A Simple Estimation Method of Weibull Modulus and Veriﬁcation with Strength Data

: This study examines methods for simplifying estimation of the Weibull modulus. This parameter is an important instrument in understanding the statistical behavior of the strength of materials, especially those of brittle solids. It is shown that a modiﬁcation of Robinson’s approximate expression can provide good estimates of Weibull modulus values (m) in terms of average strength ( < σ > ) and standard deviation (S): m = 1.10 < σ >/ S. This modiﬁed Robinson relation is veriﬁed on the basis of 267 Weibull analyses accompanied by < σ > and S measurements. Estimated m values matched normally obtained m values on average within 1%, and each pair of m values was within ± 20%, except for 11 cases. Applications are discussed, indicating that the above relation can o ﬀ er a quantitative tool based on the Weibull theory to engineering practice. This survey suggests a rule of thumb: wrought metal alloys have Weibull moduli of 10 to 200.


Introduction
Weibull first used a statistical distribution in 1939 [1] that is now known as the Weibull distribution to characterize the fracture strength of nine different materials, totaling 20 different types of samples with several loading modes. These included 2000 to 3000 cotton fiber and yarn samples, while 20 to 128 samples were typically tested for metals, with the total sample counts nearing 8000. He extended its applications to broader categories in 1951 [2]. The Weibull distribution has since been applied in wide-ranging fields in engineering and beyond [3]. While the present work is directed to the analysis of material strength, mainly from tensile and flexure testing, the lifetime prediction of engineering structures and various systems and components is another branch of statistical analysis where the Weibull distribution plays a key role [4,5]. Several recent examples of such applications can be found in [6][7][8][9][10]. It is an important tool for enhancing the precision of measurements that tend to show wide deviation.
Two-parameter Weibull distribution, the most basic form, describes the probability of failure P by [1][2][3]: where m is the Weibull modulus (also called the shape parameter), σ o is the scale parameter, and σ is the variable (fracture strength in this study), respectively. The basis of this distribution is given in terms of the weakest link theory; see Robinson and Batdorf [11,12]. From N fracture tests, a cumulative probability distribution curve is obtained as a function of (σ/σ o ). In a graphical approach, one can follow Weibull [1], plotting ln(−ln(1 − P)) against ln(σ) or ln(σ/σ o ), and obtaining a best fit line, the slope of which equals m. The scale parameter, σ o , is slightly larger than the average fracture strength <σ>, and these are related by [3,13,14]: where Γ(x) is the gamma function. This Equation (2) can be approximated by [15]: Numerous methods and software have become available, and the above procedure mainly serves as a learning tool. Still, for untrained users, the initial hurdle for conducting Weibull analysis can hardly be trivial. A survey of online Weibull calculators reveals a confusing and intimidating array of approaches. Regardless of the method used, reliable Weibull parameters can only be obtained from at least N = 10 to 20, since Robinson showed that the coefficient of variation (CV) for m is equal to 1/ √ N [11]. Ritter et al. [16] also arrived at this equation. For N = 10, CV = 0.32, and this is reduced only to 0.22, even for N = 20. For materials that have higher m values such as fiber-reinforced composites, the N value is specified in technical standards as five or more (N ≥ 5) [17][18][19]. However, at N = 5, CV = 0.45; that is, a large deviation in m needs to be anticipated. See also recent studies on this subject [20,21]. Note that in these statistical works, N is referred to as sample size instead of sample count, which is used here to differentiate the concept from physical dimensions.
Another need for Weibull modulus values is to use them as quality indicators. In ceramics fields, it has been common to indicate the brittleness of materials, since the m values of most ceramics (and cast iron, another brittle material) are 10 or lower. This is in contrast to typical metal alloys, which are generally believed to show m > 100 [22], but with a limited number of published reports of high m values. A research project on bridge maintenance [23] identified a Weibull modulus of >70 for undamaged suspension bridge cable wires, whereas an m value of 10 represents cracked, highly corroded wires with two more stages of corrosion between them. Meanwhile, an m value of 30 indicates Stage 4 corrosion, and an m value of~50 indicates Stage 3 corrosion. For this purpose, even approximate m values allow bridge inspectors to classify wires between different corrosion stages quantitatively instead of relying on visual observation. When only limited data is available, with N values less than 10, efforts of Weibull analysis may be unrewarding, and simpler approximate methods are adequate. In other cases, only average values and their standard deviation (or variance) are available, e.g., from engineering reports or from historical documents.
Simple methods for estimating m values have been discussed in the literature, and these will be examined. Strength data compiled in two interlaboratory studies [24,25] was analyzed to provide Weibull modulus values for five common metal alloys in Section 3, followed by a comparison with a database of Weibull analyses on the fracture or tensile strength of many solids. The simplest, one-parameter method will be verified to best represent more than 260 datasets. Discussion and summary conclude this study.

