386 Applications 8.2. FRACTURE MECHANICS CONSIDERATIONS The implications of the shape of the Haigh diagram, as described by the modified Jasper formulation, for example, can be examined in terms of the fracture mechanics parameter related to the threshold for crack growth, namely K th . The FLS, which can be defined as the maximum stress under which a crack will not initiate and grow, should be related to the threshold for crack propagation using the following logic. At a stress just above the fatigue limit, a crack will both initiate and grow to failure in a smooth bar. If it is assumed that crack initiation can be defined as initiation or nucleation to a defined size, then the driving force in terms of K must be equal to or exceed K th in order for failure to occur. As an illustrative example, the locus of stresses in terms of the alternating and maximum stresses corresponding to the modified Jasper equation used to fit the data on Ti-6Al-4V plate material are plotted against mean stress in Figure 8.6. The fit to experimental data is extended into the negative mean stress region in this plot. A line corresponding to R =0 is also shown for reference purposes in this Nicholas–Haigh diagram. ∗ At values of the stress ratio above R =0, all stresses are positive throughout the load cycle. One can consider two extreme cases for K th for a fixed crack size, independent of R, for crack initiation. First is the case where crack propagation, and hence the threshold for crack extension, is governed by the total stress range. In this case, K th is proportional to the alternating stress and would be represented by a horizontal line marked “K th =constant” in Figure 8.6. A second scenario, one commonly used in fatigue crack 0 200 400 600 800 1000 1200 –400 –200 0 200 400 600 800 1000 Ti-6Al-4V Plate σ alt Jasper σ max Jasper Stress (MPa) Mean stress (MPa) 10 7 cycles 20–70 Hz α = 0.287 Line of constant K max R = 0 ΔK th = ΔK 0 (1 – R ) ΔK th = constant Figure 8.6. Locus of maximum and alternating stresses corresponding to modified Jasper equation fit to experimental data. ∗ The Nicholas–Haigh diagram that plots both alternating and maximum stress was discussed in Chapter 2. HCF Design Considerations 387 0 2 4 6 8 10 12 –4 –3 –2 –1 0 R Ti-6Al-4V α = 0.287 1 ΔK th (MPa√m) Figure 8.7. Computed K th corresponding to a fixed size to crack initiation and modified Jasper equation fit to experimental data. It is assumed that K th =5MPa √ matR =0. growth modeling, is that only the positive stresses contribute to crack extension. Under this second hypothesis, the maximum stress is proportional to the threshold, K th , for a fixed crack length. For positive values of R, this is equivalent to a K max = constant criterion for threshold compared to a K = constant criterion which is represented by the horizontal line in Figure 8.6. The dotted line, shown in Figure 8.6, is the line of constant K max and is arbitrarily drawn through the same point at R = 0 as the line of constant K th . It can be seen that neither line corresponds to the lines representing the modified Jasper equation which is a best fit to the experimental data. These comparisons do not consider that K th may depend on crack length in the small crack regime (see [2], for example), or that K th is a function of stress ratio, R, as demonstrated in [2] and [3], for example. These considerations notwithstanding, the simplified analysis presented here indicates that crack initiation to a specific crack length requires either a different crack length for each mean stress, or that the value of K th is different for each value of mean stress. Pursuing the latter postulate, the value of K th can be calculated for each value of mean stress using the modified Jasper equation fit to the experimental data. Based on data for the test material that indicates the value of K th is approximately 5MPa √ mat R =0, the values of K th for other values of R are plotted in Figure 8.7. While the shape of the curve is similar to that seen often in the literature for positive values of R,itisof interest to note that the computed threshold is approximately constant as the values of R become more negative. For high values of R, however, there is no indication of the value of K th becoming constant above some critical value of R as observed experimentally in some titanium alloys [3]. Noteworthy, however, is the almost linear variation of K th with R for values of R>−1 which represents fully reversed loading. Note again that these numbers are based on a best fit to the experimental data on FLS using the modified 388 Applications form of the Jasper equation which accounts for the lesser contribution of compressive stresses to the fatigue process near the endurance limit as explained in Chapter 2. For high R values, in this case, the data are extrapolated based on the Jasper equation fit which has very little physical meaning as R approaches unity. The predicted values of threshold K based on a constant crack length for initiation, and the value of fatigue limit strength, can be computed for other shapes of the Haigh diagram. While the previous calculations were conducted for the modified Jasper equation that fits some experimental data, we examine here Haigh diagrams represented by the more common forms of equations, namely the Goodman (straight line) and Gerber (parabola) formulas discussed in Chapter 