2.9. THE PROBABILISTIC FATIGUE DAMAGE THEORY (THE K-D MODEL) 73
In this approach, a probabilistic distribution function is used to describe the fatigue test data at
the same cyclic stress level. If fatigue tests are at several cyclic stress levels such as seven stress
levels, there will be seven different probabilistic distribution functions if there are big enough
number of tests in each stress levels. However, the P-S-N curve approach has the following four
issues in its implementation for fatigue reliability design.
1. Since fatigue tests are time-consuming, there are only a few fatigue test data at each stress
levels, which are common cases, as shown in the fatigue data book [21]. In such a situation,
the P-S-N curve cannot be constructed due to the small sample size.
2. In some available fatigue test data, the total number of fatigue test might be more than
30 even though the number of fatigue tests in the same stress level is small, which is the
common case in the fatigue data book [21]. e P-S-N curve approach cannot use such
data to construct the P-S-N curves.
3. When the cyclic stress level in cyclic loading is not equal to the fatigue test stress level,
the probabilistic distribution function at this level is not available in the P-S-N curves,
which is a typical case in reality for fatigue design. So the P-S-N curves cannot be used
to solve this type of issue. e P-S-N curve approach could use the interpolation method
to obtain the probabilistic distribution function at the required stress level for reliability
fatigue design. However, this distribution function is not directly obtained from or based
on the test data, and it might induce some big error.
4. In fatigue tests, actual dimensions of fatigue specimen will be slightly different. erefore,
the actual stress level for the same nominal stress level fatigue test might be different; even
the nominal stress level is the same. However, the P-S-N curve approach ignores these
differences and use the nominal stress level to create the P-S-N curves.
e fatigue damage mechanism, which has been discussed in Section 2.2, shows that the
fatigue damage is mainly caused by cyclic loading and randomly distributed defects inside a
component such as voids and dislocations and or on the surface of a component such as man-
ufacturing scratches. is result strongly suggests that the fatigue damage mechanism for the
same type of material specimen under different cyclic test stress levels should be the same. ere-
fore, we can use all test data from every stress level to construct a probabilistic fatigue damage
model for presenting material strength, which is the topic in this Section 2.9.
2.9.2 THE MATERIAL FATIGUE STRENGTH INDEX K
0
In the traditional fatigue design, the S-N curve is typically plotted in a logarithmetic axis with
a fatigue strength S
0
f
verse the fatigue life N . S
0
f
is equal to a fully reversed stress amplitude
a
.
e fatigue life N is the number of cycles to failure at the stress level
a
. is traditional S-N
curve in logarithmic axes will typically be treated as a straight lineper Equation (2.1):
N.S
0
f
/
m
D Constant: (2.1)
74 2. RELIABILITY OF A COMPONENT UNDER CYCLIC LOAD
In the traditional S-N curve, N.S
0
f
/
m
is treated as constants. It is obvious that the N.S
0
f
/
m
cannot be a constant when the reliability of a component is used as the design parameter. e
author in 1993 proposed a probabilistic fatigue damage model [22–24] in which N.S
0
f
/
m
is
treated as a random variable and is called as the material fatigue strength index.
e material fatigue strength index K
0
is a mechanical property of a material and is solely
determined by experimental fatigue data per Equation (2.73) and can be used to indirectly rep-
resent the materials fatigue resistance to the fatigue damage or the material fatigue strength.
e material fatigue strength index K
0
is a random variable. Its sample value can be calculated
by fatigue test results of material fatigue specimen from the same type of cyclic loading stress,
which could be cyclic bending stress, or cyclic axial stress, or cyclic torsion stress.
K
0
D N
ij
.
ai
/
m
; (2.73)
where the subscript i represents the ith fatigue stress level
ai
. e subscript j represents the
j th fatigue test in the ith fatigue stress level
ai
. .N
ij
;
i
/ are the fatigue test results of the j th
fatigue test at the ith fatigue stress level
ai
. N
ij
is the number of cycles to failure or the material
fatigue life of the j th fatigue test at the ith fatigue stress level
ai
.
ai
is a fully reversed cyclic
stress amplitude or the material fatigue strength, which is equal to the fully reversed cyclic stress
amplitude of the i th fatigue stress level
ai
. m is a material fatigue property, is the slope of the
traditional S-N curve and can be calculated per Equation (2.2) which has been discussed in
Section 2.3 and repeated here as Equation (2.74):
m D
I
I
X
i
Œ
ln
.
ai
/
ln
.
N
i
/
I
X
i
Œ
ln
.
ai
/
I
X
i
Œ
ln
.
N
i
/
I
I
X
i
Œ
ln
.
ai
/
2
"
I
X
i
ln
.
ai
/
#
2
; (2.74)
where I is the number of different stress amplitude
ai
for the total fatigue tests;
ai
is the ith
stress amplitude of a fully reversed cyclic stress in the fatigue test; ln.N
i
/ is the average fatigue life
in log-scale at the fatigue test level
ai
, which can be calculated per Equation (2.3) and repeated
here as (2.75):
ln.N
i
/ D
P
J
ln
N
ij
J
; (2.75)
where N
ij
is the number of cycles to the failure of the j th fatigue test under the ith same stress
amplitude
ai
.
For the same type of cyclic loading test stress, there are three possible cyclic loading
stresses: fully reversed cyclic stress, non-zero mean cyclic stress, and notched cyclic stress.
