# Prediction of Pre- and Post-Breakage Behavior of Laminated Glass Using a Phase-Field Damage Model
## 3 Phase-Field Model  
The application of phase field model to damage has afforded considerable attention; see [8; 15; 16] (to cite a few), so only the basic relations are reviewed in this section focusing on the formulation in the context of layered plate theory, time integration, aspects of implementation of the Hertz contact, and the issue of viscoelasticity. The model is based on variational framework of a brittle fracture [17], where the bulk energy competes with the dissipated energy via energy minimization. To this end, the [PERSON]'s principle of stationary action is employed and the displacement field \(\mathbf{u}(t)\) and crack topology \(\Gamma\) are obtained by minimizing the action integral  
\[\mathcal{A}(\mathbf{u},\mathbf{u},\Gamma)=\int_{0}^{T}\mathcal{L}(\mathbf{u},\mathbf{u}, \Gamma)\,\mathrm{d}t, \tag{1}\]  
where \(t\in\langle 0,T\rangle\) is the time from a given range. Minimizing this functional may prove unfeasible as it is performed over all admissible crack topologies \(\Gamma\). To remedy this, the phase-field model brings regularization and introduces a scalar damage field \(d\), which represents a relative measure of damage as a transition from the intact material \(d=0\) to a fully damaged material \(d=1\) indicating a crack. Therefore, the action integral is dependent on two continuous fields \(\mathbf{u}\) and \(d\) instead of a discontinuous topological set \(\Gamma\), i.e.,  
\[\mathcal{A}(\mathbf{u},\mathbf{u},d)=\int_{0}^{T}\mathcal{L}(\mathbf{u},\mathbf{u},d)\, \mathrm{d}t, \tag{2}\]  
which is more suitable for finite element implementation. The kernel of this integral is called the Lagrangian \(\mathcal{L}\) and, for a linear dynamic system, it reads  
\[\mathcal{L}(\mathbf{u},\mathbf{u},d)=\mathcal{K}(\mathbf{u})-\Psi_{\mathrm{e}}(\mathbf{u},d)- \Psi_{\mathrm{s}}(d)+\mathcal{P}(\mathbf{u}), \tag{3}\]  
where \(\mathcal{K}\), \(\Psi_{\mathrm{e}}\), \(\Psi_{\mathrm{s}}\), \(\mathcal{P}\) are the kinetic energy, the elastic energy, the dissipated energy, and the external work, respectively. We assume that the first and the last terms are not affected by  
Figure 3: Final fracture patterns of 7 LG samples. Figure 2: Final fracture patterns of 5 LG samples. the phase field model. Thus, assuming an elastic domain \(\Omega\) with the density \(\rho\) loaded by external tractions \(\mathbf{t}\) over \(\Omega_{\mathrm{f}}\) provides  
\[\mathcal{K}(\dot{\mathbf{u}})=\int_{\Omega}\frac{1}{2}\rho\dot{\mathbf{u}} \cdot\dot{\mathbf{u}}\,\mathrm{d}\mathbf{x},\quad\mathcal{P}(\mathbf{u})=\int_{\Omega_{ \mathrm{f}}}\mathbf{t}\cdot\mathbf{u}\,\mathrm{d}\mathbf{x}. \tag{4}\]  
The remaining two terms, i.e., the elastic potential \(\Psi_{\mathrm{e}}\), representing the stored energy of an undamaged material, and dissipated energy \(\Psi_{\mathrm{s}}\) then allow for the transition from a purely elastic response to the phase-field damage model. The present formulation complies with the physical requirement permitting the compressive stress to be transmitted across the crack even for a fully damaged material with \(d=1\). Hence, the elastic energy is decomposed into an active (tensile) part \(\Psi_{\mathrm{e}}^{+}\), which is degraded by the damage field, and a passive (compressive) part \(\Psi_{\mathrm{e}}^{-}\), which remains independent of the damage field, to obtain  
\[\Psi_{\mathrm{e}}(\mathbf{u},d)=\int_{\Omega}\left(g(d)\psi^{+}(\mathbf{u})+\psi^{-}( \mathbf{u})\right)\mathrm{d}\mathbf{x}, \tag{5}\]  
where the strain energy densities \(\psi^{+},\psi^{-}\)  
\(\mathbf{u}_{i+1}\) at time \(t_{i+1}\) is determined explicitly. The only difficulty of this scheme is thus the decomposition of strain energy density \(\psi\).