216 Effects of Damage on HCF Properties
1.5
2
2.5
3
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
Semi-cir Tens
D/d = 1.05 Tens
D/d = 1.1 Tens
Semi-cir Bend
D/d = 1.05 Bend
D/d = 1.1 Bend
K
t
r /d
Figure 5.3. Values of k
t
for U-shaped notch.
5.4. FATIGUE NOTCH FACTOR
The fatigue behavior of a notched component is not necessarily governed solely by the
maximum stress at the notch root. In general, the fatigue life or fatigue strength is greater
in a notched component based solely on the stress at the notch root compared to a
smooth bar with that same stress distributed uniformly over the entire cross section. Many
researchers over the years have attributed this observation to the concept that fatigue
behavior depends on the stresses or strains over a critical volume of material. Thus, as the
size of a notch becomes very small, the fatigue behavior tends to be governed more by the
far-field stresses or average stresses. The notch root radius also plays a significant role
in determining the fatigue limit. In the limiting case, as the notch root radius decreases
to zero, the notch will behave like a crack as discussed later. On the other hand, as the
notch root radius becomes very large, the fatigue behavior is governed more by the peak
stress at the notch. In the latter case, in the limit, the stress is given by the far field

stress multiplied by k
t
. From these observations, and from experimental data developed
over many years, equations have been developed for predicting the fatigue behavior in
a notched component based on fatigue behavior in smooth bars where the majority of
databases can be found. The fatigue notch factor, k
f
, is defined as the ratio of the fatigue
strength in a smooth bar to the fatigue strength in a notched bar, that is
k
f
=
unnotched fatigue limit stress
notched fatigue limit stress
(5.1)
This equation is defined, in the strictest sense, for fully reversed loading R =−1 as
well as for a specific number of cycles and a specific specimen geometry. The fatigue
Notch Fatigue 217
notch factor generally covers the range 1 <k
f
<k
t
, indicating that use of k
t
rather than
k
f
can be overly conservative to describe the notch sensitivity. An alternate quantity used
to define the notch behavior of a material is the notch sensitivity, q, defined as
q =

k
f
−1
k
t
−1
(5.2)
The notch sensitivity for a material takes the range 0 <q<1. When q =0, this represents a
material and notch geometry where k
f
= 1, the case where the material is insensitive to the
presence of a notch. When q = 1, this corresponds to k
f
= k
t
, where the material is notch
sensitive and the behavior is governed exclusively by the local stress at the root of the notch.
From a historical perspective, Schütz [1] points out that A. Thum, one of the most
prominent researchers in fatigue in his era, founded the doctrine of “Gestaltfestigkeit,”
which maintains that for a high fatigue strength the shape of the component, as developed
by the designer, is much more important than the material itself. The notch fatigue stress
concentration factor k
f
was also created by Thum who, according to Schütz, correctly
called it a “crutch with which one can limp from the un-notched specimen to the notched
component.” He knew that it has to be determined anew in every case by exactly those
fatigue experiments which it is supposed to make unnecessary.
5.4.1. k
f
versus k

t
relations
There are many empirical relationships between k
f
and k
t
, which try to express the
combined effects of the gradient of stress at the notch root, the individual contributions
of notch root radius and depth of notch, the effect of volume of material being stressed,
and the multiaxial stress state which arises at the tip of a very sharp notch. A review of
formulas for k
f
is presented in [2]. The authors classify all the expressions for k
f
into
one of the three types according to their assumptions: the average stress (AS) model, the
fracture mechanics (FM) model, and the stress field intensity (SFI) model. The latter was
a development by the authors and involves determining a weighted average stress over a
volume of material ahead of a notch. It is an extension of a model based on average stress
over a volume, which, in turn, is a more complicated version of a model that averages
stress over a length at the notch tip. This, in turn, is the AS model. The basic differences
between the two general types of models, the AS and the FM model, are pointed out
in [2]. The AS model assumes that there are no cracks in the material and fatigue occurs
when an average stress over some area (or volume) exceeds the stress required for fatigue
failure in a uniformly stressed bar. The FM model, on the other hand, assumes that there
are cracks in all specimens under fatigue loading. The criterion for the propagation or
non-propagation of these cracks forms the basis of the model. Gradient stress field and
short crack behavior have to be considered in developing the criterion. For HCF, the
threshold for crack propagation is established. The applicability of such a model for finite
fatigue life is more difficult and is not a subject that will be dealt with in this book.

