226 Effects of Damage on HCF Properties out that close to the fatigue limit (under HCF), blunt notches and sharp notches behave differently in respect to their crack-growth mechanisms. When notches act like cracks, the mechanism leading to a fatigue limit is the growth of small cracks from the notch tip which may become non-propagating cracks. The criterion for the fatigue limit is the onset of crack propagation from an arrested crack and not crack initiation. Taylor [14] points out that these cracks are always short cracks and they arrest because their threshold values increase faster than the applied stress intensity. This occurs, in general, under a stress field involving steep stress gradients such as at the tip of a very sharp notch. For a plain or blunt-notched specimen, on the other hand, non-propagating cracks are not found, especially for very short crack lengths. Taylor goes on to attribute the fatigue limit in such geometries to the arrest of cracks at a grain boundary, the arrest defining the fatigue limit. This is a material-based limit according to Miller [15] rather than a limit based on the mechanics of the notch. A numerical example can be used to show the effects of notch geometry on fatigue strength. To illustrate the effect of very small notches on the fatigue strength, the formulas of Peterson for k t and Neuber for k f are plotted in Figure 5.9 for two different values of the material parameter a N and for three different values of notch depth, d (dimensions of the notch and the material parameter a N are in arbitrary units). The reciprocal fatigue strength, 0 0.2 0.4 0.6 0.8 1 1 1.5 2 2.5 3 3.5 4 4.5 5 a N = 0.2, d = 0.1 a N = 0.5, d = 0.1 a N = 0.2, d = 0.25 a N = 0.5, d = 0.25 a N = 0.2, d = 0.75 a N = 0.5, d = 0.75 1/K t 1/k f k t k t from Peterson k f from Neuber Figure 5.9. Fatigue notch factor as a function of k t for increasingly small notches. Constants a N and notch depth d are for Neuber equation for k f and Peterson equation for k t . Notch Fatigue 227 1/k f is plotted against k t and the curve representing k f =k t is shown as the bottom (thick line) curve. The higher the curve, as k f approaches unity, the less is the effect of the notch on the smooth bar fatigue strength. Curves for the various parameters show that the fatigue strength is least effected by the shallowest notch d =01 while the deepest notch d =075 comes closest to taking a fatigue strength debit equal to k t . In the limit, as the notch depth goes to zero, the fatigue strength is unaffected by the presence of the notch. Another example illustrating the complex behavior under fatigue loading of a notched specimen can be explained further with the use of Figure 5.10, which plots nominal stress against k t . The heavy solid line represents the fatigue limit and, as shown, represents a fairly large body of experimental data. The figure is based upon the work of Nisitani and Endo [16]. For low values of k t the fatigue strength is approximated by  0 /k t whereas beyond what is called the branch point [17], the fatigue strength is constant. The difference between the initiation limit curve and the curve due solely to k t is small as k t approaches one. This is recognized in many of the formulas for the true fatigue limit strength that Crack propagation limit Crack initiation limit Fatigue limit of smooth specimens, σ 0 σ 0 /K t Stress concentration factor, k t 1 0 Nominal stress Non propagating macro crack Fracture No cracking Branch point Figure 5.10. Schematic representation of fatigue strength at notch for increasing values of k t . 228 Effects of Damage on HCF Properties show k f is different than k t . The crack initiation limit is assumed to be governed by the value of notch tip stress as well as the stress gradient at the notch. It is assumed, therefore, that as k t increases, the stress necessary to initiate a crack becomes lower. However, because of the steep gradient of stresses at the notch root for high values of k t , a crack that initiates enters a rapidly decreasing stress field where the tendency to propagate decreases. From a fracture mechanics perspective, the stress intensity decreases to a value below the crack growth threshold and thus the crack arrests. Above the solid line in the figure, fracture will occur after crack initiation and continued propagation. Below the crack initiation limit, no cracks will form. In the intermediate region, shown shaded in the figure, a macrocrack will form but will not propagate. While the authors [16] refer to this region as one where macrocracks form, it is feasible that cracks of any arbitrarily small size may form and a separate crack initiation limit can be established. From a practical point of view, such cracks would have to be observed experimentally in order to establish the microcrack initiation limit and this becomes much more difficult to accomplish. 