456 Applications and an effective stress ratio can then be defined as: R eff = K mineff K maxeff (8.16) Note that in this case, K does not change due to residual stresses, but R eff decreases if K res is a negative number. Since the residual stresses are compressive, the computed K res is negative. While it should be recognized that K res being negative has no formal meaning, a negative K res is used to represent the contribution of the compressive loading across the crack surfaces. This, in turn, is added to the K of the applied loading. If closure is observed at a given crack length, then the measured value of P open corresponds to the point where K min overcomes K res . In the reported investigation, at the higher stress ratio test condition, R applied = K min K max (8.17) and K min =08K max (8.18) Since closure was absent in the load-displacement traces at R =08, with or without LSP, then, K min +K res ≥0 (8.19) and, therefore, combining Equations (8.18) and (8.19) yields: K res ≥−08K max (8.20) Equation (8.20) implies that the magnitude of K res is less than or equal to 80% K max during the R =08 test. For the case of R = 01 testing where K max is the same as the R =08 test, it follows from Equations (8.15) and (8.16) that: R eff = 01K max +K res K max +K res (8.21) and R eff = 01K max −08K max K max −08K max (8.22) or R eff ≈−35 (8.23) HCF Design Considerations 457 Note that R eff equal to −35 is a limiting value or upper bound (in magnitude) because it is based on the assumption of K res =−08K max [Equation (8.20)]. The above calculations imply an apparent closure level of ≈ 80% during crack growth with R eff ≈−35atthe crack tip. It is important to point out that closure contributions at a certain value of K, and residual stresses producing a value of K at the crack tip, are different and independent concepts. The effect of closure, quantified by Equation (8.14), reduces the range of K whereas residual stresses reduce the mean stress. The behavior of a cracked specimen that has a residual stress can be further understood with the aid of Figure 8.67 where compliance curves are shown for a material that exhibits no crack closure in (a) and for a material having a closure or opening value, K op , as shown in (b). In both cases, curves denoted by “A” represent the cracked body with no residual compressive stresses. Curves “B” and “C” depict the behavior for two different degrees of residual stress producing a calculated K res of magnitudes K res1 and K res2 . From Equation (8.15), the compliance curves are shifted in the negative K direction by different magnitudes. For the material with no closure in (a), the point where the resultant value of K is zero represents the point in the compliance curve where the slope will change. The portions of the compliance curves that are dashed, for “negative K,” have no meaning in terms of K and will have slopes in the dashed regions corresponding to a closed crack. In terms of applied load, the apparent closure load will be some fraction of the total applied load. If the material exhibits crack closure as depicted in (b), the superposition of residual K will produce compliance curves as shown. In case B, the inherent closure of the specimen is at a higher K than the closure produced by the residual stresses. In case C, however, the residual K (b) K op C A B K (a) K res,2 K res,1 C A B δ δ Figure 8.67. Schematic diagram of compliance curves for (a) closure free specimen, (b) specimen with crack closure. 458 Applications stress produces an apparent closure at K eff =0 such that the material closure is at a lower load level and would not be detected. In this case, as in the previous one, negative values of K have no physical meaning and the compliance below K = 0 is simply that of an uncracked body. It should be recognized that the superposition of K res and K applied only has physical meaning if the crack is open at all points along the crack face. If the crack is closed, the linear superposition of Equation (8.15) is no longer valid and the mathematical quantities obtained for negative values of K are incorrect. The computations of negative values of R for crack growth analysis are also of dubious value or have little physical meaning. Note also, in the compliance curves shown schematically in Figure 8.67, the top and bottom of each curve corresponds to the maximum and minimum applied load, respectively. Thus, the point where the effective value of K becomes negative has to be considered with respect to where it occurs on the applied load cycle. 