Laminated plate theories and fracture of laminated glass plate – A review

Engineering Fracture Mechanics 186 (2017) 316–330

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Engineering Fracture Mechanics journal homepage: www.elsevier.com/locate/engfracmech

Review

Laminated plate theories and fracture of laminated glass plate – A review Ajitanshu Vedrtnam a,b,⇑, S.J. Pawar a a b

Department of Applied Mechanics, Motilal Nehru National Institute of Technology Allahabad, Allahabad, UP 211004, India Department of Mechanical Engineering, Invertis University, Bareilly, UP 243001, India

a r t i c l e

i n f o

Article history: Received 14 September 2017 Received in revised form 19 October 2017 Accepted 20 October 2017 Available online 21 October 2017 Keywords: Architectural glass Impact load Laminated composites Fracture Laminated plate theories Wind shield

a b s t r a c t The designed fracture of laminated glass (LG) makes it useful for architectural, glazing, automotive safety, photovoltaic, ultraviolet ray protection, and decorative applications. The present review is divided into three sections: the first section includes a description and classification of the laminated plate (LP) theories that are used to explain fracture of LG plate, second section comprises explanation of fracture of LG samples during threepoint bending, ring on ring and ball drop impact testing using linear elastic finite element (FE) model and the last section includes numerical simulations techniques used for explaining the impact fracture of LG. The outcome of the review highlights the requirement of quantitative work on crack propagation prediction during LG fracture and critical evaluation of the numerical algorithms available for modeling the glass-ply cracking and principal damage pattern in LG. This review paper is meant for the faster comprehensive understanding of this research area. Ó 2017 Elsevier Ltd. All rights reserved.

Contents 1. 2. 3. 4. 5.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . Laminated plate theories . . . . . . . . . . . . . . Application of fem for explaining fracture Numerical simulations of LG . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A. Supplementary material . . . . References . . . . . . . . . . . . . . . . . . . . . . . . .

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1. Introduction LG comprises of two layers of glass and one or more layers of the polymer film (inter-layer) that is sandwiched inbetween using heat and pressure. Glass is a non-crystalline material that exhibits a glass transition (reversible change from a tough and brittle state from a molten/ rubber-like state) and disordered atomic structure. Glasses are hard, good electrical ⇑ Corresponding author at: Department of Applied Mechanics, Motilal Nehru National Institute of Technology Allahabad, Allahabad, UP 211004, India. E-mail addresses: [email protected] (A. Vedrtnam), [email protected] (S.J. Pawar). https://doi.org/10.1016/j.engfracmech.2017.10.020 0013-7944/Ó 2017 Elsevier Ltd. All rights reserved.