5.7. MEAN STRESS CONSIDERATIONS The reduction of fatigue strength of a notched specimen, characterized by the quantity k f (or alternately, q), is not a unique quantity for a material given by the value obtained at R =−1 for very long fatigue life for a given specimen geometry [6]. Instead, the value of k f is often found to depend on mean stress as depicted in Figure 5.11, as shown in Mean stress Alternating stress S 0 k f S 0 k f S u S u Smooth Ductile Notched Brittle Figure 5.11. Simplified representation of smooth and notched behavior for ductile and brittle materials on a Haigh diagram (after [6]). Notch Fatigue 229 [6]. For a brittle material, where local notch plasticity is absent, both the mean stress and alternating stress are reduced on a Haigh diagram since the ultimate stress point is reduced due to a notch. For a ductile material, the ultimate stress is essentially the same in a notched as in an un-notched specimen because of either local or gross-section yielding and the resulting stress redistribution. Thus, the Goodman line in Figure 5.11 goes through the same ultimate strength point as for the un-notched specimen. In reality, notched specimen behavior falls in the region between the two extremes shown in the figure. If the smooth bar fatigue strength under fully reversed loading is denoted by S 0 ,as shown in Figure 5.11, then the modified Goodman equation connecting that point with a straight line on a Haigh diagram to the static ultimate stress, S u , is written as S a =S 0  1− S m S u  (5.12) The simplest approach to modify this equation for notched specimens is to simply reduce both the mean and alternating stresses by the elastic stress concentration factor, k t , thus S a = S 0 k t  1− k t S m S u  (5.13) This is very conservative, since the reduction in fatigue limit is characterized by k f , which is less than k t . A better approach consists of replacing k t by k f in Equation (5.13) which produces the line denoted by “brittle” in Figure 5.11. In many cases, a best value of k f , not necessarily the value for R =−1, is used in such an approach. A more realistic approach is to reduce only the alternating stress by a constant factor k f while maintaining the mean stress. For the line denoted as “ductile” in the same figure, the equation that reduces the alternating stress in this manner is S a = S 0 k f  1− S m S u  (5.14) Bell and Benham [18] performed an extensive series of notch fatigue tests on stainless steel sheet covering a wide range of both notch geometries and fatigue lives. Concentrating here only on the results for long lives, corresponding to HCF, they observed that the fatigue notch factor was a function of mean stress as suggested above. Using the argument that as the mean stress increases the amount of notch plasticity increases above some critical value, and observing that the static ultimate strength of a notched specimen is somewhat higher than that of an un-notched specimen (referred to as notch strengthening), they proposed an equation which fit their endurance limit data for notched specimens very well, as shown in Figure 5.12. Their equation, used in the figure, is k f =k fav  1− S m S up  + S m S un (5.15) 230 Effects of Damage on HCF Properties 0 5 10 15 20 25 30 35 40 0 10 20 30 40 50 60 70 80 Experimental plain specimen Experimental notched specimen Empirical k f from equation Alternating stress, tons per sq in. Mean stress, tons per sq in. k f = k fav 1 – S m S up + S m S un Figure 5.12. Comparison of experimental and empirical curves on Haigh diagram for notched specimens having lives in excess of 10 6 cycles (after Bell and Benham [18]). where k fav is an average fatigue notch factor at R =−1 for a variety of experiments, S up is the tensile strength of an un-notched specimen, and S un is the tensile strength of a notched specimen for the particular notch geometry used in constructing the Haigh diagram. This equation differs from Equation (5.14) in concept because here the fatigue notch factor k f is reduced as a function of mean stress rather than reducing only the alternating stress by a constant fraction. To compare the two methods of accounting for the effect of mean stress, numerical examples are provided for a material whose smooth bar behavior can be represented as a straight line (modified Goodman equation) on a Haigh diagram. To illustrate some of the characteristics of Equation (5.15), a Haigh diagram where the un-notched behavior is represented by the straight line modified Goodman equation, Equation (5.12), is shown in Figure 5.13. In this illustrative example, the smooth bar alternating stress is arbitrarily taken as 0.5 of the ultimate strength, S u (or S up in Equation (5.15)) and all of the quantities are shown in dimensionless coordinates with respect to S u . Taking S un =105S up , values of the notch strength factor k f =15 and 3.0 are chosen. The shape of the curves shows how, as the mean stress is increased, the notch reduction factor decreases. For both cases, where either the alternating stress is decreased according to a constant value of k f , or the equation of Bell and Benham, Equation (5.15) is used, the reduction in fatigue strength due to a notch is approximately the same. The only noticeable differences are for high values of mean stress. If S un =S up for the straight line case shown here, then the reduction of alternating stress only produces the