High Cycle Fatigue: A Mechanics of Materials Perspective part 6 pdf

36 Introduction and Background extrapolating data to the required life, often 10 7 cycles. A schematic diagram of S–N data, obtained at different values of R = constant, is presented as Figure 2.11. Here, points A, B, and C represent the FLS either directly from the data or by extrapolation to 10 7 cycles in this case. In Figure 2.11, the stress is plotted as the maximum stress although the stress range or alternating stress could also be used. Each of the data points can then be used in a constant life plot such as a Haigh diagram as depicted in Figure 2.12. For each data point, the mean stress and alternating stress have to be computed for the given value of R. The Haigh diagram, Figure 2.12, represents the locus of points having a specified number of cycles to failure, for example 10 7 as illustrated in the example. The number of cycles is typically taken to be those corresponding to a “run-out” condition, perhaps 10 8 or even 10 9 , but there are few data available to demonstrate that a true run-out condition ever exists for a material. Rather, the cycles chosen can be either in a region where the S–N curves of Figure 2.11 become nearly flat with increasing number of cycles, or where the number exceeds the number of cycles that might be encountered in service. In some Maximum stress Cycles to failure (N ) R = 0.0 R = 0.7 R = –1 10 7 A B C Figure 2.11. Schematic of S–N curves obtained at constant values of R. Alternating stress R = 0 R = 0.7 R = –1 A B C N = 10 7 N = 10 4 Mean stress Figure 2.12. Schematic of Haigh diagram constructed from S–N data obtained at constant values of R and extrapolated to 10 7 cycles. Characterizing Fatigue Limits 37 cases, neither condition may be satisfied. The points A, B, and C from Figure 2.11 are shown in the Haigh diagram, Figure 2.12, corresponding to stress ratios, R, of 0.7, 0.0, and −1, respectively. In this plot, alternating stress (half of the stress range) is plotted against mean stress. Lines corresponding to these stress ratios are also shown, with the vertical axis being the R =−1 line for fully reversed cycling. Although the Haigh diagram is often presented or used with a straight line to represent data over a wide range of mean stresses, there is no physical mechanism which requires the diagram to be linear over any region. The straight line is often associated with the terminology “Goodman diagram,” as discussed later in this chapter. The diagram is a locus of points obtained from a series of experiments conducted under a range of values of R, and cases exist where it is highly nonlinear. The Haigh diagram, ∗ a constant life diagram, could also represent the locus of points at a lower life as depicted in Figure 2.12 for an arbitrary life of 10 4 cycles. This usage, however, is not very common for establishing LCF limits. The term “mean stress” is used in a Haigh diagram, although “average stress” or “steady stress” is also commonly used. The steady stress is the stress level upon which the vibratory stresses are superimposed, but the Haigh diagram represents only constant amplitude loading, and does not represent a condition where variable amplitude or LCF loading is superimposed. Further, the conventional Haigh diagram is valid only for the frequency at which the data were obtained, which is typically less than that for which it is applied in design. It is also not valid to use the Haigh diagram for random vibratory loading without the existence of a validated cumulative damage model for HCF. Although the majority of S–N data and curves are obtained under conditions at constant values of R, some experiments, particularly on rotating components, are conducted under conditions of constant mean stress. The difference between constant R and constant mean stress is illustrated schematically in Figure 2.13. Recall that R is the ratio of minimum to maximum stress. For constant mean stress cycles, each one will have a different value of R. Conversely, cycles at constant R will have different mean stresses for different values of stress range or maximum stress. Consider a material whose S–N behavior is described by an equation of the form logN = 0 + 1 logS − 2  (2.1) where  0 and  1 are fitting parameters, assumed for illustrative purposes, to be indepen- dent of R. Note that this is a deterministic model and  2 represents the endurance limit. ∗ The constant life or fatigue limit diagram that we refer to as the Haigh diagram, often incorrectly called the Goodman diagram, has been referred to in early literature as an R–M diagram [15] or, alternately, as a R/M diagram [16]. There, R refers to the range of stress (twice the alternating stress) while M refers to the mean stress. The obvious confusion with stress ratio, R, probably led to other nomenclatures such as the (incorrect) Goodman diagram, Haigh diagram, or alternating versus mean stress diagram. 