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156 Effects of Damage on HCF Properties (a) 550.0 570.0 590.0 610.0 630.0 650.0 No. pre- damage 900 MPa, 0.5R 50,000 cycles 790 MPa, 0.5R 500,000 cycles 700 MPa, 0.1R 150,000 cycles 700 MPa, 0.1R 500,000 cycles Fatigue strength (MPa) (b) 500.0 520.0 540.0 560.0 580.0 600.0 620.0 640.0 No. pre-damage 700 MPa, 0.1R 150,000 cycles 900 MPa, 0.5R 50,000 cycles Fatigue strength (MPa) Figure 4.10. HCF limit strength with and without LCF pre-damage: (a) R HCF = 05, (b) R HCF = 01, (c) R HCF =08. LCF–HCF Interactions 157 Fatigue strength (MPa) (c) 850.0 870.0 890.0 910.0 930.0 950.0 No. pre-damage 700 MPa, 0.1R 500,000 cycles 900 MPa, 0.5R 50,000 cycles Figure 4.10. (Continued). to be higher than the baseline. The observation that they were lower suggests that there might be an effect of pre-damage due to LCF, but the magnitude is not significant. In Figures 4.10b, c, results are shown where LCF was applied at a stress of 900 MPa on a titanium alloy that has a yield stress of 930 MPa. In this case, strain ratcheting occurred during the LCF portion of the test, but no ratcheting was observed in the HCF portion of the test under step-loading conditions. Morrissey et al. [16] have reported that time- dependent creep or ratcheting may play an important role at high stress ratios where a portion of each cycle is spent at stresses near or above the static yield stress of the material. They observed measurable strain accumulation dependent on the number of cycles, as opposed to time, at the stress ratio of 0.8 where the applied maximum stress was slightly above 900 MPa. The fracture surfaces for the cases where strain ratcheting was observed in the tests summarized in Figure 4.10 did not show any indication of this phenomenon. This is in contrast to the observations of Morrissey et al. [16] where ductile dimpling was observed on fracture surfaces on specimens tested at R =08 with maximum stresses above approximately 900 MPa. In fact, over the range of conditions studied in [14], there appeared to be little or no effect of prior LCF on the subsequent HCF limit stress. If any cracks formed during LCF, they were not observable through fractography and were not of sufficient size to cause any significant reduction of HCF limit stress under subsequent 158 Effects of Damage on HCF Properties testing. So, for conditions that covered a range from overloads to underloads from LCF, there was no observable effect on the subsequent HCF limit stress in a titanium alloy. A similar investigation into load-history effects on the HCF limit stress was conducted by Morrissey et al. [15] where an attempt was made to assess the damage that might occur from HCF transients when these transient stresses exceeded the undamaged HCF limit stress. They addressed a practical problem that arises in designing against HCF because of the possible occurrence of stress transients above the FLS. These stress levels can correspond to lives below the HCF limit, but not necessarily to those which produce LCF. The slope of the S–N curve is quite shallow in this region, making the scatter in the cycles to failure quite large and, consequently, making cycle counting difficult from a deterministic life prediction scheme. As shown in the previous section, the effects of prior loading under LCF seem to have little or no effect on the subsequent HCF limit stress, even when loading under LCF has covered life ranges as high as 50–75% of anticipated life in both smooth and notched specimens. To investigate effects due to transient loading, Morrissey et al. [15] used high frequency blocks of loading to represent the cumulative effect of HCF transients which might cause crack initiation to occur. They determined the effects of these transients for two reasons. First, short duration, high frequency stresses above the fatigue limit (but still within the HCF regime) are typical of what occurs in gas turbine engine applications. Second, in contrast to LCF overloads that would by definition only occur for a very small number of cycles, HCF stress transients could produce large numbers of cycles, even if they only occur for up to 20% of the expected fatigue life at that stress. The experiments in [15] involved the application of HCF transient stress levels for a certain percentage of life (ranging from 7.5 to 25%) and then using the step-loading technique to determine the subsequent fatigue strength corresponding to 10 7 cycles at R =05. Some specimens were either heat tinted or both heat tinted and stress relieved (SR) after the pre-cycling (prior to HCF testing). The results are summarized in Table 4.1 which presents the ratio of the final failure stress under HCF loading conditions, f (after applying stress transient loads) to the 10 7 cycle HCF endurance limit, e . It is clear from the table that preloading at 855 MPa (30% above the endurance limit) for 50,000 cycles (approximately 25% of life) does not reduce the subsequent HCF strength of the Table 4.1. Summary of fatigue limit stress results for specimens subjected to stress transients # of tests LCF N LCF % of life f / e 3 855 50000 25 0.99 5 925 25000 25 0.92 3 925 15000 15 0.94 1 925 7500 75 0.93 3 925 (SR) 25000 25 0.81 LCF–HCF Interactions 159 material. Table 