2. These two equations, shown again here, are  alt  y =  0  y  1−  mean  y  (8.2)  alt  y =  0  y  1−   mean  y  2  (8.3) where the mean stress axis is intercepted at the yield strength while the alternating stress axis takes on the value denoted here as  0 . The value of  0 is normally taken from experimental data, so we can examine several cases where the values of  0 are taken as a fraction of the yield stress. In particular, we arbitrarily choose  0 / y to have values of either 0.33 or 0.5 for illustrative purposes. The curves that are represented by these values for both the Goodman and the Gerber equations are shown on a Haigh diagram in Figure 8.8. Using the same logic as before, namely that these (alternating) stress levels provide values of K th for a fixed crack length, the computed values of K normalized to  y are plotted in Figures 8.9 and 8.10 for the Goodman and Gerber 0 0.2 0.4 0.6 0.8 1 0 .2 0.4 .6 0.8 Goodman, σ 0 /σ y = 0.33 Goodman, σ 0 /σ y = 0.50 Gerber, σ 0 /σ y = 0.33 Gerber, σ 0 /σ y = 0.50 σ mean /σ y 1 0 0 σ alt /σ y Figure 8.8. Haigh diagram for modified Goodman and Gerber equations HCF Design Considerations 389 0 0.2 0.4 0.6 0.8 1 1.2 –1.0 –0.50 0.0 0.50 1.0 R Modified Goodman equation ΔK / σ y σ 0 /σ y = 0.33 σ 0 /σ y = 0.50 Figure 8.9. Threshold K corresponding to modified Goodman equation. 0 0.2 0.4 0.6 0.8 1 1.2 –1.0 –0.50 0.50 1.0 Δ K / σ y Modified Gerber parabola 0.0 R σ 0 / σ y = 0.50 σ 0 / σ y = 0.33 Figure 8.10. Threshold K corresponding to modified Gerber equation. equations, respectively. These plots only cover the range of R from −1to+1 because the two equations are only valid for positive values of mean stress. Based on the shape of these curves, it is questionable whether either of these representations of data on a Haigh diagram predicts a K th curve that is more representative of actual data than the Jasper equation did (Figure 8.7), or whether the assumptions of a threshold corresponding to a fixed crack length and having a constant value are valid. An alternate manner in which to compare the implications of the shape of the Haigh diagram on the relation between K th and R is to start with the K th −R relation and see what shape the corresponding Haigh diagram takes. To do this, the threshold is assumed to be linear in R, starting at R =0 and decreases to a value at R =1 that is greater than zero. The ratio of K th at R =0toK th at R =1 is taken as a parameter . For illustrative 390 Applications 0 1 2 3 4 5 0.0 10 20 30 40 50 Alternating stress Mean stress α = 5 α = 4 α = 3 α = 2 Figure 8.11. Haigh diagram for linear K th relation. purposes, the quantity  a is taken =1 and  alt is proportional to K th for a fixed value of the crack length, a. The results of this numerical exercise are shown in Figure 8.11 for several values of  but where the numerical values of the alternating and mean stresses have no meaning. Only the general shapes of the curves are of importance. It can be seen that all of the curves are concave upward over the range in stress ratio covered by the assumptions, namely 0 <R<1. Of interest is the comparison between these results and those of the previous section where a Gerber parabola in a Haigh diagram (Figure 8.10) produced a K th curve that was nearly linear with R for higher values of R. The Jasper equation formulation, however, produces more of a linear fit for K as a function of R for a wider range of R. These numerical examples simply point out that the shapes of the curves in the Haigh diagram are quite sensitive to the shape of the K th vs R curve and vice versa. 8.2.1. Effects of defects A number of years ago, concern was raised in the design community within the turbine engine manufacturing companies when fracture mechanics concepts were applied in order to assess the vulnerability of fan or turbine blades to FOD. As one designer related to me, if fracture mechanics worked for FOD, they would have experienced failures in almost every blade that was flying. Calculations such as the ones presented in the previous section showed that there was a major issue in trying to connect threshold fracture mechanics data with endurance limit data from a Haigh diagram. To try to shed light on this problem, a major contractual effort was conducted by Pratt and Whitney (P&W) for the Air Force Materials Laboratory [4]. While the focus of the work was on defects in the form of FOD, the findings are equally relevant to actual defects in materials. HCF Design Considerations 391 When trying to connect threshold data with endurance limit data for different values of stress ratio, R, one of the findings widely reported is that the threshold K depends, to some extent, on the test technique being used. Arguments persist as to what is a real material K th as well as how closure might be taken into account in evaluating the experimental data. Note that such concerns are not as prevalent when obtaining endurance limit data if the fatigue limit corresponding to a given number of cycles is arbitrarily defined as the endurance limit. This is little different than the definition used for K th that corresponds to a specific (slow) growth rate, 10 −10 m/cycle for many cases. Thus, it is not surprising that P&W found that K th measurements varied with both test specimen and test procedure. The problem was somewhat circumvented by using closure corrections or using