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626 Appendix I from Hawkyard et al. [12] and are assumed to decrease linearly from 5MPa √ mto 225 MPa √ matR=07 and maintain values of 5 and 225 MPa √ matR<0 and R>07, respectively. The stress intensity factor solution of Raju and Newman [13] for surface cracked smooth bars is utilized, and assumptions are made concerning the geometry of the crack such that the stress intensity factor may be approximated as K = a 163 1+ a D (I.5) where a is the crack depth and D is the diameter of the bar. Due to the presence of the notch, the applied stress is multiplied by a K T of 2 in order to determine K HCF , K LCF , a crit , a gHCF , and a gLCF . This is an appropriate assumption when the crack is very short. However, as the crack extends out of the notch, it has been shown that the crack growth rate will approach that for K T =1 and a crack depth equaling the total depth of the crack and notch [14]. These assumptions using K T are conservative and will cause the model to underestimate the crack propagation life. However, under the conditions of interest, the propagation life is a small percentage of the total life. NUMERICAL RESULTS The numerical algorithm has been implemented on a personal computer and operates in one of two modes. In the first mode, the user specifies a , m , and n, and the total life corresponding to 10 7 HCF cycles is calculated in terms of CCF load blocks, N. In the second mode, the user specifies m , N, and n, and an iterative algorithm is invoked to determine a such that the calculated value of N is within a specified tolerance of the requested value of N. Solution time increases with increasing N and increasing fraction of total life spent in crack propagation, especially when the crack is actively growing in both HCF and LCF cycles. For the most computationally intensive cases considered, the solution took no more than one to two minutes on a 100-MHz personal computer. Correlation with experiment A limited number of HCF–LCF tests on notched bars were conducted by Guedou and Rongvaux [9]. Ti-6Al-4V bars were cycled at room temperature with n =1800 HCF cycles per CCF load block and R HCF =085. The resulting initiation lives are shown along with those for LCF-only tests in Figure I.4. All results are plotted as a function of maximum stress ( m + a ). A power law fit to the LCF-only initiation data is also shown. This fit was used to define the S–N curve shown in Figure I.3 which, in turn, defines the constants for use in the modified Goodman equation over intermediate values of alternating stress range. Predictions for the HCF–LCF initiation life using Equation (I.2) are also shown Appendix I 627 10 2 10 3 10 4 600 700 800 900 1000 LCF only (Experimental) LCF only (Correlation) CCF (Experimental) CCF (Predicted with endurance limit) CCF (Predicted with no endurance limit) LCF cycles (or CCF load blocks) to initiation Maximum stress (MPa) Figure I.4. Comparison of predicted and measured LCF and CCF initiation life for Ti-6-4 [9] with R HCF =085 and n=1800. for both scenarios: end =300 MPa and end =0. As expected, the predictions are identical for cases in which HCF > 300MPa. At high values of maximum stress, both assumptions overestimate the detrimental effect of HCF cycles on the initiation life. At lower maximum stress levels, the best correlation is found when end =0. Figure I.5 shows a similar comparison of experimentally inferred and numerically predicted crack propagation lives for the LCF-only and HCF–LCF tests of Guedou and Rongvaux [9]. The numerical predictions underestimate the propagation lives of the LCF- only tests. This is expected, since K T at the crack tip has been taken as two over the entire life of the crack. As discussed earlier, this is a conservative assumption which is expected to underestimate the propagation life. Predicted values for the HCF–LCF tests are, of course, the same under either endurance limit assumption. The drastic change in slope is unrelated to that of the finite endurance limit assumption shown in Figure I.4. In Figure I.5, it is a function of K th . The number next to each predicted point corresponds to the fraction of