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and thermal insulators, brittle, mostly optically transparent, chemically inert, materials without exact melting point, high compressive strength isotropic materials and mostly composed of O, Si, Fe, Al, Ca, Na, K, and Mg. Glass is not a single compound thus, the chemical formula of glass is expressed as combination of silica dioxides, alkali oxides, and metal oxides. The chemical formula of glass can be expressed as aX2O bYO 6SiO2 where, a, b are number of molecules; X is an atom of an alkali metal i.e. Na, K etc, Y is an atom of a bivalent metal i.e. Ca, Pb etc, for example, Soda-lime Glass - Na2O CaO 6SiO2 Potash-lead Glass - K2O PbO 6SiO2 and so on. The glasses can be made of the variety of materials, metallic alloys, ionic melts, aqueous solutions, molecular liquids, and polymers (acrylic glass, polycarbonate, polyethylene terephthalate). The most common glass is soda-lime-silica glass having sand (contains silica), soda ash and limestone, and also a smaller amount of various additives. Annealed glass is the fundamental flat glass product that is the first outcome of the float process. Annealed glass is also formed by heating the constituents’ up to fusion and then cooling the mixture to a rigid state in a prescribed atmosphere. Annealing causes the internal stresses to relieve slowly, However, An annealed glass tends to break into big jagged shard thus it can’t be used in heavy traffic area. An annealed glass is also an initial material for producing superior products through further dispensation such as toughening, laminating, and coating. Tempered glasses/ toughened glasses are produced by controlled chemical or thermal treatments that increas the strength of glass and modify the fracture pattern (tempered glass breaks into small, square fragments). Surface coatings are applied to glass for modified appearance, low maintenance, improved transmission, absorption properties, scratch and corrosion resistance. Heat-strengthened glass is produced by heat treating to stimulate surface compression that result-in glass with the intermediate strength between tempered and annealed glass. When broken, it breaks off into sharp pieces that are squarer and smaller than that of annealed glass but less square and bigger than that of tempered glass [1–5]. The effect of the interlayer material on the stiffness, impact strength, fracture pattern, and the load-bearing capacity of LG plate is well known. Most commonly polyvinyl butyral films (PVB) and Ethylene Vinyl Acetate (Cross-Linked EVA) are utilized as inter-layer material. However, in addition to PVB and EVA, Ionoplast Polymers, Cast in Place (CIP) liquid resin and Thermoplastic polyurethane (TPU) are also used as inter-layers and sanitary glasses are also becoming popular. Inter-layer improves mechanical properties like impact strength, fracture toughness and failure mode of LG [1]. As the area of impact increases, there is a possibility of increment of the impact resistance. The fracture of LG is designed so as to the Inter-layer keeps together the broken pieces that can possibly cause dangerous incidents or accidents. The LG dampens the energy of impact and improves the brittle fracture behavior when compared with the monolithic glass. This functionality forces designers to use LG wherever there may be an injury risk due to glass fracture. The LG has interlayers considering their end uses i.e. automotive industry, architectural industry, photo-voltaic, decorative, specialty market and fire resistance. A good number of theoretical studies [6–13], experimental studies [14–22,42–45,48], numerical studies [15,23–48], numerical studies using LS DYNA and ABAQUS [23–31,40,41], and discrete element (DE)/finite element (FE) based studies are reported to analyze impact failure of LG. The bending behavior of LG is also discussed widely [49–69]. The thermal breakage of LG during fire condition is discussed in [70]; it was found that LG prevents new vent formation during the fire. The time-temperature dependent behavior of PVB and its application to LG is also discussed recently [71]. The present work is focused on reviewing the LP theories and numerical algorithms used to model the impact fracture of LG.

2. Laminated plate theories LG plates are widely used in automotive, structural, architectural, glazing, photo-voltaic decorative, defense and other applications. LG plates are often subjected to impact loading (windshields, structural applications) which can cause serious accidents. The numerical simulations are the competent way to facilitate lesser design costs, design time, and safer structures. Thus, the numerical methods are used extensively for estimating the design parameters in LG structures. Laminated composite plate theories include First-order shear deformation theory (FSDT), Higher-order shear deformation theory (HSDT), Classical Lamination Theory (CLT), Zigzag Theory (ZZT), Layer-wise Lamination Theory (LLT), and 3D Elasticity Theory. The theories used for sandwich and LPs are reviewed by Altenbach [72]. Theories that are applicable to shear deformation modeling of hybrid LP are reviewed by Noor and Burton [73,74]. The layer-wise LP theories are discussed by Reddy and Robbins [75]. Liu and Li [76] evaluated the LP theories based on the displacement assumptions and also develop a technique [77] for convincing the prerequisite of completeness by including all the terms up-to third order in the assumed displacement field for the LP theory. The theories including the dynamic response of LPs were discussed by Sun and Whitney [78]. The prime focus of their study was to evaluate the consequence of the diverse shear deformation on the plate thickness. A generalized shear deformation (two-dimensional) theory for LPs was developed by Reddy [79]. The progress of the FE analysis for LPs from 1990 to 2008 was reviewed by Zhang and Yang [80]. Khandan et al. [81] have also reviewed different methods used for modeling of the LPs with special emphasis on the normal stresses and the transverse shear. The FE theories developed for anisotropic, multi-layered, shell structures, and the composite plate was reviewed by Carrera [82]. The progress of displacement based theories for LPs with an insight on efficiency and accuracy was discussed by Wanji and Zhen [83]. The inter-laminar stresses and displacements were analyzed by Matsunaga [84] using a global higher-order theory of cross-ply LPs. A higher order shear deformable LP theory was developed using an inverse method in 3D elasticity bending solutions [85]. The theory was utilized for the solving for the bending related problems of LPs and also for the stress analysis effectively (results were compared with existing shear deformation theories) [86,87].

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Fig. 1. Stress and strain variation using CLT.

