# A Simple Estimation Method of Weibull Modulus and Verification with Strength Data
## 4 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 [PERSON] [32], K\({}_{\mathrm{JC}}\) (K\({}_{\mathrm{lc}}\) 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\({}_{\mathrm{JC}}\)). 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 (\(<\)\(\sigma\)\(>\)), 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(r). 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; \(<\)\(\sigma\)\(>\); S; \(<\)\(\sigma\)\(>\)/S; observed m value, m\({}_{\mathrm{obs}}\); estimated m value from Equation (14) given below, m\({}_{\mathrm{est}}\); N; the ratio of m\({}_{\mathrm{obs}}\) to m\({}_{\mathrm{est}}\) (= m\({}_{\mathrm{obs}}\)/m\({}_{\mathrm{est}}\)); 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\({}_{\mathrm{obs}}\) to m\({}_{\mathrm{est}}\) 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\({}_{\mathrm{obs}}\)/m\({}_{\mathrm{est}}\) (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\({}_{\mathrm{obs}}\)/m\({}_{\mathrm{est}}\) = 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.