensile strength (MPa) & 1.3 \\ \(G_{m}\) & Maxwell shear modulus (Pa) & 5.97 \(\times\) 10\({}^{8}\) \\ \(\eta_{m}\) & Maxwell viscosity coefficient (Pa-s) & 2.9 \(\times\) 10\({}^{13}\) \\ \(G_{k}\) & Kelvin shear modulus (Pa) & 7.7 \(\times\) 10\({}^{8}\) \\ \(\eta_{k}\) & Kelvin viscosity coefficient (Pa-s) & 6.7 \(\times\) 10\({}^{10}\) \\ Contact parameter & & \\ \(k_{n}\) & Normal stiffness (Pa/m) & 2.56 \(\times\) 10\({}^{12}\) \\ \(k_{s}\) & Tangential stiffness (Pa/m) & 1.02 \(\times\) 10\({}^{12}\) \\ \(\varphi_{j}\) & Friction (\({}^{\circ}\)) & 30 \\ \(c_{j}\) & Initial cohesion (MPa) & 3.2 \\ \(\sigma_{ij}\) & Tensile strength (MPa) & 1.2 \\ Rheological setting & & \\ \(\Delta t^{cr}\) & Rheological timestep time (s) & 0.005 \\ \(F\) & Empirical time constant & 0.0196 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Rheological simulation parameters of sandy mudstone. Figure 6: Comparison of uniaxial compression simulation curves and laboratory test curves. (**a**) coal; (**b**) mudstone; (**c**) medium-grained sandstone; (**d**) fine sandstone.