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Simulating progressive failure in laminated glass beams with a layer-wise randomized phase-field solver
... The glass strength distribution was determined by separately examining all areas of the windshield using ringon ring tests and four-point bending tests. Recently, Schmidt et al. (2024) investigated the simulation of distributed and gradual fracture patterns in laminated glass beams using randomized phase field models, where tensile strength is modelled using Weibull variables. They studied progressive failure through combinatorial analysis and Monte Carlo simulations. ...
Laminated glass, known for its enhanced strength and safety features, is an advanced building material composed of multiple layers bonded together, providing superior durability and resilience compared to traditional float glass. This study investigates the dynamic response of freely hanging laminated glass to near-field blast loading. Our aim is to introduce a setup for testing unsupported specimens under blast conditions, facilitating the collection of data for future application. Experimental trials were conducted, examining two laminated glass configurations: a 5-layer setup and a 7-layer setup. Panels were designed using the sacrificial glass ply concept, comprising alternating layers of glass panels and polyvinyl butyral interlayers. The experimental methodology involved two types of explosives: semtex 1A and granulated TNT. The samples were suspended on wire rope slings to eliminate support influences, enabling comparison with numerical models while reducing the number of unknown parameters from boundary conditions in the simulations. Samples were incrementally loaded with explosive weights to estimate the minimum impact energy required for glass failure. The overpressure from the blast was recorded and the response of the panels in terms of the acceleration, as well as crack formation and propagation, was studied. Accelerometers were affixed to each panel, whereas the crack development was examined after each test. The experimental results were compared to a numerical model. The study explores cracks based on numerous parameters like number of blasts and glass configuration. It also shows how different parameters impact panel velocity under blast and how damage affects their natural frequency. Comparison with numerical models shows that the setup accurately replicates the absence of supports.
Owing to the random nature of heterogeneity, damage and fracture behavior of quasi-brittle materials exhibits a considerable degree of uncertainty. Computational modeling of stochastic fracture in quasi-brittle materials has become an indispensable tool for analysis and design of engineering structures. To this end, we present in this paper a computational framework to capture probabilistic fracture in heterogeneous quasi-brittle solids by combinging the random field theory and the phase-field cohesive zone model (PF-CZM). The spatial variation of the material strength and fracture energy is represented by a cross-correlated bivariate random field generated by the Karhunen-Loève expansion. The recently proposed PF-CZM is employed to simulate the stochastic crack nucleation and propagation in quasi-brittle solids. The objectivity of the Monte-Carlo simulation is achieved by imposing a specific condition on the phase-field length scale parameter and the correlation length of the random field. In particular, upon this condition the width of the fracture process zone (FPZ) is considerably smaller than the correlation length of the random field such that the material inside the FPZ does not exhibit significant spatial variations of mechanical properties. As the fracture energy is intrinsically incorporated in the PF-CZM, it is unnecessary in this case to explicitly consider the FPZ width. The resulting probabilistic PF-CZM together is applied to the Monte-Carlo simulations of fracture in concrete structures of different geometries. It is shown that the stochastic simulation results are insensitive to both the phase-field length scale parameter and the finite element mesh discretization as in the previous deterministic analyses. Enhanced with the specific condition on the involved characteristic lengths, the PF-CZM provides a viable tool for stochastic simulations of damage and fracture in quasi-brittle structures.
We investigate variational phase-field formulations of anisotropic brittle fracture to model zigzag crack patterns in cubic materials. Our objective is twofold: (i) to analytically derive and numerically test the fundamental behavioral aspects predicted by the two main available fourth-order models, and to guide the calibration of their unknown parameters; (ii) motivated by the pronounced non-uniqueness of the phase-field solution in the anisotropic case and uncertainties in the material model, to transition from a deterministic to a stochastic model by introducing a material-related random field in the anisotropic phase-field energy functional. We employ Monte Carlo, randomized quasi-Monte Carlo and stochastic spectral methods to estimate statistical moments of the phase-field variable. The two fourth-order phase-field models are shown to predict a very different response both in their deterministic and stochastic versions. For either modeling choice, we believe that the stochastic approach, which captures several possible zigzag crack paths, holds significant promise to enable meaningful predictions of anisotropic fracture with phase-field models.
Glass as a construction material has become indispensable and is still on the rise in the building industry. However, there is still a need for numerical models that can predict the strength of structural glass in different configurations. The complexity lies in the failure of glass elements largely driven by pre-existing microscopic surface flaws. These flaws are present over the entire glass surface, and the properties of each flaw vary. Therefore, the fracture strength of glass is described by a probability function and will depend on the size of the panels, the loading conditions and the flaw size distribution. This paper extends the strength prediction model of Osnes et al. with the model selection by the Akaike information criterion. This allows us to determine the most appropriate probability density function describing the glass panel strength. The analyses indicate that the most appropriate model is mainly affected by the number of flaws subjected to the maximum tensile stresses. When many flaws are loaded, the strength is better described by a normal or Weibull distribution. When few flaws are loaded, the distribution tends more towards a Gumbel distribution. A parameter study is performed to examine the most important and influencing parameters in the strength prediction model.
