106 Introduction and Background that random construction of a staircase sequence will provide similar results, particularly for the mean value of the FLS. In preparing data for a publication, Morrissey and Nicholas [45] used the bootstrap method to determine the FLS of Ti-6Al-4V for a cyclic life of 10 9 cycles based on available test data, some of which were obtained from tests to only 10 8 cycles or even less. In this artificial staircase procedure using the bootstrap method, every data point was treated in one of the following three manners and every data point was used once: 1. If a specimen failed at a given stress level in less than 10 8 cycles, for example, then that data point provided a failure point for 10 9 cycles at the same stress level. 2. If a specimen failed within 10 9 cycles at a given stress level, then that specimen provided a failure point at any of the higher stress levels used in the staircase sequence, that is, at any number of stress increments (fixed) above the stress level at which the original test was performed. 3. If a specimen survived for 10 9 cycles at a given stress level, then that specimen provides a survival point for any stress level lower than the one at which it was actually conducted. Using this logic, and using all of approximately 20 data points, several different “artificial” staircase test sequences were constructed from the available data pool. Surprisingly, the mean value of the FLS from each sequence used was reproducible to within 1MPa for a FLS of approximately 510 MPa. There is no rigorous mathematical basis for applying the bootstrap method to the construction of an “artificial” staircase test sequence using the three rules stated above, but the consistency of the values for the mean suggests that the method is both viable and reliable when testing time, machine availability, or limitation in number of specimens available becomes a practical limitation in conducting a real staircase test sequence. 3.5. OTHER METHODS In addition to step and staircase testing, another method for determining the fatigue limit or fatigue limit strength is referred to as the ArcSin √ P method, where a pre-selected number of stress levels are used to test a fixed number of specimens at each level [38, 39]. Similar to what is used for the staircase tests, the FLS and the associated statistics are determined from the probability of fracture at each stress level. Whereas the staircase method tends to test near the mean stress, the ArcSin √ P method deliberately spreads out the stress levels to hopefully encompass a wider range of stresses with which to determine the scatter in the distribution function that is used to describe the FLS. Both methods provide an accurate estimate of the mean, even for small numbers of tests, but Accelerated Test Techniques 107 neither method can provide an accurate estimate of the standard deviation unless large numbers of tests are conducted. When small numbers of specimens are used, both the staircase and the ArcSin √ P method underestimate the standard deviation of the FLS [39]. From observations of a number of numerical simulations, Braam and van der Zwaag [39] developed a method for evaluating the statistical properties of the FLS as determined from the ArcSin √ P method and provided insight into the planning and conducting of such tests. For the ArcSin √ P method, a fixed number of equidistant stress levels (m) are used and at each stress level a fixed number of specimens (n) is tested. For these data, a sinusoidal distribution function is used to represent the FLS. The probability density function (pdf) is f P x =  2b sin   x −a b  a ≤x ≤a+b (3.6) The probability of failure at each stress level is fitted to this distribution function using a least-squares routine, this providing the values of the parameters a and b as b = 2  ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ m  i=1 z i S i − m  i=1 S i m  i=1 z i m m  i=1 z 2 i −  m  i=1 z i  2 m ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ (3.7) a = m  i=1 S i −b  2 m  i=1 z i m (3.8) Here, S i is the stress at level i, where i = 1 is the lowest stress in the test series, and m represents the number of stress levels used. The quantity z i is the ArcSin √ P i , with P i being the probability of fracture at stress level i. The probability of fracture equals the ratio of the number of specimens broken to the total number of specimens tested at a given stress level. The quantity a+b/4 is the average value of the sinusoidal distribution and b is the interval between the two points where the pdf becomes 0. The sinusoidal pdf, unlike a normal (Gaussian) pdf, has the feature of covering a finite limit for values of the FLS. This implies that failures will not occur below some