# A Partitioned Rigid-Element and Interface-Element Method for Rock-Slope-Stability Analysis
## 2 Theory
### Nonlinear Contact Iterative Method  
The change in nodal displacement on a discontinuous interface will change the contact mode of the node pair (as shown in Figure 1d), and the change in the contact mode will lead to a change in the contact force; furthermore, the change in the contact force will affect the constitutive relation of interface material. This nonlinear problem can be solved using the contact iterative method below, and the contact model is shown in Figure 3. \(\xi\eta\zeta\) is the local coordinate system of the contact node pair, \(\mathbf{f}^{1}\) and \(\mathbf{f}^{2}\) are the contact forces of any contact node pair, and \(\delta\) is the vertical distance between the node pair, which can be written as  
\[\mathbf{f}^{1}=-\mathbf{f}^{2}=\bigl{(}f_{\xi},f_{\eta},f_{\zeta}\bigr{)} \tag{29}\]  
\[\delta=(\mathbf{u}_{1}-\mathbf{u}_{2})\cdot\mathbf{\xi} \tag{30}\]  
where \(\mathbf{u}_{1}\) and \(\mathbf{u}_{2}\) are the nodal displacements of the contact node pair. Assuming that the interface material is elastic and perfectly plastic, according to the Mohr-Coulomb criterion, the contact modes of node pairs can be divided in three modes, as shown in Table 1. When the contact force or the vertical dista  
Figure 3: Contact model of node pair between partitioned rigid elements. changes, the contact mode of the node pair will also change. The relevant parameters of the node pair can be modified using the operations in Table 2.  
Firstly, the vertical distance between the contact node pair is \(\delta\), which determines whether the contact node pair is in the close or debonding mode. When the vertical distance is greater than \(t+\sigma_{t}/k_{zz}\) before iteration and less than 0 after iteration, the contact mode is changed from debonding to close, and the vertical distance is modified to 0 because the node pairs cannot penetrate each other. Next, the normal force \(f_{\xi}\) is used to determine whether the contact node pair will debond or not. When the normal force is greater than the limit after iteration, the contact mode is changed from close to debonding. According to the principle that the debonding node pair cannot transfer the contact force, the contact force is modified to have a small value of \(\mathbf{s}\). Then, the tangential forces \(f_{\zeta}\) and \(f_{\eta}\) are used to determine whether the contact node pair will slip or not. When the node pair is in the stick mode and does not yield before the iteration and then yields after the iteration, the contact mode changes from stick to slip, and the tangential forces are modified according to the Mohr-Coulomb criterion. When the contact node pair is in the close mode, according to the equivalent deformation, the normal stiffness \(k_{zz}\) can be taken as \(E/t\), where \(E\) is the elasticity modulus of the structural plane based on the test and \(t\) is the thickness of the interface. However, the normal compression deformation is not the main cause of slope failure, and to prevent the excessive interpenetration of the node pair and ensure the normal displacement coordination of the rigid element and interface element on the contact surface, the normal stiffness can be taken as a value according to experience, such as \(k_{zz}=10^{7}\) kN/m\({}^{3}\). When the contact node pair is in the debonding mode, a small value of \(k_{zz}\) is taken as the normal stiffness, such as \(k_{zz}=10\) kN/m\({}^{3}\). Different nonlinear elastic models can be used for tangential stiffness \(k_{xx}\) and \(k_{yy}\) according to the mechanical properties of the structural plane. The following Clough-Duncan hyperbolic model [6] is used for the numerical examples in this paper (assuming that the tangential modulus is isotropic):  
\[k_{xx}=k_{yy}=k_{1}\gamma_{w}\left(\frac{\sigma_{n}}{P_{a}}\right)^{n}\left(1- \frac{R_{f}\tau}{\sigma_{n}\tan\varphi_{1}}\right)^{2} \tag{31}\]