# A Simple Estimation Method of Weibull Modulus and Verification with Strength Data
## 5 Discussion
### Estimation of m from Standard Deviation or from Coefficient of Variation
These results certainly contradict the above conclusion of [PERSON] [116] and the results in Table 4. [PERSON] concluded that no relationship exists between fracture toughness and CV, but it is plausible that the CV data he collected was unrepresentative of real ceramics behavior. Another issue in S or CV values is the rounding of the data. Some reports provided only single digit values, and such data cannot be used unless rough estimates are acceptable. The values of m\({}_{\rm est}\) are mostly within \(\pm\) 20% of m\({}_{\rm obs}\) from a Weibull plot of the base strength data. This \(\pm\) 20% limit is based on 71 datasets in the present study, which started from a listing of strength values, and the values of m\({}_{\rm obs}\)/m\({}_{\rm est}\) ratios were calculated. The average was 0.98 (S = 0.088), and the m\({}_{\rm obs}\)/m\({}_{\rm est}\) ratio ranged from 0.79 to 1.19. This limit can be used to judge the m and CV values from the literature. When these two values are off from the modified Robinson relation by more than 20%, one or the other value is likely to be in error. The most common source is the trimming of outliers to make the CV smaller. In the above direct comparison of m\({}_{\rm obs}\) and m\({}_{\rm est}\) from the same strength data, two cases were at 0.79. These were both from historic iron data. When the two old cases (listed in Table 2) are censored, the range is reduced to 0.83 to 1.15, and the \(\pm\) 20% limit is conservative. In many brittle solids, more than one type of flaw may control the fracture, leading to a bimodal Weibull distribution. An example is given in ASTM C1239 [128]. A bimodal Weibull distribution is shown in Figure 2 and in the data in Table 5 in the C1239 standard with a sample count of 79. It follows the slope of m = 6.79 on the low side, and m = 21.0 above a fracture strength of 620 MPa. When the data is replotted and a single m value is calculated, one obtains m\({}_{\rm obs}\) = 9.98, while the average strength was 659.23 MPa and standard deviation was 59.56 MPa, yielding m\({}_{\rm est}\) = 12.18. Thus, m\({}_{\rm obs}\)/m\({}_{\rm est}\) = 0.825, which fits with the normal pattern of m estimation. This shows that the present method can be used for the bimodal cases, averaging the two slope regions. However, an arbitrary cut-off of the original data on the low end will raise the m value toward the high slope. A unimodal example in ASTM C1239 [128] has m\({}_{\rm obs}\) = 6.38. Using the data given in Table 4 of C1239, m\({}_{\rm obs}\), \(<\)\(\sigma\)\(>\), and S were calculated using the methods of this study, and resulted in m\({}_{\rm obs}\) = 6.52 and m\({}_{\rm est}\) = 6.23. These three m values match well, showing that a valid S (or CV) value will lead to a satisfactory estimate of m.  
Cast iron has long been known for its brittleness, but it has been used widely despite its drawbacks. Weibull moduli of white and gray cast iron are indeed low at 2 and 6 [123]. Other cast alloys showed varied behavior. For example, an Al-Cu alloy casting [123] had m = 4, while Al-Si casting alloys had m\({}_{\rm obs}\) of 60 to 116 [54] (see Table 3). Two extensive tests of A357 Al castings confirmed high m\({}_{\rm obs}\) values for high-quality Al castings. Using 354 and 388 samples, m\({}_{\rm obs}\) values of 47.5 and 30.6 were obtained [124]. For Al-7 Si and Mg AM60 castings, three studies reported m\({}_{\rm obs}\) values from 2.5 to 38 [125; 126; 127]. It is clear that cast alloys need to be treated separately from wrought alloys and between castings made from different processes.