26 Introduction and Background scenario to explain the events that occurred, however unlikely it might seem. Incidents such as these bring to mind the words of the famous detective Sherlock Holmes who said “when you have eliminated the impossible, whatever remains, however improbable, must be the truth” [13]. Hopefully, the information contained within this book will add to the understanding of many of the aspects of high cycle fatigue material behavior. REFERENCES 1. Wöhler, A., “Über die FestigkeitsVersuche mit Eisen und Stahl” [On Strength Tests of Iron and Steel]. Zeitschrift für Bauwesen, 20, 1870, pp. 73–106. 2. Schütz, W., “A History of Fatigue”, Engng Fract. Mech., 54, 1996, pp. 263–300. 3. Crouch, J.O., “Air Force Turbine Engine Reliability”, presented at the NAS Committee of National Statistics Sponsored Reliability Workshop, Washington, DC, 9–10 June 2000. 4. Engine Structural Integrity Program (ENSIP), MIL-HDBK-1783B (USAF), 15 February 2002. 5. Engine Structural Integrity Program (ENSIP), MIL-STD-1783 (USAF), 30 November 1984. 6. John, R., Nicholas, T., Lackey, A.F., and Porter, W.J., “Mixed Mode Crack Growth in a Single Crystal Ni-Base Superalloy”, Fatigue 96, Vol. I, G. Lütjering and H. Nowack, eds, Elsevier Science Ltd, Oxford, 1996, pp. 399–404. 7. Grandt, A.F., Jr., Fundamentals of Structural Integrity, John Wiley & Sons, Inc., Hoboken, NJ, 2004. 8. Greenfield, P., and Suhr, R.W., “The Factors Affecting the High Cycle Fatigue Strength of Low Pressure Turbine and Generator Rotors”, GEC Review, 3, No. 3, 1987, pp. 171–179. 9. Hawkyard, M., Powell, B.E., Husey, I., and Grabowski, L., “Fatigue Crack Growth under Conjoint Action of Major and Minor Stress”, Fatigue Fract. Eng. Mater. Struct., 19, 1996, pp. 217–227. 10. Barenblatt, G.I., “On a Model of Small Fatigue Cracks”, Eng. Fract. Mech., 28, 1987, pp. 623–626. 11. Miller, K.J., “The Short Crack Problem”, Fatigue Engng Mater. Struct., 5, 1982, pp. 223–232. 12. Lankford, J., “The Influence of Microstructure on the Growth of Small Fatigue Cracks”, Fatigue Engng Mater. Struct., 8, 1985, pp. 161–175. 13. Sir Arthur Conan Doyle, The Sign of Four, 1890. Chapter 2 Characterizing Fatigue Limits 2.1. CONSTANT LIFE DIAGRAMS Unlike in LCF where life is a function of applied stress or strain, and stress ratio is a parameter, HCF tries to deal with infinite life, endurance limits, or FLSs. In the long-life regime, then, the issue is whether HCF failure will occur under a stress level that is exceeded, or infinite (or very long) life can be achieved if the stresses are below that level. If, in the ideal world, S–N curves had a horizontal asymptote at some reasonably achievable number of cycles, then the terminology infinite life, endurance limit, or FLS corresponding to some large number of cycles would all refer to the same information. In the early days of fatigue, it was generally felt that such an endurance limit existed and that the information on the stresses corresponding to this limit could and should be represented by some simple equation or plot. Material capability of this type is often called the run-out stress, but it is more correct to refer to it as the FLS corresponding to a given number of cycles, typically 10 7 or greater. In the 1850s, Wöhler [1] introduced the fatigue limit at 10 6 cycles because that was considered the useful engineering life for many HCF applications such as steam engine components. Further, it would appear that it was also a practical limitation based on available test techniques. While 10 6 and 10 7 have been used widely as the fatigue limit for many years in many applications, recent data indicate that FLSs for some materials may continue to decrease at cycle counts up to and beyond 10 10 cycles [2, 3]. Today, high speed rotating machinery can achieve service lives approaching and perhaps exceeding cycle counts of 10 9 –10 10 . Thus testing must include large numbers of cycles representative of potential service exposures. This, in turn, requires high frequency testing capability or extremely long testing times. 2.2. GIGACYCLE FATIGUE In the field of “gigacycle fatigue,” indicating lives of the order of 10 9 cycles or higher, data have been generated indicating that some materials do not have a fatigue limit within the range of cycles tested using ultrasonic test machines. For many materials, the behavior is as depicted in Figure 2.1 where a dual behavior is noted. For example, the observed fatigue behavior in the region between 10 7 and 10 9 cycles has shown that the S–N curve still has a slightly negative slope [4]. The duality of the S–N curves has been linked in many cases with fractographic observations that partition the behavior 27 28 Introduction and Background Stress Number of cycles Surface initiation Interior initiation Figure 2.1. Schematic of observed behavior in gigacycle fatigue. into failures that initiate at or near the surface, and failures that initiate subsurface. In the latter case, longer lives are observed as depicted in the figure. An example of such observed behavior is illustrated in Figure 2.2 for 2024 T3 aluminum [5]. In this case, two mechanisms are observed from fracture surfaces. Mode A denotes specimens that failed from broken inclusions in the material. These events occurred for tests that lasted less than 10 6 