The probabilistic estimation of fatigue crack life time under spectrum loading is complicated by load interaction effects [ 1,2 and others]. Life time estimates have recently been obtained for loading spectra consisting of random overload sequences superimposed on base-line cyclic loading [3]. For highly irregular spectra, however, the superposition model seems to be inadequate.
The analysis presented here shows that an application of the closure concept [4,5] to the probabilistic modeling of the fatigue crack growth under loads characterized by highly irregular spectra permits the development of a relatively simple procedure for crack life time assessment.
Consider a body with a growing crack, under random load as schematically shown in Fig. 1. The loading history is assumed to consist of alternating sequences of minima and maxima, the triplet c~=i,G=,~o~i+l representing an arbitrary loading cycle. The crack grows during [he a~cencling part of the cycle, so that the stress range is:
(1)
A(~i -~ (~max ,i --(~min,i
It is assumed that: (a) the loading is a stationary random process whose values (O,~,Or~,A~) for any two different cycles are noncorrelated random variables, and (b) all ~ (as well as (y= and Acy) are equally distributed.
Let U(x),V(x),W(x) be the probability distribution functions of O=x,Om~, and A(r, respectively. It is actually sufficient to know two of the above distributions, the third can be derived by means of (1). If distributions of cy~ and Ao are known (or could be deduced from the load records) the distribution of o=~ can be obtained as follows:
(2)
According to the closure concept introduced by Elber [4] the effective stress intensity factor range AKo~, which determines the crack growth rate, depends on the crack opening stress cy , which in turn depends on the maximum and o~ .
minimum stresses observed m preceding cycles.