# Prediction of Pre- and Post-Breakage Behavior of Laminated Glass Using a Phase-Field Damage Model
## 1 Introduction  
Owing to its high aesthetic value, laminated glass enjoys an ever growing interest in modern architecture. It typically consists of several layers of glass and polymeric viscous interlayers [1]. This constitution retains the functional properties of an ordinary glass, but its properties after cracking are improved [2] as interlayers hold individual fragments of fractured glass together, thus partially preventing a catastrophic failure. Although experimental work plays a key role in promoting new structures to practical applications, an effective structure can hardly be designed without numerical simulations addressing the response of a given structure to a variety of conditions not generally accessible experimentally.  
Developing a robust and accurate computational model for glass laminates is a complex task. While layered configuration may prevent collapse and improves post breakage behavior, its simulation proves complicated as glass is essentially damaged by brittle failure with a very small process zone and negligible ductility. Although a number of numerical techniques applied to float or laminated glass have been proposed, this scientific topic is still widely open to further investigation. See, for example, [3] for an extensive overview of numerical methods applied to float glass, where the standard finite element method with the Rankine failure criterion is compared to the extended finite element method, discrete element method, and their combinations. Similarly, some critical aspects and several modeling techniques applied to laminated glass structures are presented in [4, 5]. The application of peridynamic models [6] is also worth mentioning.  
At present, a very promising approach to the modeling of brittle fracture is the application of the phase-field model [7, 8]. The principal advantage of this family of models is its ability to predict crack initialization as well as crack branching [9] with no additional ad hoc criteria. This is attributed to variational consistency as the crack development arises directly from the energy minimization of the regularized functional for a brittle material.  
The present paper examines the ability of the phase field model to simulate gradual fracture of a laminated glass subjected to several consecutive low velocity impacts. This issue has already been investigated in [10] with the help of commercial LS-DYNA software where the results of an extensive experimental program were also presented. Attention is therefore afforded to theoretical and computational aspects of the phase field model in light of plates [11], while experimental measurements used to support our numerical implementation are outlined only briefly just for the sake of completeness. Proceeding in the footsteps of [10], the five (5 LG) and seven (7 LG) layer laminates consisting of three and four plies of a float glass bonded to a PVB (polyvinyl butyl) interlayers are examined, respectively. In particular, the ability of the phase field model to estimate the response of a laminated glass observed experimentally is tested by comparing the measured and simulated distributions of contact force and gradually evolving crack patterns.  
The remainder of the paper is structured as follows. We begin with a short review of the experimental part of the project in Section 2. Essential details of the spatially reduced Mindlin Reissner model with phase field damage are introduced in Section 3. Remarks on time integration and implementation of the Hertz contact and interlayer viscoelasticity are also presented. Numerical implementation including the modeling strategy is discussed first in Section 4 in the framework of a single glass ply (SG) followed by the simulation of the two selected laminates in Section 5. A thorough discussion arising from a comparative study with experimental measurements is also provided. The principal outcomes are finally summarized in Section 6.