306 Effects of Damage on HCF Properties 20 μ μ m Figure 6.42. Cross-section of a fretting fatigue cracked specimen prior to fracture [47]. Contact surface 112× 20.0 kV 100μm AMRAY Loading direction 45° 0001 Figure 6.43. Cross-section of failed fretting fatigue sample showing crack orientation [52]. fatigue experiment. Their criterion for initiation of a crack is related to the direction of maximum shear stress range, but propagation is governed by the direction of the maximum tangential stress range around the crack tip. They propose a numerical integration scheme where the crack length is incremented and the orientation continually updated. In the numerical example presented, however, they find that “the fretting fatigue crack growth curve under mixed mode agrees well with the fatigue crack growth curve under mode I, except for the initial first few data points." 6.12. A COMBINED STRESS AND K APPROACH An example of an investigation where many of the features needed for an accurate life prediction scheme are incorporated is that of Golden and Grandt [22]. There, they combined an initiation model with fracture-mechanics-based crack growth modeling to try to predict experimental data from several fretting-fatigue experiments. The crack growth Fretting Fatigue 307 0 200 400 600 800 1000 10 3 10 4 10 5 10 6 10 7 Baseline T i on T i , R = 0.0 T i on T i , R = 0.5 Inco 718 on T i σ eq (MPa) N f Figure 6.44. Smooth bar and fretting test data adjusted for stressed surface area using equivalent stress [22]. portion accounted for the three-dimensional nature of the cracks and their subsequent growth, accounting for aspect ratios. The initiation modeling accounted for multiaxial and mean stress effects while making use of an equivalent stressed surface area as done in [33] (see also Appendix E). From their initiation modeling alone, Figure 6.44 shows how the use of an equivalent stress parameter that is adjusted to account for the gradient of stress using the Fs approach can consolidate data from a number of fretting-fatigue tests but does not match smooth bar fatigue data. The reason for this is the assumption that the initiation model represents total life. The authors then introduce the crack growth life prediction to subtract from the total life observed experimentally. The computations, based on a assumed crack depth of 25m, are shown to produce crack propagation lives that are relatively insensitive to the assumed initial crack size. Table 6.1, taken from [22], illustrates the insensitivity of crack propagation lives to initial crack size, a i , for four different fretting-fatigue experiments. Note for Experiment # 4, which comes closest to the HCF region, that an initial crack size as small as 64m produces a crack propagation Table 6.1. Crack propagation lives ∗ from four experiments for different values of a i (from [22]) Exp’t Initial crack size, a i m 6.4 12.7 25.4 50.8 101.6 1 51464136 31 2 188 166 146 125 97 3 199 178 158 135 104 4 677 641 595 511 329 ∗ Lives are given in thousands of cycles. 308 Effects of Damage on HCF Properties life that is not significantly different than that for an initial crack size of 50 m. In this study, the initial crack size for all of the computations was chosen to be equal to the El Haddad transition crack length, a 0 , which was approximately 25m. The predicted lives for several sets of experiments using either the initiation model alone or the combined initiation and propagation modeling are compared with experimental lives in Figure 6.45. The figure shows how the combined predictions match reasonably closely with the experimental results up to lives in excess of 10 6 cycles. These results can be seen better in Figure 6.46 where the predictions for total life are compared with 10 3 10 4 10 5 10 6 10 7 10 3 10 4 10 5 10 6 10 7 R = 0.0, N i R = 0.0, N i + N p R = 0.5, N i R = 0.5, N i + N p Predicted (cycles) Experiment (cycles) Figure 6.45. Comparison of predicted versus experimental fretting fatigue lives for initiation and total life calculations [22]. 0 50 100 150 200 10 4 10 5 10 6 10 7 Data Prediction Data Prediction Remote σ a (MPa) N f P = 1640 N/mm R σ = 0.0 P = 1170 N/mm R σ = 0.5 Figure 6.46. Experimental fretting fatigue lives and model predictions for two data sets plotted against remote applied stress [22]. Fretting Fatigue 309 experimental results for two sets of data obtained at different applied stress ratios. To display the data better, the plot uses applied (remote) stress. 6.13. COMPARISON OF FRETTING-FATIGUE FIXTURES In [53], three experimental fixtures were compared to assess their characteristics regarding the propensity for crack propagation after initiation. The first system, denoted “A,” has been used to test relatively thin (2–4 mm) flat specimens against flat pads with blending radii at the edges and is shown in Figure 6.47 [2]. Unlike classical fixtures, the entire load applied to the specimen is transferred to the fixture through the fretting pads since the portion of the specimen beyond the contact area is stress free. As a result, gross slip throughout each test is eliminated (for valid tests), thereby simplifying calculation of the shear stress on the contact. Thin specimens are used in order to keep the shear force, Q, low compared to the clamping force, P, while still maintaining high enough values of bulk stress to represent the conditions that occur in actual components. The second system, denoted “B,” uses a flat specimen with cylindrical or short flat pads with blending radii on both sides, as is shown in