High Cycle Fatigue: A Mechanics of Materials Perspective part 42 pdf

396 Applications 0 100 200 300 400 500 0 200 400 600 800 1000 Defect Free 0.2 mm Defect 0.38 mm Defect K t = 1, a i = 110 μm K t = 3.83, a i = 13 μm K t = 4.86, a i = 13 μm σ a (MPa) σ m (MPa) R = 0.5 R = 0.1 Experimental Predicted Figure 8.17. Haigh diagram for smooth and notched specimen data compared with fracture mechanics-based computations of allowable stresses based on initial flaw size a i . 8.2.2. Application to LCF–HCF Looking at the same general problem of relating a Haigh diagram to a threshold plot, Nicholas and Zuiker [7] evaluated behavior under combined LCF–HCF loading. Using another set of data and addressing the problem from yet another perspective, an analysis was conducted to create a Haigh diagram corresponding to an initial crack size based on actual threshold data. Details of the analysis for general LCF–HCF loading conditions are presented in Appendix I. The specific problem addressed the initiation of a crack from LCF loading and the propensity for that crack to grow when subjected to HCF loading. No load history or interaction effects were assumed in this study. The analysis considered the initiation phase using the threshold stress intensity, K th and was handled with a Haigh diagram that was treated as representing a damaged material having an equivalent crack depth of 0.05 mm. This crack depth was chosen to correspond to that used on experimental data in the literature where this depth was taken to be the definition of initiation [8]. The Haigh diagram for initiation to a crack of depth of 0.05 mm in Ti-6Al-4V, obtained from data of Guedou and Rongvaux [8] and others, was represented as a straight line that went from  alt =240MPa at  mean =0to alt =0at mean =1200MPa, the ultimate strength of the material. For essentially the same alloy and heat treatment, threshold data were also taken from Hawkyard et al. [9] which show that K th decreases linearly from 5 MPa √ m at R =0to225 MPa √ matR = 07, and remains constant at that value for R>07. As a second assumption, the straight-line decrease of K th was continued beyond R =07 by Nicholas and Zuiker for comparison purposes. To compare initiation criteria from a Haigh diagram to those based on threshold fracture mechanics for this material, the alternating stress was plotted against mean stress for the HCF Design Considerations 397 two threshold stress intensity relations described above: one for K th constant for R ≥07, and one for K th linearly decreasing for all R. The calculations were carried out for three initial crack lengths, 0.05, 0.1, and 0.15 mm using, for simplicity, the approximate relation for an edge crack, K =  √ a, where a = a initial is the starting crack size. For each crack size, the threshold for crack propagation is plotted along with the Haigh diagram for initiation in Figures 8.18 and 8.19 for the two relations. Dashed lines in the figures represent constant values of R as indicated. The differences between the two sets of relationships for the threshold for crack propagation above R =07 were found to be relatively minor for these conditions. It is observed that the value for K th for high R 0 50 100 150 200 250 0 200 400 600 800 1000 1200 Goodman a initial = 0.05 mm a initial = 0.10 mm a initial = 0.15 mm Alternating stress (MPa) Mean stress (MPa) R = 0 R = 0.5 R = 0.7 R = 0.9 Figure 8.18. Damage tolerant Haigh diagram: K th constant for R>07. 0 50 100 150 200 250 0 200 400 600 800 1000 1200 Goodman a initial = 0.05 mm a initial = 0.10 mm a initial = 0.15 mm Alternating stress (MPa) Mean stress (MPa) R = 0 R = 0.5 R = 0.7 R = 0.9 Figure 8.19. Damage tolerant Haigh diagram: K th linearly decreasing. 398 Applications seems to have little effect on the damage tolerant Haigh diagram plot which shows the stress states at which an initial crack will start to propagate. It is of interest to compare the shape of the modified Goodman line (linear) with that predicted based on the assumption of an initial crack using fracture mechanics as shown in Figure 8.18. Note that these curves are essentially the same as those shown in Figure 8.11, where the assumption of a linear representation of threshold data as a function of R was used. The damage tolerant curves are seen to have a much steeper slope at low mean stresses. Taking, for example, the curve for a 0.05 mm crack (which is equivalent to the initiation criterion used to determine the modified Goodman line [8]), it can be seen that the damage tolerant curve is below the Goodman line at intermediate mean stress levels 0 <R<075. On the other hand, the curve is above the Goodman line at very high mean stresses and also at low mean stresses R < 0 if the curve is extended. When the damage tolerant curve is below the Goodman line, this implies that if a crack were present in the material, it would grow at stresses that would otherwise not cause crack initiation in 10 7 cycles. On the other hand, when the damage tolerant curve is above the Goodman line, this indicates that a crack could initiate but would not propagate. These two separate regions constitute two types of material behavior: damage intolerant when the damage tolerant curve is below, and damage tolerant when the damage tolerant curve is above the Goodman initiation line. While these specific curves are based on a number of assumptions and limited experimental observations, there appear to be trends that indicate the Haigh diagram, as constructed from smooth bar fatigue limit data, does not accurately characterize the damage tolerance of a material as a function of mean stress. 