Probabilistic modelling of Hertzian fracture of glass by flying objects impact in bad weather

https://doi.org/10.1016/j.ijimpeng.2018.03.010Get rights and content

Highlights

  • Stochastic methodology has been developed for predicting Hertzian failure load of a glass panel.

  • Loading rate effect has been incorporated into the model.

  • Predicted CPD functions of failure load of glazing panels have been verified experimentally.

  • Probability of Hertzian failure of glass panels in scenarios of hail impact has been illustrated.

Abstract

Impact of hailstone and flying debris in extreme weather conditions is a major contributor to damage to glazing panels and other types of building facades. Localized damage to the glazing panel in the form of Hertzian fracture is controlled mainly by the amount of force that is developed at the point of contact. Weibull statistics forms the foundation of contemporary probabilistic models designed to estimate the value of the limiting force to result in Hertzian fracture. This conventional approach to probabilistic modelling has significant shortcomings as the calibrated probabilistic parameters have to be specific to the type of installations. The alternative approach of predicting risks of Hertzian fracture is by stochastic simulations of Griffith flaws. This paper presents an adaptive stochastic simulation methodology for predicting risks of Hertzian fracture of glass which is subject to the impact by hail and flying debris. The proposed simulation model has been verified experimentally for annealed glass based on comparison of the simulated contact force values against results from experimental measurements. The introduced simulation procedure has been used for assessing the risk of Hertzian fracture in both annealed and toughened glass. The use of the adaptive stochastic modelling methodology for predicting Hertzian failure in scenarios of hail impact is illustrated in this paper by examples.

Introduction

Glazing panels are the most vulnerable kind of installation in a building, in terms of susceptibility to damage in bad weather. Damage to building facades and skylights caused by flying debris or hailstones may have serious consequences, that are associated with risks of electrocution and fire safety hazards in addition to disrupting the continuous functionality of the building [1]. Thus, the design of facades and skylights (including the selection of glazing products) can have important implications for stakeholders and occupants of the building in terms of safety and protection of the property and its contents.

To address the potential hazards of flying debris, and hail, current building codes of practice for the design of building facades contain provisions for testing specimens of glazing panels by accelerating (i) a timber plank of cross-sectional dimensions: 100 mm  ×  50 mm, weighing 4 kg, at an incident velocity of impact 0.4 times the regional wind velocity and (ii) an 8 mm diameter steel sphere with the same velocity [2]. Most experimental and numerical investigations on debris hazards were based on these two types of impact tests [3], [4], [5]. These impact experiments are limited to determining the qualitative behaviour (pass or fail criteria) of a glazing panel based on the provided testing guidelines. The main drawback of such regulatory practices is uncertainty as to how well these impact tests represent the performance behaviour of glazing installations of different dimensions, other than the tested specimens. There are also doubts as to how well observations from impact tests employing steel balls and timber planks as impactor objects that are accelerated on a glazing specimen as target be representative of impact by hail ?.

The well-known probabilistic model of Fischer-Cripps and Collins [6] and Fischer-Cripps [7] is based on the concept of crack growth. The probability of fracture is essentially determined by modelling by the probability of the size of the critical Griffith flaw exceeding the threshold for crack growth. However, the behaviour of any individual flaw that varies in size and is subjected to very different stress conditions is not modelled. The values of the Weibull parameters k and m can simply be obtained from calibration against results from physical experimentation. The (calibrated) values are dependent on a range of factors including the dimensions of the glass pane, the stress history and the degree of seasoning [8], [9], [10], [11], [12]. Thus, a calibrated probabilistic model has to be specific to the type of installation. Individual modelling parameters do not represent physical attributes, given that the derivation of their values is purely empirical in nature. The generality of the model is therefore uncertain because individual Griffith flaws and their behaviour are not represented in the model.

A different approach to probabilistic modelling of glazing panels is based on numerical simulations of individual Griffith flaws and their disposition within the glass pane. The random nature of such properties is represented by the stochastic process [8], [13]. In a recent development over the stochastic simulation of the strength of glazing panels the risks of triggering flexural failure when subject to a point contact style of impact was modelled by Pathirana, et al. [14]. A statistical model defining the distribution of the size of flaws was derived in that study. By stochastic simulations, multiple models of the glass pane featuring a disposition of flaws of known size and location can be generated for analyses to determine if crack growth is triggered by a flaw of known location, dimensions and stress state. Thus, the simulation parameters have their own physical meaning and are not merely mathematical artefacts. The advantage of the use of a stochastic model over the more conventional probabilistic model of Fischer-Cripps and Collins [6] and Fischer-Cripps [7] is its versatility and proven validity, based on a comparison of simulated results with experimental results involving specimens of different dimensions and types. The potential benefit of this new development is profound, as the accuracy of the simulations has been demonstrated across a range of conditions.