a technique that minimizes closure and load-history effects. P&W generated data for a Haigh diagram by conducting smooth bar fatigue tests at constant stress levels to obtain lives up to nearly 10 7 cycles and then extrapolating the data to 10 7 cycles. These tests were conducted at stress ratios of R =−1, 0.1, 0.5, and 0.7 at both room temperature, 27  C, and at an elevated temperature of 260  C for titanium alloy Ti-8Al-1Mo-1V, hereafter referred to as Ti 8-1-1. A regression analysis to all of the data at different values of R was used to establish the behavior of the material as N f =A   mean +  2  B  D (8.4) where N f is the number of cycles to failure and AB, and D are empirical constants used to fit the data. This information, in turn, provided data for the mean and −975% data points which were plotted on a Haigh diagram at 27  C and 260  C as shown in Figure 8.12. 0 100 200 300 400 500 0 200 400 600 800 1000 1200 27 °C, Mean 27 °C, –97.5% Min 260 °C, Mean 260 °C, –97.5% Min Alternating stress (MPa) Steady stress (MPa) Ti 8-1-1 Figure 8.12. Haigh diagram for Ti 8-1-1 at 27  C and 260  C. 392 Applications The minimum curves, representing the −975% lower bound from the statistical analysis of the data, are often used for HCF design [4]. The values of mean stress on the Haigh diagram at zero alternating stress were taken as the material’s monotonic ultimate and yield strengths, for the mean and minimum curves, respectively (another version of the “modified Goodman diagram”). The data show that the fatigue strength at 260  C is only slightly lower than that at 27  C. The major difference is in the yield and ultimate strengths which are used as anchor points for the Haigh diagram and may not be related directly to fatigue strength at all. To evaluate the use of fracture mechanics for construction of a Haigh diagram from threshold data, values of K th were obtained from three different methods as detailed in [5]. The methods involved both increasing and decreasing K tests and a closure corrected version of the decreasing K tests. These tests are designated as K thinc , K thdec , and K theff respectively. The threshold values at 260  CatR =01, 0.5, and 0.7 for the various methods are shown in Figure 8.13. To construct a fracture mechanics-based Haigh diagram from threshold data, an initial or reference crack size has to be defined. In the study by Pratt and Whitney, the initial material quality (IMQ) defect distribution was used as the initial crack size. The IMQ, in turn, was determined using a probabilistic approach from measurements of the size of alpha particles from which fatigue failures were observed to originate. A lognormal distribution with measured sizes ranging from approximately 7 to 17 microns with a mean slightly above 10 microns was used to represent the initial defect size. The statistics of the smooth bar fatigue strength as well as the statistics of the initial flaw sizes were used to construct the Haigh diagrams of Figures 8.14 and 8.15 where the mean and 1 2 3 4 5 6 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 ΔK th,dec ΔK th,eff ΔK th,inc R K th (MPa m ) Δ √ Figure 8.13. Threshold stress intensity as a function of stress ratio at 260  C. HCF Design Considerations 393 0 100 200 300 400 500 0 200 400 600 800 12001000 Mat'l data, mean Mat'l data, –97.5% Simulated, mean Simulated, –97.5% Alternating stress (MPa) Steady stress (MPa) Ti 8-1-1 27 °C Figure 8.14. Haigh diagram for Ti 8-1-1 at 27  C. 0 100 200 300 400 0 200 400 600 800 1000 Mat’l data, mean Mat’l data, –97.5% Simulated, mean Simulated, –97.5% Alternating stress (MPa) Steady stress (MPa) Ti 8-1-1 260 °C Figure 8.15. Haigh diagram for Ti 8-1-1 at 260  C. minimum −975% curves are labeled “predicted” to indicate the fracture-mechanics- based predictions using deterministic values of K th obtained from the increasing K th tests (see Figure 8.13). It is immediately apparent that the fracture mechanics approach to establishing a Haigh diagram does not work well because the values of K th obtained from the increasing K th tests are nearly independent of R. Further, the predicted results are conservative at zero mean stress but substantially anticonservative at higher mean stresses corresponding to higher values of R. Increasing the value of the initial crack size, with no physical basis for doing so, would lower the curve but not change its shape significantly. Although using the decreasing K th test data would provide a curve whose 394 Applications shape would be closer to the smooth bar Haigh diagram, the stress levels would be much higher. At the same time, the validity of such K th data can be questioned because of the presence of significant amounts of closure in the tests at lower values of R and the inability to reproduce such data with other test techniques. Since the primary purpose of the P&W program was to evaluate the use of fracture mechanics to determine the susceptibility of engine blades to service-induced damage in the form of FOD, the procedure described above was repeated for specimens with initial notches [4]. The effects of defects was only studied at 260  C. Simulated FOD was put into specimens with notch depths of either 0.20 or 0.38 mm. These specimens were then fatigue tested to obtain data from which a Haigh diagram was constructed. For comparison, a fracture mechanics approach was used based on initial