226 Effects of Damage on HCF Properties
out that close to the fatigue limit (under HCF), blunt notches and sharp notches behave
differently in respect to their crack-growth mechanisms.
When notches act like cracks, the mechanism leading to a fatigue limit is the growth of
small cracks from the notch tip which may become non-propagating cracks. The criterion
for the fatigue limit is the onset of crack propagation from an arrested crack and not crack
initiation. Taylor [14] points out that these cracks are always short cracks and they arrest
because their threshold values increase faster than the applied stress intensity. This occurs,
in general, under a stress field involving steep stress gradients such as at the tip of a very
sharp notch. For a plain or blunt-notched specimen, on the other hand, non-propagating
cracks are not found, especially for very short crack lengths. Taylor goes on to attribute
the fatigue limit in such geometries to the arrest of cracks at a grain boundary, the arrest
defining the fatigue limit. This is a material-based limit according to Miller [15] rather
than a limit based on the mechanics of the notch.
A numerical example can be used to show the effects of notch geometry on fatigue
strength. To illustrate the effect of very small notches on the fatigue strength, the formulas
of Peterson for k
t
and Neuber for k
f
are plotted in Figure 5.9 for two different values of the
material parameter a
N
and for three different values of notch depth, d (dimensions of the
notch and the material parameter a
N
are in arbitrary units). The reciprocal fatigue strength,
0
0.2
0.4
0.6


=

0.75
a
N
=

0.5, d

=

0.75
1/K
t
1/k
f
k
t
k
t
from Peterson
k
f
from Neuber
Figure 5.9. Fatigue notch factor as a function of k
t
for increasingly small notches. Constants a
N
and notch

Endo [16]. For low values of k
t
the fatigue strength is approximated by 
0
/k
t
whereas
beyond what is called the branch point [17], the fatigue strength is constant. The difference
between the initiation limit curve and the curve due solely to k
t
is small as k
t
approaches
one. This is recognized in many of the formulas for the true fatigue limit strength that
Crack propagation limit
Crack initiation limit
Fatigue limit of smooth specimens, σ
0
σ
0
/K
t
Stress concentration factor, k
t
1
0
Nominal stress
Non propagating
macro crack
Fracture

f
(or alternately, q), is not a unique quantity for a material given by the value obtained at
R =−1 for very long fatigue life for a given specimen geometry [6]. Instead, the value
of k
f
is often found to depend on mean stress as depicted in Figure 5.11, as shown in
Mean stress
Alternating stress
S
0
k
f
S
0
k
f
S
u
S
u
Smooth
Ductile
Notched
Brittle
Figure 5.11. Simplified representation of smooth and notched behavior for ductile and brittle materials on a
Haigh diagram (after [6]).
Notch Fatigue 229
[6]. For a brittle material, where local notch plasticity is absent, both the mean stress
and alternating stress are reduced on a Haigh diagram since the ultimate stress point
is reduced due to a notch. For a ductile material, the ultimate stress is essentially the

=
S
0
k
t

1−
k
t
S
m
S
u

(5.13)
This is very conservative, since the reduction in fatigue limit is characterized by k
f
, which
is less than k
t
. A better approach consists of replacing k
t
by k
f
in Equation (5.13) which
produces the line denoted by “brittle” in Figure 5.11. In many cases, a best value of
k
f
, not necessarily the value for R =−1, is used in such an approach. A more realistic
approach is to reduce only the alternating stress by a constant factor k

=k
fav

1−
S
m
S
up

+
S
m
S
un
(5.15)
230 Effects of Damage on HCF Properties
0
5
10
15
20
25
30
35
40
0
10 20 30 40 50 60 70
80
Experimental plain specimen
Experimental notched specimen

is
the tensile strength of an un-notched specimen, and S
un
is the tensile strength of a notched
specimen for the particular notch geometry used in constructing the Haigh diagram. This
equation differs from Equation (5.14) in concept because here the fatigue notch factor k
f
is reduced as a function of mean stress rather than reducing only the alternating stress by
a constant fraction.
To compare the two methods of accounting for the effect of mean stress, numerical
examples are provided for a material whose smooth bar behavior can be represented as
a straight line (modified Goodman equation) on a Haigh diagram. To illustrate some of
the characteristics of Equation (5.15), a Haigh diagram where the un-notched behavior is
represented by the straight line modified Goodman equation, Equation (5.12), is shown
in Figure 5.13. In this illustrative example, the smooth bar alternating stress is arbitrarily
taken as 0.5 of the ultimate strength, S
u
(or S
up
in Equation (5.15)) and all of the quantities
are shown in dimensionless coordinates with respect to S
u
. Taking S
un
=105S
up
, values
of the notch strength factor k
f
=15 and 3.0 are chosen. The shape of the curves shows

