Effect of Loading Frequency Ratio on Multiaxial Asynchronous Fatigue Failure of 30CrMnSiA Steel

Multiaxial asynchronous fatigue experiments were carried out on 30CrMnSiA steel to investigate the influence of frequency ratio on fatigue crack initiation and propagation. Test results show that the surface cracks initiate on the maximum shear stress amplitude planes with larger normal stress, propagate approximately tens of microns, and then propagate along the maximum normal stress planes. The frequency ratio has an obvious effect on the fatigue life. The variation of normal and shear stress amplitudes on the maximum normal stress plane induces the crack retardation, and results in that the crack growth length is longer for the constant amplitude loading than that for the asynchronous loading under the same fatigue life ratio. A few fatigue life prediction models were employed and compared. Results show that the fatigue life predicted by the model of Bannantine-Socie cycle counting method, section critical plane criterion and Palmgren-Miner’s cumulative damage rule were more applicable.


Introduction
In engineering practice, the multiaxial fatigue failure behavior of structures and components is affected by the phase angle, mean stress, loading sequence, stress/strain amplitude ratio, and frequency ratios of tensile to shear stress/strain components (asynchronous loadings). Compared with other multiaxial loading paths, there are limited studies on multiaxial asynchronous loadings. A lot of researches on steel materials were focused on the fatigue failure under uniaxial and multiaxial constant amplitude loadings, such as the fatigue failure mechanism of 25CrMo4 (EA4T) steel [1][2][3] and 39NiCrMo3 steel [4], crack initiation mechanism of 25CrMo4 steel [5] and 30CrMnSiA steel [6,7], short crack growth behavior of 25CrMo4 steel [8], 42CrMo4 steel [9][10][11] and 30CrMnSiA steel [6,7], the influence of chemical composition on crack initiation of 25CrMo4 steel [12], and the influence of loading paths on fatigue failure of 30CrMnSiA steel [13][14][15][16], while there are limited studies on the fatigue failure mechanism and influencing factors under multiaxial random fatigue loadings.
Multiaxial random fatigue is one of the most common forms of mechanical failure of metal alloys in engineering practice, and asynchronous loading is one of the simplest. In most studies [17][18][19], the asynchronous loading path is usually considered as one type of multiaxial loading path. The commonly used frequency ratios of tensile to shear stress or strain are 2:1 and 1:2 [20][21][22]. Experiments were conducted on S460N steel by Vormwald et al. [20,21] to verify the application of fatigue criterion based on the cyclic plasticity under asynchronous loading of four different frequency ratios, however, the prediction results were non-conservative compared with the experimental fatigue life.
The limited research above indicates that the present studies were mainly focused on fatigue limits and the direction of crack initiation under asynchronous loading. The effect of frequency ratios on fatigue life and surface crack growth paths and the failure mechanism under asynchronous loading are still insufficient and further research is needed. The present study aims to investigate the effect of frequency ratios on fatigue life and surface crack growth behavior, highlighting the crack initiation and propagation path of 30CrMnSiA steel. Multiaxial fatigue experiments under asynchronous loading paths were performed, and the surface crack growth paths were observed. The mechanisms of initiation and propagation and the transformation from stage I to stage II crack were discussed with the stress analysis. The surface crack lengths versus loading blocks under asynchronous loadings were compared and analyzed. A variety of fatigue failure model including commonly used cycle counting methods, fatigue failure criteria, and Palmgren-Miner's cumulative damage rule were used to predict fatigue life.