218 Effects of Damage on HCF Properties
Formulas relating k
f
and k
t
all have one thing in common. For very sharp notches,
they all tend to asymptote to a single limiting value of stress, independent of the notch
severity. In the limit, this would correspond to a crack which requires a certain level
of stress to cause fatigue above a threshold stress intensity factor range. On the other
extreme, for very shallow notches, the values of k
f
and k
t
merge, indicating that the lower
stress gradients or stresses away from the notch root tend to have no influence on the
fatigue strength as one approaches the case of a smooth bar.
Some commonly used equations for k
f
, which are based primarily on experimental
data, are listed below. These formulas were based on analysis of stress fields ahead of
notches and fall in the category of AS models as defined by Weixing et al. [2]. The
equation due to Neuber [3] is of the form
k
f
=1+
k
t
−1
1+


a
m
r
(5.3)
The equation of Peterson [4] is
k
f
=1+
k
t
−1
1+
a
m
r
(5.4)
The equation of Heywood [5] is
k
f
=
k
t
1+2

a
m
r
(5.5)
where, in all of the above, a
m

is a material constant
∗
and r is the notch root radius. For a
fixed value of k
t
, the fatigue notch factor, k
f
, is considered to be a function of notch root
radius only. The material constant, a
m
, is different for each of the three equations and is
most commonly used as an empirical constant to fit experimental data. There have been
attempts over the years to relate these constants to tensile properties of materials, but
these have met with only limited success. In fact, the relation between fatigue properties
and tensile properties is a lofty goal indeed, but such a relationship probably does not
exist in the opinion of this writer. The use of k
f
in fatigue design is still recommended in
some textbooks and handbooks (see, for example [6]).
5.4.2. Equations for k
f
A plot of the value of k
f
as a function of notch root radius, r, for each of the three
equations, is presented in Figure 5.4 using a dimensionless value, r/a
m
, for the root
∗
The material constants for the equations of Neuber, Peterson, and Heywood are often written as a
N

, a
P
and
a
H
, respectively. Here we use the nomenclature a
m
to represent any one of them.
Notch Fatigue 219
0
1
2
3
4
5
6
02468
Fatigue notch factor, K
f
r /a
m
10
K
t
= 5.0
K
t
= 3.0
K
t

= 1.5
Neuber
Peterson
Heywood
Figure 5.4. Comparison of fatigue notch factor equations.
radius. Here, a
m
is a material constant for each of the three equations and would not
necessarily be the same for each equation nor for each material. Figure 5.4, plotted for
three arbitrarily chosen values of k
t
= 15, 3.0, and 5.0, shows that each equation has a
trend of having k
f
approaching k
t
for large values of r/a
m
, but k
f
approaches unity for
the Neuber and Peterson equations and approaches zero for the Heywood equation as r
approaches zero. This indicates that the Heywood equation should not be used for very
small values of r because values of k
f
under unity are not physically realistic.
∗
For any
∗
While it would be easy to dismiss the Heywood equation as not being either accurate or physically meaningful

as the notch root radius approaches zero, the concepts behind the equation and the limitations of its use are
carefully pointed out by Heywood in his paper [5]. In particular, he notes that engineering materials have
inherent imperfections that can be treated as very small notches. Thus, Equation (qq05) has validity only above a
notch root radius that produces a value of k
f
of one. For smaller notches, no further reduction in fatigue strength
can occur than is already present in the material due to the inherent imperfections. The equation proposed
by Heywood, Equation (qq05), regards the material as “containing a number of stress concentrations due to
its heterogeneous nature.” For notches of extremely small depth, but for notch radii of “typical proportions,”
Heywood also proposed the following formula
k
f
=
k
t
1+2