Approximate Methods of Weibull Modulus Estimation
The most common index of data scatter assumes the normal (or Gaussian) distribution and determines the average <σ> and its standard deviation, S. Another is the Weibull distribution, as shown in Equation (1). These parameters are related. Equation (2) above connects the <σ> value to the Weibull parameters, σ o and m. Similarly, standard deviation, S, is given by [3,11,13,14]: where Γ(x) is again the gamma function. See [14] for the steps of its derivation. Note that Γ(x) is readily available in Microsoft Excel ® (version 16.16.8, Microsoft, Redmond, WA, USA, 2016). When Equations (2) and (4) are combined, the coefficient of variation, CV, is defined as [3,11,13,14]: CV is a function only of m, and is bounded by 1/m and (π/ √ 6)/m = 1.283/m [13]. In expressing m in terms of the inverse of CV, we have two bounds: m = 1/CV = <σ>/S and m = 1.283/CV = 1.283<σ>/S. (6) This implies that m can be approximated by <σ>/S or the inverse of CV. In 1972, Robinson [11] first derived this relationship between m and <σ>/S (or 1/CV), and also gave approximate expressions as: m = 1.2 <σ>/S, and: m = (<σ>/S) 1.064 , (1.1 < m < 60).
Equation (7), or the Robinson relation, has 15% error at m = 2, going down to a 6% error at m = 100. Equation (8) has 1% error for 1.1 < m < 60. When two constants are used, Equation (5) can be represented as: m = 1.0461·(<σ>/S) 1.049 , (R 2 = 0.9997), achieving an excellent match. In practice, this can be deemed exact. When numerical values are calculated for CV using Equation (5) by supplying a series of m values, one can then exchange the two sequences, making CV a variable. Thus, Equations (7), (8), and (9) can be compared to Equation (5) and it is found that the last two indeed represent the values of m well. In contrast, Equation (7) has about 7% error. A regression analysis of CV versus m from Equation (5) (7), yielding a high regression coefficient of R 2 = 0.9999. This is a slight improvement over Equation (9). Thus, Robinson's Equation (7) can be converted to an essentially exact representation using: m = 1.271 <σ>/S.
In 1981, Ritter et al. [16] derived Robinson Equation (7) as an approximation for m > 2. However, their method is questionable, since the actual constant ( √ K) from their equation (A29) varies from 1.05 at m = 2 to 1.25 at m = 20, reaching the upper limit of 1.283 at m = 60. They selected the constant to be √ 1.44 = 1.20, which is only valid at m = 9 ± 1. Thus, their value appears to be an arbitrary choice, which happened to coincide with Robinson's constant. The Ritter equation was incorporated into ASTM C1499 [26]. By 1986, it was apparently well-known, as Wetherhold [27] used it to confirm the m values he obtained using the corresponding CV values. In 1989, van der Zweig [28] presented the same Robinson relation, but without citing prior works [11,16,26]. Three more approximate expressions have been published by [29][30][31] as follows (in the order of appearance): Equations (11) and (12) were obtained using Monte Carlo simulation as the basis. All five approximations satisfy the bounds for m, except at m < 1.67 to 2.3 for Equations (11)- (13). These six approximations are close to the essentially exact Equation (10). When a systematic error of up to 10% to 20% is acceptable, any one of them can be used. However, the merit of using them has vanished, since the linear Equation (10) with R 2 = 0.9999 can easily be used. Furthermore, in Section 4, it will be shown that experimentally obtained m values can be represented well using the Robinson relation (Equation (7)) with a reduced constant of 1.1, or m = 1.1 <σ>/S.

Weibull Moduli of Five Metal Alloys
ASTM Committee E28 on Mechanical Testing conducted an interlaboratory study on automated ball indentation testing, which included controlled tests of the tensile strength of four alloys, Al 6061-T651, Al 7075-T651, steel 1018, and steel 4142 [24]. The tensile strength datasets of 30 samples each were analyzed for this study to determine the Weibull modulus values for these common metal alloys. The bias of individual laboratory results was corrected based on the average deviation with global average for four alloys. The method discussed in Section 1 was used to obtain the mobs values from the Weibull plots. The scale parameter, σo, was obtained using Equation (2). Weibull plots for the four alloys are shown in Figure 1, and mobs values of 124.0, 91.4, 73.8, and 78.1 were obtained as noted in the figure. Separately, the corresponding mest values were estimated from the values of <σ> and S using m = 1.1<σ>/S, yielding 124.9, 91.9, 73.3, and 77.8. The two sets matched well within ± 1%. Another interlaboratory study under strict test protocol was conducted in Europe [25]. It produced a special ingot, from which 200 bars of Nimonic ® 75 Ni-base alloy (Special Metals Corp, Huntington, WV, USA) were fabricated. The tensile strength data from 18 bars of this Nimonic ® 75 alloy [25] was analyzed for the Weibull modulus, and m = 125.3 was obtained, as shown in Figure 1. The corresponding mest value was 129.7, matching to +3%. Thus, the commonly cited value for wrought metal alloys (m >100) is reasonable for these Al and Ni alloys, but appears ~25% too high for ductile steels. More than 20 m values above 50 are included in the metals data below, but some, including pure copper, were below an m value of 20. More comprehensive testing is needed to better define the ranges of m values for various material types beyond those surveyed in this work. Data from [24,25]. Values of m are shown. The horizontal axis is shifted to avoid overlaps.