f
=

1.5, S
un
/S
up
=

1.05
Eqn K
f
=

3, S
un
/S
up
=

1.05
S
a
/ S
u
S
m/ S

as a function of mean stress seem to be effective
methods for accounting for the observed dependence of k
f
on mean stress in ductile metals.
5.8. PLASTICITY CONSIDERATIONS
One of the earliest and simplest attempts to treat fatigue at notches that undergo plastic
deformation at the notch root is due to Gunn [19]. He provided a simplified procedure
to use the smooth bar Haigh diagram to create a notched Haigh diagram based solely
on the elastic stress concentration factor, k
t
, of the notch. The procedure is illustrated
in Figure 5.15 for a linear representation of the smooth bar data on a Haigh diagram.
The procedure is applied in an identical manner for any shape curve. For purely elastic
behavior, the notch stresses are reduced by the same amount for both the alternating and
mean stresses. However, when the nominal stress multiplied by k
t
equals the yield stress,
Y , the notch root will undergo local plastic deformation. Then, it can be assumed that for
higher applied peak stresses, the region around the notch root will undergo cyclic plastic
deformation for one or more cycles until the entire behavior becomes elastic again. At
that point, the mean stress will be reduced from the applied value divided by k
t
since the
maximum stress at the notch does not exceed the yield stress, but the alternating stress
will remain the same. The approximation due to Gunn then uses the smooth bar curve
reduced by k
t
for stresses below yield at the notch root as indicated by the line AB in
Figure 5.15. For regions above where applied stresses cause notch root yielding, the local
mean stress remains unchanged since the maximum stress there is limited to the yield

V-notched specimens whose geometry is shown in Figure 5.16. Some of the data, reported
in [21], are presented in Figure 5.17 where three combinations of notch dimensions
h
d
D
60°
ρ
Figure 5.16. Cylindrical fatigue specimen with circumferential V-notch.
0
100
200
300
400
500
0 200 400 600 800 1000
Smooth bar
Small notch
Medium notch
Large notch
Alternating stress (MPa)
Mean stress (MPa)
Ti-6Al-4V plate
K
t
= 2.8
R = 0.1
R = 0.5
R = 0.8
Figure 5.17. Notch fatigue data on Ti-6Al-4V with k
t

Medium 0203 0254 572 521
Large 0330 0729 572 426
0
100
200
300
400
500
0 200 400 600 800 1000
k
t
= 1
k
f
= 2.8
k
f
= 2.8 (alt only)
Alternating stress (MPa)
Mean stress (MPa)
Ti-6Al-4V plate

K
t
= 2.8
R = 0.1
R = 0.5
R = 0.8
Figure 5.18. Representation of fatigue notch data with k
f

[22]. These data were obtained on smooth and notched specimens that were both stress
relieved after machining and chem milled. While this process is aimed at producing a
pristine material, the experimental data may be different than those obtained after stress
relieving only as was done for the data in [21] (corresponding to a fatigue limit at 10
6
cycles). It should also be noted that the fatigue properties of titanium alloys are very
sensitive to surface finish [23]. In the work in [22], a nominal value of k
t
= 25 was
obtained for double-edge V-notch rectangular specimens with a nominal root radius of
0.033 in. (0.84 mm). The experimental data are presented in Figure 5.19 where each data
point is the average of several tests. The Jasper equation is used to fit the smooth bar
data and produces a good representation of the data points obtained at R =−1, 0.1,
0.5, and 0.8. Three methods of representing the notch data are shown. In the first, only
the alternating stress is reduced by the value of k
f
for R =−1, which is approximately
k
f
= 20. This produces the poorest fit to these notch data. The second method uses a
value of k
f
that is linearly dependent on mean stress between a mean stress of zero and
the ultimate stress, following the suggestion of Bell and Benham discussed above. This
method produces a slightly better fit to the data and has the right general shape, but the
curve falls above the experimental data points for non-zero mean stresses. The third and
final method is just to apply the same value of k
f
to all of the smooth bar data. This
method produces a very good fit to all the data except at R =08 where both plasticity