cycles. Mode B refers to longer life specimens where failure is believed to have been initiated by persistent slip bands. If all of the data are taken together, the scatter in life is extremely large, especially at stress levels corresponding to average lives around 10 6 cycles. However, if the data are segregated according to the two observed mechanisms, the scatter for each mode is much less and the duality of the S–N curve is more easily distinguished. The authors attribute the scatter in lives about 10 6 cycles to the competition between these two mechanisms of crack initiation. In this particular material, 160 200 240 280 320 360 400 10 4 10 5 10 6 10 7 10 8 σ max (MPa) N f (Cycles) Mode A Mode B Figure 2.2. S–N curve for 2024/T3 aluminum alloy (R =01) from [5]. Characterizing Fatigue Limits 29 the two mechanisms of crack initiation are not distinguished by being on or away from the surface. Data on two materials from another source [6] illustrate the more common demarcation between surface and subsurface initiation as depicted schematically in Figure 2.1. In Figure 2.3, data on Ti-6Al-4V are shown that were obtained with an ultrasonic test apparatus operating at 20 kHz as well as with a conventional machine operating at 150 Hz. The two frequencies produced data that could not be distinguished from each other and are not separated in Figure 2.3. The first part of the curve up to 10 7 cycles appears to have a fatigue limit above 600 MPa below which infinite life could be expected to occur. It is only with the addition of the longer life data that the drop in the S–N curve is noted and a fatigue limit of approximately 340 MPa is observed corresponding to 10 10 cycles. The sharp drop in fatigue strength between 10 7 and 10 10 cycles is attributed to a change in failure mechanism whereby fatigue changes from surface to subsurface initiation as identified in the figure. The authors also point out the possibility that mean stress (these experiments were conducted at R =0) plays an important role in the decrease in fatigue strength at very high fatigue lives. Data from the same investigation [6] on a martensitic stainless steel produced results that have some similarities but some differences from that on titanium. The results, shown in Figure 2.4, show no indication of a drop in fatigue strength as longer lives are reached. This tends to validate the test procedure involving ultrasonic excitation of the specimen. On the other hand, the change from surface to subsurface initiation at very long lives is also observed in the stainless steel. The concept of material behavior at the surface of a specimen compared to that at the subsurface is discussed in Chapter 5 in conjunction with shot peening. However, the behavior at the surface being different from that at the subsurface has been a common 0 200 400 600 800 1000 1200 10 3 10 4 10 5 10 6 10 7 10 8 10 9 10 10 10 11 Surface initiation Subsurface initiation Run out Curve fit Maximum stress (MPa) Ti-6Al-4V Mill annealed R = 0 Fatigue cycles Figure 2.3. Fatigue data for Ti-6Al-4V from tests up to 20 kHz. 30 Introduction and Background 0 200 400 600 800 1000 1200 10 4 10 5 10 6 10 7 10 8 10 9 10 10 10 11 Surface initiation Subsurface initiation Run out Maximum stress (MPa) Fatigue cycles Martensitic Stainless Steel X20CrMoV121 R = 0 Figure 2.4. Fatigue data for tempered martensitic steel from tests up to 20 kHz. observation in many works dealing with gigacycle fatigue where the duality of S–N curves has been observed in some cases, as noted in Figure 2.4. Shiozawa et al. [7] point out that the fracture mode is different in steels in the gigacycle regime and can be characterized, in general, as being either surface initiation or subsurface initiation. In the latter case, while they do not specifically distinguish the internal material being different than the material on the surface as was done by [8], they distinguish the mechanisms of crack initiation as being different. Internal initiations, characterized by the presence of defects which lead to what is termed a “fish-eye” pattern, are deemed to constitute a different fracture mechanism. The two different modes are deemed to have different S–N curves, each one having its own characteristic curve based on stress level and cycle count, dependent on the probability of the dominant mode being present. Figure 2.5, after [7], illustrates the concept of each mode having a different probability of occurrence at Fatigue life Surface failure mode Probability Internal failure mode IS Figure 2.5. Schematic of probabilities for surface and internal failure modes [7]. Characterizing Fatigue Limits 31 different fatigue lives. From these concepts, the authors [7] propose that four different types of S–N behavior in steels can take place as illustrated conceptually in Figure 2.6. Each S–N curve corresponds to the relative position of the probability distributions of the internal and surface initiation modes illustrated in Figure 2.5. Type A is the common S–N curve governed by the surface fracture mode with the internal fracture mode occurring (speculatively) at very long or infinite lives as illustrated in Figure 2.4 for martensitic steel. Data on another material, forged titanium plate (Figure 2.7), also illustrate such behavior as shown by Morrissey and Nicholas for Ti-6Al-4V [9]. In this figure, the 20 kHz S I Type A Fatigue life Stress amplitude S I Type B Fatigue life Stress amplitude S < IS << I S I Type C Fatigue life Stress amplitude S 2 IS 3 I S I Type D Fatigue life Stress amplitude Figure 2.6. Classification of S–N curves using the concept of duplex S–N curves [7]. Max. stress (MPa) 0 100 200 300 400 500 600 10 4 10 5 10 6 10 7 10 8 10 9 60 Hz (servohydraulic) Not Cooled Cooled Cycles Figure 2.7. S–N fatigue data obtained at 20 kHz on Ti-6Al-4V forged plate [9]. 