Laminated glass composed of several layers of glass plies bonded to a polymer interlayer enjoys ever growing interest in modern architecture. Being often used in impact protection designs requires understanding of both pre- and post-breakage behavior of these structures. This paper contributes to this subject by examining an application of an explicit phase field dynamic model to the description of fracture in a laminated glass subjected to a low velocity impact. The achieved results indicate the ability of the proposed model to successfully describe the onset of damage and subsequent crack propagation. It has, however, been observed that a relatively fine mesh is needed to interpolate a sharp discontinuity accurately, which makes this approach computationally demanding. The model is first validated against experimental results obtained for a single-layer float glass. Next, the usability of the phase-field damage model as a crack predictor in individual layers of the composite is investigated. The dependence of the results on residual stiffness, element type, and initial tensile strength is examined and discussed.
This study aims to analyze the performance of laminated glass against ballistic loading and investigates its residual load-bearing capacity. Two groups of specimens were used in quasi-static four-point bending experiments, first without prior ballistic damage and then with it. The main objective was to compare the load-bearing capacity of these two groups to see the effect of ballistic damage. Three different layer compositions were used. The ballistic loading was conducted using an in-service 9 mm bullet fired from a semiautomatic carbine with the glass specimens hanging on steel ropes in a free boundary setup. Numerical simulation and analytical methods were used and validated against the measured response of the undamaged specimens. The simulations were in good agreement with the experimental results. All of the glass specimens were able to withstand the ballistic loading, and the subsequent performance during the quasi-static bending loading was similar to that of the undamaged specimens. The quality of the glass edges seemed to be more important than ballistic damage. The front-plate damage played a negligible role, and the back-plate damage needed to be extensive to influence subsequent performance. Provided that ballistic damage is mainly localized only to the centers of the plates, it did not affect the post-impact loading capacity.
This article covers the issues of applying structural glass in shaping all-glass architectural objects. Glass, as a transparent material, is a source of inspiration for new architectural solutions. With the development of technology and the increasing knowledge of glass’s mechanical and strength properties, the possibility of using the material for construction purposes has also been acknowledged. Structural elements and building envelope elements can create a uniform material structure of all-glass objects. This observation contributed to the analysis presented in the article. The research was mainly aimed at investigating the architectural and structural-related conditions in shaping all-glass structures in buildings. In this paper, we specify criteria and typology in terms of the applied design solutions. The criteria investigated in the study included functional-spatial aspects, the form, and the structure. All-glass objects were divided into pavilions, extensions, and links in terms of functional and spatial aspects. Architectural forms were specified and characterised as cubic, cuboid, cylindrical, and free-forms. Regarding structural solutions, frames, grillages, beam-wall, and plate-wall systems were indicated as the main load-bearing structures implemented in the buildings under study. The results have been obtained to describe the architectural and structural shaping of all-glass objects. One of the main results of the work is the indication between functional-spatial aspects, the form, and the structure. This correlation confirms the close relationship in architecture between art and engineering.
A unified framework for an impromptu switching between the coupled (i.e., the monolithic), the staggered, and the uncoupled solution strategies for the phase-field based computations of material fracture is presented. As a model problem, a 2d quasi-brittle solid phase-field formulation, which is based on the tension-compression split energy functional, is chosen. In order to fit within the proposed framework, the classical staggered approach is reformulated as a loop of uncoupled monolithic steps that we call modified-staggered. Numerical examples show that an automatic switching between the coupled and the modified-staggered solution methods (when the former fails) may shorten the computational times (compared to the pure staggered approach) for an order of magnitude.
A phase-field description of brittle fracture is employed in the reported four-point bending analyses of monolithic and laminated glass plates. Our aims are: (i) to compare different phase-field fracture formulations applied to thin glass plates, (ii) to assess the consequences of the dimensional reduction of the problem and mesh density and refinement, and (iii) to validate for quasi-static loading the time-/temperature-dependent material properties we derived recently for two commonly used polymer foils made of polyvinyl butyral or ethylene-vinyl acetate. As the nonlinear response prior to fracture, typical of the widely used Bourdin–Francfort–Marigo model, can lead to a significant overestimation of the response of thin plates under bending, the numerical study investigates two additional phase-field fracture models providing the linear elastic phase of the stress-strain diagram. The typical values of the critical fracture energy and tensile strength of glass lead to a phase-field length-scale parameter that is challenging to resolve in the numerical simulations. Therefore, we show how to determine the fracture energy concerning the applied dimensional reduction and the value of the length-scale parameter relative to the thickness of the plate. The comparison shows that the phase-field models provide very good agreement with the measured stresses and resistance of laminated glass, despite the fact that only one/two cracks are localised using the quasi-static analysis, whereas multiple cracks evolve during the experiment. It was also observed that the stiffness and resistance of the partially fractured laminated glass can be well approximated using a 2D plane-stress model with initially predefined cracks, which provides a better estimation than the one-glass-layer limit.
In variational phase-field modeling of brittle fracture, the functional to be minimized is not convex, so that the necessary stationarity conditions of the functional may admit multiple solutions. The solution obtained in an actual computation is typically one out of several local minimizers. Evidence of multiple solutions induced by small perturbations of numerical or physical parameters was occasionally recorded but not explicitly investigated in the literature. In this work, we focus on this issue and advocate a paradigm shift, away from the search for one particular solution towards the simultaneous description of all possible solutions (local minimizers), along with the probabilities of their occurrence. Inspired by recent approaches advocating measure-valued solutions (Young measures as well as their generalization to statistical solutions) and their numerical approximations in fluid mechanics, we propose the stochastic relaxation of the variational brittle fracture problem through random perturbations of the functional. We introduce the concept of stochastic solution, with the main advantage that point-to-point correlations of the crack phase fields in the underlying domain can be captured. These stochastic solutions are represented by random fields or random variables with values in the classical deterministic solution spaces. In the numerical experiments, we use a simple Monte Carlo approach to compute approximations to such stochastic solutions. The final result of the computation is not a single crack pattern, but rather several possible crack patterns and their probabilities. The stochastic solution framework using evolving random fields allows additionally the interesting possibility of conditioning the probabilities of further crack paths on intermediate crack patterns.
