Weibull Modulus Estimated by the Non-linear Least Squares Method: A Solution to Deviation Occurring in Traditional Weibu
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INTRODUCTION
THE Weibull distribution has been widely used to analyze the variability of the fracture properties of brittle materials for over 30 years. Fitting a Weibull distribution also later became a popular method in the prediction of the quality and reproducibility of castings.[1–3] The cumulative distribution function (CDF) of the Weibull distribution is given by[4] x xu m P ¼ 1 exp ; ½1 x0 where P is the probability of failure at a value of x, xu is the minimum possible value of x, x0 is the probability scale parameter characterizing the value of x at which 62.8 pct of the population of specimens have failed, and m is the shape parameter describing the variability in the measured properties, which is also widely known as the Weibull moduli. In a practical application, x could be substituted by the symbol r for the properties of materials (e.g.,
T. LI, W.D. GRIFFITHS, and J. CHEN are with the University of Birmingham, Birmingham B15 2TT, UK. Contact e-mail: [email protected] Manuscript submitted December 22, 2016.
METALLURGICAL AND MATERIALS TRANSACTIONS A
Ultimate Tensile Strength (UTS)), and the lowest possible value of property could be assumed to be 0, making xu = 0, so that Eq. [1] can be re-written as a 2-parameter Weibull function: m r P ¼ 1 exp : ½2 r0 There are several approaches to the estimation of the Weibull modulus in Eq. [2], with the most common methods being the Linear Least Squares method (LLS) and the Maximum Likelihood method (ML). Many researches focused on the bias of the estimated Weibull modulus obtained by the estimation methods. Khalili and Kromp[5] recommended the ML and the LLS methods after a comparison of the ML, LLS methods, and methods of momentum. Butikofer et al.[6] found that the LLS method was less biased than the ML method for a small sample size. Tiryakioglu and Hudak[7] and Wu et al.[8] studied the best estimators for the LLS method. However, there is still a shortcoming in the LLS method. In practice, some data points of the measured properties seriously deviate from the linear behavior in the traditional LLS method for Weibull estimation, resulting in a bad fit in the linear regression model. A good example was that of Griffiths and Lai’s[2] measurement of UTS of a commercial purity ‘‘top-filled’’
Mg casting, as shown in Figure 1. It is clear that the data points were not randomly scattered along the fitted straight line in this linear regression, and the corresponding R2 value was only 79.1 pct, both of which suggested that it was a bad linear fit. These outliers would exert much influence on the regression line, making the Weibull modulus deviate from its true value. This type of behavior (i.e., data deviation in the lower tail) in the plots of the linearized Weibull function (Figure 1) has occurred widely and resulted in estimation bias to various degrees, of which examples can be found in References 2 and 9 through 14. Keles et al.[14] made a summary of this deviation occurring in the measurement of brittle materials. When this deviation oc
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