Material Strength Data
The availability of Weibull modulus values is limited in comparison to the more commonly reported values of the average and standard deviation of the tensile (or yield) strength. Most of the data that is collected is from tensile testing for metals, fibers, and composites, whereas flexure testing Data from [24,25]. Values of m are shown. The horizontal axis is shifted to avoid overlaps.

Material Strength Data
The availability of Weibull modulus values is limited in comparison to the more commonly reported values of the average and standard deviation of the tensile (or yield) strength. Most of the data that is collected is from tensile testing for metals, fibers, and composites, whereas flexure testing is more common in ceramics and glasses. Limited fracture data is also included. Fracture toughness data that is suitable for this study is even more difficult to find. Even in the large-scale round-robin study of fracture toughness reported by Wallin [32], K Jc (K Ic by J-integral estimation) values were only given in graphical form without the average and standard deviation. However, their data clearly showed two distinct regimes for ductile and brittle fracture, implying high and low m values. This data style is understandable, as their main aim was to establish the master curve for describing the ductile-brittle transition behavior. The master curve approach is now established as ASTM E1921 standard test method [33], which incorporated a three-parameter Weibull distribution to describe the ductile-brittle transition behavior (test temperature dependence of K Jc ). A Weibull modulus of m = 4 was chosen from theoretical analysis, and it was also used to minimize the sample size effects.
In the survey of the literature for the present study, datasets consisting of the average strength (<σ>), standard deviation (S), Weibull modulus (m), and sample counts (N) have been collected. This kind of dataset is to be referred to as type V for values. When more than 10 strength values of comparable samples were available, either in numerical or graphical form, the data was digitized and analyzed. This kind will be called type D for digitized data, and the Weibull modulus will be determined using the basic method with simple linear regression available in Microsoft Excel ® . The collected datasets will be presented in five groups. These are (1) historic iron and steel, (2) metals, (3) ceramics and glasses, (4) fibers, and (5) composite materials in tabular form. The tables have the following columns: material identification; <σ>; S; <σ>/S; observed m value, m obs ; estimated m value from Equation (14) given below, m est ; N; the ratio of m obs to m est (= m obs /m est ); data type (V or D); note; and reference number.
Before the datasets were separated into five tables for publication, all 267 datasets were in a single file, and the constant for a trial approximation equation was varied to achieve the average value of the ratio of m obs to m est that was closest to unity. First, Equations (9) and (10) were used as trial functions, since both are equivalent to Equation (5). When these are inserted, the average values of m obs /m est (standard deviation in parentheses) were 0.948 (0.091) and 0.874 (0.083), respectively. These produced 5.2% and 12.6% error, indicating that the theoretical values are not suitable to represent experimental m values. Actual m values contain errors from various sources that have not been anticipated in theory. Next, the constant of linear Equation (10) was reduced to 1.2, 1.11, 1.10, and 1.09; the results are shown in Table 1. The constant of 1.11 gave the closest value of 1.001, while it increased to 1.01 using 1.10. The previously used constant of 1.2 (Equation (7)) resulted in m obs /m est = 0.926 or 7.4% error. Next, one-parameter power-law equations were inserted starting with Equation (8), changing the exponent to 1.05, 1.045, and 1.04, showing the results in Table 1. The exponent of 1.045 gave a value of 1.001 (0.095). This is comparable to the linear Equation with the constant of 1.11. Between the two groups giving similar matches in predicting m values from <σ> and S values, a simpler linear equation is preferable. Furthermore, the constant of 1.1 is selected as it provides a good fit and ease of use as well. This is given as Equation (14): m = 1.10 <σ>/S.
This will be called the modified Robinson relation hereafter. This was selected since it is linear and produced nearly the unity m obs /m est value (off by 1%) with a constant of two digits. Obviously, one can use 1.11 as the constant, but a different data population shifts the degree of agreement by 1% to 3%, as will be seen below. It is an approximate relation to represent observed Weibull moduli. It should be recalled that theoretically exact relationships are worse by 4% to 12%. The collective datasets are plotted in Figures 2 and 3 to show general behavior. Since 267 sets are included, most points overlapped, especially for lower <σ>/S values. In Figure 2, red + symbols represent m obs and blue dots represent m est , against <σ>/S values. Both symbols follow a single straight line. Figure 3 illustrates the deviation of measured m obs values from the linear estimates of Equation (14) using m obs /m est . Most data points are within ± 0.1 (or ± 10%) of the unity ratio, with 11 points outside of ± 0.2. For the collective dataset, the average of m obs /m est is 1.01, and the standard deviation is 0.09, justifying the selection of the constant, 1.1, which is used in Equation (14). Note that the corresponding average m obs /m est increases to 1.111 (0.099) for the simplest m est = <σ>/S expression. This may be useable for getting a rough idea of m values.  Figure 2, red + symbols represent mobs and blue dots represent mest, against <σ>/S values. Both symbols follow a single straight line. Figure 3 illustrates the deviation of measured mobs values from the linear estimates of Equation (14) using mobs/mest. Most data points are within ± 0.1 (or ± 10%) of the unity ratio, with 11 points outside of ± 0.2. For the collective dataset, the average of mobs/mest is 1.01, and the standard deviation is 0.09, justifying the selection of the constant, 1.1, which is used in Equation (14). Note that the corresponding average mobs/mest increases to 1.111 (0.099) for the simplest mest = <σ>/S expression. This may be useable for getting a rough idea of m values.   Table 2 lists the data of wrought iron and steel from the 19th century to the early 20th century. These are arranged roughly in the order of age. For these datasets, the average of m obs /m est is 1.007 and the standard deviation is 0.106, showing a similar deviation and a slightly larger standard deviation than the whole dataset. All the m obs data in this group, except for the last five, were calculated from digitized strength data from the literature. Figure 4a shows the values of m obs and m est against <σ>/S, as shown in Figure 2. Again, the m obs values straddle the blue dots for m est from Equation (14). Figure 4b represents the deviation from the unity line. This figure is apparently skewed upward, but these are balanced by overlapping points below the unity line. This group also has the two lowest m obs /m est , which are below 0.8, balancing the high values.    Table 2 lists the data of wrought iron and steel from the 19 th century to the early 20 th century. These are arranged roughly in the order of age. For these datasets, the average of mobs/mest is 1.007 and the standard deviation is 0.106, showing a similar deviation and a slightly larger standard deviation than the whole dataset. All the mobs data in this group, except for the last five, were calculated from digitized strength data from the literature. Figure 4a shows the values of mobs and mest against <σ>/S, as shown in Figure 2. Again, the mobs values straddle the blue dots for mest from Equation (14). Figure 4b represents the deviation from the unity line. This figure is apparently skewed upward, but these are balanced by overlapping points below the unity line. This group also has the two lowest mobs/mest, which are below 0.8, balancing the high values.    (14). N = sample counts.

