Abstract
Mean torsional stress is considered to have less effect on the torsional fatigue strength of steels, but several experimental results have been recently reported that mean torsional stress caused significant reduction in torsional fatigue strength in the very high cycle region for shot-peened spring steel. To investigate the effect of mean torsional stress on high strength steel, ultrasonic torsional fatigue tests with mean torsional stress were conducted for spring steel and bearing steel, which are used for mechanical components subjected to cyclic shear stress. Torsional fatigue strengths up to 109 cycles were obtained for fully-reversed torsional loading (R = −1) to pulsating torsional loading (R = 0). The results revealed that mean torsional stress caused a reduction in fatigue strength in the very high cycle region for both spring steel and bearing steel, and applying higher mean shear stress would result in transition of the fracture origin from a surface to an internal inclusion. The reduction in torsional fatigue strength was discussed from the viewpoint of the transition of the fatigue origin, and applicability of a $\sqrt{\textit{area}} $ parameter model was discussed for predicting the reduction in torsional fatigue strength.
This Paper was Originally Published in Japanese in J. Soc. Mater. Sci., Japan 71 (2022) 976–982. Abstract, Sec. 2.2, Sec. 4.1, and Ref. 5 are slightly modified.
1. Introduction
It is well known that mean stress has a large effect on the fatigue limit in axial loading fatigue of steel, and fatigue limit diagrams such as the modified Goodman diagram are widely used in industry to predict the influence of mean stress on the fatigue limit. In contrast, the effect of mean torsional stress on the torsional fatigue limit of steel is generally thought to be small [1]. Recently, Mayer et al. have reported that the mean torsional stress significantly reduced the very high cycle torsional fatigue strength of shot-peened spring steel by conducting fatigue tests using an in-house ultrasonic torsional fatigue testing machine capable of applying mean torsional stress [2–5]. The result is significantly non-conservative to the current understanding that the effect of mean torsional stress on torsional fatigue strength is small.
Typical mechanical elements subjected to cyclic torsion under mean torsional stress are coil springs. If the effect of mean torsional stress on torsional fatigue limit cannot be ignored, it is necessary to reconsider the conventional fatigue design method for coil springs. Clarifying the mechanism of the effect of mean torsional stress on the torsional fatigue limit will contribute to the development of steels with excellent resistance to torsional fatigue. Therefore, it is necessary to reexamine the effect of mean torsional stress on torsional fatigue strength of high strength steels such as spring steel.
In this study, fatigue tests up to the very high cycle region were conducted for spring steel and bearing steel, which are subject to cyclic shear stress, using an in-house ultrasonic torsional fatigue testing machine capable of applying mean torsional stress, and the effect of mean torsional stress on the very high cycle torsional fatigue strength of high strength steels was examined. As a result, it was found that increasing the mean torsional stress lowered the torsional fatigue strengths in the very high cycle region and resulted in the transition of the fatigue origin from a surface to an internal inclusion. The reason why mean torsional stress reduced the torsional fatigue strengths was discussed, and the prediction of the reduction in torsional fatigue strength was also discussed.
2. Ultrasonic Torsional Fatigue Test
2.1 Ultrasonic torsional fatigue testing machine
Figure 1 shows a schematic diagram of an in-house ultrasonic torsional fatigue testing machine capable of applying mean torsional stress. The machine was assembled, from top to bottom, from a torsion-type ultrasonic transducer, a half-wavelength bar with a flange for gripping, a horn for amplifying ultrasonic oscillation, a specimen, a horn for reducing ultrasonic oscillation, and a half-wavelength bar with a flange for applying mean torsional stress. A microscope was used to measure the torsional displacement amplitude at the end of the specimen. The measurement of the torsional displacement amplitude was conducted as follows. A black coating was applied near the measurement point. Small bright spots with a size of a few pixels could be seen as a result of unevenness of the black coating when the specimen surface was observed with a microscope in a stationary state. The torsional displacement amplitude was determined by measuring the length of the trajectory of a moving bright spot during oscillation, which was equivalent to twice the torsional displacement amplitude, using a microscope [6].
The flange positions in the upper mounting part and in the lower flange integrated bar for applying mean torsional stress in Fig. 1 were designed to be the nodes of the torsional oscillation. Therefore, in principle, no torsional oscillation occurs at the flange positions. By gripping the upper flange and applying a torsional moment to the lower flange with a coupled force that is provided by dead loads, a static torsional moment can be applied to the specimen without affecting the ultrasonic torsional oscillation [7]. To confirm the feasibility of applying mean torsional stress, a dumbbell-shaped specimen with a parallel section in the center was prepared. Strain gauges were attached to the center of the specimen, and ultrasonic fatigue tests were conducted with different stress ratios while the maximum applied shear stress was fixed. The applied shear stress over time is shown in Fig. 2 [8]. It reveals that the device was capable of applying cyclic torsion while applying the mean torsional stress.
