## 2 Materials and Methods
### Mechanical Behavior of Sliding Layers  
The interface between the initial support and the lining is simulated using the Coulomb friction model. As shown in Figure 2\(c\), the surfaces in contact can sustain a certain level of \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(\alpha_{\mathrm{c}}\) & \(\alpha_{\mathrm{t}}\) & \(f_{\mathrm{c},r}\)/MPa & \(f_{\mathrm{t},r}\)/MPa & \(\varepsilon_{\mathrm{c},r}\)/10\({}^{-3}\) & \(\varepsilon_{\mathrm{t},r}\)/10\({}^{-5}\) \\ \hline
1.94 & 1.25 & 19.1 & 1.71 & 1.79 & 9.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of concrete. shear resistance, known as cohesion, before sliding occurs. This shear resistance depends on the normal contact pressure. In the basic formulation, the critical shear stress is defined as a function of this pressure. Once the threshold is exceeded, sliding initiates. The maximum frictional stress at the interface is proportional to the applied normal load. Accordingly, the relationship between shear stress and shear displacement at the interface is given by  
\[\tau_{r\rho}\big{|}_{r=R_{0}+d}=\left\{ \begin{array}{cc}K_{\rm n}\Delta u_{ \theta}&\Delta u_{\theta}\leq\delta_{\rm m}\\ f\cdot\sigma_{r}\big{|}_{r=R_{0}+d}&\Delta u_{\theta}>\delta_{\rm m}\end{array} \right. \tag{19}\]  
It should be noted that \(K_{\rm n}\) and \(\delta_{\rm m}\) are not material constants but are affected by the level of normal stress. Direct shear tests were conducted using a custom-designed shear box, where the lining concrete sample was placed in the upper box and the initial support sample in the lower one. A normal load was applied vertically to the upper box, while a horizontal displacement was imposed on the lower box at a constant rate of 0.2 mm/min. This setup is illustrated in Figure 3. Before testing, concrete blocks were cast for both the lining and the initial support. The initial support samples measured 20 cm \(\times\) 10 cm \(\times\) 5 cm, and the lining samples were 10 cm \(\times\) 10 cm \(\times\) 5 cm. After seven days of curing, a 2 mm thick sliding layer was applied to the lining sample by either brushing or spraying. The entire assembly was then cured for an additional 21 days. To evaluate shear performance, a range of sliding layer materials was examined. These included plain asphalt, modified asphalt cement, geotextile-reinforced asphalt, and waterborne polyurethane. The shear mechanical behavior of each material was tested using the above-described procedure. Shear stiffness values for each sliding layer material were obtained from the measured displacement-shear stress curves, as summarized in Table 2. These tests were conducted under a constant normal stress of 10 MPa. Figure 4 presents the shear response curves of various materials used in the sliding layer. Among all materials, the highest shear stiffness--943.2 MPa/m--was observed at the concrete-concrete interface between the lining and initial support. In contrast, sliding layers significantly reduced this value. Plain asphalt yielded a shear stiffness of 33.1 MPa/m, geotextile-reinforced asphalt 18.2 MPa/m, and modified asphalt cement only 2.6 MPa/m. Waterborne polyurethane showed a relatively higher stiffness of 98.5 MPa/m. The reductions in shear stiffness, compared to direct concrete contact, were substantial: 96.49% for plain asphalt, 98.07% for geotextile-reinforced asphalt, 99.72% for modified asphalt cement, and 89.56% for waterborne polyurethane. These results demonstrate the effectiveness of the sliding layer in mitigating shear transmission across the interface. Notably, the friction coefficient was not determined in these tests. Due to the adhesive and deformable nature of the asphalt-based materials, no distinct slip occurred at the interface, and the shear deformation was primarily governed by the material's behavior rather than frictional sliding. In this study, plain asphalt was  
Figure 3: Configuration of the shear test for sliding layers.