propagation life over which the crack does not grow in HCF. (The crack grows in CCF during the remainder of the propagation life.) The four highest stress predictions begin growing in CCF immediately upon crack initiation (Path 3 in Figure I.2). In these cases, crack propagation life is underestimated by approximately an order of magnitude. At lower values of maximum stress (and constant R HCF ), K HCF is initially below K th at crack initiation and the crack must propagate for a period in LCF-only before CCF crack growth can occur (Path 6 in Figure I.2.). Better correlation with experiment is obtained at lower stresses, possibly indicating that the values of K th 628 Appendix I 10 1 10 2 10 3 600 700 800 900 1000 LCF only (Experimental) LCF only (Predicted) CCF (Experimental) CCF (Predicted) Propagation life (LCF cycles or CCF load blocks) Maximum stress (MPa) 0.00 0.00 0.00 0.00 0.57 0.88 0.92 0.93 Figure I.5. Comparison of predicted and measured LCF and CCF crack propagation life for Ti-6-4 [9] with R HCF =085 and n=1800. as a function of R used here are lower than those in the material tested. Guedou and Rongvaux [9] estimate K th =3MPa √ matR=085, whereas K th =225 at R=085 has been used here. Finally, Figure I.6 shows the correlation between the experimentally measured and predicted vales of total life. The total life is dominated by the initiation life. Thus, the assumption of no endurance limit gives better correlation with the total life. The assumptions concerning the K T at the crack tip during crack propagation have little effect on the correlation with total life. Unless otherwise noted, subsequent results are calculated under the assumption of no endurance limit. Effect of LCF on HCF capability Figure I.7 shows the computed values of allowable a as a function of m for failure in 10 7 HCF cycles (plus one LCF cycle representing the initial loading to a peak stress of a + m .) Line 1 indicates the alternating and mean stress combinations which will cause initiation in 10 7 HCF cycles with no superimposed LCF cycles N =1. If the number of cycles to initiation is increased from 10 7 to 10 8 , the line moves down as shown. Another curve (line 2) is drawn to indicate the stress states above which a crack will propagate under HCF based on an initiation crack size (50 m here) and the assumed value of K th as a function of R. At low values of mean stress, this line has a slope which increases significantly and follows a line along which a + m =constant. This value of maximum Appendix I 629 10 2 10 3 10 4 600 700 800 900 1000 LCF only (Experimental) LCF only (Predicted) CCF (Experimental) CCF (Predicted with no endurance limit) CCF (Predicted with endurance limit) Total Life (LCF cycles or CCF load blocks) Maximum stress (MPa) Figure I.6. Comparison of predicted and measured LCF and CCF total life for Ti-6-4 [9] with R HCF =085 and n=1800. 0 50 100 150 200 250 0 200 400 600 800 1000 1200 Alternating stress (MPa) Mean stress (MPa) LINE 3: N I,LCF = 1 LINE 1: N I,HCF = 10 7 LINE 2: ΔK th,HCF R = 0 R = 0.7 N I,HCF = 10 8 LINE 4: ΔK th,LCF Figure I.7. Haigh diagram as predicted by analysis showing underlying mechanisms which govern the shape of the solution curve. 630 Appendix I stress corresponds to K HCF =K th at the initiating crack length. Note that in this low mean stress region, corresponding to negative R, the crack can initiate in less than 10 7 HCF cycles at stress states above line 1 but will never propagate if the stress state is below line 2. Conversely, there is a region bounded by 200 < m < 600MPa where the crack will not initiate (below line 1), yet a 50-m crack could propagate (above line 2) based on the assumptions in this analysis. If the numerical assumptions are correct, this would imply that within this range of mean stresses, which is in a “safe” design space for initiation based on an initiation criterion represented in a Haigh diagram, the material is intolerant to small amounts of damage. For initial defects or service induced damage such as FOD equivalent to a crack of 50 m or greater, that damage would grow and eventually cause failure under HCF loading. This could occur, even though the stress states in this