76 Introduction and Background N Mean stress Alternating stress R = constant R = constant 10 7 2 × 10 7 3 × 10 7 Alternating stress σ Goodman σ Goodman Goodman diagram Loading history Figure 3.2. Schematic of step-loading procedure. others do not because the test is terminated after a large number of cycles (run-out). This results in two populations of specimens, one failed and the other unfailed, which are difficult to analyze statistically. Another justification for a non-constant load to determine the fatigue limit is, as Prot [14] points out, “in practice, fatigue loads are not regularly variable, but they are not uniform amplitude loads.” One of the main concerns in establishing material allowables for HCF is the sparse amount of data available and the time necessary to establish data points for fatigue limits at 10 7 cycles or beyond. The conventional method for establishing a fatigue limit is to obtain S–N data over a range of stresses and to fit the data with some type of curve or straight-line approximation. For a fatigue limit at 10 7 cycles, for example, this requires a number of fatigue tests, some of which will be in excess of 10 7 cycles. This is both time consuming and costly. One method for reducing the time is to use a high frequency test machine such as one of those that have appeared on the market within the last several years. In addition, the use of a rapid test technique such as one developed by Maxwell and Nicholas [22] involving step loading, described above, can save considerable testing time. It has been demonstrated that such a technique provides data for the fatigue limit of a titanium alloy which are consistent with those obtained in the conventional S–N manner [22, 26]. 3.3.1. Statistical Considerations To examine the expected outcome using the step-loading technique, consider the schematic of Figure 3.3. One can define a fatigue limit on an S–N curve arbitrarily as N f , even though there is no assurance that this is a true endurance limit corresponding to “infinite” life. At N f , there will exist an unknown cumulative distribution function (CDF) which Accelerated Test Techniques 77 Stress Number of cycles N f 01 CDF Stress Step number 0 2 4 6 B A Figure 3.3. Schematic of S–N curve and CDF for two different degrees of scatter. will define the failure function at that number of cycles over some range of stresses. The stress corresponding to CDF =0 defines the stress level below which there are no failures within N f cycles. When CDF = 1, the corresponding stress defines the condition under which all specimens fail at or below N f cycles. ∗ If there is a large amount of scatter as in curve “A,” which may occur if the S–N curve is very flat, then a larger number of steps in the step-loading technique will be required to cover all of the possible values of stress where failure may occur below the cycle count being considered, N f . If, however, there is less scatter as in curve “B” or the S–N curve is steeper, which will essentially cut off the higher values of stress which cause failure at lower numbers of cycles, then the number of steps is fewer. In either case, the larger the number of steps in a test, the higher is the expected stress. Thus, what might appear to be a “coaxing” effect is no more than the statistics of the distribution of material fatigue strength. The actual number of steps in a step-loading experiment depends on the starting stress, the distribution function or range in stress levels, and the size of the step. An alternate to the step-loading approach for determining the fatigue limit is to conduct tests at various values of stress up to the number of cycles corresponding to the fatigue limit. Two types of data are obtained. First, some specimens will fail before N f is reached, and these will provide data for a S–N curve which can be fit and extrapolated to N f . The second type of data will be stress levels for which no failure was obtained within N f cycles. These stress levels will be denoted as run-outs or lower bounds on the fatigue limit. In conducting tests under constant stress, consider the case where the S–N curve is relatively flat such as when the number of cycles, N f , is very large. As a hypothetical example, consider the fatigue behavior in the region between 10 7 and 10 9 cycles, where it ∗ It should be noted here that some mathematical representations of distribution functions can go from zero to infinity, such as a normal distribution. In those cases, we have to deal with a situation where the CDF approaches 0 or 1 within some very small probability. 