identical result to Equation (5.15). The two methods for accounting for the notch fatigue strength reduction turn out to be almost identical in this illustrative example. Notch Fatigue 231 0 0.1 0.2 0.3 0.4 0.5 0 0.2 0.4 0.6 0.8 1 smooth bar k f = 1.5 alt stress only k f = 3 alt stress only Eqn k f = 1.5, S un /S up = 1.05 Eqn K f = 3, S un /S up = 1.05 S a / S u S m / S u Figure 5.13. Comparison of two equations of Bell and Benham [18] for notched specimen on a Haigh diagram for material represented by modified Goodman equation. If a nonlinear curve is used to represent the smooth bar fatigue data, then the two equations, modified to take care of the actual shape of the un-notched curve, produce slightly different results as illustrated in Figure 5.14. For an example case where the curve represents data on titanium, shown earlier in Chapter 2, the two methods produce slightly different curves. Nonetheless, both reducing the alternating stress only by a constant factor 0 100 200 300 400 500 0 200 400 600 800 1000 Jasper equation Alt stress only Equation Alternating stress (MPa) Mean stress (MPa) k f = 3.0 Figure 5.14. Comparison of two equations of Bell and Benham [18] for notched specimen on a Haigh diagram for real material represented by Jasper equation. 232 Effects of Damage on HCF Properties k f and reducing the effective value of k f as a function of mean stress seem to be effective methods for accounting for the observed dependence of k f on mean stress in ductile metals. 5.8. PLASTICITY CONSIDERATIONS One of the earliest and simplest attempts to treat fatigue at notches that undergo plastic deformation at the notch root is due to Gunn [19]. He provided a simplified procedure to use the smooth bar Haigh diagram to create a notched Haigh diagram based solely on the elastic stress concentration factor, k t , of the notch. The procedure is illustrated in Figure 5.15 for a linear representation of the smooth bar data on a Haigh diagram. The procedure is applied in an identical manner for any shape curve. For purely elastic behavior, the notch stresses are reduced by the same amount for both the alternating and mean stresses. However, when the nominal stress multiplied by k t equals the yield stress, Y , the notch root will undergo local plastic deformation. Then, it can be assumed that for higher applied peak stresses, the region around the notch root will undergo cyclic plastic deformation for one or more cycles until the entire behavior becomes elastic again. At that point, the mean stress will be reduced from the applied value divided by k t since the maximum stress at the notch does not exceed the yield stress, but the alternating stress will remain the same. The approximation due to Gunn then uses the smooth bar curve reduced by k t for stresses below yield at the notch root as indicated by the line AB in Figure 5.15. For regions above where applied stresses cause notch root yielding, the local mean stress remains unchanged since the maximum stress there is limited to the yield stress. The point of local yielding, B, is the intersection of the smooth bar curve reduced by k t and the line representing maximum stress at the notch root equal to the yield stress, Y . This latter line is shown dotted in the diagram and goes through the point on the Mean stress Alternating stress Y C Y/k t Smooth bar Notched bar A B Figure 5.15. Notch Haigh diagram using simplified procedure of Gunn [19]. Notch Fatigue 233 mean stress axis whose magnitude is Y/k t . For stresses beyond yield at the notch root, the allowable alternating stress is that at point B since the local mean stress remains constant. This results in a predicted Haigh diagram of ABC as shown in Figure 5.15. Although Gunn [19] developed a more sophisticated model for the Haigh diagram based on the stress–strain behavior of a material, the simplified procedure illustrated by Figure 5.15 was found to be sufficiently accurate for practical purposes [20]. Lanning and co-workers have performed a large number of notch fatigue tests on Ti-6Al-4V to determine the fatigue strength at 10 6 cycles using circumferentially V-notched specimens whose geometry is shown in Figure 5.16. Some of the data, reported in [21], are presented in Figure 5.17 where three combinations of notch dimensions h d D 60° ρ Figure 5.16. Cylindrical fatigue specimen with circumferential V-notch. 0 100 200 300 400 500 0 200 400 600 800 1000 Smooth bar Small notch Medium notch Large notch Alternating stress (MPa) Mean stress (MPa) Ti-6Al-4V plate K t = 2.8 R = 0.1 R = 0.5 R = 0.8 Figure 5.17. Notch fatigue data on Ti-6Al-4V with k t =28. 