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

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 =01 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

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.

(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 = 01 for Ti-6Al-4V with and without stress relief annealing (SRA)
R
LCF
= 01.
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

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 = 01 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

, 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

The linear fit of threshold data at R = 01 was used to determine the parameters K
eff
th
and  K
eff
th
= 323 and  = 0294 for Ti-6-4 and K
eff
th
= 229 and  = 0353 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 = 01 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 = 01. The fit at R =01 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 =01 and predicting the
behavior at R =05 and R =07 (see [6], Figure 4.5).
0
2
4
6
8
10
12
0 5 10 15 20 25
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 fa (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 fa (4.25)

K
th
, is reduced for small cracks in the following manner:
K
th
=K
th

a
a +a
0

1/2
fa
fa+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

this case. The K solution is approximated in Equation (4.24) by setting fa = 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 −05, 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  =125
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

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.

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 −05 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 fa =1 corresponding to the simple
√

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 = 01. 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.