626 Appendix I
from Hawkyard et al. [12] and are assumed to decrease linearly from 5MPa
√
mto
225 MPa
√
matR=07 and maintain values of 5 and 225 MPa
√
matR<0 and R>07,
respectively. The stress intensity factor solution of Raju and Newman [13] for surface
cracked smooth bars is utilized, and assumptions are made concerning the geometry of
the crack such that the stress intensity factor may be approximated as
K =

a
163

1+
a
D

(I.5)
where a is the crack depth and D is the diameter of the bar. Due to the presence of the
notch, the applied stress is multiplied by a K
T
of 2 in order to determine K
HCF
, K
LCF
,
a

such that the calculated value of N is within a specified tolerance of the
requested value of N. Solution time increases with increasing N and increasing fraction
of total life spent in crack propagation, especially when the crack is actively growing in
both HCF and LCF cycles. For the most computationally intensive cases considered, the
solution took no more than one to two minutes on a 100-MHz personal computer.
Correlation with experiment
A limited number of HCF–LCF tests on notched bars were conducted by Guedou and
Rongvaux [9]. Ti-6Al-4V bars were cycled at room temperature with n =1800 HCF cycles
per CCF load block and R
HCF
=085. The resulting initiation lives are shown along with
those for LCF-only tests in Figure I.4. All results are plotted as a function of maximum
stress (
m
+
a
). A power law fit to the LCF-only initiation data is also shown. This fit
was used to define the S–N curve shown in Figure I.3 which, in turn, defines the constants
for use in the modified Goodman equation over intermediate values of alternating stress
range. Predictions for the HCF–LCF initiation life using Equation (I.2) are also shown
Appendix I 627
10
2
10
3
10
4
600
700
800

at the crack tip has been taken as two over the
entire life of the crack. As discussed earlier, this is a conservative assumption which is
expected to underestimate the propagation life. Predicted values for the HCF–LCF tests
are, of course, the same under either endurance limit assumption. The drastic change in
slope is unrelated to that of the finite endurance limit assumption shown in Figure I.4. In
Figure I.5, it is a function of K
th
. The number next to each predicted point corresponds
to the fraction of propagation life over which the crack does not grow in HCF. (The
crack grows in CCF during the remainder of the propagation life.) The four highest
stress predictions begin growing in CCF immediately upon crack initiation (Path 3 in
Figure I.2). In these cases, crack propagation life is underestimated by approximately
an order of magnitude. At lower values of maximum stress (and constant R
HCF
), K
HCF
is initially below K
th
at crack initiation and the crack must propagate for a period in
LCF-only before CCF crack growth can occur (Path 6 in Figure I.2.). Better correlation
with experiment is obtained at lower stresses, possibly indicating that the values of K
th
628 Appendix I
10
1
10
2
10
3
600

has been used here.
Finally, Figure I.6 shows the correlation between the experimentally measured and
predicted vales of total life. The total life is dominated by the initiation life. Thus,
the assumption of no endurance limit gives better correlation with the total life. The
assumptions concerning the K
T
at the crack tip during crack propagation have little effect
on the correlation with total life. Unless otherwise noted, subsequent results are calculated
under the assumption of no endurance limit.
Effect of LCF on HCF capability
Figure I.7 shows the computed values of allowable 
a
as a function of 
m
for failure in
10
7
HCF cycles (plus one LCF cycle representing the initial loading to a peak stress of

a
+
m
.) Line 1 indicates the alternating and mean stress combinations which will cause
initiation in 10
7
HCF cycles with no superimposed LCF cycles N =1. If the number of
cycles to initiation is increased from 10
7
to 10
8

HCF
=085
and n=1800.
0
50
100
150
200
250
0 200 400 600 800 1000 1200
Alternating stress (MPa)
Mean stress (MPa)
LINE 3: N
I,LCF
= 1
LINE 1: N
I,HCF
= 10
7
LINE 2: ΔK
th,HCF
R = 0
R = 0.7
N
I,HCF
= 10
8
LINE 4: ΔK
th,LCF
Figure I.7. Haigh diagram as predicted by analysis showing underlying mechanisms which govern the shape

