406 Applications 0 100 200 300 400 500 600 700 800 10 4 10 5 10 6 10 7 R = –1 R = 0.1 R = 0.5 R = 0.8 Effective stress (MPa) Number of cycles Ti-6Al-4V 60 Hz SWT model Figure 8.27. Representation of straight line fits using SWT parameter for various values of R. by finding the best value of the exponent m using a least squares fit to a straight line representation of all of the lines in Figure 8.26. The resulting model has a value of m = 034 and is shown in Figure 8.28. Here, all of the lines are brought together in a single band, but the degree of scatter is somewhat large. To illustrate the scatter on the original data of Figure 8.26, Figure 8.29 is drawn along with the best fit straight line shown as a solid line in the figure. While the single straight line seems to represent all of the data reasonably well, the degree of scatter is quite broad. What these plots illustrate is that when data from different values of R are represented by a single function, the 0 100 200 300 400 500 600 700 800 10 4 10 5 10 6 10 7 R = –1 R = 0.1 R = 0.5 R = 0.8 Effective stress (MPa) Number of cycles Ti-6Al-4V 60 Hz Walker model m = 0.34 Figure 8.28. Consolidation of straight line fits to data using Walker model. HCF Design Considerations 407 0 100 200 300 400 500 600 700 800 10 4 10 5 10 6 10 7 R = –1 R = 0.1 R = 0.5 R = 0.8 Effective stress (MPa) Number of cycles Ti-6Al-4V 60 Hz Walker model m = 0.34 Figure 8.29. Consolidation of all data using Walker model. resulting scatter is a combination of the true scatter as illustrated in Figure 8.26 as well as the inability of the function to consolidate results obtained at different values of R. For the particular data sets shown here, the data at R =08 are considered to be different than the other data because at this high value of R, time-dependent behavior has been observed. The data at R =−1, on the other hand, may be represented by a different value of the exponent m in Equation (8.5) because it has been found that a different exponent can consolidate data at negative R better than the one used for positive R. In general, this provides another degree of flexibility in representing data at various values of R with a single function. This becomes an exercise in curve fitting that has little or no physical significance. eff = max 1−R m (8.5) The example above illustrates the issue of distinguishing between true material scatter and the capability of a model to represent data obtained under a variety of conditions or in terms of various parameters. An extreme case of this dilemma is when fatigue data are obtained under multiaxial conditions at lives both at the endurance limit and lower. Even if load-history effects are ignored and are unimportant, the number of parameters used in testing can result in only a few tests under a single condition and a large body of data at different conditions. The evaluation of a model to fit such a data set requires the ability to distinguish between inherent material scatter and the capability of the model to represent the various conditions. As illustrated above, the scatter is due to a combination of both and it is often very difficult to distinguish between modeling capability and material scatter. While duplicate tests under a single condition are generally very useful to 408 Applications evaluate material scatter, tests covering a wide variety of conditions do not always allow for sufficient replicate tests for practical reasons such as total test number limitations. 8.4.2. Threshold considerations In dealing with a cracked component using fracture mechanics rather than a fatigue limit using stress, the concept of a threshold for crack growth has to deal with ever-decreasing growth rates and the possible existence of a true threshold. The concept of a crack growth threshold, K th , and its very existence, has been addressed in the literature for decades and will not be discussed in any detail in this book. For a brief review of the discovery of the threshold concept, see Paris and Tada [12]. There, they discuss the variables that affect the threshold at various values of mean stress or stress ratio. In much of the published literature, the variation in the threshold with different values of R is attributed to crack closure as well as small crack behavior. It is interesting to point out that in smooth bar fatigue, when establishing fatigue limits or endurance limits, closure has little or no meaning except in the crack growth region which is very small for long-life fatigue. The subject of load-history and interaction effects in fatigue crack growth, with par- ticular attention to threshold behavior and small crack effects, have been reviewed by Skorupa [13, 14]. In [13], many of the observed phenomena are reviewed, with the author noting that trends are not always consistent from