Figure 6.48. This fixture is typical of many reported in the literature and produces partial transfer of the load through the fretting pads to a nominally rigid-fixture device that controls the applied clamping load and pad alignment. The fixture compliance controls load transfer from the specimen to the pads. Gross slip occurs early in each test until surface interactions change local friction behavior, after which partial slip conditions prevail. A modified fixture where the slip distance is controlled by a independent loading device, thus controlling the shear force indirectly, has been developed [54]. The third system is a simplified laboratory dovetail fixture (Figure 6.49), denoted “C,” in which the shear and clamping loads are the reactive forces to the applied loading and, therefore, are both cyclic and in-phase with the applied load. Cylindrical or short flat P P S Q Q Figure 6.47. Schematic of full load transfer fixture “A”. 310 Effects of Damage on HCF Properties P P Q S Q Figure 6.48. Schematic of partial load transfer fixture “B”. S Figure 6.49. Schematic of laboratory dovetail fixture “C”. pads with blending radii may be used. As in fixture B, gross slip can occur early in the test until partial slip conditions prevail. The distributions of xx below the edge of contact at maximum load for experiments conducted on the three fixtures are shown in Figure 6.50 on a semi-log plot. Here, y is the coordinate into the specimen depth. For fixtures A and B, y is normal to the surface. For fixture C, y is inclined 15 away from the contact area, since this was the crack growth path. For the dovetail experiments (C1 and C2), the subsurface stresses decay the most rapidly and become compressive at a local minimum that occurs at ∼100m. The local minimums for fixtures A and B are not as well defined. They are tensile and occur closer to the surface, between ∼50 and 100m (A1, A2, B1). For the cylindrical pad (B2), the stress field does not decay as rapidly as in the other cases, and the local minimum is not present. The effective stress intensity factor range, K eff , for each experiment as well as the short crack (K thsc and long crack (K thlc thresholds for a given crack depth, a, are Fretting Fatigue 311 –500 0 500 1000 1500 2000 10 0 10 1 10 2 10 3 A1 A2 B1 B2 C1 C2 σ xx (MPa) Y Position (μm) Figure 6.50. Axial stress as a function of depth from surface for three fretting fixtures. 0 5 10 15 10 0 10 1 10 2 10 3 A1 A2 B1 B2 C1 C2 K th, LC K th, SC a (mm) K eff (MPa √m) Figure 6.51. Effective stress intensity factors for three fretting fatigue fixtures. shown in Figure 6.51. K eff for all three fixtures under nearly all tested conditions falls above K thsc for cracks below ∼100 m, suggesting that all cracks will propagate to some length, however, not always to failure. K eff for fixture A is shown to cross the short crack threshold near 10–20 m crack depths under some loading conditions. The dovetail fixture experiments, however, have a local minimum for K eff at approximately 100 m where long-crack fracture mechanics dominates, indicating a potential for these cracks to arrest. Fixtures A and B have K eff continuously increasing and without a local minimum. The most rapidly increasing K eff is shown for fixture B. K eff is similar in magnitude for fixtures A and B for very small flaw sizes, but the rate of increase of K eff for fixture A does not occur until after ∼100 m. Crack growth lives, therefore, would be expected to 312 Effects of Damage on HCF Properties be much longer for experiments using the A fixture than those for the B fixture. If the total lives are similar, then initiation would have taken place earlier in the A experiments than for those in B. The results for the dovetail fixture, on the other hand, show a distinct possibility that initiation could have taken place at stress levels below which were recorded for total failure. Some critical stress level had to be achieved, according to the K analysis, in order to prevent crack arrest because of the observed dip in the applied K. The difference in the initiation and threshold behavior of fixtures A and B compared to the dovetail fixture C can be primarily attributed to the differences in thickness. For experimental data generated corresponding to similar total lives in the range of 10 6 to 10 7 cycles, the stress intensity factor fields are somewhat different. Only the dovetail fixture appears to closely approach a condition where the effective threshold, corrected for small crack behavior, overcomes the applied stress intensity factor, thereby leading to crack arrest. In the other two fixtures, propagation appears to always be taking place for any crack length. The stress intensity factor fields are rather insensitive to the local contact stresses, which vary with . Based on the experimental conditions covered, the dovetail fixture is the only one where cracks always can initiate and arrest until the load for exceeding the crack growth threshold is exceeded. Thus, the load causing failure should relate to the threshold stress intensity factor and not to any initiation criterion. On the other hand, fixture B produces conditions where the load that produces failure should be related to a criterion for initiation since any crack that initiates will continue to propagate. In fixture A, the load for failure may be related to either an initiation criterion or a fracture mechanics threshold, depending on the specific conditions. In general, for any fixture, it would seem to be important to compute the fracture mechanics driving force, K eff , and compare it with the threshold stress intensity factor corrected for small crack behavior if the data are to be interpreted correctly. The data comparisons among the three fixtures demonstrate that a simple fracture mechanics model, involving exceeding a material threshold condition, is insufficient to explain all of the experimental results. Rather, both crack initiation and crack propagation appear to be necessary considerations for this total life range of 10 6 –10 7 cycles. 