Estimation of Weibull parameter with a modified weight function
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Estimation of Weibull parameter with a modified weight function B. K. Chandrasekhar Ceramic Technological Institute, Bharat Heavy Electricals Ltd., Bangalore-560 012, India (Received 1 August 1995; accepted 15 April 1997)
The Weibull modulus is widely used for estimating the reliability of ceramic components in engineering applications. An improvement in the evaluation of the Weibull modulus is achieved by using an appropriate weight function to the data points while fitting a straight line to the Weibull plot by the least square method. The conventional weight function is a function of the probability of failure. This paper describes an alternate method of obtaining the weight function based on first principles. This modified weight function is a function of the stress at failure rather than probability of failure. Evaluation of the two-parameter Weibull modulus was estimated on simulated strength distribution data with both the weight functions. A comparative analysis indicates that the modified weight function gives a different result than the conventional weight function. The paper also highlights the effect and importance of uncertainties in the measurement of strength on the calculated Weibull modulus.
I. INTRODUCTION
The strength and reliability of ceramic materials are required to be evaluated for use in critical engineering applications.1 These materials are characterized by a relatively large distribution in strength (scatter), and hence a number of samples have to be tested for estimation of strength. The scatter in the experimentally measured strengths arises because of the statistical distribution of the flaws in the material. The material fails when a flaw is subjected to the highest stress intensity factor. Depending on the size of the flaws, their location in the sample, and their orientation to the applied stress, one obtains different values in strength resulting in scatter. The reliability of a ceramic component is expressed by the Weibull distribution2 given by: ∑ µ ∂ ∏ s 2 su m P 1 2 exp 2V (1) , so where P is the failure probability at stress s, V is the unit volume of the specimen (dimensionless), s is the applied stress at failure, s u is the threshold stress below which the probability of failure is zero, s o is the scaling factor, and m is the Weibull modulus (dimensionless). For the highest reliability,3 s u can be set equal to zero, resulting in ∑ µ ∂m ∏ s P 1 2 exp 2 (2) , so where the volume factor is merged with s o , the characteristic strength. With this modification, the three-parameter Weibull function is reduced to a twoparameter Weibull function. The Weibull relationship 2638
http://journals.cambridge.org
J. Mater. Res., Vol. 12, No. 10, Oct 1997
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assumes a uniform, random, and unimodal flaw distribution in the samples.4,5 II. EVALUATION OF WEIBULL MODULUS
In the simplest case, the Weibull modulus is evaluated from the experimental data by taking the double logarithm of Eq. (2) leading to ∂ µ 1 y l
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