one investigator to another and from one material to another. In [14], mechanistic arguments are used to explain what controls load-interaction effects and to explain the generally observed trends. Again, the author points out that there is no consistent explanation and understanding of many of the complex phenomena observed when non-constant fatigue cycling is applied to materials. At the end of his review article on the history of fatigue, Schütz [15] gives his outlook on remaining problems. He offers the opinion that prediction of fatigue life under vari- able amplitudes remains unsolved, as much as ever, “despite innumerable claims to the contrary in the literature.” It can be stated here that this problem also remains largely unsolved when considering the FLS or endurance limit under variable amplitude loading. The interaction of LCF and HCF in terms of the observable growth rates and the results of the interaction was discussed in Chapter 4. There, emphasis was placed on the “threshold” conditions above which infinite life can be expected, particularly in terms of applied stress. In the following section, we extend the discussion in terms of the interactions when trying to determine a fatigue crack growth threshold that can be used in the design process. In this case, we are dealing with damage that can be quantified in terms of a measurable crack size. Many of the features of LCF–HCF interactions discussed in Chapter 4 are relevant to this section. Conversely, many of the aspects of the interactions of LCF with HCF in determining a suitable threshold for design could easily have been put in Chapter 4. In this discussion, we deal primarily with the history effects in determining a threshold stress intensity that is applicable to the design process HCF Design Considerations 409 for the determination of the condition that leads to the onset of crack propagation once a LCF crack has formed. The emphasis here is on the determination of the value of the threshold stress intensity. Concepts such as an inherent material threshold, a lower bound for such a threshold, the variability of such a threshold with conditions such as the value or history of stress ratio, R, and the existence of fundamental laws that govern such a threshold are discussed. Finally, the section concludes with an engineering definition of a threshold that can be used in the design process. 8.4.3. Experimental threshold measurements Reliable knowledge of the load conditions corresponding to the onset of fatigue crack growth (FCG) becomes a critical issue as damage tolerance concepts are increasingly applied to a variety of engineering applications. Such knowledge can be obtained from failure analysis of real structures and/or from laboratory tests performed on samples of a material. The latter way is usually preferable since it is more economical and effective, which points out the importance of the role which experimental techniques play in FCG studies. Further efforts to improve the accuracy of design and life estimation technologies dependent upon FCG analysis must consider and minimize, whenever possible, any substantial differences between testing and real service conditions. The existing procedure for FCG testing outlined in ASTM E 647-93a [16] provides for the special care which must be taken to approach threshold conditions in a continuous and gradual manner. As a result, the FCG threshold values are measured for the smallest cyclic plastic zone possible under respective loading parameters. In real components, especially those operating under high stress ratios, R, FCG usually commences in the presence of larger cyclic plastic zones created in the vicinity of preexisting crack-like defects by previous load excursions. 8.4.3.1. “Jump-in” method To account for the possible influence of cyclic plastic zone size on the fatigue threshold at high R, other methods for defining a threshold that can be used in design have been proposed. Döker et al. [17], Castro et al. [18], Lenets and Nicholas [19], and subsequently Mall et al. [20] used a so-called “jump-in” or “step-down” threshold test method with an abrupt load change while keeping a constant value of R [17, 20] or K max [18]. With this procedure (Figures 8.30 and 8.31), the fatigue crack is grown under certain load conditions K max 0 K 0 long enough to develop corresponding monotonic and cyclic plastic zones. Subsequently, K is instantaneously decreased to a value K 1 (slightly above the expected threshold for FCG) and then held constant until clear evidence of crack propagation is obtained. The above load sequence is repeated with successively smaller K values: K 1 >K 2 >K 3 until no crack growth can be detected after an abrupt load change within a specified number of cycles. The FCG threshold, K th ,is 410 Applications