8.3. DAMAGE TOLERANCE FOR HCF The natural tendency in the implementation of a “damage tolerant” approach to fatigue would be to relate remaining life based on predictions of crack propagation rate to inspectable flaw size. In Chapter 1, the concepts involved in damage tolerance were discussed. As pointed out there, this has been shown to work well in LCF, and such an approach was adapted by the US Air Force in 1984 as part of the ENSIP Specification for critical engine structural components [10]. For HCF, direct application of such an approach cannot work for “pure” HCF because required inspection sizes are well below the state-of-the-art in non-destructive inspection (NDI) techniques and the number of cycles in HCF is extremely large because of the high frequencies involved. Thus, crack propagation times to failure could be extremely short, and the resultant inspection intervals would be too short to be practical. The basic problem is illustrated schematically in Figure 8.20 which shows that LCF involves early crack initiation and a long propagation life as a fraction of total life as HCF Design Considerations 399 Crack length % Fatigue life Inspection limit LCF HCF 0 100 Figure 8.20. Schematic showing conceptual differences between HCF and LCF. discussed in Chapter 1. There is no attempt in this schematic to represent the complex processes in the early stages of crack nucleation and initiation, nor to distinguish between fatigue of smooth bars versus notched samples. In general, LCF cracks are typically of an inspectable size early enough in total life so that there is a considerable fraction of life remaining during which an inspection can be made. HCF, on the other hand, requires a relatively large fraction of life for initiation to an inspectable size, or the creation of damage which can be detected, to occur. This results in a very small fraction of life remaining for propagation. It is therefore impractical to apply the damage tolerant approach as used for LCF to pure HCF. While considerable research is being conducted at the present time to identify and detect HCF damage in the early stages of total fatigue life, conventional damage tolerance seems impractical at present for HCF. However, the problems that arise in the field are generally not related to material capability under pure HCF. Rather, the problems fall into two main categories. First, and foremost, is the existence of vibratory stresses from unexpected drivers and structural responses which exceed the material capability as determined from laboratory specimen and sub-component tests. Design allowables are normally obtained on material which is representative of that used in service including all aspects of processing and surface treatment and are often represented as points on a Haigh or “Modified Goodman diagram.” (This point is qualified in the following paragraph.) The second category involves the introduction of damage into the material during production or during service usage. The three most common forms of damage, either alone or in combination, are LCF cracking, FOD, and contact or fretting fatigue. These damage mechanisms and their effect on material HCF capability were discussed in Chapter 4. As pointed out there, methods appear to be available to quantify the fatigue limit of a material subjected to HCF, and to establish a threshold for a crack of an inspectable size. 400 Applications However, the main issue in design for HCF is to quantify the severity of damage induced by LCF, FOD, or contact fatigue. This requires establishing what material allowables should be used to account for such damage. Other modes of service-induced damage, such as creep, thermo-mechanical fatigue, corrosion, erosion, and initial damage from manufacturing and machining, must also be taken into account in establishing material capability and inspection intervals. To account for various forms of damage, or to design for pure HCF, the concept of a threshold below which HCF will not occur is necessary because of the potentially large number of HCF cycles which can occur over short service intervals. This is due to the high frequency of many vibrational modes, often extending into the kHz regime. In fact, current design for HCF through the use of a Haigh diagram seeks to identify maximum allowable vibratory stresses so that HCF will not occur in a component during its lifetime. The current ENSIP specification requires this HCF limit to correspond to 10 9 cycles in non-ferrous metals, a number which is hard to achieve in service and even harder to reproduce in a laboratory setting. Consider that a material subjected to a frequency of 1 kHz requires nearly 300 hours to accumulate 10 9 cycles. Superimposed on these considerations is the probability of the occurrence of high-amplitude vibratory stresses combined with the simultaneous probability of the material capability being at the design minimum or design allowable level. 