Historic Wrought Iron and Steel
The first row in Table 2 gives the results of 1810 chain links that were retrieved from the Essex-Merrimack suspension bridge when it was replaced in 1910 [34,35]. This was built by James Finley, and was one of the earliest iron suspension bridges in the West. The strength level was high for its age, but the m values were below 10, which was indicative of the brittle state expected of such an old iron. In this case, tests were done on materials after 100 years of continuous use outdoors. The next three rows provide the results of tests conducted at the Franklin Institute in 1837 [35,36]. Both strength and m values are comparable to the Finley iron. The next eight rows show the test data from Kirkaldy's 1863 book [37]. The selected six groups are given first, grouping the same or related sources of best quality iron, cast steel, and charcoal iron. One group that was noted as "Govan Ex Best" produced a high m value of 43, while cast steel had a high strength (580 MPa) and an m value of 17. Gordon selected the data of coveted Swedish iron from Kirkaldy [35,37], which gives an m value of 30, justifying its high reputation. The next row, which is noted as charcoal iron, may also be of Swedish origin. The following two rows represents most of Kirkaldy's iron and steel data, totaling 688 tests. These data were tabulated in [38] and were split into two groups by sample diameters. The smaller diameter group had 40 MPa higher strength, but both had low m values of~9, as low-quality iron samples were included. These two datasets represent the general quality level of 1860 iron in the United Kingdom (UK).
The next three rows give data from two Indiana bridges (built in 1869 and 1873) [38]. By this time, m values had doubled from the Finley iron, indicating the quality improvement of iron available in the United States (US) Midwest. Beardslee [39] conducted extensive testing at the US Navy, reporting 846 test results in 1879. For the entire tests, an m value of 26 was comparable to Indiana bridge irons, but the strength was 50 MPa higher. When the data were separated into low-strength and high-strength groups (with overlaps), the m value rose to 42 for the low group, which was comparable to the best data from the 1860 UK, albeit with 30 MPa lower strength. However, a general sampling of US wrought iron still showed the presence of low-quality iron with m values of approximately 10 to 15, as shown by Gordon [35,36]. By the 1880s, steel was widely used, and Unwin's book provided four examples of better materials [40]. These included Bessemer steel, boiler plates, and railroad rail.
The last seven rows give the data for patented high strength steel wires, mainly for suspension bridge cables. Percy's article [41] reported UK test results in the 1880s. The strength reached a level of 1 GPa, but the m value was still 13.7 [15,44]. At about the same time, wires for the Brooklyn Bridge (finished in 1883) had the strength of 1.1 GPa. Unfortunately, no test data has been located so far. Over the next 20 years, steel technology improved further and provided 1.5-GPa steel wires for the Williamsburg Bridge in New York (finished in 1903). Perry [42] provided 160 test data for the cable wires that were removed during the rehabilitation work in the 1980s, and Weibull analysis was conducted [15,44], yielding m = 16 with an average tensile strength of 1.5 GPa. The values are remarkable after more than 80 years of use, since these wires did not have galvanizing protection against corrosion. The next four datasets were commented on earlier in the Introduction [23], while the last set was from the Mid-Hudson Bridge [43]. These wires were from suspension bridges after many years of service. As noted before, m values are indicative of the state of corrosion of the suspension cable wires, and are useful in assessing the remaining service life of suspension cables. There are also many historic wrought iron bridges in need of rehabilitation, and simple Weibull analysis, which is being discussed here, will be helpful in their evaluation. Table 3 lists the data of metals and alloys. For these datasets, the average of m obs /m est is 1.034, and the standard deviation is 0.093. The m obs /m est average is 2.4% higher than that for the whole set. This comes partly from 11 data points with m obs /m est > 1.1, but these large deviations are still within the expected behavior. Most metallic alloys exhibit good ductility, and Weibull analysis is usually not needed. It is often assumed that metal alloys have m values of over 100, but lower m values have been obviously observed. The top four rows are calculated from the results of the ASTM study at established laboratories in the United States [24], and the m values were 74 to 124 (Figure 1), as discussed in Section 3 above. The data for Nimonic ® 75 Ni-base alloy from the European interlaboratory study [25] yielded m = 125 ( Figure 1). Here, another Ni alloy, Inconel ® 625 (American Special Metals, Miami, FL, USA) showed m = 55 [45]. These six datasets can be treated as the benchmarks for ductile steel, Al alloys, and Ni alloys. It is important to recognize that these studies used well-controlled sets of samples. This approach is usually replicated in research laboratories, but it is not representative of large sample data for design and reliability works, where mill practice, alloy chemistry, and structural shapes vary (see Section 5.2 for more discussion). Weibull data for metals and alloys are indeed scarce, and 22 of the 43 datasets in Table 3 showed m > 50. These were all ductile metals and alloys, including 18Ni maraging steel, stainless steels, and Mg alloys. However, some ductile metals such as copper showed low m values of 12 to 16 as well. Sintered steel showed low ductility along with low m values below 30. More data with low m values are given in Section 5 using standard deviation (or CV), where one finds that m < 30 is common for large-scale industrial datasets. Brittle fracture data for steels at sub-zero temperatures should be assessed using Weibull analysis, but even here, the test results of repeated tests are difficult to find. Several datasets for cleavage fracture were analyzed and added to Table 3 at rows 9 to 12 [46]. Surprisingly, m values were 10 to 20 in a sharp contrast to the theoretical value of 4 predicted by Wallin [32,33]. Another brittle material is nickel aluminide (NiAl). Monocrystalline NiAl showed consistently low m values of about 5 [47], while NiTi intermetallic showed a slightly higher m of 8.5 [46]. General trends can be viewed in Figure 5. No peculiar behavior is present.    Table 4 lists the data of ceramics and glasses. For these datasets, the average of mobs/mest is 1.011, and the standard deviation is 0.100. These values are similar to those of the whole set, while the general trends seen in Figures 6a and b resemble those of historic iron and steel. That is, more deviations larger than ± 0.1 exist. The maximum m value is limited to 24, and larger m values are found for high-performance ceramics such as alumina, porcelain, silicon nitride, and stabilized zirconia [47,55,56]. In the strength testing of these brittle materials, Weibull analysis is included as a rule, and many reports are available. However, the average and standard deviation data are left out from many test reports. In fact, Equation (7) is used to get CV values from m in ASTM C1499 [26]. This omission of <σ> and S precluded such tests from this study, unfortunately.  