## 3 Pre-Ie-Fe
### The Governing Equations for PRE-IE-FE
#### 3.2.1 Governing Equations of Thin-Layer Interface Elements  
The length of the thin-layer finite element is \(l\), the thickness is \(t\), and the width perpendicular to the direction of the \(xy\) plane is \(w\). A local coordinate system, \((s,\,n)\), is established on the thin-layer finite element, with \(s\) as the tangential direction of the interface and \(n\) as the normal direction of the interface. Assuming that the finite element on the upper part of the interface element is a linear element, the tangential displacement, \(u_{up}\), and normal displacement, \(v_{up}\), at any point on the upper surface of the thin-layer finiteelement can be obtained via the linear interpolation of the displacements of nodes 1 and 2 belonging to the finite element, as follows:  
\[u_{np}=\frac{1}{2}\left(1-\frac{2s}{l}\right)u_{1}+\frac{1}{2}\left(1+\frac{2s}{ l}\right)u_{2} \tag{9}\]  
\[v_{up}=\frac{1}{2}\left(1-\frac{2s}{l}\right)v_{1}+\frac{1}{2}\left(1+\frac{2s }{l}\right)v_{2} \tag{10}\]  
where \(u_{1}\) and \(v_{1}\) are the tangential and normal displacements of finite element node 1, respectively, and \(u_{2}\) and \(v_{2}\) are the tangential and normal displacements of finite element node 2, respectively. Since the lower part of the interface element is a rigid body element, the normal displacement, \(u_{down}\), and tangential displacement, \(v_{down}\), at any point on the lower surface of the thin-layer finite element can be obtained by interpolating the displacement at the centroid point, \(c\), of the rigid body element shape, as follows:  
\[u_{down}=u_{c}+\left[n_{c}-\left(-\frac{t}{2}\right)\right]\theta_{c} \tag{11}\]  
\[v_{down}=v_{c}+(s-s_{c})\theta_{c} \tag{12}\]  
where \(u_{c}\) and \(v_{c}\) are the tangential and normal of the rigid body element centroid, \(c\), respectively, and \(\theta_{c}\) is the angular displacement (positive in the counterclockwise direction) of the rigid element rotating around the centroid. \((s_{c},n_{c})\) is the coordinate of the rigid body element centroid, \(c\), in the local coordinate system. \(s\) represents the tangential coordinate of any point on the lower part of the interface element, while \(n\) represents the normal coordinate of any point on the lower part of the interface element. Equations (11) and (12) are only suitable for small angular displacements. Figure 2: Schematic diagram of the coupled interface elements between finite element and rigid body elements. Then, the tangential displacement difference, \(\Delta u\), and normal displacement difference, \(\Delta v\), between the upper and lower surfaces of the thin finite element can be expressed as follows:  
\[\Delta u=u_{up}-u_{down}=\frac{1}{2}\bigg{(}1-\frac{2s}{l}\bigg{)}u_{1}+\frac{1} {2}\bigg{(}1+\frac{2s}{l}\bigg{)}u_{2}-u_{c}-\bigg{(}n_{c}+\frac{t}{2}\bigg{)} \theta_{c} \tag{13}\]  
\[\Delta v=v_{up}-v_{down}=\frac{1}{2}\bigg{(}1-\frac{2s}{l}\bigg{)}v_{1}+\frac {1}{2}\bigg{(}1+\frac{2s}{l}\bigg{)}v_{2}-v_{c}-(s-s_{c})\theta_{c} \tag{14}\]  
Combining Equations (13) and (14) yields an expression in matrix form for the difference between the upper and lower surface displacements, U, of the thin finite element, as follows:  
\[\Delta u=\left[ \begin{array}{c}\Delta u\\ \Delta v\end{array} \right]=\left[ \begin{array}{ccc}\frac{1}{2}\Big{(}1- \frac{2s}{l}\Big{)}u_{1}+\frac{1}{2}\Big{(}1+\frac{2s}{l}\Big{)}u_{2}-u_{c}- \big{(}n_{c}+\frac{t}{2}\big{)}\theta_{c}\\ \frac{1}{2}\Big{(}1-\frac{2s}{l}\Big{)}v_{1}+\frac{1}{2}\Big{(}1+\frac{2s}{l }\Big{)}v_{2}-v_{c}-(s-s_{c})\theta_{c}\end{array} \right]\]  
\[=\left[ \begin{array}{ccc}\frac{1}{2}\Big{(}1-\frac{2s}{l}\Big{)}&0&\frac{1} {2}\Big{(}1+\frac{2s}{l}\Big{)}&0&-1&0&-\big{(}n_{c}+\frac{t}{2}\big{)}\\ 0&\frac{1}{2}\Big{(}1-\frac{2s}{l}\Big{)}&0&\frac{1}{2}\Big{(}1+\frac{2s}{l} \Big{)}&0&-1&-\big{(}s-s_{c}\big{)}\end{array} \right]\left[ \begin{array}{c} u_{1}\\ v_{1}\\ u_{2}\\ v_{2}\\ u_{c}\\ v_{c}\\ \theta_{c}\end{array} \right] \tag{15}\]  
\[=\left[ \begin{array}{cc}B_{a}&B_{\gamma}\end{array} \right]\left[