Time K Growth No growth K max 0 ΔK 0 ΔK 0 ΔK 0 ΔK 1 ΔK 2 ΔK 3 Figure 8.30. Load pattern for “jump-in” threshold method for constant K max . Time K ΔK 1 ΔK 2 ΔK 3 Growth No growth K max 0 ΔK 0 ΔK 0 ΔK 0 Figure 8.31. Load pattern for “jump-in” threshold method for constant R. considered to be bracketed by the stress intensity range corresponding to the last loading block that produces crack propagation and that corresponding to the dormant crack (e.g. K 2 >K th ≥K 3 in Figures 8.30 and 8.31). Note that the crack length at which FCG threshold conditions are approached depends on the threshold value and therefore cannot be specified prior to the test. For a nickel-based superalloy Inconel 617 and a titanium alloy Ti-6246, lower K th values have been obtained after the “jump-in” tests as compared to the gradual load shedding toward threshold conditions [18, 20]. Such findings indicate that a difference in the cyclic plastic zone size between a laboratory specimen subjected to FCG testing via existing procedures and a real component in service may result in non-conservative design solutions. Lenets and Nicholas [19] used two methods, shown schematically in Figure 8.32, where thresholds were obtained from increasing and decreasing K experiments after growing HCF Design Considerations 411 – Growth N KK N – No growth Jump A Jump B Figure 8.32. Schematic of jump loading history. the crack under large amplitude loading. Figure 8.33 presents the results which show that the decreasing K threshold (Jump A) is smaller than the increasing K threshold (Jump B), although the authors feel that the lower threshold is not realistic because service conditions usually produce loadings below threshold until a severe event occurs in which HCF becomes critical, as in Jump B. With both the “jump-in” [17, 18, 20] as well as the ASTM E647 load-shed [16] methods, the fatigue threshold value is obtained at the transition between the two condi- tions “growth – no growth.” Figures 8.30 and 8.31 depict the load (stress intensity, K) pattern used for conducting such tests under constant K max and constant R conditions, respectively. Engineering applications, however, are typically characterized by initially non-propagating (sometimes even non-existent) cracks. Therefore, fatigue threshold val- ues obtained at the transition from no growth to growth seem to be of greater practical interest. Fortunately, it appears that the “jump-in” method can easily be modified so that it better represents service conditions in terms of non-propagation of the initial defects. To achieve this, a smaller K value (insufficient for the FCG to occur after the instantaneous load change) should be applied first. For example, in Figures 8.30 and 8.31, K 3 should be applied instead of K 1 . Then, provided that the crack does not grow, successively Jump A Jump B 2 2.5 3 3.5 4 0 5 10 15 20 25 30 ΔK th (MPa m) K max (MPa m) Figure 8.33. Threshold data for jump tests. 412 Applications higher K values should be applied until crack propagation begins. Note that such modi- fication does not affect the main advantage of the method, that is the possibility to obtain FCG threshold values in the presence of cyclic plastic zones of any size. Further, unlike the approach discussed above, such a modified “jump-in” method is not limited to a threshold determination only. After the conditions for the onset of FCG are reached, the stress intensity or load can be kept constant to monitor subsequent (post-threshold) crack propagation behavior. Another advantage of this proposed technique is that, similar to real behavior in service, no crack growth occurs before threshold is reached. Thus, it should be possible to reproduce threshold conditions at certain pre-determined crack lengths. This becomes especially important when the influence of local microstructure or stress–strain state is to be taken into account (e.g. short crack, welding residual stress, or notch effect). It is obvious that with such a modified “jump-in” method, threshold conditions for FCG are preceded by rather extensive loading which produces K values smaller than or equal to K th and during which the crack remains dormant. The opinion exists [21] that such cycling at small amplitude has no influence on the subsequent FCG behavior of metallic alloys as long as the boundaries for K maxth and K minth are not surpassed and the crack does not grow. However, no attempt has been made so far to evaluate experimentally the load-history effects on the FCG threshold values furnished by the “jump-in” threshold test method. For damage in the form of cracks from FOD, fretting or LCF, for example, the use of a fracture mechanics threshold to determine the allowable vibratory stress seems to be a promising approach for HCF, and follows the concept now being used successfully