a
m
r

k
t
−1
k
t

(qq05a)
that has k
f

equal to unity when k
t
is unity. Heywood, in commenting on the complexity of a fatigue notch
factor formula, notes that “more complicated relationships, such as those requiring two or more notch sensitivity
constants.…will not be considered, as they are not likely to be used in practice, even though they may have a
more satisfactory theoretical basis for their derivation” [5].
220 Effects of Damage on HCF Properties
particular value of notch root radius, r, the curves in Figure 5.4 can be shifted horizontally
left or right by altering the material constant, a
m
. While a
m
has the dimension of length,
it has not been shown to have any real physical significance and should be treated as no
more than an empirical curve fit parameter. It is apparent that these equations were meant
to show trends for very small and very large values of r/a
m
, but were based on data for
stress concentrations common in engineering usage.
It is the opinion of the author that such equations, as listed above, which relate to
k
f
, have limited use in engineering applications unless one is dealing with the same
material under nominally identical applications where a fatigue database already exists.
Extrapolations beyond where the equations are fit to experimental data are only as good
as their validations with more experimental data. In addition to the equations for k
f
being
material as well as geometry-dependent (both k
t

and root radius, r, appear in them), the
fatigue notch sensitivity can also depend on stress ratio, R. Most pertinent to the subject
of this book, however, is the fatigue life at which the equations are valid. Figure 5.5 is
a schematic of S–N curves for a smooth bar as well as that for a notched bar of the
same material. No attempt is made to realistically represent the relative slopes, inflection
points, or spacing of the two curves. If point A represents the fatigue life of the smooth
bar at the indicated stress level, the effect of a notch can be described by either the
reduction in fatigue strength by S1 for the same fatigue life or the reduction in life by
N 1 for the same applied stress. The former is characterized by the fatigue notch factor,
while the latter is often used in design of a component operating under a given stress
field. In HCF, the only concern is the reduction in strength or the fatigue notch factor
corresponding to a long-life fatigue strength or endurance limit if such exists. For point B
Smooth bar
Notched bar with K
t
N1
S1
S2
N
S
A
B
Figure 5.5. Schematic of S–N behavior of smooth and notched specimens.
Notch Fatigue 221
on the curve, the notch reduces the fatigue strength by S2. There is generally no simple
relation between the values of S2 and S1 as either a function of the smooth bar stresses at
A and B, or of the fatigue lives corresponding to A and B. Thus, it is speculative to use a
fatigue notch factor for point B based on data obtained at lives corresponding to point A
in the diagram. It is not surprising to find, therefore, that fatigue notch factors developed
mainly for LCF are not applicable to FLSs near the endurance limit for a range of stress

concentration factors and stress ratios using a single material constant as demonstrated
by Haritos, et al. [7].
An alternate method of representing the trends of equations for k
f
as a function of
notch root radius, r, as shown in Figure 5.4, is to plot the fatigue strength as a function
of elastic stress concentration factor, k
t
, as presented in Figure 5.6. If fatigue strength is
related simply by the stress at a point, the maximum occurring at the notch root, then the
fatigue strength would be simply 
0
/k
t
as shown by the heavy solid line labeled 1/K
t
in
the figure. Experimental data show, however, that there is a size effect so that the fatigue
strength is higher than that determined strictly by stress at a point, and is given empirically
by the fatigue notch factor which was shown in Figure 5.4 for several empirical fits to
experimental data. In Figure 5.6, the normalized fatigue strength is shown for several
equations, including that of Smith and Miller [8], which represent values for k
f
as a
function of notch geometry for an elliptical-shaped notch. Smith and Miller represent k
f
in terms of notch root radius, r, and notch depth, d,as
k
f
=1+769


d
r
(5.6)
0
0.2
0.4
0.6
0.8
1
12345678910
Normalized fatigue strength
K
t
1/K
t
Smith and Miller
Neuber (a /r = 0.5)
Neuber (a /r = 5.0)
A
B
C
Figure 5.6. Fatigue strength predictions as a function of k
t
.
222 Effects of Damage on HCF Properties
which can be represented as a function of k
t
using the equation from Peterson [9], which
approximates the stress concentration factor of a circular notch of radius, r, and depth,

d,as
k
t
=1+2

d
r
(5.7)
5.5. FRACTURE MECHANICS APPROACHES FOR SHARP NOTCHES
The previous equations show that the fatigue strength increases over 
0
/k
t
as k
t
increases.
However, as k
t
increases beyond some critical value, the equations for k
f
break down and
a limiting (minimum) value of fatigue strength is reached. The theory behind this limiting
value is that a notch starts to act the same as a crack as the notch becomes sharper and
sharper. This, in fact, is the starting point for the theory of fracture mechanics. Smith and
Miller [10] have determined this limiting value of stress to be
 = 05
K
th
√
d

