High Cycle Fatigue: A Mechanics of Materials Perspective part 29 doc

266 Effects of Damage on HCF Properties the contacting surfaces. It was shown by Tomlinson et al. [4] as early as 1939 that if relative motion (slip) occurs, even at a level as small as 10 −6 in. (0025 m), fretting will result. One rather ironic example of a fretting-fatigue failure is that experienced while con- ducting fretting-fatigue experiments in the laboratory. Figure 6.6 is a schematic of the upper part of a dovetail fixture used to conduct fretting-fatigue experiments [5]. The fixture is free to rotate because it is held by a pin which, in turn, is held in a clevis which supports it as shown in the figure. Similar to the case of the riveted joint described above, small relative motion can occur between the pin and the fixture in the region shown as a thick line in the figure. After numerous tests to typical cycle counts of 10 6 or 10 7 per test, the very large number of cycles (approaching one billion) eventually produced failure of the grip as depicted in Figure 6.6. This ironic example is one where fretting-fatigue failure of the grip assembly halted the conduct of real fretting-fatigue experiments. Another scenario where fretting-fatigue failure can occur is at a bolted joint as depicted schematically in Figure 6.7. While such a failure mode is rare, it is not nonexistent. In a set of rotating components bolted together, excessive vibration of one of the components can lead to cyclic loads that produce a combination of normal, shear and transverse loads in the contact region shown as the thick line in the figure. These loads, combined with the initial static axial load in the bolts due to tightening, can lead to fretting-fatigue crack initiation in the bolt and potential failure of the bolt as depicted by a crack initiating in Figure 6.7. In this example, and in all of the prior examples, the conditions that produce fretting-fatigue cracks are assumed to be ones where partial slip occurs in the contact region. This is in contrast to the condition of total slip that is produced by higher loads and larger slip amplitudes. Although the latter is normally associated with LCF, and shows up as wear or galling, the one-to-one correlation between fretting fatigue and partial slip, and wear and galling to total slip, is only a generalization and not an all inclusive rule. Thus, while HCF is normally associated with fretting fatigue, there are questions regarding the P Figure 6.6. Schematic of fretting fatigue failure in pin-held fixture. Fretting Fatigue 267 Figure 6.7. Schematic of fretting fatigue in bolted joint. possible role of LCF in fretting fatigue and whether fretting fatigue is attributable to HCF exclusively. 6.3. REPRESENTING TOTAL CONTACT LOADS, Q AND P A common method for describing the general fretting conditions in an experiment or a component, particularly to distinguish between partial slip and total slip conditions, is to plot the total shear force, Q, as a function of the total normal force, P, in the contact region. Such a plot, particularly if it includes the first several load cycles as well as occasional cycles to trace any changes of the forces with cycles, will provide information on the evolution of the stick or slip nature of the region of contact. For most of the laboratory experiments used to study fretting fatigue, the loads P and Q are applied directly or are a direct consequence of applied bulk loads. In many of these experimental configurations, the clamping load, P, is maintained essentially constant. In such cases, the plot of Q versus P provides very little information unless the cycle goes through a stick and slip condition during the cycle. In such a case, the hysteresis loop of that plot provides evidence of this phenomenon. ∗ Real applications and simulated experiments involve geometries, where the Q versus P conditions are more complicated. Two specific cases are shown schematically in Figure 6.8, (a) a dovetail slot and (b) a simulated dovetail apparatus for laboratory testing. While the two geometries possess ∗ The hysteresis would also be evident in plots of Q against bulk load or Q against relative displacement. 268 Effects of Damage on HCF Properties F P Q Blade Disk (a) (b) P Q F Specimen Fixture Figure 6.8. Schematic of (a) dovetail slot in engine and (b) dovetail laboratory simulation experiment. many of the same features, the dovetail load F develops from centrifugal forces which also act on the disk while the laboratory apparatus only experiences the applied load F . As will be shown later, the reactive loads P and Q are different in the two cases shown. In the case of a dovetail slot in an engine, or in a simulated dovetail experiment, the Q versus P relation throughout the cyclic loading history provides much useful information and, in general, is difficult to obtain. The calculation of conditions for such a plot involves analysis of a configuration where the structure and loading are statically indeterminate, meaning that the elastic compliance of the system must be considered in the computations. The resultant Q and P values are then load-path dependent, especially when the loading conditions produce conditions of both total slip and partial slip (stick–slip) at the contact interface. Figure 6.9 shows a Q against P plot for any two bodies in contact. Two lines bound the region where all combined values of Q and P can take place. The one boundary is when the bodies slide with respect to each other in one direction while the other is