Guidelines for Two-Parameter Weibull Analysis for Flaw-Containing Materials
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WALLODI Weibull introduced an empirical distribution based on the ‘‘weakest link’’ developed by Pierce,[3] which has since been widely applied to the fracture-related mechanical properties of ceramics and metals. The cumulative probability function of the twoparameter Weibull distribution is expressed as follows: m r P ¼ 1 exp ½1 r0 [1,2]
where P is the probability of failure at a given stress (strain, fatigue life, etc.), r, or lower. The term r0 is the scale parameter, and m is the shape parameter, alternatively referred to as the Weibull modulus. The Weibull distribution has been successfully applied to materials in which fracture properties are determined overridingly by the defect distributions. For instance, Green and Campbell[4] showed that the tensile strength (ST) of A356 castings alloys follows a two-parameter Weibull distribution and that the filling system design has a strong effect on the Weibull modulus. Hence, the reliability of castings could be measured with the Weibull modulus. Since the results of Green and Campbell, the two-parameter Weibull distribution has been used extensively in the casting literature to characterize fracture-related mechanical properties such as tensile strength (ST),[5] elongation,[6] and fatigue life (Nf).[7–9] According to Campbell,[10] the Weibull modulus, m, for tensile strength is often between 1 and 10 for pressure die castings, and between 10 and 30 for many gravity-filled castings. For good quality aerospace
castings, m is expected to be between 50 and 100. Hence, the Weibull modulus has been used as a measure of casting quality and reliability. Recently, Tiryakiog˘lu and Campbell[11] provided guidelines for interpreting Weibull probability plots including the three-parameter Weibull distribution and Weibull mixtures. They stated that the two-parameter Weibull distribution is applicable only when castings have defects with large sizes that come from the same, single distribution, i.e., one source of damage during melt preparation and mold filling (the reader is referred to Reference 11 for continued discussion on how to make the distinction between the two- and threeparameter Weibull distributions). Ceramics, with their low-fracture toughness, are also expected to yield fracture properties that will follow the two-parameter Weibull distribution. The present study is intended to provide a step-by-step procedure for Weibull analysis when only two parameters are warranted and relies on the linear regression technique, which is commonly used in the analysis of fracture data.
II.
PROCEDURE FOR THE 2-PARAMETER WEIBULL ANALYSIS
A. Step 1. Assign Probability to Each Data Point Numerous probability estimators (alternatively referred to as plotting position formulas) can be found in the literature. These formulas can all be written in the following form: P¼
MURAT TIRYAKIOG˘LU, Professor of Mechanical Engineering and the Director of the School of Engineering, is with the University of North Florida, Jacksonville, FL 32224. Contact e-mail: m.tiryakioglu@
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