## 4 3D FDEM Modeling of Rough Concrete-Rock Joint Failure in a Direct Shear Test
### Results  
First, we present the results for case of the asperity angle \(\alpha\) = 40 degrees. Figure 10 presents the shear stress-shear displacement curve obtained during our direct shear test simulation. Note that the shear displacement \(d_{\rm s}\) was defined by the horizontal displacement of the top-loading platen while the bulk shear stress \(\tau_{\rm bulk}\) was computed by dividing the horizontal contact force of the top-loading platen as an area (i.e., Area = Specimen width \(\times\) Specimen length). Figure 11 shows the macroscopic crack propagation (i.e., CE6s with \(D\) =1 in Equation (3)) by white color (first column in Figure 11) and the maximum principal stress distribution (PS1) and minimum principal stress distribution (PS3) at different stages of the simulation. The tension stress is positive in PS1 and PS3. At the onset of the direct shear test, the constant \(\sigma_{\rm n}\) was initially applied in the vicinity of the C-R joint as the boundary condition (i.e., CNL condition). When \(d_{\rm s}\) starts to increase, the concrete asperity sliding along the inclined C-R asperity surface and simultaneously surface friction is applied to the contact area of C-R joint surfaces, as shown in Figure 11a-I. In this stage, the tensile stress is induced near the contact asperity, as illustrated in Figure 11b-I. As these tensile stresses reached the tensile strength of CE6 in the vicinity of the asperity, the tensile crack was first initiated and propagated in a perpendicular direction to the contact area (Figure 11a-II). As shown as Figure 11a-IV, with the minor increase in \(d_{\rm s}\), the existing tensile crack was continuously propagated in a perpendicular direction to the contact area, while a new tensile crack was created by the tensile stress reaching the strength in the vicinity of the asperity at different locations. The accumulation of damage induced by the continuous tensile and shear crack propagation significantly reduced the ability of asperity resistance by the further increase in the \(d_{\rm s}\). As the tensile crack extends into the bases of the asperity (Figure 11a-V), the asperity is completely separated from their bases. Meanwhile, during the single asperity fracturing, other asperities were also fractured by the same fracturing process (see Figure 11 III-V). When all of the asperities are completely separated from their bases as shown in Figure 11b-VI (marked as the yellow line), the overall behavior of the vicinity of the C-R joint behaves as a smooth surface. Figure 10 reflects the above-mentioned shear failure behavior of the C-R joint in terms of the bulk shear stress-shear displacement curve. Initially, the bulk shear stress increased linearly with the shear displacement (Figure 10-I), which was caused by dilation along the normal direction of the C-R joint. The dilatation was induced by sliding the C-R joint along the inclined surface and the dilatation effect of the friction force between the C-R joint by increasing normal forces to the friction area. When the tensile crack is initiated, the \(\tau_{\rm bulk}\) is slightly decreased with the \(d_{\rm s}\) increment (Figure 10-II). In this stage, the PS3 distribution changed along the tensile crack because the single asperity behaved as sliding along the crack surface rather than dilatation as the \(d_{\rm s}\) increases. However, the \(\tau_{\rm bulk}\) still increased due to local normal stress increment in the other asperities (see Figure 10 II-III), until the \(\tau_{\rm bulk}\) approached the shear resistance limit, \(\tau_{\rm lim}\) (Figure 10-III). In this stage, the fluctuation of \(\tau_{\rm bulk}\) is presented because during the single asperity fracture, the other asperities are in various fracturing processes (i.e., crack initiation and propagation). After further \(d_{\rm s}\), the accumulation of damage induced by continuous tensile crack propagation significantly reduced the ability of asperity resistance.