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176 Effects of Damage on HCF Properties K t = 1.94 K t = 2.25 c a Fracture surface 1.27 70 1.27 1.27R 2.03R 35 10.16 1.27 All dimensions in mm Figure 4.21. Double-notch-tension specimen geometry and crack shape nomenclature. Time K K th max K th min K r ΔK th eff Precrack Threshold test K pc max Figure 4.22. Schematic of LCF–HCF loading history and nomenclature. LCF–HCF Interactions 177 100 200 300 400 10 100 20 50 200 R LCF = 0.1, σ max = 430 MPa R LCF = –1.0, σ max = 265 MPa K max = 5.1 MPa√m, R = 0.1 a/c = Fit R = 0.1 Fatigue limit stress (10 7 cycles) σ max (MPa) c (μm) R HCF = 0.1 Figure 4.23. HCF thresholds at R =01 on a Kitagawa type diagram. 100 200 300 400 10 100 σ max (MPa) c (μm) R HCF = 0.5 20 50 200 R LCF = 0.1, σ max = 430 MPa R LCF = –1.0, σ max = 265 MPa K max = 5.8 MPa√m, R = 0.5 a/c = Fit R = 0.5 Fatigue limit stress (10 7 cycles) Figure 4.24. HCF thresholds at R =05 on a Kitagawa type diagram. The horizontal line in each figure represents the experimentally determined endurance limit of the uncracked specimen corresponding to 10 7 cycles. It can be seen in both figures that the data obtained using LCF at R = 01 (circles) show what appears to be equivalent to an overload effect since the threshold values of stress are consistently above the extrapolated long crack threshold. Conversely, data obtained using LCF at R =−10 (triangles) tend to fall slightly below the projected long crack threshold, the type of effect being representative of what one would expect when a material sees an underload during prior cycling. These results seemed to indicate that an overload type condition (LCF at R =01) retards the subsequent HCF propagation while an underload (LCF at R =−10) might tend to slightly accelerate the subsequent HCF propagation. The apparent overload effect from precracking at a higher stress than that required to propagate the crack 178 Effects of Damage on HCF Properties under HCF is the result of multiple “overloads” since constant amplitude loading was used to precrack the specimens. The apparent multiple overload effect is consistent with observations such as those of Frost [41] who noted that cracks formed by precracking at a higher alternating stress are all stronger than expected and that the “propagation stress” (stress to obtain K th ) is increased if the initial loading conditions are such as to induce compressive residual stresses at the crack tip. It should also be noted here that the overload ratios (OLR = ratio of precrack stress to FLS) in the works of Moshier et al. were relatively small compared to those used in typical studies of overload effects where values of OLR commonly exceed 1.5 in order to produce noticeable effects [42]. While it is not the intent of this book to discuss the governing mechanisms of retardation due to overloads, the reader is referred to the review article by Sadananda et al. [42] who address that subject. In the Kitagawa diagram used to present the threshold data above, all combinations of crack length and stress corresponding to a K solution equal to the threshold value can be plotted to establish the threshold crack-growth line. Although small crack corrections were not used here, the concept of data points below the endurance limit represents a threshold for cracks of a particular size. It is clear that such cracks could not be naturally initiated since they represent stress levels below the endurance limit. While such cracks could be preexisting in the material in the form of initial defects, material with such defects in sufficient quantities would have a lower endurance limit (by definition!). It follows, therefore, that data plotted on a Kitagawa diagram representing a crack length and stress below the endurance limit generally represent a condition where the crack was initiated above the endurance limit. The question can be raised as to whether any point on a Kitagawa diagram is unique or, instead, is dependent on the history of loading in getting to that point. The long-crack threshold, for example, represents a data point that may be dependent on loading history. While standards have been set for determining this threshold by following a predetermined loading history, the history dependence and the existence of a unique long-crack threshold still have to be questioned. This subject is discussed further in Chapter 8. Noting that prior loading history involving multiple overloads and underloads can influ- ence measured FCG thresholds in surface cracks at notches, Moshier et al. [6] conducted a series of long crack experiments to further study the load-history effect on threshold. The study was conducted on both Ti-6Al-4V (Ti-6-4) and Ti-5Al-2Sn-2Zr-4Mo-4Cr (Ti- 17). They measured the threshold under constant R, HCF loading, using an increasing K step-loading procedure as shown above (Figure 4.22). LCF precracking was done using a range of K values where the corresponding K max representing LCF loading was greater than those of the subsequently measured HCF thresholds, thereby producing the multiple overload condition. This condition is represented schematically in Figure 4.22. Data obtained at R HCF =0.1 showed a linear relation between the HCF threshold and the K max and R values of the LCF precrack for long cracks in