416 Applications a value of K th =24 MPa √ matR =08 was found. For the constant K max , increasing K min test, a value of K th = 22 MPa √ m at a final load ratio of 0.92 was obtained. They conclude that this latter value corresponds to a lower bound threshold for small crack propagation, even though the test is conducted on a long crack specimen. By comparison, for the identical material, Ti-6Al-4V, small crack growth behavior from naturally initiated and FOD-induced small cracks was not observed below a K of 29 MPa √ m. 8.4.5.1. Compression precracking The concern for history effects in determining a threshold that can be used for engineering purposes in design has led to considerations of alternate techniques. Forth et al. [24] present data using the conventional load-shedding technique recommended by ASTM at constant R and constant K max on 7075-T7351 aluminum. In addition, they tested using a compression precracking scheme to avoid development of plasticity-induced closure during the precracking. From the results for this particular material, and from other reported observations, they conclude that the threshold and near threshold growth rates obtained during the K-shedding portion of a test are not necessarily the same as those obtained under constant load, increasing K tests. In their work, constant load levels were maintained after reaching threshold. Although these tests are generally conducted at the load level (and K level) where the decreasing K tests were terminated at the defined threshold of 10 −10 m/cycle, there are many cases where the load has to be increased in order to start the crack growing again. In the work of Forth et al. [24], however, the crack grew immediately. In their case, they conclude that remote crack closure may still be present in decreasing K tests dependent on the load history, thus producing an artificially high threshold. Further, oxide buildup is also observed when plane stress conditions occur at the specimen surface. By comparison, the authors introduced a compression– compression precracking scheme and a K-increasing test procedure at constant R to determine a threshold that they consider to be independent of load history. In the case of the aluminum alloy that they tested using middle through-crack M(T), specimens, they observed that the crack growth curve generated using the constant R load-reduction method can be non-conservative by more than an order of magnitude in growth rates near threshold. In summarizing this work [25], the authors conclude that tests with history effects do not accurately describe the material threshold. They observed the constant R load-reduction test generated artificially high threshold values when compared to steady-state data. 8.4.5.2. Load-shed rates The ASTM standard for threshold testing defines the threshold as the asymptotic value of K at which da/dN approaches zero. The operational definition recommends that the growth rate be 10 −10 m/cycle and that the rate of decrease of K be limited. To HCF Design Considerations 417 explore the possibility of obtaining a realistic threshold for design, Bain and Miller [26] used a surface flaw specimen (Kb bar). This specimen geometry represents the (high) stress levels as well as the surface flaw geometry constraint expected in many highly stressed turbine engine components. This work was a continuation of earlier work by Sheldon et al. [27] where the effects of high load shed rates were explored. The very gradual shed rate recommended by ASTM, C =−008mm −1 [see Equation (8.6)], cannot be used with the surface flaw geometry in the Kb bar because the specimens are not physically large enough to allow the stress intensity factor to be shed at this rate before the crack would break through the specimen at the back face. Larger specimen sizes would require extremely high loads to achieve the required K levels. With these constraints on specimens and testing, much higher shed rates than those recommended by ASTM were explored. Experiments were conducted using shed rates from C =−0295 mm −1 to C =−1181mm −1 on Ti-6Al-4V titanium alloy. The threshold measured from these tests was similar to that obtained using a compact tension specimen with a conventional shed rate of C =−008 mm −1 [27]. Statistical analysis of the data showed that the results using high shed rates had similar scatter and threshold values as compared to those obtained using either compact tension or single-edge-notched specimens and, most significant, resulted in a much faster determination of threshold. The results for the value of threshold from this study are summarized in Table 8.1. Threshold data such as the type described above can be used in design for both the threshold conditions when a crack exists or to compute the crack growth rate in the near threshold regime. To do this, a crack growth model that relates da/dN to K is needed. Since data on threshold and low growth rates normally exist for different values of R, and design conditions also exist at various values of R, a method of consolidating data at various values of R is also required. The threshold data discussed above, which were obtained at values of R from −10 to 0.8 [28], were incorporated into a sigmoidal crack growth model using an effective K defined using a dual Walker exponent relation [29]: K eff =K 1 −R m−1 m = m + for positive values of R m = m − for negative values of R Table 8.1. Summary of threshold values on Ti-6Al-4V at room temperature, R =01, from study by Bain and Miller [26]. (K in MPa √ m Specimen type Shed rate, C mm −1 Number of tests Low value High value Average value −3 minimum SEN −1181 13 383 598 4.58 318 C(T) −0236 to −1181 11 414 534 4.70 367 Kb-bar −0295 to −1181 5 387 460 4.23 346 418 Applications For the Ti-6Al-4V titanium alloy, a values of K eff at threshold of 3.829 ksi √ in. was used with values of m + = 072 and m − = 0275. This approach seems to provide an engineering method for representing threshold data over a wide range of values for R. The curve representing this dual Walker model, shown in Figure 8.36 (without data), was found to represent data on the same material from a variety of sources using several different specimen geometries. The shape of this curve is discussed later. An extensive review of fatigue thresholds as well as their relationship to fatigue limits, including an extensive bibliography, can be found in the paper by Lawson et al. [30]. 