I
IL LIN I
S
UNIVERSITY OF ILLINOIS AT URBANACHAMPAIGN
PRODUCTION NOTE
University of Illinois at
UrbanaChampaign Library
Largescale Digitization Project, 2007.
UNIVERSITY OF ILLINOIS ENGINEERING EXPERIMENT STATION
Bulletin Series No. 898
A CRITICAL REVIEW OF THE CRITERIA FOR NOTCHSENSITIVITY
IN FATIGUE OF METALS
C. S. YEN
Formerly Research Associate in
Theoretical and Applied Mechanics
T. J. DOLAN
Research Professor of Theoretical
and Applied Mechanics
Published by the University of Illinois, Urbana
305035247279 U S,
:: PRESs :,
ABSTRACT
It is the purpose of this bulletin to summarize and to appraise critically
the numerous interpretations or correlating methods that have been
proposed in the technical literature to compare the endurance limits of
notched rotating beam fatigue specimens with those of unnotched speci
mens. The interrelation of the ideas proposed by several investigators
was studied. The discrepancy between theoretical and effective stress
concentration factors is attributed to the fact that the structural action
in real materials is different from that of a homogeneous, isotropic,
elastic, "idealized material" commonly assumed in the theoretical
analysis of stresses.
After surveying all relevant hypotheses it was summarized that
notchsensitivity of a metal member depends upon three different factors,
namely: (a) the basic material characteristics, of which the localized
workhardening (or strain strengthening) capacity may be considered as
an index; (b) the degree of material homogeneity, which is influenced by
inherent defects, tensile residual stresses, heat treatments, etc.; (c) the
geometry of the specimen (including overall size), the radius at the
root of the notch being of prime importance in this geometric factor. It
was concluded that the criterion for fatigue failure or for endurance limit
should include not only the peak stress at a critical point as is conven
tionally assumed, but also the conditions existing in a critical region
surrounding the point. A rational approach and procedure for attacking
the problem of notch effect as well as size effect is suggested.
CONTENTS
I. INTRODUCTION 9
1. Importance of Notches 9
2. Purpose and Scope 10
3. Comparison of Theoretical and Effective Stress
Concentration Factors 10
4. Comparison of Notchsensitivity in Fatigue and
Static Tension 12
5. Comparison of Fatigue and Impact Notchsensitivity 12
II. INTERPRETATIONS BASED ON CONCEPTS OF
MATERIAL BEHAVIOR 14
6. Plasticity 14
7. Cohesive Strength 16
8. Tensile Strength 17
9. Workhardening Capacity 18
III. ANALYSES OF STRESS CONDITIONS 20
10. Change of Notch Radius 20
11. State of Stress and Shear Energy Theory 20
IV. ELEMENTARY STRUCTURAL UNIT 23
12. Stress at the End of an Elliptic Crack 23
13. Neuber's Formula for Sharp Notches 24
14. Morkovin and Moore's Application to Fatigue
Test Data 26
15. Moore's Formula for Values of p' 26
V. STRESS GRADIENT AND STRESS CONCENTRATION 28
16. Relative Stress Gradient 29
17. Stress Gradient, State of Stress, and Amount of
Stress Concentration 29
18. Influence of Grain Size 31
19. The Effect of a Stress Gradient 33
20. Extent of Stress Concentration 34
VI. FAILURE BELOW SURFACE 40
21. Moore and Smith's Formulas 40
22. Peterson's Formulas 42
23. Lack of Evidence Regarding the Basic Assumptions 42
VII. STATISTICAL THEORIES OF FATIGUE 44
24. Freudenthal's Equation 44
25. Aphanasiev's Equation 45
VIII. HOMOGENEITY OF MATERIALS 48
26. Cast Metals 48
27. Heat Treatments 48
IX. CLOSURE 50
FIGURES
1. Lowering of Peak Stress by Plastic Action (from reference 16) 15
2. General Relation Between Tensile Strength and Endurance Limit for Steel
Specimens (from reference 21) 16
3. Relation Between Tensile Strength and Endurance Limit for Grooved
Steel Specimens (from reference 23) 17
4. Average Stress on Elementary Structural Unit in Terms of the Relative
Sharpness of the Notch (from reference 32) 24
5. Definition of Notch Geometry: Notch Angle = w, Notch Radius = r
(from reference 5) 25
6. Relation Between Notchsensitivity Index and Stress Gradient (from ref
erence 39) 30
7. Notchsensitivity Index vs. Number of Grains in Region of Peak Stress
(from reference 36) 31
8. Notchsensitivity Index vs. Relative Decrement in Stress Across One
Grain (from reference 36) 32
9. Notchsensitivity Index vs. Radius at Root of Notch (from reference 37) 33
10. Gain in Fatigue Strength vs. 1//r for Shouldered Shaft Specimens of
Plain Carbon Steels Tested in Reversed Bending: r = fillet radius in
inches (from reference 41) 35
11. Gain in Fatigue Strength vs. 1/V r for Shouldered Shaft Specimens of
Heat Treated Alloy Steels: r = fillet radius in inches (from reference 41) 35
K. 1
12. Plotting of Notchsensitivity Index q = _ 1 vs. 1// r (for the same
data as in Fig. 10) 37
13. Plotting of Kt  K. vs. 1/y r (for the same data as in Fig. 10) 38
14. Plotting of K K vs. 1// r (for the same data as in Fig. 10) 39
15a and b. Distribution of Longitudinal Stress in Rotating Beam Specimens 41
16. Agreement of Aphanasiev's Equation with Test Data for Shouldered Shaft
Specimens (from reference 45) 47
I. INTRODUCTION
1. Importance of Notches
It has been well known(1)* for many years that most failures in
machine parts (and in some structural members) are progressive frac
tures resulting from repeated load; these "fatigue failures" nearly always
start at an imposed or accidental discontinuity, such as a notch or hole.
For example(21), fatigue fracture developing from a small hole wrecked
a huge costly turbine; a sharp fillet on an axle caused a serious accident
in a school bus; stamped marks of inspection on a propeller resulted in
the crash of an airplane. Thus a tiny "notch" is frequently a potential
nucleus of fatigue failure which may lead to serious damage.
The term "notch" in a broad sense is used to refer to any discon
tinuity in shape or nonuniformity in material. A notch is frequently
called a "stress raiser" because it develops localized stresses that may
serve to initiate a fatigue crack (or reduce the loadcarrying capacity).
Notches are hardly avoidable in engineering practice; they may occur as
(a) a metallurgical notch, which is inherent in the material due to metal
lurgical processes (as inclusions, blowholes, laminations, quenching
cracks, etc.); (b) a mechanical notch, of some geometrical type which
usually results from a machining process (as grooves, holes, threads,
keyways, fillets, serrations, surface indentations); (c) a service notch,
which is formed during use (as chemical or corrosion pits, scuffing,
chafing or fretting, impact indentations, and so on). Hence the potential
loadcarrying capacity of a material under repeated stress can seldom be
attained in actual machine parts because of the presence of these imposed
or accidental notches. The fatigue "notchsensitivity," or susceptibility
of a member to succumb to the damaging effects of stressraising
notches (this susceptibility varies with different materials) is therefore
an important consideration in almost every branch of machine design
involving the proportioning of members for service under repeated stress.
Usually the term notch as used in its narrow sense refers only to
notches of type (b)  i.e., the mechanical notch. The pronounced reduc
tion of fatigue strength due to a sudden change in crosssection of a
loadcarrying member has been noted in many experimental investiga
tions ever since the classical tests of Wohler(3". Many investigators have
* Parenthesized superscripts refer to correspondingly numbered entries in the Bibliography.
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proposed numerous methods of interpreting the available fatigue test
data obtained from notched specimens. The large number and variety of
these hypotheses, interpretations, and correlating methods employed to
compare the endurance limits for notched specimens with those of un
notched specimens have led to confusion and unsatisfactory results from
the viewpoint of factual knowledge for the machine designer.
2. Purpose and Scope
It is the purpose of this report to review briefly each of these inter
pretations and to appraise them critically in terms of physical signifi
cance and of agreement with experimental test data. The interrelations
of these different methods and the general basic concepts are analyzed.
The probable fundamental factors involved in the "notch effect" and a
rational procedure for attacking the problem of notchsensitivity are
suggested.
In general, most of these previous analyses were primarily qualitative
and based only on a single concept of basic material behavior, such as
plasticity, damping capacity, cohesive strength, workhardening capac
ity, elementary structural unit, or statistical theories of fatigue. Several
of these approaches are discussed in Chapters II and III. Interpretations
based upon static tensile strength or impact strength are only indirect
and accidental, and have not led to an accurate and functional correla
tion. Some attempts at rationalization have been based on the stress
conditions as affected by the geometry of the notch, such as state of
stress, shear energy theory, stress gradient and extent of stress concen
tration. These are discussed in Chapters IV and V. Other quantitative
approaches, such as correlations based on notch radius, or of stress at a
given depth below the surface (discussed in Chapter VI), represent a
type of empirical approach deduced from speculative thinking which
lacks evidence in the light of recent fatigue tests or theories. Other
developments based on statistical effects and the homogeneity of metals
are presented in Chapters VII and VIII.
