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246 Effects of Damage on HCF Properties 0 100 200 300 400 500 10 4 10 5 10 6 10 7 R = –1 R = 0.1 R = 0.5 Predicted, R = –1 Predicted, R = 0.1 Predicted, R = 0.5 Alternating stress (MPa) Cycles to failure k t = 2.68 ρ = 0.53 mm Figure 5.25. Comparison of predictions and experimental data for notch fatigue data in Ti-6Al-4V [29]. distance, a c , was 0.074, 0.086, and 0.094 mm for R =−1, 0.1, and 0.5, respectively. What makes this approach work so well is the fact that G F turns out to be a constant for a large body of data. This is analogous to finding a critical distance which is a constant for a material. In this case, the critical distance is computed from Equation (5.28) and is dependent on both stress ratio, R, and notch root radius, . It is also shown in [29] that the fatigue notch factor can be obtained from the following equation. k f =k t A 0 +s A 0 3/2 1+s (5.30) 5.11. ANALYSIS METHODS In the work of Nisitani and Endo [16], the stress field ahead of an elliptical hole in an infinite plate subjected to remote tension, as shown in Figure 5.26, is reported as y x = m 4 3 +m 2 m 2 −m−3 +m +1 m −1m 2 −1 (5.31) Notch Fatigue 247 b y x 2a σ ∞ σ ∞ ρ Figure 5.26. Schematic of elliptical hole in tension. = a +x a +2ax +x 2 a m= a (5.32) k t =1+2m (5.33) The authors note that the branch point in Figure 5.10, where a notch starts to behave like a crack, has a constant root radius, , independent of notch depth for a given material. This constancy is attributed to the fact that the relative stress distribution near a notch root is determined by the notch root radius alone. From experimental results and the analysis of a crack at the root of an elliptical hole, they demonstrate that the crack initiation limit curve and the crack propagation limit, and their intersection at the branch point (see Figure 5.10), can be determined by using only max at the notch root and the notch root radius, . The determination of these quantities, however, is based on fitting of experimental data to the theoretical curves. The unified treatment proposed in [16] can be summarized in a schematic of their approach shown in Figure 5.27. The limiting values of the nominal stress times k t for crack initiation and propagation are shown as a function of 1/ for the elliptical notch. Points A and B correspond to the fatigue limit of a smooth bar and the branch point, respectively. The curves B–C and B–D bound the region where fatigue limit stresses for a sharp notch will fall whereas milder notch results will be on curve A–B. The experimentally observed dependence of the fatigue limit of an elliptical notch solely on notch root radius, , and k t , allows the construction of the curves in Figure 5.27 from results of only a limited number of fundamental experiments. An alternate method for bridging the gap between fatigue limit of a notch (initiation limit) and the threshold stress intensity for a very sharp notch or crack has been developed 248 Effects of Damage on HCF Properties Deeper notch Shallower notch Initiation limit Propagation limit A B C D Stress × k t 1/ρ Figure 5.27. Schematic figure of crack initiation limit and crack propagation limit [16]. by Atzori and Lazzarin [30]. They extended the Kitagawa diagram for cracks to include blunt cracks (i.e. U-shaped notches) as illustrated schematically on Figure 5.28. Note the similarity to Figure 5.7 and the accompanying discussion earlier. For a very blunt notch, where a becomes large, the fatigue limit stress is simply 0 /k t , where 0 is the smooth bar fatigue limit and k t is the elastic stress concentration factor. As the length factor a, which would be the depth of a U-shaped notch, decreases, this approximation becomes more conservative (line CD) so that below some critical value at a = a ∗ , the notch becomes a crack and the Kitagawa diagram becomes applicable as shown. It is easily shown from the definitions of the intersections of the K th line with 0 and Log Fatigue limit Δ σ th ΔK th Δσ 0 K t Δσ 0 Log a Short cracks Long cracks Classic notches a 0 a* CD Figure 5.28. Fatigue behavior of a material weakened by notches or cracks [30]. Notch Fatigue 249 0 /k t being a = a 0 and a = a ∗ , respectively, that the following expression provides the value of a ∗ k 2 t = a ∗ a 0 (5.34) where a 0 is the standard definition used in a Kitagawa diagram for an edge crack characterized by K th = th √ a a 0 = 1 K th 0 2 (5.35) If the theoretical elastic stress concentration factor for a notch is used, that is k t =1+2 a (5.36) then for a very deep notch, where a , the notch root radius corresponding to a = a ∗ is defined as ∗ and is found to be ∗ =4a 0 . Then, the threshold stress intensity for such a notch becomes K th = 0 √ ∗ 2 (5.37) which the authors relate to the generalized stress intensity factor suggested by Tanaka [31] and Glinka [32] for rounded notches. These results are interpreted as bridging the gap between the concepts of sensitivity to defects and notch sensitivity. The former refers to crack-like