The complex failure mechanisms of glass laminates under in-plane loading conditions is modelled within the framework of phase-field strategy. Laminated glass is widely used for structural purposes due to its safe post-glass-breakage response. In fact, the combination of several glass plies bonded together with polymeric interlayers allows overcoming the brittleness of the glass and to reach a pseudo-ductile response. Moreover, the post-breakage behaviour of the laminate is strictly correlated by the mechanical properties of the constituents. Ruptures may appear as cracks within the layers or delamination of the bonding interface. The global response of a glass laminate, validated against experimental results taken from the literature, is carried out by investigating a simplified layup of two glass plies connected by cohesive interfaces through an interlayer. Delamination of the adhesive interface is described, and crack patterns within the materials are fully described. Finally, the proposed approach put the basis for future comparisons with results of experimental campaign and real-life applications.
Glass edges result from cutting glass sheets and a further optional finishing. The mechanical interference into the brittle material glass causes flaws and cracks at the edge surface. Those defects have an influence on the strength of the whole glazing. Within the scope of a research project at the Institute of Building Construction from the Technische Universität Dresden, the grinding and polishing process is examined in terms of characteristic visible effects on the glass edge and the edge strength. Thereby a special focus of the research project is the impact of various polishing cup wheels for the chamfer surface of annealed glass. The article presents some basics about the processing steps of glass edges surfaces, introduces the considered grinding and polishing cup wheels and gives an overview of the performed experimental examinations. A microscopic analysis enables a characterisation of typical defects at the surfaces. Furthermore, four-point bending tests are performed to determine the bending tensile stresses at failure. The combination of both methods enables an analysis of the fracture-causing defect before destruction and a correlation between the optical surface quality and the bending tensile stresses. Additionally, the microscopy could be used to support the adjustment of a grinding machine and control reproducible edge qualities. The evaluation shows that a special development of polishing cup wheels for the chamfer can improve the surface quality and consequently increases the edge strength.
Because of their characteristic high slenderness ratios, laminated glass elements are frequently subjected to buckling phenomena. Here, simple analytical formulae for the evaluation of the effective thickness for the compressive buckling verification of laminated glass beams are proposed, based on the Enhanced Effective Thickness model, widely used for the design of laminated glass. The model applies also to multilaminates. Tables for the calculation of the relevant coefficients in the most common cases have been added for ease of reference and to facilitate the practical use. Comparison with numerical results, performed by considered paradigmatic cases, confirm the accuracy of the proposed formulae.
Fracture is one of the most commonly encountered failure modes of engineering materials and structures. Prevention of cracking-induced failure is, therefore, a major constraint in structural designs. Computational modelling of fracture constitutes an indispensable tool not only to predict the failure of cracking structures but also to shed insights into understanding the fracture processes of many materials such as concrete, rock, ceramic, metals and biological soft tissues. This manuscript provides an extensive overview of the literature on the so-called phase field fracture (PFF) models, particularly, for quasi-static and dynamic fracture of brittle and quasi-brittle materials, from the points of view of a computational mechanician. PFF models are the regularised versions of the variational approach to fracture which generalises Griffith's theory for brittle fracture. They can handle topologically complex fractures such as intersecting and branching cracks in both two and three dimensions with a quite straightforward implementation. One of our aims is to justify the gaining popularity of PFF models. To this end, both theoretical and computational aspects are discussed and extensive benchmark problems (for quasi-static and dynamic brittle/cohesive fracture) that are successfully and unsuccessfully solved with PFF models are presented. Unresolved issues for further investigations are also documented.
Laminated structures are parallel (or redundant) systems, i.e. failure occurs when all the elements (glass plies) reach, in cascade, the ultimate limit state. Following the failure mode approach, the reliability analysis is consequent to the identification of all possible rupture modes of the glass plies, where each mode is identified by the sequence of collapse, synthetically schematized as an event-tree. The event “structural failure” is the union of all the possible failure modes. The static theorem of limit analysis guarantees that the more the structure is divided into load bearing elements acting in parallel, the safer it is, but this conclusion holds only for ideal ductile systems. For brittle glass it is often assumed that lamination gives a beneficial contribution in all cases, but glass strength is affected by a size effect in terms of area, because surface micro-cracks govern the overall capacity of the material. Taking this into account, we show through the failure mode approach, under some simplifying assumptions, that lamination can decrease the strength of a plate made of annealed glass, since the higher the number of plies is, the larger is the surface area under tensile stress. This finding focalizes the attention on the importance of an accurate characterization of the size effect in glass strength.