region indicate that cracks would not initiate under either LCF or HCF. Following the same series of assumptions, additional curves can be drawn for the boundaries below which LCF cycles will not cause initiation in a specified number, N , of cycles, or below which a 50-m crack will not propagate. The initiation curve for N =1 LCF cycle is shown as line 3 in Figure I.7 and represents a + m = ult . Any stress state above or to the right of this line will cause tensile failure on the first cycle. Finally, above and to the right of line 4, crack growth will occur under LCF loading once a crack initiates. Note that line 4 coincides with line 2 at low values of m , above which HCF crack growth will occur (for a 50-m crack). This is due to the assumption that the crack is closed at =0 so that for R HCF < 0, K LCF =K HCF . Under the assumptions of this analysis, the region where failure can occur due to either HCF or LCF is above the heavy line in Figure I.7. The safe design space defined for N =1 in Figure I.7 can be determined for other values of N. Figure I.8 shows solutions for various values of N and n where Nn=10 7 . As N increases, the allowable alternating stress decreases at higher values of mean stress. As expected, a finite number of LCF cycles reduces the maximum mean stress that may safely be applied to the structure. Comparison of Figures I.3 and I.8 indicates that the limiting value of mean stress for a given value of N corresponds to the allowable stress range for a specified value of N in Figure I.3. Parametric studies Inherent in the previous analyses were several assumptions concerning the behavior of the material. Among these were the variation in K th with R and the form of the initiation damage relationship. In each case, alternate assumptions can be used and lead to different results. Here, we consider the sensitivity of the results to such changes. First, consider the form of the K th versus R behavior. Experimentally measured values of K th as a function of R are shown in Figure I.9 [12] for Ti-6-4. Two fits to the experimental data are also shown. A piecewise linear fit is shown which can be Appendix I 631 0 50 100 150 200 250 0 200 400 600 800 1000 1200 N = 10 0 , n = 10 7 N = 10 2 , n = 10 5 N = 10 3 , n = 10 4 N = 10 4 , n = 10 3 N = 10 5 , n = 10 2 Alternating stress (MPa) Mean stress (MPa) R = 0 R = 0.5 R = 0.818 R = 0.667 N = 1 (Tensile overload) N = 1 (n = 10 7 ) Initiation line Figure I.8. Effect of number of superimposed LCF cycles on the HCF capability of Ti-6-4 as estimated by analysis. 0 1 2 3 4 5 6 0 0.2 0.4 0.6 0.8 1 Closure model CTOD model (Taylor, 1988) Experimental (Hawkyard et al., 1996) Δ K th (MPa √m) R Figure I.9. Comparison of CTOD-based and closure-based K th vs R models with experimental data on Ti-6-4 [12]. 632 Appendix I rationalized through closure arguments and is incorporated into the previous analyses. As an alternative, K th may be defined as K th =K tho 1−R 1+R 05 (I.6) which is based on cyclic crack-tip opening displacement (CTOD) arguments [15]. In each case, a least squares approach has been used to obtain an optimal fit to the experimental data. Although in this material, the closure-based approach clearly correlates better with experiment, data obtained on other materials have been shown to correlate well with the CTOD approach [15, 16]. Figure I.10 indicates how the different variations in K th with R affect the form of the solution. The dashed lines indicate the magnitude of alternating stress required to cause crack propagation in a 50 -m-deep crack using K th variation based on both closure and CTOD models. Differences in the solution for allowable a versus m are significant and are particularly noticeable at high R. Thus, the variation in K th at high R appears to be important in predicting HCF life as a function of mean stress. Figure I.11 repeats the results shown in Figure I.8 which show the allowable a versus m for various combinations of HCF and superimposed LCF. Here, however, predictions are made using both CTOD and closure-based K th R values. Note that while the differences between the methods are significant for small values of N at high mean stress, when N