78 Introduction and Background CDF Number of cycles 0 1 10 9 10 7 A B C D E F (b) 01 CDF Stress 10 9 10 7 A B C D E F (a) Figure 3.4. Schematic of CDF (a) for two different values of N f , (b) as a function of N . has been shown that the S–N curve still has a slightly negative slope for some materials [27]. For illustrative purposes, the CDF for failure within a given number of cycles is shown schematically in Figure 3.4(a) for either 10 7 or 10 9 cycles. At 10 7 cycles, there is no failure for stresses below level “C” and all samples will fail at or above “F.” Similarly, at 10 9 cycles, no failure occurs below “A” and all samples will fail at “E” or above. Clearly, “A” corresponds to the fatigue limit at 10 9 cycles. Consider, however, what happens in a typical experimental investigation. The CDF is shown as a function of number of cycles in Figure 3.4(b) for several stress levels depicted in Figure 3.4(a). As shown, there are no failures at “A” while at “F” most samples will have failed below 10 7 and none will reach 10 9 . At “E” there is a higher probability of survival beyond 10 7 but all fail by 10 9 . At some intermediate level “D,” some will fail by 10 7 and most will have failed by 10 9 , but as the stress level decreases to “C” or “B,” the likelihood of failure before 10 9 decreases. Considering the time and cost of conducting such long life tests, the likelihood of determining the probability density functions for a number of stress levels and, in turn, defining the fatigue limit, is poor. In this situation, the step-loading procedure may provide an equally good answer with fewer tests. Tests conducted at constant levels of stress, separated by equal increments, are discussed later in this chapter (see Section 3.6) along with the statistics for determining fatigue limits and the corresponding scatter. 3.3.2. Influence of number of steps Experimental data using Ti-6Al-4V forged plate material and employing the step-loading procedure [28] are shown in Figure 3.5. In that investigation, the values of the fatigue limit for four different values of R were not known a priori. Thus, the initial stress value in the step-loading procedure was highly variable. The results, plotted against the number of steps, show no indication of a systematic increase with number of steps and, therefore, Accelerated Test Techniques 79 no evidence of coaxing. On the other hand, experimental results which show an increase in stress with number of steps are shown in Figure 3.6 where the starting stress for any of the four conditions was either the same or very similar. The tests are on a notched sample with k t =22 with one batch untested and the other subjected to LCF cycling as indicated on the plot [21]. The plots of stress versus number of steps show a linear increase. Since the starting stresses are the same for each condition, the slope is related to the size of the step. Thus, this increase with number of steps is not necessarily coaxing, it is probably no more than the scatter in material behavior as described above. 200 300 400 500 600 700 800 900 1000 1 R = –1 R = 0.1 R = 0.5 R = 0.8 Maximum stress (MPa) Number of steps Ti-6Al-4V plate 60 Hz 23456789 Figure 3.5. Fatigue limit stress vs. number of steps. 250 300 350 400 450 500 123456789 Baseline R = 0.1 Baseline R = 0.5 LCF–HCF R = 0.1 LCF–HCF R = 0.5 Maximum stress (MPa) Number of steps LCF 30 cycles 430 MPa Figure 3.6. Fatigue limit stress vs. number of steps. 80 Introduction and Background 3.3.3. Validation of the step-test procedure Data for a Haigh diagram were obtained using the step-loading procedure for both the bar and plate forms of Ti-6Al-4V [21]. The data are shown in Figures 3.7 and 3.8. In each of the figures, the number of steps that were used for each specimen is indicated in the legend. All steps within an individual step-loading test were conducted with a constant value of R. Careful study of the data shows that there does not appear to be any systematic trend which would lead one to believe that the number of steps has any 0 100 200 300 400 500 600 700 800 0 200 400 600 800 1000 2 steps 3 steps 4 steps 5 steps 11 steps Alternating stress (MPa) Mean stress (MPa) Ti-6Al-4V bar 70 Hz Figure 3.7. Haigh diagram for bar material. 