The fracture strength of float glass (FG) is generally difficult to predict, since it is governed by a number of different factors. To bypass the influence of the above-mentioned variables, ruling out any possible source of uncertainty, a reliable testing procedure should be able to produce an equibiaxial stress state in the central part of the specimen. The Coaxial Double Ring (CDR) test configuration is able to achieve such an ideal condition, but only if geometric nonlinearities are of lesser importance. A number of alternative test set-ups have been investigated in recent years, but none of these offers such advantages as to be preferred a priori. To this end, this study aims at finding out, through experimental investigations, an alternative CDR configuration to predict the failure strength of glass plates as a starting point for the
development of a design-by testing procedure. A total of 393 FG square-shaped specimens, with different edge lengths (100 or 400 mm) and thicknesses (4, 5, 6, 8 or 10 mm), were tested using four different types of CDR test set-ups (obtained by varying the dimensions of the loading area). Laboratory outcomes, after being corrected via FEM simulations, were then interpolated using a Weibull-type statistical distribution to derive the best-fit parameters of probability density functions, which were finally used to provide the characteristic values of the glass failure strength.
The edge strength of annealed float glass is an essential aspect in engineering design. A common problem in engineering applications is the failure of the edge due to thermally induced stresses. The edge strength of annealed glass is affected by cutting, further processing and handling. Thermally treated glass exhibits higher edge strengths, but may show optically negative effects such as anisotropy and roller waves. In case of covered edges, e.g. insulating glass covered by framing, an enhanced strength of the cut edge is desirable. This paper presents results showing that enhanced strength of the cut edge is feasible and reproducible using regular cutting technology. A large part of this contribution deals with the crack system resulting from the cutting process. A deep understanding of this is essential for the relationship between cutting parameters, damage and edge strength. In addition to optical microscopic analyses, confocal microscopy and indentation tests have been carried out. The influence of different cutting process parameters on the edge strength was investigated within the scope of extensive analyses by the Fachverband Konstruktiver Glasbau e.V. This allowed to adjust selected process parameters so that the edge strength could be reproducibly enhanced. In this contribution the latest results of these investigations are presented. In addition, the correlation between microscopic optical characteristics and the mechanical strength as expressed by the macroscopic fracture stress is described.
A powerful numerical instrument, which reproduces the complex failure mechanisms of hybrid laminates under in-plane loading conditions, is developed within the framework of phase-field modelling. The ruptures, strongly influenced by geometrical and mechanical properties of the plies and affected by the state of stress, are arranged as delamination of the adhesive interface and intricate crack patterns within the layers. Therefore, the mechanical response of a hybrid laminate is obtained by studying the simplified layup of two elastic-brittle solids connected by a cohesive interface. Explicit and well detailed simulations illustrate peculiar failure mechanisms, validated, when possible, against experimental results taken from the literature and compared to simplified analytical models. Different in-plane loading conditions are explored together with the possibility to include material anisotropy. The proposed model is a first attempt to provide an effective design tool for the understanding of the intriguing failure of hybrid laminates and the enhancement of their mechanical properties like ductility.
An accurate material representation of polymeric interlayers in laminated glass panes has proved fundamental for a reliable prediction of their response in both static and dynamic loading regimes. This issue is addressed in the present contribution by examining the time–temperature sensitivity of the shear stiffness of two widely used interlayers made of polyvinyl butyral (TROSIFOL BG R20) and ethylene-vinyl acetate (EVALAM 80-120). To that end, an experimental program has been executed to compare the applicability of two experimental techniques, (i) dynamic torsional tests and (ii) dynamic single-lap shear tests, in providing data needed in a subsequent calibration of a suitable material model. Herein, attention is limited to the identification of material parameters of the generalized Maxwell chain model through the combination of linear regression and the Nelder–Mead method. The choice of the viscoelastic material model has also been supported experimentally. The resulting model parameters confirmed a strong material variability of both interlayers with temperature and time. While higher initial shear stiffness was observed for the polyvinyl butyral interlayer in general, the ethylene-vinyl acetate interlayer exhibited a less pronounced decay of stiffness over time and a stiffer response in long-term loading.
The use of glass and, in particular, laminated glass (LG) in the building industry is a continuously rising trend. Thus, in order ensure a safe fracture and a post-breakage load bearing capacity but still be economical, the consideration of the shear transfer by the polymeric interlayer between the glass panes is gaining importance in the structural design. This also relates to future European harmonised product and design standards in glass construction. In addition to simple models using particular shear modulus values for certain load situations and temperatures, linear viscoelastic material models for the interlayer can be applied for more complex design situations. We present the basics of mechanical modelling of the time- and temperature dependent material behaviour of polymers used in LG. The procedure for the experimental determination of the material parameters is explained for the most common product polyvinyl butyral. For practical application in glass design, different standards (national and international) and other building regulations are compared. The analysis of the compliant composite shows that already low shear modulus values lead to a significant reduction in glass stresses. Finally, a comparative example shows the design of laminated glass considering the shear transfer with different methods.
In this paper, the complex failure process of unidirectional hybrid laminates under uniaxial loading condition is reproduced and investigated by a one-dimensional phase-field model. The key ingredients of the approach, describing the mechanical response of a hybrid composite made of two different layers, are: (i) a phase-field method, based on a variational formulation of brittle fracture with regularised approximation of discontinuities for the two layers, (ii) cohesive law for the adhesive interface that connects the layers and (iii) robust and consolidated numerical strategy for the solution of the non-linear discretised problem. Explicit and well detailed simulations are shown for four peculiar failure mechanisms and the outcomes validated against experimental results available in literature. The model is able to discriminate among these different failure mechanisms according to the geometrical and mechanical properties of the hybrid composite. Both delamination of the adhesive interface is followed and crack patterns within the materials are fully determined. Finally, the proposed approach opens new perspective studies in higher dimension settings.
