## 2 Basic Theory  
### Thin-Layer Element  
A common interface element based on the node element was proposed by [PERSON], [PERSON] and [PERSON] [21]. The element formula is derived from the relative node displacement of the solid element around the interface element, as shown in Figure 1.  
In two-dimensional analysis, the constitutive relation or stress-relative displacement relation of the interface behavior is expressed as  
\[\left\{ \begin{array}{c}\sigma_{n}\\ \tau\end{array} \right\}=\left[ \begin{array}{cc}k_{n}&0\\ 0&k_{s}\end{array} \right]\left\{ \begin{array}{c}v_{r}\\ u_{r}\end{array} \right\}=\left[ \begin{array}{c}C\\ \end{array} \right]_{i}\left\{ \begin{array}{c}v_{r}\\ u_{r}\end{array} \right\} \tag{1}\]  
where \(\sigma_{n}\) is the normal stress, \(\tau\) is the shear stress, \(k_{n}\) denotes the normal stiffness, \(k_{s}\) = shear stiffness, \(v_{r}\) and \(u_{r}\) are the relative normal and the relative shear displacements, respectively, and \(\left[ \begin{array}{c}C\\ \end{array} \right]_{i}\) is the constitutive matrix of the interface or joint element. In soil-structure interaction problems, it is usually assumed that the thickness \(e\) of the element is zero. In most problems, this formulation can provide satisfactory solutions for stick and slip modes for which the normal stress remains compressive. For some other modes such as debonding, the solutions are often unreliable. An analysis shows that the planar approximation of the element treated as a solid element can provide a satisfactory simulation of the finite-sized interface zone, and at the limit its results approach those of the zero-thickness element [22]. By defining the elements between adjacent contact bodies, simulation of the contact equivalent mechanical properties of the interface using thin layer elements can be implemented [23]. The proposed element essentially represents a solid element of small finite thickness. It is assumed that the thin layer element is generated by the solid element with eight nodes, as shown in Figure 2, and the displacement of any point of the element \(p(x,y,z)\) can be expressed as the following by the knowledge of the FE. \[\left\{ \begin{array}{c}x=\sum\limits_{i=1}^{8}N_{i}.x_{i}\\ y=\sum\limits_{i=1}^{8}N_{i}.y_{i}\\ z=\sum\limits_{i=1}^{8}N_{i}.z_{i}\end{array} \right. \tag{2}\]  
In the formula, \(x_{i}\), \(y_{i}\), \(z_{i}\), is the coordinate of the nodes, while \(N_{i}\) is the shape function. The relationships between the element stress, the strain, and the node \(p\) are shown as follows:  
\[\varepsilon=\textbf{B}\cdot\textbf{p} \tag{3}\]  
Figure 1: Interface element with relative nodal displacement. Figure 2: Thin-layer element. \[\sigma=\mathbf{D}\cdot\mathbf{B}\cdot\mathbf{p} \tag{4}\]  
\(\mathbf{B}\) and \(\mathbf{D}\) are the geometric matrix and elastic matrix, respectively. \(l_{1}\) and \(l_{2}\) are the length and the width of the thin-layer elements, and \(e\) is the thickness in \(z\)-direction, which is shown in Figure 2. The thickness \(e\) is much smaller than the sizes \(l_{1}\) and \(l_{2}\) of the other two directions of the element. The in-plane strain components (\(\varepsilon_{x}\), \(\varepsilon_{y}\), \(\gamma_{xy}\)) are ignored under the circumstances, as well as the stress components (\(\sigma_{x}\), \(\sigma_{y}\), \(\tau_{xy}\)) of the element. The values of partial derivatives of the form functions of thin layer elements of different sizes with respect to the local coordinates at Gaussian points are analyzed and compared in Table 1.  
As can be seen from the data in Table 1, \(\partial N_{i}/\partial z\) is greater than \(\partial N_{i}/\partial x\) and \(\partial N_{i}/\partial y\). When the size in the \(z\) direction is much smaller than the size in the \(x\) and \(y\) directions, it can be considered that \(\partial Ni/\partial x=\partial N_{i}/\partial y\approx 0\) and the strain component \(\varepsilon_{x}=\varepsilon_{y}=\gamma_{xy}\approx 0\). In other words, only three strain components are not zero at the Gauss point.