Abstract
The Maximum Likelihood method and the Linear Least Squares (LLS) method have been widely used to estimate Weibull parameters for reliability of brittle and metal materials. In the last 30 years, many researchers focused on the bias of Weibull modulus estimation, and some improvements have been achieved, especially in the case of the LLS method. However, there is a shortcoming in these methods for a specific type of data, where the lower tail deviates dramatically from the well-known linear fit in a classic LLS Weibull analysis. This deviation can be commonly found from the measured properties of materials, and previous applications of the LLS method on this kind of dataset present an unreliable linear regression. This deviation was previously thought to be due to physical flaws (i.e., defects) contained in materials. However, this paper demonstrates that this deviation can also be caused by the linear transformation of the Weibull function, occurring in the traditional LLS method. Accordingly, it may not be appropriate to carry out a Weibull analysis according to the linearized Weibull function, and the Non-linear Least Squares method (Non-LS) is instead recommended for the Weibull modulus estimation of casting properties.
Highlights
THE Weibull distribution has been widely used to analyze the variability of the fracture properties of brittle materials for over 30 years
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 widely known as the Weibull moduli
This observation was consistent with the results of References 5 and 7
Summary
THE Weibull distribution has been widely used to analyze the variability of the fracture properties of brittle materials for over 30 years. 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: r m P 1⁄4 1 À exp À. The Weibull modulus can be determined according to the slope of a simple linear regression, (i.e., ordinary least squares) of Ln [ÀLn (1 À P)] against Ln(r), where the P value is assigned by a probability estimator.
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