Being one of the most promising candidates for the modeling of localized failure in solids, so far the phase-field method has been applied only to brittle fracture with very few exceptions. In this work, a unified phase-field theory for the mechanics of damage and quasi-brittle failure is proposed within the framework of thermodynamics. Specifically, the crack phase-field and its gradient are introduced to regularize the sharp crack topology in a purely geometric context. The energy dissipation functional due to crack evolution and the stored energy functional of the bulk are characterized by a crack geometric function of polynomial type and an energetic degradation function of rational type, respectively. Standard arguments of thermodynamics then yield the macroscopic balance equation coupled with an extra evolution law of gradient type for the crack phase-field, governed by the aforesaid constitutive functions. The classical phase-field models for brittle fracture are recovered as particular examples. More importantly, the constitutive functions optimal for quasi-brittle failure are determined such that the proposed phase-field theory converges to a cohesive zone model for a vanishing length scale. Those general softening laws frequently adopted for quasi-brittle failure, e.g., linear, exponential, hyperbolic and Cornelissen et al. (1986) ones, etc., can be reproduced or fit with high precision. Except for the internal length scale, all the other model parameters can be determined from standard material properties (i.e., Young’s modulus, failure strength, fracture energy and the target softening law). Some representative numerical examples are presented for the validation. It is found that both the internal length scale and the mesh size have little influences on the overall global responses, so long as the former can be well resolved by sufficiently fine mesh. In particular, for the benchmark tests of concrete the numerical results of load versus displacement curve and crack paths both agree well with the experimental data, showing validity of the proposed phase-field theory for the modeling of damage and quasi-brittle failure in solids.
We are concerned with finite element modeling of geometrically non-linear laminated glass beams consisting of stiff elastic glass layers connected with compliant polymeric interlayer of temperature-sensitive viscoelastic behavior. In particular, four layerwise theories are introduced in this paper, which differ in the non-linear beam formulation used at the layer level (von Kármán/Reissner) and in constitutive assumptions used for interlayer (viscoelastic solid with time-independent bulk modulus/Poisson ratio). We show that all formulations deliver practically identical responses for simply-supported and fixed-end three-layer beams. For the most straightforward formulation, combining the von Kármán model with the assumption of time-independent Poisson ratio, we perform detailed verification and validation studies at different temperatures and compare its accuracy with simplified elastic solutions mostly used in practice. These findings provide a suitable basis for extensions towards laminated plates and/or glass layer fracture, owing to the modular format of layerwise theories.
We present an approach for phase-field modeling of fracture in thin structures like plates and shells, where the kinematics is defined by midsurface variables. Accordingly, the phase field is defined as a two-dimensional field on the midsurface of the structure. In this work, we consider brittle fracture and a Kirchhoff–Love shell model for structural analysis. We show that, for a correct description of fracture, the variation of strains through the shell thickness has to be considered and the split into tensile and compressive elastic energy components, needed to prevent cracking in compression, has to be carried out at various points through the thickness, which prohibits the typical separation of the elastic energy into membrane and bending terms. For numerical analysis, we employ isogeometric discretizations and a rotation-free Kirchhoff–Love shell formulation. In several numerical examples we show the applicability of the approach and detailed comparisons with 3D solid simulations confirm its accuracy and efficiency.
Stochastic phase-field models can reproduce the nonlinearity and randomness of the mechanical behavior of quasi-brittle materials within a unified macroscopic continuum framework. To study the probabilistic characteristics of microscopic and macroscopic strength within the stochastic phase-field model and their relationship, a series of numerical results of a specimen subjected to uniaxial tension using a stochastic phase-field model is presented. The probabilistic characteristics of the macroscopic responses and the underlying mechanisms are discussed. Based on the numerical results, a probabilistic model is developed, which can effectively determine the probability information of the macroscopic structural strength of quasi-brittle uniaxial tensile specimens. In particular, the interaction of several characteristic scales involved in the stochastic phase-field model can be demonstrated.
The prediction of fracture in thin‐walled structures is decisive for a wide range of applications. Modelling methods such as the phase‐field method usually consider cracks to be constant over the thickness which, especially in load cases involving bending, is an imperfect approximation. In this contribution, fracture phenomena along the thickness direction of structural elements (plates or shells) are addressed with a phase‐field modeling approach. For this purpose, a new, so called “mixed‐dimensional” model is introduced, which combines structural elements representing the displacement field in the two‐dimensional shell midsurface with continuum elements describing a crack phase‐field in the three‐dimensional solid space. The proposed model uses two separate finite element discretizations, where the transfer of variables between the coupled two‐ and three‐dimensional fields is performed at the integration points which in turn need to have corresponding geometric locations. The governing equations of the proposed mixed‐dimensional model are deduced in a consistent manner from a total energy functional with them also being compared to existing standard models. The resulting model has the advantage of a reduced computational effort due to the structural elements while still being able to accurately model arbitrary through‐thickness crack evolutions as well as partly along the thickness broken shells due to the continuum elements. Amongst others, the higher accuracy as well as the numerical efficiency of the proposed model are tested and validated by comparing simulation results of the new model to those obtained by standard models using numerous representative examples.