The materials used in this study were spring steel (equivalent to JIS SUP12) and bearing steel (JIS SUJ2). The spring steel was prepared using a laboratory-scale vacuum induction melting furnace, and its chemical composition is shown in Table 1. The bearing steel was purchased from the market. The materials were quenched and tempered. The spring steel was heated at 860°C for 5 min in a salt bath, followed by oil quenching, and tempering at 420°C for 30 min in air. The bearing steel was heated at 840°C for 20 min in air, followed by oil quenching, and tempering at 165°C for 150 min in air. The tensile strength, yield stress, and Vickers hardness of each steel are shown in Table 2.
Table 1 Chemical composition of spring steel (mass%).
Table 2 Tensile strength, yield stress and hardness.
The specimens used were hourglass-shaped, which is a typical specimen shape for ultrasonic fatigue testing. Specimens were designed to resonate in the first-order torsional oscillation mode, in which the center of the specimen is the node and both ends twist in opposite directions at a resonance frequency of 20 kHz. The specimen shape and dimensions are shown in Fig. 3. The specimen surface was mirror-finished before testing.
In the hourglass-shaped specimen, cyclic shear stress reaches the maximum at the necked position in the center of the specimen. The shear stress amplitude during fatigue testing was determined from the measured torsional displacement amplitude at the specimen end using the premeasured relationship between the shear stress amplitude at the necked position and the torsional displacement amplitude. The relationship was obtained as a relationship between the shear modulus of material (G = 78 GPa) multiplied by the shear strain amplitude at the necked position measured using a bi-axial strain gauge (FCA-1-11, Tokyo Measuring Instruments Laboratory Co., Ltd.) and the torsional displacement amplitude at the specimen end under continuous fatigue loading.
2.3 Fatigue testing
Fatigue tests were performed at three different stress ratios: R = −1, −0.33 and 0. The fatigue tests were stopped when the resonance frequency decreased by 20 Hz due to fatigue crack initiation or the number of cycles reached 109 cycles. The former stopping condition was determined from preliminary tests that a 20 Hz decrease in resonant frequency corresponded to the presence of a surface fatigue crack longer than 1 mm. Excessive heating of specimens was prevented using intermittent loading and air cooling. For intermittent loading, the ramp-up time was fixed at 105 ms, the loading time was 200 ms or 300 ms, and the dwelling time was set appropriately so that the specimen surface temperature did not exceed 40°C. The effective cyclic frequency considering intermittent loading was 700 Hz to 2.5 kHz. After fatigue tests, the fracture surfaces of failed specimens were observed using a scanning electron microscope (VE-9800, Keyence Corporation), and elemental analyses at fatigue origins were conducted using an electron probe micro analyser (JXA-8530F, JEOL Ltd.). For an internal origin, the inclination of the fracture surface from the specimen axis was measured using a laser microscope (VK-X100, Keyence Corporation).
3. Fatigue Test Results
3.1 S-N property and fracture origin
The S-N diagrams for spring steel and bearing steel are shown in Fig. 4. In the figures, the open symbols represent surface origins, the solid symbols each represent an internal origin from an inclusion and the arrows indicate that the fatigue tests were discontinued at 109 cycles.
The figures show that the very high cycle fatigue strengths of the spring steel and bearing steel decreased under the effect of the stress ratio. It can also be seen that the ratio of specimens that failed from an internal origin increased as the stress ratio increased.
3.2 Constant life diagram
Based on the above results, torsional fatigue strengths at 109 cycles are plotted as functions of mean torsional stress in Fig. 5. In the figures, the plotted points on the horizontal axes are the estimated torsional strength by assuming that the torsional strength $\tau_{B} = \sigma_{B}/\sqrt{3} $ where σB is the tensile strength. For reference, a modified Goodman relation and a Gerber parabola are also plotted in each figure, where tensile stress is replaced by shear stress in both diagrams.
The results for spring steel and bearing steel show that the trends of reduction in torsional fatigue strength at 109 cycles under the effect of mean torsional stress are almost the same as the trend of reduction in fatigue limit of steels subject to axial loading, that is, the fatigue strengths are on the conservative side from the modified Goodman relation and on the non-conservative side from the Gerber parabola.