The CLT proposed by Kirchhoff [88], Love [89], and Rayleigh [90] is an extension to the classical plate theory for homogeneous and isotropic material. The in-homogeneity of the laminate material was taken into account by certain modifications, but the scope of the developed theory is limited to thin plates with no shear deformation. The CLT ignores the 3transverse strain and normal stress components. The laminate is considered a two-dimensional single layer system (called equivalent single layer) in CLT. The CLT is effectively utilized for designing fairly thin plates. Fig.1 shows the number of laminate, followed by the modulus of laminate, further the strain and stress variation using CLT. FSDT is developed by the contributions of many authors (Reissner [91,92], Hencky [93] and Mindlin [94]) for considering the effect of shear deformation in thin and moderately thick (thickness 1/10 of the planar dimensions) LPs. FSDT includes shear correction factors for effectively determining the stresses and deformations in a plate [95–97]. The shear deformation shell and plate theories are reviewed by Reddy and Arciniega [98]. The theories based on stress and displacement for the anisotropic and isotropic LPs were reviewed by Ghugal and Shimpi [99]. A variational formulation for the FSDT without shear correction factors was proposed by Auricchio and Sacco [100]. The suggested theory could be used to evaluate out-of-plane shear stresses without the post-processing procedures. HSDT includes 2-dimensional PTs with superior order in-plane displacements with a constant deflection along the LP thickness. The shear deformation theories were compared by Aydogdu [101] for the static and dynamic analysis of LPs. The LP theory for fiber reinforced composite materials was developed by Whitney and Sun [102] for the impact loading. The HSDT of LPs was discussed by Reddy [103] that includes same dependent unknowns as were considered in FSDT of Whitney and Pagano [104], but the presented HSDT included parabolic distribution (along with the thickness of plate) of the crossways shear strains. The HSDT developed by Phan and Reddy [103] can be used for analysis of anisotropic LPs without the requirement of shear correction coefficients. 3rd order shear deformation theory was developed by Lan and Feng [105] for calculating the stresses and deflections of the simply supported LPs. Wu and Hsu [106] and Matsunaga [107] have also developed the HSDT for LPs applicable for the thick elastic plate. Senthilnathan et al. [108] have developed the HSDT for evaluating the natural frequency of the isotropic/orthotropic, simply-supported LPs. Khdeir and Reddy [109] have also discussed the HSDT for obtaining exact solutions of simply supported rectangular LPs for a range of loading conditions. Kim and Cho [110] have explained the utility of HSDT displacement fields based on thickness direction for improving the predictions of deformations and stresses. Mantari et al. [111] have also developed the HSDT for LPs that gives results closer to 3D-elasticity bending solutions without using the shear correction factor. A nonlinear dynamic analysis is performed for thick LPs by Ganapathi et al. using HSDT [112]. LLT is best suited for the closed-form analysis of layered plates having no-edge effects. LLT includes the division of the laminate layers into a random number of mathematical layers along the plate thickness. A set of displacement functions is constituted for each interface and the in-plane functions are calculated using Euler–Lagrange differential equations. The development of LLT was reviewed by Carrera [113]. Plagianakos and Saravanos [114] have developed the LLT for predicting the static response of thick LPs. Carrera [115,116] evaluated a mixed LLT for calculating the out-of-plane and in-plane responses of thick LPs. Carrera and Demasi [117] have further evaluated the accuracy of the FE mixed LLT using the Reissner variational theorem. LLT FE model was used by Desai et al. [118] for dynamic analysis of LPs. LLT was also utilized for free vibration analysis by Nosier et al. [119]. Mantari and Soares [120] have discussed LLT based on HSDT for the bending analysis of LPs. The LLT is computationally difficult because unknown functions depend on the number of the random layers. For overcoming the shortcoming (expensive computational cost) of LLT, ZZT having high-order or linear functions was introduced. Initially, stable transverse displacement along the thickness which results in zero transverse deformation was assumed in ZZT, further, Di Sciuva [121] has developed an improved ZZT which consider the unknowns for the in-plane displacements for each layer in terms of the reference plane while across the plate thickness the displacement was considered constant. An improvement in the theory of Sciuva [121] was proposed by Murakami [122] and Liu and Li [77]. Sahoo and Sing [123] proposed an inverse trigonometric ZZT for static analysis of LPs by proposing shear strain shape function with the assumption of nonlinear distribution of in-plane displacement across the thickness which enables the no shear stress in the boundary conditions. A higher-order theory applicable to LPs was discussed by Cho and Parmerter [124–126] in which the zigzag linearly varying displacement was superposed by a displacement field having cubical variation. A refined ZZT having no requirement shear correction factors was developed by Versino et al. [127] for an extensive range of material systems including LPs. Many authors [128–133] have a significant contribution in the development of exact 3-dimensional elasticity solutions for the problems related to the vibration in the LPs. Srinivas et al. [129] developed a 3D linear deformation theory of elasticity

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Table 1 Summary of LP theories. LP theory

Advantages

FDST

(a) Suitable for analyzing the thin plate; for moderate thick and thicker LPs (except for LPs subjected to the transverse shear effects). (b) Less complex equations and computation.