However, these approaches all seem to indicate a partial truth re
garding the behavior of a notched specimen. They are of value in record
ing various attempts at correlating data, and may guide our attempts to
predict the notched fatigue strength of a member until the knowledge on
this problem is further advanced.
3. Comparison of Theoretical and Effective Stress Concentration Factors
When a notch is introduced in a specimen subjected to an elastic
static load, the stress at the root of the notch is markedly increased. The
ratio of the value of the peak stress in the notched member to that in a
corresponding unnotched member is called the "theoretical stress concen
Bul. 398. CRITERIA FOR NOTCHSENSITIVITY
tration factor," Kt*. The peak stress in the notched specimen may be
determined either mathematically"' 5), photoelastically(2, 6), or by Xray
measurement(7), while the peak stress in the unnotched specimen is al
ways calculated from the elementary stress formulas (such as S = P/A,
S = Mc/I, and S = Tc/J for axial, bending, or torsional loads respec
tivelyM1)).
Since the peak stress is raised by the factor Kt, it might be expected
that the strength of the notched specimen would be reduced by the factor
Kt. Experimentally it is observed that the amount of reduction of load
carrying capacity due to a notch roughly tends to increase with (but
always is smaller than) the factor Kt. The ratio of the endurance limit
of an unnotched specimen to that for a notched specimen is called the
"strength reduction factor" or "effective stress concentration factor," Ke.
The endurance limits are determined by fatigue testing in which only
elementary stress formulas are used for calculating the stresses in the
specimens, whether notched or unnotched.
The discrepancy between theoretical and effective stress concentra
tion factors Kt and Ke varies not only for different metals but also for
different sizes of specimen and different types of notch; the lack of a
rational explanation for these variations has led to much confusion and
speculation. The fundamental cause for this discrepancy may be attrib
uted to the fact that the response of a material subjected to a repeated
loading is quite different from the behavior of the same material when
subjected to an elastic static loading. The analyses upon which the
theoretical factors are based depend on the assumptions of an isotropic
material which is perfectly elastic and homogeneous and whose stress,
conditions and strength properties are not influenced by time or tempera
ture. However, when dealing with fatigue tests of metals, the small
localized spots (crystals, slip bands or grain boundaries) in which fatigue
failure initiates are anisotropic and far from homogeneous. Localized in
elastic readjustments which are sensitive to time and temperature and
which alter the stress and strength occur in the material at stress levels as
low as its endurance limit, or even lower. Better understanding of the
mechanism of deformation in polycrystalline metals under repeated load
ing will therefore help to clarify the reason for the discrepancy between
Kt and Ke.
When a metal piece is said to be notchsensitive, it is inferred that.
the ratio Ke/Kt for that piece is relatively high; that is, the value of
strength reduction factor Ke is relatively high with respect to the value
of Kt. In most cases the value of Ke lies between 1 and Kt, but there are
* In many instances the value of Kt is defined as the ratio of the peak stress in the notched
member to the nominal stress computed from the dimensions of the minimum section at the notch.
For comparative purposes either definition is acceptable, but the values of Kt are slightly different..
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occasional exceptions. For example, for some stainless steels the value of
Ke may be less than one (Section 9) and for some quenched and tem
pered steels Ke it may sometimes be greater than Kt8). The value of Ke
is directly proportional to the value of Kt only when the "notchsensi
tivity" is constant.
4. Comparison of Notchsensitivity in Fatigue and Static Tension
The problem of notchsensitivity of metals has been investigated in
experiments employing three different types of loading  static tension,
Charpy impact, and rotatingbeam fatigue tests. From results of static
tension tests9', 10, 11, 12) it has been shown that for ductile steels the ulti
mate strength and yield strength increase with notch depth and sharp
ness of the notch angle; the breaking stress (load per unit of actual area
at fracture) also increases, moderately or remains approximately con
stant. For brittle metals such as cast iron, cast brass, and magnesium
alloys (which may have many internal defects or high residual stresses
but little capacity for plastic flow) there often is little difference between
the strength values of notched and unnotched bars.
For cast iron and some cast aluminum alloys under repeated loading
there is also little difference between the values of the endurance limit
for notched and unnotched barso13). For ductile metals, however, the
presence of a notch usually reduces the fatigue strength. Under repeated
loading a ductile material does not undergo largescale plastic flow at
the notch root or large extension and rounding out of the notch, both of
which tend to relieve the stress concentration. Experimentally it is gen
erally found that soft steels (which have higher tensile strength in
notched specimens than in unnotched ones) are not highly notch
sensitive in fatigue tests; on the other hand, hard steels which she w
reduced static tensile strength due to a sharp notch are also very
notchsensitive in fatigue. These observations of the behavior of soft and
hard steels resulted in one attempt at a correlation of fatigue notch
strength with static tensile strength, as is discussed in Section 8.
5. Comparison of Fatigue and Impact Notchsensitivity
It is generally believed that hard steels are more notchsensitive than
soft steels either in a fatigue test, a static tension test, or an impact test.
Some test data have indicated that the stronger the steel the lower is the
Charpy impact value1"3 and the greater the fatigue notchsensitivity;
hence, one might infer the possibility of a relation between impact values
and fatigue notchsensitivity. However, no direct correlation between
these two types of test has ever been reported14, 21) and some contrary
Bul. 398. CRITERIA FOR NOTCHSENSITIVITY
evidence indicating that there is no reason to expect a correlation has
been presented(81.
In a study by Dolan and Yen"8*, experimental data were presented
from fatigue and Charpy tests on two alloy steels and one carbon steel,
heattreated in several different ways to approximately the same hard
ness level. It was concluded that no direct functional relationship was
evident between the concepts of notchsensitivity in a fatigue test and
the notchsensitivity evidenced in a Charpy impact test. A rough quali
tative correlation was indicated for comparisons of the same material at
the same hardness and tensile strength level; but even this qualitative
relationship was inaccurate (and the relative order of notchsensitivity
was reversed in the two types of test) when comparing materials of
different chemical analyses.
The Charpy test develops a higher rate of strain at peak stress than
the fatigue test. Fracture under a single impact is not dependent upon
the cumulative chance effects developed during the repetitions and re
versal of load which are of paramount importance in the submicroscopic
phenomena leading to failure in fatigue. Therefore, direct correlations
between fatigue properties and Charpy values do not seem feasible.
II. INTERPRETATIONS BASED ON CONCEPTS
OF MATERIAL BEHAVIOR
6. Plasticity
Moore in 1931 reported"5) a new localized "plasticity" property of
metals, that is, the ability to stand occasional overstress in localized
zones without developing a crack. He called this "crackless plasticity"
and described it as a property unique to the conditions encountered in
repeated loading, and one which could not be measured by the ductility
in static tests. For example, the ductility of an alloy steel was much
higher than that of hard spring steel, but neither steel showed a high
degree of crackless plasticity (or notchinsensitivity) under repeated
stress. Coppernickel alloys exhibited good elongation and reduction of
area in a static test, but in fatigue tests did not resist localized plastic
deformation without starting a crack. Pure metals and very finegrained
metals appeared most sensitive to the effect of notches.
Thum in 1932(16) and several others"(49 claimed that the theoretical
peak stress is lowered by plastic action which redistributes the stress
and decreases the effective stress concentration factor. It was implied
that differences in notchsensitivity of various metals were due to the
relative degrees of lowering of the peak stress. If Sn denotes the nominal
stress in a notched specimen as found by the elementary formula, then
KS. will be the value of the theoretical peak stress and KeS. will be
considered as that of the "actual" peak stress as shown in Fig. 1. The
ratio of the increase of the "actual" peak stress, KeS,, over the nominal
stress S., to the increase of the theoretical peak stress, KtSn, over the
nominal stress Sn, was regarded as a material property. It has been called
the "notchsensitivity index," q. That is:
K,&S  S& Ke  1
q   (1)
KtS.  S& Kt  1
whence Ke may be found if q for a material and Kt for any notch are
known: Kq
The values of q vary between 0 and 1 as the values of K, vary be
tween 1 and Kt, but test values are occasionally found beyond these
limits, as has been discussed in Section 3. The greater the value of q, the
greater the notchsensitivity of the material.
Bul. 398. CRITERIA FOR NOTCHSENSITIVITY
Peterson"17 showed that the notchsensitivity index q depended not
only on the properties of the material itself, but also on the shape and
dimensions of the test piece. This peculiarity complicates the estimation
of fatigue strength of notched members, and indicates that q is not a
fundamental "material property."
Foppl018) suggested that damping capacity, a property of the material
independent of the dimensions of the test piece, was a measure of plastic
strain and hence was likewise a measure of notchsensitivity. Several
other investigators"49 also tried to correlate damping capacity with
notchsensitivity on the basis of the observation that some metals of low
notchsensitivity, such as cast iron, possess high damping capacity. How
ever, there are exceptions to this statement; and since the complicated
phenomena and structural actions involved in either fatigue or damping
are not fully understood, any direct quantitative correlation does not at
present appear feasible(49).
