# A Partitioned Rigid-Element and Interface-Element Method for Rock-Slope-Stability Analysis
## 2 Theory
### Equations of Interface Element  
Since the use of non-thickness joint elements may cause the adjacent rigid elements to overlap, PRE-IE uses thin-layer elements as interface elements [15; 16] (as shown in Figure 1c). In this paper, we take the eight-node hexahedral thin-layer element shown in Figure 2 as an example. The thickness of the interface element is \(t\), and the mass force is negligible. According to the principle of virtual displacement, the equivalent integral weak form of equilibrium equations and boundary conditions of the interface element can be obtained, which has the following matrix expression:  
\[\int_{\Omega}\delta\mathbf{\varepsilon}^{T}\mathbf{\sigma}\mathrm{d}V-\int_{S_{i}} \delta\mathbf{u}^{T}\mathrm{F}\mathrm{d}S=0 \tag{6}\]  
where \(\Omega\) is the integral region of the interface element, \(\mathbf{\varepsilon}\)  
where \(\mathbf{N}\) is the shape function matrix of the four-node quadrilateral isoparametric element, \(\mathbf{u}\) is the nodal displacement, \(\mathbf{a}_{up}^{e}\) is the nodal displacement array on the upper surface of the interface element, and \(\mathbf{a}_{down}^{e}\) is the nodal displacement array on the lower surface of the interface element. Then, the displacement difference between upper and lower surfaces \(\Delta\mathbf{u}\) can be expressed as  
\[\Delta\mathbf{u}=\left[ \begin{array}{c}\Delta u_{x}\\ \Delta u_{y}\\ \Delta u_{z}\end{array} \right]=\mathbf{u}_{up}-\mathbf{u}_{down}=\mathbf{B}\mathbf{a}^{e} \tag{11}\]  
where \(\mathbf{B}\) is the shape function matrix of the interface element and \(\mathbf{a}^{e}\) is the nodal displacement array of the interface element, which can be expressed as follows:  
\[B=\left[ \begin{array}{cc}\mathbf{N}_{1}&\mathbf{N}_{2}&\mathbf{N}_{3}&\mathbf{N}_{4}\\ \end{array} \right]=\left[ \begin{array}{c}\mathbf{a}_{up}^{e}\\ \mathbf{a}_{down}^{e}\end{array} \right] \tag{12}\]  
\[\mathbf{a}^{e}=\left[ \begin{array}{c}\mathbf{u}_{1}\\ \mathbf{u}_{2}\\ \mathbf{u}_{3}\\ \mathbf{u}_{4}\\ \mathbf{u}_{5}\\ \mathbf{u}_{6}\\ \mathbf{u}_{7}\\ \mathbf{u}_{8}\end{array} \right]=\left[ \begin{array}{c}\mathbf{a}_{up}^{e}\\ \mathbf{a}_{down}^{e}\end{array} \right] \tag{13}\]  
Since the element thickness \(t\) is small compared to the whole structure, there are only tangential strains \(\gamma_{zx}\) and \(\gamma_{zy}\) parallel to the interface and normal strain \(\varepsilon_{z}\) perpendicular to the interface in the interface element. The strain \(\varepsilon\) is related to the displacement difference \(\Delta\mathbf{u}\) and is constant in the thickness direction, which can be expressed as follows:  
\[\varepsilon=\left[ \begin{array}{c}\gamma_{zx}\\ \gamma_{zy}\\ \varepsilon_{z}\end{array} \right]=\left[ \begin{array}{c}\frac{\Delta\mathbf{u} _{x}}{t}\\ \frac{\Delta\mathbf{u}_{y}}{t}\\ \frac{\Delta\mathbf{u}_{z}}{t}\end{array} \right]=\frac{\Delta\mathbf{u}}{t} \tag{14}\]  
the stress \(\mathbf{\sigma}\) can be expressed as:  
\[\mathbf{\sigma}=\left[ \begin{array}{c}\tau_{zx}\\ \tau_{zy}\\ \sigma_{z}\end{array} \right]=\left[ \begin{array}{ccc}tk_{xx}&0&0\\ 0&tk_{yy}&0\\ 0&0&tk_{zz}\end{array} \right]\left[ \begin{array}{c}\gamma_{zx}\\ \gamma_{zy}\\ \varepsilon_{z}\end{array} \right]=\mathbf{D}\mathbf{\varepsilon} \tag{15}\]  
where \(\mathbf{D}\) is the elastic matrix of the structural plane, \(k_{zz}\) is the normal stiffness, and \(k_{xx}\) and \(k_{yy}\) are the tangential stiffnesses. Figure 2: Interface element (eight-node hexahedral thin-layer element). Substituting Equations (11), (14) and (15) into Equation (8), the potential energy of the interface element can be written as follows:  
\[