For a fatigue test under a fully reversed cyclic stress level
a
,
ai
will be just equal to the
fully reversed cyclic stress amplitude
a
:
ai
D
a
for a fully reverse cyclic stress: (2.76)
2.9. THE PROBABILISTIC FATIGUE DAMAGE THEORY (THE K-D MODEL) 75
If a fatigue test is conducted under a non-zero mean cyclic stress level .
a
;
m
/, it should be
converted into fully reversed cyclic stress. e modified Goodman approach can be used to
consider the effect of mean stress per Equation (2.21), which has been discussed in Section 2.5
and repeated here as Equation (2.77):
ai
D
8
ˆ
<
ˆ
:
a
when
m
< 0
a
S
u
.
S
u
m
/
when
m
0;
(2.77)
where S
u
is the material ultimate tensile strength and will be treated as a deterministic value
because it is only used for considering the effect of mean stress.
For a notched fatigue test under the cyclic stress level (
a
;
m
/ with a fatigue stress con-
centration factor K
f
, the cyclic stress level needs to be transferred into a fully revered cyclic
stress. It is typically that K
f
will be multiplied with the stress amplitude.
ai
can be calculated
by the following equation:
ai
D
8
ˆ
<
ˆ
:
K
f
a
when
m
< 0
K
f
a
S
u
.
S
u
m
/
when
m
0;
(2.78)
where K
f
is the fatigue stress concentration factor, which has been discussed in Section 2.6. S
u
is the material ultimate tensile strength.
For each fatigue test, we can obtain one sample value of the material fatigue strength
index K
0
per Equation (2.73). When the number of fatigue test is big enough such as more
than 30 tests, we can plot its histogram. Based on the shape of the histogram, we can assume its
type of distribution function and then conduct the goodness-of-fit test to verify the assumption.
Finally, we can determine its type of distribution and corresponding distribution parameters.
ese topics have been discussed in Section 2.13 of Le [8]. e material fatigue strength index
K
0
is typically a lognormal distribution [22–24]. Since the sample value of K
0
is calculated per
Equation (2.73) and is related to the value of m, m will be one parameter for describing the
material fatigue strength index K
0
. If we assume that the material fatigue strength index K
0
is a
lognormal distribution with a log-mean
ln K
0
and a log standard deviation
ln K
0
, the material
fatigue strength index K
0
will be a three-parameter distribution as shown in Table
2.33.
Table 2.33: e material fatigue strength index K
0
with three distribution parameters
Slope of the Traditional S-N
Curve
Log-normally Distributed K
0
The Log Mean The Log Standard Deviation
m μ
lnK
0
σ
lnK
0
76 2. RELIABILITY OF A COMPONENT UNDER CYCLIC LOAD
Now, we will use the fatigue test data listed in Table 2.11 in Section 2.8.1 to show how
to get the distribution parameters of the material fatigue strength index K
0
.
Example 2.21
A sheet-type flat fatigue specimen designed per ASTM STM E466-15 is shown in Figure 2.8.
e material is aluminum 6061-T6 10 Gauge sheet. Its chemical composition is shown in Ta-
ble 2.9. Its mechanical properties are shown in Table 2.10. For all fatigue test specimen, the
nominal dimensions of the middle section of the specimen are the width b D 0:600 ˙ 0:005
00
and the thickness t D 0:100 ˙ 0:005
00
. e fatigue test loading is cyclic axial loading with a load-
ing frequency 20 (Hz), and the loading ratio S
r
D
F
min
F
max
D 0. e test conditions and results of
these five different cyclic stress levels are listed in Table 2.11. Stresses in Table 2.11 are calculated
by using the nominal dimensions, that is, the width b D 0:600
00
and the thickness t D 0:100
00
.
Use those data to determine the type of distribution of the material fatigue strength index K
0
and its three distribution parameters.
Solution:
(1) e slope of the traditional S-N curve.
To calculate the sampling value of each fatigue test, we need to use Equation (2.74) to calculate
the slope of the traditional S-N curve. Since the cyclic stresses in these fatigue tests are no-
zero-mean cyclic stresses, we need to use Equation (2.67) to convert them into equivalent fully
reversed cyclic stresses. We also need to calculate the average fatigue life in log-scale at the
fatigue test level
ai
per Equation (2.75). e equivalent fully revered cyclic stress amplitude
ai
and the average fatigue life in log-scale ln.N
i
/ at the ith fatigue test level
ai
per the test
data are listed in Table 2.34. In Table 2.34, the first column is the cyclic stress level #i. e
2nd and 3rd columns are the mean stress and stress amplitude of the cyclic axial stress at the
corresponding cyclic stress level. e fourth column is the number of tests at the same cyclic
stress level. e fifth column is the equivalent fully revered cyclic stress amplitude. e sixth
column is the equivalent fully revered cyclic stress amplitude in a log-scale. e seventh column
is the average fatigue life in log-scale at the same cyclic stress level.
Use the data in the 6th and 7th columns in Table 2.34 per Equation (2.74) to conduct the
linear regression in Excel. m value is displayed in Figure 2.16:
m D 3:8812: (a)
(2) e sampling data of the material fatigue strength index K
0
.
In this example, we have five different stress levels. Since all cyclic stresses are the same
type of cyclic stress, that is, the cyclic axial stress, we can use all of 195 fatigue tests in all 5 stress
levels to calculate the sampling values of the material fatigue strength index K
0
. Based on the
test data .N
ij
;
ai
/ listed in Table 2.11, we can get 195 sampling data of K
0
per Equation (2.73).
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