56 Introduction and Background
0
0.5
1
1.5
2
–1 –0.5 0 0.5 1
Haigh equation
k
1
= 0.0, k
2
= 1.0
k
1
= 0.25, k
2
= 0.75
k
1
= 0.5, k
2
= 0.5
k
1
= 0.75, k
2
= 0.25
k
1
= 1.0, k

Of both practical and historical significance is the observation that Jasper [32], in 1923,
proposed that fatigue life is related to the stored energy density range per cycle in
a material when evaluating data obtained earlier by Haigh. Applying this concept to
HCF conditions, it can be assumed that all stresses and strains are elastic, thus all
equations represent purely elastic behavior. For purely uniaxial loading, the shaded area
in Figure 2.31 illustrates schematically the stored energy for the cases where loading is
purely tensile R > 0. The energy for the case where R<0, which involves tension and
compression in a single cycle, is illustrated in Figure 2.32 by the shaded area. The stored
energy density range per cycle is then given for uniaxial loading by
U =


max

min
d=
1
E


max

min
d (2.16)
Characterizing Fatigue Limits 57
maxmin
σ
ε
Figure 2.31. Stored energy (shaded) for elastic loading under tension fatigue R > 0.
min

=
m
−
a
(2.18b)
where 
m
and 
a
represent the mean and alternating stresses, respectively.
For the specific case of fully reversed loading, R =−1, the energy is written as
U =
1
2E
2
2
−1
 (2.19)
where 
−1
represents the alternating stress (= maximum stress) at R =−1. For any other
case of uniaxial loading, the following equation is easily derived and can be used to
obtain the value of the alternating stress on a Haigh diagram in terms of stress ratio, R:

a
=

−1
√
2

as the Jasper equation curve of Figure 2.33 for positive mean stresses and have a shape
Characterizing Fatigue Limits 59
0
0.2
0.4
0.6
0.8
1
1.2
0 0.5 1 1.5 2 2.5 3 3.5 4
σ
alt
/σ
–1
σ
mean
/σ
–1
Normalized Haigh diagram
Jasper equation
Figure 2.33. Normalized Haigh diagram representing Jasper equation for positive mean stresses.
0
100
200
300
400
500
600
700
800

Mean stress (MPa)
10
7
cycles
60–70 Hz
Figure 2.35. Haigh diagram for Ti-6-2-4-6.
0
100
200
300
400
500
600
700
800
–400 –200 0 200 400 600 800 1000
ML data
ASE data
Jasper, α = 0.287
Alternating stress (MPa)
Mean stress (MPa)
Ti-6Al-4V plate
10
7
cycles
20–70 Hz
Figure 2.36. Haigh diagram for Ti-6Al-4V plate including modified Jasper equation fit.
and 1917, noted that “this series of tests  was probably the first that ever revealed
any difference between the actions of pull and push in relation to fatigue.”
∗

7
cycles, samples were fatigue tested using the step-loading procedure of Maxwell and
Nicholas [33] that is discussed in the next chapter.
The experimental values for the FLS obtained for a broad range of values of R are
plotted in the form of a Haigh diagram in Figure 2.36. The data are shown as ML in the
figure, and are combined with previously unpublished data from Allied Signal Engines
(ASE), now Honeywell, on the same material. A best fit of the data using the modified
Jasper equation is also shown in the figure. The constant, , in the modified Jasper
equation (2.22) was obtained as 0.287 by fitting to the experimental data obtained from
0
100
200
300
400
500
600
–400 –200 0 200 400 600 800 1000
α = 1
α = 0.75
α = 0.5
α = 0.25
Alternating stress
Mean stress
Constant life diagram
modified Jasper
Figure 2.37. Haigh diagram for modified Jasper equation for various values of .
62 Introduction and Background
R =08toR =−35. The weighted energy was obtained for each data point as a function
of the variable  in Equation (2.22) and the percent least squares error between the
average energy of all the data points and the individual energy values was minimized.