HDST

(a) More accurate calculation of transverse shear than FDST. (b) Satisfy all boundary conditions. (c) Effectively analyze the behavior of complicated thick LPs under different loads. (d) Capable of representing the section warping in the deformed configuration. (a) The simplest Equivalent single layer LP theory Based on the displacement field. (b) The quick and simple predictions especially for the behavior of thin plated structures. (c) Suitable for structures that consist of a symmetric and balanced laminate subjected to pure tension or pure bending.

CLT

ZZT

LLT

3D elasticity theory

(a) Suitable for thick LPs having the free edges, corners or holes. (b) Suitability to through-the-thickness piece-wise behavior of stresses and displacement. (c) The compatibility of the displacements and the interlaminar equilibrium of the transverse stresses in the thickness direction are assured (a) Best suited for the closed-form analysis of layered plates having no-edge effects. (b) Comparatively accurate results than earlier developed theories. (c) Accurately captures the local state of stress for all LPs, for different plate thickness. (a) Suitable for problems related to the vibration in the LPs. (b) Usable for calculating bending and buckling of simply supported thick LPs.

Assumptions/limitations (a) The displacement w is considered constant through the thickness and displacements u and v vary linearly through the thickness of each layer. (b) Assumes constant transverse shear stress. (c) Requires shear correction factor to satisfy the plate boundary conditions. (d) Shear correction factor determines accuracy of the results. (e) The normal lines to the mid-plane before deformation remain straight and normal to the plane after deformation. (f) Unable to predict edge defect. (a) Complex equation and more computation than FDST. (b) Based on an assumption of nonlinear stress variation through the thickness.

(a) The shear strains across the interfaces between adjacent laminate are not continuous. (b) Ignores the effects of the transverse shear strains on the deformation of the elastic 2D structure. (c) Ignores some of the deformation mode constraints by reducing the model to a single degree of freedom results. (d) Neglecting shear stresses leads to a reduction or removal of the normal force, bending moment and twisting couple along free edges. (e) Neglects transverse shear strains, under estimate the values of deflections and over predict the natural frequencies and buckling loads. (f) The bending, buckling stresses are not predicted well. (a) Computationally expensive but lesser than LLT.

(a) Require many different unknowns for multilayered plates. (b) Computationally time-consuming and expensive.

that is further extended by Srinivas and Rao [130] for the vibration, bending and buckling of simply supported thick LPs. Noor [131] has also developed a theory for free vibration in the multilayered LPs. Recently effort is made for the exact solution of vibration related problems by Kulikov and Plotnikova [134] and Loredo [135]. But the expensive computation is still a foremost concern for 3D elasticity theories that can give an exact solution of the inter-laminar stress of the LPs. Table 1 summaries the merits and limitations of all the LP theories discussed in the present section.

3. Application of fem for explaining fracture of LG The FEM includes division of the continuum in a finite number of elements specified by the finite number of parameters and the solution is obtained by assembly of the elements according to the standard procedures followed [136]. A good number of authors [137–158] have utilized FEM for the analysis of LPs. The nonlinear temporary response of the LPs was reviewed by Mallikarjuna and Kant [137]. A classification of computational methods used for LPs and shells is given by Noor et al. [138] and Dawe [140]. Further, Wang et al. [141] and Chen and Dawe [142] and developed a method for the linear transient analysis of rectangular LPs. Liew et al. [147] Xiang et al. [148,149] and Sladek et al. [150] have discussed mesh-less methods for LPs, shells and functionally graded plates. The transient response of LPs [152] and tapered orthotropic plates

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Fig. 2. (a) Fracture of glass samples (Photograph) with 0.76 mm thick EVA interlayer. (b) Total deflection of LG-EVA 0.76 mm at fracture 582.00 N. (c) Force– Extension diagram of all the five LG-EVA (0.76 mm).

