## 3 Theoretical Analysis and Programming of Zero-Thickness Interface Elements
### Application Principle and Operation Process of Zero-Thickness Interface Elements  
In analyzing the mechanical behavior of conventional thickness interface elements, three primary states are typically considered: adhesion, relative displacement (sliding), and separation. Therefore, the finite element calculation process must include the following steps: (1) identifying the contact area at each load increment; (2) distinguishing between the sliding zone, where relative displacement occurs, and the adhesion zone, where no relative displacement is observed; and (3) evaluating whether the interface element stress reaches a failure criterion, leading to either relative displacement or separation during each incremental step. When implementing zero-thickness interface elements in finite element analysis and programming, the procedure must first address the mechanical constraints and deformation behavior at the contact surface, as illustrated in Figure 5.  
Figure 5: Schematic diagram of the contact surface under the influence of an external force. It can be seen from Figure 5 that \(a\) and \(b\) represent two contacting objects, and \(M\) represents the contact point. When point \(M\) is relatively displaced due to an external force, the displacement of its global coordinate system can be given as  
\[ \begin{array}{l}x^{*}=x+\Delta u\\ y^{*}=y+\Delta v\end{array}  \tag{36}\]  
where (\(x^{*}\), \(y^{*}\)) are the coordinates after displacement. The volume increment of the initial contact volume of the interface element after conversion can be given as  
\[dV^{*}=\left| \begin{array}{cc}\frac{\partial x^{*}}{\partial x}&\frac{ \partial x^{*}}{\partial y}\\ \frac{\partial y^{*}}{\partial x}&\frac{\partial y^{*}}{\partial y}\\ \end{array} \right|dxdy \tag{37}\]  
This equation can also be expressed as \(dV^{*}=(detF^{*})dV\), where \(detF^{*}\) is the partial differential of the volume increment. When \(detF^{*}>0\), it indicates that the two objects are separated and displaced. When \(detF^{*}=0\), this indicates that the two objects are attached, and when \(detF^{*}<0\), this indicates that the objects are squeezed and deformed, or relative displacement occurs. Therefore, its displacement can be expressed as  
\[ \begin{array}{l}x_{a}=x+\alpha\Delta u\\ y_{a}=y+\alpha\Delta v\end{array}  \tag{38}\]  
It can be seen from the above equation that when \(detF_{\alpha=0}=1>0\), it indicates that no deformation occurs (the initial contact area is equal to the contact area after the force is applied); when \(detF_{\alpha=1}=detF^{*}<0\), it indicates that the two objects produce relative displacement but are not completely separated. Assume that the external force acting on the interface element, \(\sigma_{n}\), is normal stress. When the interface element is in the startup state, the relationship can be given as  
\[\sigma_{n}=\sigma_{n}^{0}+\Delta\sigma_{n}\geq R_{T} \tag{39}\]  
where the relative coefficient \(\lambda\) is  
\[\lambda=\frac{R_{T}-\sigma_{n}^{0}}{\Delta\sigma_{n}} \tag{40}\]  
When the finite element analysis is used to calculate with the zero-thickness interface element, the stress is mainly calculated by using relative displacement. Therefore, the principle of virtual work can be applied to simulate the situation where the interface elements cannot withstand shear force when they are in a sliding state. The numerical calculation steps are as follows: (1) Calculate the stiffness matrix of the interface elements [\(K_{i}\)] and [\(K_{i+1}\)], and triangulate the stiffness matrix [\(K_{i}\)]; (2) Calculate [\(K_{i}\)] {\(\Delta U_{i}\)} = {\(R_{i-1}\)}; (3) Calculate the stress increment vector {\(\Delta\sigma_{i}\)} and the displacement increment vector {\(\Delta U_{i}\)}; (4) Calculate the force vector {\(R_{i}\)}; and (5) Test the convergence of the equilibrium force. The numerical calculation process for two-dimensional finite element analysis in this study is divided into three main components: (1) pre-processing, (2) main calculation and analysis, and (3) post-processing.