when they slide in the other direction. All other points in the enclosed region represent partial slip conditions. The boundaries are given by Q =P and Q =−P (6.1) where  is the average COF. All sliding must take place along these lines. The lines are denoted as the “slide out line” and the “slide in line” as described by Gean [6] in characterizing the conditions in a dovetail slot in an engine where the blade can tend to slide out of the disk or back into it. In Figure 6.9, conditions are also shown for a common laboratory experiment conducted under constant clamping load. The clamping load is normally applied first, so line O–A represents the initial loading. Then, a typical experiment will involve partial slip conditions Fretting Fatigue 269 P Q 0 0 slide in line slide out line A B C D E Figure 6.9. Plot of Q against P for any two bodies in contact. Possible paths for simple laboratory experiment with constant P are also shown. as the specimen is subjected to cyclic shear loading between points B and C, for example. In some experiments, the initial shear load may produce slip, such as at point B. If the COF increases, then the slide out line will take on a higher slope and the remainder of the test will be conducted in partial slip. If the conditions are such that slip takes place at one point in the cycle for each cycle, such as in cycling through D–C, then a ratcheting condition will occur. If slip takes place at both maximum and minimum shear load, then the cycle would be represented by D–E. In either of the last two cases, the hysteretic nature of the cycle does not show as a hysteresis loop in a Q–P plot such as Figure 6.9. Rather, it has to be recognized that once a point in the cycle is reached that touches either of the slide lines in the figure that sliding will take place. The amount of sliding is not reflected in such a plot. A third dimension would have to be added to the diagram to account for the magnitude of the slip. A Q–P plot for an engine mission that shows the paths that Q and P can follow in a dovetail slot due to changes in engine rpm is shown in Figure 6.10. Starting at rest with zero-applied load, the first increase in engine rpm produces sliding to point A. A decrease in rpm from that point produces a further increase in crush load, P, even though the centrifugal load from the blade is decreasing [6]. The reason for this is the compliance of the disk changes with rpm (or centrifugal load) so that the net effect is to increase the load P with decrease in rpm. If the engine goes through small throttle excursions after the maximum rpm at point A, then the Q–P path will involve points somewhere along the line A–B with the direction of movement due to increase or decrease in rpm as shown in the figure. If the decrease in rpm is large, the path can go from maximum rpm at A to point B, where sliding-in starts to take place, to point C where minimum rpm is reached. An increase in rpm from that point will take the path from C to D and then along the line D–A. Complete shut down will follow the path from A to B to the origin. Thus, for small throttle excursions, the behavior will be somewhere along the line A–B while for 270 Effects of Damage on HCF Properties P Q 0 0 slide in line slide out line A B D C decr rpm incr rpm Figure 6.10. Q–P plot for an engine dovetail slot. large excursions, where low rpms are reached, the path can follow A–B–C–D–A. Note again that unless points A or B are reached during some throttle excursion, the dovetail slot remains in partial slip. The behavior in a dovetail slot, Figure 6.10, is different than that in a simulated dovetail fixture in a laboratory, described later in this section. In a laboratory experiment, centrifugal loading is absent so that the compliance of the fixture does not change due to the loading. The external loading is that applied by a specimen representing a blade. The Q–P path followed during a laboratory experiment on a simulated dovetail fixture is depicted in Figure 6.11 [7]. Here, the specimen slides along the fixture as load is increased until point A is reached. A decrease in load will then follow a path from A in the direction of B. Continual cycling with no sliding will then take place somewhere along the A–B line, never reaching A or B during the test. Increasing and decreasing P Q 0 0 slide in line slide out line A B D C decr load incr load Figure 6.11. Q-P plot for laboratory dovetail fixture. Fretting Fatigue 271 load will produce motion as indicated in the figure. If the load decrease is very large, however, sliding will take place at point B and continue along B–C until minimum load is reached at point C. When the load is increased, the lines C–D (partial slip) and D–A (total slip) will be followed until maximum load is reached at point A. In the dovetail experiment, a complete hystersis loop A–B–C–D–A can be followed for low values of load ratio, R, while for large R a path somewhere along the A–B line will be followed. In making comparisons, the three plots, Figures 6.9, 6.10, and 6.11, show the condi- tions involving only partial slip. The line followed on a Q–P plot for the constant load experiment, the engine dovetail slot, and the laboratory dovetail specimen has a slope that is zero, negative, or positive, respectively. In developing models for fretting fatigue based on stress and stress histories, the load and load histories play an important role and the differences between laboratory simulations and actual usage, as demonstrated above, should be taken into account. 