Fig. 1. Lowering of Peak Stress by Plastic Action
(from reference 16)
It has been a general concept that plastic action mitigates peak stress
in fatigue loading. However, as discussed by Yeni(o, the strainhardening
accompanying repeated loading gradually reduces the subsequent plastic
deformation; this tends to make the range of peak stress in each cycle
approach the value calculated by elastic theory. This may be related to
the observation that no lowering of stress at notches was observed by
Xray measurements under alternating stresses not exceeding the yield
limit(20, 49). Consequently the hypothesis that lowering of the range of
peak stress is due to plasticity is not confirmed either by theoretical
study or by experiment.
Since the fatigue phenomena are initiated on an atomic or submicro
scopic scale(19), it is probable that only the microscopic inelastic adjust
ments in localized regions are important in determining the notch
sensitivity. These minute inelastic deformations (which constitute the
property Moore called crackless plasticity) probably do not relieve the
ILLINOIS ENGINEERING EXPERIMENT STATION
macrostresses imposed by external load at an ordinary notch, but may
develop localized strainhardening effects which raise the fatigue strength
and thus reduce the notchsensitivity.
7. Cohesive Strength
Kuntze(20) doubted whether Thum's hypothesis of lowered peak stress
as set forth above conformed to the experimental results, and propounded
his own theory to explain notchsensitivity by the mechanism of plastic
~2
1::
'4.J
1U/timate Tensile Strength in /000 I/b per sq. n.
Fig. 2. General Relation Between Tensile Strength and Endurance Limit for Steel Specimens
(from reference 21)
Bul. 398. CRITERIA FOR NOTCHSENSITIVITY
deformation. According to Kuntze, plastic deformation involved not only
slip but also a loosening of small particles of materials in the test piece.
In notched specimens, if the metal offered sufficient cohesive resistance,
plastic slip was found to take place simultaneously throughout all the
section, and the stress concentration would not influence the strength.
But if the cohesive resistance was not sufficient, sliding occurred in a
Ultimate Tensile Strength in /000 /lb per sq, in.
Fig. 3. Relation Between Tensile Strength and Endurance Limit for Grooved Steel Specimens
(from reference 23)
portion of the crosssection only, and the material "loosened" in the
remaining portion where there was no slip. Thus, according to Kuntze,
notchsensitivity of a material was explained by its relative weakness
in cohesive strength.
8. Tensile Strength
Investigators(21, 22, 23, 24, 49) have shown that notchsensitivity in fa
tigue of metals depends on the severity of notch and on the nature of
the material, which in turn is often appraised in terms of the tensile
strength. In general, the higher the tensile strength the greater is the
notchsensitivity, as shown in Figs. 2 and 3. The increased strength
reduction for the stronger metals may be due mainly to their lowered
capacity for the minute plastic flow which results in localized work
hardening. Consequently, the amount gained in fatigue strength for
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notched specimens by changing to a higher tensile strength steel varies
from zero to about 100 percent depending upon the geometry of the
notch; the more severe the notch, the less is the percentage gain in
fatigue strength.
For steels of the same tensile strength the notchsensitivity is altered
somewhat by differences in metallurgical structure produced by different
heat treatments. It was found(8' that for the same tensile strength a
drastically quenchedandtempered steel was less notchsensitive than
the same steel in a slowly quenchedandtempered or normalized condi
tion. Some apparent differences may also be due to residual stresses as
discussed in Section 27.
9. Workhardening Capacity
McAdam and Clyne(22) used the percentage decrease in the endurance
limit due to a notch as an index of notchsensitivity, and discussed the
relationship between the notchsensitivity and other mechanical proper
ties of metal, namely ductility, mechanical hysteresis, tensile strength,
and workhardening capacity. It was emphasized that notchsensitivity
depended considerably on workhardening capacity and it was proposed
that the quantity
1 tensile strength
true breaking stress/
might be used as an index of tensile workhardening capacity. However, a
direct comparison between these indices of notchsensitivity and work
hardening capacity from experimental test results showed a wide scatter
ing and apparently did not indicate a consistent relation for different
materials or for specimens of different geometry.
Gillett"25) reports that for 18:8 stainless steels the endurance limit on
a polished unnotched bar in the fully soft condition was found to be less
than that of a similar bar with a notch. In general, austenitic stainless
steels all had surprisingly good notched fatigue strengths and were not
notchsensitive. This fact has been attributed to their remarkable work
hardening capacity as revealed in static tension tests. Some manganese
austenitic steels which showed similar workhardening behavior in static
tension tests have been expected by some investigators to have the same
low notchsensitivity in fatigue, though no data have been reported.
If the workhardening capacity of a material is really responsible for
low notchsensitivity, one may wonder why the fatigue strengths for
unnotched specimens (of these austenitic steels for example) are not
increased by workhardening in the same proportion as the notched
Bul. 398. CRITERIA FOR NOTCHSENSITIVITY
specimens. That is, do the unnotched specimens have the same work
hardening capacity as the notched specimens of the same material? This
question can be answered by considering the effect of the stress gradient
and the workhardening capacity in a localized zone. The notched spec
imen (with a steep stress gradient) requires workhardening of only a
small localized volume to resist the peak stress applied. Inelastic de
formation at the root of the notch has the effect also of reducing the
peak stress and of bringing a larger volume of material into play in
resisting the load; this is equivalent to a "supporting effect" from the
elastic material surrounding the hardened volume. The material has the
same workhardening capacity, but the notched specimen shows higher
fatigue strength than that predicted from elastic theory, due to the
restraints and readjustments in the surrounding "understressed" metal.
Since the mechanism of workhardening is not completely understood,
the term "workhardening capacity" (which refers to the maximum
amount of cold work which a material can receive without fracture) is
essentially a hypothetical concept which assumes that when a material
reaches its workhardening capacity fracture ensues. The nature of work
hardening is vague in simple static tension tests; in repeated loading the
mechanical readjustments are presumed to be even more complicated.
III. ANALYSES OF STRESS CONDITIONS
10. Change of Notch Radius
Any elastic or plastic strain at the root of a notch tends to change
the root radius. Moore and Jordan(26) assumed that fatigue loading
might produce a lengthening of notch radius r over a very small length
of arc at the bottom of the notch. Lengthening of the notch radius from
r to r' would reduce the concentration factor Kt to an "effective" stress
concentration factor Ke. From test data on two steels (SAE 1020, and
quenched and tempered SAE 2345) they obtained an empirical expres
sion for the effective radius r' which if substituted for r in Neuber's
diagramý5) gave values of stress concentration factor equal to Ke as
found from actual fatigue tests. This empirical expression was:
r' = cid4  c2t + r (2)
where cl and c2 were material constants determined by fitting the equa
tion to test data; d was the net diameter of the test section; and t was
the notch depth. However, tests have shown"51 that there is no appreci
able change in the actual notch radius in the usual fatigue test.
S11. State of Stress and Shear Energy Theory
Several investigators(27, 28, 49) considered that the notch effect might
be partly or entirely due to the state of combined stress existing at the
notch root; hence different theories of elastic failure which had been
used to explain the effect of combined stress in static tests were applied
to the notched fatigue specimens. Experimental data on the effect of
combined stresses on the endurance limits of unnotched specimens showed
that for 0.1 percent and 0.34 percent carbon steels(29, 30) and a 2 percent
Ni1 percent Cr0.35 percent Mo steel'31) the shear energy theory agreed
with the test results quite well, but for a 3% percent NiCr steel(29) only
the total energy theory fitted the test results. For a cast iron(29), as in
static tests, the principal stress theory indicated the best agreement with
fatigue test data. For notched fatigue specimens of highstrength steels
it was found(27, 28) that the shear energy theory agreed most closely with
the test results, as is discussed later.
An application of the shear energy theory to the stress conditions
existing in a notched rotating beam specimen (in which the extreme fiber
at the root of the notch is subjected to completely reversed stress) may
be developed using the following notations:
Bul. 398. CRITERIA FOR NOTCHSENSITIVITY
Let S. = endurance limit of an unnotched specimen under uniaxial
stress state (p.s.i.)
Si, S2, Sa = three principal stresses at the bottom surface of the notch
(longitudinal, circumferential, and radial, respectively,
when loaded to the endurance limit) (p.s.i.)
Sn = endurance limit (nominal flexural stress as found from the
ordinary flexural formula Mc/I at the notch root) (p.s.i.)
K, = theoretical stress concentration factor = Si/S,
Kr = theoretical strength reduction or "shear energy" factor;
i.e., theoretical value of the ratio S,/Sn
u = Poisson's ratio
The shear energy criterion for failure of material may be formulated
in terms of the three principal stresses Si, S2, S3 as follows:
(S1  S2)2 + (S  S3)2 + (S3  S)2 = 2S,2 (3)
From Neuber's theory(') we have
S = longitudinal stress = KtS,
S2 = circumferential stress = u (S .  Sn) = uS (K, 1
S3 = radial stress = 0 (at the surface)
Substituting these relations in Eq. 3 and simplifying we have
K,1 Ke1 2
K = K 1 u .  + u K "(4)
If we assume u = 0.3 and Kt = 2, then from the above equation K, =
0.966Kt. For extremely large values of K, the value of the theoretical
strength reduction factor approaches 0.954K1 when calculated from
Eq. 4.
