# Prediction of Pre- and Post-Breakage Behavior of Laminated Glass Using a Phase-Field Damage Model
## 5 Damage of Laminated Glass Plate  
The results presented in the previous section now open the door to the simulation of laminates exploiting the theoretical framework outlined in Section 3.3. Similar to the experimental program discussed in Section 2, we address both types of laminates with their geometrical details provided in Table 1. The material properties of the impactor and glass layers are taken from Table 5. For the polymer interlayer, we set the density \(\rho_{\text{int}}=1100\) kg/m\({}^{3}\). The Poisson ratio \(\
u=0.49\) is considered to approach a volumetrically incompressible material while avoiding a significant shear locking. Given the room temperature during experiments in the range of \(24.9\)\({}^{\circ}\)C to \(26.0\)\({}^{\circ}\)C, we set \(T_{\text{act}}\) in Equation (40) to \(25\)\({}^{\circ}\)C. As suggested in Section 3.6, the viscoelastic properties of the interlayer are included only empirically via Equation (38). No other material parameters of the PVB interlayer are needed as damage is expected to occur in the glass layers only. To validate implementation of the layered plate theory, we begin again with nondestructive tests. For the sake of brevity, attention is limited to 5 LG-3 and 7 LG-2 samples as these show a similar evolution of gradual damage due to repeating impacts of variable intensity; see Table 2. The response of both types of laminates is examined through the distribution of contact force for two impact heights causing no damage. The results plotted in Figure 12 indicate a satisfactory agreement between numerical predictions and experimental measurements, thus excluding any systematic error potentially linked to numerical implementation. The observed differences are merely associated with the 8 th-pole Butterworth low-pass filter (2\(\times\) CFC 1000 filter) we used to filter out the frequencies associated with vibrations of the impactor from the measured accelerations [10]. Apart from eliminating the initial data before the largest amplitudes (approximately 0.3-0.4 ms of a signal) and the final part (after 20 ms) with small oscillations affected significantly by the experimental noise, the filter was applied to the entire time domain associated with experimental measurements. The eliminated data could have affected the initial slope of the experimentally observed contact force so its direct comparison with numerical predictions might be misleading. Moving to damage brings to mind the key role of the initial tensile strength we suggested in the previous section. Its value influences the amount of energy available in the glass to drive the fracture process. This is illustrated in Figure 13 assuming two different values of the initial tensile strength \(f_{\mathrm{t}}^{\mathrm{in}}\). Similarly to the result presented in Figure 11, the strength was reduced to \(f_{\mathrm{t}}=45\) MPa according to Equation (41). The damage patterns at the onset of cracking and at the time when the damage fully localized into isolated cracks are shown for the back glass layer (the furthermost from the impacted one) of a 5 LG-laminate loaded by the impact energy of 236 J (impact height of 50 cm). It is evident that the value of the initial tensile strength qualitatively changes the result of the simulation. One may therefore consider this value as another material parameter albeit depending on a given computational model. For \(f_{\mathrm{t}}^{\mathrm{in}}=300\) MPa, the damage initiated at \(t=0.22\) ms, whereas, for \(f_{\mathrm{t}}^{\mathrm{in}}=400\) MPa, it was delayed to \(t=1.2\) ms. While the time to stabilize the crack growth was in both cases about the same, the larger amount of energy dissipated in the latter case resulted in a significantly denser crack pattern. It follows that choosing the value of \(f_{\mathrm{t}}^{\mathrm{in}}\) sensitively the model is able to control the branching and development of multiple cracks with no additional ad hoc criteria.