Metal fatigue. A low-cycle fatigue test of 384 specimens of a superalloy at a particular strain range and high temperature yielded the following data. Data are grouped by the log (base 10) of the number of cycles to failure into intervals of 0.05. The data show the number of specimens failing from each cause. S denotes failure due to surface defect, I denotes failure due to interior defect. The following analyses indicate the potential improvement in life that would result if a research program yields a new manufacturing method that eliminates surface defects.

(a)    On normal paper (for log cycles), make a hazard or probability plot of the distribution when both failure modes act. Take the grouping of the data into account.

(b)   Is the usual assumption of lognormal fatigue life of metal specimens adequate?

(c)    Estimate u and the 50 and 1% points of this fatigue life distribution. Usually one is interested only in low percentage points of a fatigue life distribution. Do you recommend fitting a line to the entire sample to estimate the 1% point? Why?

(d)   On normal paper, make a plot of the distribution of log cycles to failure from interior defect, assuming that failure due to surface defect is eliminated

(e)   Are failures due to interior defect adequately described by a lognormal fatigue distribution?

(f)     Estimate u and the 50 and 1% points of this "potential" fatigue life distribution.

(g)    Do (d), (e), and (f) for surface defects.

(h)   Some specimens may not have surface defects and would be prone only to failure from interior defects. How would this affect preceding results?

(i)      The assumption of independent competing failure modes may not hold. How does dependence affect the preceding results?

(j)     A common mistake in estimating the distribution for a particular mode is to treat the failures from that mode as if they were a complete sample, ignoring failures from any other modes. Plot the interior failures as if they were a complete sample of 60 failure times. Compare this plot with the correct plot, especially the 1% and 50% points.

(k)    Repeat 0) for the 324 surface failures.