High Cycle Fatigue: A Mechanics of Materials Perspective part 40 pdf

376 Effects of Damage on HCF Properties 6. Haritos, G., Nicholas, T., and Lanning, D., “Notch Size Effects in HCF Behavior of Ti-6Al-4V”, Int. J. Fatigue, 21, 1999, pp. 643–652. 7. Nicholas, T., Barber, J.P., and Bertke, R.S., “Impact Damage on Titanium Leading Edges from Small Hard Objects”, Experimental Mechanics, 20, 1980, pp. 357–364. 8. Gallagher, J. et al., “Advanced High Cycle Fatigue (HCF) Life Assurance Methodologies”, Report # AFRL-ML-WP-TR-2005-4102, Air Force Research Laboratory, Wright-Patterson AFB, OH, July 2004. 9. Birkbeck, J.C., “Effects of FOD on the Fatigue Crack Initiation of Ballistically Impacted Ti- 6Al-4V Simulated Engine Blades”, Ph.D. Thesis, School of Engineering, University of Dayton, Dayton, OH, August 2002. 10. Peters, J.O. and Ritchie, R.O., “Influence of Foreign-Object Damage on Crack Initiation and Early Crack Growth During High-Cycle Fatigue of Ti-6Al-4V”, Eng. Fract. Mech., 67A, 2000, pp. 193–207. 11. Ruschau, J.J., Nicholas, T., and Thompson, S.R., “Influence of Foreign Object Damage (FOD) on Fatigue Life of Simulated Ti-6Al-4V Airfoils”, Int. Jour. Impact Engng, 25, 2001, pp. 233–250. 12. Ruschau, J.J., Thompson, S.R., and Nicholas, T., “High Cycle Fatigue Limit Stresses for Airfoils Subjected to Foreign Object Damage”, Int. J. Fatigue, 25, 2003, pp. 955–962. 13. Martinez, C.M., Birkbeck, J., Eylon, D., Nicholas, T., Thompson, S.R., Ruschau, J.J., and Porter, W.J., “Effects of Ballistic Impact Damage on Fatigue Crack Initiation in Ti-6Al-4V Simulated Engine Blades”, Mat. Sci. Eng., A325, 2002, pp. 465–477. 14. Thompson, S.R., Ruschau, J.J., and Nicholas, T., “Influence of Residual Stresses on High Cycle Fatigue Strength of Ti-6Al-4V Leading Edges Subjected to Foreign Object Damage”, Int. J. Fatigue, 23, 2001, pp. S405–S412. 15. Peterson, R.E., “Stress Concentration Factors”, John Wiley & Sons, Inc., New York, 1974, p. 22. 16. Peterson, R.E., “Notch-Sensitivity”, Metal Fatigue, G. Sines, and J.L. Waisman, eds, McGraw- Hill, New York, 1959, pp. 293–306. 17. Lanning, D.B., Nicholas, T., and Haritos, G.K., “On the Use of Critical Distance Theories for the Prediction of the High Cycle Fatigue Limit in Notched Ti-6Al-4V”, Int. J. Fatigue, 27, 2005, pp. 45–57. 18. Nicholas, T., Thompson, S.R., Porter, W.J., and Buchanan, D.J., “Comparison of Fatigue Limit Strength of Ti-6Al-4V in Tension and Torsion after Real and Simulated Foreign Object Damage”, Int. J. Fatigue, 27(10–12) (Special Issue on Fatigue Damage of Structural Materials), 2005, pp. 1637–1643 presented at International Conference on Fatigue Damage of Structural Materials V, Hyannis, MA, 19–24 September 2004 (to be published in Int. J. Fatigue). 19. Nalla, R.K., Altenberger, I., Noster, U., Liu, G.Y., Scholtes, B., and Ritchie, R.O., “On the Influence of Mechanical Surface Treatments – Deep Rolling and Laser Shock Peening – on the Fatigue Behavior of Ti-6Al-4V at Ambient and Elevated Temperatures”, Mat. Sci. Eng., A355, 2003, pp. 216–230. Part Three Applications In the previous chapters, some of the technical aspects of HCF have been addressed, many as a result of research efforts in the particular areas. Emphasis was placed mainly on the scientific basis and fundamental principles of observed phenomena related to HCF. In the following Chapter 8, an attempt is made to address some of the more practical aspects related to HCF design. Many of the issues are directly related to topics covered in the prior chapters, whereas some of the considerations are not previously addressed. Although the aspect of HCF design could be an entire book in itself, it is hoped that the following chapter provides guidance and stimulates consideration of the issues that a designer faces when dealing with HCF. Some research findings that are relevant to HCF design are discussed in detail here, particularly those dealing with fatigue-crack-growth-thresholds and some dealing with the beneficial effects of compressive residual stresses. This page intentionally left blank Chapter 8 HCF Design Considerations 8.1. FACTORS OF SAFETY Factors of safety are applied in structural design to account for both variability and uncertainty. In HCF, the variability in the fatigue strength of a material is accounted for by considering the statistical variation and employing a −2 or −3 (for a normal distribution) or some other factor to account for the lowest value of a material variable within some statistical limits. These allowable values can then be plotted on a Haigh diagram or incorporated into some other design tool. The ignorance associated with not knowing the value of the applied load, such as from stresses due to some vibratory condition, can also be accounted for by further reducing the allowable stresses due to expected values of the applied loads. For example, if the vibratory stresses are not known very well and the maximum vibratory stress is projected to be up to 50% higher than the calculated design value, then the material capability can be taken to be 1/1.5 or 0.67 of the original design allowable. For vibratory loading, one must be careful to distinguish whether the reduction is taken on the maximum stress or only on the vibratory portion of the cycle. On a Haigh diagram, the vibratory portion is clearly distinguished since it is the vertical axis, the alternating stress. One of the main reasons why the Haigh diagram has become