defects whereas the latter represents geometric stress raisers imposed into a material. From a phenomenological point of view, these can be substantially different. The modeling concepts tend to bridge this gap and provide a way of looking at the two as essentially the same problem. The work in [30] has been extended to a more general geometry than the elliptical notch in an infinite plate by Atzori et al. [33] through the introduction of a shape factor as is done in fracture mechanics. Using a more general form of the stress intensity factor K = √ a (5.38) the K th curve of Figure 5.28 is shifted down and to the left. The transition points a =a 0 and a = a ∗ are replaced by a = a D and a = a N , respectively, where the new points are defined as a D = a 0 2 a N = a ∗ 2 = k 2 t a 0 2 (5.39) 250 Effects of Damage on HCF Properties where a D is the intrinsic defect size. If the small crack correction of El Haddad is introduced, then the fatigue limit behavior of a component in the presence of a defect size close to a D is given by th 0 = 1 2 a a 0 +1 (5.40) If k t is kept constant, the depth of a completely sensitive notch is now a N . This extension of previous work in [30] is presented as a “universal” diagram able to summarize experimental data related to different materials, geometry, and loading conditions. The diagram is applied both to the interpretation of the scale effect and to the surface finishing effect. Ciavarella and Meneghetti [34] reviewed some of the empirical formulas developed for the fatigue strength of notched components that were based on concepts involving the relation of the fatigue strength with the stress at a certain distance or averaged over a certain distance ahead of a notch. This distance, often thought to be a microstructure- based quantity, has generally been used more as a fitting parameter for experimental notch fatigue data. Extrapolation to extremely sharp notches, where fracture mechanics controls, is often found to be inaccurate. The use of averaging of the stress ahead of a notch including very sharp notches (cracks) or using stress at some distance ahead of the notch is no better than the accuracy of the stress field solution for the particular notch. In comparing the formula for crack-like behavior as cracks get small due to El Haddad, often shown on a Kitagawa diagram, a modified version of the notch formula of Neuber is proposed in [34]. This involves simply using the definition of k t for the elliptical notch in an infinite plate, resulting in a “Neuber modified” equation k f =1+ k t −1 1+ k t −1 a/a 0 (5.41) which compares favorably with the El Haddad formula K th = th a +a 0 (5.42) The modified Neuber formula is shown to be slightly more conservative in the small crack regime. Further, it has the correct asymptotic behavior for blunt notches as →, namely that k f →k t . Tanaka [31] first noticed that for a sharp notch, averaging the stress over a characteristic distance l 0 then, in order to match the fatigue limit for arbitrarily small cracks, l 0 must be equal to 2 a 0 , where a 0 is the El Haddad transition point on the Kitagawa diagram (see also Taylor [13]). From this, it is shown that the following expression holds true: k f = 1+ a a 0 (5.43) Notch Fatigue 251 This is referred to as the El Haddad formula. Ciavarella and Meneghetti then proposed two possible formulations for describing the notch fatigue limits covering the entire range from sharp notches, which act like cracks, to blunt notches. In the dual models proposed, the two curves covering different parts of the diagram shown earlier in Figure 5.28 are made continuous at a transition point. The first model proposes Equation (5.43) for a<a c and the following equation from Lukas and Klesnil [35] for a>a c k f = k t 1+ k t −1 2 a/a 0 (5.44) where the transition point is found as: a c =a 0 k t −1 (5.45) The second proposed formulation uses Equation (5.43) for a<a ∗ and simply k f =k t as illustrated in Figure 5.28 where a ∗ is defined in Equation (5.34). Both formulations were found to provide a reasonably good representation of FLSs from a large body of notch fatigue data. What the results illustrate is that there is no one single formulation that is able to represent the wide range of notch data available in the literature. By representing portions of the data, a better fit can be achieved. Notice that in both cases, the authors use the El Haddad representation of sharp notch, crack-like data, which appears to provide a good fit in the sharp notch region. The difference between the two formulations for this region is in the transition point to a blunt notch model. In the first method, the transition is at a c , while in the second, it is at a ∗ . These quantities can be compared through the following formula, using Equations (5.34) and (5.45): a ∗ =k 2 t a 0 >a c =a 0 k t −1 (5.46) which illustrates that the two criteria coincide when the notch depth is lower than a c . For a>a c , the second criterion using k f = k t is more conservative but it lacks the smooth transition of the other. 