## 2 Mathematical Modelling
### Theoretical Modeling
#### 2.1.2 Delamination Formulation  
The deformation field for the 2D delamination laminated structure is obtained based on the FSDT model, as discussed earlier. The structure with delamination was divided into the intact region (1), lower delamination region (2), and upper delamination region (3), as shown in Figure 2. The lower and upper delaminated layers are assumed to undergo deformation without contact with each other at the center delamination boundary, as shown in Figure 2b. Global coordinates (\(X\), \(Y\), \(Z\)) are assigned to the center delamination plate, as shown in Figure 2a. The mid-plane deformation field (\(u_{0}\), \(v_{0}\), \(w_{0}\)) of the intact region (Segment 1) is considered to be different from the mid-plane deformation field (\(u_{0}\)\({}^{L}\), \(v_{0}\)\({}^{L}\), \(w_{0}\)\({}^{L}\)) of the lower delamination region (Section 2) and the mid-plane deformation field (\(u_{0}\)\({}^{U}\), \(v_{0}\)\({}^{U}\), \(w_{0}\)\({}^{U}\)) of the upper delamination region (Section 2). The rotations of transverse normal of intact, lower delamination, and upper delamination regions are represented as \(\varnothing_{x}\) and \(\varnothing_{y}\), \(\varnothing_{x}\)\({}^{L}\) and \(\varnothing_{y}\)\({}^{L}\), and \(\varnothing_{x}\)\({}^{U}\) and \(\varnothing_{y}\)\({}^{L}\), respectively. The thickness of the intact, lower delamination, and upper delamination regions are considered to be \(h\), \(h_{1}\), and \(h_{2}\), respectively.  
The deformation field of the intact region (Segment 1) is given by  
\[u=\left\{u_{0}\right^{l}\}+z\{\varnothing\} \tag{3}\]  
The deformation field of the delamination region (Segments 1 and 2) is given by  
\[u^{L,U}=\left\{u_{0}\right^{L,U}\}+z\Big{\{}\varnothing^{L,U}\Big{\}} \tag{4}\]  
Figure 2: Geometry of laminate composite structure with center delamination.  
\[\text{where }{u_{0}}^{i}=\left\{\begin{array}{c}{u_{0}}\\ {v_{0}}\\ {w_{0}}\\ \end{array}\right\},{u_{0}}^{L,U}=\left\{\begin{array}{c}{u_{0}}^{L,U}\\ {v_{0}}^{L,U}\\ {w_{0}}^{L,U}\\ \end{array}\right\},\text{and}\ \varnothing=\left\{\begin{array}{c}{\varnothing_{x}}^{L,U} \\ {\varnothing_{y}}^{L,U}\\ 0\end{array}\right\}\]  
The upper and lower delamination region boundary elements should be connected to single elements of the laminated region along the interface of intact and delaminated regions. It is essential to impose continuity conditions between the elements of intact and delamination regions. To ensure the continuity of the displacement field across the thickness of the plate at such element nodes, the displacement field of the neutral plane of the lower or upper delaminated portion at the interface boundary should be essentially identical to that of the element nodes adjacent to the planes in the intact region. Therefore, the continuity equations at the interface are given by  
\[\begin{array}{c}{u_{0}}^{L,U}=\left\{{u_{0}}\right\}+e\{\varnothing\},\\ {v_{0}}^{L,U}=\left\{{v_{0}}^{L,U}\right\}+e\{\varnothing\},\\ {w_{0}}^{L,U}={w_{0}},\\ {\varnothing_{x}}^{L,U}={\varnothing_{x}},\\ {\varnothing_{y}}^{L,U}={\varnothing_{y}}\end{array} \tag{5}\]  
where \(e=\frac{{h_{-}h_{12}}}{2}\).  
Therefore, from the Equation (5), the deformation field of the lower and upper delamination region at the delamination boundary can be written in terms of the mid-plane deformation of the intact laminate structure.  
\[\left\{\begin{array}{c}{u_{0}}^{L,U}\\ {v_{0}}^{L,U}\\ {w_{0}}^{L,U}\\ {\varnothing_{x}}^{L,U}\\ {\varnothing_{y}}^{L,U}\end{array}\right\}=\left[\begin{array}{cccccc}1&0&0&e& 0\\ 0&1&0&0&e\\ 0&0&1&0&0\\ 0&0&0&1&0\\ 0&0&0&0&1\end{array}\right]\left\{\begin{array}{c}{u_{0}}\\ {v_{0}}\\ {w_{0}}\\ {\varnothing_{x}}\\ {\varnothing_{y}}\end{array}\right\} \tag{6}\]  
The governing differential equations of motion for the FRP composite plate with delamination are obtained similarly to Equation (2).