a design tool for vibrating components or structures, particularly rotating machinery, is because the mean stresses are normally reasonably well known or easily determined, while the vibratory stresses are much less known and subject to larger uncertainties. For the particular example of a 50% error in vibratory stress, a Haigh diagram based on the straight line-modified Goodman equation is shown in Figure 8.1 for a titanium alloy, Ti-6Al-4V. The straight line representing the allowable alternating stress goes from a value of 500 MPa at R =−1 (fully reversed loading, zero mean stress) to the ultimate strength of the material, 980 MPa. The line is drawn for positive mean stress only in this discussion. Behavior at negative mean stress values and the associated uncertainty was discussed earlier in Chapter 2. In addition to the alternating stress, the maximum allowable stress is also shown as a function of mean stress on this Haigh diagram. These two lines, shown thick, represent the material capability. While the values shown are nominal, not statistically minimum values, these will be considered material allowables for the sake of illustration here. Two cases can be considered in reducing the material allowables to account for uncertainties in the applied loads. First, the alternating and maximum stress values are reduced to account for an uncertainty of 50% only in the alternating stress. The curves in Figure 8.1 representing this case for alternating and maximum stress are designated as “FS on alt” denoting that the factor of 379 380 Applications 0 200 400 600 800 1000 1200 –400 –200 0 200 400 600 800 1000 Alt stress, mat'l Max stress, mat'l Alt stress, Fs on alt Max stress, Fs on alt Alt stress, Fs on max Max stress, Fs on max σ alt (MPa) σ mean (MPa) Goodman equation Figure 8.1. Haigh diagram showing factor of safety (FS) on alternating or maximum stress for Goodman equation. safety (uncertainty) is applied only to the alternating stress. Second, similar reductions are shown to account for uncertainty in the maximum value of the loading and are denoted by “FS on max.” In this case, the uncertainty is on both the mean stress and the alternating stress. From Figure 8.1 it is seen that the reduction due to uncertainty in alternating stress only results in a proportional reduction of the alternating stress capability that goes to zero at the maximum value of mean stress on the plot. At the same time, the reduction in maximum stress is numerically the same as for the alternating stress. For this case, the effects at high mean stress which also correspond to high values of R is less than at lower mean stresses or lower values of R. If, on the other hand, the total stress is reduced due to uncertainty, then the reduction of either the alternating stress or the maximum stress is a constant value at all values of the mean stress as shown on the plot. Note that a fixed value of R is represented by a radial line from the origin on a Haigh diagram. The reduction is a proportion of the total along a radial line, but shows as a constant value when plotted here against mean stress. If something other than a straight line is used on a Haigh diagram, the results of reducing the material capability to account for uncertainty in the applied loading show up in a slightly different fashion. For illustrative purposes, the Haigh diagram described by the compression modified Jasper equation described in [1] (see Chapter 2) is used to represent the material capability including behavior under negative mean stress as shown in Figure 8.2. The heavy solid line is the conventional alternating stress while the heavy dashed line is the corresponding maximum stress. As in the previous case, material capability is decreased to account for uncertainty in either the alternating stress or the maximum stress. While the alternating stress reduction is a fraction of the material HCF Design Considerations 381 0 200 400 600 800 1000 1200 –400 –200 0 200 400 600 800 1000 Alt stress, mat'l Max stress, mat'l Alt stress, Fs on alt Max stress, Fs on alt Alt stress, Fs on max Max stress, Fs on max σ alt (MPa) σ mean (MPa) Jasper equation Figure 8.2. Haigh diagram showing factor of safety (FS) on alternating or maximum stress for Jasper equation. capability at all values of mean stress due to uncertainty in the applied alternating stress (“Alt stress, FS on alt”), the reduction in alternating stress capability is not proportional to mean stress when the maximum stress capability is reduced due to uncertainty in loading (“Alt stress, FS on max”). The reason for this is the non-linear nature of the curve representing material capability on the Haigh diagram. Similarly, the maximum stress capability reduction reflects the non-linear nature of the material capability curve. It can also be seen in this Haigh diagram, as well as in Figure 8.1 above for the Goodman equation, that when the maximum stress is reduced due to uncertainty in applied loading, the corresponding reductions in material capability cover a new (reduced) range of mean stress on the plot. 8.1.1. Modeling errors In establishing factors of safety, representing design allowables or just presenting data, the parameters used to represent the data can have an influence on the perceived error between experimental data and model representation. Most fatigue and FLS data are normally presented as stress. However, where other variables are involved, for