Table 4 lists the data of ceramics and glasses. For these datasets, the average of m obs /m est is 1.011, and the standard deviation is 0.100. These values are similar to those of the whole set, while the general trends seen in Figure 6a,b resemble those of historic iron and steel. That is, more deviations larger than ±0.1 exist. The maximum m value is limited to 24, and larger m values are found for high-performance ceramics such as alumina, porcelain, silicon nitride, and stabilized zirconia [47,55,56]. In the strength testing of these brittle materials, Weibull analysis is included as a rule, and many reports are available. However, the average and standard deviation data are left out from many test reports. In fact, Equation (7) is used to get CV values from m in ASTM C1499 [26]. This omission of <σ> and S precluded such tests from this study, unfortunately.    Table 5 lists the data of fibers, which constitute the largest group in this Weibull modulus survey. Datasets for over 90 types of fibers have been collected. About half of them are for carbon fibers, reflecting the high interest in their properties. For these datasets, the average of mobs/mest is 1.007, and the standard deviation is 0.072. The value of mobs/mest is close to unity. Data trends on Figure 7 show less data scatters than other similar plots. This trend is better seen in Figure 7b, and less than 20% of the data points showed a deviation higher than ± 0.1. The m values are confined to a range of 2 to 11, implying more brittle behavior even compared to ceramics, and reflecting the higher strength levels of fibers. Only four datasets exceeded m ≈ 10 [70][71][72]. Natural fibers included showed mostly low m values below 5, while those above 8 were either carbon or ceramic fibers made in recent years. Data for glass fibers became scarce after the 1970s, while early data lacked some of the parameters that were needed here, and only 10 datasets were found. Again, reported Weibull modulus studies often omitted <σ> and S data.  Table 5 lists the data of fibers, which constitute the largest group in this Weibull modulus survey. Datasets for over 90 types of fibers have been collected. About half of them are for carbon fibers, reflecting the high interest in their properties. For these datasets, the average of m obs /m est is 1.007, and the standard deviation is 0.072. The value of m obs /m est is close to unity. Data trends on Figure 7 show less data scatters than other similar plots. This trend is better seen in Figure 7b, and less than 20% of the data points showed a deviation higher than ± 0.1. The m values are confined to a range of 2 to 11, implying more brittle behavior even compared to ceramics, and reflecting the higher strength levels of fibers. Only four datasets exceeded m ≈ 10 [70][71][72]. Natural fibers included showed mostly low m values below 5, while those above 8 were either carbon or ceramic fibers made in recent years. Data for glass fibers became scarce after the 1970s, while early data lacked some of the parameters that were needed here, and only 10 datasets were found. Again, reported Weibull modulus studies often omitted <σ> and S data.    One of the reasons that fiber m values are low is the variation of fiber diameter along the length. Some studies have considered diametral effects [86,87], but it is difficult to separate them in general. Due to the high strength levels that fibers achieve, extremely small flaws can induce fracture [88,89]. That is another source of low m values, making Weibull analysis an essential part of fiber studies. Table 6 lists the strength and Weibull modulus data of composite materials. For these datasets, the average of mobs/mest is 0.992, and the standard deviation is 0.088. The average mobs/mest value is 2% lower than that of the entire data, while the general trends seen in Figure 8 appear to skew slightly to lower mobs as the m values increase. However, high mobs/mest values are mostly populated at low m values in Figure 8b. The m value of composites reached 44, exceeded only by ductile metals. Many Weibull studies were conducted earlier, but typically no values of <σ> and S were included. Two articles are useful in finding Weibull modulus values for many composites not included here [27,90]. Another article to be noted is [91], as it included many lay-ups and tested at different loading rates. However, the sample counts were three for each condition, so it is hardly worth calling it a "statistical" study. However, no similar tests appear to exist, and it may serve as a preliminary guide. In regard to small sample counts, four datasets with N = 5 are included in Table 6 for glass fiber composites. They used samples of large diameter (12 to 18 mm), and the tests followed an industrial guideline [17] for concrete-reinforcing bars. Since the m values were from 20 to 40, the sample count of five or more was deemed adequate for quality control purposes. One of the reasons that fiber m values are low is the variation of fiber diameter along the length. Some studies have considered diametral effects [86,87], but it is difficult to separate them in general. Due to the high strength levels that fibers achieve, extremely small flaws can induce fracture [88,89]. That is another source of low m values, making Weibull analysis an essential part of fiber studies. Table 6 lists the strength and Weibull modulus data of composite materials. For these datasets, the average of m obs /m est is 0.992, and the standard deviation is 0.088. The average m obs /m est value is 2% lower than that of the entire data, while the general trends seen in Figure 8 appear to skew slightly to lower m obs as the m values increase. However, high m obs /m est values are mostly populated at low m values in Figure 8b. The m value of composites reached 44, exceeded only by ductile metals. Many Weibull studies were conducted earlier, but typically no values of <σ> and S were included. Two articles are useful in finding Weibull modulus values for many composites not included here [27,90]. Another article to be noted is [91], as it included many lay-ups and tested at different loading rates. However, the sample counts were three for each condition, so it is hardly worth calling it a "statistical" study. However, no similar tests appear to exist, and it may serve as a preliminary guide. In regard to small sample counts, four datasets with N = 5 are included in Table 6 for glass fiber composites. They used samples of large diameter (12 to 18 mm), and the tests followed an industrial guideline [17] for concrete-reinforcing bars. Since the m values were from 20 to 40, the sample count of five or more was deemed adequate for quality control purposes. Appl. Sci. 2019, 9, x 2 of 41