8.6.3.2. Crack arrest In the discussion above of crack growth behavior, the load levels applied to obtain continuous crack growth were of a sufficient magnitude to avoid crack arrest as depicted earlier in Figure 8.48. In many of the experiments performed by Ruschau et al. [64, 65], the crack growth behavior of an LSP sample indicated that cracking initiated under similar loading levels as those of the baseline samples, grew a short distance, then arrested as shown in Figure 8.68. This crack arrest behavior for LSP samples required that the maximum cyclic stress had to be increased (5–10%) in order to re-initiate cracking. Again, this re-initiated crack often arrested after minimal growth, forcing the continual process of increasing the peak cyclic stress to re-propagate the crack until steady state crack growth eventually took place and the crack propagated to failure. 0 1 2 3 4 5 6 7 0 2,50,000 5,00,000 7,50,000 10,00,000 no LSP LSP Crack length, a (mm) Cycles Crack arrest Failure R = 0.1 P max = 2615 N Figure 8.68. Crack length vs cycles data for baseline and LSP samples. HCF Design Considerations 459 The crack arrest behavior can be attributed to large compressive residual stresses in the LSP region which display a steep gradient near the leading edge, increasing in intensity a short distance from the leading edge as illustrated in Figure 8.63. For a propagating crack, as discussed above, this non-uniform stress field results in a residual compressive stress field at the crack tip and hence an effective compressive stress intensity, K res , that increases in absolute value with increasing crack length, thus producing crack arrest. For crack growth to continue, the applied maximum stress intensity, K max , at the crack tip must exceed K res as well as the inherent threshold stress intensity range, K th , of the material. The sequential growth and arrest behavior can be explained with the aid of Figures 8.69 and 8.70 which were constructed using the stress intensity solution for the airfoil geometry, Equations (8.12) and (8.13), and an expression for K res . The latter could be derived, in principal, from the residual stress profiles shown in Figures 8.63 and 8.64 and a lot of interpolation, taking into account the three-dimensional nature of the problem due to the increase in specimen thickness with increasing crack length (see Figure 8.61). Instead, Figures 8.69 and 8.70 are constructed using some numerical simulations that include a crude approximation of the residual stress profile deduced by Lykins and John [67] for the same airfoil geometry and the same material used by Ruschau et al. These figures are discussed later. In [67], as in [64], crack arrest occurred when the crack was in the LSP processed region, requiring an increase in load to continue the crack growth. The sequence of crack growth and arrest that was obtained with incremental increases in the applied load is shown in Figure 8.71, where the dashed line represents the experimental observations. From numerous measurements of the applied load and crack length where arrest occurred, Lykins and John [67] were able to calculate the value of the applied K at that point. Then, having data for the threshold stress intensity for the corresponding values of R, the –3 –2 –1 0 1 2 3 4 5 0246810 K applied K residual K total K * Crack length (mm) σ * = 1.0 Figure 8.69. Dimensionless K for airfoil specimen for applied stress  ∗ =1. 460 Applications 0 0.5 1 1.5 2 2.5 01 2 453 σ* = 1.0 σ* = 1.1 σ* = 1.2 σ* = 1.5 K* Crack length (mm) Figure 8.70. K values for several applied load levels in airfoil geometry. 0.0 1.0 2.0 3.0 4.0 0 1 × 10 7 2 × 10 7 3 × 10 7 4 × 10 7 5 × 10 7 Crack length, a (mm) Cycles, N Measured Calculated P max = 2.04 kN 4.6 kN R = 0.1 Figure 8.71. Crack growth behavior in airfoil geometry with LSP. value of K res was computed at that threshold condition. From this, a curve of K res against crack length was deduced and subsequently used to predict the growth rate behavior using a da/dN curve having the appropriate threshold value. The approach taken and the numerical results are shown in Figure 8.72 which utilizes a value of K th from other experiments. The curve indicated as “calculated K res ” is a visual best fit to the computed values of K res over the range in crack lengths for which data were obtained. Using this curve and a fit to da/dN data, the growth rate/arrest behavior, shown in Figure 8.71, was well predicted. Using a very crude approximation to the shape of the K res profile from [67] and the K solution for the airfoil, Equations (8.12) and (8.13), some numerical simulations are presented to illustrate the growth/arrest behavior that can occur within a compressive residual stress field. In Figure 8.69 the shape of the applied, residual, and total stress HCF Design Considerations 461 –60 –40 –20 0 20 40 60 1.5 2.0 2.5 3.0 3.5 4.0 K (MPa √m) Crack length, a (mm) K max,a K th Deduced K res Calculated K res Figure 8.72. Maximum applied and residual K versus crack length. intensity are shown. The values of stress and K are presented in a dimensionless form and no attempt is made to use real numbers in this illustrative example. Additionally, a