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

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

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

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

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

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

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
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

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

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