Consider the s-N curve in Figure 1 again.
In the LCF region (upper left, N<107 cycles) it has long been sound
engineering practice to read the curve at some operating level (stress, strain, SWT, or
similar metric) and count cycles until the lower bound cycle count was exceed and the
component would be retired from further service. With the recognition of the random
fatigue limit comes the necessary, but unpopular, realization that similar cycle counting
in the HCF region of the s-N curve (lower right, N>>107 cycles) isn't
feasible because the behavior of a given specimen, or component, depends on its own unique
fatigue limit. For example, reading the curve at SWT=50 shows the behavior of some
specimens with lower individual fatigue limits to have failed, while others, tested at
the same stress level, ran out, producing censored observations, where actual failure
time is unknown other than being greater than the censoring cycle count. Counting the
cycles exhausted at SWT=50, then, would depend on knowing the individual RFL, which of
course, being random, is unknowable a priori. Thus what is sound engineering
practice in the LCF region is not feasible for HCF. This is only a modest setback, however
because the RFL model proves a method of estimating the probability of failure,
even if the cycle count is intractable.
The RFL Model is a Precursor to a "Probabilistic Goodman" Diagram.
The next step is to consolidate the s-N behavior described by the RFL model and
illustrated in Figure 1, into a "Probabilistic Goodman(1)" diagram.
It should be recognized that while this step will be a welcome advance, it will not be the
ultimate destination. There is much more to the behavior of a material responding to HCF
than can be described using only the mean stress, alternating stress measure of the
HCF data are characterized by
more than just runouts (described elsewhere). Tests of 107 108 and even 109 cycles often
display an asymptotic behavior, illustrated in figure 1.
these data produced unsatisfactory results as the usual assumption of uniform variance
(data scatter), over the many orders of magnitude in fatigue life, proved untenable.
Attempts to deal with this ballooning variance by various variance-stabilizing
transformations were similarly unsuccessful, as were attempts to model the behavior with a
single-valued fatigue limit asymptote.
The difficulty is that the fatigue limit exhibited in a
given s-N test is not a constant, as had been previously thought,
but rather a random variable, just as the individual fatigue life of a specimen tested at
a given stress is random. While this may seem self-evident in retrospect, it was
only very recently recognized and described mathematically.
Alas, the devil IS
in the details. While understanding the idea of a random fatigue limit is simple
enough, estimating the parameters of such a model is a challenging task indeed.
Professor William Q. Meeker has graciously made his S-Plus adjunct
software available to anyone using S-Plus. It includes the codes he used in his
recent book (below). To get the software, which requires S-Plus to use, visit his website.
|Pascual and Meeker, "Estimating
Fatigue Curves with the Random Fatigue-Limit Model," TECHNOMETRICS
Vol. 41, No. 4, p.277-302, November 1999
Stigler, Statistics on the Table, a history of statistical concepts and methods,
Harvard University Press, Cambridge, MA, 1999, p277-290.
Meeker and Escobar, Statistical Methods for
Reliability Data, Wiley, 1998. This is an excellent book. If
your bookseller is out of stock, you can get a copy quickly from amazon.com.