finite stress and, further, that all samples will fail above another value of stress. The Gaussian probability density function is of the form f G x = 1 √ 2 exp  −  x −  2 2 2  (3.9) The two distribution functions are very similar and can be compared for the same value of the function at the mean value, . For the averages of each function to be equal,  =a +b/2 (3.10) 108 Introduction and Background 0 0.1 0.2 0.3 0.4 0.5 –3 –2 –1 0 1 2 3 Gaussian Sinusoidal f(x) x μ = 0 b = 4 a = –2 Figure 3.30. Comparison of Sinusoidal and Gaussian pdf’s. and the two functions are compared for a mean value m =0 [39]. The resulting curve is shown in Figure 3.30, where the similarities are demonstrated. For the numbers chosen,  =1016 for the Gaussian distribution. Braam and van der Zwaag [39] performed a number of simulations for staircase and ArcSin √ P methods of testing to evaluate the effects of the numbers of tests on the ability of the methods to produce the correct distribution function inherently assumed in the simulation. Note that the staircase method has two parameters that can be varied in the test sequence, namely the total number of tests and the step size. In all the simulations, the starting value was at the mean value, similar to what was shown above. For the ArcSin √ P method, three parameters can be varied in the test simulation: the number of steps, the step size, and the number of tests per stress level. For the staircase test simulations, total numbers of tests (N ) of 20, 50, and 100 were used. In each case, the mean was well predicted and the distribution function became more narrow as N increased. As N decreased, the standard deviation was underestimated since for small numbers of tests, the probability of a failure at a low stress level is small. Their simulations for the staircase test method also showed that the influence of the step size on the mean is only significant if N is very small (<30). However, the step size was found to have a large influence on the standard deviation. A similar observation can be made from the results of the numerical simulations summarized in Table 3.3 where the “true” normal distribution of FLS has a standard deviation of 5. For 10 numerical simulations of a series of 10, 30, or 50 tests, the average value of the computed standard deviation is 0.70, 1.32, and 3.33, respectively. Worse yet, the individual values for the standard deviation have a significant amount of variability from one test series to another, even when a total of 50 specimens is used in a staircase test series (see Table 3.3). Accelerated Test Techniques 109 For the ArcSin √ P simulations in [39], results showed that a larger number of total tests leads to a narrower distribution function, as expected, while the conclusion that the mean is well predicted for most cases was verified. From simulations of both methods, it was concluded that the error in standard deviation depends on the ratio of s, the step size, to , and the standard deviation (which is never really known!). The difficulty of this problem can be summarized in the following conclusion drawn by Schütz [7] in his review article on the history of fatigue: “A probably unsolvable problem is the scientifically and practically correct calculation of fatigue life at very low probabilities of failure, because the type of distribution would have to be known.” 3.6. RANDOM FATIGUE LIMIT (RFL) MODEL Berens and Annis [24 (Appendix B)] point out that it has been common procedure to determine FLSs of a Haigh diagram corresponding to some HCF life from fits to constant amplitude S–N data. At a constant value of R, for example, fatigue strength at N cycles can be defined statistically in terms of the stress level for which p percent of a population of specimens will survive for at least N cycles. This definition, however, does not provide for the characterization of the fatigue limit in terms of an observable stress property since the result of a fatigue test is the random number of cycles to failure for the fixed stress level that was used in the test. As a result, the distribution properties of fatigue strengths at a fixed, large N have generally been inferred from the distribution of cyclic lives from specimens that have failed or have reached some defined run-out life at a fixed stress level. There are at least three practical problems in this traditional approach to estimating fatigue limits: First, at the stress levels of interest, S–N curves are relatively flat. While accurate mean values can be determined from step or staircase methods as pointed out earlier in this Chapter, extremes of the distribution curve are hard to obtain. It is the extremes that are important for design. A second problem is the large variation in fatigue