# A Simple Estimation Method of Weibull Modulus and Verification with Strength Data
## 1 Introduction  
[PERSON] 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 near 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]:  
\[1-\mathrm{P}=\exp\{-(\sigma/\sigma_{0})^{m}\}, \tag{1}\]  
where m is the Weibull modulus (also called the shape parameter), \(\sigma_{0}\) is the scale parameter, and \(\sigma\) is the variable (fracture strength in this study), respectively. The basis of this distribution is given in terms of the weakest link theory; see [PERSON] and [PERSON] [11; 12]. From N fracture tests, a cumulative probability distribution curve is obtained as a function of (\(\sigma/\sigma_{0}\)). In a graphical approach, one can follow [PERSON] [1], plotting \(\ln(-\ln(1-\mathrm{P}))\) against \(\ln(\sigma)\) or \(\ln(\sigma/\sigma_{0})\), and obtaining a best fit line, the slope of which equals m. The scale parameter, \(\sigma_{0}\), is slightly larger than the average fracture strength <\(\alpha\)>, and these are related by [3; 13; 14]:\[<\sigma>\ =\sigma_{\rm o}\ \Gamma(1+1/{\rm m}), \tag{2}\]  
where \(\Gamma({\rm x})\) is the gamma function. This Equation (2) can be approximated by [15]:  
\[<\sigma>\ =\sigma_{\rm o}\ (1+0.276\ {\rm m}^{-0.776}). \tag{3}\]  
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 [PERSON] showed that the coefficient of variation (CV) for m is equal to 1/\(\sqrt{\rm N}\)[11]. [PERSON] 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 \(\geq\) 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.