C(T) specimens. Figures 4.25 LCF–HCF Interactions 179 0 2 4 6 8 10 12 0 5 10 15 20 25 R = 0.1 R = 0.1 SRA ΔK th (MPa√m) ΔK Precrack (MPa √m) ΔK th = 0.303 ΔK Precrack + 2.999 4.6 MPa √m Load shed threshold Figure 4.25. Threshold data at R = 01 for Ti-6Al-4V with and without stress relief annealing (SRA) R LCF = 01. and 4.26 show the observed linear relationship between K of the (LCF) precrack and the K threshold for testing at R HCF = 0.1 for Ti-6-4 and Ti-17, respectively. A similar plot to Figure 4.25 that includes small-crack specimen data is presented in Figure 4.27 and shows that small-crack data follow the same linear trend. A number of the precracked specimens were SRA to remove any residual stresses developed in the precracking. It can be observed that for SRA, the threshold obtained is independent of precrack history in both materials. The data also agree fairly well with the long-crack threshold obtained under conventional load shed techniques, shown in the figures as “Load Shed Threshold.” However, the data from SRA lie slightly below the long crack threshold, more for Ti-6-4 than for Ti-17. The authors speculated that the reason for the slightly lower threshold is that SRA completely eliminates load-history effects and residual stresses whereas the 0 2 4 6 8 10 12 0 5 10 15 20 25 R = 0.1 R = 0.1 SRA ΔK th (MPa√m) ΔK Precrack (MPa√m) ΔK th = 0.366 ΔK Precrack + 2.137 3.5 MPa√m Load shed threshold Figure 4.26. Threshold data at R =01 for Ti-17 with and without stress relief annealing (SRA) R LCF =01. 180 Effects of Damage on HCF Properties 0 2 4 6 8 10 12 0 5 10 15 20 25 Long crack R = 0.1 Long crack R = 0.1 SRA Small crack R = 0.1 LCF ΔK threshold (MPa √m) ΔK Precrack (MPa √m) R = 0.1 HCF ΔK = 4.6 MPa√m Long crack threshold Figure 4.27. Experimental relationship between precrack and threshold stress intensity for R = 01 threshold tests in long crack C(T) specimens and small surface flaws. long crack threshold retains whatever residual stresses or closure is present at the end of the conventional load-shed procedure for threshold testing. The resulting threshold could even be considered to be a “true” material threshold, although no such definition of a true threshold seems to exist. To represent the linear relation between K th of the HCF testing and the K Precrack ,a simple model was developed to fit the data and to use in follow-on analytical modeling of other load-history-dependent threshold test results. The model is purely empirical in nature and was not intended to be a general overload model, many of which already exist in the literature. 4.3.1.1. An overload model While overload models to estimate FCG retardation abound in the literature, there are few to estimate thresholds. A model for estimating threshold from the LCF precrack level was developed using an approach similar to those used in closure and crack retardation modeling. From the data shown in Figures 4.25 and 4.26 for long cracks in C(T) specimens in both Ti-6-4 and Ti-17, an effective threshold was found to be a material constant. This was used to formulate a relationship between K max threshold and K r , where K r is defined as the stress intensity factor level above which K is effective and below which it is not. This reduces the stress intensity factor range to an effective range: K eff th =K max th −K r (4.21) where the various terms are defined in the loading schematic, Figure 4.22. K r is analogous to the opening stress intensity factor K op used in closure-based models, and K pr , the crack LCF–HCF Interactions 181 propagation intensity factor [42]. After many iterations to consolidate the experimental data, K r was found to be best represented in the form K r = K max pc +K min th (4.22) where is a fitting parameter and subscripts “pc” and “th” refer to the LCF precrack and HCF threshold, respectively. The effective stress intensity factor range, K eff th , is assumed to be a material constant representing the minimum effective K to propagate a crack. The linear relation between K th and K pc , observed experimentally, is then written as K th = 1−R th K eff th 1−R th + 1−R th 1−R th 1−R pc K pc (4.23) The linear fit of threshold data at R = 01 was used to determine the parameters K eff th and K eff th = 323 and = 0294 for Ti-6-4 and K eff th = 229 and = 0353 for Ti-17). This model was then applied to data obtained at different values of R for the HCF threshold testing. The ability of this equation to represent threshold data obtained at these other values of R after precracking at R = 01 is illustrated in Figure 4.28 for long cracks in C(T) specimens for Ti-6-4. The empirical fit to the data is made only for the data obtained at R = 01. The fit at R =01 provides the constants which, in turn, produce the model predictions shown for other values of R in the figure. It can be seen that the model provides an excellent representation of the entire data set. An equally good correlation was obtained for T-17 by fitting the data at R =01 and predicting the behavior at R =05 and R =07 (see [6], Figure 4.5). 