Earlier attempts at modeling the stress-life
(s-N) behavior of cyclic fatigue in the long life regime used a linear equation relating
log(cycles) and log(stress), modified with a constant runout stress, or fatigue limit: log N_{i} = b_{0} + b_{1} log (S_{i} – b_{2})
+ e_{i}
equation (1)
where, for specimen *i*, N_{i} represents cycles to failure, S_{i}
is the applied stress parameter, and b_{2} is a *constant* fatigue limit (S_{i}>b_{2}), and e _{i} is a
random variable representing the scatter in cycles to failure about the predicted life.
Typically, the life random variable, e , would be represented
by a lognormal distribution with zero mean. For this assumption, e
_{i} is the difference between the log life of specimen *i* and the log
median life at the test stress S_{i}. The parameters of the median life
prediction, b_{0}, b_{1},
and b_{2} are estimated from test data and b_{2} is interpreted as the fatigue limit stress condition.
Since b_{2} is an asymptote, the s-N curve flattens as
S approaches the fatigue limit. This model is only marginally adequate for the median
behavior in the long life regime but it is not consistent with the commonly observed
increase in the standard deviation of lives as S approaches the constant fatigue limit
(see Figure 1). But the main shortcoming of a constant fatigue limit is that it doesn't
work. Since it is a single-valued constant, the fatigue limit, b_{2}, must be less than the lowest
stress tested (so that the logarithm of (S_{i} - b_{2}) is defined) weather the
specimen failed at that stress or not. This causes the b_{2} asymptote to be so low as to produce an unrealistic
material model that had to be continually revised downward to accommodate newer, low
stress data.
The random fatigue limit model is a generalization of Equation (1) in which the fatigue
limit term is modeled as a random variable that can be considered to result from inherent,
but unknown, quality characteristics of each specimen in the population. Thus the value of
the fatigue limit is *not* a single constant, but rather an individual characteristic
of each specimen (or component). The RFL model for test specimen *i* is given by:
log N_{i} = b_{0} + b_{1} log (S_{i} – g_{i})
+ e_{i} equation
(2)
where g_{i} is the random fatigue limit for specimen
*i* (S_{i} > g_{i}) and is expressed in
units of the stress parameter. In this model, e is the random
life variable associated with scatter from specimens that have the same fatigue limit.
The RFL model produces probabilistic s-N curves that have the characteristics commonly
seen in HCF data. This is illustrated in Figure 1 which presents the 1^{st}, 5^{th},
10^{th}, 50^{th}, 90^{th} 95^{th} and 99^{th}
percentile s-N curves as would be determined from the distribution of fatigue limits. The
percentile s-N curves display the commonly observed shape in the HCF regime. Further, it
is easily seen in Figure 1 that a difference in test lives from two specimens with
slightly different fatigue limits could be quite large. The increased scatter in fatigue
lives is explained by different specimens having different fatigue limits and this is true
regardless of the scatter in life at higher stresses. Thus, the RFL model accommodates not
only the flattening of the s-N curve but also the increased scatter that is typical of HCF
lives. Experience to date indicates that the fatigue limit scatter dominates in the HCF
regime when S is close to g_{i} while the scatter in
life is more significant when S is large compared to g_{i}.
There are two random variables in the RFL model for which probability distributions are
needed. Experience again suggests the conventional lognormal distribution is appropriate
for e_{i},
the scatter in cyclic lives. Thus, the conditional distribution of cycles to failure given
g will be a lognormal distribution with mean equal to b_{0} + b_{1} log (S
– g ) and standard deviation equal to s_{e}
. Then e is lognormal (0,s_{e}). The Weibull distribution does well describing the skewed
downward behavior of the random fatigue limit, g_{i}.
The Weibull parameters, h, b,
represent the 63.2^{th} percentile runout stress, and shape parameter,
respectively. Thus h has the same units as the stress metric. Specimens will have inherent, but
unknown, quality differences that result in distinct fatigue resistance in the long life
regime. The upper limit of runout stress is observed to be more restrictive than the lower
limit. That is, while a very low quality specimen is sometimes observed, albeit
infrequently, extremely high runout stresses are never observed. (And it is these
infrequent lower performers that are at the crux of the HCF problem.) The RFL model
provides a means to measure the propensity for this life-limiting behavior. |