Summary
It is shown in this section that Equation (14) provides the best correlation between the observed Weibull modulus and estimated values from the average and standard deviation (or the coefficient of variation) of experimentally determined strength datasets; that is, m = 1.1 <σ>/S = 1.1/CV (modified Robinson relation). This conclusion is based on a comparison of these values from more than 260 datasets. Estimated m values matched to normally calculated m obs values on average within 1%, and each pair of m values was within ± 20% except for 11 cases.

Estimation of m from Standard Deviation or from Coefficient of Variation
Using the modified Robinson relation, it is now possible to estimate the Weibull modulus when only the average and standard deviation of a strength dataset are known. Two interlaboratory studies on mechanical testing standards that were cited earlier contain base data for many materials [24,25]. These studies aimed to define the reproducibility, R, of mechanical tests conducted at different laboratories. The reproducibility includes interlaboratory deviations of strength calibration, deviations due to sample chemistry and process variables, test conditions, etc., since the main aim of these studies was to establish the accuracy of mechanical test results. The ASTM study [24] separately reported the standard deviation for within-laboratory precision, s r , that for between-laboratory precision, s R , and R. ASTM standard E8 provided these parameters for six alloys (two Al alloys, three steel alloys, and one Ni alloy) [107]. This ASTM E8 shows that the reproducibility R is three to six times larger than the standard deviation of a single series of tests at one location, s r , while National Physical Laboratory (NPL) report [108] put the factor at four on average. Thus, the s R values can be used for estimating m values when samples come from a single set. When many sets of samples from multiple sources are tested at different sites, R values are appropriate. These two reports provided the average tensile strength and standard deviation for 20 different alloys (six for Al alloys, 12 for steel alloys, and three for Ni alloys). The m est values from the ASTM E8 [107] and NPL report [108] are shown in Table 7. While the m est values for Al (45 to 82) and A105 steel (75) are comparable to the previously known values in Table 3, the m est values for 316 and 51410 steels and Inconel ® 600 are higher (91, 174, and 151). From NPL collected data, the m est of Al alloys are again comparable at 47 to 122, while steel and Ni alloy resulted in m est values of 44 to 169. Again, the m values reported fit the range found in Table 3, where the m est values of structural steels were from 44 to 110, while those of stainless steels and Ni alloys were from 37 to 175.    [110]; Note 7-4: NA = not available for [114]; Note 7-5: NA = not available for [118]. CNT = carbon nanotubes.
Two groups of steel in Table 3 have lower m values of 15 to 29, but these were sintered materials that were expected to contain numerous voids. The following six rows in Table 7 are the data from laser-sintered maraging steel [51]. The first row was listed in Table 3 as it included nine strength values. Its m obs value was calculated as 99.2, which agreed reasonably with the m est of 113.5 with the modified Robinson relation. Other m est values given here were between 25-179, reflecting the variability of the selective laser melting process used. However, the last row with an m value of 179 is clearly out of the range. The estimated m values should lead to a better selection of process parameters, since normally sintered medium carbon steel showed m values within the range of 15 to 27 [52], as shown in Table 3. The seven datasets that follow are from dual-phase steel with ferrite and martensite phases [109]. The carbon level is low (0.11-0.12%), producing a uniform strain of more than 10%. Estimated m values ranged from about 20 to 113, and most were in the 30 to 50 range. The case of the highest m value was for quasi-static loading, and the yield strength also showed a low scatter, making the high m value plausible (one dataset was not included since only a one-digit S value was given).
The next 15 rows in Table 7 covered various alloys with a wide range of m values. Most of them are roughly comparable to the similar alloys given in Table 3. Ti-6Al-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 ® steel (Carpenter Tech. Corp, Philadelphia, PA, USA), which has high strength and fracture toughness [115]. It is of composition, 0.23C-13.4Co-11.1Ni-3.1Cr-1.2Mo, 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 Ic or J 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 Ic 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 Wallin et al. [32,116], who theoretically predicted m = 4 for K Ic 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 obs = 2.3 and m 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 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 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, Salem [47] [116] and the results in Table 4. Salem 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 est are mostly within ± 20% of m obs from a Weibull plot of the base strength data. This ± 20% limit is based on 71 datasets in the present study, which started from a listing of strength values, and the values of m obs /m est ratios were calculated. The average was 0.98 (S = 0.088), and the m obs /m 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 obs and m 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 ± 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 obs = 9.98, while the average strength was 659.23 MPa and standard deviation was 59.56 MPa, yielding m est = 12.18. Thus, m obs /m 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 obs = 6.38. Using the data given in Table 4 of C1239, m obs , <σ>, and S were calculated using the methods of this study, and resulted in m obs = 6.52 and m 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 obs of 60 to 116 [54] (see Table 3). Two extensive tests of A357 Al castings confirmed high m obs values for high-quality Al castings. Using 354 and 388 samples, m obs values of 47.5 and 30.6 were obtained [124]. For Al-7Si and Mg AM60 castings, three studies reported m 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.