. Any stress
state above or to the right of this line will cause tensile failure on the first cycle. Finally,
above and to the right of line 4, crack growth will occur under LCF loading once a crack
initiates. Note that line 4 coincides with line 2 at low values of 
m
, above which HCF
crack growth will occur (for a 50-m crack). This is due to the assumption that the crack
is closed at  =0 so that for R
HCF
< 0, K
LCF
=K
HCF
. Under the assumptions of this
analysis, the region where failure can occur due to either HCF or LCF is above the heavy
line in Figure I.7.
The safe design space defined for N =1 in Figure I.7 can be determined for other
values of N. Figure I.8 shows solutions for various values of N and n where Nn=10
7
.
As N increases, the allowable alternating stress decreases at higher values of mean stress.
As expected, a finite number of LCF cycles reduces the maximum mean stress that may
safely be applied to the structure. Comparison of Figures I.3 and I.8 indicates that the
limiting value of mean stress for a given value of N corresponds to the allowable stress
range for a specified value of N in Figure I.3.
Parametric studies
Inherent in the previous analyses were several assumptions concerning the behavior of
the material. Among these were the variation in K
th
with R and the form of the initiation


=

10
2
, n

=

10
5
N

=

10
3
, n

=

10
4
N

=

10
4
, n

analysis.
0
1
2
3
4
5
6
0 0.2 0.4 0.6 0.8 1
Closure model
CTOD model (Taylor, 1988)
Experimental (Hawkyard et al., 1996)
Δ
K
th

(MPa √m)
R
Figure I.9. Comparison of CTOD-based and closure-based K
th
vs R models with experimental data
on Ti-6-4 [12].
632 Appendix I
rationalized through closure arguments and is incorporated into the previous analyses. As
an alternative, K
th
may be defined as
K
th
=K

versus 
m
for various
combinations of HCF and superimposed LCF. Here, however, predictions are made using
both CTOD and closure-based K
th
R values. Note that while the differences between
the methods are significant for small values of N at high mean stress, when N becomes
sufficient to cause reductions in the allowable alternating stress (versus pure HCF), the
results become relatively insensitive to the K
th
model in use.
0
50
100
150
0 200 400 600 800 1000 1200
Alternating stress (MPa)
Mean stress (MPa)
ΔK
th
by
CTOD theory
Initiation life
= 10
7
ΔK
th
by
closure theory

R = 0 R = 0.5
R = 0.905
R = 0.818
R = 0.967
ΔK
th
by closure theory
ΔK
th
by CTOD theory
Figure I.11. Effect of K
th
vs R model on the CCF Haigh diagram as predicted by analysis.
0
50
100
150
200
250
0 200 400 600 800 1000 1200
N = 1
N = 10,000
N = 1
N = 10,000
Alternating stress (MPa)
Mean stress (MPa)
R = 0 R = 0.5
R
= 0.905
R

an endurance limit effectively raises the initiation line (line 1 in Figure I.7). Note that
the intersection of the solution for 
end
= 300MPa and the R = 0 line corresponds to
half the endurance limit stress range which again matches the 10
7
initiation life point in
Figure I.3.
CLOSURE
Discussion
The numerical results presented here are based on simple models of crack initiation and
propagation. Many potentially important phenomena are neglected such as the possi-
ble non-linear accumulation of initiation damage, the effect of previous cycling on the
instantaneous endurance limit, acceleration in the HCF crack growth rate due to periodic
underloads (LCF cycles), reduction in K
th
as a function of the number of LCF cycles,
small crack effects, hold time effects on LCF cycles, and many more. Therefore, these
results must be viewed as preliminary, giving only a qualitative indication of how LCF
and HCF cycling interact to reduce overall life. Although simple, the method satisfacto-
rily predicts the effects of combined HCF–LCF loading (see, e.g. Figure I.5) despite the
limited amount of experimental data available for calibration. Additional experimental
results should allow for better calibration and will pave the way for incorporation and
assessment of many of the above phenomena.
Although the Goodman assumption is used in accounting for mean stress effects in crack
initiation, the resulting Haigh diagram obtained differs from the Goodman assumption
for total life in two regions as shown, for example, in Figure I.7. At very low mean
stress, the predicted response curve follows lines 2 and 4, and is significantly steeper
than that displayed by the Haigh diagram. At high mean stress, the allowable alternating
stress follows line 2 and remains constant up to very high mean stress. In both high and

