46 Introduction and Background 1.0 0.8 0.6 0.4 0.2 –0.2 –0.4 –0.4 –0.2 0 0.2 0.4 0.6 0.8 1 Static breaking stress Maximum stress Minimum stress Zero stress Compression Tension Figure 2.22. Goodman diagram as shown in Goodman, 1899 [20] on p. 455. 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 σ 0 = 1 σ 0 = 0.75 σ 0 = 0.5 σ 0 = 0.25 Alternating stress/ultimate stress Mean stress/ultimate stress R = 0 Figure 2.23. L–W equation for R>0 plotted on a Haigh diagram. Characterizing Fatigue Limits 47 from the original Goodman equation at R =0 where  max =05 u . The curve is closer to a Gerber parabola than a straight line in that region but may not have any more physical significance than the other two representations of experimental data. It can be seen that the development of methods to represent fatigue limit data was based on a combination of mathematically simple forms as well as graphical representations. While there seemed to be a physical basis of some sort for all of the approaches, combined with simplicity, it should be pointed out that the choices of methods of representing data were scrutinized throughout their development. A particularly insightful example of such a critique was made in 1880 [26], although similar words would be equally valid today. Commenting on the W–L formula, used to fit Wöhler’s data, Smith writes, “In fact, the formula seems to be more the product of pure imagination than to be based on the experiments.” He further points out that “the formula can, of course, be made to agree with the extreme results of any particular set of experiments. What I, and I dare say some others, would like to know, is whether this formula agrees with even a rough degree of approximation with the intermediate results.” He goes on to question “whether the parabolic curve . is the correct curve ” and sums up: “I submit that the proper thing to do is not to coin rules out of imagination, but to persevere in careful and patient experiments, and to watch narrowly and to study and analyze closely the results until the true intimate laws of the stress and fatigue of metals are revealed.” 2.6. HAIGH DIAGRAM AT ELEVATED TEMPERATURE In the final program on HCF material behavior [27], a Haigh diagram was constructed for a single crystal alloy, PWA 1484 at a life of 10 7 cycles based on 1900  F S–N test data. The stress-life behavior was characterized for each stress ratio, R, in terms of the alternating stress  alt using a power law relationship: N f =k m alt (2.8) Data were used only from <001> oriented specimens with the exception of eight spec- imens oriented at < 001+15> that were tested at R = 08. The <001 +15> oriented data fall on top of the < 001> data and consequently were included in the data set to characterize the R = 08 stress-life behavior. Values for the constants k and m for each tested value of R are shown in Table 2.1. Using the empirical S–N relationships, the curve was extrapolated to longer lives and a Haigh Diagram was constructed corresponding to a life of 10 7 cycles, as shown in Figure 2.24. The Haigh diagram shows the HCF capability of PWA 1484 at 1900  F. The shape of the diagram is fairly conventional for low values of mean stress. A gradual decrease 48 Introduction and Background Table 2.1. Constants for S–N empirical fits at 1900  F Rmk −1 −12420E +27 −0333 −3019E+11 01 −5956E+14 05 −4910E+12 08 −7519E+11 0 10 20 30 40 50 01020304050 59 Hz HCF data Alternating stress (ksi) Mean stress (ksi) R = –1 R = –0.33 R = 0.8 R = 0.5 R = 0.1 46 Hr Rupture capability Figure 2.24. 1900  F Haigh diagram for PWA 1484, 10 7 cycles. in alternating stress capability is accompanied by an increase in allowable mean stress. However, above R =05, the alternating stress capability drops off rapidly as the stress rupture capability is approached. The stress rupture capability can be represented as an asymptote at R = 1 at 46 hours, the time for 10 7 cycles at 60 Hz (which was the original planned frequency). For this material, there are both cycle-dependent and time-dependent modes of failure, the former normally associated with the value of the alternating stress and the latter associated primarily with the mean stress. To account for this more complex fatigue behavior, a Walker model was used to represent the behavior of the material. In the Walker model, an equivalent alternating stress is defined to take into account different mean stress conditions. This equivalent alternating stress,  equivalent_alt , is then used in the stress-life power law relationship as shown below: N f =k equivalent_alt m (2.9) where the equivalent alternating stress is given by:  equivalent_alt = alt 1−R W −1 (2.10) Characterizing Fatigue Limits 49 The Walker exponent, w, was determined by taking data from several values of R and iterating until the standard error in predicted life was minimized. The Walker model collapsed the S–N response over a range of stress ratios. While the Walker exponent is normally determined over the full range of stress ratio data that are available, in this case R =−1toR = 08, the goal here was to capture only those specimens failing in a pure fatigue mode. Fractography showed that failure was dominated by fatigue only at low stress ratios or low mean stress loading conditions. Two approaches were considered