example stress ratio, parameters are introduced which help to consolidate the data in terms of a single parameter such as effective stress, Smith–Watson–Topper (SWT) parameter, pseudostress, or some other quantity. In more complicated cases, such as for multiaxial loading or complicated stress states such as in the contact region for fretting fatigue (Chapter 6), the parameters used to represent data from a wide range of conditions can take on more complicated forms. Data obtained in terms of stress can then be converted into a parameter that does not necessarily have the units of stress. The units of the model parameter can then have an effect on the perceived or apparent error between the 382 Applications experimental data and the model. This, in turn, can have an effect on the true factor of safety or allowable stress. As a simple example, consider an S–N curve of a material which covers a range in cycle count from 10 4 to 10 9 . While real numbers are used here, they have no intended meaning other than for illustrative purposes. To represent the S–N curve, we use the following: P =3500N −05 +37 N 00007 (8.1) where P represents a parameter such as stress. If stress is the actual parameter, then the S–N curve is as shown in Figure 8.3 by the solid line. The constants in the equation and the curve are taken to represent a material that has an endurance limit for a very large number of cycles as shown in the diagram. If it is assumed that experimental data are obtained that have values of stress that are 25% different than the average curve, then a number of points at different values of N can be computed and are plotted for values of N covering the range of the curve plotted in Figure 8.3. These points represent either 125P or 075 P and show what a 25% spread in experimental data points would look like in a plot when the behavior is represented by Equation (8.1). If a reduction of 25% from the average were to be used as a design allowable, the lower set of points in Figure 8.3 would constitute the design allowable curve. If we now assume that the model does not use a parameter in terms of stress, but depends instead on the square root of stress, then the exact same data plotted as square root of stress are those shown in Figure 8.4. A quick glance at this plot and comparison with Figure 8.3 seems to indicate that the parameter does a better job of representing or correlating the data than the linear representation using stress. In a similar fashion, if the allowable stress were taken as 0.75 of the average model behavior based on a parameter 0 20 40 60 80 100 10 4 10 5 10 6 10 7 10 8 10 9 Model linear Experiment linear Model parameter N model 25% variation Figure 8.3. Representation of model using actual stress with 25% error. HCF Design Considerations 383 0 2 4 6 8 10 12 10 4 10 5 10 6 10 7 10 8 10 9 Model sqrt Experiment sqrt Model parameter N model 25% variation Figure 8.4. Representation of model using square root of stress with 25% error. that uses square root of stress, the allowable curve (not shown) would be lower (on the basis of percentage of the parameter) in Figure 8.4 than the data representing 0.75 times the actual stress. Following this same logic, the use of a parameter that involves the square of the stress would appear to produce much larger scatter as shown in Figure 8.5. Here, a design curve for 0.75 of the model behavior would be higher than the data with a true error of 25% in stress. This example illustrates that care has to be taken when dealing 0 2,000 4,000 6,000 8,000 10,000 10 4 10 5 10 6 10 7 10 8 10 9 Model square Experiment sqrt Model parameter N model 25% variation Figure 8.5. Representation of model using square of stress with 25% error. 384 Applications with parameter representations of real data in determining allowable stresses when the parameter is not in the units of stress. ∗ If data are represented by a model that consolidates data from, for example, different values of R, then the scatter in the data and the fidelity of the model both have to be taken into account when establishing a design allowable. This subject is discussed later in Section 8.4.1 dealing with “Representing fatigue limit data.” 8.1.2. Material variability Another concern in using the Haigh (Goodman) diagram is the statistical variability of fatigue data, particularly in the long-life regime where S–N curves tend to be nearly horizontal. A Haigh diagram could represent average data or a statistical minimum such as a −2 or −3 confidence limit (for a normal distribution). For the latter cases, a large amount of data are required, along with some physical or statistical evidence that the scatter follows any assumed distribution trend. While most of the data in many cases, particularly for endurance limits, are obtained at R =−1, representing fully reversed loading with no mean stress, the application of the Haigh diagram to HCF often involves conditions at high values of R such as 0.7–0.9. These conditions represent high mean stresses with small, superimposed vibratory stresses. There are often very few data available in this region, and the extrapolation of data from low values of R and the assumption that statistical distributions of data are the same at high R as they are at low R are questionable. It is not surprising, therefore, to note that (straight line plot) Modified Goodman Diagrams (MGD) were replaced in MIL-HDBK-5 with equivalent stress