196 Effects of Damage on HCF Properties 4.4.1. LCF–HCF nomenclature Combined LCF–HCF loading has caused some confusion over the years because of nomenclature and the differences that occur when dealing with actual usage versus mathematical formulations involving simple linear summation concepts. Consider the schematic of Figure 4.39 that shows the major cycles (LCF) with a hold or dwell time in between them during which HCF cycles can occur. The problem we deal with is the superposition of the HCF cycles on the LCF behavior. The nomenclature is that of Powell and coworkers at University of Portsmouth, formerly Portsmouth Polytechnic. LCF loading alone produces a stress intensity level K major and represents the contribu- tion of the major throttle excursions in an engine. During dwell times, at maximum K for the LCF cycle, a vibratory loading occurs whose total amplitude is denoted by K minor . If the total contribution of the individual loading components is considered, then a linear summation law would require that the total growth rate be the sum of K total +K minor , not K major +K minor . This is because in the combined case, the effective amplitude of the LCF cycles is K total , not K major , since the LCF cycle now goes from minimum to maximum through an amplitude K total . This concept, while fairly clear, has not been followed consistently in the literature over the years and has led to some confusion when evaluating the applicability of a linear summation concept for crack growth under com- bined LCF–HCF loading. The confusion arises because the minor cycle amplitude also contributes to the major cycle amplitude as shown in the schematic of Figure 4.39. Another point to consider in discussing combined LCF–HCF, referred to here as CCF, is the manner in which the data are both recorded and discussed. While the stress or load ratio, R, is a common and well-understood quantity, there are other parameters with which to describe the entire load sequence in CCF, not including spectrum loading where overloads, underloads, and sequencing also have to be considered. For the load ratios we can use R minor and R major to represent the value of R for the HCF and LCF cycles, respectively. The number of HCF cycles per LCF cycle is usually given by n. Unless the value of n is sufficiently high, the contribution of the HCF cycles may not be detected since the HCF is generally applied at a much smaller value of K than are the LCF ΔK major ΔK total ΔK minor Figure 4.39. Schematic of combined major/minor cycle loading. LCF–HCF Interactions 197 cycles. A parameter that is often used to represent the relative magnitude of the combined stress cycles is the amplitude ratio, Q Q is defined as the ratio of the amplitude of the minor cycles to the magnitude of the major cycles. It can also be written in terms of the K values. Q = K minor K major (4.36) Note again that the major cycle amplitude refers to K major as shown in Figure 4.39, not to the value of K total even though the latter is used in linear damage summation. There is no single, yet simple, way of relating the major and minor cycle ratios, other than through the definition of Equation (4.36). The following useful expressions are easily derived and are found frequently in the literature, particularly in the papers coming from University of Portsmouth. R minor = 2 −Q 1−R major 2 +Q 1−R major (4.37) Q = 2 1−R minor 1−R major 1+R minor (4.38) In the example cited above, Figures 4.36 and 4.37, Q = 012 for R major = 01 and R minor = 09. For parallel experiments with the same major cycle, higher amplitude vibratory loading corresponds to Q =022 and R minor =082. The crack-growth rate data from CCF are normally presented with respect to the entire cycle block. Thus, as illustrated in Figures 4.36 and 4.37, the growth rate is shown as da/dBlock where the block consists of a single LCF cycle with n HCF cycles superimposed. For the horizontal axis, K total or K max can be used to describe the block loading. As an alternative, K major can be used. In the latter case, the HCF data cannot be shown superimposed on the block loading. A similar comment can be made for the use of K total . If it is desired to represent the LCF, HCF, and CCF data on the same plot, the use of K max is preferable. However, if a linear superposition concept is attempted graphically, both a vertical shift to account for n HCF cycles per LCF cycle, and a horizontal shift to account for the different maximum values in HCF and LCF has to be used. The reader is cautioned that there is no easy way to show LCF, HCF, and CCF data on the same (log) plot while also demonstrating linear summation graphically. 