8.4.6. Crack closure Crack closure is mentioned often in the discussion of crack growth at rates anywhere from threshold and near threshold to rapid propagation. The concept of crack closure is attributed to Elber [31, 32] and relates to the phenomenon of a crack closing above the minimum cyclic applied load due to some obstacle in the crack wake. Closure in the early days of the concept was attributed mainly to plastic deformation at the crack tip leaving a wake of deformed material behind as the crack advances. This is one reason why small cracks are commonly assumed to be open since they have not propagated a sufficient distance for closure to develop in the wake of the crack. In subsequent years, closure has been observed and attributed to many different phenomena including debris such as oxides forming behind the crack tip [33], roughness from the irregular deformation and fracture patterns (crack tortuosity) forming on the mating surfaces [34], and a number of other phenomena [35, 36]. If closure is present, the effective crack driving force is considered to be K eff as shown schematically in Figure 8.37 rather than the total applied 0 1 2 3 4 5 6 7 –1 –0.5 0 0.5 1 R Ti-6Al-4V Room temperature Threshold ΔK (ksi in.) Figure 8.36. Representation of crack growth threshold with Dual Walker model. HCF Design Considerations 419 K max K min K cl ΔK = K max – K min ΔK eff = K max – K cl Time K Figure 8.37. Schematic showing definitions of K and K eff . driving force, K, obtained from LEFM. The rationale behind this formulation is that the crack tip sees no driving force when it is closed. 8.4.6.1. K max –K concept Crack-growth-threshold data reported earlier in this chapter and many other sets of data reported elsewhere (see [37, 38], for example) follow a trend described and modeled by Schmidt and Paris [39] which is shown schematically in Figure 8.38. The model is based, in part, on the crack closure concept using the notion that two threshold quantities, K ∗ maxth and K ∗ th , both defined in the figure, are constant and independent of R. Threshold conditions for various values of R are illustrated schematically in Figure 8.39. If the concept of crack closure is introduced, a constant value of K cl can be assumed (as shown) in order to produce a condition where K eff = constant (see Figure 8.37) is K max,th Propagation region ΔK th ΔK K max * * Figure 8.38. Schematic of threshold boundary of Schmidt and Paris [39]. 420 Applications K max,th K cl ΔK th Low R High R K * * Figure 8.39. Schematic of threshold conditions at various values of R for K max –K threshold concept. the threshold condition. It is significant to point out that closure does not have to be introduced, measured, or assumed when applying the K max –K threshold concept. As seen in Figure 8.38, it predicts that the measured values of thresholds, K th and K maxth , will be load-ratio independent, respectively, above and below a transition R. In the work of Ritchie [36], this value was estimated to be approximately R =05. Thus, while the arguments explaining the observed behavior of the values of threshold (and near-threshold growth rate behavior) have commonly been linked to the concept of crack closure, the data can also be interpreted in terms of intrinsic behavior governed by two distinct parameters, a threshold for the range of stress intensity, K ∗ th , and a threshold for the maximum stress intensity, K ∗ maxth , both shown in Figure 8.38 [40]. The K max –K threshold concept simply requires that two thresholds have to be exceeded before crack growth can occur. Instead of the “ideal” or simple behavior of the threshold stress intensity depicted by the right angle in the K max –K plot of Figure 8.38, the behavior in some cases cannot be described by a constant value of K above some critical value of R. Rather, the threshold is found to decrease in many cases with increasing R [41] as depicted by curve B in Figure 8.40(a) rather than being constant as depicted by the curve A in the same figure. The threshold data of [41], as well as those from other sources [28] were well represented by the dual Walker curve shown above in Figure 8.36. Note that the data include values obtained at negative R. Noting the simple relations among K max K min K, and R, K =K max −K min =K max 1−R R = K min K max (8.7) and noting that the loading condition can be completely described by two independent quantities, we find that an alternate manner of representing threshold data is the often- used plot of the threshold stress intensity as a function of R as in Figure 8.40(b) for the two cases of the data representation shown in Figure 8.40(a). Note that the linear plot for case B in Figure 8.40(a) using K max as the x-axis is nonlinear in the plot against R in Figure 8.40(b). To further add confusion to the representation of threshold data, often HCF Design Considerations 421 1 1.0 0.5 R = 0 R = 0.5 R = 0.7 R = 0.9 A B 3 5 K max /K max,th ΔK /K max,th (a) ΔK /K max,th A B 0 0.2 R 0.4 0.6 0.8 1 0 0.5 1 (b) * * * Figure 8.40. Schematic of two-parameter threshold in plots of (a) K versus K max and (b) K versus R. in an attempt to explain governing mechanisms, the threshold can be represented as a plot of K max as a function of R as shown in Figure 8.41 for the same cases A and B from Figure 8.40. The data represented by case B in the above are often interpreted as representing an additional mechanism in addition to being governed by fixed values of K max and K as governing parameters. Boyce and Ritchie [41], for example, raised the possibility of high R thresholds being associated with sustained load cracking because of the high values of K max associated with high R threshold values as shown in Figure 8.41. In their case, from experiments on Ti-6Al-4V, they concluded that in the observed behavior, similar to that represented by case B above, the intrinsic threshold behavior is itself K max -dependent and such sensitivity to K should be considered in developing threshold models. In a similar manner, a constant value of K th for small values of R may only be an approximation and the actual shape of the curve should be taken into account. 