3.3 Fatigue fracture surface
As already mentioned, all the specimens failed from the surface for fully-reversed torsional loading (R = −1), whereas some specimens failed from an internal origin for the specimens subjected to mean torsional stress (R = −0.33, 0). Furthermore, as the mean torsional stress increased, the ratio of specimens that failed from an internal origin increased.
Figures 6 and 7 show typical fatigue fracture surfaces from a surface origin for R = −1 and from an internal origin for R = −0.33, 0 in spring steel and bearing steel, respectively.
In the case of a surface origin, a mode II fatigue crack initiated and then propagated, followed by bending to a mode I fatigue crack, and the propagation of the mode I crack led to final failure, as is generally observed in torsional fatigue of smooth specimens.
In contrast, in the fatigue fracture surface for an internal origin which occurred under mean torsional stress, a so-called fisheye was observed with an inclusion as the fatigue origin. In the origins, some had an spherical or square-shaped inclusion (Fig. 6(b) and Fig. 7(b)), while others had a step although no inclusion was clearly observed (Fig. 6(c) and Fig. 7(c)).
Furthermore, the angle between the specimen axis and the normal of the ODA region of the fish-eye (the dark area near the inclusion when the origin is observed with an optical microscope) was measured for all internal origin failures using a laser microscope. It was confirmed that the angles fell within a range of 45° ± 2°. This suggests for all specimens of internal origin failures that the fatigue crack initiated and propagated along the plane of the maximum principal stress from the debonding between the internal inclusion and the matrix or from the crack inside the inclusion.
In addition, elemental analyses of the fatigue origins were conducted using an electron probe microanalyzer. For spring steel, the results indicated that inclusions that could be clearly identified were an oxide of Al, Ca, etc., and if a step was observed, MnS was likely to exist in the step. For bearing steel, the results indicated that inclusions that could be clearly identified were TiN or an oxide of Al, Ca, etc., and if a step was observed, MnS was likely to exist in the step. Sandaiji et al. [9, 10] have reported that internal origin failure would occur if fully-reversed torsion (R = −1) was applied to bearing steel (SUJ2) using an ultrasonic torsion fatigue machine, and not only a hard inclusion such as an oxide but also MnS, which is a soft inclusion, could be the origin of torsion fatigue failure. A cross-sectional observation of the origin revealed that mode I fatigue cracks originated or bent from both ends of a mode II crack caused by cyclic shear stress acting on axially elongated flat MnS, resulting in the step of the origin. The findings support that the steps in the internal origins observed in spring steel and bearing steel were also considered to originate from MnS as in the study by Sandaiji et al. [9, 10].
4. Discussion
4.1 Effect of mean torsional stress on torsional fatigue strength
From the fracture surface observation results, fatigue crack initiation can be classified into Mode II crack initiation from the specimen surface and Mode I crack initiation from an inclusion inside the specimen. Therefore, the fatigue crack initiation mode must be determined by the competition of the fracture strengths of both modes. In the following, the effect of mean torsional stress on torsional fatigue strength will be explained using the results of spring steel shown in Fig. 8 as an example.
First, in the absence of mean torsional stress, a Mode II fatigue crack initiates at the surface of a specimen if the sizes of inclusions are small enough, as in the specimens used. In this case, the effect of mean torsional stress is thought not to be significant as previously known (the line (a) in Fig. 8). Next, let us consider the case where mean torsional stress is applied. Focusing on small inclusions in spring steel and bearing steel, the inclusions are subjected to cyclic axial loading with mean tensile stress in the direction of the maximum principal stress (45° from the specimen axis). As is well known, fatigue strength decreases with increasing mean stress in tension under cyclic axial loading (line (b) in Fig. 8). Therefore, if internal origin is assumed, the fatigue strength decreases as the mean torsional stress increases, and when the fatigue strength falls below the fatigue strength in the case of a surface origin, the failure mode transition will occur to the mode I fatigue crack initiation from an internal inclusion. Furthermore, when the mean torsional stress is sufficiently large, the specimen surface yields; the fatigue limit is supposed to be almost determined by the yield stress as in the case of axial loading (line (c) in Fig. 8). As a result, the mean torsional stress dependence is considered to be as shown by the solid line in Fig. 8.