fall along a line approximating 
a
+
m
=
ult
. In any case, the resulting Haigh diagram
diverges significantly from the Goodman assumption, especially at high values of mean
stress.
It is interesting to note that this behavior of the model can qualitatively predict the exper-
imental observations of Suhr [6]. In this work, a 12% CrNiMo blading alloy was cycled
at a mean strain of 0.01 and variable alternating strain amplitude of 0–600 microstrain.
Tests were conducted in HCF-only; HCF with superimposed periodic underloads to zero
strain every 10
5
HCF cycles (combined HCF–LCF loading); and with a fixed number
of LCF cycles preceding HCF-only cycling to failure (or runout). The results indicate
that at high values of alternating strain, the number of cycles to failure is independent
of whether LCF cycles are distributed throughout the HCF loading or all applied prior
to HCF loading. As the alternating strain amplitude is reduced, a transition in behavior
occurs such that specimens with all LCF cycling applied prior to HCF cycling exhibit
significantly longer lives than specimens subjected to the same number of LCF cycles
distributed periodically between blocks of HCF cycles. The explanation for this behavior
is as follows. At high 
HCF
, K
HCF
is above K
th
and once a crack initiates, it grows in

a
=0 is, by definition, the
636 Appendix I
0
50
100
150
0 200 400 600 800 1000 1200
Alternating stress (MPa)
Mean stress (MPa)
Safe design space for HCF/LCF conditions
(N =10
4
, n =10
3
)
N = 1 (n = 10
7
) Initiation line
(GOODMAN ASSUMPTION
)
(Line A
)
σ
max
CORRESPONDS TO
Δ
σ (R = 0)
FOR N
LCF

line
(line B), the safe design space proposed here is somewhat larger than the safe design space
predicted numerically. The discrepancy is greatest at the intersection of the two lines and
its magnitude is dependent upon the details of the numerical analysis. For example, if
we consider the existence of an endurance limit for initiation damage, then, as shown in
Figure I.12, the “knee” of the curve (at 
m
≈600MPa) has much less curvature and the
safe design space proposed here is more accurate. That there is less curvature in the knee
for the case of an endurance limit is easily explained. For such a case, any stress point
Appendix I 637
on the Haigh diagram below the Goodman initiation line (line A in Figure I.13), HCF
cycles will cause no damage of any form. Thus below this line initiation is brought about
only through LCF cycling. To the left of the line defined by 
a
+
m
=
LCF
(line B in
Figure I.13), initiation in LCF requires in excess of 10
4
cycles and the part is safe for
the required life. However, if there is initiation damage attributable to HCF cycles below
the endurance limit, then the actual safe design space will lie within the proposed safe
design space, as the numerical solution does in Figure I.13. Prediction of the exact form
of the safe design space will require further refinement of the numerical model.
Conclusions
Predictions have been made for the safe design space under combined HCF–LCF loading
in terms of allowable values of mean and alternating stress using data from the literature

8. Walker, K., “The Effect of Stress Ratio During Crack Propagation and Fatigue for 2024-T3
and 7075-T6 Aluminum”, Effects of Environment and Complex Load History for Fatigue Life,
ASTM STP 462, American Society for Testing and Materials, Philadelphia, 1970, pp. 1–14.
9. Guedou, J Y. and Rongvaux, J M., “Effect of Superimposed Stresses at High Frequency on
Low Cycle Fatigue”, Low Cycle Fatigue, ASTM, Philadelphia, 1988, pp. 938–969.
10. Chesnutt, J.C., Thompson, A.W., and Williams, J.C., “Influence of Metallurgical Factors on
the Fatigue Crack Growth Rate in Alpha-Beta Titanium Alloys”, AFML-TR-78-68, Wright-
Patterson AFB, OH, May 1978 (ADA063404).
11. Grover, H.J., Fatigue of Aircraft Structures, Government Printing Office, Washington, DC,
1966.
12. Hawkyard, M., Powell, B.E., Husey, I., and Grabowski, L., “Fatigue Crack Growth under Con-
joint Action of Major and Minor Stress”, Fatigue Fract. Eng. Mater. Struct., 1996, pp. 217–227.
13. Raju, I.S. and Newman, J.C., “Stress-Intensity Factors for Circumferential Surface Cracks in
Pipes and Rods Under Tension and Bending Loads”, Fracture Mechanics: Seventeenth Volume,
ASTM STP 905, J.H. Underwood, R. Chait, C.W. Smith, D.P. Wilhem, W.A. Andrews, and J.C.
Newman, eds, American Society for Testing and Materials, Philadelphia, 1986, pp. 789–805.
14. Dowling, N.E., “Notched Member Fatigue Life Predictions Combining Crack Initiation and
Propagation”, Fatigue of Engineering Materials and Structures, 2, 1979, pp. 129–138.
15. Taylor, D., Fatigue Thresholds, Butterworths, London, 1989.
16. Taylor, D., A Compendium of Fatigue Thresholds and Crack Growth Rates, EMAS, Warley,
UK, 1985.
17. Bell, W.J. and Benham, P.P., “The Effect of Mean Stress on Fatigue Strength of Plain and
Notched Stainless Steel Sheet in the Range From 10 to 10
7
Cycles”, Symposium on Fatigue
Tests of Aircraft Structures: Low-cycle, Full-scale, and Helicopters, ASTM STP 338, ASTM,
Philadelphia, 1963, pp. 25–46.
18. Madayag, A.F., Metal Fatigue: Theory and Design, John Wiley and Sons, Inc., New York,
1969.
Index