0 100 200 300 400 500 600 700 800 –200 0 200 400 600 800 2 steps 3 steps 4 steps 6 steps 10 steps Alternating stress (MPa) Mean stress (MPa) Ti-6Al-4V plate 70 Hz Figure 3.8. Haigh diagram for plate material. Accelerated Test Techniques 81 influence on the results. In fact, it is rather remarkable that the expected trend of higher strength versus number of steps from a purely statistical point of view is not observed. This is probably due to the choice of starting stress for each test which was very variable because each test covered a different value of R compared to the prior test. Conventional S–N tests conducted at 420 Hz on plate material were used to determine the fatigue strength corresponding to 10 7 cycles by least squares fit to the S–N data obtained at lives close to 10 7 cycles. The results are shown in Figure 3.9 for tests conducted at a number of values of stress ratio, R, from 0.5 to 0.8. It can be seen that the data lie right on top of the data from step-loading tests in the same range of R. Further, there seems to be no effect of frequency in going from 70 Hz in earlier tests to 420 Hz in the present tests. Data were also obtained at R = 05 and R = 08 using the step-loading procedure to compare with the interpolated S–N data (horizontal line) as shown in Figure 3.10. Different values of stress in the first loading block, shown on the x-axis, were used to evaluate the effect of number of blocks for the two values of R. Numbers in parenthesis in the figure indicate the number of load blocks used to determine the stress corresponding to 10 7 cycles. In both the plate material used here and the bar material used elsewhere, the failure at R = 05 is purely fatigue, while at R =08, it is observed that the fracture surface shows no indication of fatigue, but rather, ductile dimpling [29]. This issue is discussed later. In both cases, however, Figure 3.10 shows no indication of a trend with number of blocks or starting stress for the step-loading procedure. Data obtained at 1.8 kHz are presented in Figure 3.11. Three types of tests are repre- sented, conventional S–N to failure, terminated S–N producing run-outs, and step loading at either 10 7 or 10 8 cycles. While the vertical scale is blown up significantly, it can be 0 100 200 300 400 500 600 700 800 0 200 400 600 800 1000 Ti-6Al-4V Plate 10 7 cycles ML 70 Hz Step ASE 70 Hz Step ML 420 Hz S-N Alternating stress (MPa) Mean stress (MPa) Figure 3.9. Haigh diagram for plate material comparing step test and S–N data. 82 Introduction and Background (a) 500 550 600 650 400 450 500 550 600 Ti-6Al-4V 10 7 cycles R = 0.5, 420 Hz Step tests Fatigue strength at 10 7 cycles (MPa) Block 1 stress (MPa) From S –N curve ( ) = # steps (3) (8) (5) (6) (3) (2) Fatigue strength at 10 7 cycles (MPa) (b) 800 850 900 950 600 650 700 750 800 850 900 Ti-6Al-4V R = 0.8, 420 Hz Step tests Block 1 stress (MPa) From S –N curve (9) (12) (4) ( ) = # steps Figure 3.10. Influence of block 1 stress on FLS at 10 7 cycles in step-loading fatigue limit strtess; (a) R =05, (b) R =08. noted that there is very little scatter at R =08 where all the tests were conducted, and no influence of a history effect due to the step-loading procedure. The lower step-test data point at 10 8 cycles represents two independent tests which had a maximum stress within 1 MPa of each other. The data obtained at R =08 are of particular interest in the evaluation of the validity of the step-loading procedure. In an investigation on the bar material, Morrissey et al. [29] noted that at high values of R, the material accumulated strain under fatigue loading. Accelerated Test Techniques 83 950 1000 1050 1100 10 5 10 6 10 7 10 8 10 9 10 10 Failure Run-out Step test Maximum stress (MPa) Number of cycles Ti-6Al-4V bar 1800 Hz R = 0.8 Figure 3.11. Fatigue limit stress results at R =08 1800 Hz. Tests conducted at different frequencies