becomes sufficient to cause reductions in the allowable alternating stress (versus pure HCF), the results become relatively insensitive to the K th model in use. 0 50 100 150 0 200 400 600 800 1000 1200 Alternating stress (MPa) Mean stress (MPa) ΔK th by CTOD theory Initiation life = 10 7 ΔK th by closure theory (line 2 from Fig. 7) Solution (closure ΔK th ) Solution (CTOD ΔτK th ) N = 1 (Tensile overload) Figure I.10. Effect of K th vs R model on the HCF-only Haigh diagram as predicted by analysis. Appendix I 633 0 50 100 150 0 200 400 600 800 1000 1200 N = 1 N = 100 N = 1,000 N = 10,000 N = 1 N = 100 N = 1000 N = 10,000 Alternating stress (MPa) Mean stress (MPa) R = 0 R = 0.5 R = 0.905 R = 0.818 R = 0.967 ΔK th by closure theory ΔK th by CTOD theory Figure I.11. Effect of K th vs R model on the CCF Haigh diagram as predicted by analysis. 0 50 100 150 200 250 0 200 400 600 800 1000 1200 N = 1 N = 10,000 N = 1 N = 10,000 Alternating stress (MPa) Mean stress (MPa) R = 0 R = 0.5 R = 0.905 R = 0.818 R = 0.667 N = 1 (n = 10 7 ) Initiation line (No endurance limit ) 2 σ end = 0 MPa 2 σ end = 300 MPa N = 1 (n = 10 7 ) Initiation line (finite endurance limit ) Figure I.12. Effect of endurance limit assumption on the HCF-only and CCF (N = 10000) Haigh diagrams as predicted by analysis. In Figure I.12, Haigh diagrams are shown for two different initiation phase assump- tions: (a) the case of “no endurance limit” such that HCF cycles of infinitesimal stress amplitude cause finite damage (as incorporated in the previous analyses) and (b) the 634 Appendix I case of end =300MPa. Results are shown for N =1 and 10,000. The presence of an endurance limit effectively raises the initiation line (line 1 in Figure I.7). Note that the intersection of the solution for end = 300MPa and the R = 0 line corresponds to half the endurance limit stress range which again matches the 10 7 initiation life point in Figure I.3. CLOSURE Discussion The numerical results presented here are based on simple models of crack initiation and propagation. Many potentially important phenomena are neglected such as the possi- ble non-linear accumulation of initiation damage, the effect of previous cycling on the instantaneous endurance limit, acceleration in the HCF crack growth rate due to periodic underloads (LCF cycles), reduction in K th as a function of the number of LCF cycles, small crack effects, hold time effects on LCF cycles, and many more. Therefore, these results must be viewed as preliminary, giving only a qualitative indication of how LCF and HCF cycling interact to reduce overall life. Although simple, the method satisfacto- rily predicts the effects of combined HCF–LCF loading (see, e.g. Figure I.5) despite the limited amount of experimental data available for calibration. Additional experimental results should allow for better calibration and will pave the way for incorporation and assessment of many of the above phenomena. Although the Goodman assumption is used in accounting for mean stress effects in crack initiation, the resulting Haigh diagram obtained differs from the Goodman assumption for total life in two regions as shown, for example, in Figure I.7. At very low mean stress, the predicted response curve follows lines 2 and 4, and is significantly steeper than that displayed by the Haigh diagram. At high mean stress, the allowable alternating stress follows line 2 and remains constant up to very high mean stress. In both high and low mean stress cases, the differences between the predicted response and the expected Goodman-type response (i.e. a straight line) are due to regions in the Haigh diagram in which a crack is predicted to initiate but not grow. For the experimental data considered here, this phenomenon may be due to the small crack size considered (a i =50 m) which may exhibit increased crack growth rates due to small crack effects. Such effects are not considered in this analysis. If a longer initial crack size were considered, both lines 2 and 4 in Figure I.5 would move down and to the left resulting in a solution curve which looks more like the Haigh diagram. Note also that considering a K th versus R curve for which K th ⇒ 0asR ⇒ 1 also