# Prediction of Pre- and Post-Breakage Behavior of Laminated Glass Using a Phase-Field Damage Model
## 6 Conclusions  
The paper described application of the phase-field model to the modeling of gradual evolution of damage in laminated glass structures subjected to low-velocity impact of increasing intensity. To reduce computational complexity, the theory was introduced in the framework of the layered Mindlin plate theory with piecewise constant distribution of  
Figure 17: Evolution of cracks in 5 LG laminate due to impact height \(h_{\text{imp}}=25\) cm; the 1 st layer is the impacted one.  
Figure 18: Evolution of cracks in 7 LG laminate due to impact height \(h_{\text{imp}}=40\) cm, the 1 st layer is the impacted one.  
out-of-plane shear stresses. Further simplifications included the transfer of kinetic energy from the impactor into the glass layer through the liner Hertz law (36), calculation of the instantaneous shear modulus via the \(t/2\) rule (38), and the use of \(f_{\rm t}\to G_{f}\) relation (33) in 3D simulations, albeit strictly valid for 1D setting only. Such formulated model allowed us to perform all calculations very efficiently with the help of a fully explicit dynamic solver. The principal observations are:  
* Introducing a properly selected initial tensile strength to accommodate sufficient energy prior to the onset of cracking appears crucial to correctly track the evolution of damage, thus supporting our observations presented in [10]. This was demonstrated by comparing the distribution of the contact force and gradually evolving crack pattern with the experimental results. The phase field model then proved its ability to predict the expected sequence of fracture of individual glass layers as well as crack branching within these layers. Owing to a random behavior of glass laminates, partially associated with random nature of initial flaws in the glass, this parameter, hardly deterministic, reduces the predictive capability of any damage model applied to the modeling of impact resistance of such structures. To promote the current approach in predictive or parametric studies, the simulations should be potentially presented in the stochastic framework once data from a sufficiently broad experimental program are available.
* Given the presence of the length scale parameter to provide regularization, these models are mesh dependent both in terms of the element size and mesh orientation. However, because the leading Equation (23) follows from the minimization of total energy, the convergence of the solution can be expected upon a sufficient mesh refinement. In this regard, an unstructured mesh made of 3-node triangular elements with degraded contribution of out-of-plane shear was found to be optimal.
* Unlike the predicted density of cracks, the crack orientation and path, turning towards the plate edges, is fully compatible with experimental observations. To arrive at improved prediction of the crack pattern might require more accurate, physically sound, contact rule. The linear model based on the Hertz law is too localized and cannot accurately describe the contact between the glass fragments and the impactor head and thus to ensure proper transition of the kinetic energy into the plate.  
Conceptualization: [PERSON], [PERSON] and M.S.; Methodology: [PERSON] and T.J.; Experimental work: M.S.; Numerical implementation: [PERSON] and T.J.; Writing--original draft: [PERSON]; Writing--review and editing: [PERSON]; Supervision: [PERSON]; Project administration: [PERSON]. All authors have read and agreed to the published version of the manuscript.  
We are thankful for financial support provided by the Czech Science Foundation, project No. 22-15553S.  
Not applicable.  
All computer codes implemented in open source FEniCS are publicly available in gitlab repository [29].  
The authors declare no conflict of interest.