422 Applications 0 1 2 3 4 5 0 0.2 0.4 0.6 0.8 1 K max R A B Figure 8.41. Schematic of two-parameter threshold in plot of K max versus R. So, when plotting threshold data or examining the consequences of such data for design purposes, the possible governing mechanisms or consequences of extrapolating to other conditions should be considered. It is important to recognize the various ways in which the data can be represented and some of the interpretations of the meaning of the shape of the various plots. A review of some of the key considerations that have evolved over many years in the use of the two parameter approach to fatigue thresholds involving the use of K max and K is given in [42]. 8.4.6.2. Crack propagation stress intensity factor Lang [43] introduced an alternate to a crack-closure-based model for threshold using a crack propagation stress intensity factor, K PR , which is defined in the equation K eff =K max −K PR +K T (8.8) where the term K PR +K T has the same mathematical significance as the K cl term in closure-based threshold models. In the interpretation of the Lang formulation, K T is an inherent material threshold and K PR is an experimentally derived quantity that is determined from the onset of crack propagation in an increasing K test started below the material threshold until the onset of crack growth is detected. The mathematical description and the experimental determination of K PR are not related to closure in any manner and no measurement of closure is either defined or attempted. While K T is defined as an inherent threshold or a material constant, K PR is a variable with R, for example, under constant amplitude cycling. The variation of K PR with R is determined experimentally and produces a threshold plot of K versus K max which is not necessarily HCF Design Considerations 423 linear nor does it represent a constant value of K above some critical value of R. After certain load sequences involving overloads or underloads are applied, “master curves” can be obtained for a given material, but only after extensive testing. Essentially, K PR must be determined as a function of the prior loading history which includes K max , unloading ratio, and number of overloads. A notable difference of the K PR approach relative to crack closure modeling is that the K PR approach predicts that even under very high RR>08 steady-state conditions, K PR is found to be greater than K min . Further, the observed behavior is not attributable to crack closure. Lang [43] hypothesizes that K PR is controlled primarily by the residual stress state ahead of the crack tip, and that it represents the point in the loading cycle where these stresses become tensile. As in crack-closure modeling where closure is difficult to measure and those measurements are subject to interpretation, the crack tip stress field is even more difficult if not impossible to measure. In both methods, however, finite element modeling can be used to justify the approach and to correlate with experimental observations. 8.4.7. An engineering approach to thresholds We introduce here an engineering approach to the determination of a threshold stress intensity that can be used in design when a spectrum that involves a combination of LCF and HCF is present in the operating environment. In many cases, a component is subjected to such a spectrum during qualification testing. The test sequence is often used to determine the condition under which the HCF starts to affect the LCF growth rate using step-testing procedures. The condition where the HCF starts to have a significant affect is deemed to be the threshold for HCF crack propagation, namely the threshold stress intensity for HCF. For complex spectra as well as complex geometries where the K analysis is not fully validated, an engineering approach can be used. This approach is nothing more than the threshold approach described in Chapter 4 for simple LCF– HCF load spectra. Here, it is applied to any complex spectrum that contains an HCF component that may lead to failure when the crack length due to crack propagation from the spectrum exceeds the critical threshold value for the HCF component. Overload and retardation effects are automatically considered even though they are not specifically identified. This approach involves the use of a Kitagawa type diagram for the specific geometry and the specific load spectrum. Applied load (stress) levels at which the HCF part of the loading spectrum starts to have an appreciable effect on the crack growth rate can be plotted as a function of the crack length at which this occurs on a Kitagawa diagram as shown schematically in Figure 8.42. The K solution for a crack in the specified component geometry can then be fit to the experimental data on the Kitagawa diagram. The best-fit curve can then be used to infer the crack growth threshold stress intensity, without having to resort to a laboratory sample test procedure to obtain that same quantity. 