High Cycle Fatigue: A Mechanics of Materials Perspective part 46 pps

436 Applications surface treatment may be different. Specifically, in HCF, cracks may take a long time to initiate but, with the aid of compressive residual stresses, they may arrest. This may provide a measurable improvement in the endurance limit or the fatigue limit strength at a large cycle count. 8.6.1. Application to notches As noted above, the effect of residual compressive stress is to move the mean stress in the negative direction. For a smooth bar, this simply shifts the Haigh diagram, a plot of alternating stress against mean stress, to the left. In most cases, for a smooth bar, the compressive residual stress near the surface produces a tensile residual stress field in the interior, the magnitude of which is dependent on the compressive stress profile and the cross-sectional dimensions. In most cases, under tensile loading, fatigue failures will occur in the interior rather than at the surface. Under bending, however, failure can occur at the surface or interior to the surface depending on the geometry of the specimen (applied stress gradient) and the residual stress distribution profile. If there is a notch in a specimen or component, a residual compressive stress field will also reduce the local stresses in the vicinity of the notch. The residual stress field, produced by a surface enhancement procedure such as conventional shot peening (CSP), low plasticity burnishing (LPB), laser shock processing (LSP) or other, will depend on whether the surface treatment is applied before or after the notch is introduced. The notch depth as well as the depth of the residual compressive stresses must also be considered. For illustrative purposes, the assumption will be made in the following that the residual stress field extends beyond the notch depth so that the local notch stress field due to applied loading is reduced uniformly by the residual stresses that are assumed to be uniform over this depth. The net effect will be to shrink, rather than just shift, the Haigh diagram as illustrated in the following examples. The first case examined will be a modified Goodman equation that connects the fully reversed alternating stress with the UTS on the mean stress axis. For the examples presented here, the fully reversed alternating stress (corresponding to R =−1  −1 , is taken to be 500MPa while the ultimate strength is assumed to be 1000 MPa. This is somewhat representative of Ti-6Al-4V plate material where  ult =980 MPa. Figure 8.51 is a Haigh diagram where the modified Goodman equation (straight line between  −1 and  UTS  is arbitrarily extended into the compressive mean stress region. For stress concentration values of k t = 15, 2, and 3, the FLSs are reduced as shown in the figure and are still straight lines. If the reduction is due to the effect of a notch on the fatigue limit, then k f would replace k t in the diagram. Also shown in the diagram are the ultimate and yield stress lines which limit the smooth bar response in both tension and compression. Another equation that can be used to represent the fatigue limit is the Smith Watson Topper (SWT) equation that, if extended into the negative mean stress regime, has the HCF Design Considerations 437 0 200 400 600 800 1000 –500 0 500 1000 k t = 1 k t = 1.5 k t = 2 k t = 3 Alternating stress (MPa) Mean stress (MPa) Ti-6Al-4V plate UTS YS Figure 8.51. Haigh diagram showing allowable regions for several values of k t using modified Goodman equation extended into negative mean stress region. 0 200 400 600 800 1000 –500 0 500 1000 k t = 1 k t = 1.5 k t = 2 k t = 3 Alternating stress (MPa) Mean stress (MPa) Ti-6Al-4V plate UTS YS YS for k t = 3 Figure 8.52. Haigh diagram showing allowable regions for several values of k t using Smith–Watson–Topper (SWT) equation. shape shown in Figure 8.52 for purely elastic behavior which is generally representative of HCF. Prevey et al. [47] have used this formulation to define what they term “safe” regions. If the reduction for an equivalent k t is considered, then the curves are as shown in the figure. Since the local stress at the notch root is k t times the applied stress, the applied stress to produce yielding is also reduced by the value of k t , producing a reduced yield zone as shown in the figure for the case of k t =3. 438 Applications 0 50 100 150 200 250 300 –300 –200 –100 0 100 200 300 k t = 1 k t = 1.5 k t = 2 k t = 3 Alternating stress (MPa) Mean stress (MPa) YS for k t = 3 R = 0.1 R = 0.1 A B C Figure 8.53. Blowup of SWT equation plot showing safe region and example of benefit of compressive residual stress. Following the approach of Prevey et al. [47], a blowup of Figure 8.52 is shown in Figure 8.53 where the behavior for applied loading is indicated along the line denoted by R =01. Assuming the allowable load in an unnotched specimen is at point A, a notch with an equivalent k t (actually an equivalent k f ) reduces the allowable stresses to those indicated by point B in Figure 8.53. If the introduction of compressive residual stresses seeks to retain the allowable alternating stress to be that of the undamaged specimen without a notch, then the stress state would have to be along the k t = 3 line, namely at point C in the figure. To accomplish this, a residual compression of magnitude CA in the figure would have to be introduced. The net effect of the residual compression is to shift the Haigh diagram to the left by the amount of the residual compressive stress. Lines of constant R remain parallel to their original orientation, but are all shifted to the left. In