4.2 Prediction of fatigue strength for internal origins using the $\sqrt{\textit{area}} $ parameter model
It takes a longer time to obtain a large number of data sets of longer fatigue lives with lower amplitudes in torsional fatigue tests. Thus, it is convenient for engineering applications if the solid line (b) in Fig. 8 can be estimated without conducting fatigue tests under mean torsional stress. In the case of fatigue failure originating from the inside, a Mode I fatigue crack was initiated from a small inclusion in the inside. Based on this experimental fact, the fatigue strength was estimated using the $\sqrt{\textit{area}} $ parameter model [11], which was used to predict the fatigue strength when a fatigue crack initiated from a small defect.
Murakami et al. proposed the following equation to predict the fatigue limit for axial fatigue loading under mean stress in the case of internal fatigue failure from a small inclusion [11].
| \begin{equation}
\sigma_{w} = \frac{1.56(Hv + 120)}{\sqrt{\textit{area}}^{\frac{1}{6}}} \cdot \left(\frac{1 - R}{2} \right)^{\alpha}
\end{equation}
| (1) |
where σw is the fatigue limit (MPa), Hv is the Vickers hardness, $\sqrt{\textit{area}} $ is the square root of the projected area of an inclusion on the plane perpendicular to the loading direction (µm), R is the stress ratio, and α is a material constant for mean stress correction. Murakami et al. proposed the following equation for estimating α [11].
| \begin{equation}
\alpha = 0.226 + Hv \cdot 10^{-4}
\end{equation}
| (2) |
It should be noted, however, that eq. (2) is an experimental equation obtained from the results of axial fatigue loading.
Next, it is necessary to modify eq. (1) to take into account torsional loading, which is a combined stress state, since eq. (1) was proposed for axial fatigue loading. Experimental and analytical studies have already been conducted on the effect of a combined stress state when mean stress does not act. In the case of fatigue failure originating from a small inclusion, the relationship between the fatigue limit τw for pure cyclic torsion and the fatigue limit σw for fully-reversed axial loading is given in the following equation [12, 13].
| \begin{equation}
\frac{\tau_{w}}{\sigma_{w}} = 0.83{\sim}0.87
\end{equation}
| (3) |
Karr et al. [5] assumed that the relationship in eq. (3) was applicable to the case with mean torsional stress and proposed the following equation by substituting eq. (1) for eq. (3), providing a prediction equation for the fatigue limit when fatigue failure occurs from a small inclusion in the specimen due to cyclic torsional loading with mean torsional stress.
| \begin{equation}
\tau_{w} = \frac{0.86 \cdot 1.56(Hv + 120)}{\sqrt{\textit{area}}^{\frac{1}{6}}} \cdot \left(\frac{1 - R}{2}\right)^{\alpha}
\end{equation}
| (4) |
Figure 9 shows eq. (4) with our experimental results. In the figure, two cases are plotted: the material constant α for mean stress correction was estimated from eq. (2) (the cross-hatched region in Fig. 9) and the material constant α was determined to fit the experimental results (the shaded area in Fig. 9), where the minimum and maximum values of $\sqrt{\textit{area}} $ of the inclusions found on the actual fatigue fracture surface were used as the upper and lower limits for the two cases.
As a result, the predictions were non-conservative if the material constants α of spring steel and bearing steel were estimated using the eq. (2) obtained from the experimental results of axial loading fatigue tests, but the predictions by the $\sqrt{\textit{area}} $ parameter model were in close agreements with the experimental results if α = 0.5. The results imply that the torsional fatigue strength for internal origin can be predicted with sufficient accuracy for practical application if the material constant α for mean stress correction in the case of a combined stress state can be determined based on experimental results. The above discussion also strongly suggests that it is possible to improve the torsional fatigue strength in the case of an internal origin and to decrease the scatter of the torsional fatigue strength by controlling the size of inclusions, as in the case of axial fatigue loading of high strength steels.
5. Conclusions
Very high cycle torsional fatigue tests were conducted on spring steel and bearing steel using an in-house ultrasonic torsional fatigue testing machine capable of applying mean torsional stress, and the effect of mean torsional stress on very high cycle torsional fatigue strength was evaluated. As a result, it was found that the very high cycle fatigue strengths of both spring steel and bearing steel decreased with increasing mean torsional stress, and the transition of the fatigue origin from the surface to an internal inclusion was also observed.
Furthermore, a model was proposed to explain the effect of mean torsional stress on fatigue strength by focusing on the transition of the fracture origin. It was also shown that the torsional fatigue strength in the case of fatigue failure originating from an internal inclusion under mean torsional stress is probably predicted by the $\sqrt{\textit{area}} $ parameter model.
Acknowledgments
This research was supported in part by Japan Science and Technology Agency (JST) Grant Number JPMJTM15H3 and the Suzuki Foundation “Grant for Scientific Technology Research Project”.
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