0 2 4 6 8 10 12 0 5 10 15 20 25 R = 0.1 Expt R = 0.3 Expt R = 0.5 Expt R = 0.7 Expt R = 0.3 Model R = 0.5 Model R = 0.7 Model R = 0.1 precrack Δ K Precrack (MPa√m) Δ K threshold (MPa√m) Figure 4.28. Experimental data and model predictions for long crack data. 182 Effects of Damage on HCF Properties 4.3.1.2. Analysis using an overload model For the general case of an overload model applied to any geometry, the K solution for the specific geometry is written in the form K = √ a fa (4.24) Denoting the precrack condition with subscript pc, the K used to produce an overload condition, which we refer to as the LCF precrack, is K pc = pc √ a fa (4.25) For the particular overload model used here, the linear relation between K th and K pc , Equation (4.23), requires K th = K pc when there is no overload as in the limiting case of steady-state crack growth near threshold. In this case, K is the long crack threshold, K lc th , a material constant, and Equation (4.23) can be rewritten as K th =K pc +K lc th 1− (4.26) where is an empirical constant. In these equations, K represents the maximum value of K at a given value of R, but could also represent K throughout. To extend the modeling to small cracks, the concept of El Haddad et al. [3] is employed to produce the endurance limit stress for arbitrarily small cracks, and the K solution, Equation (4.24) for long cracks. Following the procedure of modifying the material capability (threshold stress intensity) instead of the crack length, the threshold value of K, denoted by K th , is reduced for small cracks in the following manner: K th =K th a a +a 0 1/2 fa fa+a 0 (4.27) where a 0 , as defined by El Haddad, and modified for the more general K solution of Equation (4.24), is given by Equation (4.4) previously, but repeated here: a 0 = 1 K th e fa 0 2 (4.28) which can be solved for a 0 either graphically or iteratively. The crack length a 0 is usually not considered in the fa term of Equation (4.24) when deriving the effective K in Equation (4.27). For each crack length, the threshold stress is obtained from a combination of Equation (4.27) and the K solution, Equation (4.24), in the following equation: = K th √ a fa (4.29) LCF–HCF Interactions 183 4.3.2. Examples of LCF–HCF interactions As an illustrative example, a SEN specimen is cracked at a stress corresponding to the endurance limit, e , and another one at a stress above the endurance limit, =125 e in this case. The K solution is approximated in Equation (4.24) by setting fa = 1. From the K solution, and the small-crack correction, Equation (4.27), the threshold condition for a baseline chosen arbitrarily as K th = 5 MPa √ m and e = 300 MPa is represented in the form of a Kitagawa type diagram in Figure 4.29. The diagram shows the smooth transition from a fracture mechanics dominated long-crack threshold, where the slope on a log-log plot is −05, Equation (4.24), to a crack-length independent endurance limit, e = 300 MPa, for arbitrarily short cracks. If, however, the crack is initiated at a stress equal to or above the endurance limit, = e and =125 e for the two cases here, then the longer the crack developed under this “LCF” or precrack condition, the higher will be the value of K OL as determined from Equation (4.25). For each case modeled, pairs of values of a and K pc are obtained. Then K th can be calculated from Equation (4.26) and modified for short cracks using Equation (4.27) to get an effective value of K th . The relation between and a is obtained from Equation (4.29) to determine points on the Kitagawa diagram as illustrated in Figure 4.29. The divergence of the curves for = e and = 125 e is due to a combination of the increase in K pc due to increase in LCF crack length, as well as the form of the specific model used here, Equation (4.26), which shows that the threshold increases with increase in K pc . The assumption implied in this illustrative analysis, which could be applied to any actual experimental geometry, is that the endurance limit is a material constant at a given R and does not depend on any prior load history which does not produce cracks. This assumption has been validated under several experimental conditions described at the beginning of this chapter. The net result, 20 40 60 80 100 300 10 –1 10 0 10 1 10 2 10 3 10 4 Overload model σ e = 300 MPa K th = 5 MPa√m Baseline σ = σ e σ = 1.25σ e Stress (MPa) Crack length (μm) Figure 4.29. Kitagawa diagram for theoretical SEN specimen behavior. 184 Effects of Damage on HCF Properties as shown in Figure 4.29, is that the stress required to produce growth of a crack under HCF, which developed under constant load at or above the endurance limit, is below the endurance limit stress but above the baseline stress calculated from fracture mechanics with a small crack correction. The difference is attributed to the load-history effect. The concept and model described above was applied to an expanded database from experiments on notched specimens in which very small cracks could be detected [4]. The K solution was obtained for the notch geometry with a surface flaw, and the overload effect on the threshold for K was obtained from the model, Equation (4.23). To account for a small crack effect, K was then modified in accordance with Equation (4.27). The specific value of the a/c ratio in the K solution was used for each data point in the calculation of a 0 which is where the endurance limit stress and K th equation cross on a Kitagawa diagram. The data and the model fit are shown in a Kitagawa diagram in Figure 4.30 where the model prediction is based