In 1943 both Moore and Morkovin(27) and Peterson'28" tried to em
ploy various theories of failure of elastic action, especially the shear
energy theory, to explain the observed differences between theoretical
and experimentally observed strength reduction due to notches. The
results of Moore and Morkovin's investigation on three SAE steels (1020
asrolled, 1035 asrolled, and X4130 quenched and tempered) indicated
that the shear energy theory correlated with the test results better than
the principal stress theory or the shearing stress theory for specimens
not smaller than 1/2 in. in diameter. However, there were tendencies for
the small specimens to behave differently from the predictions of any of
these theories.
22 ILLINOIS ENGINEERING EXPERIMENT STATION
Peterson(28) analyzed Moore and Jordan's data(26" on two steels and
found that the data on the quenched and tempered SAE 2345 steel could
be predicted fairly accurately by the shear energy theory, but that for a
low carbon steel the reduction of fatigue strength due to the notch was
generally less than that indicated by the theoretically computed factors.
Comparison of the different theories of elastic failure has indicated
that the principal stress theory or the shearing stress theory predict the
highest values of effective stress concentration factor, i.e., these theories
require that K,= Kt. In order of decreasing magnitude the total energy
theory requires lower values of Kr, the principal strain theory gives
still lower values, and the shear energy theory predicts the lowest value
for K,. However, when actual test data for ordinary steels are compared,
even the values of K, predicted by shear energy theory are very often
too high (except for high strength heattreated steels and for large
specimens which respectively are more notchsensitive than low strength
asrolled steels or small specimens). For brittle metals of low fatigue
notchsensitivity like cast iron, none of these theories are adequate to
explain the insensitivity to the stressraising effects of a notch.
IV. ELEMENTARY STRUCTURAL UNIT
Extensive analytical work has been done to evaluate the localized
stresses at notches of various shapes according to the classical theory
of elasticity, and the results found were assumed to be directly appli
cable to mild notches or to those regions where the stress variation was
not drastic(5". As the radius of the notch approaches zero, however, the
stress concentration factor theoretically approaches infinity, which is
not true for actual materials.
The classical theory of elasticity assumes the material to be per
fectly homogeneous and infinitely divisible, and does not recognize the
structure to consist of finite particles as in the case of actual engineering
materials. The fact that the actual materials are made of a finite num
ber of particles as atoms or crystal grains of definite dimensions for
each kind of material has been recognized by some mathematicians in
their stress analyses; the concept was introduced that these particles
might be represented by many small cubic blocks of uniform size called
the "structural elementary unit." The size of the structural elementary
unit was assumed as a property of the material.
If the values of the theoretical stresses in the region of peak stresses
are averaged over the surface of an elementary structural unit, the value
of the effective maximum elastic stress would be reduced due to the steep
stress gradients existing over the unit; hence, the stress concentration
factor would also depend upon the size of such a particle when the notch
is sharp.
12. Stress at the End of an Elliptic Crack
Gurney(32) derived an equation mathematically for computing the
average elastic stresses over the area of an elementary structural unit at
the end of an elliptical hole or crack. The results are expressed in terms
of the ratio p'/r of the length of the structural unit to the radius of curva
ture of the end of the elliptic crack, which has axes of lengths a and b.
In Gurney's equation if the ratio a/b remains constant, the average
elastic stress (and therefore the stress concentration factor) decreases
as the ratio p'/r increases, as is shown in Fig. 4. When r is equal to
p', the average stress is 1.1 a/b = 1.1 \/ a/p' , whereas the peak stress
computed by the theory of elasticity is 2 a/b. When r approaches zero,
ILLINOIS ENGINEERING EXPERIMENT STATION
Kano  O ; 
o Rafi/us of Curvature of the End of the Eliptic Crack r
Fig. 4. Average Stress on Elementary Structural Unit in Terms of the Relative Sharpness
of the Notch (from reference 32)
the average stress becomes 1.8 /a/p' whereas the peak stress would
be infinite according to the classic theory of elasticity. Therefore, the
effect of an elementary structural unit of constant length p' = r is
roughly to halve the stress concentration, whereas reducing r from a
value of p' to zero (increasing the value of Kj from 2 a/b to infinity) only
increases the effective stress concentration factor by about 70 percent.
13. Neuber's Formula for Sharp Notches
Neuber(15 took the effect of the size of elementary structural unit
into consideration by selecting the following equation for predicting the
effective stress concentration factor K, from the theoretical value of K,:
Kt  1
K. = 1 + (5)
1+ r
in which r = notch radius and p' = half the width of the elementary
structural unit. Eq. 5 may be rewritten as:
Ke  1 1
 =  (5a)
K1 1 p'
K1+
The left member in the above equation represents notchsensitivity
index q defined by Thum and Peterson'171 (see Section 6). Hence q
Bul. 398. CRITERIA FOR NOTCHSENSITIVITY
appears to be a function of both the notch sharpness r and the size p'
of the structural unit.
Neuber mentioned that for sharp notches there was a relatively large
deformation at the notch root which the basic equation of elasticity did
not take into consideration. Since this deformation lowered the stress
concentration in the same sense as did the concept of individual struc
tural units, to include only one of these two factors was sufficient if an
empirical material constant was evaluated to fit experimental results.
That is, the effect of the deformation was also included in selecting a
value of p' in Eq. 5 to fit the actual material behavior.
Fig. 5. Definition of Notch Geometry: Notch Angle = u,
Notch Radius = r (from reference 5)
The effect of notch angle w (Fig. 5) was considered negligible for
relatively blunt notches but not for sharp notches (i.e., those with
relatively small notch radius). Neuber therefore derived the following
equation from Eq. 5 to take care of the effect of notch angle:
K, Kt  1 (6)
Ke = 1 + (6)
1+
or
K.  1 1
q  _ (6a)
Kt  1 p'
1+
By using 0.48 mm (= 0.019 in.) as the empirical value of p', Neuber
found that Eq. 6 agreed with the results of bending tests (probably
static tests) and photoelastic measurements. Since Neuber's formulas are
not entirely based upon rigid mathematical analysis, but result from
empirical interpolation between theoretical limiting values, the extent of
its application as an exact relation is likely limited.
Furthermore, basically both Gurney's and Neuber's equations were
derived only for the case of static loading, and only for explaining the
notch effect (not including size effect). Hence it is a question how far
these theories can be generalized to explain notch effect and size effect
in repeated loading.
ILLINOIS ENGINEERING EXPERIMENT STATION
14. Morkovin and Moore's Application to Fatigue Test Data
Morkovin and Moore(331 found Neuber's value of p' = 0.019 in. to
agree well with fatigue test results for SAE 1020 and SAE 1035 steels
asrolled. However, p' = 0.0014 in. was found by trial and error to give
a better correlation with the results of fatigue tests of annealed SAE
1035 steel; a value of p' = 0.00068 in. was obtained for quenched and
drawn SAE X4130 steel.
They mentioned that the test data available did not seem to justify
any attempt to determine a quantitative correlation between the value of
p' and grain size.
15. Moore's Formula for Values of p'
In order to determine the value of p', Moore(34) presented an empiri
cal formula which, when used with Eq. 6, gave computed values which
agreed fairly well with the strength reduction factor Ke from actual
fatigue tests of six steels. This relation is:
( \ 0.05\
p'= 0.2 1  0.05 (inch) (7)
S.) d
where
S, = yield strength of the steel, p.s.i.
S. = tensile strength of the steel, p.s.i.
d = critical diameter at the root of the notch, in.
The basis for selection of the above equation was as follows. It was
thought that the increased strength indicated by the fact that K, was
less than Kt was due in large part to the resistance to fracture of a metal
under repeated plastic strain. This increased resistance was arbitrarily
assumed to be some function of the ratio between yield strength S, and
tensile strength Su. The assumption that
( S.  S 3 3 (7a)
p = c()= c(1 (7a)
was tried first; by empirical selection of the constant c it gave favor
able agreement with experimental results. Then the probable tendency
of small specimens to be weaker because of the proportionally larger
area occupied by a single crystal grain was considered, and a modified
formula was tried: / c
By a process of trial and error the values of c and a were obtained
as 0.2 in. and 0.05 in. respectively, to give best agreement between com
Bul. 398. CRITERIA FOR NOTCHSENSITIVITY 27
puted values and the values of Ke determined directly from fatigue tests.
Moore found it preferable to regard p' as an inverse measure of
notchsensitivity only, instead of the dimension of a structural unit as
Neuber originally conceived it.
Moore's empirical formula (Eq. 7) and his method of applying it
agreed reasonably well with the fatigue data for six steels he studied;
a correlation with data on SAE 1045, 3140 and 2340 steels quenched and
drawn to a structure of tempered martensite(8) also has been attempted
during the course of this study. However, for quenched and drawn SAE
4340 steel having a ratio of S,/SU equal to 0.95(35), the prediction of the
value of Ke by Moore's method yielded a deviation of about +20 per
cent; for slowly quenchedanddrawn or normalized SAE 1045, 3140 and
2340 steels having ratios of S,/Su from 0.62 to 0.68, and for an austem
pered SAE 2340 steel for which Sy/S. = 0.79, the predictions deviated by
about 22 to 42 percent of the observed value of Ke.
V. STRESS GRADIENT AND STRESS CONCENTRATION
The stress gradient, as represented by the slope of the stress distri
bution curve at the root of a notch, has been established as an important
factor in notchsensitivity20', 6, 3). The maximum stress gradient de
noted by m p.s.i. per in. may be estimated by the following formulas.