6.14. ROLE OF COEFFICIENT OF FRICTION Fretting-fatigue experiments normally result in the degradation of the mechanical proper- ties of material in a region of contact where cyclic tangential loads occur in the presence of normal or clamping loads. Such a process involves cyclic material damage accrual where many mechanical factors affect the degradation process. Included in these are the contact pressure, the slip amplitude at the interface, the COF and the magnitude of the tangential shear stress, and bulk stress at the contact interface. In fretting fatigue, as opposed to wear or galling, the contact region is one where the two bodies experience Fretting Fatigue 313 relative motion only near the edges of contact (“slip”) while the central region remains in full contact, normally referred to as “stick.” Of the many variables associated with the phenomenon [28], one parameter, the relative COF between the contacting bodies, has been shown numerically to have a significant influence on the magnitude of the stresses and relative slip displacements in the contact region [8]. In many modeling efforts it is assumed that fretting-fatigue life prediction criteria are predicated on contact stress state. For example, in a critical plane stress criterion, (see, for example, [21, 31, 55]), the crack is assumed to initiate along the plane with the maximum value of some stress-based parameter. Other criteria, for example Nicholas et al. [47] and Naboulsi and Mall [38], also require the determination of the contact stress field. Hence, considering various pad configurations and load conditions with nominally identical experimental fretting-fatigue life, in these cases, it is sufficient to compare their maximum stress. Also, the history dependence of fretting fatigue during cycling causes the stress state to undergo changes, for example Naboulsi and Mall [55]. Such a consideration is beyond the scope of what has been investigated. Experiments have shown that the magnitude of the average COF changes with number of cycles in fretting of titanium against titanium [56]. While no direct measurements have been made of COF in the slip region where relative motions are of the order of only tens of microns, several computational efforts have deduced values by matching results from models to experimentally observed phenomena. In fretting-fatigue experiments on a stainless steel material, elastic–plastic finite element modeling using a higher COF, =15 was found to better model the experimentally observed extent of local plasticity than using = 05 or 1.0 [57]. In another study, Goh et al. [58] found that in crystal plasticity calculations for Ti-6Al-4V fretting-fatigue experiment simulations, a value of =15 provided the best simulation of the experimentally observed dimensions of the slip regions of the contact. It is to be noted that these values of COF are considerably higher than average values generally measured in contact experiments and subsequently used in modeling of contact geometries. For example, Farris et al. [59] have reported a saturated value of =05 after a large number of fretting-fatigue cycles in the same titanium alloy. In other fretting-fatigue experiments, a saturated value of an average COF =05–07 was obtained using flat pads with a blend radius against 1mm thick specimens of identical Ti-6Al-4V [60]. The saturated value was reached after approximately 1000 cycles and an initial value of about =025. In another study, McVeigh et al. [56] analytically modeled the interfacial conditions in nominally flat contacts and obtained good correlation of computations with experimentally observed amounts of interfacial damage in Ti-6Al-4V specimens using an average value of 0.4 for the COF. In the same material, = 033 has been used in one investigation as an average value for computations of contact stress fields [61] while =03 has been used in another [8]. In Nicholas et al. [62], the K solution for a Mode I crack at the edge of the region of contact where stresses are maximum was obtained. Although the local stress fields were 314 Effects of Damage on HCF Properties different for two cases of fretting geometries and loads corresponding to the same life of 10 7 cycles, there was also no correlation of the K fields from the two cases compared. From this observation, the possibility that initiation took place at stress levels lower than the final one obtained in the step test procedure in two cases, and that the final stresses were based on a threshold for crack propagation, was eliminated. Subsequently, Naboulsi and Nicholas [48] investigated the possibility that the COF is not a constant in the region of slip but, rather, that f may depend on the relative slip amplitude. While the results of their numerical simulations indicated that a variable COF might reproduce the experimental conditions in [8] better than a constant COF, the findings pointed clearly toward the need to have a higher COF, perhaps greater than =10, in order for the stress fields from several different experimental