lives near the endurance limit. Third, there are requirements to determine fatigue limit strengths at longer and longer lives, making the testing even more time consuming. In addition to step and staircase testing, an innovative approach to estimating the FLS distribution was proposed by Pascual and Meeker [46]. A random fatigue limit (RFL) model is postulated in which each specimen has its own fatigue limit strength in much the same way that each specimen has its own finite fatigue lifetime if tested at a sufficiently high stress above the endurance limit. This random fatigue limit is explicitly included in the S–N model. Past attempts at modeling the stress-life (S–N ) behavior of cyclic fatigue and extending it into the long life regime often used an equation of the form: ln N i = 0 + 1 lnS i – 2  + i (3.11) 110 Introduction and Background where, for specimen i N i represents cycles to failure, S i is the applied stress parameter,  2 is a constant fatigue limit (S i > 2 ), and  i is a random variable representing the scatter in cycles to failure about the predicted life [47]. The parameters of the median life prediction,  0  1 , and  2 are estimated from test data and  2 is interpreted as the FLS condition. Since  2 is an asymptote, the S–N curve flattens as S approaches the fatigue limit. This model may be 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 where the S–N curve becomes nearly horizontal. But its main shortcoming is that it does not work. Since the single-valued, constant, fatigue limit,  2 , must be less than the lowest stress tested (so that the logarithm of (S i − 2  is defined) it must be less than even the lowest run-out stress tested. This causes the  2 asymptote to be so low as to produce an unrealistic material model. The RFL model [46] is a generalization of Equation (3.11) 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 fatigue limit is not a single constant, but rather an individual characteristic of each specimen, namely, a statistical variable. The RFL model for test specimen i is given by: ln N i = 0 + 1 lnS i – i  + i (3.12) where  i is the random fatigue limit for specimen iS i > i  and is expressed in units of the stress parameter. In this model,  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 com- monly seen in fatigue data. An example of this can be found in [48] and is illustrated in Figure 3.31 which presents the 1st, 25th, 50th, 75th and 99th 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 even though the majority of the data were obtained in the lower life regime. Further, it is easily seen in Figure 3.31 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. The two random variables in the RFL model require probability distributions. In [48], the conditional distribution of cycles to failure, given , was assumed to have a lognormal distribution while the random fatigue limit, , was assumed to have an SEV distribution. The equations for the cumulative distribution and probability density function of the SEV distribution are Fz =1 −exp−expz (3.13) Accelerated Test Techniques 111 0 20 40 60 80 140 120 100 1.E + 03 1.E + 04 1.E + 05 1.E + 06 1.E + 07 1.E + 08 1.E + 09 1.E + 10 Cycles Stress parameter (ksi) p = 0.99 p = 0.75 p = 0.50 p = 0.25 p = 0.01 Run-outs RFL distribution Figure 3.31. Example S–N curves calculated from percentiles of the random fatigue limit distribution. fz =  1    exp z −expz (3.14) where z =  −     (3.15) The SEV distribution was selected as a model for fatigue limits because it has a basis in extreme value theory and it is skewed to the left, that is, to values smaller than the median. Berens and Annis [48] note that if the random variable Y has a Weibull distribution then log Y will have an SEV distribution – the SEV is to the Weibull as the lognormal is to the normal. From a set of S–N data including run-outs, the five parameters of the RFL model can be estimated using maximum likelihood methods [46]. Maximum likelihood estimates have known, desirable statistical properties and confidence bounds can be calculated when wanted [49], but these calculations require sophisticated software. ∗ An example application of the RFL model approach is presented in [48] that demon- strates that the model can produce a valid description of S–N data in the HCF regime. It also demonstrates one use for the model by investigating the necessity of having long ∗ Computer software that works with the S-PLUS statistical analysis program can be obtained from Dr. Meeker (wqmeeker@iastate.edu or http://www.public.iastate.edu/∼wqmeeker/other_pages/wqm_software.html). 