## 3 Numerical Simulation of Direct Shear Test of Bonded and Naturally Rough Concrete-Rock Interfaces
### Cohesive Frictional Model  
Experimental research (see for example [6; 38; 39; 52]) on the shear behavior of concrete-rock interfaces under a relatively low normal loading shows that the evolution of the shear stress can be idealized as composed of three phases (Figure 15). In the first phase (between A and B), the shear stress increases linearly with the applied displacement up to the peak shear strength. The shear behavior is mainly controlled by the properties of the concrete-rock bonds. In the second phase (between B and C), the shear stress decreases gradually to reach the residual shear strength. The shear behavior is driven by the progressive failure of the bonds and the progressive mobilization of friction. In the third phase (between C and D), the shear stress does not change significantly with displacement. There is a complete failure of the bonds and friction becomes the main mechanism of the shear behavior. The three phases are very similar to the four periods identified by [PERSON] et al. [52]. Indeed, [PERSON] et al. [52] evaluated the acoustic emission signals during the direct shear test of rock-rock, concrete-rock, concrete-concrete interfaces. It was observed that the shear behavior of these interfaces could be separated into four periods, the linear pre-peak, the non-linear pre-peak, the post-peak, and the residual period. For the specific case of bonded concrete-rock interfaces, [PERSON] et al. [52] reported different levels of acoustic emission for each period. There were no acoustic emissions in the linear pre-peak period, only very few acoustic emissions signals at the end of the non-linear pre-peak period, a drastic increase in the acoustic emissions signals and then fast decrease in these signals in the post-peak period and minimum value of acoustic emissions signals at the residual period. This observation corroborates the idealization of the shear stress evolution hereby presented. It is worth mentioning that in the behavior law the interface is not modeled as a physical entity, but as a behavior of individual constraints between concrete and rock. In this way, the constitutive relations presented below are calculated for individual constraints. In the first phase (between A and B), the cohesive behavior rules the overall shear response through a linear elastic traction separation relationship (stress displacement relationship). The elastic behavior is expressed in terms of an elastic constitutive matrix that correlates the normal and shear stresses (\(\sigma\), \(\tau\)) to the normal and shear displacements across the interface (\(\delta_{n}\), \(\delta_{s}\)) according to Equation (1) [53]. \[t= \begin{cases}\sigma\\ \tau\end{cases} =\begin{bmatrix}K_{n}&0\\ 0&K_{s}\end{cases} \begin{cases}\delta_{n}\\ \delta_{s}\end{cases}  \tag{1}\]  
where \(t\) is the stress vector, (\(\sigma\), \(\tau\)) are the normal and shear stress components, (\(K_{n}\), \(K_{s}\)) are the normal and the shear stiffness, and (\(\delta_{n}\), \(\delta_{s}\)) are the normal and the shear displacement. In the second phase (between B and C), damage mechanics is used to simulate the progressive degradation and failure of the cohesive bonds at the interface. Damage initiation and damage evolution are used to control the progress of the failure mechanism. Damage initiation is used to indicate the beginning of the degradation of the cohesive response at a local point in the interface [53]. The damage initiation criterion used in the model is the maximum stress criterion, in this criterion damage initiates at a local point when the stress ratio in Equation (2) becomes equal to 1. \[max\Big{\{}\frac{\langle\sigma\rangle}{\sigma^{\phi}}+\frac{\tau}{\tau^{\phi} }\Big{\}}=1 \tag{2}\]  
Figure 15: Idealization of the evolution of the shear stress of concrete–rock interface. where \(\langle\rangle\) is the Macaulay bracket, (\(\sigma^{\circ}\), \(\tau^{\circ}\)) are the normal and shear strength of constraints at the local point of the interface (individual bonds) and are specified as parameters of the model.