There are two common failure modes of glass when subjected to the impact of hail or flying debris: (a) flexural failure and (b) Hertzian fracture (Fig. 1a and b). The governing mode of failure depends on the thickness and size of the glass panel, as well as the amount of force developed at the point of contact (referred to herein as “contact force” which was first introduced by Yang, et al. [15] and Sun, et al. [16]). Thicker or larger size glass panels potentially generate higher inertial resistant to counter the impulsive action of the impact which is responsible for failure by flexure, and more so when subjected to higher impact velocity. This phenomenon has been incorporated into the calculation of resistance to impulsive action as illustrated in the literature [14]. The required value of quasi-static force to fail a thicker piece of glass in flexure is also higher because of its higher bending capacity. In contrast, Hertzian fracture which is the subject matter of interest in this study is a localised phenomenon and hence is not sensitive to changes in the values of parameters controlling inertial resistance, namely thickness and size of panel. Hertzian fracture is instead characterised by the formation of a hole at the point of contact together with radial and circular cracks surrounding the hole. For this reason, flexural failure caused by the impulsive action of the impact, which involves deflecting the glass pane, is more likely to occur in slow impact conditions and particularly thinner glass panes than Hertzian fracture. The latter mode of failure is more likely to occur in a thicker glass panel when impacted by a hard projectile at a high velocity [17], [18], [19].

Intuitively, the stochastic model developed for defining the distribution of flaw size may well be used to predict the probability of failure of glass irrespective of the mode of failure (i.e. flexural or Hertzian) provided that suitable computations for predicting crack growth for both failure modes have been incorporated into the simulation. Thus, the existing model defining flaw size distribution (originally derived from the testing of annealed glass failing in flexure) may well be adapted for the probabilistic modelling of Hertzian fracture. However, this modelling concept can only be a proposition until the accuracy of predicting both types of failure using the same stochastic model has been validated by physical experimentation.

The objective of this paper is to present a stochastic model for simulating Hertzian fracture in glass as well as to validate the accuracy of the model by comparison of the simulated results against results recorded from physical experimentation. The key original contribution to knowledge in this paper is the validation of a stochastic model that is sufficiently versatile to cover both types of failure mechanisms for both annealed and toughened glass by the use of a single model. The remainder of this paper is structured as follows:

  • (a)

    Section 2 - illustrates the use of existing models to predict Hertzian fracture deterministically

  • (b)

    Section 3 - adapts an existing probabilistic function for flaw size distribution (in conjunction with the deterministic model presented in “(a)”) to develop the stochastic model for Hertzian fracture in annealed glass

  • (c)

    Section 4 - presents results from physical experimentation of annealed glass specimens to validate the accuracy of the stochastic model developed in “(b)”

  • (d)

    Section 5 - demonstrates the validity of the stochastic simulation model for the prediction of Hertzian fracture in toughened glass

  • (e)

    Section 6 - applies the validated methodology to predict the conditional probability of Hertzian fracture of annealed and toughened glass when impacted by hail in projected storm scenarios.

Section snippets

Deterministic modelling for Hertzian fracture

The contact force bearing capacity (Pc) for Hertzian fracture (resulting in punching failure) on a glass pane when loaded by a spherical indenter can be predicted deterministically by the use of the model developed by Mouginot and Maugis [22] as defined by Eq. (1).Pc=[a3ϕ(c/a)]0.5[E2γπ34(1υ2)]0.5where a is the radius of the circle representing the contact area, c is the size of the crack, E is Young's modulus of glass, γ is the fracture surface energy, υ is Poisson's ratio of glass, and ϕ(c/a)

Stochastic simulations of Hertzian fracture in annealed glass

The previous section presented a deterministic model for predicting Hertzian fracture on the flat surface of a glazing panel. The model enables the limiting contact force for initiating fracture to be found for a considered flaw length (c) and the proximity of the “starting point” of the flaw from the centre point of contact (ro). The deterministic model on its own has limited utility in practice because the controlling parameters in relation to the number, length and location of the flaws are

Experimental validation of stochastic model for annealed glass

The development of the stochastic model to trigger Hertzian fracture in annealed glass is based on tests involving use of spherical specimens (and simulations of impact by a spherical indenter) in order that the same tests can be repeated easily in the future should further verification be deemed necessary. It has been demonstrated that the impact action generated by a spherical indenter is representative of the mean of the results generated by impactor of random irregular shapes (with an

Deterministic modelling of Hertzian fracture of toughened glass

The same iterative procedure introduced in Section 2 for determining the value of the failure load (Pcr) of annealed glass was extended to toughened glazing panels. In the case of toughened glass panels, pre-compressive stresses are introduced to the surface for increasing the failure strength well above that of annealed glass. The typical value of residual compressive stress of the surface of a toughened glass panel varies in between specimens in the range: 80 MPa–180 MPa [24]. The effects of

Contact force as controlling parameter

Studies on Hertzian fracture as presented in the earlier parts of this paper were based on experiments employing a steel indenter object that allowed an accelerometer to be attached for measuring acceleration (hence, impact force). The effects of the hardness of the indenter object have not been taken into consideration in these experiments. Thus, the amount of impact intensity required to initiate Hertzian fracture could have been misrepresented for cases where the impactor was not as hard as

Conclusion

In a previous study, stochastic simulations were undertaken to derive the probability of failure of glass. The calibrated flaw density and statistical distribution of flaw size were central to the stochastic model, which was proven to be sufficiently robust in view of the good accuracy of predictions when applied to glazing panels of varying dimensions. However, only results taken from quasi-static tests to flexural failure had been incorporated into the modelling. Hertzian fracture was not

Acknowledgement

The authors would like to acknowledge the support of Mr. Suresh Sutrave (Managing Director at Atlite Skylights Pty. Ltd.) in terms of providing toughened glass specimens for conducting impact experiments forming part of the research program.

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