Nonlinear analytical study of structural laminated glass under hard body impact in the pre-crack stage

Highlights

• A nonlinear analytical model was proposed for calculating the pre-crack impact response of LG panel.

• The indentation dominating the failure observed in the impact test was introduced.

• The key features of experimental results were validated with the analytical results.

• The effect of safety windows film on reducing the impact response was examined.

• Effective thickness of LG based on the indentation equivalence was proposed and calibrated for hard body impact.

Abstract

Emerging glass structures, which frequently use laminated glass (LG) as load bearing elements, see a significant rise in recent decade. Existing analytical solutions for LG under impact present limitation when introduced into structural LG products, as structural LG having more glass plies and soft polymeric interlayers requires more accurate nonlinear analytical model. In this study, a nonlinear analytical model was proposed for the simply supported square structural LG subjected to hard body impact. The motion equations were established based on a third order shear deformation theory and von Kármán nonlinear strain–displacement relationship. Based on a two-step perturbation method, the solutions of the motion equations were obtained. The fourth-order Runge–Kutta method was used to capture the impact force variation. Drop weight impact tests with increasing impact velocity, were conducted to record the impact force of LG panels before breakage. Eighteen LG panels with PVB or SG interlayers were tested. Through analysing the fracture initiation from high speed photos as well as the impact force variation in the impact attempt causing fracture, certain feature of the experimental impact force response was determined to be validated with analytical prediction. The validation results show that the proposed model can well reproduce the examined feature and achieve satisfactory impact force response. Case study was then designed to investigate the influence due to the safety windows film on reducing the pre-crack impact response. The effective thickness of LG based on the equivalence of indentation was also proposed for the hard body impact.

Introduction

Structural use of glass, in particular, those glass components acting as load bearing elements commonly require greater redundancy to survive the glass fracture [1], [2]. Therefore, structural laminated glass (LG) products, which comprise more than two glass plies and one polymeric interlayers to achieve better post-fracture performance, have been increasingly used in glass buildings [3]. The most commonly used polymeric interlayers in glass buildings are Polyvinyl Butyral (PVB) [4] and ionoplast interlayers [5]. The impact resistance is one essential factor in the design of glass structures, as glass materials (even thermally toughened glass) present high vulnerability under impacts [6], [7]. Impacts on the structural glass commonly have two types: soft body impact such as manual hit or falling [8], [9], [10], or hard body impact like axe attack or windborne debris hit [11], [12]. The structural calculations or experimental study on the soft body impact on the glass products have reached a high level of accuracy [13], [14], which can be found in the existing design code and specifications [15]. However, the hard body impact which yield greater threat to the glass products has less attention.

Most works of the hard body impact on the glass products are carried out by laboratory tests or numerical simulation, which demands the speciality of engineers and facilities [16]. In the laboratory tests, the hard body impactor might be small missiles to simulate windborne debris [17], [18], [19], large weight impactor with steel hemispherical head to simulate objects falling [20]. In the automotive engineering, a headform impactor comprising aluminium sphere and Polyvinyl chloride (PVC) skin is designed to simulate human head [21], [22], [23]. Wang et al. conducted a series of experiment on testing both pre- and post-fracture behaviour of laminated glass (LG) panels using ionoplast interlayer [20]. The results show the growth trend of the energy dissipation feature and the transverse stiffness under impacts with increasing impact velocities. However, the laboratory tests are expensive and cannot cover as many design variables as the numerical models can. Finite element method (FEM) can be frequently seen in modelling the impact failure of LG products. Most works using FEM focus on developing applicable failure criterion for glass materials [24] or glass-interlayer interface [25], [26]. Other numerical models such as combined finite-discrete element method have also been used in the related topic [27]. Complex mechanical models were developed and implemented into the numerical model [16] to improve the computation accuracy, however, this also brings more difficulties for the engineers to conduct a concise simulation. To have a quick assessment of the impact resistance of glass products, an analytical model might be more practical for designers.