the cited example, the point C is just inside the reduced compressive yield line, indicating that yield at the notch root would not yet have taken place in the specimen subjected to residual compressive stresses of the amount CA. It is of interest to note that the SWT equation reaches a limiting value in the compressive region as the value of k t is increased [47]. Figure 8.54 shows the Haigh diagram for values of k t = 3, 5, and 10, the latter representing the extreme value of a very sharp notch or a crack. While the line of constant life appears to asymptote to a fixed value, the compressive yield stress line, shown for the value of k t =3, would continually shrink toward the origin in the Haigh diagram. Other equations such as that due to Jasper can be used to represent the constant life curve in a Haigh diagram as shown in Figure 8.55. Here, the equation was modified to fit data in the compressive mean stress regime, and applying values of k t to that equation HCF Design Considerations 439 0 50 100 150 200 250 300 –300 –200 –100 0 100 200 300 k t = 3 k t = 5 k t = 10 Alternating stress (MPa) Mean stress (MPa) YS for k t = 3 Figure 8.54. SWT equation plot for high values of k t . 0 100 200 300 400 500 600 700 –500 0 500 1000 k t = 1 k t = 1.5 k t = 2 k t = 3 Alternating stress (MPa) Mean stress (MPa) Ti-6Al-4V plate Figure 8.55. Haigh diagram showing allowable regions for several values of k t using compression modified Jasper equation. produces the plot shown. If the curve for k t =3 is extrapolated beyond the range where it was fit to the data corresponding to k t = 1 (smooth bar), the curve starts to bend down as shown for the curve of k t = 3. There are no experimental data nor is there a theoretical basis for the downward trend in the curve. To extrapolate the curve for notches, a modified version of the modified Jasper equation is shown where the curve is extrapolated at a constant value of alternating stress into the negative mean stress regime. The extrapolation is for illustrative purposes only, and no data are available to 440 Applications 0 200 400 600 800 1000 –500 0 500 1000 k t = 1 k t = 1.5 k t = 2 k t = 3 Alternating stress (MPa) Mean stress (MPa) Ti-6Al-4V plate UTS YS Figure 8.56. Haigh diagram showing allowable regions for several values of k t using further modified version of compression modified Jasper equation. indicate that this might be the actual trend of data. Nonetheless, the extrapolated curves are shown in Figure 8.56 and provide the reader with an idea of how the allowable stress states at a notch shrink with increasing k t or k f . As in the example above for the SWT equation, the addition of compressive residual stresses would provide a region where the allowable alternating stresses for positive values of R could be expanded because of the higher allowables in the compressive mean stress region. The possible shrinking of the allowable alternating stress in the compressive mean stress region would also have to be taken into account. The examples provided above show that the allowable region for HCF design at a stress concentration, which is below the specific constant life curve chosen and below the yield surface, depends heavily on the curve chosen. This, in turn, depends on the shape of the curve, particularly in the negative mean stress region where data are generally not available. Further, if the curves for given values of k t or k f are shifted to the left due to the presence of compressive residual stresses, the resultant curves should be based on experimental values of the endurance limit. In most cases, these values would be based on tests using the appropriate geometry with a k t value tested in the negative mean stress region. Shrinking of smooth bar data by the appropriate value of k t will often not cover a sufficient range into the negative mean stress region if compressive residual stresses are added to the applied stress state. However, in the cases cited above, the extended range of stresses available in the negative mean stress region, compared to those in the positive mean stress region (values of R>−1), clearly shows the potential benefits available from the introduction of compressive residual stresses for either smooth bars or those containing a defect in the form of an equivalent k t or k f . HCF Design Considerations 441 0 100 200 300 400 500 600 700 –400 –200 0 200 400 600 Failure Run-out Stress amplitude (MPa) Mean stress (MPa) R = –2 R = –0.5 R = 0 R = 0.5 Figure 8.57. Fatigue limit stresses for maraging steel with drilled hole [48]. An example of the limited data available on notched fatigue limit behavior into the neg- ative mean stress regime is shown in Figure 8.57 [48]. Here, small drilled holes of depth and diameter of 100 m were used to produce stress concentrations in a maraging steel. Although the data cover only a limited range of mean stresses, the lowest corresponding to R =−2, there does not appear to be any significant trend of an increase in alternating stress with decrease in mean stress in the negative mean stress region. These and other similar data, combined with the general lack of data of this type, make it difficult to accurately predict, analytically, the potential beneficial effects of surface enhancement procedures that produce compressive residual stresses. For design, consideration must also be given to the possible variation of k f with k t for different values of stress ratio, R, since many materials do not follow a single formula like the Neuber equation [49] ∗ k f =1 + k t −1 1+  a m r (8.10) where the dependence is based solely on notch geometry and not on R. 8.6.2. Shot peening The subject of the effect of residual stresses on fatigue, including the effects on threshold which are important for application to HCF, has been studied for many years. A general ∗ In this equation for a notch, a m is an empirical material constant and r is the notch root radius (see Section 5.4.2). The value of k t can be determined from formulas developed for common notch geometries or from FEM analyses for more complex geometries. 