The glass used in this study is a sodium aluminosilicate glass. Three thicknesses (0.4, 0.7, and 1.0 mm) of glass sheets are cut to 50 mm × 50 mm in size, and strengthened in refined grade KNO₃ (Sigma-Aldrich) salt baths for three different durations of time in order to achieve three distinct CT levels for each thickness. The salt bath treatment treats are adjusted depending on the glass thickness but are on the order of several hours. The surface compression stress (CS) and depth of layer (DOL) of the compressive stress are measured using a front surface stress measuring device FSM-6000 (manufactured by Orihara Manufacturing Co., Ltd.). The CT is calculated using the following equation: CT×(t−2×DOL)=CS×DOL where t is the thickness of the sample. It is worth noting that this equation uses a linear approximation of the real stress profile shape, as assumed by the FSM-6000 device. Since the frangible/non-frangible transition range is of the greatest interest for this study, the ion-exchange process is designed to target stress profiles near the CT threshold of frangibility (Allan et al., 2010; Barefoot et al., 2012).

The fracturing of the sample is more difficult than many would expect. One can break a glass by striking with a hammer, but with this technique the resulting fracture pattern would be dominated by the external impact force and not by release of the internally stored elastic tensile energy in the glass. Usually, large impact force will create multiple radial cracks from the impact point and the fracture will follow a “star-burst” pattern, where the number of radial cracks is positively correlated with the impact force (Vandenberghe et al., 2013). In the work of Tandon and Glass (2005), the main difference between the fracture patterns of 450°C–3 h and 450°C–6 h is actually the number of radial cracks, which can be attributed primarily to the increased indentation loads that are required to break samples with longer ion-exchange time. It is important to separate the effect of impact force from the effect of CT. The only straightforward way to do this is to rely on a delayed fracture mechanism where the final Stage III crack propagation is governed by the release of internally stored energy in the glass. To induce a delayed fracture event, a well-controlled flaw is introduced just deep enough to barely penetrate the CT region, i.e., just after the zero crossing point of the stress profile. The flaw initially grows slowly leading to failure in a delayed manner. Such delayed fracture has been thoroughly discussed in our previous paper (Tang et al., 2014). With delayed fracture, the impact energy is dissipated long before the final fracture event; hence, the final fracture pattern will be governed by the release of internally stored tensile energy rather than by the externally applied impact energy. In contrast, when fracture occurs immediately after the external impact event, both impact energy and stored elastic energy are released during fracture, making it difficult to separate the effects of these two sources of energy on the final fragmentation pattern.

In this study, both automated frangibility testing (Tang et al., 2014) and manual testing (Barefoot et al., 2012) methods are used. Also, a modified version of manual testing is employed involving multiple strikes. The purpose of using three testing methods is (a) to demonstrate consistency among different techniques when performing a rigorous statistical analysis of fracture data and (b) to determine the effect of multiple strikes vs. single strikes before fracture.

As described in previous sections, there are three different thicknesses, three different ion-exchange conditions, and three different testing methods in this experiment. So, there are 27 different conditions in total. Table 1 shows the raw data of 1 of the 27 different conditions as an example. The chosen condition is 0.7 mm thick glass using the manual multiple strike method detailed above.

To determine whether a sample is frangible, it is necessary to have a set of criteria, which might be application specific. We begin with the same criteria that has been published previously (Allan et al., 2010; Barefoot et al., 2012), viz., a frangible sample is a specimen sized 50 mm × 50 mm that fractures into at least five fragments, and at least one fragment is ejected outside a radius of 50 mm from the center of the sample. This criterion is referred to as C1. This definition can be simply summarized in Table 2, where n₁ is the number of non-frangible (N-F) samples and n₂ is the number of frangible samples (F).

There is an obvious problem with including ejection of glass fragments as part of the criteria for frangibility, since that depends strongly on the friction between the glass and the testing surface. Hence, the previously used criteria for frangibility should be modified to exclude any requirements on fragment ejection distances. As indicated in our previous publication (Tang et al., 2014), the true frangibility of a glass is an intrinsic property of the sample. Hence, the criteria C1 are modified to C2, in which the ejection distance requirement is removed (Table 3).

