## 4 3D FDEM Modeling of Rough Concrete-Rock Joint Failure in a Direct Shear Test  
### Modelling Procedure  
In this section, the 3D FDEM was employed to model the asperity dilatation, sliding, and degradation during the direct shear test of a rough C-R joint. As shown in Figure 9a, the 3D FDEM model has been generated and originally adopted as the direct shear test specimen. To compare the numerical simulation result with the direct shear test result, a simulation model was generated as an equivalent dimension on the experimental specimen. To consider the effect of asperity roughness \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Parameter**} & \multirow{2}{*}{**Unit**} & \multicolumn{2}{c}{**Value**} \\ \cline{3-4}  & & **35 MPa Concrete** & **Granite** \\ \hline Density (\(\rho\)) & \(kg/m^{3}\) & 2200 & 2700 \\ Young’s modulus (\(E\)) & \(GPa\) & 30.0 & 46.0 \\ Poisson’s ratio (\(\
u\)) & - & 0.20 & 0.21 \\ Microscopic tensile strength (\(T_{s}\)) & \(MPa\) & 2.10 & 20.0 \\ Microscopic cohesion (\(c\)) & \(MPa\) & 6.70 & 56.0 \\ Microscopic internal friction angle (\(\phi\)) & _Degrees_ & 35.0 & 59.0 \\ Microscopic mode I fracture energy (\(G_{lf}\)) & \(N/m\) & 30.0 & 60.0 \\ Microscopic mode II fracture energy (\(G_{IIf}\)) & \(N/m\) & 90.0 & 120.0 \\ Artificial cohesive penalty (\(P_{open}\), \(P_{tan}\)) & \(GPa\) & 3000 & 4600 \\ Artificial cohesive penalty (\(P_{overlap}\)) & \(GPa\) & 30000 & 46000 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Calibrated FDEM input parameters for the simulation of fracture behaviors of concrete and rock elements in the direct shear tests. Note that the loading platens are rigid in Section 3 while elastic in Section 4. Figure 8: Comparison of FDEM numerical simulation and experiment of the BTS test of concrete under quasi-static loading: (**a**) plot of axial stress versus axial displacement and (**b**) final fracture pattern from the numerical simulation and experimental test. \begin{table}
\begin{tabular}{l c c} \hline \hline \multicolumn{1}{c}{**Parameter**} & \multicolumn{1}{c}{**Unit**} & \multicolumn{1}{c}{**Value**} \\ \hline Normal contact penalty between rock and rock surface (\(P_{R,can}^{\text{RR}}\)) & \(GPa\) & 460.0 \\ Normal contact penalty between concrete and concrete surface (\(P_{R,can}^{\text{CC}}\)) & \(GPa\) & 300.0 \\ Normal contact penalty between concrete and rock surface (\(P_{R,can}^{\text{CR}}\)) & \(GPa\) & 300.0 \\ Normal contact penalty between rock and plate surface (\(P_{R,can}^{\text{RP}}\)) & \(GPa\) & 460.0 \\ Normal contact penalty between concrete and plate surface (\(P_{R,can}^{\text{CP}}\)) & \(GPa\) & 300.0 \\ Friction coefficient between rock and rock surface (\(\mu_{\text{ric}}^{\text{RR}}\)) & - & 0.6 \\ Friction coefficient between concrete and concrete surface (\(\mu_{\text{ric}}^{\text{CC}}\)) & - & 0.6 \\ Friction coefficient between concrete and rock surface (\(\mu_{\text{ric}}^{\text{CR}}\)) & - & 0.6 \\ Friction coefficient between specimen and plate surface (\(\mu_{\text{ric}}^{\text{RP}}\), \(\mu_{\text{ric}}^{\text{CP}}\)) & - & 0.1 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Calibrated FDEM input parameters for simulating contact behavior in the simulation of direct shear tests. of the C-R joint, we prepared three direct shear test models having the asperity angles of \(\alpha=40\), 25, and 10 degrees (see Figure 9a). Note that the asperity angle is defined as the angle from the specimen base to the inclined plane. There is no suggested method or standards for the direct shear test of the C-R joint. The size of the direct shear test specimen was determined by considering the International Society for Rock Mechanics and Rock Engineering (ISRM) suggested method for the direct shear test of rock, which is presented in this literature [56]. Based on the ISRM suggested method, the test plane of the rock specimen should be exceeded by 10 times the maximum of the asperity height.