8.3.1. Material allowables The diagram most used for design purposes in HCF is a constant life diagram (see Chapter 2) as illustrated in Figure 8.21, where available data are plotted as alternating Mean stress Alternating stress R = –1 R = 0.1 R = 0.5 R = 0.8 Safe life region Average Lower bound Safety factor UTS Max vibratory allowable Figure 8.21. Constant life diagram. HCF Design Considerations 401 stress as a function of mean stress for a constant design life, usually 10 7 or higher. As explained in Chapter 2, in the absence of data at a number of values of mean stress, it is often constructed by connecting a straight line from the data point corresponding to fully reversed loading, R =−1, with the ultimate tensile strength (UTS) or yield strength of the material. Data at R =−1 can be obtained readily from a number of techniques using shaker tables to vibrate specimens or components about a zero mean stress, while data at other values of mean stress are often more difficult to obtain, particularly at high frequencies. Alternatives to the straight line approximation in Figure 8.21 involve various curves through the yield stress or UTS point on the x-axis, or through actual data if available, to represent the average behavior. Scatter in the data can be handled by statistical analysis which establishes a lower bound for the data as discussed in Chapter 3. On top of this, a factor of safety for vibratory stress can be included to account for the somewhat indeterminate nature of vibration amplitudes, particularly those of a transient type. Finally, design practices or specifications may limit the allowable vibratory stress to be below some established maximum value, independent of the magnitude of the mean stress. The safe life region, considering all of these factors, is shown as the shaded area in Figure 8.21. What the shaded region provides, therefore, is an allowable threshold vibratory stress as a function of mean stress, the latter being fairly well defined because it is closely related to the rotational speed of the engine. If the vibratory stress is maintained within the allowable region on the Haigh diagram, there should be no failure due to HCF and, further, no periodic inspection required for HCF. Provided that the maximum number of vibratory cycles experienced in service does not exceed the number for which the Haigh diagram is established, 10 9 for example, then such a design procedure is one of “infinite” life requiring no periodic inspection. The implication of this procedure is that cycle counting is not necessary because it is inherently assumed that an unlimited number of cycles will not produce failure if stresses are maintained below the HCF design limit. Further, it is assumed that there is no damage accumulation due to HCF if this design or endurance limit is not exceeded. As pointed out in Chapter 2, there are some pitfalls in the use of a Haigh diagram in design, particularly when basing it only on data at R =−1. For example, Figures 8.22 and 8.23 show such diagrams for the same material, Ti-6Al-4V, processed into two different product forms, hot rolled bar, and forged plate, respectively. In addition to the alternating stress, the peak or maximum stress is also shown. The data in Figure 8.23 are obtained from two independent sources (denoted as ML and ASE) on the same material, yet an unusual feature of the data is the relatively large amount of scatter which occurs under R =−1 (fully reversed, zero mean stress) loading. This phenomenon has been observed in several other alloys and is under further investigation. Note also that a straight line (see Figure 8.22) does not provide a good representation of the alternating stress data. Note, further, that for high values of mean stress, the maximum stress is quite high, approaching the static ultimate stress of 1030 and 980 MPa for the bar and plate material, respectively. 402 Applications 0 200 400 600 800 1000 1200 0 200 400 600 800 1000 1200 Alternating stress (MPa) Max stress (MPa) Mean stress (MPa) R = 0 R = 0.5 Ti-6Al-4V Bar 70 Hz Stress (MPa) Figure 8.22. Haigh diagram for Ti-6Al-4V bar. 0 200 400 600 800 1000 1200 –200 0 200 400 600 800 1000 Alt stress ML Max stress ML Alt stress ASE Max stress ASE Mean stress (MPa) 10 7 cycles 60–70 Hz Stress (MPa) Figure 8.23. Haigh diagram for Ti-6Al-4V plate. Research on fatigue life at high mean stresses in a titanium alloy [11] has shown that at high mean stress, the fracture mode changes from one of fatigue to one of creep. A plot of maximum and minimum stress, Figure 8.24, shows the range of vibratory stresses (min to max) at each mean stress tested. The stress above which creep occurs is shown along with the line denoted as “Cyclic creep limit” which delineates the region of fatigue, at low mean stresses, from the region of creep, at high mean stresses. Thus, in the creep regime, consideration has to be given to the amount of time during which such vibrations occur, not only to the number of cycles. Allowable vibratory stresses, while very low HCF Design Considerations 403 0 200 400 600 800 1000 1200 0 200 400 600 800 1000 1200 Max stress (MPa) Min stress (MPa) Stress (MPa) Mean stress (MPa) Ti-6Al-4V Bar 70 Hz Creep limit Cyclic creep limit Figure 8.24. Haigh diagram showing region of transition in mechanism for high mean stress. in this region, should also be supplemented with consideration of maximum stresses. It is for these reasons in some cases that designers shy away from the high mean stress regime, often for reasons that cannot be quantified. 