Role of third order statistics in discriminating among models of fatigue crack growth in metals
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I.
INTRODUCTION
O N E of the important points made in this paper is that available data support the physical hypothesis that fatigue crack growth is controlled by the strong points encountered as the crack progresses. We call this the "strongest link" hypothesis in contradistinction to the "weakest link" hypothesis. The coefficient of skewness "/l(1r4)* of the probability distributions (~h) of the time to reach a given crack length employed in fatigue crack growth constitutes a quantity that can be employed to discriminate among such distributions and between these hypotheses. We shall use yl, which is a third order statistic, for their purpose. Of necessity, statistical concepts must be employed; we provide in Appendix 3 a short explanation of some of these concepts. A few introductory comments are in order before embarking on details. A variety of probability distributions (~r0* have been *See Appendix 3 for a short explanation of probablhstic and statistical terms employed hereinafter. These will be noted only in Sections I and II.
employed to describe the statistics of the random time for a crack to increase by a specified amount in fatigue, Among these might be mentioned the gamma, lognormal, inverseGaussian, Birnbaum-Saunders, and Weibull distributions. 5'7 Plausible physical arguments can be advanced in support of some of these distributions. In particular, the "weakest link" concept of crack growth can be put forward in support of the Weibull distribution, 7 etc. In the course of our analyses of fatigue crack growth data (rr2), we have found that three extreme value distributions (Ir~) merit consideration) Further, we atso have noted the history dependence of the fatigue crack growth process, 1but it will not be discussed here. The purpose of this paper is to compare the coefficient of skewness (rr4) as given by the above distributions with estimates of this coefficient obtained from data, and to discuss how skewness has implications on the nature of crack
E KOZIN is Professor of Systems. Polytechnic Institute of New York, Route 110, Farmingdale, NY 11735. J.L. BOGDANOFF is Professor Emeritus, School of Aeronautics and Astronautics, Purdue University, West Lafayette. IN 47907. This paper is based on a presentation made at the symposium "Stochastic Aspects of Fracture" held at the 1986 annual AIME meeting m New Orleans, LA, on March 2-6. 1986, under the auspices of the ASM/MSD Flow and Fracture Committee. METALLURGICAL TRANSACTIONS A
growth models. The location of the toe of a distribution relative to the mean is important in reliability and maintainability studies where attention is focused on early failures in a fleet of large size. Since the coefficient of skewness has a substantial influence on the toe location for standardized distributions (7r5, 7r6), it is important to discuss it in detail.
II.
BASIS FOR C O M P A R I S O N
Standardized rv's (random variables) have mean (location parameter) zero and variance (scale parameter) equal to one (7r6). All rv's with finite variance can be standardize
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