dimensionless value of the threshold stress intensity is shown as a dashed line correspond- ing to an arbitrarily chosen value of K ∗ = 07. For the numbers chosen, the total stress intensity barely exceeds the threshold at a crack length in the vicinity of a =1mm. At a slightly longer crack length, if the crack could have been started at that crack length with a precrack or starter notch, the crack would be expected to arrest since the applied stress intensity then dips below the threshold. The same scenario is depicted in Figure 8.70 in a slightly expanded scale. Here, in addition to the total stress intensity at a dimensionless stress (or applied load) of  ∗ = 1, the total K is shown for values of  ∗ = 11, 1.2, and 1.5. At  ∗ =1, crack arrest is expected at a length of approximately 0.8 mm. If the load is increased to  ∗ = 11, the crack should resume propagation until it reaches a length “a” approximately 1.6 mm. Further increase in load to  ∗ = 12 should produce continued crack propagation until a length of about 3mm where the crack barely arrests. If the applied load is  ∗ = 15, no crack arrest should occur once the crack has been started. This is equivalent to the case cited above where crack growth data were obtained throughout a test on an LSP sample. 8.6.3.3. Crack growth retardation Another example of the use of compressive residual stresses to retard crack growth is found in the work of Kang et al. [68] who used residual stresses from welding to retard crack growth in edge cracked specimens of a structural steel. They evaluated two methods for evaluating K eff in the presence of a residual stress field. The first, defined as the R  method, considers K and R to be determined from the applied K values and the 462 Applications computed K res as shown in Equations (8.15) and (8.16). The value of K res as a function of crack length, a, is found from the conventional weight function method as K res a =  a 0  res xpx a dx (8.24) where px a is the weight function for the specific geometry used and  res x is the initial residual stress field in an unflawed specimen. It is important to point out that the computations are valid only when the entire crack is open under the total loading, that is the stress field due to the applied load plus the residual stress field. The second method used in [68] involved the use of experimental closure (or crack opening) measurements to determine the effective value of K. The two methods, based on calculations of K res and K op , were used to compare experimental crack growth rate measurements. It was found that, in the region where the residual stresses changed sign from compression to tension, the R  method gave non-conservative results in comparing experimental crack growth rates to predicted values. The predictions, based on measured values of K op , were also not very good unless the load-displacement traces, used to determine K op , were interpreted to distinguish between the opening and the closing portion of the cycle. The authors concluded that the behavior of the crack, in the region where the residual stress field went from compression to tension, involved a partial opening of the crack. This would violate the assumption of a fully open crack in the calculations of the value of K res as indicated above. Their speculation centers around the correct determination of the opening load when the load-up and unload traces are slightly different in the compliance measurements. In determining thresholds for high cycle fatigue, it is important to recognize the limitations of the approaches for determining the effective value of K in crack growth analysis. This, in turn, creates concerns in applying these approaches to the threshold condition for long life (crack arrest) based on experimental measurements on a specimen or component that has been subjected to a residual compressive stress field. 8.6.3.4. A numerical example We conclude the discussion of the use of compressive residual stresses in retarding or arresting cracks with some simple numerical examples of a two-dimensional body with a crack growing from the edge from applied far-field loads. In addition to the applied K, a residual K will reduce the total K depending on the residual compression induced in the specimen. We specifically examine the effects of the depth of the residual stress field and the shape. For this purpose, we take two depths, 0.1 and 1 mm, and two shapes, uniform and triangular, as depicted in Figure 8.73. The shallow depth is representative of a typical magnitude imparted by conventional shot peening while the deeper residual stresses are representative of methods such as LSP or LPB or deep roller burnishing. The triangular profile is a more realistic representation of what has been measured for the HCF Design Considerations 463 0.1 mm 1 mm Shallow Deep Uniform Triangular Depth Depth Figure 