## 3 Numerical Simulation of Direct Shear Test of Bonded and Naturally Rough Concrete-Rock Interfaces
### Identification of the Parameters of the Model  
As mentioned in Section 3.1, the parameters of the model were obtained using the results of direct shear tests of bush-hammered interfaces. The parameters related to the first phase (between A and B; see Figure 15) and the third phase (between C and D; see Figure 15), namely, the normal and the shear stiffnesses (\(K_{n}\), \(K_{s}\)), and the residual friction (\(\mu\)), are based on the experimental data. The parameters related to the second phase (between B and C; see Figure 15), namely, the normal and shear strengths of the bonds (\(\sigma^{\circ}\), \(\tau^{\circ}\)) the plastic displacement (\(\delta_{B}\)-\(\delta_{C}\)) and the damage evolution parameter \(\alpha\), were obtained through calibration. This calibration is justified since, in the second phase, different events are occurring simultaneously, locally, progressively, and in a very short time interval such that it is challenging to design experiments to quantify the influence of each event in the shear stress evolution. The aforementioned events are damage initiation, damage evolution, friction mobilization, the competition of strength between the interface and the tips of concrete to determine whether the failure proceeds along with the interface or through the tips of concrete.  
For the calibration, numerical simulations of direct shear tests of bush-hammered interfaces were carried out (Figure 18). In these simulations, the normal and shear strengths of the bonds (\(\sigma^{\circ}\), \(\tau^{\circ}\)), the plastic displacement (\(\delta_{B}\)-\(\delta_{C}\)) and the damage evolution parameter \(\alpha\) were iterated, aiming to characterize the strength of the cohesive bonds and their progressive degradation, thus calibrating the cohesive part of the model (Figure 20).  
Since two types of failure were identified, the parameters of the model were also obtained separately for these two types of failures. For the first type of failure, the parameters of the model are such that the properties of the interface capture all the energy associated with a failure that occurred mostly along with the interface and with very little damage in concrete. For the second type of failure, the parameters of the model are such that the interface captures all the energy associated with failure along with the interface and failure of the tips of concrete that remains attached to granite.  
The two sets of parameters of the model are presented in Table 6.  
Figures 21 and 22 show both the experimental tests and the numerical simulation results of the direct shear test of bush-hammered interfaces. Bearing in mind that these simulations were performed on the sole purpose to calibrate the cohesive part of the  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Cohesive Frictional** & **Parameters \({}^{*}\)** & **Type I Failure (0.5** & **Type II Failure** & **Unit** \\
**Model** & & **MPa)** & (**L0-1.5 MPa)** & \\ \hline \multirow{4}{*}{Cohesive part} & \(K_{s}\) & 15.0 & 15.0 & GPa/m \\  & \(K_{s}\) & 7.5 & 7.5 & GPa/m \\ \cline{1-1}  & \(\sigma^{\circ}\) & 1.0 & 1.4 & MPa \\ \cline{1-1}  & \(\tau^{\circ}\) & 1.2 & 1.6 & MPa \\ \cline{1-1}  & \(\delta_{B}\)-\(\delta_{C}\) & 0.9 & 2.25 & mm \\ \cline{1-1}  & \(\alpha\) & 4 & 4 & - \\ \hline \hline Frictional part & \(\mu\) & 0.70 & 0.65 & - \\ \hline \hline \end{tabular} \({}^{*}\)For legend refer to Section 3.2.  
\end{table}
Table 6: Parameters of the model.  
Figure 20: Calibration of the model.