For a shaft with a transverse hole and loaded in pure bending the
maximum stress gradient"36) is as follows:
m = 2.3 (8)
r
And for a bending shaft with a fillet(36 it is
m = 2.6 (9)
r
in which
K,S, = theoretical maximum stress, where S. is the nominal
stress as found by elementary stress formula, p.s.i.
r = radius of hole or fillet, in.
For a circular shaft with circumferential grooves, the following
equations(53) may be used:
t
±+1
For direct tension: m =   6 (10)
r t
3  +2
\ 2r
F i t
KS, " \ 2r 2
For pure bending: m = 6 + (10a)
Ir ' t d
L \ 2r j
where
S& = nominal stress on net section as found by elementary
stress formula, p.s.i.
r = radius of groove, in.
t = depth of notch, in. = Y2 (diameter of gross sectiond)
d = diameter of net section at root of notch, in.
Bul. 398. CRITERIA FOR NOTCHSENSITIVITY
It will be noted that the value of m in Eq. 10 ranges only from 2/r to
3/r times the theoretical stress KtS. for a wide range in values of the
ratio t/r. For pure bending the stress gradient is also a function of the
actual diameter of the specimen, as indicated by the last term in
Eq. 10a.
16. Relative Stress Gradient
In correlating fatigue data on notched specimens, AphanasievN201
derived an empirical relation between the notchsensitivity as measured
by the ratio Ke/Kt and the relative stress gradient as measured by the
ratio m/Sn: K m
_K, m U
g =  = + 1 (11)
where a and b are material constants; the other symbols are as defined
previously. From this equation it is seen that the notchsensitivity as
measured by the ratio Ke/Kt increases as the relative stress gradient
m/Sn decreases. An irrational shortcoming of the formula is that the
relative stress gradient m/S, (and consequently the material constant
a) have the dimension 1/mm or I/in.; a fundamental constant for the
material probably should not have such a dimension without physical
meaning.
17. Stress Gradient, State of Stress, and Amount of Stress Concentration
Roedel139' made a direct comparison of notchsensitivity index q
(Eq. 1) and stress gradient m for rotating beam specimens with three
different types of notches; namely, with transverse holes"17'; with fil
lets('17; and with semicircular grooves2', 27). His results showed that
for the same material the notchsensitivity index depended upon three
factors the stress gradient; the state of stress; and the ratio r/d (i.e.,
the ratio of notch radius to net diameter, which was considered as an
index of the stress concentration). Part of his results are shown in Fig. 6.
If the ratio r/d remained constant, Roedel found that the following
results were obtained: (a) for shafts with transverse holes, the notch
sensitivity index dropped rapidly at a decreasing rate and became nearly
zero for large stress gradients; (b) for a more biaxial state of stress,
such as shafts with fillets, q dropped with increased stress gradient but
seemed to reach a minimum value well above zero for high stress grad
ients; (c) for shafts with grooves, q dropped at a decreasing rate, but
the minimum value of q when determined for high stress gradients was
above that for shafts with fillets.
In all cases it was concluded that: (a) for very low values of stress
gradient, q always approached a value of 1.0; (bl for intermediate
ILLINOIS ENGINEERING EXPERIMENT STATION
'ý0.8
0.6
I 0.4
3
0.e
/9
All Specimens With jI =0125
\SAE /0?0 SteelGrooves
04S% Carbon /Steel Fillets
0.45% Carbon Steel
Transverse Holes
0 / 2 3 4 5 6
Stress Gradient, (10'/lb. per sq. //. per in.)
Fig. 6. Relation Between Notchsensitivity Index
and Stress Gradient (from reference 39)
values of stress gradient, q decreased as the stress gradient increased,
but was also influenced by the biaxiality of the stress and by the acuity
of the notch; q was higher for lower values of r/d*.
Roedel explained his results by a reasoning similar to Thum's (see
Section 6) which assumed that some small amount of local plastic yield
ing took place to lower peak stress during repeated loading. Conditions
combining high stress gradients with a small degree of triaxiality of
stress led to local slip that relieves stress concentration and hence per
mits greater loads than those calculated from theoretical stress con
centration, i.e., lower values of notchsensitivity.
According to his analysis of plotted data, the geometric quantities
found to give closest correlation with values of q for a range of values of
these variables were:
r2/d or K,/r for shafts with grooves
r/d or Kt/r for shafts with transverse holes
r or r/d2 for shafts with fillets.
He gave no physical meaning to these arbitrary geometric quantities.
SSince m is roughly proportional to KS,n/r (according to Eqs. 8 to 10 which are based upon
the theory of elasticity) if m remains constant and r decreases, K, or S. must also decrease. Because
K. 1 Se/S. 1
S K 1 Kt 1
its value will increase when Sn or Kt decreases. Therefore, the statement that q is higher for lower
values of rid is logically necessary if d remains a constant.
Bul. 398. CRITERIA FOR NOTCHSENSITIVITY
18. Influence of Grain Size
Peterson obtained test data from specimens of carbon and alloy
steels with fillets or transverse holes and tried analyzing the following
two types of criteria for notchsensitivity; both criteria are based on the
stress gradient and some arbitrary measure of grain size136).
The Number of Grains in Regions of Peak Stress. In considering
the geometric effect and size effect of specimens, he contemplated that
the region of peak stress in a notched specimen was of prime importance,
and that a different result would be expected when one or two grains
were contained in the region than would be the case if 10,000 were con
tained in the same region. This lead to the idea of a criterion based on
the number of grains in an arbitrarily selected volume at peak stress.
If (say) the region stressed to within 5 percent of peak stress was
selected, the volume theoretically affected could be determined from
photoelastic fringe photographs, and the number of grains per unit
volume could be roughly estimated from a photomicrograph of the metal.
A plotting was made on semilog paper (Fig. 7) using experimentally
determined values of notchsensitivity index as ordinates and using the
number of grains within 5 percent of peak stress as abscissas on a log
scale. In this manner a straight line with its scatter band was obtained
which may be represented approximately by the following relation:
q = 0.378 + 0.141 log g ± 0.2 (12)
where g = number of grains within five percent of peak stress. The last
term + 0.2 indicates the approximate scatter band.
N
(9
0)>
Ho/es I I
1.2 CCarbon Steel  o0 ____ _
A/lloy Steel  *
F /llets Ni _
Carbon Steel  A 
Alloy Steel  A A Cr *
0.8 A '
0.6 0 __ __  
0.4 _' * I  
0.        _ _   
0.4^
.0O/ 0./ 1.0 /0 100 /000 /0000 /00000
Number of Gra,71s Within 65% of Peak Stress
Fig. 7. Notchsensitivity Index vs. Number of Grains
in Region of Peak Stress (from reference 36)
ILLINOIS ENGINEERING EXPERIMENT STATION
Decrement in Stress Across One Grain. Another criterion for notch
sensitivity studied by Peterson was based on the thought that the stress
gradient at the surface may be an inverse measure of the tendency of
initial damage to propagate across the section. The notchsensitivity
II
Gradi'ent X Crai/n Diameter
Endurance Lim'i
Fig. 8. Notchsensitivity Index vs. Relative Decrement
in Stress Across One Grain (from reference 36)
index was plotted against the relative decrement in stress across one
grain, and a curve as shown in Fig. 8 was obtained which may be ap
proximated by the following equation:
q = 0.250  0.358 log Sd + 0.2 (13)
stress gradient at peak stress X grain diameter
where Sd =
endurance limit of the material
The above two criteria do not differ markedly in numerical results;
that is, both show about the same deviation with respect to the scatter
of actual test data. The second criterion, however, was preferred by the
author.
Later Development. In 1945 Peterson modified his manner of pre
senting similar test data on notchsensitivityv~'. It was observed that
over a considerable range, the stress gradients for stress raisers such as
fillets, grooves and holes were approximately proportional to 1/r. There
fore r was used as a parameter representing stress gradient and the test
data were plotted on charts of q versus r (Fig. 9); the notchsensitivity
index was defined in a slightly different manner as shown by the relation:
Bul. 398. CRITERIA FOR NOTCHSENSITIVITY
K.  1
q = (14)
in which K, is the strength reduction factor calculated directly from
test data as defined before, and K, is the theoretical shear energy con
centration factor as defined by Eq. 4.
As shown in Fig. 9, a family of curves were obtained representing
different kinds of steel with different grain sizes. The top curve repre
sents finegrained quenched and tempered steels; the middle pair of
Notch Raodius, r, in Inches
Fig. 9. Notchsensitivity Index vs. Radius at
Root of Notch (from reference 37)
curves represent normalized steels of medium grain size; and the bottom
curve (which carries a question mark because of insufficient data) repre
sents a very coarsegrained medium carbon steel.
In considering Moore's relation between p' and Sy/Su (Eq. 7), Peter
son suggested that a family of curves of q versus r similar to those
shown in Fig. 9 could also be obtained using Sy/Su instead of the grain
size as a parameter. This chart would also tend to take care of the effects
of cold work during repeated stressing and would be simpler to apply in
designo40).
