2.3. FATIGUE TEST, S-N CURVE, AND MATERIAL ENDURANCE LIMIT 11
M
A
A A
B
A
B
M M M
(a) (b) (c) (d)
Figure 2.1: Schematic of a gradually crack propagation.
area under a normal service cyclic load will be larger than the ultimate material strength,
causing the component to rupture due to static loading.
4. Fatigue damage is irreversible and will be gradually accumulated. e fatigue damage is due
to crack propagation. When the cyclic loading stops, the propagated microscopic cracks
still exist. erefore, the fatigue damage is irreversible and will be gradually accumulated
on the continuous cyclic loading.
2.3 FATIGUE TEST, S-N CURVE, AND MATERIAL
ENDURANCE LIMIT
A cyclic load applied on a component can be any type of cyclic load and can be described by
six models of cyclic loading spectrum [4], which has been described in Section 1.2. But, lots
of material fatigue strength data is typically obtained from a stress-life method. In the stress-
life method, a specimen is subjected to cyclic stress with a constant stress amplitude until it
fractures and fails. ere are many different types of fatigue specimen and fatigue test procedures.
Fatigue test specimen will be designed and manufactured according to corresponding fatigue
standards such as ASTM standards, and the test procedure will also follow the procedure defined
by corresponding fatigue test standards. e cyclic stress for a fatigue test could be cyclic bending
stress, cyclic axial stress, or cyclic shear (torsion) stress. e cyclic stress in a stress-life method is
typically a fully reversed cyclic stress, that is, a constant stress amplitude with zero-mean stress.
e main reasons for this are as follows. (1) Lots of fatigue test data are from rotating bending
fatigue test, in which the cyclic stress is a fully reversed cyclic stress. (2) In fatigue theory for
fatigue design, non-zero mean cyclic stress will typically be converted into fully reversed cyclic
stress with an equivalent stress amplitude by including the effect of mean stress. (3) Even though
fatigue tests are under cyclic stress with non-zero mean stress, it might be still presented as
fatigue test data with an equivalently fully reversed cyclic stress for the purpose that the fatigue
test data can be used for fatigue design. In the following, we will assume that cyclic stress in the
stress-life method is a fully reversed cyclic stress.
In a stress-life method with a fatigue test specimen under a fully reversed cyclic stress,
test results are the stress amplitude S
0
f
and the number of cycles at the failure N . Both S
0
f
and
N is material fatigue strength data. is stress amplitude S
0
f
in a fatigue test is called as the
12 2. RELIABILITY OF A COMPONENT UNDER CYCLIC LOAD
material fatigue strength at the given number of cycles n
L
D N . e physical meaning of this
fatigue strength S
0
f
is that when the number of cycles of a fully reversed cyclic stress is n
L
D N ,
the fatigue specimen will fail if the stress amplitude
a
of a fully reversed cyclic stress is more
than S
0
f
. In other words, the maximum stress amplitude of a fully reversed cyclic stress cannot
exceed S
0
f
to avoid a fatigue failure when the service life is specified as n D N . e number of
cycles at failure N in the fatigue test is called as the material fatigue life at this specified stress
level
a
D S
0
f
of a fully reversed cyclic stress amplitude. e physical meaning of the material
fatigue life N is that if the fatigue test specimen is under a fully reversed cyclic stress with a stress
amplitude
a
D S
0
f
, the fatigue specimen will fail when the service life n
L
of such cyclic stress
is more than N . erefore, the fatigue test results .S
0
f
; N / is a pair of fatigue strength data in a
fatigue test.
After fatigue tests on the fatigue specimen of the same material are continuously con-
ducted at different stress amplitudes (stress levels), a group data .S
0
f
; N / will be collected and
can be depicted as an S-N curve, as shown in Figure 2.2. e S-N curve is typically plotted in
a Cartesian coordinate with a log-log scale.
Low-Cycle
Fatigue
High-Cycle
Fatigue
Infinite Life
Number of Cycles at Failure N
Fatigue Strength S' (ksi)
f
S
ut
S'
e
10
0
10
3
10
6
Figure 2.2: Schematic of an S-N curve.
In Figure 2.2, the small dots are a pair of fatigue test data. ere are three different fatigue
regimes, as shown in Figure 2.2. e fatigue failure form N D 1 to N D 1000 (cycles) is generally
classified as low-cycle fatigue. In low-cycle fatigue, the stress level at the number of cycles at
failure N D 1 is the ultimate material strength S
ut
. For low-cycle fatigue, the test method is
typically through strain-life method or linear-elastic fracture mechanics method, in which, the
strain or the crack growth will be controlled or measured. is low-cycle fatigue is not the
concerned topics of this book.
e fatigue failure with N > 1000 (cycles) is generally called as high-cycle fatigue. e
high-cycle fatigue will be the focus of this book and is the typical case for most of the fatigue
design in the industry.