6.4. LOAD AND STRESS DISTRIBUTIONS To derive criteria for fretting-fatigue failure, based solely on a mechanics analysis, requires determination of the stress–strain behavior in the contact area. A complete analysis of the fretting-fatigue process should include deriving criteria for both the initiation of cracks in the contact region, and criteria for the propagation or non-propagation of fretting fatigue–induced cracks. A major issue with such an approach, whether for fretting fatigue or fatigue in general, is the size of a crack associated with the concept of initiation. Criteria for crack propagation, particularly with the onset or threshold, are fundamentally dependent on the crack size chosen in any definition of initiation. It is not clear whether such a crack length is independent of load level, number of cycles, or other parameter such as stress ratio, R, in developing a model for fretting fatigue. Further, while HCF should normally be associated with a threshold condition, the behavior at stresses corresponding to a finite life are also of engineering importance and, more importantly, are easier to obtain in experiments than those corresponding to a fatigue limit corresponding to a very large number of cycles. Experiments from which fatigue models are developed for fretting fatigue can be clas- sified under two categories: those associated with a finite number of cycles to failure, and those associated with the determination of a FLS. The latter can be obtained correspond- ing to some cycle count, typically 10 7 or higher, and usually involve either step testing or staircase testing (see Chapter 3). For the first category of test types, typically LCF tests or tests at constant stress (or strain) corresponding to a finite life, data can be obtained on the stress conditions in the contact region corresponding to the conditions to develop a given initiation crack size. The total life of the test, obtained experimentally, consists of both the initiation and the propagation phase. For modeling purposes, it is important 272 Effects of Damage on HCF Properties to subtract out the propagation life. This can be done using an accurate representation of the da/dN – K behavior of the material at the appropriate value of R, the K solution for the crack geometry which emanates from the fretting region, and a suitable choice for an initial crack length corresponding to initiation. The criteria developed for initiation will depend on this choice of crack length. What cannot be determined from a test of this type is the threshold for crack extension under the type of stress fields typical of those in the fretting region. What typically happens is that a crack will initiate and then continue to propagate because the initiated crack will have a stress intensity range above the threshold value. Thus, the crack propagation calculation will cover a range of K values from above threshold to either a critical value or one corresponding to a reasonably large crack where further propagation life will be very small. The second type of test, corresponding to the determination of conditions producing a threshold for fretting-fatigue failure at a large number of cycles, usually involves some type of step testing. In this situation, the stresses associated with the equivalent of a fatigue limit provide information on the threshold for fatigue crack propagation. It is usually easy to produce a stress field which will initiate a crack because of the high intensity of local stresses in the contact region, but the K field has to be of sufficient magnitude to produce continued crack propagation. Tests of this type, corresponding to a fatigue limit test, can provide information on the threshold for crack propagation under fretting fatigue. One of the great misuses of this type of test is to assume that the stresses corresponding to failure are those that lead to crack initiation. It is possible that cracks formed at much lower stresses during the step testing and the final stresses are the ones that produce stress intensity values above the threshold. Examples of this condition are presented later in this section. Again, as in the previous scenario, the crack size corresponding to initiation has to be chosen. The reverse situation can also take place where the crack does not initiate until the maximum load is reached during a step test, and then the crack will continue to propagate because it is already above the threshold value of K. This appears to be the less likely of the two scenarios because of the very high stresses in the region of contact. Because of this, it is anticipated that there may be many case of fretting fatigue where tiny cracks are initiated, but then arrest. This can be a dilemma for those owners of systems who do not like to have cracked parts in service, where the cracks are below the inspectable level, and the analysis and testing indicates that the cracks will not propagate unless the stresses are higher than those that initiated the cracks. 