for fitting the Walker model. The first (termed “Walker model A” in what follows) used 59 Hz HCF data at all R ≤ 01. A second approach (Walker model C) was examined because despite the fatigue-based appearance of specimens at R = 01 tested at 59 Hz, previous work showed that a time-dependent process is also present at this test condition [27]. At a constant stress level at R =01, fatigue life in cycles increased as the frequency was increased from 59 to 900 Hz. A linear line with a slope of 1:1 approximated the data fairly well indicating a fully time-dependent process up to the highest frequency that was tested, 900 Hz. A transition to time-independent behavior with cycles to failure maintaining a constant level may exist just beyond 900 Hz or could occur well beyond that frequency. As a result, the estimate of the Walker exponent for pure HCF may be affected by using the lower 59 Hz data. Therefore the second approach used only high frequency data (370–400 Hz) at R =01in combination with 59 Hz data at R =−1 and R =−0333 to represent time-independent behavior. Walker model constants for each subset of HCF data are shown in Table 2.2. In Figure 2.25, Walker model A approximates the 59 Hz HCF data fairly well up to a value of R = 01. Walker model C shows a benefit in alternating stress capability at R =−0333 and R = 01 compared to the other models. Both Walker models deviate from the 59 Hz Haigh diagram above R = 01 since the mean stress increases and time- dependent failure mechanisms reduce the cyclic capability of the material. To account for the time-dependent behavior of the material, two approaches were used in modeling the rupture behavior of PWA 1484 at 1900  F. The first approach, the simpler of the two, assumes that only the applied mean stress contributes to rupture damage. This approach is referred to as the Mean Stress Rupture Model. The second method, the Cumulative Rupture Model considers the summation of rupture damage from applied stress over the entire fatigue cycle. Table 2.2. Constants for 1900  F Walker models Walker model HCF Data subset Number of tests k m Walker exponent, w A R ≤−01 @ 59 Hz 24 5.83E16 −717 0165 C R ≤−0333 @ 59HzR= 01 @ 370–400 Hz 20 7.63E16 −698 03817 50 Introduction and Background 0 10 20 30 40 50 01020304050 59 Hz HCF data Walker model A Walker model C Alternating stress (ksi) Mean stress (ksi) Figure 2.25. Walker models A and C with 59 Hz 10 7 cycles Haigh diagram. The Mean Stress Rupture Model is represented by the expression in Equation (2.11) that relates mean stress to time to rupture. The expression was derived by fitting a power law relationship to four tests that were conducted until rupture. The resulting equation is t f =219×10 9  −5069 mean (2.11) where  mean is the applied mean stress in ksi and t f is the time to failure in hours. In the Cumulative Rupture Model, the rupture damage due to the applied stress is integrated over the cyclic load and the applied stress is expressed as a sinusoidal function using  as the period of the cycle in hours. To do this, the load cycle is divided into small time increments, t. At each time increment, the applied stress is calculated and the corresponding rupture life is determined using Equation (2.10). The rupture damage for the time increment is calculated and damage fractions for t are summed over the loading cycle. The number of cycles to failure can be calculated using the assumption that failure occurs when the rupture damage equals one. When R is less than zero, a portion of the loading cycle is compressive. Two scenarios were considered when applying the Cumulative Rupture Model: (a) compressive stress is neither damaging nor beneficial to life, and (b) compressive stress is damaging to life. Thus, in total, three rupture models were considered: Mean Stress Rupture Model, Cumu- lative Rupture without compressive damage, and Cumulative Rupture with compressive damage. Figure 2.26 shows the rupture model predictions for a constant life of 10 7 cycles at 59 Hz compared to the HCF test data. Both cumulative rupture models approximate the shape of the Haigh diagram based on the experimental data. The mean stress model predicts a mean stress of 32.5 ksi independent of R for 10 7 cycles at 59 Hz since the model is purely time-dependent and does not consider any cyclic contribution to the damage process. It is worth noting that the cyclic models, Figure 2.25, seem to do a good job Characterizing Fatigue Limits 51 0 10 20 30 40 50 0 1020304050 59 Hz HCF data Cumulative rupture, no compressive damage Cumulative rupture with compressive damage Mean stress rupture model Alternating stress (ksi) Mean stress (ksi) R = –1 R = –0.33 R = 0.8 R = 0.5 R = 0.1 Figure 2.26. 1900  F10 7 cycle Haigh diagram at 59 Hz with rupture model predictions. of fitting the data at low values of mean stress. They are derived, however, from fitting those very same data. The rupture model predictions, on the other hand, are derived strictly from rupture data with no cyclic content. The ability of the rupture models to represent data over the entire range of mean stresses, Figure 2.26, seems to indicate that the behavior of PWA 1484 at 1900  F is purely time-dependent, or, at least for modeling purposes, can be treated as such. The example cited above points out some of the considerations that are encountered when plotting data on a Haigh diagram and then trying to interpret the data or model the material fatigue limit when the material behavior includes time dependence. Factors such as cyclic frequency become important considerations and, at a minimum, should be indicated in Haigh diagram plots. 