In this study ring bending tests on plates are performed to analyze the tensile strength of tin and air sides of float glass. Fracture stress values computed from experimental data applying the plate theory are analyzed by means of Weibull distribution. Probability of failure vs. fracture stress plots show much higher scatter for the air side than for the tin side. The linear peridynamic solid constitutive model coupled with damage is applied to simulate the ring bending, up to the critical state of crack initiation and early stage of crack propagation. A convergence study with respect to the number of nodes is performed. The simulated sequence of initial damage patterns, including the bond breakage inside the load ring and the formation of ring damage zone followed by initiation of radial cracks agrees with experimental observations for most specimens with tin side subjected to tensile deformation. A novel procedure to identify the critical bond stretch in the damage model based on experimental data for the critical ring force is proposed. The corresponding characteristic values are presented for both tin and air sides of the float glass.
In this paper, a stochastic phase-field model is presented to simulate the nonlinearity and randomness of the mechanical behaviors of quasi-brittle materials within a unified macroscopic continuum framework. By characterizing the spatial variations in material failure strength and fracture toughness for fully correlated random fields, a complete theoretical framework of the stochastic phase-field model is established, and a corresponding numerical analysis program is proposed. In particular, this model is used with the probability density evolution theory to derive the probability density function of the mechanical response. Numerical results show the effect of material randomness on the failure patterns and macroscopic responses of quasi-brittle media under tension. Comparisons between the experimental and numerical results demonstrate the effectiveness of the proposed model in replicating the probabilistic information of macroscopic responses of random quasi-brittle structures. Furthermore, the effects of autocorrelation length on probabilistic failure of quasi-brittle media are investigated by the stochastic analysis of a three-point bending beam.
Continuum finite element (FE) modeling of damage and failure of quasibrittle structures suffers from the spurious mesh sensitivity due to strain localization. This issue has been addressed for deterministic analysis through the development of localization limiters. This study proposes a mechanism-based model to mitigate the mesh sensitivity in stochastic FE simulations of quasibrittle fracture. The interest is placed on the analysis of large-size structures, where the mesh size is conveniently chosen to be larger than the width of the fracture process zone as well as the correlation length of the random fields of constitutive properties. The present model is formulated within the framework of continuum damage mechanics. Two localization parameters are introduced to describe the evolution of the damage pattern of each finite element. These parameters are used to guide the energy regularization of the constitutive law, as well as to formulate the mesh-dependent probability distributions of constitutive properties. Depending on the prevailing damage pattern, different energy regularization schemes and mesh dependence of the probability distribution functions are used in the constitutive law. The model is applied to simulate the stochastic failure behavior of quasibrittle structures of different geometries featuring different failure processes including damage initiation, localization, and propagation. It is shown that using fixed probability distribution functions of constitutive properties could lead to strong mesh dependence of the prediction of the mean and variance of the peak load. The probability distribution functions of constitutive properties must be linked to the damage pattern, which may evolve during the failure process. Such a mechanism-based modeling of the probability distributions of constitutive properties is essential for mitigating the spurious mesh sensitivity in stochastic FE analysis of quasibrittle fracture.
In this paper, a phase-field model is presented for the description of brittle fracture in a Reissner-Mindlin plate and shell formulation. The shell kinematics as well as the phase-field variable are described on the midsurface of the structure. Non-Uniform Rational B-Spline basis functions are used for the discretization of both the displacement/rotations and the phase-field. The spectral decomposition for the tension-compression split is applied on the total strain tensor, which varies through the thickness. Thus, the plane stress condition has to be enforced numerically. Various numerical examples are presented in order to verify the accuracy and effectiveness of the method and a detailed comparison to existing formulations is performed.
We present a phase field model to simulate brittle fracture in an initially straight Euler–Bernoulli beam, with generalization to curved beams. We start from formulating the problem with the principle of minimum potential energy in a 3D solid, with the displacement field and the phase field as primary arguments. We then select, for each cross section, representative fields that characterize the said cross section, including the beam deflection and rotation, and two independent ansatz variables within the cross section to represent the phase field. The problem then reduces to a minimization with only one-dimensional field variables. A feature of the proposed method is, without discretizing the phase field within the cross section, it can represent its variation within the cross section, allowing to simulate cracks partially going through the thickness due to bending as well as axial loads.
Laminated glass, as a typical kind of sandwich-structure composite, is widely used in various fields such as safety, vehicle and transportation engineering. Laminated glass mainly suffers from dynamic/impact loadings, and glass-ply cracking is the main failure pattern of laminated glass. This paper presents a non-local ordinary state-based peridynamic modeling and numerical approach for simulating the dynamic failure process of sandwiched laminated glass under low-velocity impact loading. The mechanical behavior of the PVB layer is simulated by reformulating classical visco-elastic model under the framework of ordinary state-based peridynamic theory, and the adhesion between the glass and PVB interlayer is described by using a penalty-based method. Fracture patterns of a laminated glass plate under drop-weight loading were investigated, and the numerical results compared well with experimental observations. Furthermore, a series of numerical simulations were conducted to in-depth analyze the effect of the thickness of the PVB interlayer and the glass layers on the fracture mode, initial locations and crack propagation speed of the laminated glass when subjected to impact.
It is a consolidated belief that the reliability of brittle plates is increased using laminated layers instead of a monolith: redundancy permits stress redistribution after partial breakage. To quantify this effect, the retro-cumulative function of plate strength is calculated for inflected laminates with a variable number of plies using the failure modes approach. The material is elastic perfectly-brittle and its strength may be characterized by different statistics, implying either volume-size-effect, area-size-effect, or no-size-effect. The plate is analysed in limit conditions for what concerns the capability of the adhesive layers to transfer shear stresses. Robustness is discussed by assuming that an accidental event may break one of the plies.