defects of the size of the notches, a distribution of fatigue strengths as characterized by Equation (8.4), and deterministic values of K th as obtained from the decreasing K th tests as shown in Figure 8.13. The choice of these threshold values apparently stems from the general shape of the curve which shows the threshold to decrease with increasing values of R since the Haigh diagrams show the same trend. The distribution of initial flaw sizes was determined from a database of FOD sizes mea- sured in the field. The data were separated into repairable FOD and non-repairable FOD. The 0.20 mm deep notch was considered representative of repairable FOD whereas the 0.38 mm deep notch was representative of non-repairable FOD. For each type of FOD, a distribution function of FOD depths was constructed from 162 measurements of repairable and 104 measurements of non-repairable damage. This resulted in a Weibull distribution as best representing the repairable FOD and a lognormal distribution representing the non-repairable FOD. The notch depths measured in the field covered a range from 0.075 to 0.40 mm (mean of 0.16 mm) for the repairable damage and from 0.075 to 2.5mm (mean of 0.60) for the non-repairable damage. The notch depths chosen for testing, 0.20 and 0.38 mm, were somewhat representative of these two distributions. For the mathematical procedure to determine the allowable stresses in a Haigh diagram, the notch depths used in the testing were chosen as the mean values of the initial flaw size while the distribution function of initial sizes was taken from the FOD data. Following the statistical procedure outlined above, mean values of the stresses for the Haigh diagram were obtained. These results, as well as the baseline notch behavior, are plotted in Figure 8.16. It can be seen that for the FOD simulations with the accompanying assumptions, particularly the choice of threshold values, the fracture mechanics procedure appears to work fairly well for data obtained in the range from R = 01toR =05. In a review article, Cowles [6] summarizes the findings of the P&W study by noting that the fracture mechanics approach to constructing a Haigh diagram for defect-free material does not work very well whereas the same approach, using a different set of K th values, seems to work much better for large initial defects from FOD. The challenge, he notes, is in the smaller size range for initial damage or defects where small crack behavior HCF Design Considerations 395 0 100 200 300 400 0 200 400 600 800 1000 Mat’l data, defectfree Mat’l data, 0.20 mm defect Mat’l data, 0.38 mm defect Simulated, 0.20 mm defect Simulated, 0.38 mm defect Alternating stress (MPa) Steady stress (MPa) Ti 8-1-1 260 °C Figure 8.16. Haigh diagram for defect containing specimens at 260  C. and crack closure become important considerations. Further, he notes, the results are very sensitive to the threshold stress intensity factor and that crack initiation might also have to be considered. Following this thought, Larsen et al. [5] also conducted a reassessment of the work per- formed by P&W. They performed fracture mechanics calculations based on the decreasing K th data of Figure 8.13. For the computations, they treated the notch not as an equiv- alent crack as had been done previously [4], but as a notch with an initial semi-circular surface flaw of depth a i at the root of the notch. Further, they chose the value of a i for the notch specimens, and the initial flaw size for the baseline smooth bars, to best match the experimental data as shown in Figure 8.17. For the notch data, this crack size was 13 microns which is close to the mean size of initial defects found from fractographic observations of failed specimens. However, for the smooth bar data, an initial crack size of 110 microns had to be assumed to match the experimental data. Use of 13 microns for the initial crack size would produce stresses far above those on the smooth-bar Haigh diagram. They also concluded that the anomalously high value of initial crack size of 110 microns could be associated with lack of consideration of the combination of initiation and propagation making up the total life in smooth bar tests as well as the possible fast growth of small cracks. For the notches that were produced by the shearing action from the impact of a sharp tool, the assumption that initiation life is small or negligible seems reasonable. The same, however, cannot be said about the smooth specimens. Even if calculations are performed for smooth bars with no initiation life, there does not seem to be a way of consolidating fracture mechanics-based FLS levels with those obtained for total life tests on smooth bars when the total life is in the endurance limit range of 10 7 or higher cycles. . statistical analysis of the data, are often used for HCF design [4]. The values of mean stress on the Haigh diagram at zero alternating stress were taken as the material’s monotonic ultimate and yield. the assumptions of a threshold corresponding to a fixed crack length and having a constant value are valid. An alternate manner in which to compare the implications of the shape of the Haigh diagram. linear in R, starting at R =0 and decreases to a value at R =1 that is greater than zero. The ratio of K th at R =0toK th at R =1 is taken as a parameter . For illustrative 390 Applications 0 1 2 3 4 5 0.0