diagrams. MIL- HDBK-5 (a military design handbook) used the term “Modified Goodman Diagram” to be consistent with the older nomenclature that referred to any linear relation (on a Haigh diagram) as an MGD. The main reasons they were replaced were that the old diagrams showed no actual data, and were constructed from S–N curves that were extrapolated, often from limited data. Many times these original S–N curves covered only a small portion of the range of stress ratios and/or mean stresses represented on an MGD; thus extensive extrapolation was sometimes involved. Further, once the MGDs were constructed and placed in the Handbook it became impossible for anyone to assess the adequacy of the original data, the reasonableness of the extrapolated curve fits, or the extent of scatter/variability in the data. One classic case was Ti-6Al-4V mill annealed sheet, where it was found that the entire MGD was constructed based on 12 highly scattered data points, covering only a small range of maximum stresses and stress ratios. ∗ A similar situation arises in LCF where the number of cycles to failure is determined for a given applied stress. Use of a parameter based on the square root of stress, for example (Figure 8.4), makes the S–N curve flatter and hence, the uncertainty in life for a given stress, larger. On the other hand, the model gives the appearance of providing a better fit to actual data! HCF Design Considerations 385 A statistically significant diagram would be expected to have a minimum of 4–5 points for each of approximately 10 stress ratios. The current equivalent stress fatigue data presentations have their limitations too, but at least these limitations are evident and can be evaluated by any concerned user of MIL-HDBK-5. Material quality is another parameter that has to be considered when using the Haigh diagram in the design process. If the data are obtained in the laboratory on smooth spec- imens, then the heat treatment, processing, microstructure, surface finish, and specimen size must all be considered when applying the data to actual components. Perhaps the single most critical issue in the use of a Haigh diagram, however, is the degree of initial or service-induced damage which the material in a component may contain when such damage is not present in the material used for the database. If, for example, microcracks or other damage develops in service, then the Haigh diagram has no meaning for this material because it represents “good” or undamaged material. A design methodology which considers the development of damage from any source other than the constant amplitude HCF loading must be used to account for the different state of the material. For example, if cracks develop due to LCF, then the tolerance of the material might be evaluated using fracture mechanics and the concept of threshold stress intensity factor. The Haigh diagram, which does not represent fatigue life in the presence of other damage mechanisms, has no validity under this situation. The effects of damage on HCF strength were reviewed in Chapters 4–7 for several common types of damage. One method for developing a Haigh diagram which represents material in a real component is to develop the data on actual hardware. In this case, a combination of LCF and HCF loading, or general spectrum loading, could be used to obtain a single point on the diagram. Extrapolating this point to any other region on a Haigh plot based on straight line or other assumptions has little or no physical basis and meaning, and most probably will be misleading. It is also important to note that plotting a mix of fatigue life data from combinations of LCF and HCF loading on a single diagram is dangerous because the physical processes are different and extrapolations and interpolations are difficult. LCF usually involves high amplitude, low frequency loading which leads to crack initiation early in life; a considerable fraction of life is spent under crack propagation. It is this aspect that has allowed for the successful application of damage tolerance using fracture mechanics in design. HCF, on the other hand, generally involves low amplitude, high frequency loading, with a considerable portion of life spent in crack initiation. Interaction of the two types of loading, therefore, can involve the processes of crack initiation and growth under LCF, and subsequent propagation under HCF. The Haigh diagram which represents total life, most of which constitutes initiation under HCF, represents material behavior in the absence of initial cracks and, therefore, has no meaning under combined LCF–HCF loading. While the subject of LCF–HCF interactions was discussed extensively in Chapter 4, the same concepts have to be applied for any other type of damage that may affect the HCF behavior such as creep or corrosion or other damage mode. . represent a material that has an endurance limit for a very large number of cycles as shown in the diagram. If it is assumed that experimental data are obtained that have values of stress that are. HCF have been addressed, many as a result of research efforts in the particular areas. Emphasis was placed mainly on the scientific basis and fundamental principles of observed phenomena related. with “Representing fatigue limit data.” 8.1.2. Material variability Another concern in using the Haigh (Goodman) diagram is the statistical variability of fatigue data, particularly in the long-life