38 Introduction and Background Time Stress Constant R Constant mean stress Figure 2.13. Schematic illustrating constant R and constant mean stress cycles. To be consistent with fatigue thresholds typically being linearly decreasing functions of R, we arbitrarily choose the following relationship to represent the endurance limit:  2 =150−100R (2.2) In this numerical simulation, S is taken as the stress amplitude in arbitrary units. Using constants  0 =10 and  1 =−2S–N curves are constructed as shown in Figure 2.14. If the same equation is used to construct S–N curves at constant values of mean stress, the resulting curves are as shown in Figure 2.15. The numerical values were taken only for values of R between 0 and 1. Thus the range of mean stresses where a complete S–N curve over the range of log N between 5 and 10 for this numerical exercise, is somewhat limited. For this particular illustration, the constant mean stress curves have the same general shape as the constant R curves but seem to be bunched together more. However, this aspect of presenting the curves is highly dependent on the shape of the S–N curves for the particular material. 0 100 200 300 400 500 5678910 R = 0 R = 0.5 R = 0.8 Stress amplitude Log N Figure 2.14. S–N curves for a hypothetical material at constant R. Characterizing Fatigue Limits 39 0 100 200 300 400 500 5678910 σ mean = 150 σ mean = 300 σ mean = 500 Stress amplitude Log N Figure 2.15. S–N curves at constant values of mean stress. An example of data obtained at several values of mean stress is presented in Figure 2.16 for a SKH51 tool steel [17]. While extrapolation and interpolation of data sets are often necessary, the data in Figure 2.16 illustrate that single parameters with which to consolidate data sets are not always viable because the individual groupings of data, at constant mean stress in this case, do not follow well-behaved trends. In this case, the slopes on the S–N curves do not change continuously with increase or decrease in mean stress. Further, the cycles to failure where the curves become horizontal, indicating a fatigue limit, also do not follow a consistent trend. Converting this data set to one where stress ratios, R, are constant, is not very practical if formulae exist for interpolating or extrapolating constant R data. This subject is discussed further in Chapter 8. 0 500 1000 1500 2000 10 3 10 4 10 5 10 6 10 7 10 8 Failure Run-out Stress amplitude (MPa) Cycles to failure σ m = –784 MPa σ m = +784 MPa σ m = 0 MPa Figure 2.16. S–N data on SKH51 tool steel obtained at constant mean stresses [17]. 40 Introduction and Background While most of the data in many cases 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. Material quality is another parameter which has to be considered when using the Haigh diagram in the design process. If the data are obtained in the laboratory on smooth specimens, heat treatment, processing, microstructure, surface finish, and specimen size must all be considered when applying the data to structural components. Perhaps, the single most critical issue in the use of a Haigh diagram is the degree of initial or service-induced damage which the material in the 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 High 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 other source 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 has no validity under this situation. It is these type of issues that are addressed in the remainder of this book. One method of developing a Haigh diagram which represents material in a 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. Connecting this point to any other point on a Haigh plot has no physical basis and no meaning, and most probably will be misleading. Unless the data are obtained under purely HCF loading, with no other contributing damage mechanisms, the Haigh diagram has no basis for use in design and the data would be better used in tabular fashion for specific design conditions. It is relevant to note that trying to mix LCF and HCF data on a single diagram is dangerous because the physical processes are different. LCF usually involves high amplitude, low frequency loading, which leads to crack initiation early in life and a considerable fraction of life 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 a process of crack initiation and growth under LCF, and subsequent propagation under HCF. The Haigh diagram only represents the HCF initiation portion in the absence of initial cracks and, therefore, has no meaning under combined LCF–HCF loading. This topic is discussed in detail later in Chapter 4. Characterizing Fatigue Limits 41 2.5. EQUATIONS FOR CONSTANT LIFE DIAGRAMS To illustrate the different methods of representing data graphically, we consider several models or equations that have been used traditionally for constant life diagrams [18]. The first and probably most popular is the Goodman line or equation, more commonly referred to as the straight line in the modified Goodman diagram. It was actually proposed by Haigh [19] in 1917 as a method for representing constant life, so