112 Introduction and Background run-out lives in the analysis. The data for this application, shown in Figure 3.31, consisted of 95 S–N test results on Ti-6Al-4V plate with 15 of these being run-outs at 10 7  10 8 ,or 10 9 cycles and two that failed between 10 7 and 10 8 cycles; the others had shorter lives. Results showed that the scatter in life of the smooth bar specimens was adequately described by the distribution of the fatigue limit parameter. Further, it was shown that the added information from testing to lives greater than a run-out life of 10 7 did not significantly change the fatigue limit distribution. However, it was illustrated that using only lives less than 10 7 in the analysis produced a significantly lower and unreasonable fatigue limit distribution. Thus, the assumption that shorter life data provide useful information about the HCF limit strength distribution can be seriously questioned. Annis and Griffiths [50] used the results from the RFL model to generate sample staircase results for the FLS with a Monte Carlo simulation. Unlike the normal distribution used in much of the earlier work on scatter in staircase test results, a Weibull distribution was used for the scatter in the RFL (stress axis), while a lognormal distribution was used to represent the scatter in lives (cycles axis). Their preliminary results, using a Bayesian- based likelihood estimation technique to analyze the randomly generated staircase data, showed that the mean value of the RFL was well predicted for small numbers of tests ranging from a total of 10 to a total of 31. The scatter, as observed previously for staircase testing, was not as well represented from such a small population. Their results are summarized in Table 3.7 which provides computed values for the mean, where the probability of failure is p =05, and p =001 which is near the tail end of the distribution where the probability of failure is only 1%. While the scatter is not well represented, the values at p = 001 seem to be better than those obtained using a normal distribution for such small sample sizes and where only scatter in FLS is considered. In the RFL model, scatter in lives as well as stress is considered. As noted by Annis and Griffiths [50], the centering nature of the staircase method results in a limited range of stress levels with a corresponding clustering of run- out life within 1–2 orders of magnitude. The RFL model, which contains 5 parameters, requires substantial data over a range of stress levels and lives to accurately estimate the parameters. Using Bayesian methods, the analysis of the parameters makes use of prior knowledge of the behavior. A further advantage of this method, where Weibull statistics are used to represent fatigue strength, is that in the HCF regime the upper limit of the Table 3.7. Numerical simulations of staircase tests using RFL model [50] Number of tests FLS p =05 FLS p = 0 01 “Truth” 53.84 40.31 10 53.98 42.08 20 54.28 42.98 31 53.38 40.54 Accelerated Test Techniques 113 fatigue strength is more restrictive than the lower limit. The Weibull distribution reflects this behavior where very low quality specimens are sometimes, although infrequently, observed, but extremely high run-out stresses are never observed. The infrequent low run-out stresses are the crux of the problem in determining lower bound stresses for design purposes and the RFL model is felt to provide a means to measure the propensity for this life-limiting behavior [50]. 3.6.1. Data analysis In addition to the staircase data obtained on titanium under the National HCF program, step testing was used to determine the FLS at a life of 10 7 cycles [24]. In the staircase testing, the results of which are shown in Figures 3.27 and 3.28 above, the number of cycles to failure for the tests that failed before 10 7 cycles were also recorded. These additional data can be used to evaluate statistical approaches for representing S–N data and different methods for experimentally determining the fatigue limit strength as well as minimum HCF capability. The additional step tests were conducted because they result in a failure point for each specimen tested. This can be an advantage when material is costly and test failures are required for needed assessments. A summary of the step-test matrix in this evaluation is provided in Table 3.8. The step tests were run with 10 7 fatigue blocks and 4 ksi maximum stress steps. The average number of cycles/specimen was ∼52 ×10 7 cycles for a total of 260 ×10 6 cycles for the step test program. The average of the five step tests gave a FLS of 79.8 ksi whereas the analysis of the staircase tests totaling 26 specimens resulted in a mean value of 79.4 ksi (see Table 3.1). For comparison purposes, the RFL model (see Table 3.7), which is based on a combination of LCF and HCF data as well as an empirical fit to the S–N curve, provided a value for the SWT parameter of 53.84 which, in