for LCF. Provided that an inspection can be made, and crack lengths measured, knowledge of the threshold for crack propagation can be used to assess the susceptibility of the material to HCF crack propagation from the damaged area. If the stresses are maintained below this limit, and the limit corresponds to a sufficiently low growth rate, perhaps 10 −10 m/cycle or lower, then safe HCF life is assured. The potential growth of such cracks under LCF, and the time interval where such growth produces a crack where HCF might occur, must also be considered. This would establish the required inspection interval. One key issue in this proposed scenario is the determination of a suitable threshold for the types of cracks which may occur in service, some of which could be quite small. Various types of loading conditions can be used to determine a threshold, most of which are for long cracks. The effects of loading history on the threshold for HCF under spectrum type loading has already been discussed in Chapter 4. The determination of a threshold condition applicable to design is still a subject of some controversy. 8.4.4. Mechanisms in threshold testing The study by Lenets and Nicholas [19] mentioned above, for example, revealed a con- sistent difference between the FCG thresholds obtained via two loading patterns, both HCF Design Considerations 413 utilizing an abrupt change in the K value at constant K max (see Figure 8.32). Increasing K applied to the initially dormant crack resulted in higher FCG thresholds as compared to the situation when gradually decreasing K values were applied to the growing crack. The discrepancy was larger for lower K max and much less significant at the highest value of K max . It is important to understand the possible reasons for the observed differences in threshold due to different loading or testing histories. Since the loading pattern itself cannot change an intrinsic material’s resistance against crack propagation, one should assume that the difference in the FCG threshold behavior revealed in that study originates from the loading history and was attributed by the authors to an extrinsic (crack tip shielding) mechanism. A close similarity can be pointed out between threshold behavior in the referenced study by Lenets and Nicholas and observations from previous studies of the influence of fatigue underloads on FCG close to threshold in a pressure vessel steel [22]. In the latter case, an underload (i.e. cyclic loading at stress intensities below the fatigue threshold) was shown to retard subsequent crack propagation at K>K th . A proposed mechanism for the influence of such subthreshold cycling was associated with a fatigue crack closure phenomenon arising from the crack flank fretting-oxidation. Thus, it can be argued that the initial loading block during the Jump B tests, while being insufficient to propagate the crack could, nonetheless, result in excessive fretting-corrosion deposits on the fracture surfaces. Such deposits would wedge the crack tip open, thereby decreasing the effective driving force for its propagation. As a result, the crack would not propagate even at the next load level, K i = 32 MPa √ m, that was sufficient to maintain crack propagation during the Jump A test (see Figure 8.32) where conditions for extensive crack flank oxidation (i.e. subthreshold cycling) were absent. However, at least two reasons exist which do not allow one to accept the above explanation. First, unlike a low-alloyed pressure vessel steel used in [22], the Ti-alloy of the referenced study is not prone to fretting-oxidation. Second, the fracture surface contact behind the crack tip was observed in [22] for a low stress ratio R = 005 whereas in the referenced study, the values of R were much higher, varying from 0.4 (the lowest R at K max = 6MPa √ m) to 0.98 (the highest R at K max =25 MPa √ m). For this range of R values, the fracture surface contact as well as all related crack shielding processes, if any, should be less significant. Lenets and Nicholas concluded that the FCG threshold behavior that they observed can only be explained if some other crack tip shielding or load-history induced mechanisms, not associated with fracture surface contact, are involved. An observation, somewhat helpful in this respect, was reported by Mall et al. [20] where crack-tip opening was recorded with extremely high accuracy using an interferometric displacement gage (IDG). During a varying K test at R =05 and K max 0 =20 MPa √ m, they observed a higher closure level than that obtained from either crack opening dis- placement (COD) or back face strain (BFS) methods. Further, immediately after a sudden load change, the IDG measurements indicated some crack tip shielding even though 414 Applications conventional crack closure measurements via