8.73. Profiles of residual stress fields used in numerical examples. residual stress field rather than the uniform stress which is a mathematical approximation in some analyses. The residual K values for the stress profiles of Figure 8.73 are shown in Figures 8.74 and 8.75 for a residual stress amplitude (maximum) of 100 MPa for the shallow and deep profiles, respectively. Both stress and K are shown here as positive values but for residual compressive stresses, the stresses are negative as are the values of K. These are only mathematical values of K and although they are negative, they are to be subtracted from the values of K for the applied stresses. Figure 8.74, for the shallow residual stresses, shows that the residual K peaks at about the depth of the stress profile (0.1 mm) for the uniform stress while K peaks at a value of about half of that for the triangular profile. For the deep stress profile, the identical trends are observed. In fact, if the depth is made dimensionless by using a/t, where t is the total thickness, or by using a/d where d is 0 0.5 1 1.5 2 0 0.2 0.4 0.6 0.8 1 Uniform Triangular K (MPa √a) Crack depth, a (mm) σ res = 100 MPa Shallow Figure 8.74. Residual K from the residual stress fields of Figure 8.73 for shallow residual stress profile,  res =100 MPa. 464 Applications 0 1 2 3 4 5 6 7 8 0246810 Uniform Triangular K (MPa √a) Crack depth, a (mm) σ res = 100 MPa Deep Figure 8.75. Residual K from the residual stress fields of Figure 8.73 for deep residual stress profile,  res = 100 MPa. the depth of the residual stresses, then the plots are identical. For K, the numerical values are proportional to √ a. Figures 8.74 and 8.75 are identical, then, if K values are reduced by √ 10 and “a” is normalized to a/d. In both cases, the residual stresses continue to have an effect on the residual K for depths far beyond the depth of the stress profile. It is often (mistakenly) assumed in the literature that the crack is retarded only when it is in the residual stress field. These computations illustrate that it is the value of K, not the stress field at the crack tip, that retards crack growth. As noted previously, the K calculations are valid only for a fully open crack and become invalid when the residual K (compression) exceeds the applied K (tension). 8.6.4. Autofrettage We conclude the discussion of residual stresses with some comments and observations on autofrettage, a metal fabrication technique in which the stock is momentarily subjected to enormous pressure. Also known as strain hardening, the goal of autofrettage is to increase the resistance to fatigue, either crack initiation, or crack propagation, of the part. The technique is commonly used in the manufacture of high pressure pump cylinders, battleship cannon barrels, and fuel injection systems for diesel engines. Autofrettage became an important consideration during World War II when it was first used in the production of large numbers of cannon by the US Army. The pressurization of a cylinder can provide beneficial effects in fatigue by either inducing residual compressive stresses at the inner diameter or through the process of strain hardening the material in the same region. In both cases, the material has to be deformed beyond the elastic limit. The stresses in a cylinder due to internal pressure are easily computed for the elastic case from well-known equations in the theory of elasticity HCF Design Considerations 465 a b r p i Figure 8.76. Schematic diagram of hollow cylinder with nomenclature. such as found in [69]. Denoting the internal pressure by p i , the radial position by r, and the dimensions of the cylinder as shown in Figure 8.76, the hoop stresses are   = a 2 p i b 2 −a 2  1+ b 2 r 2  (8.25) Theses stresses are plotted in Figure 8.77 for several different ratios of b/a, the ratio of the outer to inner diameter of the cylinder. From both the plot as well as from the equation, it is readily seen that the maximum hoop stress is at the inner diameter. The magnitude of this maximum stress is  max = p i a 2 +b 2  b 2 −a 2  at r =a (8.26) 0 5 10 15 20 25 1 1.2 1.4 1.6 1.8 2 b /a = 1.05 b /a = 1.10 b /a = 1.25 b /a = 1.5 b /a = 2 Hoop stress / internal pressure r /a Figure 8.77. Normalized hoop stresses for cylinder with internal pressure. . the stresses are negative as are the values of K. These are only mathematical values of K and although they are negative, they are to be subtracted from the values of K for the applied stresses open at all points along the crack face. If the crack is closed, the linear superposition of Equation (8.15) is no longer valid and the mathematical quantities obtained for negative values of K are. enormous pressure. Also known as strain hardening, the goal of autofrettage is to increase the resistance to fatigue, either crack initiation, or crack propagation, of the part. The technique