[153] was discussed by Tuna and Türkmen. The transient analysis of LPs is also conducted by Maleki et al. [154] using the generalized differential quadrature method. Blast loading of LPs was investigated by Shokrieh and Karamnejad [155]. PhungVan et al. [157] have utilized an isogeometric analysis using NURBS basis functions and HSDT for determining the dynamic behavior of rectangular and circular composite plates. A small investigation was conducted during the present review to evaluate the adequacy of FEA in determining the fracture pattern of LG. Firstly; the LG specimens were prepared by combining two glass beams (treated soda-lime glass) of thickness 5 mm each with an inter-layer of EVA of 0.76 mm thicknesses in between. The LG-EVA samples are prepared using the lamination heat box by vacuum bag de-airing (at Mehr Image Pvt. Ltd., Delhi, India). Fig. 2a shows the photograph of fractured LG-EVA (0.76 mm) samples after the 3-point bending testing. The experimentation was modeled using the linear elastic model of ANSYS 14.5 for obtaining deflection at the load the LG sample fractured during experimentation. Table 2 shows the properties of glass, PVB, and EVA used for simulation [159]. Fig. 2b represents the deformation obtained by the LG sam-

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A. Vedrtnam, S.J. Pawar / Engineering Fracture Mechanics 186 (2017) 316–330 Table 2 Properties used for simulation.

Young’s modulus [N/mm2] Poisson ratio [–] Density [kg/m3]

Glass

PVB

EVA

70,000 0.23 2500

220 0.495 1100

80 0.41 995

Fig. 3. (a) Fractured LG sample after ring on ring test. (b) Total displacement in LG-EVA (0.76 mm) sample.

ples. A comparison between Fig. 2a and b clearly shows that the dense fracture is observed at the higher deformation zone. The Fig. 2c shows the force–extension diagram of all the five LG-EVA (0.76 mm) samples that clearly shows that multiple fractures are experienced by the samples during experimentation. The crack propagation and multiple fractures in LG could not be captured by the FE model effectively; however, the fracture zone is clearly identified by the linear elastic model. Fig. 3a shows the photograph of fractured LG-EVA (0.38 mm) sample after the Ring on Ring test, during the test it is observed that the crack initially generated in the lower glass plate. It is also observed that the first crack was generated in the middle of the LG plate where actual loading was done. It is seen that the crack patterns on two glass plates are nearly overlapped. In all the LG samples the first crack was generated in the lower glass plate and on increased loading fracture similar to lower glass plate occurs in the upper glass plate. Fig. 3b represents the output of the FE analysis using COMSOL for calculating the total displacement at the fracture load during experimentation. It shows that the displacement is having the maximum value at the center and is least after the diameter of the larger ring. The displacement increases significantly in the zone where the load is applied by the smaller ring. The fracture of LG sample shown in Fig. 3a can be compared with the Fig. 3b. The comparison clearly shows that the zone having higher displacements have the denser fracture in tested LG sample. But the crack propagation in the LG sample could not be captured by the FE model. Destructive ball drop testing (EN 356) were performed on LG samples. This test method is intended for use as an in-plant quality control test to evaluate the impact performance of LG. In the present work, a 2.3 kg, 83 mm diameter smooth solid steel ball is dropped from 1200 mm height. Test specimen size is 305 mm  305 mm. Fig. 4a shows the fractured LG-EVA sample, highly dense spider web pattern on the upper glass plate with significant deformation in the upper surface of interlayer was observed. Fig. 4b shows the output of FE analysis simulated the experimentation. The transient analysis in ANSYS 14.5 (Explicit Dynamics Module) is utilized to simulate experimentation. Surface to surface (STS) and node to the surface (NTS) is used as contact condition of the ball and LG plate. As in case of 3-point bending and Ring on ring test, the denser fracture zone can be identified by the FE output, but the present capability of the numerical analysis could not trace the crack propagation in the LG. However, a later section of the review discusses the various numerical models followed by the researchers for impact simulation of the LG, still, it is observed that adequate explanation of crack propagation tracing in LG requires significant attention of the researchers.