However, neither of these sets of criteria account for the possibility of having an artificially large fragment count due to excess applied energy, i.e., they do not distinguish between immediate and delayed failure events. If enough extra energy is applied when impacting the glass, even an unstrengthened sample without any CT may break into many fragments. Hence, if a glass fails immediately upon impact, the externally applied energy may be contributing to the fragmentation pattern. In our case, if a glass fails immediately into ≥5 fragments, the true frangibility is unknown, since it is not known to what degree that final fracture pattern is due to the release of internally stored tensile energy. On the other hand, if a sample fractures immediately into <5 fragments, it is certain to be non-frangible, since excess applied energy can only lead to “false positives” and not false negatives for frangibility. Accounting for this factor, the criteria of frangibility becomes C3, as shown in Table 4, where n₁ and n₃ cases are both known to be non-frangible, n₂ is known to be frangible, and the frangibility of n₄ is unknown.

Another unknown factor is the possible effect of surface flaws as a result of multiple strike attempts before the glass exhibits failure. Such surface flaws might influence crack branching and lead to a false positive for frangibility. Accounting for this uncertainty, the criterion of frangibility becomes C4, as shown in Table 5. Here, there are two additional unknown cases, n₆ and n₈, where multiple attempts are required to induce what is an apparently frangible fracture.

Finally, we should consider variation in the threshold for fragment count, since the critical number 5 is a somewhat arbitrary choice. Criteria 5 are similar to Criteria 3, except that the tested samples are grouped into <4 fragments or ≥4 fragments, instead of <5 fragments or ≥5 fragments, as shown in Table 6.

As described in the previous section, the calculation of frangibility probability for criteria C1 and C2 is quite straightforward. The frangibility probability (p^) can be estimated as the proportion of frangible samples: p^C1=p^C2=n2∑i=12ni For C3, C4, and C5, as shown in Tables 3–5, some cases fall into the “unknown” category. In two extreme cases, one might consider the “unknown” samples to be entirely non-frangible (N-F) or entirely frangible (F) such that the frangibility is calculated, for C3 and C5: p^C3min=p^C5min=n2∑i=14ni,p^C3max=p^C5max=n2+n4∑i=14ni For C4: p^C4min=n2∑i=18ni,p^C4max=n2+n4+n6+n8∑i=18ni

The above calculation gives a range for the actual frangibility probability, p^, i.e., p^min≤p^≤p^max. However, depending on the relative fraction of “unknown” samples, this range can be too big to provide any meaningful information. A more sophisticated approach is to partition the “unknown” samples into frangible or non-frangible groups based on Bayes rule (James et al., 2013), where we estimate the posterior frangibility probability for I (Immediate) and D (Delayed) groups as the proportion of frangible samples in each group.

Then, we assume that this probability remains the same across the I and D groups, i.e., for C3 and C5, n2n1+n2=n4′n3+n4 For C4: n2n1+n2=n4′n3+n4=n6′n5+n6=n8′n7+n8

Therefore, the final frangibility probability estimated for C3 and C5 is: p^≈n2+n4′∑i=14ni For C4: p^≈n2+n4′+n6′+n8′∑i=18ni

From Table 8, we can see that test method does not have significant impact on frangibility for most criteria and thicknesses. This indicates that all three methods are good for assessing the frangibility probability. In this paper, the manual testing is performed by a well-trained technician, who managed to deliver consistent impact force that ensures a yield rate (percentage of delayed fracture to all fracture) of more than 50%. If the manual testing generates too many immediate fractures, it will lead to artificially high-frangibility probability, especially if the frangibility criterion is like C1 and C2, which do not require the fracture to be delayed. Using frangibility criterion like C3, C4, and C5 (require fracture to be delayed) will eliminate the effect of different yield rate generated by different testing methods.