## 2 Dam Sliding Model
### Interface Contact-Sliding Model  
A contact and sliding model for the dam-foundation interface is implemented in the context of the finite element mesh. In this study, it is assumed that the pre-existing or newly generated fracture propagates along the dam-foundation interface plane and the propagation criterion is dominated by the shear mode (mode II) fracture. In addition, the dam base and its corresponding rock-foundation surfaces are assumed to contact and separate vertically along the fractured region.  
The dam side of the interface is defined as a slave surface and the foundation side is defined as a master surface. The fracture extension starts from the heel of a dam, since the crack initiation occurs from the heel of a dam due to the hydraulic pressure. The contact surface is defined by the vertical direction and the horizontal direction. As shown in Figure 2, in the vertical direction, mesh penetration between the master and slave surface is controlled [23]. The principle of minimum total potential energy, which is the basis of the finite element method, is applied to the dam-foundation contact model. The basic form of the principle can be represented as:  
\[\delta\Pi\equiv\delta U-\delta W=0 \tag{4}\]  
where \(\Pi\) is the total potential energy, \(U\) is the strain energy, \(W\) is the work done by the external force. Equation (4) is applied to the contact constraint in the vertical direction that controls mesh penetration. The contact constraint is formed by the Lagrange multiplier method and virtual work as:  
\[\delta\Pi=\delta ph+p\delta h \tag{5}\]  
where \(h\) is the clearance between contact interfaces, \(p\) is the pressure delivered through the interfaces. Equation (6) shows that the pressure delivered through the interface is zero when there is a positive clearance between the foundation (master) and dam base (slave) surfaces. In case of zero clearance, it is defined that the foundation and dam base surfaces are regarded as contacted and the contact pressure can be delivered through the interface as shown in Equation (7).  
\[h>0,\ p=0 \tag{6}\]  
\[h=0,\ p\
eq 0 \tag{7}\]In the horizontal direction, the fracture progression along the dam-foundation interface is modeled by the stick-slip phenomenon. The Lagrange multiplier method is applied to the minimum total potential energy principle to enforce exact stick-slip conditions [23]. In the \(i\)-th iteration, the augmented virtual work can be expressed as:  
\[\delta\Pi=\int_{S}\tau_{i}\delta\gamma_{i}+\Delta\gamma_{i}\lambda dS \tag{8}\]  
where \(\tau_{i}\) is the friction force at the interface, \(\gamma_{i}\) is the sliding displacement, \(\lambda\) is the Lagrange multiplier, and \(S\) is the interface surface region. In the stick state, when the external shear force \(\tau\) exceeds the friction force \(\tau_{i}\), the state is changed to sliding as shown in Equation (9). Meanwhile, Equation (10) shows that if the direction of the friction force and the direction of sliding are opposite in the sliding state, the state is changed to stick.  
\[\tau>\tau_{i} \tag{9}\]  
\[\Delta\gamma_{i}\tau_{i}<0 \tag{10}\]  
By this mechanism, the dam-foundation model follows exact stick-slip conditions. The relationship between stress and strain is depicted in Figure 3. Here, the critical shear force \(\tau_{cri}\) is the sum of the cohesive forces and the frictional forces of the interface, which is the same as the numerator in Equation (3). There is no additional sliding in the stick state (\(\Delta\gamma=0\)). When the absolute value of an external shear force exceeds the critical shear force, the existing state is changed to sliding.  
Figure 3: Stick–slip motion with Lagrange multiplier in \(\Delta\gamma-\tau\) plane.  
Figure 2: Schematic contact pressure–clearance relationship in contact interface [23].