4.4.2. Example of anomalous behavior An illustration of the type B behavior of Figure 4.35 observed in combined LCF–HCF testing is taken from [52] where the fracture mechanics of a nickel-based single crystal alloy was studied. The focus was on the interaction of HCF at high stress ratio and at 198 Effects of Damage on HCF Properties high frequency combined with LCF at low stress ratio and low frequency in combined cycle loading. The test plan utilized previous threshold data generated at 1100 F, for R =01 and 0.8 in the <001/010> orientation on alloy PWA 1484. After precracking, the specimen was run at a stress ratio of 0.1 at a frequency of 10 CPM (0.167 Hz) until a crack length of a i was achieved. The target for a i was selected to be well below the K max threshold value for the R = 08 test data. A block of 1000 cycles of R = 08at60Hz was then performed between each R =01 LCF cycle. This block loading was continued until the calculated value for threshold at R = 08 was superseded at crack length a f . After growing beyond the calculated value of the R = 08 threshold, the loading was returned to LFC loading only at R =01. The test scheme was designed so that all loading blocks could be performed on a single sample so that specimen-to-specimen variation differences would not be included in the results. A schematic of the test approach is shown in Figure 4.40. For reference purposes, some specimens were tested under constant load to allow stress intensity K to increase with increasing crack length to detect the onset of crack-growth rate behavior of the R =08 cycles. The results of the first test under combined LCF–HCF loading are presented in Figure 4.40 which shows that a sharp increase in crack-growth rate occurred at crack length a i corresponding to the change in waveform from LCF to LCF + HCF indicating that the R =08 (HCF) cycles did affect the LCF crack-growth rate. This was in direct contradiction to the anticipated response based on the earlier R =08 threshold result that indicated the testing was below the HCF threshold. A return to pure LCF loading resulted in the anticipated return to the original R = 01 trend. A second specimen was then run to confirm that the data was repeatable and the same result was achieved. The results of the second test sequence are shown in Figure 4.41. The open circles indicate the HCF only test results showing the threshold region for HCF. As in da /dN LCF a o a f a i LCF only LCF + HCF LCF only σ max σ max 0.1 0.1 0.8 0.1 σ max σ max σ max σ max σ max K max (ksi√in.) Figure 4.40. HCF–LCF testing approach and prediction. LCF–HCF Interactions 199 10 –8 10 –7 10 –6 10 –5 10 –4 10 –3 6 8 10 20 LCF only LCF only HCF only HCF + LCF LCF only da /dN LCF (in./cycle) 30 R = 0.1 Threshold 97 K max (ksi in.) Figure 4.41. LCF–HCF Interaction test results the first test, the interaction between HCF and LCF produces an accelerated growth rate below the HCF threshold. To further study this unexpected crack-growth rate acceleration, five loading schemes were devised and performed at a constant K LCF of 10ksi √ in. The frequency was 10 cycles per minute (CPM) for the LCF portion and 60 Hz for the HCF portion. The loading schemes are shown in Figure 4.42 as Case (a) through Case (f). A single sample was again used to perform all six cases in succession with each case consuming up to 0.020 in (0.5 mm) of specimen ligament before proceeding to the next case. The results of the test sequence shown in Figure 4.42 are summarized in Figure 4.43. Comparison of the various case loading blocks suggests the higher mean stress or dwell is the significant contributor to the accelerated crack-growth behavior as opposed to the number of HCF cycles. As shown in Figure 4.43, the application of dwell, Case (b), increased the growth rate from the baseline without dwell, Case (a). The dwell produced nearly the same acceleration as adding 1000R = 08 cycles, Case (c). The similarity between Case (c) and Case (d) (1000 versus 500 HCF cycles) suggests that the sensitivity to time-dependent behavior is inclusive of very small differences in hold times and 200 Effects of Damage on HCF Properties = 0 N HCF N LCF 16.7 s Dwell 1000 500 1000 500 Case (a) (b) (c) (d) (e) (f) σ max σ max σ max 0.8σ max 0.8σ max 0.64σ max 0.1σ max 0.1σ max 0.1σ max Figure 4.42. Constant K testing to identify crack-growth rate acceleration drivers. 10 –7 10 –6 10 –5 10 –4 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 (a) (b) (c) (d) (e) (f) da /dN (in./cycle) Crack length (in.) Figure 4.43. Test plan 2 results. Legend refers to loading blocks shown in Figure 4.42. or cyclic frequency. In fact, threshold testing done previously on that program clearly showed a pronounced frequency effect between 20 Hz, 1 Hz and 10 CPM (0.167 Hz). An interesting aspect of these findings is that the results indicate that there is a dwell effect present in a temperature regime well below what has typically been called the creep regime. The significance of this finding is that very small decreases in frequency that is 10 CPM alone versus 10 CPM with as little as an 8-second dwell at maximum load, cause up to a 2X–3X acceleration in crack-growth rate. One possible explanation proposed for the LCF–HCF interaction effects was oxidation at the crack tip. Tests in vacuum were recommended. 