424 Applications Log stress, s Log flaw size, a Threshold stress intensity, K th a 0 Figure 8.42. Schematic of Kitagawa diagram for component with complex geometry and complex K solution. Even if the K solution is not of the highest fidelity, but that same solution is used in design, the threshold for the onset of crack growth due to the HCF portion of the loading can be accurately determined. For the endurance limit stress, the same spectrum applied to the same component geometry is used to establish the FLS of the uncracked body, typically using step testing or staircase testing as described in Chapter 3. Without consideration of small crack effects, this approach will provide an accurate assessment of the threshold conditions without having to resort to laboratory testing and considerations of history effects in the method used for threshold determination as described in the section above. 8.4.8. Observations from field failures One of the methods for the evaluation of the conditions leading to HCF failures under combined LCF–HCF loading, as described in Chapter 4, is examining the fracture surfaces of failed parts. In examples cited, LCF often precedes the onset of HCF, the latter consuming only a small fraction of life. The determination of LCF life is typically much easier than determining the HCF threshold, which may be loading-history dependent, since LCF loading in rotating components is related to rotational speeds and the resultant centrifugal load-induced stresses which are well characterized in many instances. HCF loading, on the other hand, is usually related to some type of vibratory condition that is not always well defined, particularly in cases where the vibration is somewhat random. If a crack initiates in LCF, and then accelerates in HCF, the crack length at which this transition takes place can sometimes be determined from examination of the striations on a fracture surface of a failed part. Several instances of this situation have occurred in field incidents on US Air Force engines. The fracture surfaces revealed the crack length at which the transition took place. Knowing, or speculating, on the HCF stress ratio present when the vibratory stresses started contributing to the crack extension, and assuming the HCF Design Considerations 425 threshold stress intensity at that value of R was the one corresponding to long crack laboratory data with no load-history effects, the vibratory stress magnitude could be deduced from either a Kitagawa diagram (including a short crack correction, if necessary) or a fracture mechanics analysis. In some cases, the HCF markings on the fracture surface were not easily discernable because the HCF growth rates were extremely small and the resulting striations were observable only under very high magnification under SEM. Nonetheless, LCF–HCF interactions can produce markings on a fracture surface which allow an indirect determination of the HCF stresses that were present. In the presence of a complicated mission cycle, such an approach may be the only way to determine the role of HCF and HCF threshold on the failure conditions when LCF–HCF conditions prevail. Several limitations exist on the use of fracture surface markings to establish thresholds for HCF under combined LCF–HCF loading conditions. First is the fact that striations can be distinguished on a fracture surface only for certain combinations of material and growth rate regime. Materials such as aluminum show striation markings much better than other materials such as titanium. Second, under loading spectra where some of the cycles go into compression R < 0, the fracture surfaces come into contact and the features may be obliterated because of the smearing due to compressive loading on the surfaces. What appears at first to be a very useful tool for analyzing threshold behavior from fractography then turns out, unfortunately, to be non-applicable in certain cases. 8.5. PROBABILISTIC APPROACH TO HCF/FOD DESIGN The technical issues of the FOD problem in terms of experimental and analytical deter- mination of the severity of damage were discussed in Chapter 7 and the accompanying Appendix G. Here we consider some of the aspects of getting information of that type into the design process. We can note that probability-based methods are becoming increasingly important in the design process. The present ENSIP document contains requirements dealing with the probability of failure due to HCF in terms of the acceptable number of failures per 10 7 engine flying hours whereas the original version did not deal at all with probability of failures. Of the many aspects of HCF design, the problem of FOD is cer- tainly one that lends itself to a probabilistic design approach. FOD not only involves the probability of incurring damage of a certain severity at a certain location in a component, but it also involves the probability of a vibratory stress of a given amplitude occurring at the location of the FOD site as well as the variation in the endurance limit of undamaged material. An attempt to incorporate the various aspects involved in the probability of failure from FOD is presented conceptually in Figure 8.43. “Material failure strength” in HCF is shown for an undamaged material. The mean or median fatigue strength, repre- sented by the dashed line, is usually the quantity used in design. Additionally, a −3 . 416 Applications a value of K th =24 MPa √ matR =08 was found. For the constant K max , increasing K min test, a value of K th = 22 MPa √ m at a final load ratio of 0.92 was obtained threshold models. In a similar manner, a constant value of K th for small values of R may only be an approximation and the actual shape of the curve should be taken into account. 422 Applications 0 1 2 3 4 5 0. crack length at which this transition takes place can sometimes be determined from examination of the striations on a fracture surface of a failed part. Several instances of this situation have