This last condition differs from most previous diagrams or laws that attempt to pin the
data to the ultimate strength for theoretical reasons, or to yield stress for engineering
purposes. One of the main reasons for avoiding an intercept such as the ultimate strength
is that ultimate strength depends on the strain rate at which a test is conducted, and
since most metals exhibit some degree of strain-rate hardening [34]. Data plotted on a
Characterizing Fatigue Limits 63
Nicholas-Haigh diagram are normally conducted at a single frequency which produces a
different strain rate for each different amplitude of alternating stress. For increasing values
of mean stress, as R approaches 1, the strain rate approaches zero since the alternating
stress amplitude is normally found to decrease in this region. This is equivalent to having
a cyclic stress–strain curve near the ultimate stress at an arbitrarily slow rate of loading.
Thus, a fourth condition can be suggested for a completely defined Nicholas–Haigh
diagram, namely, that the frequency of loading for all data points be specified on the
diagram.
It should be noted that data for Haigh or other constant life diagrams for large cycle
counts, generally in excess of 10
7
for HCF applications, are obtained at high frequencies.
Thus, the strain rates associated with such tests are not quasi static. For this reason,
maximum stress values obtained under fatigue testing at high values of R and high
frequency can often be above the quasi-static ultimate strength of the material. This, in
turn, will not produce a smooth plot on a Haigh diagram if a quasi-static ultimate strength
is used to anchor the plot to the x-axis.
As a further consideration, it is recommended that all data points be included, partic-
ularly those obtained at high values of RR→1. In this region, many materials tend
to exhibit creep behavior which may lead to a creep rather than a fatigue failure (see the
discussion in Chapter 3 on testing at high stress ratios as well as the earlier discussion
on Haigh diagrams at elevated temperature). Nonetheless, if the frequency is specified,
the time to failure can be easily deduced. Data of this type should be appended with a
footnote if necessary, but they are still valid design data, even though the mode of failure

max
Alternating stress (MPa)
Mean stress (MPa)
Ti-6Al-4V plate
10
7

cycles
20–70 Hz
Yield stress
Figure 2.38. Nicholas–Haigh diagram for Ti-6Al-4V plate with modified Jasper equation fit shown for alter-
nating stress (solid line) and maximum stress (dashed line).
limit strength. Noting again that compressive residual stresses do not alter the range of
stress or alternating stress applied to a material or structure but, rather, reduce the mean
stress, the Haigh diagram provides data on the reduction of the allowable alternating
stress as a function of mean stress. Using the Jasper equation as a measure of the
allowable alternating stress, if a material is subjected to a vibratory stress at R =0, with
an alternating stress magnitude denoted by 
0
, then the mean stress is also 
0
. Addition
of a compressive residual stress of an equal magnitude 
0
will then result in a mean stress
of zero which, in turn, will increase the allowable alternating stress by a factor of
√
2, as
seen from Equation (2.21). Yet if the original stress state is at R =−1, and a compressive
residual stress is added, then the modified Jasper diagram representing Ti-6Al-4V plate

the crack propagates only due to positive stresses (positive K), then the cracks formed
at lower values of R must be larger than those at higher R because the maximum stress
(positive portion of cycle) decreases with decreasing R (see Figure 2.38). The indirect
evidence of small numbers of cycles in the last loading block is discussed further in
Chapter 3 when dealing with specimens that are deliberately precracked before endurance
limits are determined using step loading.
The speculation about crack initiation and subsequent propagation at negative R can
be explained with the aid of Figure 2.40, where the conditions for initiation compared
to propagation are shown schematically. The initiation of a crack, if associated with the
0 × 10
0
2 × 10
6
4 × 10
6
6 × 10
6
8 × 10
6
1 × 10
7
–4 –3 –2 –1 0 1
ML data
ASE data
Last block cycles
Stress ratio, R
Ti-6Al-4V plate
Smooth bar step tests
10
7