4.4.3. Another example of anomalous behavior Another observation of the interaction of LCF and HCF and the subsequent growth rate behavior is that of Russ [45] who conducted studies on Ti-17, a processed titanium alloy. In that work, LCF cycles with R =01 were superimposed on what are referred to LCF–HCF Interactions 201 as “baseline” HCF cycles having R =07. Both cycles have the same value of maximum load (or stress) for a given crack length. Data were obtained by decreasing the load to obtain a pure HCF threshold and then increasing the load with the LCF–HCF spectrum shown schematically in Figure 4.44. While the study uses the nomenclature of the HCF cycles being the baseline, and the LCF cycles being the equivalent of an underload (also called a negative overload in the literature), the phenomenology is identical to that observed in the works described above (see [32–35] and Figure 4.39, for example). In the work of Russ, the number of HCF cycles per LCF cycle was varied between 10 and 1000. Figures 4.45 and 4.46 show the observed growth rate behavior in the region near the HCF threshold for HCF cycle counts of 200 and 1000 per LCF cycle, respectively. Both figures show the trend expected if no interaction occurs, that is, if linear summation of LCF and HCF growth rates is assumed. The experimental data appear to demonstrate an accelerated growth rate under the combined loading as well as growth rate acceleration below the expected HCF threshold. Although the value of R =07 for the HCF cycles is high enough to expect the complete absence of closure, the author finds that a combination of closure during the application of the R = 01 cycle and residual deformation ahead of the crack may account for the interaction effect. From plane strain finite element modeling, the author shows that closure develops immediately behind the Time K Step down block R = 0.7, 500,000 cycles Step up block R = 0.7, 500,000 cycles R = 0.1, 50 cycles Figure 4.44. Schematic of load history for determining threshold for crack propagation under combined LCF–HCF loading. 202 Effects of Damage on HCF Properties 10 –6 10 –5 10 –4 10 –3 10 –2 68910 P max = 1.77 kN P max = 0.86 kN Linear summation da /dB (mm/block) 157 K max (MPa m) Figure 4.45. Fatigue-crack-growth rate results for 200 HCF cycles per LCF cycle. 10 –6 10 –5 10 –4 10 –3 10 –2 6810 P max = 2.85 kN Linear summation da /dB (mm/block) 1579 K max (MPa m) Figure 4.46. Fatigue-crack-growth rate results for 1000 HCF cycles per LCF cycle. LCF–HCF Interactions 203 crack and compressive residual stresses develop ahead of the crack tip after the R =01 underloads. It should be pointed out that, consistent with FEM results by Sehitoglu et al. [53], compressive residual stresses were observed even in the absence of closure during the R =07 cycles. The distances over which the closure and residual stresses develop are only a few microns, inviting speculation as to whether such phenomenology could affect the net growth rate behavior. However, as in the previous examples of Type B behavior as defined in Figure 4.35, the observed behavior is non-conservative when compared to the predictions of a linear summation model. Another observation by Russ [45] was on the threshold for crack propagation and the differences in the growth rate behavior in the near-threshold regime. For a constant value of K max = 8 MPa √ m, Figure 4.47 shows the combined-cycle growth rate for different numbers of HCF cycles per block (1 LCF cycle) as well as the results from a simple linear summation of the individual contributions of LCF and HCF. For the maximum number of HCF cycles per block, 1000, the growth rate is dominated by the HCF cycles. For all conditions other than 10 HCF cycles per block, the experimentally measured growth rates are higher than those predicted by the linear summation model. These differences are approximately a factor of two or more. Equally important from a life prediction point of view, the threshold for combined cycle loading is lower than predicted by linear summation. For the spectrum illustrated in Figure 4.44, where the ratio of HCF to LCF cycles is 10,000, the threshold value of K max for pure HCF is 727 MPa √ m whereas, during increasing loading for the combined cycle, the threshold at which crack growth first appears is 699 MPa √ m. While these differences in growth rate and threshold do not appear to be very significant, Russ points out that the differences in fatigue lifetimes can be significant when the combined cycling is in the region of the HCF threshold where a significant portion of the total time for crack growth to failure takes place. 