5.12. EFFECTS OF DEFECTS ON FATIGUE STRENGTH While much analysis has been conducted on what may be described as ideal cracks, many applications requiring use of a fatigue threshold in HCF design deal with real defects in the form of voids or inclusions, for example. For these irregular shapes, an engineering stress intensity has been developed that is based on the “area” of the crack. The area is defined as the projection of the actual area of the crack or defect onto a plane 252 Effects of Damage on HCF Properties normal to the applied stress. The maximum value of the stress intensity can be written, approximately, as K Imax =C 0 √ area (5.47) where 0 is the maximum applied tensile stress and “area” is defined as the projected area. The constant C has been estimated to be 0.5 for surface defects and 0.65 for internal defects [36]. These approximations are a good engineering tool but are subject to limitations on crack size, crack geometry, and material microstructure. The evolution and limitations of these equations and their application to engineering problems is discussed thoroughly in the book by Murakami [36]. These equations have seen extensive use in the work on gigacycle fatigue, discussed in Chapter 2, where failure at ultra-long fatigue lives and establishment of fatigue limit stresses or endurance limits deal with crack initiation from internal defects. If a fracture mechanics approach using a threshold value is not adapted for small inclusions in a material, then the FLS approach can be used. Problems such as foreign object damage (see Chapter 7) can be addressed with a threshold for an equivalent crack. Many other problems in HCF center around the issue of the effect of odd-shaped inclusions on the FLS of a material. The effects of small defects in materials, particularly steels, is discussed in detail by Murakami [36]. In that book, he provides formulas for the fatigue limit strength, w , of a steel with a non-metallic inclusion as w = CH V +120 √ area 1/6 (5.48) where H V is the Vickers hardness in units of kgf/mm 2 , area is in m, and C is a constant depending on the location of the inclusion. For an inclusion in contact with the surface, C =141, while for an internal inclusion, C =156 [36]. Murakami also provides an empirical formula for the upper bound to the FLS of a steel having no inclusions as wu =16H V (5.49) For the two cases of internal and surface connected inclusions, the maximum size of an inclusion that will have no effect on the fatigue strength of a steel is easily calculated from Equations (5.48) and (5.49). The results, in terms of the parameter √ area, are presented in Figure 5.29 which shows that for all but the softest of steels, as measured by the Vickers hardness, inclusions have to be smaller than several microns in order not to affect the FLS. On the other hand, as the Vickers hardness decreases below approximately H V =200, the material is very intolerant to inclusions of much larger sizes. It can be noted that the form of the Murakami Equation (5.48) involves an inclusion dimension to the 1/6 power in the denominator. Contrast this with the equations for a Notch Fatigue 253 0 10 20 30 40 50 0 100 200 300 400 500 600 700 800 C = 1.41 C = 1.56 Sqrt “Area” (microns) Vickers hardness, H V Surface connected inclusion Internal inclusion Figure 5.29. Critical dimensions for inclusions in steels (formulas from [36]). long crack that have the crack length raised to the −1/2 power, denoting a square root singularity. If the Kitagawa diagram is used to compare FLSs to a corresponding crack length, the resultant diagram in dimensionless form is shown in Figure 5.30 where the El Haddad short crack correction has been introduced. As shown in the diagram, long cracks corresponding to a/a 0 1 have a slope =−1/2 while short cracks approach zero slope corresponding to the normalized endurance limit stress = 1. At a/a 0 =1, the slope on this log–log plot is −025. The Murakami equation provides for a slope of −1/6 which is indicated in the figure. This slope is tangent to the Kitagawa diagram curve for a range of crack lengths slightly below the region where a =a 0 , the El Haddad short crack parameter. The actual tangent point where the slope =−0167 −1/6 is at a/a 0 =046. The values shown in Figure 5.29 correspond to where this curve would intersect the endurance limit for given values of Vickers hardness number. These computations and 10 –2 10 –1 10 0 10 –3 10 –2 10 –1 10 0 10 1 10 2 10 3 Normalized endurance stress a /a 0 1 1 2 6 Figure 5.30. Normalized Kitagawa diagram showing slope of Murakami equations. 254 Effects of Damage on HCF Properties plots indicate that the simplified formula of Murakami, Equation (5.48), for different values of H V , seems to follow the same trend as the El Haddad equation for the endurance limit stress for small cracks. 