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In the linear regression method for estimating parameters of a Weibull distribution, multiple flaw distributions may be further evidenced by derivation form the linearity of data from a single Weibull distribution. In this paper, a new technique of estimating multiple Weibull parameters is conducted, compared to commonly used regression probability estimator.

Weibull statistics has become a well-established characterization tool in the field of fracture strength of ceramics. Experiences have shown that mechanical and fracture properties of brittle materials are strongly affected by the component size and the Weibull weakest link model properly considers the size effect of the components [6]. Based on physical assumptions [7], the Weibull equation describes the relationship between the probabilities of failure PfPf of a brittle material under uniaxial tensile stress of σ.σ. It thus predicts the inherent dispersion in fracture strength of brittle materials. The simplified two parameters Weibull equation equation(1) View the MathML sourcePf≡F(σ)=1-exp-σσ0m, Turn MathJax on has been widely used in estimating failure chance of ceramic components. The two parameters in Eq. (1), the so called Weibull parameters, determine the shape and location of the cumulative distribution function F(σ).F(σ). The Weibull modulus m,m, sometimes called the shape parameter, has a value between 5 and 20 for technical ceramics [4]. On a normalized scale, a higher mm leads to a steeper function and thus a lower dispersion of fracture stresses. The scale parameter σ0σ0 is closely related to the mean fracture stress, influences the variance of the fracture stress, i.e. the steepness of the function: a smaller σ0σ0 means – on an absolute scale – a lower dispersion. Once a set of NN experimentally measured fracture stresses are obtained, it is desirable to fit the Weibull equation (Eq. (1)) to these observations, i.e. to determine the two parameters mm and σ0,σ0, knowledge of which leads to a complete characterization of the material for the given volume. There are several methods available in the literature for the determination of these two parameters [2] and [3]. The most widely used is the linear regression method due to its simplicity. The linear regression method is based on the fact that Eq. (1) can be written as a linear form when take the logarithm twice: equation(2) View the MathML sourcelnln11-Pf=mlnσ-mlnσ0. Turn MathJax on So, the measured fracture stresses are ranked in ascending order, and a probability of failure PiPi is assigned to each stress σiσi. The Weibull modulus can be obtained directly from the slope term in Eq. (2), and the scale parameter can be deduced from the intercept term. Several prescribed probability estimators has been used as the PiPi-value in different literatures. The following four expressions are often applied to define the probability estimator [2] and [3]: equation(3) View the MathML sourcePi=i-0.5n,Pi=in+1, Turn MathJax on View the MathML sourcePi=i-0.3n+0.4,Pi=i-0.375n+0.25, Turn MathJax on where Pi is the probability of failure for the ith-ranked stress datum. The relative merits of these estimators have been investigated by several authors with actual or computer-generated strength data. It has been shown that the first of Eq. (3) gives the least bias and is therefore preferred. However, advanced ceramics typically contain two or more active flaw distributions each with an independent set of parameters [1]. The objective of this paper is to model these conditions and estimate the multiple flaw distributions using linear regression method.

This paper outlined the application of parameter estimation based on linear regression method when a sample of test specimens yields two or more flaw distributions, when strength data shows a quadratic, cubic, or higher degree pattern. However, for the data sets with two or more flaw distributions which one of them occurs in a small number of specimens, it is sufficient to just report the existence of this flaw distribution. It is not necessary to estimate Weibull parameters because they would be potentially biased. Generally, when Weibull probability plot of some specimens shows the possibility of existing two or more flaw distributions, experimenters use costly and time consuming fractography methods to categorize the tested specimens. The described linear regression method provides a much easier way to subgroup such samples and gives basic estimates for further analysis.