# Prediction of Pre- and Post-Breakage Behavior of Laminated Glass Using a Phase-Field Damage Model
## 4 Damage of Single Glass Ply
However, when degrading the the whole out-of-plane shear contribution by setting \(\xi=1\) (the shear contribution now taken entirely by \(\psi^{+}\) is fully degraded by function \(g(d)\)) in the analysis with a triangular mesh, we arrive at the expected response as shown by the dashed line. The next example addresses a non-destructive test allowing us to compare the experimentally measured contact force with numerical predictions up to the impact height of 30 cm. The results for various types of meshes, recall Table 6, are presented in Figure 8a for \(h_{\text{imp}}=10\) cm. As seen, all meshes provide a comparable response matching well the experimental measurements. However, it has been observed that four node quadrilateral elements, albeit using one point integration in shear, experience a zero energy mode manifested by local vibration from element to element. For that reason and given the results in Figure 7, the triangular mesh in Figure 6 with shear degradation option was used in all subsequent analyses. Its applicability is further promoted by the comparative study displayed in Figure 8b, where a reasonable agreement with experimental results is suggested even for a relatively high impact energy. The last experiment is concerned with adjusting the onset of damage. This is because the stress calculated in the contact zone between an impactor and glass exceeds the glass  
Figure 8: Comparison of evolution of measured and predicted contact force for nondestructive test: (**a**) influence of finite element type—impact height of 10 cm; (**b**) application of triangular mesh for impact heights of 5 cm and 30 cm. Figure 7: Initial damage distribution on quarter plate (**left**) and deflection distribution along the \(x\)-axis for triangular and quadrilateral elements and for triangular elements with assumed shear damage with (\(\xi=1\)) denoted as triangles-full (**right**). strength \(f_{\rm t}\) earlier than the glass breaks in experiments [13,28]. This is quite important also with the phase-field analysis. For illustration, we consider a destructive test where the tested sample broke already at the impact height \(h_{\rm imp}=20\) cm. The results appear in Figures 9-11. Figure 11 indicates that assuming the actual static tensile strength \(f_{\rm t}^{\rm in}=f_{\rm t}=45\) MPa promotes the damage initiation too early in comparison to the experimental results, while  
Figure 10: Damage evolution for single glass ply with static tensile strength. Figure 9: Comparison of evolution of measured and predicted contact force for destructive test: (a) constant tensile strength; (b) variable tensile strength. delaying the onset of damage via artificially increasing the tensile strength to \(f_{\text{t}}^{\text{in}}=145\) MPa shows a considerable improvement in the prediction of the contact force. Figure 10 further suggests that controlling the crack initiation by \(f_{\text{t}}^{\text{in}}=45\) MPa does not generate sufficient energy to drive a rapid crack evolution and crack branching as observed experimentally [10,14]. A remedy is provided by restoring the initial high tensile strength to its physically correct value at the onset of damage [10], i.e., \(f_{\text{t}}^{\text{in}}=145\to f_{\text{t}}=45\) MPa. The former value ensures accommodation of sufficient energy, whereas the latter value allows its rapid dissipation leading to more realistic crack pattern. While seemingly simple, this step still deserves attention, particularly in the framework of the presented phase field model. Remind that the evolution of damage is driven by fracture energy. However, the relationship between the tensile strength \(f_{\text{t}}\) and fracture energy \(G_{\text{f}}\) provided by Equation (33) is valid for 1D analysis only. When used with a general 3D analysis, such calculated fracture energy may lead to damage initiation for stresses, which do not exceed the prescribed tensile strength.