234 Effects of Damage on HCF Properties producing the same value of k t =28 were used. The dimensions of the notches are listed in Table 5.1. For reference purposes, typical values of a 0 for this titanium alloy are around 0.05 mm while the grain size is between 0.015 and 0.020 mm. From the dimensions of the small notch it appears that none of the specimens should be expected to show small notch behavior. Data for these specimens, shown in Figure 5.17, show a reasonably large amount of scatter which the authors attribute to sampling a small volume of material in each test. At the highest value of R = 08, however, the small notch data seem to lie above the data from the other notch sizes. Calculations show that for values of R above approximately 0.5 that plastic deformation takes place at the notch root. Creep ratcheting, discussed earlier, is also a consideration when the stresses approach the yield stress in this material. This would occur at values of R in the vicinity of R =08. To try to quantify the notch effect for the data in Figure 5.17, we can follow the suggestion of Bell and Benham [18], for example, that the Haigh diagram for a notched specimen may better represent a material if the alternating stress is reduced by k f but the mean stress is left unchanged. In Figure 5.18, a value of k f of 2.8 is chosen to represent Table 5.1. Dimensions for circumferential V-notches with k t =28 Notch type  (mm) h (mm) D (mm) d (mm) Small 0127 0127 572 547 Medium 0203 0254 572 521 Large 0330 0729 572 426 0 100 200 300 400 500 0 200 400 600 800 1000 k t = 1 k f = 2.8 k f = 2.8 (alt only) Alternating stress (MPa) Mean stress (MPa) Ti-6Al-4V plate K t = 2.8 R = 0.1 R = 0.5 R = 0.8 Figure 5.18. Representation of fatigue notch data with k f reductions. Notch Fatigue 235 the data of Figure 5.17. The Jasper equation, used earlier in this book (see Chapter 2), is used to represent the smooth bar data. For comparison purposes, the smooth bar curve is reduced by k f for the maximum stress in one case. This amounts to reducing both the mean and alternating stresses by the same amount. The value of k f = 28 is taken from Figure 5.17 based on the zero mean stress R =−1 data points. In this particular case, k f = k t for the R =−1 condition. This same value of k f is used to reduce only the alternating stress and the resulting curve is also shown in Figure 5.18. Comparing Figure 5.18 with the data in Figure 5.17 shows that the reduction of only the alternating stress, as suggested in [18], produces a more reasonable representation of the notch data for this particular material, Ti-6Al-4V plate. While it is always convenient to have a simple formula to predict notched behavior from smooth bar behavior, even if it is just for mildly notched specimens, the scatter in fatigue limits and the tendency for real materials not to follow the laws we create for them makes predicting notch behavior a cumbersome task. As an example, taking the same titanium alloy as discussed above in [21] and the data presented in Figure 5.17, we look at data obtained at a fatigue limit of 10 7 cycles from a recent Air Force program [22]. These data were obtained on smooth and notched specimens that were both stress relieved after machining and chem milled. While this process is aimed at producing a pristine material, the experimental data may be different than those obtained after stress relieving only as was done for the data in [21] (corresponding to a fatigue limit at 10 6 cycles). It should also be noted that the fatigue properties of titanium alloys are very sensitive to surface finish [23]. In the work in [22], a nominal value of k t = 25 was obtained for double-edge V-notch rectangular specimens with a nominal root radius of 0.033 in. (0.84 mm). The experimental data are presented in Figure 5.19 where each data point is the average of several tests. The Jasper equation is used to fit the smooth bar data and produces a good representation of the data points obtained at R =−1, 0.1, 0.5, and 0.8. Three methods of representing the notch data are shown. In the first, only the alternating stress is reduced by the value of k f for R =−1, which is approximately k f = 20. This produces the poorest fit to these notch data. The second method uses a value of k f that is linearly dependent on mean stress between a mean stress of zero and the ultimate stress, following the suggestion of Bell and Benham discussed above. This method produces a slightly better fit to the data and has the right general shape, but the curve falls above the experimental data points for non-zero mean stresses. The third and final method is just to apply the same value of k f to all of the smooth bar data. This method produces a very good fit to all the data except at R =08 where both plasticity and time-dependent ratcheting or creep are known to occur. In this particular example, the ability to fit the notch data with a model produces results somewhat different than in the earlier example where the surface finish and notch geometry were different, but the material was identical. . for a material whose smooth bar behavior can be represented as a straight line (modified Goodman equation) on a Haigh diagram. To illustrate some of the characteristics of Equation (5.15), a Haigh. different values of the material parameter a N and for three different values of notch depth, d (dimensions of the notch and the material parameter a N are in arbitrary units). The reciprocal fatigue. propagate. While the authors [16] refer to this region as one where macrocracks form, it is feasible that cracks of any arbitrarily small size may form and a separate crack initiation limit can