442 Applications overview of the effects of residual stresses on fatigue, particularly in the use of conven- tional shot peening, can be found in the book by Almen and Black [50], for example. On the history of the process, they write: The discovery of the improvement in fatigue properties from shot peening stemmed from efforts in 1928 to 1929 by the Buick Motor Division of General Motors Corporation to remove spots of scale from valve springs by blasting them with grit. It was found that the fatigue properties of the springs were markedly improved by grit blasting. This provided the impetus for an extensive program which led to the development of modern shot-peening methods and applications. In the early days it was thought that the improvement in fatigue properties secured by shot peening was due to work hardening of the surface material, but it was soon learned that the major part of the benefits lay in the compressive stress induced at the surface of the shot-peened part. In the preface to their book, Almen and Black [50] point out the simplifying assumptions that must be considered in the application of surface treatments such as shot peening to the fatigue life or fatigue limit of a material. Specifically, they note that it is generally assumed that “practically all machine-part fractures occurring in normal service are fatigue fractures,” and “metal surfaces are weaker in fatigue than sound subsurface material.” The latter, in particular, is used and justified as a basis for developing methods for imparting surface residual compressive stresses in a material through methods such a shot peening. There has been much research conducted over many years on the effects of shot peening on the fatigue properties of materials. Only a limited number of those studies are cited here to illustrate some of the significant findings. In a study to optimize the parameters to improve the HCF strength of a titanium alloy, Hanyuda and Nakamura [51] evaluated the fatigue strength of titanium alloys as a function of the two dominating parameters, surface roughness and residual stress. Wagner et al. [52], in studying the effect of shot peening on the fatigue strength of Ti-6Al-4V, had found that an increase in the surface roughness promotes fatigue crack initiation. In [51], however, they found that surface residual stress has a dominating influence on fatigue strength in comparison with other factors such as surface roughness, depth of work hardening layer, and depth of the compressively stressed layer. The conclusions on the latter two factors were made solely in the context of the constraints of the shot-peening process. A more important finding in their work, one that reinforces a recurring theme in much of the research on the beneficial effects of compressive residual stresses, is that the resistance to crack propagation at the surface is more effective on fatigue strength as compared with the resistance to crack initiation or propagation below the surface at an early stage. Hasegawa et al. [53] found that in 0.5% carbon steel, the increase in fatigue strength at room temperature due to shot peening was due to both work hardening as well as compressive residual stress at the surface. The benefit, however, was minimal in the LCF regime and increased as the HCF regime is approached as seen in Figure 8.58. HCF Design Considerations 443 Annealed Shot peened 200 250 300 350 400 450 500 10 3 10 4 10 5 10 6 10 7 Stress amplitude ( MPa) Number of cycles to failure Room temperature Figure 8.58. S–N curves for annealed and shot-peened specimens of 0.5% carbon steel from rotating bending tests at room temperature (from [53]). When the same tests were conducted at elevated temperatures, however, the magnitude of the residual stresses was reduced which resulted in a decrease in the fatigue strength benefits. The shot-peened specimens still had higher fatigue strengths than the annealed specimens. This was attributed to the existence of work hardening which remained after cyclic loading at elevated temperature. Dorr and Wagner [54] studied the effects of shot peeening on the fatigue strength of Timetal (Ti-1100) in both fully lamellar and duplex microstructural conditions. Their work was aimed at evaluating optimum shot-peening conditions for fatigue life improve- ment in both smooth and notched components under fully reversed R =−1 conditions, using electro-polished (EP) specimens as the baseline. They noted that among the factors affecting the fatigue strength were local residual tensile stresses at a nucleation site (sub- surface), the mean stress sensitivity of the material due to the microstructural condition, and the sensitivity of the material/microstructural condition to environment for subsurface initiations. These factors, in combination, could increase or decrease the fatigue strength after application of shot peening. Results for two of the microstructural conditions tested under axial loading at R =−1 are illustrated in Figure 8.59 for (a) duplex and (b) lamellar microstructures. In these cases, particularly in (a), the benefits of the shot peening appear to be concentrated