Analytical studies on the impact response of composite laminates can be frequently found. Due to low transverse shear moduli relative to the in-plane Young’s moduli, transverse shear deformation plays a much important role in the kinematics of composite laminates. Choi et al. [28] proposed a modified displacement field of plate theory for carbon/epoxy laminates to consider the effect of in-plane pre-load. The analytical contact force history in the low-velocity impact was compared with that from a pendulum impact test. The results show that as the impact energy increases, the analytical result will present higher difference from the experimental result, indicating the impact velocity or impact energy variation needs to be considered in analytical solution. Singh et al. [29] improved a spring–mass system to represent the contact, shear, bending and membrane stiffness of composite laminates. The comparison between FEM result and analytical result shows that the local indentation at impact point should also be considered in a low velocity large mass impact. In Singh’s model, the modified contact stiffness was proposed to consider the damage caused by external impact. Wang et al. [30] reviewed eighteen theoretical contact models reported in existing works. It can be found that local indentation is frequently used to characterize the contact behaviour between the impact body and the plate. Therefore, adopting a suitable contact model is a key step in the low-velocity impact model of the laminated structures. Analytical impact model of other composites such as functionally graded carbon nanotube-reinforced composite (FG-CNTRC) [31], [32], carbon fibre reinforced plastics (CFRP) [33], and laminate consisting of polymethyl methacrylate (PMMA) and thermoplastic polyurethane (TPU) [34] under dynamic load can also be found. In contrast, the report on the analytical solution for the LG under low-velocity hard body impact is limited so far. A recent work of Yuan et al. [22] aimed to develop an analytical model for thin automotive LG subjected to low velocity impact of headform impactor. The first-order shear deformation plate theory incorporating the effect of bending, membrane and transverse shear was employed in this model. The peak transverse displacement and contact force from analytical model were compared with that from experimental test. The difference in the contact duration between analytical and testing result was found to be evident, whereas the trend of transverse displacement was satisfactory. Amabili et al.  [35] proposed a theoretical model to study the nonlinear dynamic response of laminated glass plates subjected to blast load. In this model, the geometrically nonlinear damping model  [36] was adopted to capture the variation of damping values associated with large-amplitude oscillations of the plate. Other analytical models commonly focus on the static load [37] or blast load [38].

However, as above mentioned, the structural LG is produced via lamination with multiple glass layers, which might be up to 19 mm (e.g. fully tempered glass) for each glass layer. The first order shear deformation theory, which is widely used in the existing analytical works, is very likely to be not applicable in such product, as the multiple “stiff glass - soft polymer” interlaminar deformation is complex. Due to great difference in the material characteristics between the glass layers and the polymeric interlayers, the effect of shear deformation is commonly significant. Therefore, high-order shear deformation plate theories should be used to study the dynamic behaviour of laminated glass panel. At the same time, the contact law used in the analytical model for the laminated glass panel containing the soft-core layer should be modified based on the experimental results. In addition, existing research shows that the increase of impact energy might generate greater deviation of analytical result from realistic one, which has not been well examined. Therefore, in this paper, three novelty points are primarily introduced: (1) the third order shear deformation theory is adopted to consider complex interlaminar deformation, which is more appropriate for structural LG than existing works; (2) the local indentation, which exhibits significant effect on the impact failure due to hard body impact but has not been considered in the existing analytical efforts, is introduced in the proposed model; (3) impact tests are conducted with varying its impact energy, which is then used to examine the potential deviation from the experimental data in the proposed analytical model. Finally, a nonlinear dynamic analytical model can be built for the structural LG under hard body impact with low velocity.

In this work, the impact responses of LG under different impact velocities were investigated theoretically and experimentally. An analytical model considering geometric nonlinearity and shear deformation was presented. The motion equations of the LG and impactor were obtained by combining TSDT with larger deflection theory and framework of Newton’s second law of motion, respectively. Furthermore, the second-order differential equation (SODE) including the perturbation solution of the dynamic equation of LG and dynamic equation of impactor can be solved by a fourth order Runge–Kutta method (RK4). The relationship between impact force or indentation and time can be obtained. Drop weight impact tests were subsequently carried out to record the pre-crack impact force data of LG panels. Followed by a procedure of determining the validated feature of impact force response, the comparison between the analytical and experimental results was finally conducted to examine the applicability of the proposed model. Two case studies: (1) the investigation into the influence of the safety windows film on reducing the pre-crack impact response; (2) the validation of the proposed effective thickness of LG based on the indentation equivalence, were then conducted.