2.3. FATIGUE TEST, S-N CURVE, AND MATERIAL ENDURANCE LIMIT 13
Material endurance limit: For some materials like steels as shown in Figure 2.2, there is a value
below which fatigue specimen will not fail with a very large number of cycles such as more
than 10
6
(cycles). Material endurance limit S
0
e
is usually defined as the maximum fully reversed
stress amplitude that a material can withstand infinitely without fracture. For some materials,
the fatigue strength S
0
f
at the fatigue life N D 10
6
(cycles) is named as material endurance limit
S
0
e
.
In high-cycle fatigue, a fatigue life between N D 10
3
(cycles) and N D 10
6
(cycles) is
defined as a finite-life region, and a fatigue life N 10
6
(cycles) is defined as an infinite life.
In a finite-life region, when there are fatigue tests on at least three different stress am-
plitude levels of fully reversed cyclic stress, the average fatigue life N at the same fatigue stress
level vs. the fatigue strength S
0
f
in a log-log scale coordinate system can be typically simplified
as a linear line, as shown in Figure 2.2. e material fatigue strength S
0
f
and the fatigue life N
on this linear line has the following relationship:
N
S
0
f
m
D Constant; (2.1)
where S
0
f
and N are the material fatigue strength and the corresponding fatigue life on the
simplified linear line. m is the slope of the traditional S-N curve and is a material mechanical
property determined by fatigue test data. m can be determined through the linear least-squares
regression by using the fatigue test results:
m D
I
"
X
I
.
ln
ai
ln N
i
/
#
X
I
ln
ai
!
X
I
ln
N
i
!
I
"
X
I
.
ln
ai
/
2
#
P
I
ln
ai
2
; (2.2)
where I is the number of different stress amplitude levels
a
for fatigue tests;
ai
is the ith stress
amplitude level in fatigue tests; ln N
i
is the average fatigue life in a log-scale at the fatigue test
level
ai
. If there are a total J fatigue tests at the fatigue test stress level
ai
, the ln N
i
will be:
ln N
i
D
P
J
ln N
ij
J
; (2.3)
where N
ij
is the number of cycles at the failure of the j th fatigue test under the fatigue test stress
level
ai
.
When there are only a few fatigue tests for the S-N curve, Equation (2.1) is the design
equation for the traditional fatigue design approach. e case with plenty of fatigue tests will be
discussed in Sections 2.8 and 2.9 and will be the focus of this book.
Example 2.1
Fatigue tests of steel specimens under a fully reversed cyclic bending stress are listed in Table 2.1.
Determine the material property m on a log-log scale.
14 2. RELIABILITY OF A COMPONENT UNDER CYCLIC LOAD
Table 2.1: A group of fatigue test data
Stress Amplitude σ
a
(Mpa)
Sample Size
Fatigue Life N (cycles) × 10
3
392.40 4 34, 42, 43, 48
372.78 6 36, 47, 48, 53, 62, 65
353.16 6 60, 70, 77, 84, 89, 116
333.54 4 111, 114, 145, 197
313.92 5 171, 253, 254, 301, 309
Solution:
In this group of fatigue tests, there are five different stress amplitude levels. So, I D 5. In each
fatigue stress levels, the fatigue tests are repeated for several times. We will use Equation (2.3) to
calculate the average fatigue life in a log-scale at each stress amplitude level. For example, there
are six repeated fatigue tests on the third stress level
a3
D 353:16 (Mpa), the average fatigue
life in a log scale log N
3
will be:
ln N
3
D
ln
.
60000
/
C ln
.
70000
/
C ln
.
77000
/
C ln
.
84000
/
C ln
.
89000
/
C ln
.
116000
/
6
D 11:30523:
e stress amplitudes and corresponding average fatigue life on a log-log scale for this example
are listed in Table 2.2.
By using the data from Table 2.2 with Equation (2.2), the material property m is:
m D 8:303:
Table 2.2: Average log stress amplitudes and fatigue life
Stress level # 1 2 3 4 5
lnσ
ai
5.972282 5.920989 5.866922 5.809763 5.749138
lnN
i
10.63186 10.8372 11.30104 11.83417 12.43832
For fatigue design with an infinite fatigue life, the material endurance limit S
0
e
is one
fatigue strength data for the material. When there is a lack of fatigue test data for material
endurance limit S
0
e
, it can be estimated by using the ultimate strength of the same material [5].
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