Fuzzy Regression Analysis for Fatigue Crack Growth

This paper presents a method based on fuzzy regression to analyze fatigue crack growth data, where the variations of the parameters are not only due to measurement errors but also system errors. A membership function is used to describe the system errors. Crack growth data under constant and random amplitude stress are analyzed and the results are compared with conventional least-squares regression methods. Introduction Fatigue is an important failure mechanism of many electrical and mechanical components and numerous methods have been developed to estimate fatigue damage. In the traditional approaches, uncertainty and ambiguity involved in fatigue phenomena have been treated with probability theory [l31. However, the calculated probability of failure is often much smaller than that observed in the field [4], due to the exclusion of failure resulting from subjective uncertainties and system error including subjective extrapolation of data. Fuzzy set theory provides a useful tool to handle subjective uncertainties in a quantitative way. Fuzzy set theory has been used to analyze the S-N curves and predict fatigue life [5-71. In this paper, a method to analyze the fatigue crack growth rate using fuzzy regression is proposed. The aim of applying fuzzy regression analysis is to provide a possible solution to bridge the gap between the calculated and observed probability of fatigue failure. The method assumes that the variations of the parameters are not due to measurement errors but system errors including subjective uncertainty which can be described by means of a membership function. The method is applied to crack growth data tested at both constant and random amplitude stress and the results are compared with conventional least-squares regression methods. Fuzzy Regression A fuzzy set is a collection of objects without clear boundaries. If A is a fuzzy set on the set Z, then an element represented by a of Z can partially belong to A. A membership function, mA(u), is used to describe the grade of a belonging to A. The value of the membership function is between 0 and 1. If mA(u)=O, element U does not belong to A. If ma(a)=l, the element a is clearly belong to A. Ordinary sets are a special case of fuzzy sets. A fuzzy set A on the set of real numbers is called a L-R fuzzy parameter [SI if the membership function of ais calculated as a > a where a, c1>0, c2>0 are constants, c1, c2 represent vagueness measurement and L(y) and R(y) are continuous strictly monotonically decreasing functions on [0,1] which satisfy L(y) = R(y) = 1 if y I O L(y) =R(y)=O if y 2 1 (2) Examples include L(y) = 1-y, and R(y) = l-yz. The L-R fuzzy parameters are written symbolically as A = [ a , c1, c2). If c1 = c2 = c, then A=( a, c). In fuzzy regression the parameters of a function are considered as fuzzy parameters and estimated to fit the empirical data. Consider the linear regression function with n variables Y = Alx l+ A 2 ~ 2 + .... + Anxn (3) If A,, A,, ..., A, are fuzzy parameters, then Y is a fuzzy set. Define the fuzzy parameter vector 8 = ( A I , ..., a,) (4) where Aj = (a j , cj) is a L-R fuzzy parameter. With 3 = (xl, x2, ..., x,)T, the fuzzy linear function can be written as where a is denoted as A membership function of a fuzzy parameter vector Z is [61: TH0334-3/90/0000/0437$01 .OO Q 1990 IEEE 431 m g (4 = Min [m4aj ) l (9) where mAi(9) is the membership function of the jth fuzzy parameter,"which has the form of equation (1). For simplicity, let L(y) and R(y) be linear functions (figure l), such that