2.7. ROLE OF MEAN STRESS IN CONSTANT LIFE DIAGRAMS In dealing with data on FLSs, it is common to plot these stresses as a function of stress ratio (R = ratio of minimum to maximum stress), or, more commonly, of mean stress. The Haigh diagram, incorrectly referred to as a Goodman diagram, is a common method of representing the fatigue limit or endurance limit stress of a material in terms of alternating stress, defined as half of the vibratory stress amplitude. Thus, the maximum dynamic stress is the sum of the mean and alternating stresses. For many rotating components, the mean stress is known fairly accurately, but the alternating stress is less well defined because it depends on the vibratory characteristics of the component. Thus a Haigh diagram represents the allowable vibratory stress as the vertical axis as a function of mean or steady stress as the x axis. While attempts have been made to define the equation which best represents the data on a Haigh diagram, as described earlier in this chapter, 52 Introduction and Background variability from material to material, scatter in the data, and lack of sufficient data in many cases prevent the fitting of an equation to such data. When mean stress values are negative, or for values of R less than minus one, there are very few data and no general guidelines for extrapolating equations which were meant to represent data on a Haigh diagram only for positive values of mean stress. In cases such as contact fatigue, very high compressive stresses can be present, necessitating knowledge of fatigue behavior or fatigue limits for negative mean stresses. One of the most important areas where negative mean stresses can occur is in the case of the introduction of compressive residual stresses into a material or component. Shot peening, for example, is commonly used as a surface treatment to improve the fatigue properties of a material by introducing residual compressive stresses into the material up to depths typically no greater than 0.1 mm. While compressive stresses in the vicinity of the surface reduce the maximum stress from vibratory loading at the surface, they do not reduce the vibratory amplitude. Thus, in effect, they drive the mean stress lower, often into the compressive regime. While these residual compressive stresses are known to improve the fatigue characteristics in many materials and geometries, they are generally not taken into account in design and are used, instead, to improve the margin of safety. If such a condition is to be taken into account in design, a thorough understanding of material behavior and fatigue limits under negative mean stresses is required. The subject of residual stresses and accounting for them in design is discussed in Chapter 8. Forrest [15] has assembled a large body of fatigue limit strength data on ductile metals and plotted them in dimensionless form as indicated in Figure 2.27. The thick straight Mean stress /yield stress –1 Alternating stress/s –1 2.0 1.5 1.0 0.5 –1.0 –0.5 0 0.5 1.0 Figure 2.27. Schematic of observed behavior at negative mean stress. Characterizing Fatigue Limits 53 line represents the data (not shown) quite well and illustrates that extending the fit into the mean stress regime produces alternating stress values that continue to increase with decreasing mean stress. The data chosen by Forrest [15] for this type of plot were filtered from available data to meet a criterion to accept only those results where special precautions were taken to insure axiality of loading. The data covered a range of aluminum and steel alloys. As noted earlier, when discussing the effects of compressive residual stress on fatigue limit strength, Forrest [15], in 1962, already recognized the importance of the shape of the curve fit to the data by stating that the “behaviour is particularly significant with regard to the effect of residual stresses on fatigue strength.” Representing compressive mean stress data on Haigh or other types of diagrams described above was never much of a consideration because data obtained at mean stresses below zero, corresponding to fully reversed loading or R =−1, essentially did not exist. We now examine the capability of equations to represent negative mean stress data. In addition to the modified Goodman equation and the others described above, there have been many variations of straight lines and other curves to try to represent fatigue limit data for HCF design. Additionally, there are fatigue equations that are used mainly to fit data in the LCF regime, which try to account for the effects of mean stress or stress ratio by introducing a single parameter to consolidate such data. Two such equations are the one due to Smith, Watson, and Topper (SWT) [28] and the commonly used Walker equation. For the SWT equation, an effective stress is given in terms of maximum stress and strain range. In HCF, elastic behavior is assumed, thus strain range and stress range can be used