conditions to approach each other. These stress fields which resulted in similar fretting-fatigue lives of 10 7 cycles were postulated to be similar since they produced similar fretting-fatigue lives. Only a high COF could produce such trends for the experimental conditions modeled. In their simulation of experiments with short and long fretting pads, both the static and the dynamic COF were assumed to have a range of values, but the static coefficient, s , was assumed to be larger than the dynamic coefficient, d . Based on the cases analyzed, the static coefficient of friction, s = 20, and the dynamic coefficient of friction, d = 10, produced the best match of stress state for the long and short pad configurations. In Hutson et al. [63], experimental observations in fretting-fatigue experiments similar to those in [8] and [47] indicated that for a low average clamping stress (200MPa), the propagation life should be short since no cracks were found at a life of 10 6 cycles which was 10% of the expected life under the imposed stress conditions. Calculations using =03 indicated that initial cracks with depths of 80 m or less would have infinite or very long propagation lives. Changing the coefficient of friction to = 10, however, produced the short propagation lives that were expected. This indirect evidence that the local coefficient of friction is higher than the average values measured experimentally is consistent with that of the computations of Naboulsi and Nicholas [48] described above. One of the problems in modeling fretting fatigue with a single value of the coefficient of friction, , is that is not constant throughout the experiment and may take many cycles to reach a stabilized value. If an average value of is obtained too early in the test, it may not adequately represent the phenomena and stress field calculations that represent what happens later in the test when initiation takes place. Friction experiments where the shear load is increased until the onset of total slip is reached are commonly used to obtain average values of the coefficient, ave = Q/P. Figure 6.52 shows a best fit to actual data for determining , the actual data are not shown. In this particular case involving titanium, over 10 4 cycles are required before equilibrium is achieved. Similar findings on the evolution of the coefficient of friction in fretting-fatigue exper- iments are reported by Hills et al. [20]. In work on an aluminum −4% Cu alloy, they Fretting Fatigue 315 0 0.1 0.2 0.3 0.4 0.5 0.6 0 5000 10,000 15,000 20,000 25,000 Average coefficient of friction Fretting cycles Ti-6Al-4V pads and specimens Figure 6.52. Evolution of coefficient with number of fretting cycles (from [33]). report the value of to go, in one case, from an initial value of 0.2 to 0.55 in only 20 cycles. In another case, at lower contact pressure, is observed to go from 0.2 to 0.75 in 150 cycles. Based on these examples, the number of cycles to reach a stable value of depends on both the materials and the contact conditions, both geometry and loads. If the two bodies in contact are subjected to fretting under partial slip conditions, the surfaces may undergo modification resulting in an increase in the COF, f [64]. Values of the average COF have been reported to increase over the first numbers of cycles as illustrated above. However, the relation between the average coefficient and the local coefficient is not known. The local coefficient may have a large effect of the local stresses in the contact region. Following this line of reasoning, Dini and Nowell [65] conducted an analysis of the contact region between a flat pad with a blending radius and a half plane to establish the relationship between the average (mean) COF, m , and the COF that exists within the slip zone after n cycles of loading, n . Their assumption is that the local relative displacements in the slip zone will result in a surface modification and will produce an increase in the local value of . The average value of is determined experimentally by increasing the shear load, Q (Figure 6.53) until the onset of slip. Using the nomenclature of Figure 6.53, calculations were carried out for geometries that ranged from Hertzian contact where the flat contact length a is zero to an almost sharp punch where a/b =099. For various combinations of normal to shear load ratios, the calculated values of m and n (denoted f m and f n are shown in Figure 6.54. Two important conclusions can be drawn from these results. The first is that the local value of is higher than the average value. The second is that the local value is very sensitive to the measured average value for the sharp punch case, and less sensitive for the Hertzian (rounded punch) case. However, the numerical results show that fairly large values of the local coefficient, n , up to 1.0 are predicted for some cases where the average coefficient, m , is in the range closer to 0.3–0.5 for nominal values of Q/P in the same range 0.3–0.5. . have reported a saturated value of =05 after a large number of fretting-fatigue cycles in the same titanium alloy. In other fretting-fatigue experiments, a saturated value of an average COF. was obtained using flat pads with a blend radius against 1mm thick specimens of identical Ti-6Al-4V [60]. The saturated value was reached after approximately 1000 cycles and an initial value of. observed amounts of interfacial damage in Ti-6Al-4V specimens using an average value of 0.4 for the COF. In the same material, = 0 33 has been used in one investigation as an average value for