## 3 Theoretical Analysis and Programming of Zero-Thickness Interface Elements
### Establishment of Zero-Thickness Interface Elements in Finite Element Analysis
\(H_{k}\) (\(\xi\), \(\eta\)) is the interpolated shape function (\(-1\subseteq\xi\), \(\eta\subseteq 1\)) on node \(k\). Assuming that the thickness of the interface element is \(e\), the single-side curve M of nodes 1, 5, and 2 corresponds to the interpolation coordinate system. (\(\xi\), \(\eta\)) and can be expressed as  
\[M \begin{Bmatrix}x\\ y\end{Bmatrix} = \begin{cases}x(\xi)=g_{1}(\xi)x_{1}+g_{2}(\xi)x_{2}+g_{5}(\xi) x_{5}\\ y(\xi)=g_{1}(\xi)y_{1}+g_{2}(\xi)y_{2}+g_{5}(\xi)y_{5}\end{cases}  \tag{21}\]  
where \(g_{1}(\xi)\), \(g_{2}(\xi)\), and \(g_{5}(\xi)\) are all shape functions and can be defined by Lagrange polynomial equations. If (\(T\), \(N\)) are vector components on the curve, \(M\), and are perpendicular to each other, then T\(\cdot\)N = \(-1\), which can be expressed as  
\[N(\xi)= \begin{cases}\frac{\partial x}{\partial\xi}=\frac{1}{2}(-x_{1}+x_{2})+ \xi(x_{1}+x_{2}-2x_{5})\\ -\frac{\partial y}{\partial\xi}=-\frac{1}{2}(-y_{1}+y_{2})-\xi(y_{1}+y_{2}-2y_ {5})\end{cases}  \tag{22}\]  
Considering that the interface element is an element with a very small thickness, \(e\), as shown in Figure 4, the coordinates of nodes 2, 1, and 5 corresponding to nodes 3, 4, and 6 can be expressed as  
\[ \begin{cases}x_{3}\\ y_{3}\end{cases} = \begin{cases}x_{2}\\ y_{2}\end{cases} +e\frac{N(\xi=+1)}{\|N(+1)\|} \tag{23}\]  
\[ \begin{cases}x_{4}\\ y_{4}\end{cases} = \begin{cases}x_{1}\\ y_{1}\end{cases} +e\frac{N(\xi=-1)}{\|N(-1)\|} \tag{24}\]  
\[ \begin{cases}x_{6}\\ y_{6}\end{cases} = \begin{cases}x_{5}\\ y_{5}\end{cases} +e\frac{N(\xi=0)}{\|N(0)\|} \tag{25}\]  
where the definition is that \(e=10^{-nL}\) and \(L=0.5\) [\((x_{max}-x_{min})-(y_{max}-y_{min})\)]. Since \(L\) is a deformable function, to maintain the thickness, \(e\), at a minimum value, the ratio of \(e/L\) should be in the order of \(10^{-4}\) to \(10^{-6}\). (3) Equilibrium: Energy conservation equations  
Based on the displacement analysis of the finite element method, the element node displacement is used as the unknown number of the problem. This unknown number can be obtained using the node equilibrium equation group and the total potential energy principle. The energy conservation equation can be given as  
\[\delta W_{int}+\delta W_{ext}=0 \tag{26}\]  
where \(\delta W_{int}\) is the internal strain energy; \(\delta W_{ext}\) is the work performed by the external potential energy. In the range of domain, \(\Omega\), it can also be expressed as  
\[\delta W_{int}=-\int_{\Omega}(\sigma)\{\delta e\}d\Omega \tag{27}\]  
According to the displacement-strain relationship, it is expressed in the matrix form as  
\[\{\epsilon\}=[B]\Big{\{}U^{l}\Big{\}} \tag{28}\]or  
\[\{\epsilon\}=[A][T][L]\Big{\{}U^{l}\Big{\}} \tag{29}\]  
where \(\{U^{l}\}\) is the displacement of the element in the global coordinate. \([B]\) is the displacement-strain transformation transpose matrix. \([A]\) is the strain-displacement transformation relationship matrix. \([L]\) is the node displacement function. Finally, the stress matrix can be expressed as  
\[\{\sigma\}=[E]\{\epsilon\}=[R]\Big{[}E^{l}\Big{]}\Big{[}R]^{T}\{\epsilon\} \tag{30}\]  
or  
\[\{\sigma\}=\Big{\{}U^{l}\Big{\}}\Big{[}B]^{T}[E] \tag{31}\]  
In addition, according to the strain energy principle, the internal strain energy can be expressed as  
\[\delta W_{int}=-\Big{\langle}U^{l}\Big{\rangle}\int_{\Omega}\Big{[}B]^{T}[E][ B]\Big{\{}\delta U^{l}\Big{\}}d\Omega \tag{32}\]  
or  
\[\delta W_{int}=-\Big{\langle}U^{l}\Big{\rangle}\Big{[}K^{l}\Big{]}\Big{\{} \delta U^{l}\Big{\}} \tag{33}\]  
where \([K^{l}]\) is the element stiffness matrix, which can be expressed as  
\[\Big{[}K^{l}\Big{]}=\int_{\Omega}\Big{[}B]^{T}[E][B]d\Omega \tag{34}\]  
In the case of extremely low thickness, it can be expressed as  
\[\Big{[}K^{l}\Big{]}=\int\int_{-1}^{+1}\Big{[}B]^{T}[E][B]|det\big{]}|d\xi d\eta \tag{35}\]  
where \(J\) is the Jacobin matrix and \(|det|\) is the Jacobin determinant.