5.13. NOTCH FATIGUE AT ELEVATED TEMPERATURE In Section 2.6, the construction of a Haigh diagram at elevated temperature was illustrated for a single crystal material that showed evidence of creep behavior at certain maximum or mean stress levels. In this chapter, Section 5.8, the effects of plastic deformation of notches on the shape of a Haigh diagram for notched components were discussed. Here, that discussion is expanded to include the effects of creep, typical of elevated temperature behavior, on the construction of a Haigh diagram for a notched component. There are two main considerations in this problem: the stress gradient present at a notch or stress concentration and the redistribution of stress due to inelastic deformation (creep), particularly at a region of high stress and stress gradients such as a notch. There are many analytical and empirical methods for tackling such a problem and the specific material, geometry, and loading conditions will dictate which approach is better. For illustrative purposes, we refer to the work of Harkegard [37] who provides details for a rather simple approach. The starting point for development of a Haigh diagram for notched components in the creep regime is to consider the smooth bar behavior at temperatures below the creep regime. Such behavior can be easily represented as a straight line in a Haigh diagram that goes from the fully reversed R =−1 alternating stress to the true fracture stress, f . This slight variation of the Goodman equation is accurate only if there is no yielding at the notch root. This differs from some of the previous formulations for notch behavior because it considers only the local stresses at the notch root as opposed to average stresses and the use of the elastic stress concentration factor, k t . Because of stress gradients, this is a conservative approach based on k t , not k f . The use of true fracture stress changes the Goodman equation for smooth bar behavior [Equation (5.3) in Chapter 2], a = −1 1− m u (5.50) that ends at the ultimate stress, u , to the modified form a = −1 1− m f (5.51) where the true fracture stress, f , is defined as f = u 1−Z (5.52) Notch Fatigue 255 and Z is the reduction in area in a tension test. It is recognized in the construction of the Haigh diagram for use in notched components that at the root of a notch, localized plastic flow can take place upon the first load application. Thus, the maximum stress that takes place is limited to the yield strength of the material if there is little strain hardening or small plastic strains. To account for this, the “fictitious” mean stress is used in plotting the Haigh diagram as depicted in Figure 5.31. The fictitious or nominal stress is computed assuming purely elastic behavior. However, the notch root stress is limited by the yield strength, thus the curve is limited by the line denoted by s max =s y as shown in the figure (see also Figure 5.15 and the discussion accompanying it). For elastic behavior at the notch root, the smooth bar and notched behavior are assumed to be governed by the local stress, with gradients not considered. As the local stresses increase with increasing mean stress, the behavior at the notch root becomes inelastic, but the stresses are assumed to not exceed the yield strength. For nominal (fictitious) stresses above the yield stress, the maximum local stress is assumed to be the yield stress. Thus, in Figure 5.31, for fictitious mean stresses above that where yielding first takes place, the allowable alternating stress is the same as that at the yield condition, as shown by the horizontal line in the figure. If the behavior of the material is only known at room temperature, it can be extended to higher temperatures below the creep regime through the use of the fatigue ratio, V w , V w = 0 T u T (5.53) by assuming V w to remain constant as temperature increases. In this definition, 0 is the alternating stress at zero mean stress and u is the ultimate stress. For the case where creep occurs in the material at high temperatures, a fatigue limit no longer exists [20]. Now, the behavior is time-dependent and number of cycles is eventually replaced by time as illustrated in Chapter 2 when discussing the Haigh diagram for a single crystal material at elevated temperature (see Section 2.6). In the present approach, the fatigue ratio, Equation (5.53), is used in the following form: V r = A t T u t T (5.54) s max = s y s y s f s –1 Fictitious mean stress Alternating stress Figure 5.31. Haigh diagram of a notch component below the creep regime (after [37]). . ELEVATED TEMPERATURE In Section 2.6, the construction of a Haigh diagram at elevated temperature was illustrated for a single crystal material that showed evidence of creep behavior at certain maximum or. experimental results and the analysis of a crack at the root of an elliptical hole, they demonstrate that the crack initiation limit curve and the crack propagation limit, and their intersection at. representation of FLSs from a large body of notch fatigue data. What the results illustrate is that there is no one single formulation that is able to represent the wide range of notch data available
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