4.1 also shows that cycling at 925 MPa for up to 25% of life has little effect on the HCF strength. However, the HCF strength is reduced by an average of 19% when subjected to prior cycles with a maximum stress of 925 MPa followed by a stress relief process. This implies that there are residual stresses present after the stress transient cycling that act to reduce any loss in HCF strength. The nature of these stresses might be attributed to crack closure, residual compressive stresses ahead of the crack tip, or other phenomena. The net effect is one similar to what is commonly seen in crack growth studies under spectrum loading and is referred to as an overload effect, where prior cracking at a high load level tends to retard crack extension at lower loads. To analyze the experimental data, a Kitagawa diagram was constructed for a circular bar, the geometry used in the experiments. The resulting diagram is shown in Figure 4.11 where the parameters are given in the figure and no correction is shown for short cracks. The crack growth analysis for the curve is presented below. For comparison, the diagram shows the line for a simple edge-crack geometry for a material having the same endurance limit as well as long-crack threshold stress intensity. This simple example also illustrates the dependence of the shape of the Kitagawa diagram on the specific geometry and value of R being used. The crack growth analysis in [15] was also made in order to distinguish the initiation stage from the total life observed experimentally. The approach was to subtract the crack propagation time from an initial crack size a 0 from the total life to failure, where a 0 is the transition crack size in the Kitagawa diagram. When repeated over the range of stress levels used for the coupon fatigue tests, the result is a stress-life curve for nucleation life compared to the stress-life curve for total life. 100 1000 10 1 10 2 10 3 Circular bar Edge crack Endurance limit Maximum stress (MPa) Crack depth (μm) Ti-6Al-4V plate R = 0.5 K max,th = 6.38 MPa σ e = 660 MPa a 0 = 29.7 μm a 0 = 68.1 μm 200 400 600 800 Figure 4.11. Kitgawa diagram for a circular bar. 160 Effects of Damage on HCF Properties The Forman and Shivakumar [17] stress intensity factor for a surface crack in a rod can be used for this analysis. This solution assumes that cracks have a circular arc crack front whose aspect ratio starts as 1.0 for small cracks. As the crack propagates, the aspect ratio changes according to experimental observations that are embedded in the K solution. Therefore, only the crack depth, a, is used in the solution. The crack depth is defined as the radial distance from the circumference to the interior of the crack. Equation (4.11) is the general form of the stress intensity factor solution. The geometry correction factor F 0 , defined by Equations (4.12) and (4.13), is dependent only on the ratio of crack size to rod diameter, [Equation (4.14)]. K I = 0 F 0 √ a (4.11) F 0 =g 0752 +202 +037 1−sin 2 3 (4.12) g =092 2 tan 2 2 1/2 sec 2 (4.13) = a D (4.14) To calculate the propagation life of the fatigue tests from an initial crack size, a crack growth law must be defined. The crack growth rate da/dN versus the stress intensity factor range K was based on experimental data for the same Ti-6Al-4V used in this study [18]. da dN =465×10 −12 K 389 (4.15) The propagation analysis procedure is, for each stress level, to start with the defined initial crack size a 0 , calculate K, and then calculate the crack growth rate da/dN . A small increment of crack growth, a, is then applied and the corresponding increment of cycles, N , is calculated. This procedure is repeated until failure when K max equals the fracture toughness K c that is approximately 60MPa √ m for this material. As discussed above, a Kitagawa type diagram can be useful in evaluating the potential for a crack to reduce the HCF capability of a material. Since stress transients above the HCF endurance limit were being introduced to the material, the possibility that cycling at these stress levels resulted in cracking was evaluated numerically. For this purpose, a Kitagawa type diagram was constructed for the experimental conditions under consid- eration. The endurance stress, e , was taken as the value that had been experimentally determined and the crack growth threshold curve was developed using the K solution above for a crack on the surface of a cylindrical bar. The resulting diagram is shown in Figure 4.12, along with the El Haddad short crack correction. The crack length, a 0 , used LCF–HCF Interactions 161 100 1000 11010 2 10 3 Kth,LC Kth,SC KOL,LC KOL,SC Stress (MPa) Crack length, a (μm) σ e 0.81σ e 0.93σ e a 0 Figure 4.12. Kitagawa diagram with model for overload and short crack corrections. Data points correspond to fatigue limit stresses of LCF–HCF samples with and without stress relief. SRA samples should follow non-model curve, non-SRA should have overload effect. Dots show what fatigue limit stresses imply initial crack lengths might be. No such cracks were found. for the El Haddad short crack correction was determined from the intersection of the threshold curve with the endurance limit stress as a 0 =57m. The crack sizes corresponding to the threshold for crack propagation