Correction Factor for Unbiased Estimation of Weibull Modulus by the Linear Least Squares Method
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INTRODUCTION
THE Weibull distribution is widely applied for the modeling and analysis of strength data in the material field, such as brittle materials,[1,2] metals,[3] polymers[4] and dental materials.[5] This distribution is based on a ‘‘weakest link theory’’, which means that the most serious flaw in the material will control its strength, similar to a chain breaking if the weakest link fails.[6] For the two-parameter Weibull distribution, when the strength of material is at or below a stress S, the cumulative probability of failure F is m S ½1 F ¼ 1 exp ; S0 where S0 is the scale parameter and m is the Weibull modulus, which is also called the shape parameter.
XIANG JIA is with the College of Systems Engineering, National University of Defense Technology, Hunan 410073, P.R. China. Contact e-mail: [email protected] GUOGUO XI is with the School of Materials Science and Engineering, Beihang University, Beijing 100191, P.R. China. SARALEES NADARAJAH is with the School of Mathematics, University of Manchester, Manchester M13 9PL, UK. Manuscript submitted October 20, 2018.
METALLURGICAL AND MATERIALS TRANSACTIONS A
There are different forms of failure, such as flaw and fracture. The Weibull modulus m determines the scattering extent of failure strength. Given a set of stress data such as experimentally measured fracture stresses, numerous methods have been proposed to estimate the parameters S0 and m, including method of moments, maximum likelihood (ML) method, and linear least squares (LLS) method. The method of moments conforms to the assumption that the mean and variance of the experimental data are equal to those of the Weibull distribution.[7] This method can not obtain a closed-form estimation of Weibull modulus. In addition, the precision of estimation is also poor.[8] The ML method[7,9] is derived from the likelihood function describing the possibility of the experimental data. The likelihood function is the product of the probability density function (PDF) m m S m1 S ½2 exp f¼ S0 S0 S0 over all the experimental data. The values of m and S0 that make the likelihood function largest are the ML estimates. The ML method has been claimed to be capable of achieving the highest estimation precision.[6,7] However, this method usually overestimates m,[8,10] which would, in turn, yield overestimates of the reliability at low stresses, i.e. the values of stresses are small, and lead to lower safety in reliability prediction,[6,8,10] as the overestimate of reliability brings misleading judgement making the usage of material insecure. From an
engineering perspective, safety must be prioritized over estimation precision. Hence, overestimation and decreased safety are disadvantages in the use of ML method. Given its straightforwardness and simplicity, LLS is by far the most popular method used to estimate Weibull parameters for stress data. The LLS is especially favored in material science with applications in bulk metallic glass,[11] borosilicate glass,[12] nuclear materials,[13] rock grains
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