# Prediction of Pre- and Post-Breakage Behavior of Laminated Glass Using a Phase-Field Damage Model
## 5 Damage of Laminated Glass Plate
As the solution depends on the length scale parameter \(l\), which in turn is estimated from the underlying finite element mesh, it appears useful to briefly address this issue. Proceeding with the previously studied 5 LG laminate, we keep the initial tensile strength \(f_{\mathrm{t}}^{\mathrm{in}}=300\) MPa and assume the impact energy of 236 J (impact height of 50 cm). The first example compares structured and unstructured meshes sketched in Figure 14 (bottom figures). Considering a comparable mesh density, the length of the process zone is set to \(l=7\) mm for all tested meshes. The damage patterns for the time instant \(t=0.4\) ms are compared in Figure 14 (top figures) for two examples of structured meshes and one example  
Figure 12: Comparison of evolution of measured and predicted contact force for nondestructive test: (**a**) 5 LG-3 laminate; (**b**) 7 LG-2 laminate. of the unstructured mesh; recall Figure 6. The evolution of damage along preferential direction associated with the structured meshes is evident. The second example addresses the effect of mesh refinement considering two mesh densities, a coarser mesh with 9425 elements and the process zone \(l=6\) mm (left) and a finer mesh with 19,119 elements and the process zone \(l=4\) mm (right) being essentially the one in Figure 6. It has been found that the mesh density influences the onset of damage, i.e., the transition point \(f_{\mathrm{t}}^{\mathrm{in}}\to f_{\mathrm{t}}\). The damage initiation is delayed with increasing the element size with the time at the onset of damage \(t^{\mathrm{in}}=0.31\) ms and \(t^{\mathrm{in}}=0.22\) ms for the coarser and finer mesh, respectively. The damage patterns plotted in Figure 15 correspond to time \(t=t^{\mathrm{in}}+0.02\) ms. While the onset of damage and partially also the crack pattern differ, the crack growth velocity appears similar for both meshes. A potential reason for decreasing the initiation time \(t^{\mathrm{in}}\) with increasing the mesh refinement can be attributed to the reduction of the fracture process zone leading to a brittle failure when approaching this parameter to zero. This issue is under current investigation. Figure 14: Damage distribution for three different meshes. Figure 13: Damage evolution in back glass layer of 5 LG laminate for two values of initial tensile strength \(f_{\mathrm{t}}^{\mathrm{in}}\). The above findings are finally exploited in the simulation of gradual damage caused by several consecutive impacts for the selected laminates. The finer element mesh in Figure 15, recall also Figure 6, was used providing a satisfactory agreement between the measured and predicted contact force. Point out that, with repeating impacts, the new calculation always started from the damage state associated with the previous loading step. We begin with the 5 LG-3 sample. To correctly predict the onset of damage for a given impact height \(h_{\text{imp}}=20\) cm, the initial tensile strength \(f_{\text{t}}^{\text{in}}=250\) MPa was used. The distribution of the contact forces is presented in Figure 16a. In agreement with experiment, the back layer fractured first. To force the damage to progress to another layer required increasing the impact height to 25 cm. Similarly to experiments, the layers cracked gradually from the back layer towards the impacted one. Unlike experiments, however, both remaining layers cracked at this simulation step. The evolution of damage in individual layers is displayed in Figure 17 for the selected time instances. It is worth mentioning that the same value of \(f_{\text{t}}^{\text{in}}\) was prescribed to all layers, whereas, in [10], each layer was assigned a different value of \(f_{\text{t}}^{\text{in}}\) to follow the experimentally observed cracking sequence as close as possible. Since this was not the principal objective of this study, primarily focusing on the potential of phase field to model fracture in laminated glass, we did not investigate this issue any further.