the diagram that plots alternating stress as a function of mean stress is not really a Goodman diagram but a Haigh diagram and is thus referred to as such throughout this book. While some of the equations to be discussed later are used to represent FLS data in Haigh diagram plots, many have generally been established from empirical fits to data and have no scientific basis. However, some of the earliest works tried to establish physical principals for fatigue [18]. Goodman [20] proposed use of the “dynamic theory,” which is explained in detail by Fidler [21], where it is assumed that vibratory loads produce the same effect on a material as a suddenly applied load. Thus, vibratory loads are considered to instantaneously produce stresses that are double those as if they were applied slowly. The consequence of this is that the sum of the mean load and twice the alternating load should not exceed the static ultimate strength of a material in order to have infinite fatigue life. This results in a straight line on a Haigh diagram when the allowable alternating stress is taken equal to half the ultimate stress (see Figure 2.17). Of interest is that Goodman proposed such an equation as one that is “very easy of application and is, moreover, simple to remember” [20]. He also noted that “whether the assumptions of the theory are justifiable or not is quite an open question.” While the dynamic theory was simple, it was observed that experimental data on all materials did not agree with the theory. To better 0 0.25 0.5 0.75 1 0 0.25 0.5 0.75 1 Stress/ultimate stress Mean stress/ultimate stress R = 0 Ultimate stress σ vib = σ ult σ vib = σ ult /2 Goodman line Figure 2.17. Schematic of Goodman diagram showing allowable vibratory stress. 42 Introduction and Background address true material behavior, the fully reversed alternating stress was adopted as an empirical constant rather than half of the ultimate strength. What was originally termed the “modified Goodman diagram” involved using straight line fit on a Haigh diagram based on the experimentally determined fully reversed loading data point(s). The so-called Goodman equation (or Modified Goodman equation) represented in the following plots is of the form  a = −1  1−  m  u  (2.3) where subscripts a, m, and u refer to the alternating, mean, and ultimate stresses, respec- tively, and  −1 represents the experimental value of the alternating stress under fully reversed cyclic loading (R =−1). An alternate form of the Goodman equation is a straight line that intersects the mean stress axis at the yield stress,  y , as opposed to the ultimate stress. Such a representation of a fatigue limit constant life diagram was suggested by Soderberg [22] and the equation is named after him in the literature.  a = −1  1−  m  y  (2.4) To possibly better represent experimental data, a parabolic equation attributed to Gerber [23] has been used which is of the form  a = −1  1−   m  u  2  (2.5) Summarizing available fatigue limit data on a Haigh diagram, Forrest [15] observed from a large body of data on ductile metals that 90% lie above the Goodman line and two- thirds between the Goodman line and the Gerber parabola. He concluded that “although not 100% safe, the Goodman line can therefore be recommended as a useful working rule in design, when the fluctuating stress fatigue data are not available.” He cautioned, however, that these conclusions do not necessarily apply to parts containing stress raisers. There are a number of other representations of constant life data points [18], but one more that achieved some popularity in the 1870s is known as the Launhardt–Weyrauch (L–W) formula, developed by Launhardt [24] for positive values of R and extended by Weyrauch [25] for negative values of R. Using  u as the ultimate stress, the equations are  max = 0 + u − 0 R R > 0 (2.6)  max = 0 + 0 − −1 R R < 0 (2.7) where subscript max refers to the maximum stress and  0 and  −1 refer to the maximum stress at R =0 and R =−1, respectively. For the three formulas, the Modified Goodman Characterizing Fatigue Limits 43 equation, the Gerber equation, and the L–W formulas, experimental data of Wöhler on Krupp steel were used to establish a best-fit curve. The first method of representing data is the diagram where maximum stress is plotted against minimum stress, as shown in Figure 2.18. It can be seen that the Launhardt– Weyrauch (L–W) formula and the Gerber parabola are almost identical in their repre- sentation of the data (not shown). The straight line attributed to Goodman does a lesser job of fitting data. The line representing minimum stress was often drawn to indicate the lower stress boundary for admissible stress. The distance between the minimum and maximum curves is then the allowable vibratory stress. In this and subsequent plots, the curves are shown in terms of normalized stresses with respect to the ultimate stress. It was Goodman [20] who stated that fatigue behavior should depend on the ultimate stress, not the elastic