(5.8)
where K
th
is the threshold stress intensity range, the value of K below which a crack
will not grow at a selected arbitrary value of very slow growth rate, commonly taken as
10
−10
m/cycle. A horizontal dashed line in Figure 5.6 is used to depict this limiting value
of stress, whether it be given by Equation (5.8) or any other equation or fit to experimental
data. With the establishment of the existence of this limiting stress for high values of k
t
,
the plot of Figure 5.6 can be broken into three regimes. In regime A, cracks will always
initiate and propagate to failure. This defines stress states above the endurance limit for
any notch geometry. In region C, cracks will not initiate nor propagate, so this region
can be defined as the safe region for HCF. In region B, cracks may initiate because
they are subjected to local stresses beyond 
0
/k
f
, but they will not propagate because
the driving force is insufficient to cause subsequent propagation. The physical reason for
this behavior is that there are sufficiently large enough stresses at the notch root, and
over a sufficient distance or volume, to cause crack initiation. However, for very large
values of k
t
, which normally correspond to very small root radii [see Equation (5.7)],
the local notch stress field dies out very quickly. The far-field stress field, in addition, is
insufficient to cause the initiated crack to continue to propagate because the local stress
intensity is below threshold. Part of the reasoning behind this is the fact that the stress

intensity for a small crack may be insufficient, or below threshold, if the far-field stress
field is low, as would normally be the case for a very high value of k
t
.
The equation of Neuber, (5.3), shown in Figure 5.4 for fatigue notch factor can be
plotted as FLS, which by definition is equal to 1/k
f
, as a function of the notch root
Notch Fatigue 223
0.1
1
0 0 0.1 1 10 100
Normalized fatigue strength, σ/σ
0
r /a
0
K
t
= 5.0
K
t
= 3.0
K
t
= 1.5
K
th
Endurance limit
El Haddad eqn
Figure 5.7. Kitagawa diagram for notches with various k

t
values.
radius, r, which can be taken as a characteristic length parameter. In this plot, Figure 5.7,
logarithmic coordinates are used as was done by Kitagawa and Takahashi [11] to illustrate
the dilemma associated with small cracks. For the three particular values of k
t
, the FLS
is shown as a function of notch root radius normalized with respect to a material constant
which will be called a
0
. The concepts illustrated in Figure 5.7 involve both fatigue
crack initiation and threshold fracture mechanics for the case of smooth bar fatigue. The
initiation concept is represented by the endurance limit which is a stress level below
which cracks will not initiate and propagate and above which will produce fatigue failure
under HCF conditions. The fracture mechanics threshold stress intensity factor is the
quantity below which a crack will not propagate and above which it will propagate. For
an edge crack in an infinite body, the stress intensity is given as
K = 112
√
a
c
(5.9)
which, if plotted as stress against crack length, a
c
, is a straight line of slope −05on
the log–log plot of Figure 5.7, which is equivalent to a Kitagawa diagram. As discussed
earlier in Chapter 4, the two limiting cases, that of a smooth bar and that of a bar with
crack, have been consolidated in various ways, most notably by El Haddad et al. [12].
They introduce a pseudo-crack length, a
0

, which produces a non-zero stress intensity for
a smooth bar with no real crack. As the crack length increases, the contribution of the
a
0
term becomes small and the long crack fracture mechanics solution is approached.
Ignoring the factor of 1.12 in Equation (5.9), the El Haddad relationship has the form

0c
=
K
th

a +a
0

(5.10)
224 Effects of Damage on HCF Properties
where 
0c
is the critical stress range for propagation of a crack of length a. This equation
is shown in Figure 5.7 to be asymptotic to the endurance limit and the threshold stress
intensity line as discussed previously. Here, a is the actual crack length and a
0
is a
material constant which, for the edge crack described by Equation (5.9), and ignoring the
factor 1.12, is
a
0
=
1



