In this section, the results are explained first for the contact state of the fretting contact zone. Then, the maximum Von Mises equivalent stress and contact surface stress of the specimen are analyzed. Finally, the critical plane method combining two multiaxial damage parameters, MD and SWT, is used to analyze the effect of the governing parameters on the fretting damage initiation. The results of Exp. 1 to Exp. 7 are used to analyze the effect of the mode angle θ. The results of Exp. 8 to Exp. 10 in combination with Exp. 4 are used to study the effect of oblique loads λ, and the results of Exp. 11 to Exp. 16 in combination with Exp. 4 are used to analyze the effect of stiffness ratio γ. The analysis of these two parameters is based on the mode angle of 45°, and this is the most typical load setting for tension–bending mixed-mode fretting fatigue. Furthermore, to present the computational results more distinctly, certain data are displayed in normalized form, with normalization based on the computational results from Exp. 4.
4.1. Contact Analyses
The experimental results of many researchers have indicated that the fretting fatigue crack initiation at the contact edge and subsequently rapid propagation is the main reason for reducing the fatigue life of structures [
13,
18,
42].
Table 3 shows that the stick–slip ratio of the contact zone decreased with the increase in the dimensionless oblique load, while the contact width remained almost constant. The reasons can be explained as follows. Since the oblique loading,
λ, increased, its normal component,
Fsin
θ, and tangential component,
Fcos
θ, increased as well. The tangential traction of the contact surface, which causes the slip zone to expand, increases with the increase in
Fcos
θ, and the magnitude of
Fsin
θ is much smaller than the normal force,
P; the resultant force in the normal direction is slightly reduced, so the contact width is almost unchanged. From
Table 4, it is evident that with the increase in stiffness ratio, the contact width decreases. This trend aligns with the findings of fretting fatigue experiments conducted by Lee et al. [
43] using different mating materials and is consistent with the Hertz contact theory [
29].
Table 5 shows the effect of the mode angle on the contact zone, which reveals that the mode angle has no obvious effect on the contact width, while it has the most significant effect in adjusting the stick–slip ratio of the contact zone. The ratio of the stick–slip width decreases with the increase in the mode angle, and the decreased gradient is large when the mode angle is between 0° and 45°. The contact condition of
θ = 75° is close to the result of
θ = 90°. It means that the slip regime is mainly affected by the mode angle under the tension-dominated (0° <
θ < 45°) tension–bending mixed-mode fretting fatigue.
4.2. Stress Analysis
The maximum Von Mises equivalent stress of the specimen
σeff,max, the distribution of normal stress
p(
x), the horizontal component of stress
σxx, and the tangential stress
q(
x) on the contact surface are examined in this section to understand the fretting fatigue behavior of the material under tension–bending mixed-mode loading. The
σeff,max produced in any loading case is less than the yield stress,
σy, for the material (
σy =310 Mpa [
13]), as shown in
Figure 7, and it demonstrates that the linear elastic constructive relation is reasonable.
Figure 7 also shows that the
σeff,max increases with the increase in the dimensionless stiffness ratio,
γ, and oblique load,
λ, but the slope of the curve related to the mode angle
θ decreases rapidly. The curve of
σeff,max vs.
θ grows rapidly when
θ < 45° but slowly increases when
θ > 45°, as shown in
Figure 7c, which indicates that the risk of material yielding is the highest for the bending-dominated (45° <
θ < 90°) mixed-mode loading and it can be conservative and simple to estimate the bending-dominated fretting fatigue by the bending fretting fatigue strength.
Figure 8 shows the dimensionless normal stress,
, distribution on the contact surface. As shown in
Figure 8a,c, the peak value and the curve profile of the normal stress are almost the same for different oblique loads and mode angles, but the location of the peak normal stress (i.e., contact center) moves towards the trailing edge of the contact as the two variables increase. These phenomena indicate that the peaks of the contact stress and the contact width are dominated by the normal load,
P, but the position of the contact center and the slip regime of the contact zone will be affected by the mode angle and oblique load. Unlike
λ and
θ, the peak value of contact pressure increases with the increase in stiffness ratio
γ, but the position of the contact center remains constant (shown in
Figure 8b), which indicates that the variable,
γ, can function in adjusting the preload between two contact bodies (such as bolts and other fasteners).
Figure 9 demonstrates the effects of
θ,
λ, and
γ on the dimensionless surface stress component
. As shown in
Figure 9,
is compressive at most of the contact surface but transforms into tensile stress at the leading edge, and the fretting fatigue crack propagation is dominantly by the high gradient of tensile stress [
39]. As expected, when a larger oblique load is applied, the peak of tensile stress is higher, and its position moves toward the contact center, as shown in
Figure 9a. Different stiffness ratios on the surface stress component,
, are shown in
Figure 9b, in which the peak tensile stress increases as the stiffness ratio increases, and the changing
γ from 0.4 to 1 has a more powerful adjustment capability on
than that from 1 to 2.5. This indicates that when the stiff specimen is pressed by the compliant pad, the crack propagation driving force is effectively reduced by adjusting the stiffness ratio,
γ. With the increase in the mode angle, the peak of tensile stress increases, and its position moves toward the contact center, as shown in
Figure 9c. This trend is obvious in the tension-dominant mode, but it is not pronounced in the bending-dominant mode, and the stress curves almost coincide with
θ = 75° and
θ = 90°, which indicates that the mode angle should be controlled in the range of 0° ≤
θ ≤ 45° to reduce the driving force of fretting fatigue crack propagation.
Figure 10 shows the distribution of dimensionless shear stress on the contact surface. As shown in
Figure 10a, as the oblique load increases, the shear stress gradient in the vicinity of the contact edge is reduced, where the peaks increase and positions move closer to the contact center, which is related to the stick–slip ratio. If
λ continues to be applied, the fretting regime will change from partial slip to gross slip, and the mode of fretting damage will also change from fretting fatigue to fretting wear. The two peaks of the shear stress increase with the increase in the stiffness ratio as well, but the distance between them decreases, as shown in
Figure 10b, which can be explained by the narrowing of the contact width. When the stiffness of the pad increases, the variable amplitude of the peak at the leading edge is smaller than that at the trailing edge, and the position of the maximum shear stress is shifted from the leading edge to the trailing edge. It is seen in
Figure 10c that the
vs.
θ curve shows that peaks appear on both sides of the contact center, and the peak of the leading edge is slightly larger than that of the trailing edge. The shear stress increases as the mode angle increases, and its two peaks move towards the trailing edge, but the stress profiles of
θ = 75° and
θ = 90° almost coincide.