limit. Another way of representing constant life data is a plot of maximum and minimum stress against stress ratio, as shown in Figure 2.19. Note that the Gerber and L–W equations look almost identical again while the Goodman line is somewhat different. In this particular plot, the linearity of the Goodman line disappears because of the particular axes being used. This may be one reason why this type of plot has never retained any popularity in the fatigue community. A third type of plot, where maximum and minimum stresses are plotted against mean stress, is shown in Figure 2.20. This plot probably contains the most information of use to a designer of rotating machinery because mean stresses are not only well determined from calculations based on centrifugal loads, but can also be well controlled in usage. Alternating stresses that result from the vibration characteristics of the machinery are much less controllable and not as well known in most cases. At high mean stresses (high 0 0.2 0.4 0.6 0.8 1 –0.5 –0.25 0 0.25 0.5 0.75 1 Goodman Gerber L–W Maximum stress/UTS Minimum stress/UTS Minimum Figure 2.18. Maximum and minimum stress as a function of minimum stress. 44 Introduction and Background –0.5 0 0.5 1 –1 –0.5 0 0.5 1 σ max Goodman σ min Goodman σ max Gerber σ min Gerber σ max L–W σ min L–W Stress/UTS Stress ratio, R Figure 2.19. Maximum and minimum stress as a function of stress ratio, R. –0.5 0 0.5 1 0 0.2 0.4 0.6 0.8 1 σ max Goodman σ min Goodman σ max Gerber σ min Gerber σ max L–W σ min L–W Stress/UTS Mean stress/UTS Figure 2.20. Maximum and minimum stress as a function of mean stress. values of R), the maximum stress can become the governing stress for design because it can approach the yield or ultimate strength of the material. The difference between the two curves is the stress range, or twice the alternating stress. In this as in the previous plots, the Gerber and L–W equations look nearly identical while the Goodman line, which is now straight, differs a little from the other two equations. The final method of plotting data is a plot of alternating stress against mean stress, commonly but incorrectly referred to as the Goodman diagram, Figure 2.21. What should Characterizing Fatigue Limits 45 0 0.1 0.2 0.3 0.4 0.5 0 0.2 0.4 0.6 0.8 1 Goodman Gerber L–W Alternating stress/UTS Mean stress/UTS Figure 2.21. Alternating stress as a function of mean stress. be properly called a Haigh diagram has evolved into the most commonly used method of plotting FLS or endurance limit stress data, but the use of a straight line fit to the data still seems to be popular. The diagram most closely related to these methods of representing fatigue limit or constant life data is the actual Goodman diagram shown in Figure 2.22, taken directly from Goodman’s book [20]. The two curves shown in the figure are labeled maximum and minimum stress. They are apparently plotted against minimum stress, as shown in Figure 2.18, although there is no label on that plot or on the y axis. Regions of tension and compression are shown above and below a zero stress line. Of greatest significance in this plot, closest to what may be called a true Goodman diagram, is that the lines used to represent the fatigue limit are straight lines on this particular plot. This is consistent with the straight line formulation of Goodman shown in Figure 2.18 by comparison. The L–W equations, shown in previous plots, can also be examined on a Haigh diagram. For positive values of R, Equation (2.6) the shape of the curve depends on the value of  0 , the maximum stress at R =0. In a non-dimensional plot, where stresses are divided by  u , the curves obtained for several values of  0 / u are plotted in Figure 2.23. The label “ 0 ” refers to the normalized value of  max at R =0. When  0 = 1, the Haigh diagram represents the points where  max = u for all values of R and is a straight line as shown. For other values of  0 , the lines are curved. Only the equation for positive values of R is plotted in the figure. The line representing R =0 is shown, so the curves are valid only to the right of that line. It is now obvious why the equation, valid now for positive R, was modified to fit data at −1 <R<0 as given in Equation (2.7). For this range of R, the points representing R =0 and R =−1 are connected by a smooth curve with slight curvature (not shown here). The curve in Figure 2.23 for  0 =05 goes through the point . that statistical distributions of data are the same at high R as they are at low R are questionable. Material quality is another parameter which has to be considered when using the Haigh diagram. fatigue limit diagram that we refer to as the Haigh diagram, often incorrectly called the Goodman diagram, has been referred to in early literature as an R–M diagram [15] or, alternately, as a. example of data obtained at several values of mean stress is presented in Figure 2. 16 for a SKH51 tool steel [17]. While extrapolation and interpolation of data sets are often necessary, the data in