# Prediction of Pre- and Post-Breakage Behavior of Laminated Glass Using a Phase-Field Damage Model
## 4 Damage of Single Glass Ply  
The explicit dynamic model is first validated for a single glass ply focusing on both the state before and after the crack initiation. A nondestructive step allows us to check the ability of the computational model and the Hertz contact to predict the dynamic response of glass, whereas the damage step identified the capability of the phase field model. To this end, both the evolution of contact force and displacements at points recorded experimentally were monitored to gain information on how the energy is transferred from \begin{table}
\begin{tabular}{c c c c} \hline \(\tau_{p}\) **(s)** & \(G_{p}\) **(kPa)** & \(\tau_{p}\) **(s)** & \(G_{p}\) **(kPa)** \\ \hline \(10^{-5}\) & 1,782,124.2 & \(10^{1}\) & 1663.8 \\ \(10^{-4}\) & 519,208.7 & \(10^{2}\) & 587.2 \\ \(10^{-3}\) & 546,176.8 & \(10^{3}\) & 258.0 \\ \(10^{-2}\) & 216,893.2 & \(10^{4}\) & 63.8 \\ \(10^{-1}\) & 13,618.3 & \(10^{5}\) & 168.4 \\ \(10^{0}\) & 4988.3 & - & - \\ \hline \end{tabular}
\end{table} 
Table 4: Parameters of generalized Maxwell chain model representing PVB foil for reference temperature \(T_{0}=20\)\({}^{\circ}\)C.  
the impactor into the glass laminate. While the experiment is described in detail in [14], the presented experimental results have not been published yet. With reference to tested samples, a glass plate with dimensions 0.5 m \(\times\) 0.5 m, recall also Table 1, is examined. Because of symmetry, only a quarter of the model with appropriate symmetry boundary conditions is considered. The material parameters of both the glass and impactor needed in numerical simulations are summarized in Table 5. As already mentioned in Section 2, the radius \(R\) of the impactor head entering Equation (35) is set to 50 mm. An example of the finite element mesh made of triangular elements is presented in Figure 6. To support the use of these elements in numerical analysis of laminates, the application of bilinear quadrilateral plate elements with reduced integration of out-of-plane shear is also tested and compared with a full three-dimensional (3D) calculations exploiting higher order tetrahedral elements. Details of the adopted finite element meshes are listed in Table 6.  
The impact of element type and the corresponding mesh on the ply response is tested first considering a simple example of a plate weakened by an initial crack. The plate is impacted with the energy of 148 J (impact height of 30 cm). The initial crack is introduced by enforcing the damage \(\tilde{d}(\mathbf{x}_{d})=0.99\) for coordinates \(\mathbf{x}_{d}=\{(x,y)|\xi>|x-0.125|\}\) to spread over the width of two elements (Figure 7 left) for a given value of the shear parameter \(\xi\); recall Equations (14) and (15). In accordance with Section 3.4, we set \(l=2h_{\text{min}}\), where \(h_{\text{min}}\) is the edge length of the smallest element in the mesh. The variation of deflection across the crack at the selected time \(t=0.003\) s is plotted in Figure 7 on the right side. The solid lines represent the case of no damage in shear with \(\xi=0\); thus, only the bending part in Equation (15) is affected by damage and the shear strains contribute entirely to \(\psi^{-}\). While the mesh with quadrilateral elements provides \begin{table}
\begin{tabular}{l c c c} \hline \hline  & **Triangle** & **Quadrilateral** & **3D Tetrahedron** \\ \hline Mesh size & - & 100 \(\times\) 100 & 20 \(\times\) 20 \(\times\) 4 \\ Characteristic length (mm) & 3/(min 2) & 3.5 & 15 \\ Number of degrees of freedom & 48,626 & 51,006 & 45,388 \\ Element type & Linear & Bilinear & Quadratic \\ Out-of-plane shear integration & Reduced & Reduced & Full \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Details on finite element mesh. Figure 6: Example of finite element mesh consisting of a random grid of 3-node triangular elements. reasonable response, we see that, for a triangular mesh, the wave passes across the crack. This can be attributed to spurious shear strain transfer.