## 3 Theoretical Analysis and Programming of Zero-Thickness Interface Elements
### Establishment of Zero-Thickness Interface Elements in Finite Element Analysis  
In the finite element method for solving solid mechanics problems, the analysis of interface elements typically involves three main components: (1) constitutive equations,  
Figure 4: Schematic diagram of local coordinates of six-node isoparametric interface elements. Figure 3: Schematic diagram of the contact surface between objects. (2) compatibility equations, and (3) equilibrium equations. These components are used to derive the stiffness matrix that governs the force-displacement relationship, which is obtained through numerical computation. The implementation of multi-segment, zero-thickness interface elements within the finite element analysis framework is described as follows. 1. Constitutive law: stress-strain relationship  
The stress-strain relationship of general interface elements can be expressed as the relationship between the global and local coordinate systems, including (i) the global coordinate system, \(\{\sigma\}=[E]\{\epsilon\}\); (ii) local coordinate system, \(\left\{\sigma^{l}\right\}=\left[E^{l}\right]\left\{\epsilon^{l}\right\}\); and (iii) the relationship between the two coordinate systems, \(\{\sigma\}=[R]\left\{\sigma^{l}\right\}\) and \(\left\{\epsilon^{l}\right\}=[R]^{\mathrm{T}}\{\epsilon\}\). Finally, the result can be expressed as \(\{\sigma\}=[R]\left[E^{l}\right][R]^{\mathrm{T}}\{\epsilon\}\), where \([E]\) and \([E^{l}]\) are the stress-strain physical equation matrices of the global and local coordinate systems, respectively. \([R]\) is the relationship transformation matrix between the global coordinate system and the local coordinate system; the superscript \(l\) is the interface unit of the local coordinate system. In the case of a zero-thickness attached interface (within the elastic range), the continuity of the vertical and tangential relative displacements between the two objects must be ensured, and the stress-strain relationship matrix can be expressed as  
\[\left[E^{l}\right]= \begin{bmatrix}0&0&0\\ 0&kE&0\\ 0&0&G\end{bmatrix}  \tag{17}\]  
For simulating the mechanical behavior of imaginary materials, the interface element parameters \(E\) and \(G\) have no real physical meaning, so the minimum \(E\) value of the two contacting solids is generally used as the \(E\) value of the imaginary material. As for the Poisson's ratio \(\upnu\) value, since the interface element has no thickness, it is assumed that \(\upnu=0\). If the Mohr-Coulomb friction interface is considered, when there is relative displacement between the two solids, that is, when a complete sliding condition occurs (\(G=0\)), the stress-strain relationship matrix can be expressed as  
\[\left[E^{l}\right]= \begin{bmatrix}0&0&0\\ 0&kE&0\\ 0&0&0\end{bmatrix}  \tag{18}\]  
2. Compatibility: strain-displacement relationship  
When considering zero-thickness interface elements in the finite element analysis, soil elements are generally considered to be parametric elements, such as six- or eight-node triangles, T6, or quadrilaterals, Q8. Because there are three nodes on each side of two adjacent soil elements, it can be assumed that the contact surface interface element is a quadrilateral Q6 parametric element, as shown in Figure 4. Assuming \(x,y\) are the global coordinate system, \(\xi\), \(\eta\) are the local coordinate system. The relationship between the coordinate systems and the displacement and shape function of each node in the element can be expressed as follows. \[\left\{ \begin{aligned} u\\ v\end{aligned} \right\}=\sum_{k=1}^{N}H_{k}(\xi,\;\eta)\left\{ \begin{aligned} u_{k}\\ v_{k}\end{aligned} \right\} \tag{19}\]and  
\[ \begin{cases}x\\ y\end{cases} =\sum_{k=1}^{N}H_{k}(\xi,\ \eta) \begin{Bmatrix}x_{k}\\ y_{k}\end{Bmatrix}  \tag{20}\]  
where \(x_{k}\), \(y_{k}\) are the coordinates on node \(k\). \(u\), \(v\) are the node displacements on the element. \(u_{k}\), \(v_{k}\) are the displacements on node \(k\). \(N\) is the element node number.