K
th

0

2
(5.11)
where 
0
is the endurance limit stress for the uncracked body. The quantity a
0
has been
interpreted over the years to be anything between a fundamental parameter representing
some microstructural aspect of a material to a purely empirical curve fitting parameter.
Mathematically, it provides a smooth transition from initiation in an uncracked body to
long crack behavior and is claimed to represent much of the small crack threshold data
reported in the literature. For almost all materials, however, a
0
is found to be a function
of stress ratio, R.
Of significance, in Figure 5.7, is the behavior of notched specimens represented by the
equation of Neuber, which provides the FLS as a function of notch root radius, using a
m
as a material constant and k
t
as a parameter. In the plot, r is normalized with respect to
an arbitrary quantity a
0

which is not necessarily the same as a
m
, although Taylor [13]
indicates that these two quantities, which both have units of length, have been noted to
be of the same order of magnitude for many materials. For a notched component, the
curves for various k
t
values show that as k
t
decreases and approaches unity (smooth
bar), the FLS approaches the endurance limit. As k
t
increases, however, the shape of the
curve becomes more like that of El Haddad for small notch root radii. The notch curve
for k
t
= 50 can be seen to be almost parallel to the threshold line and, by the proper
choice of values of a
m
, could be made to be coincident with it. This is another way of
showing that very sharp notches tend to produce fatigue limits which can be determined
from fracture mechanics by treating the notch as a crack, independent of k
t
, as shown in
Figure 5.6 and by Equation (5.8). In fact, that equation is simply the fracture mechanics
equation for an edge crack, Equation (5.9), where the crack length is replaced by the
notch depth, d, for very sharp notches with high values of k
t
. Plotted on Figure 5.7, this
equation would be parallel to the threshold line. For small values of notch root radius,

the region between the fatigue limit curves and the El Haddad fracture mechanics curve
corresponds to a region where cracks can initiate but will not propagate because they are
below the threshold for crack growth, similar to region B in Figure 5.6. For large values
of notch root radius, on the other hand, cracks will both initiate and propagate because
the FLS comes directly from the definition of k
f
given by Equation (5.1) as shown by
the equations in Figure 5.7 and the stresses and root radii are above the crack growth
threshold condition.
Notch Fatigue 225
Evaluation of Figures 5.6 and 5.7 points out some of the difficulties in making generaliza-
tions about the transition from crack initiation to crack propagation, particularly as related
to the small crack issue in both smooth and notched bars. The transition from a sharp notch
to a crack does not depend solely on k
t
, as shown in Figure 5.6, nor does it depend solely on
notch root radius, r, as shown in Figure 5.7. Instead, it depends on both, even though very
sharp notches having large values of k
t
are normally associated with geometries that have
very small notch root radii. The transition from long to short cracks is also an issue which
has to be addressed. Note again, as pointed out previously, that the extrapolation of equa-
tions for k
f
to either very small notch root radii or very high values of k
t
are speculative at
best. Figure 5.6 shows the breakdown of those equations for high k
t
by a fracture mechanics

based parameter. The shape of the curves in Figure 5.7 for very small values of r is nothing
more than an extrapolation that eventually approaches the smooth bar behavior.
5.6. CRACKS VERSUS NOTCHES
Taylor [14] has attempted to summarize the scope of the notch problem in fatigue which
encompasses a wide range of geometries from a very mild notch to a very sharp notch,
the latter acting essentially as a crack. In addition, he incorporates the concept of both
very shallow notches and cracks which, in the extreme, have no effect on smooth bar
fatigue behavior because of their small size. To do this, a semi-elliptical notch at a
surface having depth D and notch root radius r is considered. Using the El Haddad
definition of the critical crack length a
0
, the behavior of a notch can be broken up into
three regimes as shown in Figure 5.8, where the regimes are defined for values of notch
size D/a
0
 and notch shape D/. The three regimes are designated “blunt notches,”
“crack-like notches,” and “short notches,” and the boundaries between the regimes are
shown schematically in Figure 5.8. The boundary below which a notch is considered to
be short is found to be approximately at D/a
0
= 3, while crack-like notches occur for
notch shapes where D/ is greater than approximately 0.25. Taylor goes on to point
∼0.25
~3
D/a
0
D/ρ
Short
notches
Crack-like

notches
Blunt
notches
Figure 5.8. Schematic illustration of three regimes of notch behavior [14].