showed that the strain accumulation was depen- dent primarily on number of cycles, not on time, so that the phenomenon could not be considered to be cyclic creep. Rather, the strain accumulation is due to ratcheting. A similar phenomenon has been observed in the Ti-6Al-4V plate material, where cycling at stress ratios higher than approximately 0.7 leads to strain accumulation. Micrographs of the fracture surface at various magnifications taken with a scanning electron micro- scope (SEM) are presented in Figures 3.12 and 3.13 for stress ratios, R, of 0.7 and 0.8, 00-A-95, Ti-6-4, σ = 840 MPa, R = 0.7, a = 0.4 mm Figure 3.12. Fractographs at R =07. 84 Introduction and Background 00-A-91, Ti-6-4, σ = 920 MPa, R = 0.8 Figure 3.13. Fractographs at R =08. respectively. It can be observed that at R = 07 (Figure 3.12), the fracture surface looks like fatigue with well-defined faceted features and evidence of striations. At R = 08 (Figure 3.13), the features are those of a tensile test with ductile dimpling in evidence and no indications of cleavage or striations. The crossover point, at about R = 075, is nominally the same as in the bar material as reported by Morrissey et al. [29]. Data obtained over a range of frequencies from 30 to 1000 Hz under the Air Force HCF program at various laboratories are presented in Figure 3.14 for R = 08. Including 800 850 900 950 1000 10 5 10 6 10 7 10 8 All data ML 420 Hz Maximum stress (MPa) Cycles R = 0.8 Figure 3.14. S–N data obtained from 30 to 1000 Hz. Accelerated Test Techniques 85 20 30 40 50 10 4 10 5 10 6 10 7 10 8 Ti-6Al-4V Plate R = 0.8 60 Hz 60 Hz run-out 60 Hz step test 200 Hz 200 Hz step test Stress range (ksi) Cycles to failure Figure 3.15. Honeywell data at 60 and 200 Hz. the Materials Laboratory (ML) data at 420 Hz, there is very little scatter over the fatigue cycle range from 10 5 to 10 8 cycles, and no effect of frequency although frequencies of each data point are not shown. Additional data from Honeywell are shown in Figure 3.15 at R =08 at both 60 and 200 Hz. No frequency effect is apparent, the scatter is minimal, and data using the step-test procedure at 10 7 cycles fall right on top of the other data. From these results, as well as from the data in Figure 3.11 at 1800 Hz, it is concluded that step testing produces an accurate estimate of FLS in the 10 7 –10 8 life regime for R = 08 in the titanium plate where strain ratcheting is the dominant fatigue failure mechanism. 3.3.4. Observations from the last loading block An interesting observation was made by Moshier et al. [30] when evaluating the data from the step-test method on specimens with LCF cracks compared to data on specimens with no cracks. The last loading block, defined as the block of 10 7 cycles during which failure occurred, can have a cycle count anywhere from 1 to 10 7 . The data for number of cycles to failure in this block are normalized with respect to 10 7 to show at what fraction of the block failure occurred. The results, presented in Figure 3.16, show that for specimens with no prior cracks, the failure can occur anywhere in the block. When cracks are present, however, failure always occurred early in the loading block. These data show that there appears to be a very well defined HCF threshold for a cracked specimen for which failure occurs within a short time, typically under one million cycles, or does not occur at all for a given applied stress (or K). Alternately, these data show that when a crack is present, we are dealing only with the propagation phase of fatigue which is small compared to the nucleation phase which dominates the HCF life in an . that at high values of R, the material accumulated strain under fatigue loading. Accelerated Test Techniques 83 950 100 0 105 0 1100 10 5 10 6 10 7 10 8 10 9 10 10 Failure Run-out Step test Maximum. Ti-6Al-4V plate material, where cycling at stress ratios higher than approximately 0.7 leads to strain accumulation. Micrographs of the fracture surface at various magnifications taken with a scanning. regularly variable, but they are not uniform amplitude loads.” One of the main concerns in establishing material allowables for HCF is the sparse amount of data available and the time necessary to establish