applied sciences Review A Simple Estimation Method of Weibull Modulus and Verification with Strength Data Kanji Ono Department of Materials Science and Engineering, University of California, Los Angeles (UCLA), Los Angeles, CA 90095, USA; [email protected]; Tel.: 1-310-825-5534 Received: 17 March 2019; Accepted: 12 April 2019; Published: 16 April 2019 Abstract: 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 offer 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. Keywords: Weibull modulus; estimation methods; modified Robinson relation; strength data; observed datasets; large-scale data 1. 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 [6–10]. 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 [1–3]: 1 - P = exp{-(σ/σ o ) m }, (1) 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]: Appl. Sci. 2019, 9, 1575; doi:10.3390/app9081575 www.mdpi.com/journal/applsci
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