at a given FLS, due to an initial flaw which may have been produced during initial cycling, can be represented on a Kitagawa diagram for the specific geometry used here, a circular bar. For the specimens that were stress relieved to eliminate any prior load-history effects, this diagram shows that, with the small crack correction and the experimentally observed FLS of 081 e , a crack of length approximately 30 m in length would produce that threshold stress. On the other hand, specimens that saw no stress relief might exhibit an overload effect due to the initial cycling which occurred at a maximum stress level above the stress obtained during the HCF FLS tests. The magnitude of the overload effect on the threshold for subsequent crack propagation was estimated from a simple overload model developed in [6] that assumes the overload effect is caused by the prior loading history and depends on K max of the prior loading. The model prediction for the history dependent threshold is K maxth = K eff th 1−R th + 1−R th K maxpc (4.16) where = 0 294 and K eff th = 323, are fitting parameters taken from [6]. Subscripts “th” and “pc” refer to the threshold test and the precrack from the prior LCF loading, respectively. 162 Effects of Damage on HCF Properties The numerical results using the overload model for a stress of 925 MPa are plotted in Figure 4.12 along with the small-crack correction corresponding to a value of a 0 = 92 m , as determined from the model and the endurance limit stress. This model should correspond to the experimental observations where no stress relief was used, where the fatigue limit for prior loading history at 925 MPa for a range of cycle counts was approximately 093 e (see Table 4.1). For this value of stress, the overload model shows that a crack of approximately 20 m in length would be expected to produce the threshold stresses obtained experimentally. In the case of no stress relief, or with stress relief as indicated above, the initial cycling is expected to produce a crack of length 20–30 m in order to explain the observed reduction in FLS. In both cases, the size of any crack developed during LCF is near the limit of detection, using either SEM or heat tinting, but more importantly, does not have any serious detrimental effect on the subsequent FLS. In order to determine the expected flaw size that might occur due to prior cycling, the crack growth life was calculated as noted above. The crack growth life for any value of applied stress is subtracted from the total life to produce a curve corresponding to an initiation life to a crack of length a 0 = 57m. The results are plotted in Figure 4.13 in terms of percent life spent in nucleation, where it can be seen that the initiation of a crack to the size of a 0 should not be expected after the number of cycles imposed in the experiments as shown in Table 4.1. For example, at the highest stress of 925 MPa, the expected total life is approximately 10 5 cycles. At that stress, the percent of life spent in nucleation to a crack length of a 0 is expected to be about 50%. Thus, initiation corresponds to approximately 50,000 cycles, twice as much as the 25,000 imposed in the experiments. From this it is concluded that fatigue cycling for fractions of life below 50% should not produce a detrimental effect on the subsequent HCF endurance limit. 0 20 40 60 80 100 N f (cycles) Percent of life in nucleation 10 4 10 5 10 6 10 7 Figure 4.13. The percent life spent during nucleation as a function of the coupon fatigue test failure life. LCF–HCF Interactions 163 Since the cyclic life is a statistical variable, a factor of 2 in life should be within expected statistical scatter and the design life should be typically much less than the lives plotted from a limited number of experiments that determine average values. LCF–HCF interactions have also been studied by applying loading from other than smooth bars subjected to uniform stress. One example is where C-shaped specimens were precracked during fretting fatigue studies and were subsequently evaluated to determine the threshold for fatigue crack propagation [19]. Data for cracks in C-shaped specimens are plotted on a Kitagawa diagram as shown in Figure 4.14 for HCF limit stresses obtained at R =01. A similar plot for data at R =05 is shown in Figure 4.15. The small crack data for a<a 0 seem to be well represented by the El Haddad line and endurance limit stress, while the longer crack data are well represented by the long crack threshold fracture mechanics parameter, K = K lc th . A large number of data points for small cracks were obtained because, under fretting fatigue conditions, the contact stress field extends only tens of microns into the contacting bodies. Thus, cracks can initiate at the surface due to very high contact stresses there, but the cracks arrest quickly as they propagate into the fretting pad from which the C-shaped specimens were cut. Fretting fatigue is discussed in more detail in Chapter 6. Some of the specimens were stress relieved (denoted by Stress relief annealing [SRA] in the figures) to eliminate residual stresses and corresponding load-history effects from the fretting fatigue testing. Each of the figures presents three lines, one is the long crack solution, presented next, one is the short crack corrected solution (dashed line), and one is the FLS for this particular C-shaped geometry obtained experimentally for each value of R. The maximum threshold stress averaged 552 MPa for R = 01 and 684 MPa for R =05. 