in the LCF regime with perhaps even a debit in the HCF regime. Based on the known increase in fatigue strength in vacuum versus that in air in titanium alloys, the lack of benefit or deterioration of fatigue strength in the HCF regime was attributed to the “marked tensile residual stresses at the crack nucleation site” in the shot-peened specimens. They summarize their findings by noting that the fatigue crack nucleated below the residual compressive stress field in the shot-peened specimens. This crack 444 Applications (a) 400 500 600 700 800 900 10 4 10 5 10 6 10 7 10 4 10 5 10 6 10 7 Stress amplitude (MPa)Stress amplitude (MPa) Cycles to failure Cycles to failure SP EP TIMETAL 1100-D20 axial loading (R = –1) (b) 400 500 600 700 800 900 SP EP TIMETAL 1100-LF axial loading (R = –1) Figure 8.59. S–N curves for Ti 1100 in (a) duplex and (b) lamellar microstructural condition under axial loading (from [54]). SP = shot peened, EP = electro polished. does not have to interact with the residual compressive stress field because it can first propagate rather easily deeper into the specimen interior. So, in addition to mechanical effects, environmental effects should be considered since subsurface cracks nucleate and propagate under vacuum conditions until they reach the surface. Further light in the study of subsurface initiation in shot-peened specimens was shed by Shao et al. [55] in a study on 300 M steel under rotating bending conditions. Their work extended a concept proposed earlier in [56] where the terminology for another HCF Design Considerations 445 kind of fatigue strength called “internal fatigue strength” was introduced. The concept was based on observations of internal initiations below the shot-peened layer where tensile residual stresses were present and there was no work-hardening effect due to the shot peening. Through calculations of the tensile residual stress field where the failure occurred, they found that in a 40 Cr steel tested under three-point bending at R =005, the internal fatigue strength corresponding to 5×10 6 cycle life was approximately 35% higher than the “surface fatigue strength” obtained on unpeened specimens under identical test conditions. There was no attempt to relate this increase to environmental effects. In the later work of Shao et al. [55], the concept of an internal fatigue limit was extended and verified with tests under rotating bending R =−1 on 300 M steel. Using calculated values of the tensile residual stress field at the failure location, the authors found that the ratio of the internal fatigue limit to the surface fatigue limit was about 1.39. The results were obtained for a fatigue life of 10 7 cycles using the up-and-down (staircase) test method (see Chapter 3). The beneficial effects of shot peening can be attributed, in some cases, to the retardation of crack initiation at the surface due to the compressive stress field that is developed there. In other cases, the retardation of crack growth is credited with fatigue improvements. In this latter case, the beneficial effect can be attributed to the residual stress field which, in turn, affects the closure behavior of the cracks. Song and Wen [57] studied fatigue crack growth in AISI 304 SS in C(T) specimens. The surfaces of the specimens were shot peened in various locations and closure measurements were made as the crack was grown. This produces a complicated 3-dimensional stress state where closure can be related to surface distortion. Under these complex conditions, they concluded that the location of the shot-peened region has an influence on the re-initiation life and growth-rate behavior. When a crack is embedded within a peened region, crack closure develops. This appears to be another way of saying that residual compressive stresses affect the crack growth rate or threshold through a change in the effective stress intensity as will be discussed later in an example related to laser shock processing. Drechsler et al. [58] conducted a study to compare fatigue improvements from shot peening to those of roller-burnishing where, in the latter, the depth of the affected region is greater. Using various titanium alloys under rotating bending fatigue, they attempted to sort out the effects of surface roughness and work-hardening characteristics of a material on fatigue property enhancement when surface treated. For the two processes used, the shot peening produced much higher surface roughness than roller-burnishing. With several different alloys, they covered a range of cyclic hardening/softening characteristics in the test materials. This would be less significant in the HCF regime than in LCF since cycling in the HCF regime is mainly elastic. The results from the surface-treated specimens showed slope changes in the S–N curves compared to baseline electro-polished specimens. Very marked improvements in fatigue performance were observed in Ti-2.5 Cu while the improvement in the S–N curve of Ti-10-2-3 was significantly less pronounced. . that “practically all machine -part fractures occurring in normal service are fatigue fractures,” and “metal surfaces are weaker in fatigue than sound subsurface material.” The latter, in particular,. the parameters to improve the HCF strength of a titanium alloy, Hanyuda and Nakamura [51] evaluated the fatigue strength of titanium alloys as a function of the two dominating parameters, surface roughness. Haigh diagram for values of k t = 3, 5, and 10, the latter representing the extreme value of a very sharp notch or a crack. While the line of constant life appears to asymptote to a fixed value, the