Industrial Strength Data
When one examines strength data from metal industry, the deviation of data is often given in terms of standard deviation. While the normal distribution can represent the data well, it is often useful to describe the data with Weibull distribution, as it will allow advanced data analyses, such as failure and lifetime prediction. A short list of large-scale studies of steel strength are collected and summarized in this section, providing representative m values estimated, as listed in Table 8. One notable feature is that the sample counts are high, making the outcome more reliable. Two related works are also added.  a.
Hot-rolled steel [129] Evaluating the strength of 703 coils of this high-strength low-alloy (HSLA) steel, S355MC, the average and S value provided an m est of 48.4. See Table 8. b.
Shipbuilding steels [130] The published data of nine groups of shipbuilding steels was tabulated, from which m values were estimated. Steels are types A, B, and C of the American Bureau of Shipping (ABS) and ASTM A7. These grades were superseded by newer grades in the 1960s, but all were weldable low-carbon steels. The m est values were from 12 to 48.7.
c. Chinese HSLA steels [131] Q235 and Q335 (corresponding to S235 and S355) steels from four steel mills were tested. Data for a total of 20,086 plates was listed, and their m est values for six groups each are given in Table 8. The results are tightly distributed between 19-24.6. These steels were made for the penstocks of hydropower stations. The weighted m est average was 22.3.
d. Plain carbon and HSLA steels, S235, S355, and S550 [132] Standard grade steels had m est ranging from 10 to 22. The high S (low m est ) value for S235 steel was attributed to the mill practice of mixing subgrade steels, but the tests included four different structural shapes, and different processing may also be a factor. In addition, the minimum strength value was 50 MPa below the required strength for equivalent ASTM A283 steel.
e. Hot-rolled steels [133] This study provided strength data for various shapes, and those for hot-rolled plates are shown here. Samples counts are large (290 to 4095) and yielded m est values of 14 to 27 for five grades of steel.
f. Steel shapes [134] This work summarized a collection of Canadian steel data of 34,453 samples. The strength values were mostly collected from mill certificates. For steels of 300, 350, and 450 MPa (nominal) tensile strength, m values were found to range from 18 to 37. The weighted m average was 28.2. The Canadian and Chinese studies [131,134] used large sample counts and produced the most consistent and representative m values of contemporary HSLA steels, that is, m est is 22.3 to 28.2. These values are approximately one-half of the lower limit of m obs observed in the laboratory studies as discussed in Sections 4.2 and 5.1. As noted previously, this reduction in m (or increase in CV) is caused by additional deviation due to chemistry variations, process differences between steel mills, and test procedures, among others. In terms of the parameters used in the ASTM E28 study [24], it is the R parameter that governs the deviation for large-scale studies. Thus, the observed reduction in m est is expected. g.
Reinforcing bars [135,136] Two studies examined steel-reinforcing bars and showed m values of 18 to 34.
h. High strength suspension cable wires [15] Two i. Cast iron pipes [137] In spite of its known lack of ductility, cast iron pipes have been used for water distribution systems at many cities. This study reported results of a systematic examination of buried cast iron pipes in and around London, UK. Samples were excavated from 119 locations that were known to be in four different stages of deterioration: undamaged, lightly damaged, moderately damaged, and heavily damaged. The number of pipe samples, which were 0.5 to 1 m in length, was 34, 43, 36, and 36 at each stage, and about 15 samples were tested in flexure for each pipe sample. Nearly 1800 flexure strength tests were conducted. The results were analyzed, and most of the Weibull modulus values were found to be below 10, as indicated in Table 8. The damage stages and median m obs values appear to be correlated, but the m obs value is so low, even in the sound state. Thus, improved nondestructive testing methods may be more beneficial for identifying the damage states of buried pipes [139].
j. Graphite [138] A large-scale testing of nuclear-grade graphite examined the Weibull moduli of 2000 samples and listed the results in eight groups. A summary is given in Table 8, showing low m obs values of 6.8 to 13.4. These are lower than those for NBG18 in Table 4. In these studies, sample counts were within a factor of 2.3, and quality differences may be the cause. k.