COD or BFS methods gave negative results. In another study, using a completely different indirect method for the crack opening load measurement, Döker and Bachmann [21] showed that a certain amount of crack tip shielding may exist even if the COD or BFS measurements give no indication of the fracture surface contact. They found that after a sudden load change, the crack driving force does not immediately follow an externally applied load, but retains, at least for a certain number of load cycles or certain crack length increment, a memory about the shielding level characteristic of the previous load regime. In other words, the FCG process depends on a load history even in the absence of the fracture surface contact. This line of reasoning brings one to the conclusion that, immediately after a load change, the Jump A test should provide a higher driving force for FCG than does the Jump B method, even under conditions where the same K is applied in both cases. A delayed transition in the crack tip shielding after a sudden load change (jump) can also explain an influence of the load history applied to an initially dormant crack (see Figure 8.32). In this case, under the Jump B test conditions, the driving force acting at the crack tip after a sudden load change increases toward its stable (nominal) value characteristic of the constant amplitude loading. Therefore, the onset of crack propagation at a certain K value depends additionally on the number of previously applied cycles. For example, at K max =6MPa √ m, no crack growth occurred at K i =34MPa √ m when this loading block was preceded by 8 million cycles. However, when applied after about 20 million cycles, the same K i =34MPa √ m immediately resulted in crack propagation. 8.4.5. Load-history effects Other studies, too numerous to mention, have demonstrated the effect of loading history on the experimentally observed threshold, a number that is critical for design under combined LCF–HCF cycling. For example, Ritchie et al. [23] addressed the problem of determining the threshold for initially small cracks whose behavior has been shown to produce growth rates in excess of those observed for long cracks at the same values of K and R. Further, they note that small cracks have been shown to have thresholds which are below the long crack threshold using LEFM analysis. Possible reasons for this anomalous behavior are discussed in [19], for example. To accomplish this, Ritchie et al. evaluated two procedures for obtaining a lower bound threshold by conducting tests where crack closure is absent, a condition believed to be present and responsible for the behavior in small cracks. The standard load-shedding procedure recommended by ASTM E-647 was applied to tests at various load ratios, R. The load shedding follows the sequence described by the following equation: K = K initial expCa −a initial (8.6) where subscript “initial” refers to the crack length and K values after precracking where the load-shed sequence is started. The K-gradient, C, is recommended by ASTM E-647 HCF Design Considerations 415 Time K Closure Figure 8.34. Schematic of conventional load-shedding history. to be −008 mm −1 . As depicted in Figure 8.34, if tests are conducted at low values of R, crack closure may influence the behavior and, thus, the measured threshold. To avoid closure considerations, only data at high values of R are of interest. The second procedure employed by Ritchie et al. consisted of testing under a constant value of K max , and increasing K min , until the crack is arrested. This test procedure, shown schematically in Figure 8.35, maintains conditions where the loading stays above any possible closure level if a high enough value of the starting (constant) K max is chosen. The test is conducted under varying (increasing) R conditions, unlike the standard load-shed test, thereby making it hard to obtain the threshold at a specific targeted value of R. From these two test sequences, both designed to obtain a threshold value in the absence of closure, results were obtained at 1 kHz frequency. Under standard, high R load shedding, Time K Closure K max = constant Figure 8.35. Schematic of constant K max , decreasing K test history. . issue of distinguishing between true material scatter and the capability of a model to represent data obtained under a variety of conditions or in terms of various parameters. An extreme case of. process. In this case, we are dealing with damage that can be quantified in terms of a measurable crack size. Many of the features of LCF–HCF interactions discussed in Chapter 4 are relevant to this. value characteristic of the constant amplitude loading. Therefore, the onset of crack propagation at a certain K value depends additionally on the number of previously applied cycles. For example, at