The type of size-effect plays a decisive role. Under the same maximum tensile stress in the sound state, lamination is always beneficial only for volume-size-effect. For area-size-effect, the negative effect of the new surface from lamination is counterbalanced by stress redistribution only for the lower quantiles of the strength distribution. The same is true under an accidental event. With no-size-effect, the gain from lamination is again appreciable only at low failure probabilities. These findings raise questions about the positive effects of lamination for structural glass, strongly characterized by an area size effect.
The phase-field approach to fracture has been proven to be a mathematically sound and easy to implement method for computing crack propagation with arbitrary crack paths. Hereby crack growth is driven by energy minimization resulting in a variational crack-driving force. The definition of this force out of a tension-related energy functional does, however, not always agree with the established failure criteria of fracture mechanics. In this work different variational formulations for linear and finite elastic materials are discussed and ad-hoc driving forces are presented which are motivated by general fracture mechanical considerations. The superiority of the generalized approach is demonstrated by a series of numerical examples.
The subject of the paper is modelling of complex damage behaviour of laminated glass, which is a sandwiched polymer composite. Last 20 years have witnessed extensive growth in computational modelling of complex nonlinear behaviour of laminated glass panels with viscoelastic interlayers. Present review article is an attempt to highlight applications of finite element approach in failure analysis of laminated glasses. Evolution of modelling theories such as equivalent single layer theories, layerwise theories and zig-zag models have been presented to insight the fundamental concepts used in modelling. Finite element techniques such as erosion, cohesive zone and extended finite element methods available in FE packages are briefly reviewed alongwith material models for glass and interlayer. The systematic growth in FE models in numerical simulation of human head impact on windshields and blast loading is specifically presented. Finite element models have been proposed as useful guides to engineers for fabrication of laminated glasses with more realistic data to be used in real world situations without going for expensive and time consuming experimental set ups. Overall 223 research articles have been included exhibiting the research performed in context of laminated glass.
Any reliable use of glass for structural purposes cannot neglect that its breakage may be provoked by imponderable events, like impacts at critical spots or thermal shocks. Laminated glass, composed by glass plies sandwiching polymeric interlayer sheets, is used in architectural application thanks to its safe post-glass breakage response. When glass breaks, the interlayer retains the glass shards, and the cracked element maintain a certain residual load-bearing capacity, strongly influenced by the tension stiffening of the polymeric interlayer due to the adhesion with the glass shards, which depends upon the size of the shards and of the debonded zone. Here, we review the most recent experimental results on the post-glass breakage response of laminated heat-treated glass elements providing charts for the evaluation of such a stiffening effect. Based on this, simple formulas to analyze and interpret the experimental findings under both in-plane and out-of plane bending are proposed, providing analogies with the bending of bimodulus materials and the load-bearing mechanism of reinforced concrete, respectively.
This paper focuses on the modal analysis of laminated glass beams. In these multilayer elements, the stiff glass plates are connected by compliant interlayers with frequency/temperature-dependent behavior. The aim of our study is (i) to assess whether approximate techniques can accurately predict the behavior of laminated glass structures and (ii) to propose an easy tool for modal analysis based on the enhanced effective thickness concept by Galuppi and Royer-Carfagni. To this purpose, we consider four approaches to the solution of the related nonlinear eigenvalue problem: a complex-eigenvalue solver based on the Newton method, the modal strain energy method, and two effective thickness concepts. A comparative study of free vibrating laminated glass beams is performed considering different geometries of cross-sections, boundary conditions, and material parameters for interlayers under two ambient temperatures. The viscoelastic response of polymer foils is represented by the generalized Maxwell model. We show that the simplified approaches predict natural frequencies with an acceptable accuracy for most of the examples. However, there is a considerable scatter in predicted loss factors. The enhanced effective thickness approach adjusted for modal analysis leads to lower errors in both quantities compared to the other two simplified procedures, reducing the extreme error in loss factors to one half compared to the modal strain energy method or to one quarter compared to the original dynamic effective thickness method.
Glass has been overwhelmingly used for windows and facades in modern constructions, for many practical reasons, including thermal, energy, light and aesthetics. Nevertheless, due to the relatively low tensile strength and mostly brittle behaviour of glass, compared to other traditional materials, as well as to a multitude of interacting structural and non-structural components, windows/facades are one of the most fragile and vulnerable components of buildings, being representative of the physical line of separation between interior and exterior spaces. As such, multidisciplinary approaches, as well as specific fail-safe design criteria and analysis methods are required, especially under extreme loading conditions, so that casualties and injuries in the event of failure could be avoided and appropriate safety levels could be guaranteed. In this context, this paper presents a review of the state of art on analysis and design methods in use for glass facades, with careful consideration for extreme loading configurations, including natural events, such as seismic events, extreme wind or other climatic exposures, and man-made threats, i.e. blast loads and fire. Major results of available experimental outcomes, current issues and trends are also reported, summarising still open challenges.