19. The Effect of a Stress Gradient
Roedel explained the effect of stress gradient by Thum's hypothesis
of lowering of peak stress by plastic action. Dehlinger(2") in contesting
Thum's hypothesis explained the nonsensitivity to notching as a slowing
ILLINOIS ENGINEERING EXPERIMENT STATION
down in the development of fatigue cracks owing to the rapid decrease
of stress from the surface towards the center of the specimen. Peter
son"37 also suggested that the specimen with large decrement in stress
across one grain (coarsegrained material and/or steep stress gradient)
was less notchsensitive because the fatigue crack was more difficult to get
started and to propagate across the grain, and stated that this trend was
confirmed by testing experience.
On the other hand, however, Aphanasiev(20" contends that the possible
retardation in the development of cracks might influence only the shape
of SN curves, but not the endurance limit.
20. Extent of Stress Concentration
The endurance limit for notched specimens is usually greater than
that predicted by the theoretical stress concentration factor; therefore
the endurance limit is increased from the point of view of elastic theory.
Heywood141" explained that this increase or gain in fatigue strength was
due to the fact that the region of stress concentration was very localized
in extent. The increase in strength can be represented as the difference
between the endurance limit S, obtained from fatigue tests of notched
specimens, and S,' predicted by using the theoretical stress concentration
factor. That is, the relative gain in fatigue strength is
Sn  Sn' S,/Ke  Se/K, Kt
S 1 (15)
Sn' Se/K, Ke
where Se is the endurance limit of unnotched specimens.
Heywood expressed the extent of stress concentration (irrespective
of size of specimen) in terms of N/ where r was the root radius of the
notch and n was a coefficient depending on the type of notch. Since
the increase in fatigue strength was assumed to be a function of the ex
tent of stress concentration, he proposed this formula:
Kt M
 1 =  (16)
K, A/ nr
where M was a material constant directly proportional to the relative
gain in fatigue strength. Therefore, the greater the value of M, the
larger was the relative notched fatigue strength (or the smaller was
the notchsensitivity).
Heywood collected and plotted the test data on charts showing
K,/K,  1 versus 1/V r, and a straight line was obtained for each
kind of material and each type of notch, with a scatter band of about
± 10 percent. Figure 10 for plain carbon steels is shown here as a typical
example. The data in Fig. 11 for heattreated alloy steels showed wide
Bul. 398. CRITERIA FOR NOTCHSENSITIVITY
Fig. 10. Gain in Fatigue Strength vs. 1/Vr for Shouldered Shaft Specimens of Plain Carbon
Steels Tested in Reversed Bending: r = fillet radius in inches (from reference 41)
N
~i ~,
n.. +~
C~)
K
N
6
Fig. 1I. Gain in Fatigue Strength vs. 1/Vr for Shouldered Shaft Specimens of Heat Treated
Alloy Steels: r = fillet radius in inches (from reference 47)
ILLINOIS ENGINEERING EXPERIMENT STATION
scatter. Apparently the heattreated steels could not be regarded as
one class of material; both chemical composition and heat treatment
influenced their behavior when judged by this hypothesis. The values
of M and n in Eq. 16 determined from data (such as those in Figs.
10 and 11) were as follows:
Class of Material Values of M
Aluminum alloys
Copper alloys
Plain carbon steels
Magnesium alloys
Heattreated alloy steels
0.090
0.070
0.065
0.044
0.030 (Average)
Types of Notch Values of n
Shouldered shaft 1.0
Shaft with transverse hole 0.35
Vee groove in shaft 0.26
It is interesting to note the similarity between Heywood's formula
(Eq. 16) and Neuber's formula (Eq. 5a), though each was derived inde
pendently and based on somewhat different concepts. To facilitate
comparison, Heywood's formula and its transformations are grouped
on the left and Neuber's corresponding relations are grouped on the
right in the tabulation below:
Heywood's (Eq. 16)
Kt
1=
K.
K,
Kt
1+
M
Vnr
M
Vnr
M
V\nr
K,1
q t  1
K,  1
1
Sp'
1+I r
K,=I+
K = 1+K  1
+ r
Both Heywood and Neuber interpreted the relation between Kt and
K, by a single parameter V\/r for a given material and a given type
of notch, but they used different relations between K, and K, to express
notchsensitivity. A series of six plottings were therefore made of the
test data of Fig. 10 on charts of notchsensitivity versus 1/ /7, using
six different expressions respectively for notchsensitivity, such as:
Neuber's (Eq. 5)
Kt  K,
K, 1 /r
Bul. 398. CRITERIA FOR NOTCHSENSITIVITY
K, 1 Kt  Ke K, Kt  K  K, K, 1
, 1  ;K K,; K X
K,  1 Ke  1 ' Kt KtKe K, Kt  1
Examples of several of these diagrams are shown in Figs. 12, 13, and
14. All these plots had similar scatter bands except points (Fig. 12)
representing the relations derived from Neuber's formula. Though it fits
the test data for grooved specimens of certain steels fairly well, Neuber's
formula does not closely group this entire set of test data.
Heywood's formula is remarkably simple, but certain wide deviations
with respect to actual test data lead to doubt of the accuracy of the
formula for a wide variety of conditions. For example, the value of M is
not a constant when applied to cast iron for which Ke is approximately
equal to 1. Moreover, the formula would require that the gain in fatigue
strength should be infinite when the notch radius approaches zero, which
is contrary to actual experience.
0.8
0.6
0.4
0.2
0
0
0
0
0
0
0
0
0
0
0
0
0 0
)____
0
0
0
1
Fig. 12. Plotting of Notchsensitivity Index q 
vs. I/VT (for the same data as in Fig. 10)
Io
0

ILLINOIS ENGINEERING EXPERIMENT STATION
1
Fig. 13. Plotting of K,  K. vs. 1/ r (for the some data as in Fig. 10)
Bul. 398. CRITERIA FOR NOTCHSENSITIVITY
Kt  K
Fig. 14. Plotting of  vs. 1/Vr (for the some data as in Fig. 10)
KiK.
VI. FAILURE BELOW SURFACE
It was assumed in a hypothesis presented by Peterson(28) and
Moore(34) that the fatigue strength in a bending test behaved as though
the stress of significance was not that at the surface of the polished
specimens but was that computed at a small distance h' below the sur
face. The value of h' and the computed fatigue strength Sec at depth h'
were assumed to be constants of the material, being independent of the
size of specimen.
21. Moore and Smith's Formulas
In Fig. 15a the stress Se at the surface denotes the endurance limit
for unnotched specimens, and Sec represents a critical stress at a distance
h' below the surface. Both Sec and h' were assumed to be basic fatigue
properties for unnotched specimens independent of the diameter d of the
round specimens. The following equation for unnotched specimens was
derived in a simple mathematical procedure by Moore034), based upon the
geometry of Fig. 15a:
Se  (17)
2h'
1
d
In a similar way Smith(35) derived the following equation for notched
specimens from the geometry of Fig. 15b:
St.
St  (18)
Rh"
1  
1
dn
where St is the theoretical maximum stress at the surface, equal to Kt
times endurance limit Sn of the notched specimen (i.e., St = KSn);
St,, is the critical stress at a distance h" (different from h') below the
surface; the value of Sic is assumed to be a basic fatigue property for
notched specimens independent of the diameter d. of the specimen at
the net section. R is defined as the maximum stress gradient times the
diameter of the specimen divided by the peak stress (R = md./S,).
It is a dimensionless quantity and may be called the "relative stress
gradient." The value of R may be computed from Eqs. 8 to 10, or may
Bul. 398. CRITERIA FOR NOTCHSENSITIVITY
d 21h
(a) Unnoftched SpeciVen (b)Alotched Spec/imen,'
Fig. 15. Distribution of Longitudinal Stress in Rotating Beam Specimens
be derived from Neuber's theory of elasticity(5). For unnotched rotating
beam specimens R = 2.
Since St = KS., Eq. 18 may be rewritten:
Sn = Rh (19)
Kt (1 Rh"
From fatigue test data on six steels(34) h' was found to have an
average value of 0.008 in. and Sec/Stc to have an average value of 0.88,
so Eqs. 17 and 19 may be rewritten approximately as follows:
See
S=  (20)
0.016
1
d
S& =  (21)
0.88 Kt (1  )
By dividing Eq. 20 by Eq. 21 and letting d. = d we have
Se 1  Rh" d
K  = 0.88Kt  (22)
S, 1  0.016/d
from which the strength reduction factor Ke may be approximated.
In the above equations S,,. and h" are two different constants for any
given material. In order to evaluate Se. and h" for a new material, values
of two endurance limits must be determined from tests of unnotched
and notched bars of any size, or from tests of notched bars of two
ILLINOIS ENGINEERING EXPERIMENT STATION
different diameters. After Sec and h" have been determined by substitut
ing the test results in the above equations, the values of the endurance
limits of notched or unnotched specimens of any size may be predicted
from Eqs. 20, 21, or 22.
The procedure outlined above was used to calculate the strength
reduction factors from Eq. 22 for grooved specimens of six steels(22) and
an aluminum alloy"42) and for specimens of two steels with fillets or
transverse holes"17). The deviations of the calculated values of K, from
those obtained experimentally from the ratio Se/Sn ranged approximately
from +14 to 14 percent.
