Reply to Comment on: Flexural Strength by Fractography in Modern Brittle Materials

WE thank Mecholsky et al. 1 for their intense interest in our paper and their appreciation of the great importance and novelty of our results. We are also grateful for this opportunity to place our findings in a more comprehensive context to better discuss their comments. Orr is normally credited with developing the now commonly used equation relating the strength of brittle materials to the mirror radius at which ‘mist’ forms on fracture surfaces, r√Rm= A, where Rm is the mirror radius, r the strength of the sample, and A the ‘mirror constant’. The idea of investigating the limitations of Orr’s equation came to us during the course of a decade of fractographic analyses performed on a variety of brittle materials. It became apparent to us that, for fracture surfaces of submillimeter-thick glass plates, the mirror radius does not accurately predict the experimentally measured stress at failure when using published bulk values for ‘A’. In addition, flexural tests by fourpoint bending (4PBT) and ring-on-ring (RoR) consistently showed a relatively large, nonzero “residual stress”: the thinner the sample, the larger the apparent residual stress. We were surprised by this unexpected result until reading Quinn’s paper on fracture of glass plates subject to bi-axial stresses. Quinn independently confirmed that annealed glass plates fractured by RoR displayed a clear, positive y-intercept (i.e., apparent residual stress) when plotted against 1/ √Rm. We recognized that the tangent to the ‘r vs. √Rm curve’ would intersect the y-axis to produce a false apparent “residual stress” if the mirror constant were not truly a constant. It is well-known that mist forms when a critical stress intensity factor (SIF) is reached at the crack front. This SIF depends on the local stress, the length of the crack (Rm), and a shape factor, Y, related to the geometry of the crack front. At the limit, a crack very shallow compared to the sample’s thickness, H, in a sample loaded in bending (Rm/H 1) will have a shape factor that is relatively uniform along the nearly semicircular crack front. Importantly, a nearly identical shape factor applies to samples tested in uniaxial tension, as in these cases the crack front is also nearly semicircular. Conversely, a very long through-the-thickness crack in bending (Rm/H ≫ 1) will have a significantly different shape factor due to the elongated geometry of the crack front. For instance, Sherman et al. observed a crack in bending with an aspect ratio of c/a 3.125. It is therefore clear that the mirror constant of a given material, A ffi KIm/Y, must be nearly equal for shallow cracks fractured in bending (Rm/ H < 1) and for samples fractured in uniaxial tension. On the other hand, ‘A’ is expected to be larger for long through-thethickness cracks fractured in bending. Until recently, the majority of glass applications utilized relatively thick geometries (> 1 mm). The strength of glasses with untreated edges is normally on the order of 100 MPa (i.e., Rm 0.3–0.4 mm) and hence Rm/H 1. In these cases, no significant difference between the values of ‘A’ in tension and ‘A’ in bending is expected, explaining the apparent thickness independence of ‘A’ historically observed. Interestingly, the 1966 work of Kerper and Scuderi explored the possibility that the sample’s thickness might have an effect on ‘A’, but regretfully the hypothesis was tested on glass rods. This choice of sample geometry was unfortunate, because the condition Rm/H < 1 necessarily always applies. Having placed this discussion in a greater context, we can directly address the comments by Mecholsky et al. Their main concern appears to be the validity of the data reported in Fig. 2. The astute reader will note that our findings are not in any way based on these data. As these data merely provide a historical frame of reference, we did not emphasize the minutiae of how the values were extracted from the literature. Nevertheless, we are pleased to now be given the opportunity to provide the details needed for addressing this apparent discrepancy. Since the literature spans various authors and many decades of evolving knowledge, the mirror constants were originally extracted with a range of fitting equations presented in an incoherent fashion. In order to provide the consistency needed to compare across works, we therefore regressed the raw data for each reference by using a single fitting equation. In particular, these standardized values of ‘A’ were obtained as recommended by both ASTM C-158 and the NIST Recommended Practice Guide (NIST-RPG), Appendix D using the equation