Haigh diagram for, 47–51
notch fatigue at, 254–9
Endurance limit, 27, 123
notches, 241–2
see also Constant-life diagrams, Jasper
Energy considerations:
in FOD, 345–7
ENSIP (Engine Structural Integrity Program):
HCF issues in, 5, 10, 17, 21, 499–516
see also JSSG (Joint Service Specification
Guide)
Factor of safety, 379–81
Fatigue notch factor, 216–22, 347–52, 531
relations with SCF, 217–22, 441
Field experience, 13–16, 25–6, 204–12,
324–9, 336–8, 424, 493–6
see also Foreign object damage (FOD)
Findley parameter, 244, 296
Foreign object damage (FOD), 12, 322–75,
558–99, 600–16
analytical/numerical modeling, 368–71,
582–91
life prediction, 592–8
perturbation study, 371–4
JSSG requirements, 323, 600–16
bird ingestion, 600–7
ice ingestion, 607–10
sand and dust ingestion, 610–16
laboratory simulation, 338–44, 570–82
residual stress, 344–5

600–16
Kitagawa diagram, 145–8, 159, 163–5, 223,
248–51, 253, 303, 423
Low Cycle Fatigue, interactions with, 11,
145–212, 396–8, 504–5, 617–38
combined cycle fatigue, 204–12, 619–37
erroneous behavior, 197–207
nomenclature, 196–7
notched specimens, 166–7
Material quality, 40, 384
Modeling errors, 381–4
Notch fatigue, 213–60, 531–41
crack-like behavior, 222–8
mean stress effects, 228–38
stress gradients, 242–51, 532–6
critical distance approaches, 242–6
see also Stressed surface area (FS)
Worst Case Notch (WCN) approach,
536–40, 595–8
Probabilities and statistics, 6–7, 76–80,
91–105, 120–2, 517–30
application to FOD design, 425–30
bootstrapping, 527–30
Dixon and Mood method, 95–9, 120,
518–27
material quality considerations, 384–5
SEV distribution, 110–12
Random fatigue limit (RFL) model,
109–22, 362
Ratcheting, 83–5

“artificial staircase”, 105–6
step testing, 65–70, 75–89, 355
last loading block, 85–8
Thresholds for HCF:
engineering approach for determination, 423
experimental considerations, 409–23
compression precracking, 416
“jump-in” method, 409–12
load-shed method, 411, 414–16
Index 641
fatigue limit strength, 9, 27, 35–41, 405–7
effects of defects on see Defects,
effects of
role of residual stresses, 63–4, 344,
430–62
fracture mechanics approach, 66, 170–83,
386–90, 403–4, 508
crack closure, 418–22, 454–8, 630–4
K
max
–K concept, 419–22
K
PR
concept, 422
overloads and load-history effect, 12,
170–82, 190–3, 414
mechanisms, 412–14
Wöhler, A., 3, 472–3
Wöhler diagram (S–N curve), 4, 7, 18, 47,
382–4, 405–7