The relations developed agreed with all the test data fairly well. The
physical concepts or evidence and assumptions upon which this analysis
was based are appraised later (Section 23).
22. Peterson's Formulas
Based upon a similar assumption of failure corresponding to the
stress at constant depth h below the surface, Peterson suggested the
formula<28, 40).
Kormula = K, (1  Ch/r) (23)
in which K, is the strength reduction factor, and K, is the shear energy
concentration factor as defined by Eq. 4a; C is a stress gradient con
stant (using C = 3 for grooves and fillets); and r is the notch radius.
This relation appears to be a simplified empirical form of Eq. 22.
Peterson(37 40) also presented a relation similar to Eq. 23 that will
give Ke = 1, when r = 0:
K, , K[1 Ch (K,  1) (24)
K. = Kf 1  (24)
r (K,  1) + ChK, I
It can be seen that Eq. 23 and Eq. 24 are not identities. By fitting Eq.
24 to Moore's test data on SAE 1020 steels asrolled and strainrelieved,
Peterson found h = 0.002 in., whereas according to Eq. 22, to fit the
same data, h' = 0.008 in. and h"  0.004 in.
23. Lack of Evidence Regarding the Basic Assumptions
Several investigators have thus discussed notchsensitivity and its
correlation with test data based on the assumption that the specimen
behaves as though failure is initiated below the surface; however, no
evidence has been presented to show that a fatigue fracture necessarily
starts below the surface. It was suggested that the usual polishing may
coldwork and strengthen a thin surface layer of the specimen; thus, the
fracture might actually start in the weaker metal under the coldworked
layer. If this inference were correct, an SAE 1035 steel, brightannealed
Bul. 398. CRITERIA FOR NOTCHSENSITIVITY
after polishing'34), should not have exhibited the same size effect or notch
effect; hence, this is a conclusion contrary to fact. Moreover, as pointed
out by Peterson40o), if coldworking of the surface due to machining were
of importance, it could often be subject to considerable variation, and
the depth of failure cannot therefore be assumed constant. Furthermore,
the different states of stress developed by notches of different sharpness
would probably affect the depth at which failure was initiated.
It was also pointed out(34) that in all metals there are many inherent
defects whose strengthreducing effect might be equivalent to that of a
small notch in the surface of the specimen. Thus the fracture might start
at the bottom of some inherent defect imbedded in the surface, or at
some such defect slightly below the surface. However, if the inherent
defects are important enough to be seriously considered, they also vary
considerably in their stressraising effects. On the basis of this concept
the depth of failure can hardly be regarded as a constant, and the ele
mentary stress formulas which lead to the above equations should not
lead to useful or accurate relationships.
Some experimental evidence has been provided(43) that fatigue failure
initiates in a very thin surface layer less than 0.002 in. thick and pro
gresses inward gradually. Hence, the assumption of constant failure depth
seems to be only an empirical method of correlation of a limited amount
of test data for convenience in design. The assumptions made and the
relations deduced are arbitrary and lack physical significance. The em
pirical nature of this approach was recognized by Peterson and Moore'o).
VII. STATISTICAL THEORIES OF FATIGUE
Both Freudenthal"44) and Aphanasiev(54 used statistical approaches
to explain fatigue phenomena and the notch effect. While Freudenthal
derived only theoretical equations for brittle metals, Aphanasiev also
offered equations based on theory but modified in the light of experi
mental test data.
24. Freudenthal's Equation
Freudenthal considered the fatigue phenomenon as an expression of
the repetitive action of an external load. This progressive destruction
had the typical features of a mass phenomenon; both the cohesive bonds
and the load repetitions were treated as collectives in a statistical sense.
A general statistical theory was then developed by him for only truly
brittle materials, for which no plastic deformation would occur during
repeated loading so as to change the internal structure of the metal.
If a nonuniform stress distribution due to a notch or due to flexural
action in a beam is approximated by a discontinuous step function with
two intensities, it can be assumed that a percentage i of the total Q
bonds is subjected to the high stress intensity in the vicinity of the notch,
while the majority of bonds are in the field of comparatively low homo
geneous stress. If p is the probability of destruction of a bond in the
field of low stress intensity, and pi is this probability in the immediate
vicinity of the stress concentration, then it was shown that the proba
bility P2 of rupture of all Q bonds under nonuniform stress distribution
for N load repetition is:
P2 = 1  (1  P) eiQov(pp) (25)
in which e is the base of natural logarithms, and P is the probability of
rupture of all Q bonds under the homogeneous low stress, without stress
concentration.
According to this equation, the probability P, increases rapidly with
increasing value of the exponent on e, that is: (a) with increasing
number of bonds i Q in the volume affected by the stress concentration;
(b) with increasing load repetitions N; and (c) with intensity of stress
concentration (p,  p). These three influences are of the same order of
importance; a similar increase in the probability of rupture will be ob
Bul. 398. CRITERIA FOR NOTCHSENSITIVITY
tained either by increasing the volume affected by stress concentration
or by increasing the number of load repetitions. Therefore, the effective
strength reduction due to a notch should increase considerably at the
lower stress levels (for which the number of load repetions is large).
This agrees with Bennett's observation(46' that for SAE X4130 steel,
fatigue strength reduction factors based on the endurance limits were
about the same as the relative slopes of the log Slog N curves; it also
agrees with Almen's suggestion(47) that the slope of an SN curve is a
relative measure of the strength reduction factor.
Freudenthal's statistical theory for brittle materials dealt only with
the finite fatigue life. Therefore the endurance limit of notched or un
notched specimens, or strength reduction factor for ductile metals, cannot
be predicted directly from his theory or from Eq. 25. However, his later
theories"19 of the structural readjustments that occur have contributed
concepts which may help to predict"0' the endurance limit of notched
specimens.
25. Aphanasiev's Equation
Aphanasiev's statistical theory of fatigue"45) was based on the as
sumption that the metal was an aggregate of crystal grains regarded as
elementary structural units having identical yield limit and cohesive
strength in the direction of the external force. However, these grains
were subjected to different stresses due to the inhomogeneity of the
material, porosity, inclusions, influence of the grain boundaries, etc.
Thus, the frequency of the occurrence of any particular stress value
acting on an individual grain might be expressed as a function of the
value of the imposed stress.
For fatigue loading, the critical condition or criterion for the value of
the endurance limit was formulated by assuming that no grain was sub
jected to a stress exceeding the cohesive strength. The parameters of the
probability function in the equations were evaluated from static tension
test curves for the metal, which also included the effect of strain
hardening.
For the purpose of determining the fatigue strength reduction factor,
the general solution presented considerable difficulties. Finally the follow
ing tentative expression for the strength reduction factor was proposed:
2d2
K. = Kt" (26)
k R2k + 4
2d
ILLINOIS ENGINEERING EXPERIMENT STATION
where: K. = SeI/Se2 = the ratio of endurance limits of specimens of two
different sizes or shapes
B and k = constants of the material
di and d2 = diameters of specimens
RI and R2 = relative stress gradients as defined in Section 21.
(R = 2 for unnotched beam specimens)
This equation is adaptable to cases involving either notch effect or
size effect. If applied to notched members, Sei, d, and RB are for un
notched specimens, and Kt, Se2, d2 and R2 are for notched specimens. If
only the size of specimen is varied, R,  R2 = constant; Kt = constant
for both large and small specimens; and Ke represents only the ratio of
endurance limits of specimens of two different sizes.
As an example, Fig. 16 was shown by Aphanasiev to indicate a good
agreement of his equation with Peterson's test data"7) on bending of
shafts with fillets. The curves were drawn according to the following
equation as a special case of Eq. 26:
/ 1.67
1+ 2.2 .67
K, = Kt  (26a)
1 2.2 R2 + 1.33
1 + 2.2
2d22
Generally, the statistical theories of material strength apply to brittle
materials only, since the complicated effect of plastic deformation
changes the material structure and cannot readily be included in the
analysis. Aphanasiev's theory has arbitrarily considered the effect of
workhardening, but the final relation thus depends on empirical co
efficients.
There are more chances of finding a few of the weaker elements in a
large specimen than in a small specimen of uniformly heterogeneous
material. Hence a large specimen should be weaker than a small one
according to the "weakestlink" theory (i.e., assuming the specimen to be
equivalent to a chain which is no stronger than its weakest link). This
conclusion is apparently in rough agreement with the test data on size
effect. Another prediction from this theory is that a group of large
specimens should show less deviation in their strengths (or fatigue life)
than a similar group of the small specimens; however, this is apparently
not in complete agreement with the limited test data available 33, 35)
Bul. 398. CRITERIA FOR NOTCHSENSITIVITY
:o K,. 2='.Z
/.4 
I..
0
,' 0 /0 20 30 40 ,50 60 70 3 0o
Diameter of Specimen in mm.
Fig. 16. Agreement of Aphanasiev's Equation with Test Data for
Shouldered Shaft Specimens (from reference 45)
Apparently, the statistical or "chance effect" may be only one of the
factors influencing the behavior; other contributions from inelastic defor
mation on a microscopic scale must be given consideration before the
"notch effect" and the "size effect" can be quantitatively evaluated.
