4.1. Fatigue Notch Factor of Rough Specimens
Surface topographies nearby the fillets along the axial direction of these stepped shafts were measured by TR300 stylus roughness measuring instrument, the raw data of the measured surface profiles were extracted and analyzed by Fourier transform.
Figure 6 shows the machined surface topographies near the fillets along the axis direction of three stepped shafts and their amplitude-frequency analysis results. Results show that surface topographies with higher
are predominant due to their low frequency components.
The machined surface topography can be assumed to be a stationary stochastic process, and it can be modelled based on superposing a series of cosine components through Fourier transform [
36,
37]. The corresponding Fourier series can be formulated as below:
where
is the amplitude of the
i-th wave,
is the wavelength of the
i-th wave and
is the phase of the
i-th wave.
Surface morphologies can be considered as multi-notches, which can introduce stress concentrations and affect the fatigue behaviors of specimens. The SCFs induced by surface topography are as follows [
12]:
Real surface topography can be thought of as a series of notches of varying sizes and shapes, so the TCD can be used to estimate the fatigue limit of rough specimens. The reference stress used to predict the FNF of notched components is not the stress at the root of the notch but a point stress at a given distance ahead of the notch, corresponding to the point method (PM) of TCD [
38]. According to the PM of TCD, notched parts are in their fatigue limit condition when the effective stress,
, which depends on the maximum principal stress at a distance from the notch tip of
, equals the material fatigue limit,
[
38]. The FNFs of rough specimens were derived by combing the analytical solutions of the stress distribution induced by surface topographies and the PM of TCD, which are as follows [
13]:
where
is the notch’s bottom position of surface topography,
is the material characteristic length, which is defined by the fatigue crack growth threshold,
, and the material fatigue limit,
,
;
is a material parameter which is only dependent on the types of materials and load ratios [
39].
The material characteristic length
varies considerably for different materials, commonly encountered values range from microns to millimeters. High-strength steels, including 42CrMo, were recognized to have an
value of about 10 μm [
40,
41]. According to Equation (12), the high frequency cut-off of machined surface topography can be defined as follows [
13]:
where the surface frequency components,
, surface wave components with frequencies higher than
can be removed, because they make no contributions to the stress raiser at the reference point. Besides, to model the effect of surface roughness on the fatigue behavior of metals, more high frequency components have to be considered to reconstruct the machined surface topographies for high strength steels compared to some materials with less sensitivity to surface roughness.
Based on Equations (10)–(13), the high frequency cut-off is defined as
the machined surface topographies around the fillets of these stepped shafts were reconstructed by Fourier series, and the SCFs and FNFs of these reconstructed surface topographies were calculated.
Figure 7 shows the machined surface topography and the reconstructed surface topography for specimen B1.
Figure 8 shows the SCFs induced by the reconstructed surface topography and the FNFs by PM for each notch extracted from the reconstructed surface topography for specimen B1. It was found that the maximum SCF reaches 2.25 while the maximum FNF is only about 1.39.
Surface topographies act as multi-notches to introduce stress concentration and affect the fatigue performance of specimens. By analogy with the characterization of surface roughness, the maximum SCF,
, the effective SCF,
by averaging the top 10 values, the maximum FNF,
, and the effective FNF,
, by averaging the top 10 values are defined as the characteristic parameters to represent the fatigue behavior of rough specimens, where
,
.
Figure 8 shows that the maximum FNF,
, and the effective FNF,
.
According to Equations (10)–(13), the SCFs and FNFs imposed by surface topographies of each specimen are calculated and shown in
Table 6. It is clear that the SCFs are much higher than the FNFs of each specimen. Besides, the maximum FNF,
, and the effective FNF,
, increase gradually with the increase of the average roughness,
. To quantitatively evaluate the effect of surface topography on the fatigue strength and fatigue life of rough specimens, the maximum FNF,
, and the effective FNF,
, for the same degree of surface roughness were averaged, which are shown in
Table 6.
4.2. Fatigue Life Prediction of Rough Specimens
It is well established that the surface roughness has little influence on the fatigue behaviors in the low-cycle fatigue region, but affects the high-cycle fatigue regime more, where elastic strain is dominant. Therefore, only the elastic portion of the strain–life curve is modified to account for the effect of surface topography [
1]. Based on the 10
7 run-out reversals considered in this study, the fatigue limit of rough specimens is estimated with the method of dividing the fatigue limit of polished specimens by the FNF induced by surface topographies. Here, the fatigue limit of 42CrMo is predicted by subtituting 10
7 reversals into Equation (2).
Alternatively, the fatigue life of the rough specimen is predicted by adjusting the slope of the strain–life curve to capture the effect of surface roughness for parts that are not polished. With the consideration of surface topography, the total strain–life equation is revised as follows [
21]:
where
,
,
and
are obtained from
Table 5. The only information needed in this approach are the FNFs calculated by surface topographies from rough specimens.
To quantitatively estimate the effect of surface topography on the fatigue life behaviors of stepped shafts, according to Equation (14), the elastic portion of the total strain–life curve of 42CrMo was solely adjusted based on the average
and the average
in
Table 6.
Figure 9a shows the strain–life curve for 42CrMo steel based on the prediction parameters in
Table 5, and the estimated strain–life curve for rough stepped shafts based on adjusting the slop of elastic strain–life line using the average
listed in
Table 6. The black dashed line and dash-dot line represent the elastic and plastic strain–life lines of 42CrMo steel. The red dashed line and solid line represent the estimated elastic and total strain–life curves of specimens with
; the blue dashed line and solid line represent the estimated elastic and total strain–life curves of specimens with
; the cyan dashed line and solid line represent the estimated elastic and total strain–life curves of specimens with
. The predicted versus the experimentally observed fatigue lives of rough stepped shafts with scatter bands of two are shown in
Figure 9b. All data points fall within the scatter band of two, demonstrating the effectiveness of estimating the fatigue life of rough specimens based on the procedure proposed above.
Figure 10a describes the strain–life curve for 42CrMo steel and the estimated strain–life curve for rough specimens by revising the fatigue strength exponent using the average
listed in
Table 6. The estimated versus the fatigue test results of rough stepped shafts within a scatter band of two is shown in
Figure 10b. All estimated fatigue lives again fall within the scatter bands of two compared with the observed fatigue test results. The proposed two methods demonstrate that surface topographies with
make little contribution to the fatigue behavior of stepped shafts. In terms of the fatigue life prediction of rougher stepped shafts, fatigue strength estimation using the effective FNF shows higher prediction accuracy than another approach by employing the maximum FNF. The comparison shows that the effective FNF is more likely to determine the fatigue performance of rough specimens than the maximum FNF induced by surface morphologies.