The failure of a laminated glass beam is investigated by two full discrete numerical approaches: a Rigid Body-Spring Model (RBSM) and a mesh-free numerical method arising from bond-based Peridynamics (PD). The brittle nature of the failure has been modelled and investigated by exploiting the discrete nature of these models, and specifically the PD which allows the bond/spring strengths to be explicitly related with the size and orientation of the defects in the structure. Strength values have been assigned randomly, within the beam, by a Monte Carlo simulation, according to Weibull statistical distributions calibrated on experimental results ob- tained from literature. For the first time, the differences and analogies of the two discrete approaches are shown and discussed together with the analysis of variability of the load capacity of the beam related to the statistical presence of flaws in the structure. Results show that, due to the heterogeneous strength properties of the numerical models and mechanical features of the inter-layer, multiple cracking stages can be distinguished for the structural element, thus different cumulative distribution function of limit load can be obtained.
Finite element analysis Discrete element method Extended finite element method Combined finite discrete element method a b s t r a c t This paper presents a comparative study on the available numerical approaches for modelling the fracturing of brittle materials. These modelling techniques encompass the finite element method (FEM), extended finite element method (XFEM), discrete element method (DEM) and combined finite-discrete element method (FEM/DEM). This study investigates their inherent weaknesses and strengths for modelling the fracture and fragmentation process. A comparative review is first carried out to illustrate their fundamental principles as well as the advantages for the modelling of cracks, followed by the state-of-the-art trial application in the example cases. An example of a glass beam subjected to low velocity hard body impact is examined as a plane stress problem. By evaluating the applicability of different models, the most desirable model for the entire dynamic fracture response is identified, and this is found to be the FEM/DEM. The FEM/DEM model is further examined by comparing results with the experimental data from high velocity and oblique impact tests. The study reveals that the FEM/DEM yields the most satisfactory results when modelling the dynamic fracture process of brittle materials such as glass.
Tempered glass panes are subjected to high eigenstresses that induce a state of compression along the surfaces and a state of tension in the inner part. Whenever a crack reaches the tensile region, it rapidly propagates and branches in all directions driven by the eigenstress. These mechanisms induce dynamic fragmentation. The present work contains a numerical investigation of this phenomenon on panes with different thicknesses, using massively parallel simulation based on FEM with the dynamic insertion of cohesive elements. Simulations are first validated by comparing the obtained number of fragments with experimental data. Then, the resulting energy fields are examined and they show that the dissipated energy is significantly underestimated by the existing analytical models. Finally, an extended analytical model that includes the influence of the plate thickness is proposed to correctly estimate the number of fragments for high eigenstresses.
An enhanced non-local failure criterion for laminated safety glass under low velocity impact for explicit finite element simulations with element erosion is presented. Hereby, the fracture strength of the glass depends on the stress rate and is approximated by a power law for the crack velocity resulting from the sub–critical crack growth. In dependence of the size of the finite elements, the singularity of the stress field in the vicinity of the crack tip is underestimated by the element erosion technique. In contrast to classical linear elastic fracture mechanics, the model uses a non-local approach for a decrease of strength in the direction of a crack perpendicular to the maximum in–plane stresses, instead of an increase of the stress field in the vicinity of the crack tip. The failure criterion is implemented in the explicit solver LS–DYNA as an user defined material model. In order to investigate the accuracy of the model, the results are compared to head impact tests on windscreens used in the context of crash simulations.
Laminated glass is a simple sandwiched composite structure, while being widely used in the automotive industry as windshield glazings. It is considered to be safety glass due to its excellent performance in absorbing impact energy and bonding glass fragments. Meanwhile, the impact failure patterns of an automotive windshield glazing contribute to the traffic accident reconstruction. In recent decades, a growing interest has been devoted to the impact failure analysis of automotive laminated glass by means of numerical simulations. The purpose of this work is to present a comprehensive review concerning this aspect. We start by introducing six numerical algorithms for the modeling of the principal damage pattern, glass-ply cracking, followed by the introduction of material models for the plastic interlayer, PVB, and then address three numerical techniques for the adhesion modeling. Three kinds of laminated glass models are summarized. Finally, the performance of the numerical algorithms on the impact failure analysis of laminated glass in terms of glass-ply cracking and acceleration history is thoroughly discussed.
A numerical investigation on the load carrying capacity of a laminated glass beam modelled as a material with a random strength distribution is presented. The strength values were distributed randomly within the beam by a Monte Carlo simulation, according to statistical distributions calibrated on experimental results obtained from literature. A preliminary computational analysis based on the weakest link in the chain-model was conducted to study the dependence of the beam estimated limit load on the adopted discretization.
Then, after determining the optimal size of the mesh, the elastic-plastic problem has been solved by a Rigid Body-Spring Model (RBSM) discrete approach. Finally, the variability of the load capacity of the structural element is evaluated as a function of the statistics of the strength related to the size of the defects. One thousand simulations were performed to obtain statistically significant quantitative results.
Laminated glass, a composite made by bonding glass plies with polymeric interlayers, can maintain a significant load bearing capacity even when glass is broken, because of the adhesion of the glass shards to the interlayer. The post-breakage response is strongly associated with the safety performance, but it is not as well studied as the pre-glass breakage stage. Here, the tension stiffening of the interlayer due to the adhesive contact with the shards is considered as a stress perturbation, determined though a variational approach by minimizing the complementary energy functional. This allows to reach a simple but accurate estimate of the effective stiffness of the cracked laminate, which results to be strongly dependent, besides the interlayer elastic properties, upon the fragment size and the glass-to-polymer adhesion. An energetic competition à la Griffith determines the expected load vs. displacement response in a tensile test. Comparisons with numerical experiments confirm the good accuracy of the proposed approach.