100 200 300 400 500 600 700 1 10 100 Experiment Experiment with SRA Prediction w/o correction Prediction with a 0 correction Maximum stress (MPa) Crack depth, a ( μm) 800 500 Fatigue limit R = 0.1 ΔK th = 4.56 MPa√m Figure 4.14. Kitagawa type diagram for HCF threshold stresses in C-specimens, R =01. 164 Effects of Damage on HCF Properties 100 200 300 400 500 600 700 1 10 100 Experiment Experiment with SRA Prediction w/o correction Prediction with a 0 correction Maximum stress (MPa) Crack depth, a (μm) 800 500 Fatigue limit R = 0.5 ΔK th = 3.46 MPa√m Figure 4.15. Kitagawa type diagram for HCF threshold stresses in C-specimens, R =05. The values of long crack K th used were 4.6 and 29MPa √ m for R = 01 and 0.5 respectively. The R = 0.1 data followed the predictions quite well. The R = 05 data were more scattered but also followed the same trend. A majority of the data fell into the long crack region where LEFM clearly predicted the threshold. Some of the specimens, however, had cracks on the order of a 0 and these data clearly deviated from LEFM and followed the trend of the small crack threshold model. It is also apparent, by observing the trend of the curves in Figures 4.14 and 4.15 that the small crack correction not only represents the trend of the experimental data, but shows that for crack lengths much below a 0 that the FLS is not appreciably below that of an uncracked body. Thus, if assessing the potential damage of a fretting induced crack, the FLS of a body with a “small crack,” below a 0 , is not much lower than that of an uncracked body even though the calculated value of K may be much reduced from that in a long crack experiment. For small cracks, therefore, stress rather than K should be considered as the governing parameter when assessing potential material degradation from fretting fatigue. Overall, the Kitagawa diagram, using the El Haddad transition modification, provides a reasonable representation of all of the experimental data including those in the small crack regime. A stress intensity factor K analysis of this specimen was required for the fatigue crack growth threshold analysis. The fretting cracks in the C-specimens were modeled as semi-elliptical surface cracks in a plate as drawn in Figure 4.16. The stress gradient of the uncracked specimen was used to calculate K for the cracked specimen by the well-known superposition method. Since this stress gradient was not linear, a generalized weight-function solution was used as presented in Shen and Glinka [20]. Equations (4.17) and (4.18) were used to evaluate K at the depth and surface points of the semi-elliptical crack. In these equations, K A and K B are the mode I stress intensity factors at the depth LCF–HCF Interactions 165 2W t a 2c x A B Figure 4.16. Surface crack in plate geometry used for K analysis in C-shaped specimens. and surface points respectively, x is the stress gradient along the crack depth, and m A and m B are the weight functions. The weight functions are defined in Equations (4.19) and (4.20). The six coefficients, M iA and M iB , were derived with the use of known K solutions for tension and bending. The coefficients depend on the crack size, crack shape, and plate geometry. K A = a 0 x m A x a a c dx (4.17) K B = a 0 x m B x a a c dx (4.18) m A x a = 2 2a −x 1+M 1A 1− x a 1/2 +M 2A 1− x a +M 3A 1− x a 3/2 (4.19) m B x a = 2 x 1+M 1B x a 1/2 +M 2B x a +M 3B x a 3/2 (4.20) In evaluating the results of this investigation, a result of particular interest is the fact that for very small cracks, the stress levels at the threshold are those at or slightly below the endurance limit stress for uncracked specimens. However, the location of failure is found to be at the pre-existing crack as observed from fracture surfaces that were heat tinted before threshold testing in order to provide the dimensions of the pre-existing crack. A legitimate question could be raised about why failure did not occur at a location other than the precrack since the applied stresses were approximately those corresponding to the endurance limit stress. In other words, if very small cracks do not degrade the stress carrying capability of the material, why do they provide the location for subsequent failure? Note that we are dealing with a fatigue limit or threshold condition, not a number of cycles to failure where number of cycles to initiation is an important part of the . loading and is referred to as an overload effect, where prior cracking at a high load level tends to retard crack extension at lower loads. To analyze the experimental data, a Kitagawa diagram. Forman and Shivakumar [17] stress intensity factor for a surface crack in a rod can be used for this analysis. This solution assumes that cracks have a circular arc crack front whose aspect ratio. or ratcheting may play an important role at high stress ratios where a portion of each cycle is spent at stresses near or above the static yield stress of the material. They observed measurable
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