10 –6 10 –5 10 –4 10 –3 10 1 10 2 10 3 Experiment Linear summation da /dB (mm/block) HCF cycles per block K max = 8 MPa m R HCF = 0.7 R LCF = 0.1 Figure 4.47. Fatigue-crack-growth rates for LCF–HCF spectrum of Russ [45]. 204 Effects of Damage on HCF Properties 4.5. COMBINED CYCLE FATIGUE CASE STUDIES There are other examples of combined cycle loading where HCF at stress levels below their constant amplitude threshold appear to contribute to the combined cycle growth rate. An example is the study by Zhou and Zwerneman [54] where the cycle block contains small amplitude cycles with periodic overloads. Denoting the major cycles as “overload” (ol) cycles and the other as minor cycles, the ratio K minor /K ol was chosen as 0.3 and R ol = 01. A schematic of the block loading is shown in Figure 4.48. Using ASTM A588 steel as the test material, values of the number of minor cycles per block, n, were chosen as 0, 4, 9, 49, 99, and 999. The results, presented in terms of growth rate per block, are shown in Figure 4.49. The number of cycles per block includes one Time K n cycles Minor cycles R = 0.27 Overload cycle R = 0.1 Figure 4.48. Schematic of block loading with periodic overload cycles. 10 –7 10 –6 10 –5 10 –4 1 10 100 1000 da /dBlock (in./cycle) Cycles per block Conventional baseline ASTM baseline Figure 4.49. Fatigue-crack-growth rate for cycles with periodic overloads. LCF–HCF Interactions 205 major cycle and n minor cycles. Since there is only one overload (major) cycle per block, the increase in growth rate per block with increasing values of n can only be attributed to the minor cycles. Yet the minor cycles are applied at K levels that are below their constant amplitude threshold. The “conventional baseline” shown in the figure indicates that there is no contribution due to the minor cycles. The assumed threshold for the minor cycles was taken from an empirical formula for this material. If, instead, the ASTM recommended value for threshold is taken as a growth rate equal to 10 −10 m/cycle, then this small, but non-zero, contribution to the minor cycle growth results in the curve labeled “ASTM baseline” in the plot. Even under this assumption, the data show that the experimentally observed block growth rate is higher than predicted by a linear summation rule. It is clear that the threshold is reduced by the overload cycles, an effect similar to the acceleration of crack growth due to underload cycles observed by Russ [45], described above. Further evidence of the growth due to the minor cycles was obtained from acoustic emission (EM) monitoring. The intensity of the AE signals was observed to increase with increasing numbers of minor cycles per block. The authors concluded that the increase in AE activity indicated that “the minor cycles below the threshold contribute to fatigue damage.” It certainly is possible that the threshold obtained under constant amplitude loading is not applicable to the block-loading situation, an example of load-history effects. Alternatively, the determination of the threshold could be incorrect, also due to load-history effects. This latter condition is discussed in Chapter 8. In both cases, the growth rate of the minor cycles under combined LCF–HCF block loading is affected by the major cycles, whether they are of the overload or underload variety. In both cases, the linear summation approach is non-conservative. Another example of an LCF–HCF interaction is the overload study of Byrne et al. [55] on a slightly more complicated load spectrum as shown in Figure 4.50. In this case, overloads were superimposed on a baseline LCF–HCF block which combined a major cycle with a number of minor cycles having the same maximum K value. In this block, Time K Kol Kss Baseline block Overload block Figure 4.50. Schematic of block loading with superimposed overload cycles. . where the fracture mechanics of a nickel-based single crystal alloy was studied. The focus was on the interaction of HCF at high stress ratio and at 198 Effects of Damage on HCF Properties high frequency. orientation on alloy PWA 1484. After precracking, the specimen was run at a stress ratio of 0.1 at a frequency of 10 CPM (0.167 Hz) until a crack length of a i was achieved. The target for a i was selected. interaction occurs, that is, if linear summation of LCF and HCF growth rates is assumed. The experimental data appear to demonstrate an accelerated growth rate under the combined loading as well as