32 Introduction and Background data were obtained with and without cooling applied to the specimen. The temperature rise of under 100  C and the data show that temperature effects had no effect on the fatigue behavior. The data obtained at 20 kHz are also compared with 60 Hz data obtained on a conventional test machine and again show no difference. In this figure, the data at both frequencies at cycle counts of exactly 10 7  10 8 , and 10 9 are all run-outs. Further, the staircase test results provide an FLS of 510 MPa at both 10 8 and 10 9 cycles, but these points are not shown in the figure. In this case, long-life data points at 10 8 and 10 9 cycles, not shown, were obtained using statistical methods to establish the mean of the long-life fatigue strength at specified cycle counts (see Chapter 3 for a discussion of these methods). Type B is the well-known step-wise S–N curve for which the probability distributions for the surface and subsurface modes are separated. Type C is called a duplex S–N curve and occurs when the probability distributions of Figure 2.5 are close to each other. Type D is the S–N curve governed only by the internal fracture mode because the probability distribution for the internal mode is at a shorter lifetime than that for the surface mode. In these figures, the dashed lines represent the hypothetical curves that are never obtained experimentally because failure is dominated by another mode present at a lower cycle count for the given stress level. Whether the duality of the S–N curves is attributed to different mechanisms or internal versus surface behavior, the observations and proposed explanations available in the literature point to a conclusion that there are two different S–N curves. Further, the data in hand seem to indicate that there is no general relationship between the two curves. One of the main distinctions between internal versus surface initiation, particularly in the long-life HCF regime where initiation constitutes a major portion of life, is the often overlooked role of environment. While surface initiation occurs in the laboratory or operational test environment, subsurface initiation is representative of vacuum or an inert environment that can have a major role in extending the fatigue life compared to behavior in air. Gigacycle fatigue, often conducted using ultrasonic test machines, has also been per- formed rather extensively on rotating bending apparatus operating at nominal frequencies under 60 Hz, thereby taking much longer times to reach the very high cycle regime. Compared to axial resonance testing where uniform stresses are achieved, under rotating bending the maximum stress occurs at the surface. For subsurface initiations, the stress is lower than at the surface but can be corrected for the actual stress at the location of the fatigue origin. This is not always done in the literature. Nonetheless, comparison of short- and long-life behavior and mechanisms can be performed using this technique. S–N curves from rotating bending tests, demonstrating the dual mechanism behavior of surface versus subsurface initiation, are shown in Figures 2.8 and 2.9 for two high- strength steels [10]. SUJ2 is a high-carbon-chromium-bearing steel while SNCM439 is a nickel chrome molybdenum steel. The data shown are for specimens that were ground during the machining process and contained surface residual stresses. The lines are those of the authors whereas the actual data may or may not really demonstrate a plateau in Characterizing Fatigue Limits 33 800 1000 1200 1400 1600 1800 10 3 10 4 10 5 10 6 10 7 10 8 10 9 10 10 Surface initiation Subsurface initiation Stress amplitude (MPa) Cycles to failure SUJ2 R = –1 Figure 2.8. Fatigue behavior of SUJ2 steel under rotating bending. 600 800 1000 1200 1400 1600 10 3 10 4 10 5 10 6 10 7 10 8 10 9 10 10 Surface initiation Subsurface initiation Run out Stress amplitude (MPa) Cycles to failure SNCM439 R = –1 Figure 2.9. Fatigue behavior of SNCM439 steel under rotating bending. the 10 6 –10 7 life regime after which the curve drops. A single continuous curve without a plateau could easily be drawn to fit the data. However, the data show the dual behavior where surface initiation occurs at shorter lives whereas subsurface initiation from an inclusion occurs for longer lives in both materials. This seems to contradict the notion that a mean stress contributes to the observed decrease in fatigue strength, certainly not for all materials. Further, the behavior under bending fatigue is similar to that observed under axial loading though the values for fatigue strength are generally different. Based on these observations, when presenting data in the form of S–N curves into the gigacycle regime, it is important to note the conditions under which the tests were conducted