Large-scale testing
In most of the large testing projects reviewed here, Weibull analysis was not included. The simple estimation method improved in this study can easily add Weibull modulus data to elaborate data collection and analysis conducted in metal and construction industries and elsewhere. The m values will be beneficial in subsequent analyses, such as those conducted in the structural health monitoring of structures [140].

1.
Methods of estimating Weibull modulus (m) of an experimentally obtained dataset were examined. These utilized the average (<σ>) and standard deviation (S) (or coefficient of variation, CV) based on the normal distribution. Several approximate relationships have been proposed starting from Robinson [11], but all of them deviate from the exact expression given with the gamma function.
In order to obtain m values that fit with the actually observed material strength datasets, a reduction of the constant from 1.271 to 1.10 is found to be optimal. This produces the modified Robinson relation of m = 1.10 <σ>/S = 1.10/CV, which can estimate m values that are in good agreement with the m values obtained from Weibull analyses. This agreement was verified by over 260 datasets of the strength of metals, ceramics, fibers, and composite materials, with most of the data from tensile or flexure testing.

4.
Applications of this simple estimation method are discussed. A common notion that ductile metals always have high m values must be discarded. Causes of m reduction need to be considered as material variation, and test accuracies can affect the outcomes. The method can add a quantitative tool based on the Weibull theory to engineering practice.
Funding: This research received no external funding.

Acknowledgments:
The author is grateful to Robert Gordon and Stephen Walley for providing historical strength documents, to Bruce Dunn for a statistics document and to K. Naito, G.F. Guo and L. Huang for supplying unpublished strength data.

Conflicts of Interest:
The author declares no conflict of interest.