# 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
Most of them are roughly comparable to the similar alloys given in Table 3. Ti-6 Al-4V [111] is the only Ti alloy in this work, and is similar to 316L stainless steel in the NPL study above [108]. Copper and Cu-Ni [112] showed comparable m values (except as received Cu) to the reported m values of 12 to 16 for pure Cu in [48] in Table 3. The Al 2030 results showed high m values, but also indicated sensitivity to test conditions. As noted earlier, more elaborate tests are needed to verify m values over 100. The last three rows in this group are for AerMet100(r) steel (Carpenter Tech. Corp, Philadelphia, PA, USA), which has high strength and fracture toughness [115]. It is of composition, 0.23C-13.4 Co-11.1 Ni-3.1 Cr-1.2 Mo, and is used in age-hardened martensitic state (or maraging steel). The tensile strength data gives an m value of 43, which can be expected for a low C, high-strength Co-Ni steel. While the fracture toughness (K\({}_{\text{lc}}\) or J\({}_{\text{IC}}\)) levels are high, the estimated m values are low (2 to 4) using an N of 5 or 6. The Weibull plot of the K\({}_{\text{lc}}\) data also showed m = 3. This level of m is consistent with the m values of 2 to 10 that are generally obtained in brittle fracture as reported by [PERSON] et al. [32; 116], who theoretically predicted m = 4 for K\({}_{\text{lc}}\) of generic brittle solids. This should make designers cautious, despite its high fracture toughness. The next group of 12 datasets is for fibers made from carbon nanotubes (CNT) [117]. Only averaged strength data is available, giving estimated m values between 5-30. A few of them were higher than any m value for the fibers in Table 5, while the majority fitted to the range of regular fibers. A previous work on CNT bundles showed m values of 1.7 to 2.7 [118] and 4.3 to 6.8 [119], while a single CNT has an m value of about 3 [120; 121]. When the strength data of a single CNT in [120] was analyzed, m\({}_{\text{obs}}\) = 2.3 and m\({}_{\text{est}}\) of 2.9 were obtained, implying that the present method works at the nanoscale as well. However, deviation is higher, as the CNT has the tensile strength of 11 to 63 GPa. In two engineering studies of suspension cable wires, the average and standard deviation of strength data of suspension cable wires were reported. One was from the Mid-Hudson Bridge [44], containing a dataset with m\({}_{\text{obs}}\) = 32.07. This was included in Table 2. Remaining datasets of only the average plus standard deviation are listed in Table 7. These give m values of 23 to 35, matching the m\({}_{\text{obs}}\) value for one of the samples. These represent Stage 4 corrosion according to [23]. Visually, these wires were judged to be Stage 2 to 3. Another report gave results from three suspension bridges: X, W, and Z [122]. This study included wires of various corrosion stages, 1 to 4 plus cracked. Roughly half of the wires showed m values that were in agreement with visual inspection, but others showed more damages according to the m values observed. Since tensile testing is part of the standard procedures in maintenance inspection, the present simple m estimation method provides a quantitative tool to evaluate the wire inspection results. In estimating m with the data on standard deviation or CV values, it is necessary to use caution, since some sources apparently discard low strength values in the tail part of distribution. This is especially true for the data without giving a sample count, N. Some testers used N values of 5 to 6 for CV values and need added scrutiny, as results can be unreliable. When this critical information is unavailable, it is best to avoid them. For example, [PERSON] [47] collected 19 CV values on commercial ceramics with low K\({}_{\rm IC}\) values in the range of 2.2 to 6.1 MPa\(\sqrt{\rm m}\). Corresponding CV values were 0.008 to 0.104, yielding m values of 10.8 to 134 with an average of 31.7. Nine of 19 exceeded m = 20.