including the 34 Introduction and Background maximum number of cycles attempted (definition of run out). Additionally, for failed specimens, information about the mechanism such as surface versus subsurface initiation should be provided. The general subject of gigacycle fatigue has been addressed in the Euromech Collo- quium 382, held in Paris in June 1998, the papers of which were published in a special issue of a journal [11]. A second conference was held in Vienna in July 2001. Papers from the International Conference on Fatigue in the Very High Cycle Regime were published in another special issue [12]. A third conference was in Japan in September 2004. A sum- mary of a large amount of experimental data on gigacycle fatigue as well as a description of the test apparatus used in such studies can be found in the book by Bathias and Paris [13]. To date, there have been numerous attempts made to understand the very long-life of materials and the apparent lack of a fatigue limit using ultrasonic test machines [5]. They show that in higher strength materials such as spring steel and martensitic stainless steel there is no fatigue limit up to 10 9 cycles whereas in carbon steel and cast iron a fatigue limit may exist somewhere beyond 10 8 cycles. While the lack of a fatigue limit is seen in many of the reported tests in the literature, materials such as cast 319 aluminum show a definite fatigue limit beyond 10 8 to 10 9 cycles [14]. The data of Morrissey and Nicholas [9] shown earlier in Figure 2.7 for a forged titanium alloy also show evidence of a definite fatigue limit beyond 10 9 cycles. Ultrasonic test devices are not yet common laboratory equipment and require skill and experience for their proper use. There are not a large number in existence, so the generation of data in the long-life regime is still fairly limited. As an engineering compromise, cycle counts of the order of 10 7 using conventional testing machines operating at their maximum frequencies are often used as the definition of run-out or an endurance limit. The gigacycle fatigue community certainly takes issue with such a low number based on data that continue to be generated on many structural materials in the longer-life regime. 2.3. CHARACTERIZING FATIGUE CYCLES Before the twentieth century was very old, there were already several methods for repre- senting endurance limit data. Under constant stress-controlled conditions, there are five variables that can be used to characterize the fatigue cycle that is shown schematically in Figure 2.10, only two of which are independent parameters:  max , the maximum stress,  min , the minimum stress,  mean = max + min /2, the average or mean stress,  alt = max − min /2, the alternating stress or half of the stress range, and R = min / max , the stress ratio. Characterizing Fatigue Limits 35 Stress Time Max Mean Min R = min/max (a) (b) Figure 2.10. Schematic of (a) portion of a fatigue cycle included within (b) portion of a more complicated combined LCF–HCF cycle. The stress range is defined as the difference between  max and  min . An alternate nomen- clature, used at times in industry, is the amplitude ratio defined as A =  alt  mean = 1−R 1+R It is somewhat disappointing that, to this day, no general agreement has been reached in the technical community regarding which pair of parameters should be used to char- acterize a fatigue cycle. The reason for this is that each parameter has some physical or mathematical significance, or is convenient to the user. For example, stress ratio, R, may have no real physical significance, but many tests are conducted where R is the parameter that is varied from one test to another and appears in the resultant database as the constant under which the test was conducted. It was probably this type of thinking and the number of available parameters that led the pioneers of HCF research to adopt many different methods for representing endurance limit data. The diagrams on which the data are represented can be classified as constant life diagrams, even though the intent may have been for them to represent endurance limits. For practical purposes, tests were often carried out to some reasonably long life, depending on the machines available, the required number of tests or parameters to be varied, or the patience or available resources of the investigator. 2.4. FATIGUE LIMIT STRESSES The terminology “fatigue limit stress” or “strength” refers to the stress at a constant (long) life that is normally used in place of the endurance limit (infinite life) in a constant life diagram. Methods, particularly accelerated methods, for obtaining such stress values are commonly obtained from S–N plots either by having data at the desired life or . 2004. A sum- mary of a large amount of experimental data on gigacycle fatigue as well as a description of the test apparatus used in such studies can be found in the book by Bathias and Paris [13] and the data show that temperature effects had no effect on the fatigue behavior. The data obtained at 20 kHz are also compared with 60 Hz data obtained on a conventional test machine and again. the machining process and contained surface residual stresses. The lines are those of the authors whereas the actual data may or may not really demonstrate a plateau in Characterizing Fatigue