4. Numerical simulations of LG Impact failure analysis of LG is commonly performed experimentally followed by numerical simulations. The failure processes of LG due to impact analyzed during pre-failure, failure and post-failure stages. Analytical models are commonly employed to describe the mechanical performance of LG for the pre and post-failure stages [160–163]. The post-failure response of LG is also explained by numerical simulations [164,165] and experiments. The literature frequently focuses on the principal damage pattern and glass-ply cracking. A recent review by Chen et al. [166] discussed six commonly used numerical algorithms sighted in the literature for impact simulation i.e. element deletion

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Fig. 4. (a) Fractured LG sample after Ball drop impact test (from 1200 mm height). (b) Total displacement in LG-EVA (0.76 mm) sample.

method (EDM), the continuum damage mechanics (CDM), the discrete element method (DEM), the combined discrete/finite element methods (DEM/FEMs), the extended finite element method (XFEM), and the cohesive zone model (CZM). The adhesion modeling is usually simulated using shared node method, the penalty-based methods, and the intrinsic CZM. In addition to the numerical technique discussed by the Chen et al. [166], intrinsic cohesive modeling (ICM) was also utilized by Gao et al. [167] for modeling impact fracture of LG. The present review includes an overview of all the commonly employed numerical algorithms for impact simulation of the LG. The EDM consists of removal of the mass of elements from the global mass matrix or set the stress of elements to zero, so as to represent the failure of these elements (used in LS-DYNA). This method is used in [168–170]. Xu et al. developed a numerical model on LG-PVB subjected to low-speed impact. The LG was impacted by a standard head form impactor at the speed of 8 m/s based on the LS-DYNA platform. The results were compared with the dynamic experiments of LG-PVB under head form impact to find the most accurate FE model [171]. Pyttel et al. have found failure benchmark norm for the LG when subjected to impact loading. It has been stated that a hazardous energy threshold should be reached during a finite expanse before failure took place. To regulate the norm and estimate the precision of revisions in norms, various tests with curved and plain samples of LG were performed. The comparison between simulated and measured results showed that the norm works sound [66]. The consequence of EDM during impact simulation includes when an element experiences significant distortion due to impact, front time step decreases and in principle simulation does not proceed. Thus, it is required to delete distorted elements. The elements may also be deleted due to failure but when the element is removed due to the distortion, it may lead to under-prediction of impact strength as elements still can take the load. Thus, the removal of the distorted elements means to the removal of some energy from the system that causes impact analysis under-predicted. The CDM includes the depiction of the damaging effect of a material by its stress-strain behavior [172]. Sun et al. constituted a model considering the contribution of damage to the constitutive law for the modeling of glass-ply cracking

Fig. 5. DE models for LG [177].

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[173,174]. Zhao et al. used this model to simulate the damage of LG panel under the impact loading. The web-shaped, starshaped and shear damage fracture pattern were observed during simulation [175]. Sun et al. also used this model to numerically find the windshield design parameters (thickness and curvature) on its stone-impact resistance [176]. The CDM is adequate when the response is mainly brittle, or it should be applied in combination with the theories such as plasticity if the response is neither perfectly brittle nor ductile. The DEM includes discretizing the solving domain into a cluster of discrete elements (DEs), and describe the movements of DEs by Newton’s second law. The interactions between DEs are usually handled by using contact algorithms. Zang et al. used 3D discrete element method to study the impact fracture of LG. The glass and PVB of LG planes were discretized to uniform rigid spherical elements (Fig. 5). This investigation showed that the accuracy of the 3D model and numerical analysis were significantly validated in the elastic range in comparison to FEM [177]. DEM assumes that the material includes of discrete particles. DEM can be adequately used to analyze granular flow, micro-dynamics of powder flow and rock mechanics situation but considering a large number of particles and duration of an implicit simulation is restricted by computational power. The combined discrete and FE methods overcome the inability of DEM to describe the large deformation behavior of the interlayer. Therefore, as shown in Fig. 6, Lei et al. proposed a DEM/FEM based on a penalty method. As shown in Fig. 6 the solving domain is decomposed into two parts (Ua and Ub) with a common interface. Sab Sua Ssa, Sub and Ssb are correspondingly the displacement and traction boundaries of sub-domain Ua and Ub [178]. Xu and Zang developed four-point combined FE/discrete element (DE) algorithm for the brittle fracture study of the LG. In this study, the penalty method was applied to calculate the interface force between two sub-domains, the FE and the DE subdomains. It was found that on comparing the impact fracture tests with the simulation results, there were changes in a number of the cracks of upper and lower glass, but the location of cracks and propagation paths are similar to the investigation outcomes. Hence, it is concluded that the theory stated was effective to forecast some macroscopic fracture features such as the crack location and the crack diffusion [179]. Four algorithms were suggested to combine DEs and FEs (Nodel combine algorithm [179], surface-center algorithm [180], freely combined algorithm [181] and four-point combined algorithm [182]) as shown in Fig. 7(a)–(d). The XFEM includes cracking simulation by extending the shape functions of the conventional FEM with extra enriched functions in the cracking area [183–187]. The cracks can be represented independently of FE meshes and crack growth modeling doesn’t require re-meshing in XFEM. Jas´kowiez performed numerical modeling for delamination in LG by XFEM. XFEM was used to input discontinuity to the approximation field. The developed model was found suitable for three-dimensional delamination analysis [188]. A further improved XFFM has the capability of explaining the crack propagation in the LG adequately. The CZMs includes idealizing fracture as a gradual process of separation in the small region ahead of the crack front. CZM is governed by a phenomenological traction-separation law and used frequently for performing failure analysis [189–191]. Tu and Pindera have simulated the impact fracture of the LG under drop-weight impact. The common glass-ply crack patterns were captured in the simulations with a special designed FE meshes (Fig. 8). It is reflected from the Fig. 4 that the propagation of radial cracks is somewhat constrained in the areas away from the impact point when brick elements are adapted for the LG model [192].