turn, for R = 01 provides a FLS mean value of 80.3 ksi. In that application of the RFL model, neither the staircase-test nor the step-test results had been used. Unlike in the step tests, a single load level on each specimen is used for the staircase approach. The first staircase test was run at S max = 67ksi without failure. Stresses were increased on additional specimens until failure occurred within 10 7 cycles. Stresses on Table 3.8. Ti-6Al-4V Step test matrix at 900 Hz, R =01, and 75  F Spec ID Starting s max (ksi) Final s max at failure (ksi) # steps Last step N f Interpolated s max (ksi) 121-2 61.5 77.5 4 8202785 76.78 124-4 61.5 89.5 7 1125814 85.95 47-10 61.5 81.5 5 5617819 79.75 173-2 65.5 77.5 3 5627646 75.75 47-9 65.5 81.5 4 8252952 80.80 114 Introduction and Background subsequent tests were continually increased or decreased based on N f of the last tests versus 10 7 target life. Roughly 1/2 of the specimens (12 out of 26, see Table 3.1) failed within the targeted life regime. Six of the staircase tests were long-life failures that continued overnight or through the weekend awaiting setup of the next test specimen. Given that over-night or weekend test time is run without additional costs, the staircase matrix of 26 tests was similar in cost to the step matrix (Table 3.8). More staircase tests can be run at similar costs to step tests which are conducted using multiple steps/specimen. While fail/no fail methods can be used to evaluate the statistics of the FLS, additional information is available in the form of number of cycles for each failed specimen, as well as cycle counts for specimens that were not terminated when they reached the number of cycles for survival. For the Ti-6Al-4V material whose results from staircase testing are shown in Figures 3.27 and 3.28 and in Table 3.1, the data involving cycles to failure can be presented in the form of an S–N curve, as shown in Figure 3.32. The step test results, Table 3.8, as well as the staircase run-outs, are also shown on a log–log plot. The results are qualitatively very similar. The fatigue capability of only the staircase tests where failure occurred is analyzed in Figures 3.33 and 3.34. A 1D scatter in the stress direction is assumed in Figure 3.33. A 1D scatter in life is assumed in Figure 3.34. The scatter assumption in life is similar to the approach typically used in analyzing LCF results. Because the tests involve failures in the long life regime, the statistics using scatter in life, Figure 3.34 show much more scatter than that in Figure 3.33 where scatter in stress was used. At a fatigue life of 10 7 cycles, both methods produce a mean value of log S max of about 1.9 (about 79 ksi), but the −3 value of log S max goes from about 1.83 (67.6 ksi) when scatter in stress is used to about 1.7 (50.1 ksi) when scatter in life is used. In the 1.7 1.8 1.9 2 2.1 579 Ti-6Al-4V HCF Tests at 75 °F, R = 0.1 Step test failures Staircase run-outs Staircase failures log (S max ) log (cycles) 68 Figure 3.32. Step and staircase test results for Ti-6Al-4V at 75  F and R =01 (step at interpolated S max using the step interpolation approach). Accelerated Test Techniques 115 1.7 1.8 1.9 2 2.1 Ti-6Al-4V HCF Tests at 75 °F, R = 0.1 (stress error) Staircase failures Average fit Average –3s fit log (S max ) log (cycles) 57968 Figure 3.33. Staircase failures for Ti-6Al-4V at 75  F and R =01 with average and −3s predictions assuming a 1D stress scatter. 4 5 6 7 8 9 1.7 1.8 1.9 2 2.1 Ti-6Al-4V HCF Tests at 75 °F, R = 0.1 (N f error) Staircase failures Average fit Average –3s fit log (cycles) log (S max ) Figure 3.34. Staircase failures for Ti-6Al-4V at 75  F and R =01 with average and −3s predictions assuming 1D life scatter. latter case, the statistics are almost meaningless because they are trying to fit an almost horizontal S–N curve (Figure 3.32) based on scatter in the life direction only. To further evaluate the statistics of the S–N behavior of this material, predictions were made with a combination of the step and staircase tests with baseline tests from an earlier program (N f < 10 6 cycles) previously analyzed as shown in Figure 3.31. For these . contains 5 parameters, requires substantial data over a range of stress levels and lives to accurately estimate the parameters. Using Bayesian methods, the analysis of the parameters makes use of. underestimate the standard deviation of the FLS [39]. From observations of a number of numerical simulations, Braam and van der Zwaag [39] developed a method for evaluating the statistical properties of. analyze the randomly generated staircase data, showed that the mean value of the RFL was well predicted for small numbers of tests ranging from a total of 10 to a total of 31. The scatter, as observed