6.5. EFFECTS OF LOCAL AND BULK STRESSES ON STRESS INTENSITY While it is generally thought that crack initiation is governed primarily by the local contact stresses, and propagation is more related to the far-field or bulk stresses, the local stresses can have a significant effect on the K field over much larger distances than where Fretting Fatigue 273 0 200 400 600 800 1000 1200 0 2 4 6 8 10 12 0 0.05 0.1 0.15 0.2 0.25 Stress K Axial stress (MPa) K (MPa √ m) Depth or crack length (mm) μ = 0.3 Short pad t = 1 mm Figure 6.12. Stress distribution and resulting K solution for short pad specimen case. the local stresses are high. An example of this is the stress and K fields for two fretting pad geometries in an experimental configuration where the bulk stresses are high on the trailing edge of contact and go to zero on the leading edge [8]. The numerical results for stress intensity, K, for a crack normal to the contact surface are shown in Figures 6.12 and 6.13 for two different pad lengths referred to as the “short” and “long pad cases,” respectively. The results are based on analysis at the deformed edge of contact. Also shown is the stress distribution,  x , at the same location into the depth of the specimen. The values of K were found to be relatively insensitive to the exact location in the x direction when going away from the edge of contact 10m in either direction [8]. For these particular cases, where the values of  x become constant at large distances from the surface, the values of K are seen to be larger for the short-pad configuration than for the 0 200 400 600 800 1000 1200 0 2 4 6 8 10 12 0 0.05 0.1 0.15 0.2 0.25 Stress K Axial stress (MPa) K (MPa√m) Depth or crack length (mm) μ = 0.3 Long pad t = 4 mm Figure 6.13. Stress distribution and resulting K solution for long pad specimen,  =03 case. 274 Effects of Damage on HCF Properties long pad, consistent with the magnitudes of the local stress fields as shown in the figures. At much larger crack lengths, the K fields appear to be converging. However, for short crack lengths, the higher local stresses which occur over the first 0.1 mm in the short pad case, and over a much shorter distance in the long pad case, produce much higher values of K for distances beyond those at which the stresses are high. The higher local stresses in the short pad case clearly produce higher values of K over a distance beyond 0.25 mm. This problem can be looked at in a more general manner. For a very simplified approach, one can treat the local contact stresses as a point load at the mouth of a crack, and treat the far-field or bulk stresses as a uniform stress field far from the crack as illustrated in Figure 6.14. The K solutions are written individually for the bulk load and the concentrated load [9]: K  I = √ aF 1 a/b (6.2) F 1 a/b =112 −0231a/b +1055a/b 2 −2172a/b 3 +3039a/b 4 (6.3) K P I = 2P √ a F 2 a/b (6.4) F 2 a/b = 352 1−a/b 3/2 − 435 1−a/b 1/2 +2131−a/b (6.5) where a is the crack length, b is the total width of the plate, P is the concentrated load representing the local contact stresses parallel to the edge of the plate, and  is the far-field uniform stress. Only Mode I stresses on a crack normal to the contact surface P a b P σ σ Figure 6.14. Schematic of equivalent stresses in contact problem. Fretting Fatigue 275 are considered here. ∗ Simple numerical examples are solved for the far-field stress, , assuming values of 10, 100, or 1000 and the concentrated load is assumed to be P =1. A value of  =10 represents a situation where the contact stresses are very large and the bulk stress is very small. The case where  = 1000 represents the other extreme where the bulk stress dominates and/or the local contact stresses are very small. The calculated values of K are normalized with respect to the bulk stress, so graphs are presented for a normalized stress intensity K/. Figure 6.15 compares the total K, the sum of K  s due to  and P, for the three cases. The solution is shown up to a value of a/b =02. It can be seen that the case for  =10 has very high values of K for small crack lengths, governed mainly by the local fretting-induced contact stresses. In this case, it would appear to be a situation where initiation could be easily followed by crack arrest since the driving force, K, decreases by such a large magnitude. Conversely, for  = 1000, the far-field stress dominates and any crack that initiates can be expected to continue to propagate because of the continually increasing value of K with increase in crack length. The intermediate case of  = 100 shows a small decrease in K at very small crack lengths followed by a continual increase. Here, there may be a slight chance for crack arrest when the combination of contact stresses and far-field stresses is just right. In Figure 6.15, this would occur at a crack length corresponding approximately to a/b =001. The individual contributions of  and P, the far-field, and local contact stresses, respectively, can be seen in Figures 6.16, 6.17, and 6.18, corresponding to  =10 100, 0 1 2 3 4 5 0 0.05 0.1 0.15 0.2 K σ + K P , σ = 10 K σ + K P , σ = 100 K σ + K P , σ = 1000 Stress intensity/σ a /b b = 1.0 P = 1.0 Figure 6.15. Total K solution for three different values of applied far field stress. ∗ In this simple numerical example, the thickness of the plate is taken as 1. For this reason, the equations do not appear to be dimensionally correct (the thickness does not appear in them explicitly). . Effects of Damage on HCF Properties to subtract out the propagation life. This can be done using an accurate representation of the da/dN – K behavior of the material at the appropriate value of R,. fretting fatigue or fatigue in general, is the size of a crack associated with the concept of initiation. Criteria for crack propagation, particularly with the onset or threshold, are fundamentally dependent. and larger slip amplitudes. Although the latter is normally associated with LCF, and shows up as wear or galling, the one-to-one correlation between fretting fatigue and partial slip, and wear