Fig. 6. Schematic diagram of the DEM/FEM proposed in [178].

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Fig. 7. (a–d) Four DEM/FEMs proposed [179–182].

Fig. 8. The impact fracture patterns of a LG plate by using the extrinsic CZM [192].

A good number of authors [193–198] have discussed various cohesive models. Gao et al. [167] have discussed recently a bilinear cohesive model for simulating the fracture process of LG under low-velocity impact. Fig. 9 shows the fracture of LG simulated using ICM. It was claimed by the author that the presented model can be adequately used to predict the fracture

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Fig. 9. The fracture of LG simulated using ICM [167].

pattern of curved LG plates as well. Samieian et al. [199] have recently discussed the dependence of post-cracking response of LG on the thickness of PVB interlayer and adhesion between glass and PVB. 5. Conclusion It is clear from the review that a lot of research work is done in the field of laminated composite glass that includes analytical models, experimental studies, and numerical simulations. FSDT, HSDT, CLT, ZZT, LLT, and 3D Elasticity Theory can be used for explaining the fracture of LG plates. The application of linear elastic model can effectively utilize to complement the fracture pattern of LG plates during different loadings. The principal damage pattern, glass-ply cracking is analyzed using seven numerical algorithms (EDM, CDM, DEM, DEM/FEMs, XFEM, CZM and ICM). The DEM/FEMs, XFEM, and CZM used for small-size LG specimens for the glass-ply cracking modeling whereas EDM can be applied to the impact failure analysis of the bigger size of LG. Further, it is suggested that quantitative works should be carried out to validate the capacity of numerical algorithms. Appendix A. Supplementary material Supplementary data associated with this article can be found, in the online version, at https://doi.org/10.1016/j.engfracmech.2017.10.020. References [1] Morgan WL. Manufacture and characteristics of laminated glass. Ind Engng Chem 1931;23(5):505–8. [2] Beall GH. Synthesis and design of glass ceramics. J Mater Ed 1992;14:315. [3] Brostow W, Castaño VM. Voronoi polyhedra as a tool for dealing with spatial structures of amorphous solids, liquids and gases. J Mater Ed 1999;21:297. [4] Medvedev NN, Geiger A, Brostow W. Distinguishing liquids from amorphous solids, Percolation analysis on the Voronoi network. J Chem Phys 1990;93:8337. [5] Kalogeras IM, Hagg Lobland HE. The nature of the glassy state, structure and transitions. J Mater Ed 2012;34:69–94. [6] Naumenko K, Eremeyev VA. A layer-wise theory for laminated glass and photovoltaic panels. Compos Struct 2014;112:283–91. [7] Eisenträger J, Naumenko K, Altenbach H, Meenen J. A user-defined finite element for laminated glass panels and photovoltaic modules based on a layer-wise theory. Compos Struct 2015;133:265–77. [8] Foraboschi P. Analytical model for laminated-glass plate. Compos B Engng 2012;43:2094–106. [9] Galuppi L, Royer-Carfagni GF. Effective thickness of laminated glass beams: new expression via a variational approach. Engng Struct 2012;38:53–67.

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