interchangeably when dealing with FLS conditions. The SWT equation for elastic behavior is in the form  eff = −1 =   max  2  1/2 = max  a  1/2 (2.12) where  eff can be treated as a constant,  is the stress range,  =2 a , and  max is the maximum stress. The fully reversed R =−1 stress, both the maximum and alternating value, is denoted by  −1 . This equation is plotted in the form of a Haigh diagram in Figure 2.28 using a value of  eff =200 MPa which is representative of data on a forged titanium bar material [29]. The exponent in Equation (2.12) is taken as the one used most commonly, namely 0.5. For reference purposes, the Jasper equation, discussed later, is plotted because it is found to describe the shape of the Haigh diagram for positive mean stress quite well. Of greatest interest in Figure 2.28 is the shape of the curve for the SWT equation for negative mean stress, which shows an ever increasing alternating stress as mean stress becomes further negative. As an alternative, if we choose to take half the strain range (or half the stress range) as that corresponding to positive stresses only ( =+in the figure), this changes the curve only slightly in the negative mean stress regime. Note, finally, the symmetric shape of the Jasper equation about zero mean stress. This also will be discussed later. 54 Introduction and Background 0 200 400 600 800 1000 –400 –200 0 200 400 600 800 1000 SWT ² σ = + Jasper Alternating stress (MPa) Mean stress (MPa) Figure 2.28. Haigh diagram representation of SWT and Jasper equations. A similar treatment can be given to the Walker equation which, as for the SWT equation, is commonly used to consolidate LCF data obtained at different stress ratios, R. The equation is similar to the SWT equation, but adds a degree of flexibility through the exponent w. It has the form  eq =2 w  −1 = w  max  1−w (2.13) where w is the Walker exponent. With the exception of the value of the coefficient in Equation (2.13), it is identical in form to the SWT equation when w =05. Figure 2.29 is a plot of the Walker equation for various values of the exponent w, including extension of the equation to account for negative mean stress or values of R<−1. The curves are all forced to go through the same point at zero mean stress. While some data are handled by changing the value of w for negative values of R, it can be seen that the Walker equation has the same general characteristics as the SWT equation for negative mean stress, namely that alternating stress continues to increase as mean stress goes further negative. Further, for both equations, the shape of the curves for positive mean stress is concave up over the entire region. Several attempts were made in the early days of fatigue modeling to account for the observed behavior of the FLS when the mean stress was negative. In 1930, Haigh [30] pointed out that experimental data indicate that the constant life diagram is not symmetric Characterizing Fatigue Limits 55 0 200 400 600 800 1000 –400 –200 0 200 400 600 800 1000 w = 0.2 w = 0.3 w = 0.4 w = 0.5 w = 0.6 w = 0.7 Alternating stress (MPa) Mean stress (MPa) Constant life diagram Walker equation Figure 2.29. Haigh diagram representation of Walker equation. with respect to  m as required by the Gerber and generalized Goodman formulas. ∗ He suggested that the data can be represented by the generalized parabolic relation  a = −1  1−k 1   m  u  −k 2   m  u  2  (2.14) where the constants k 1 and k 2 are selected to give the best fit of the data. A plot of this equation is presented in Figure 2.30 for several combinations of k 1 and k 2 while constraining the curves to go through the ultimate stress point on the x axis and the alternating stress value at R =−1onthey axis. The case where k 1 =0k 2 =1 represents the Gerber parabola, Equation (2.5), symmetric about the y axis. When k 1 =1 and k 2 =0, the modified Goodman line is obtained, extrapolated for negative mean stress. More complex equations have been proposed, such as that by Heywood [31], who used an empirical cubic equation for representing constant life data. His equation has the form  a =  1−   m  u   −1 + u − −1   (2.15a) ∗ While the generalized Goodman equation [3] does not indicate symmetry with respect to  m , the formulation in the first Edition of Goodman’s book [20] presents equations for the static load capability in terms of the dynamic theory. If it is assumed that the strength is equal in tension and compression, the equations indicate that there is symmetry with respect to mean stress. The lack of data for negative mean stresses prevented any substantial debate on this issue of symmetry. The symmetry of the Goodman equation and its history are discussed in [18]. . data on a Haigh diagram, as described earlier in this chapter, 52 Introduction and Background variability from material to material, scatter in the data, and lack of sufficient data in many cases. fatigue of metals are revealed.” 2.6. HAIGH DIAGRAM AT ELEVATED TEMPERATURE In the final program on HCF material behavior [ 27] , a Haigh diagram was constructed for a single crystal alloy, PWA 1484 at. fatigue limit data was based on a combination of mathematically simple forms as well as graphical representations. While there seemed to be a physical basis of some sort for all of the approaches,