results in a solution curve which approaches the Goodman assumption as shown in Figure I.10. Interestingly, the solution curve shown in Figure I.7 is similar in form to the experimental results reported by Bell and Benham [17] for stainless steel sheet (see, also, [18]). In that work notched (K t = 244) Appendix I 635 and unnotched stainless steel sheets (18Cr-9Ni) were fatigued under loads leading to lives of 10 1 –10 7 cycles and over the range −10 <R≤ 091 at frequencies of 0.1 to 50 Hz. The resulting Haigh diagrams for the notched specimens exhibited a steep decline in allowable alternating stress as R increased from −1to033 followed by a region of relatively constant allowable alternating stress until the allowable stress began to quickly fall along a line approximating a + m = ult . In any case, the resulting Haigh diagram diverges significantly from the Goodman assumption, especially at high values of mean stress. It is interesting to note that this behavior of the model can qualitatively predict the exper- imental observations of Suhr [6]. In this work, a 12% CrNiMo blading alloy was cycled at a mean strain of 0.01 and variable alternating strain amplitude of 0–600 microstrain. Tests were conducted in HCF-only; HCF with superimposed periodic underloads to zero strain every 10 5 HCF cycles (combined HCF–LCF loading); and with a fixed number of LCF cycles preceding HCF-only cycling to failure (or runout). The results indicate that at high values of alternating strain, the number of cycles to failure is independent of whether LCF cycles are distributed throughout the HCF loading or all applied prior to HCF loading. As the alternating strain amplitude is reduced, a transition in behavior occurs such that specimens with all LCF cycling applied prior to HCF cycling exhibit significantly longer lives than specimens subjected to the same number of LCF cycles distributed periodically between blocks of HCF cycles. The explanation for this behavior is as follows. At high HCF , K HCF is above K th and once a crack initiates, it grows in both HCF and LCF. As all HCF and LCF cycles cause finite damage in both the crack initiation and propagation phases, there is little dependence on the order of the cycles. At lower HCF , when all LCF cycles are applied prior to HCF cycles, most or all LCF cycles act to initiate the crack. After the last LCF cycle has been completed, the crack is still sufficiently small such that K HCF <K th . Therefore, the crack will not grow and the specimen will exhibit very long life, as shown by the runouts in Suhr’s data. At lower HCF , when LCF cycles are distributed periodically between HCF cycles, HCF cycles play an active role in initiating the crack, and at the point of crack initiation there are sufficient LCF cycles remaining to grow the crack to a length sufficient for crack growth to occur in HCF cycles. Despite any shortcomings, the analysis provides insight into designing for HCF–LCF loading. It has been common practice to use a form of the Haigh diagram to design for allowable vibratory stress in the presence of a mean stress in metal components. For high frequency, low-amplitude fatigue, the crack propagation life is generally observed to be a small fraction of the total life. (In the numerical simulations presented above, crack propagation life was generally less than 1% of total life.) Thus, the N =1, n =10 7 initiation line is a good approximation to the Haigh diagram for the Ti-6Al-4V material under investigation. This is shown in Figure I.13 along with the numerical predictions from Figure I.8 for N =10 4 . The allowable mean stress at a =0 is, by definition, the [...]