on Equation (4.23) with the appropriate fit to data from long crack experiments. The data, summarized in Figure 4.30, include tests where both R =−1 and R = 01 were used for the LCF precracking and both SRA and no stress relief were applied prior to HCF testing for both cases. LCF precracking produced cracks with depths covering a range from c =16 to 370m. For modeling predictions, three curves are shown. In one case, the threshold from long cracks (K max =51 MPa √ m) is used without any overload or small crack correction. This corresponds to Equation (4.24) with K being a constant using the appropriate aspect ratio in the K solution. The second analytical curve corresponds to a prediction without any small crack correction. This simply incorporates the overload effect, Equation (4.23), in the calculation of K th . The third prediction is identical, but the small crack effect is taken into account through the use of Equation (4.27). Several 100 200 300 1 10 100 LCF R = 0.1 LCF R = –1.0 LCF R = 0.1 with SRA LCF R = –1.0 with SRA Long crack ΔK th Long crack ΔK th with a 0 Prediction without a 0 Prediction with a 0 Maximum stress (MPa) Crack depth, c (μm) 400 500 Fatigue limit Figure 4.30. Experimental data and model predictions on a Kitagawa diagram. LCF–HCF Interactions 185 features of the analytical predictions should be noted. First, the curve for constant K is not a straight line of slope −05 as is often noted in a Kitagawa diagram. The reason for this is that the K solution is of a more complex form, Equation (4.24), than for the simple case where fa =1 corresponding to the simple √ a dependence of K on crack length. The second feature for the cases of a constant load to produce an overload effect is that the model predicts behavior quite different than the constant K solution as illustrated for the simple illustrative example shown in Figure 4.29. The correlation of the data with the analytical models, as shown in Figure 4.30, shows the ability to extend a long-crack overload model to small surface flaws in a notch geometry. The cracks developed under R =01 loading have thresholds at R =01 which are predicted, somewhat conservatively, by the linear overload model of Equation (4.23) with the small crack (a 0 ) correction of El Haddad. It is clear from the data that the threshold is increased when there is an overload history applied during the LCF stage, similar in concept to the retardation effect observed in growing cracks. LCF cracks formed under R =−1 loading, on the other hand, produce thresholds which are predicted by the (constant) long-crack threshold value of K max =51MPa √ m. The maximum stress or K level in the threshold testing exceeded that of the LCF precracking only for the smallest crack formed in tests at R =−1 where precracking was at a maximum stress of 265 MPa. The remainder at R =−1 and all at R =01 had lower values of K max in threshold tests at R =01 than those in precracking. Thus all tests with the exception of one data point could be expected to demonstrate some type of overload effect. However, unlike precracking at R = 01, precracking at R =−1 seemed to have no overload effect on the subsequent threshold. The compression portion of the fully reversed loading cycle appears to negate any beneficial effect of the tension portion of the cycle upon the subsequent threshold. For both prior loading at R =01 and R =−1, stress relief annealing produced a condition where the subsequent threshold corresponded to the long crack threshold as shown by the solid symbols in Figure 4.30. These data, as well as the data from precracking at R =−1 without SRA, show no history-of-loading effect on the subsequent threshold determined at R = 01. The best representation of these threshold values would be the long crack threshold with a small crack correction for the smallest of cracks in these tests. The small crack correction can be seen in Figure 4.30. Another important observation from the experimental data of Figure 4.30 is that for cracks having depths, c, less than 30–40 m (the approximate value of a 0 as determined by the intersection of the long crack threshold curve with the constant endurance limit stress) is the value of the stress to produce crack extension. For these small cracks, while calculated values of K might be considerably below the long crack threshold, the values of stress at threshold are only slightly below or at the endurance limit stress within a reasonable scatter band. This indicates that very small cracks are not very detrimental in reducing the fatigue threshold stress, even though the location of the failure corresponds to the location of the initial cracks. . overloads, the reader is referred to the review article by Sadananda et al. [42] who address that subject. In the Kitagawa diagram used to present the threshold data above, all combinations of crack. in the calculation of a 0 which is where the endurance limit stress and K th equation cross on a Kitagawa diagram. The data and the model fit are shown in a Kitagawa diagram in Figure 4.30 where. condition for a baseline chosen arbitrarily as K th = 5 MPa √ m and e = 300 MPa is represented in the form of a Kitagawa type diagram in Figure 4.29. The diagram shows the smooth transition from a fracture
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