8.4. THRESHOLD CONCEPT FOR HCF As explained earlier, because of the large number of HCF cycles that may occur in service due to high frequency vibrations in a component, a threshold concept is necessary to insure structural integrity. Whether this threshold involves crack growth rates below some operational threshold, or the use of a fatigue limit corresponding to a large but fixed number of cycles, the concepts involve “infinite life,” at least within the number of cycles that may be assumed to occur during the lifetime of a component. These could be either from steady-state vibrations or from transient phenomena and would correspond to a different number of total expected cycles in general. The engineering approximation to an “infinite life” concept is illustrated schematically in Figure 8.25 for crack growth and fatigue. In crack growth, a preexisting crack or one that develops during the service life of a component should not grow at a rate beyond some threshold rate so that crack extension can be neglected during the expected life of the material. Similarly, for a material without a crack, the stress level has to be maintained below a fatigue limit such that fatigue failure does not occur within an expected number of cycles. If the number of expected cycles can be established, and the crack growth rate and S–N curves can be estimated, then a threshold growth rate and a fatigue limit can be determined which are consistent with each other and account for “infinite” life for the expected number of 404 Applications S N da dN ΔK FCGR threshold or endurance limit Finite growth rate or fatigue life Figure 8.25. Schematic illustrating threshold concept in fatigue and crack growth. cycles. In establishing such limits for fatigue, the shape of the S–N curve has to be taken into consideration. As an example, Figure 8.26 shows S–N data for Ti-6Al-4V at several values of stress ratio, R. It is to be noted that the slopes of the curves are different, and the transition point where the curves become nearly flat differs depending on the value of R. Therefore, stress ratio becomes an important parameter in establishing the fatigue limit, just as is shown in the Haigh diagram where mean stress is used as the variable to establish allowable alternating stress. The concept of consolidating data from smooth bar fatigue limits and from threshold crack growth rates in cracked bodies was addressed in Section 4.1 through the use of a Kitagawa diagram. There it was pointed out that such a diagram is a valid method of representing data only for a specific value of R and only for one specimen geometry for a single plot. In this chapter we explore further the methods for obtaining data that can be used in a threshold design approach, particularly with respect to obtaining useful crack growth thresholds for use in engineering design. HCF Design Considerations 405 0 200 400 600 800 1000 10 4 10 5 10 6 10 7 R = –1 R = 0.1 R = 0.5 R = 0.8 Maximum stress (MPa) Number of cycles Ti-6Al-4V 60 Hz Figure 8.26. S–N data at different values of R. 8.4.1. Representing fatigue limit data In determining a threshold for HCF for different values of R, parameters are often introduced which attempt to consolidate data obtained at different values of R at different numbers of cycles of fatigue life. Such an approach was discussed in Chapter 3 using the RFL model with an effective stress parameter. Parameters such as the SWT, Walker, or Jasper equation, discussed in Chapter 2, are commonly applied to data such as those presented in Figure 8.26. That figure shows, however, that any attempt to consolidate data with a single function of R, for all values of cyclic lives, will never be able to bring all of those data into a single curve because of the varying shape of each curve at a different value of R. Certainly, the data can be brought closer together, but never with a very good fit. In evaluating models for threshold stresses, using a function of stress to consolidate data from different values of R, the accuracy of the consolidating parameter as well as the ability of the model to represent the consolidated data both have to be taken into account. In many, perhaps most cases, the amount of data available is insufficient to be able to distinguish between capability to consolidate data with a single function and the ability of the model to represent the entire database including the endurance limit. The concept of representing a body of data with a single function can be illustrated using the data of Figure 8.26. Assuming that the linear or bilinear fits to the individual data sets at different values of R represent a good approximation to the actual data, the fits can be consolidated using the SWT equation. The SWT model is a simplified version of the Walker equation, Equation (8.5), where m =0 5. Applying this model to the straight lines in Figure 8.26 results in the plot of Figure 8.27. There it can be seen that the lines are consolidated into a single band with the exception of the line representing the data at R = 08. To better represent the data, the Walker model, Equation (8.5), was used . trends that indicate the Haigh diagram, as constructed from smooth bar fatigue limit data, does not accurately characterize the damage tolerance of a material as a function of mean stress. 8.3. DAMAGE. crack of depth of 0.05 mm in Ti-6Al-4V, obtained from data of Guedou and Rongvaux [8] and others, was represented as a straight line that went from  alt =240MPa at  mean =0to alt =0at mean =1200MPa,. the initiation phase using the threshold stress intensity, K th and was handled with a Haigh diagram that was treated as representing a damaged material having an equivalent crack depth of 0.05