VIII. HOMOGENEITY OF MATERIALS
26. Cast Metals
Another important consideration is the effect of the homogeneity or
heterogeneity of the metal on notchsensitivity. It has been mentioned
that cast iron and some cast aluminum alloys exhibit little reduction of
fatigue strength due to ordinary notching of a specimen(21' 48). This is
probably related to the fact that these materials are not homogeneous
even on a macroscopic scale. Cast iron, for example, may be regarded as
steel full of stress raisers in the form of holes or cavities which are oc
cupied by graphite flakes. Therefore an external notch, which is usually
not as severe as the internal notches, cannot develop further pronounced
reduction of strength. This case is typical for nonhomogeneous materials
(or uniformly heterogeneous materials). Metals with inherent defects or
internal notches like nonmetallic inclusions, blowholes, pores in sintered
powders, microcracks, tensile residual stresses, etc., are weaker than the
same metal without internal notches, but also they may not be as notch
sensitive. More experimental data are needed to confirm this general
statement.
27. Heat Treatments
From the above reasoning it may be generally assumed that to make
a material less homogeneous is to lower the unnotched fatigue strength
relatively more than the notched fatigue strength, and hence to make the
material less notchsensitive. The more severe the unfavorable residual
stresses, the less homogeneous the metal; hence it may be expected to be
less notchsensitive if residual stresses are present. This hypothetical
statement offers a possible explanation for changes in notchsensitivity
due to differences in heat treatments.
In a study of the effect of metallurgical structure on fatigue strength
and notchsensitivity of steel"), it was found that when the tensile
strengths were kept constant, the steels rapidly quenched and tempered
(principally temperedmartensitic structures) generally exhibited slightly
higher unnotched fatigue strengths and less notchsensitivity than the
same steels slowly quenched and tempered (structures of pearlite plus
ferrite). It is difficult to determine whether the drastically quenched and
Bul. 398. CRITERI,% FOR NOTCHSENSITIVITY
tempered steels or the slowly quenched and tempered steels had higher
residual stresses or higher nonhomogeneity, when these pertinent factors
are considered:
(a) the higher tempering temperature given the drastically quenched
steels relieves more residual stress, but probably would not heal possible
minute microcracks;
(b) the drastically quenched steels have higher flow stress which
makes the relief of residual stress more difficult;
(c) the cooling process from the high tempering temperatures may
also induce some microresidual stresses, additive to or diminishing the
previous stress state.
Since the drastically quenched and tempered steels were less notch
sensitive, one possible explanation might be the presence of inherent
residual stresses which made the material less homogeneous.
However, the drastically quenched and tempered steels exhibited
higher notched and unnotched fatigue strengths than the slowly quenched
and tempered groups. As an explanation for this fact it was suggested("4O
that the tempered martensitic steels had relatively smaller agglomera
tions of weak ferrite and more uniformly dispersed hardening constitu
ents of iron carbides which offer higher resistance to slip and crystal
fragmentation and hence higher fatigue strength. Larger crystals of free
ferrite in the slowly quenched steels offered a smaller resistance to slip.
The net result on the rapidly quenched steels was an increase in the
fatigue strengths of both unnotched and notched specimens (especially
the marked increase in the strength of notched specimens).
For the same quenched steels tempered at different temperatures or
the same metals coldworked to different degrees, the tensile strengths
and the workhardening capacity will be different. The workhardening
capacity probably exerts more influence upon the notchsensitivity than
the presence of residual stresses. The higher the tempering temperature
for the same quenched steels, the higher is the workhardening capacity
and the lower is the notchsensitivity, though the residual stresses are
also lower.
IX. CLOSURE
After reviewing these interpretations by numerous investigators it
appears evident that the fatigue notchsensitivity of a metal member
depends upon three different factors  the basic material characteristics,
the degree of material homogeneity, and the geometry of the member.
As regards the basic material characteristics, attempts have been
made by investigators to formulate a quantitative expression for this
"constant." The concepts of tensile strength, plasticity, ductility, damp
ing capacity, cohesive strength, workhardening capacity, grain size,
ratio of yield strength to tensile strength, and the elementary structural
unit have all been used in attempts to interpret or explain notch
sensitivity; all of these properties are probably indirectly related. For
example, a given material with a small grain size may be expected to
have relatively high values of notchsensitivity, tensile strength, ratio of
yield strength to tensile strength, and correspondingly low values of
plasticity, ductility, damping capacity, and workhardening capacity.
The intrinsic physical nature of the mathematician's concept of an
"elementary structural unit" itself is not known; hence there is a ques
tion as to whether the structural unit is related to the size of (a) the
space lattice; (b) a crystallite; (c) mean free ferrite path"2"; (d) some
kind of crystal grain; or (e) some other type of constituent. If not
represented by a physical constituent, the structural unit has no signifi
cance in terms of the properties of actual structural units, and can only
be an empirical material constant with the dimension of length (com
pare the material constant a in Eq. 11).
The concept of readjustments due to plastic action has provided a
useful tool to explain differences in notchsensitivity of materials quali
tatively, but there is no direct quantitative evidence for this explanation
from macroscopic or microscopic measurement of deformation or stress
in the vicinity of an ordinary notch (Sections 6 and 10). Since fatigue
phenomena originate basically on a submicroscopic scale, it is probable
that only the very minute plastic or inelastic adjustments associated
with workhardening in a localized region (and hardly detectable in
ordinary direct measurements) are important factors in determining the
notchsensitivity. The insensitiveness of austenitic stainless steel to the
damaging effects of a notch may be considered as an extreme case illus
trating the effect of workhardening capacity.
Bul. 398. CRITERIA FOR NOTCHSENSITIVITY
No method of evaluation of the workhardening capacity of a metal
under repeated stress has been widely acceptable for use. However, for
some steels, such as plain carbon steels or low alloy steels, the tensile
strength (Section 8) may perhaps be regarded as the simplest and the
most convenient quantity for use as a rough index of workhardening
capacity. The higher the tensile strength of a metal due to severe cold
work or heat treatment, the lower is the workhardening capacity, and
hence the greater is the notchsensitivity.
For the same tensile strength there is an effect of the heat treatment
on the notchsensitivity; this effect may be due to the difference in
degree of homogeneity induced by the different heat treatments. A
greater degree of material homogeneity usually results in higher strength
but also is accompanied by higher notchsensitivity. The effect of a
heterogeneous structure is best illustrated by cast iron which is insensi
tive to the effect of a notch because it has already been drastically
weakened by internal defects.
As regards the geometry of the member, the notch radius is the most
important single parameter in determining the notchsensitivity as well
as in controlling the theoretical stress concentration factor (see Chapter
V). The larger the radius the greater is the notchsensitivity and the
smaller are the theoretical as well as the effective stress concentration
factors. The second important parameter is usually the diameter at the
net section for a rotating beam specimen. Thus, notch radius and the
minimum diameter are the most important geometric dimensions that
govern the notchsensitivity, strength reduction, and theoretical stress
distribution. Since both "size effect" and "notch effect" in fatigue are
mainly the result of geometric parameters, the essential nature of both
effects is probably the same.
The material properties that have been used in attempts to predict
the notchsensitivity of a material are evaluated by "largescale" engi
neering measurements, whereas fatigue behavior is originally related to
or governed by atomic or submicroscopic readjustments in the structure.
Therefore, any predictions of notchsensitivity directly from existing
largescale properties, such as those obtained from the tension test, can
hardly be expected to be accurate in predicting fatigue behavior. For the
same material, however, the relation between geometric parameters and
notchsensitivity may be found experimentally and perhaps formulated
analytically. It might then be possible by analytical or empirical study
to express the difference in such relations quantitatively by introducing
an empirical constant for each individual material. This constant might
be a better criterion for notchsensitivity than the largescale properties
ILLINOIS ENGINEERING EXPERIMENT STATION
thus far investigated, especially if a rational physical significance can be
assigned to the constant. This would appear to be a useful procedure for
attacking the general problem of fatigue notchsensitivity.
When the fatigue strength of an unnotched specimen is considered, it
has been the general practice to disregard the effect of a stress gradient
and to assume that the specimen must fail or break if the repeated stress
imposed at any point in the specimen is above the endurance limit. But
it is believed by recent investigators that the stress gradient at peak
stress, or the relative extent and amount of stress concentration, is also
an important factor in determining the fatigue strength. Therefore, not
only the stress and strength conditions at the point of peak stress (or the
point of maximum shear energy according to the shear energy theory)
but also the conditions existing in the region surrounding this critical
point are important factors in determining the fatigue strength. The
hypothesis of the elementary structural unit assumed this region to be as
large as the structural unit, and the empirical concept of failure below
the surface assumed this zone to be deep enough to initiate failure below
the surface. Whenever a steep stress gradient occurs in a specimen, it is
essential that the criterion for fatigue failure include not only the peak
stress but also the conditions existing in a region surrounding the point.
The discrepancy between theoretical and effective stress concentra
tion factors is attributed to the basic fact that the structural action in
the materials under repeated loading is different from that under static
elastic loading. However, very few studies have been based upon concepts
of the fundamentally distinctive structural readjustments which occur
during repeated stressing. It is felt that an approach to the problem of
notchsensitivity starting from recent knowledge of the character of
fatigue damage might yield an interpretation that is at once more
satisfactory and more dependable.
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