... T.W., and Allison, J.E., “Subsurface Crack Initiation in High Cycle Fatigue in Ti-6Al-4V and in a Typical Martensitic Stainless Steel”, Scripta Metallurgica, 17, 1983, pp 601–606 2 Nishida, S., Urashima, C., and Suzuki, H.G., “Fatigue Strength and Crack Initiation of Ti-6Al4V”, Fatigue 90, Materials and Component Engineering Publications Ltd, Birmingham UK, 1990, pp 197–202 3 Palmgren, A. , “Die Lebensdauer... us to understand the behavior of Kth at high R; understand the accumulation of initiation damage; understand and quantify the synergistic interactions in CCF such as reduction in Kth and crack growth acceleration; and confirm the findings presented here Many of these activities were conducted as part of the USAF initiative on HCF and are reported on elsewhere REFERENCES 1 Atrens, A. , Hoffelner, W.,... The data for N = 104 appears to follow a line of constant maximum stress as the allowable alternating stress increases That is, for a given value of N a + m= LCF (I.7) where LCF is the stress range causing failure in N cycles at R = 0 This same relationship was found to hold in both Ti-6Al-4V and Inconel 718 smooth bar specimens at low values of alternating stress [9] Thus, it is hypothesized that the... Stresses at High Frequency on Low Cycle Fatigue”, Low Cycle Fatigue, ASTM, Philadelphia, 1988, pp 938–969 10 Chesnutt, J.C., Thompson, A. W., and Williams, J.C., “Influence of Metallurgical Factors on the Fatigue Crack Growth Rate in Alpha-Beta Titanium Alloys”, AFML-TR-78-68, WrightPatterson AFB, OH, May 1978 (ADA063404) 11 Grover, H.J., Fatigue of Aircraft Structures, Government Printing Office, Washington,... Growth Rates, EMAS, Warley, UK, 1985 17 Bell, W.J and Benham, P.P., “The Effect of Mean Stress on Fatigue Strength of Plain and Notched Stainless Steel Sheet in the Range From 10 to 107 Cycles”, Symposium on Fatigue Tests of Aircraft Structures: Low -cycle, Full-scale, and Helicopters, ASTM STP 338, ASTM, Philadelphia, 1963, pp 25–46 18 Madayag, A. F., Metal Fatigue: Theory and Design, John Wiley and Sons,... Marsh, K.J., and Pook, L.P., Metal Fatigue, Clarendon Press, Oxford, 1974 638 Appendix I 8 Walker, K., “The Effect of Stress Ratio During Crack Propagation and Fatigue for 2024-T3 and 7075-T6 Aluminum”, Effects of Environment and Complex Load History for Fatigue Life, ASTM STP 462, American Society for Testing and Materials, Philadelphia, 1970, pp 1–14 9 Guedou, J.-Y and Rongvaux, J.-M., “Effect of. .. the safe design space for combined HCF–LCF loading is below the Goodman line, and to the left of the line of constant maximum stress corresponding to the appropriate number of LCF cycles For a significant number of LCF cycles, this removes a sizable region at high values of R from the safe design space Note that near the intersection of the Goodman line (line A) and the LCF line (line B), the safe design... W .A Andrews, and J.C Newman, eds, American Society for Testing and Materials, Philadelphia, 1986, pp 789–805 14 Dowling, N.E., “Notched Member Fatigue Life Predictions Combining Crack Initiation and Propagation”, Fatigue of Engineering Materials and Structures, 2, 1979, pp 129–138 15 Taylor, D., Fatigue Thresholds, Butterworths, London, 1989 16 Taylor, D., A Compendium of Fatigue Thresholds and Crack... much less curvature and the safe design space proposed here is more accurate That there is less curvature in the knee for the case of an endurance limit is easily explained For such a case, any stress point Appendix I 637 on the Haigh diagram below the Goodman initiation line (line A in Figure I.13), HCF cycles will cause no damage of any form Thus below this line initiation is brought about only through... exact form of the safe design space will require further refinement of the numerical model Conclusions Predictions have been made for the safe design space under combined HCF–LCF loading in terms of allowable values of mean and alternating stress using data from the literature on Ti-6Al-4V The numerical procedure provides satisfactory correlation with limited experimental data on HCF–LCF loading The . for allowable a versus m are significant and are particularly noticeable at high R. Thus, the variation in K th at high R appears to be important in predicting HCF life as a function of mean. diagram, the material is intolerant to small amounts of damage. For initial defects or service induced damage such as FOD equivalent to a crack of 50 m or greater, that damage would grow and eventually. can qualitatively predict the exper- imental observations of Suhr [6]. In this work, a 12% CrNiMo blading alloy was cycled at a mean strain of 0.01 and variable alternating strain amplitude of 0–600
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