Materials
30CrMnSiA steel is the experimental material. Its chemical composition, weight percent, and mechanical properties have been given in a previous study by the authors and are also shown in Tables 1 and 2 [6]. The geometry and dimensions of solid cylindrical bar specimens are shown in Figure 1. Materials 2021, 14, 3968 3 of method and Smith-Watson-Topper (SWT) criterion [51] by the experimental results of S 333 Gr. 6 steel under asynchronous loading of four different frequency ratios, howeve the prediction results were non-conservative compared with the experimental fatigue lif The limited research above indicates that the present studies were mainly focused o fatigue limits and the direction of crack initiation under asynchronous loading. The effe of frequency ratios on fatigue life and surface crack growth paths and the failure mech nism under asynchronous loading are still insufficient and further research is needed. Th present study aims to investigate the effect of frequency ratios on fatigue life and surfac crack growth behavior, highlighting the crack initiation and propagation path o 30CrMnSiA steel. Multiaxial fatigue experiments under asynchronous loading paths wer performed, and the surface crack growth paths were observed. The mechanisms of initi tion and propagation and the transformation from stage I to stage II crack were discusse with the stress analysis. The surface crack lengths versus loading blocks under asynchro nous loadings were compared and analyzed. A variety of fatigue failure model includin commonly used cycle counting methods, fatigue failure criteria, and Palmgren-Miner cumulative damage rule were used to predict fatigue life.

Materials
30CrMnSiA steel is the experimental material. Its chemical composition, weight pe cent, and mechanical properties have been given in a previous study by the authors an are also shown in Tables 1 and 2 [6]. The geometry and dimensions of solid cylindrical ba specimens are shown in Figure 1.

Experimental Methods
The multiaxial fatigue tests were performed under load control mode and conducte on an MTS 858 testing system at room temperature and atmosphere. The loading fr quency of a loading block used in the test was 0.5 Hz.
The tension-torsion stress components of multiaxial asynchronous fatigue loadin are as follows

Experimental Methods
The multiaxial fatigue tests were performed under load control mode and conducted on an MTS 858 testing system at room temperature and atmosphere. The loading frequency of a loading block used in the test was 0.5 Hz.
The tension-torsion stress components of multiaxial asynchronous fatigue loading are as follows σ x (t) = σ x,a sin(ξ 1 ωt), τ xy (t) = τ xy,a sin(ξ 2 ωt), (2) where σ x (t) and τ xy (t) are the cyclic axial and shear stress, σ x,a and τ xy,a are the axial and shear stress amplitude. The loading frequency is controlled by ξ 1 and ξ 2 . In the tests, σ x,a = τ xy,a = 350 MPa, that is, the stress amplitude ratio was 1.0 without a phase angle.
where σx(t) and τxy(t) are the cyclic axial and shear stress, σx,a and τxy,a are the axial and shear stress amplitude. The loading frequency is controlled by ξ1 and ξ2. In the tests, σx,a = τxy,a = 350 MPa, that is, the stress amplitude ratio was 1.0 without a phase angle. The tests were conducted under four different frequency ratios (ξ1:ξ2) including: 2:1, 4:1, 1:2, and 1:4. The multiaxial asynchronous fatigue loading paths in a loading block with different frequency ratios are shown in Figure 2.   Each specimen was tested up to six times under the same loading conditions, and the surface crack morphologies were observed using a Zeiss metallurgical microscope with a maximum amplification capacity of 500×. The lens can be moved transversally and vertically. During the test process, the specimen was removed from the testing machine after a certain number of blocks each time, the crack morphology was observed and recorded by using the metallurgical microscope. To ensure the consistency of the test, marks were made on the clamping elements of the testing machine and the clamped end of the specimen, the specimen gage region (the central portion with equal cross section) should be avoided. In addition, cracks may initiate anywhere on the surface of the smooth tubular specimens, consequently, it is necessary to observe each position of the specimen gage region. The observation of the axial direction of the specimen was achieved by moving the lens vertically, and the observation of the circumferential direction of the specimen was achieved by rotating the specimen.

Multiaxial Fatigue Life
Under multiaxial asynchronous loadings, the fatigue failure life (or blocks) corresponds to the complete failure of the specimens and is defined as the number of loading blocks. The logarithmic mean fatigue life can be expressed as: where N f is the fatigue failure life (or blocks). A total of 10 specimens were tested to ensure that there are at least 2 valid test results under each loading path. The experimental fatigue lives under different asynchronous loadings are listed in Table 3, and the test results fall in the 2 times scatter band of fatigue life. The effect of frequency ratios on the multiaxial fatigue life is given in Figure 3. For ξ 1 : ξ 2 = 1:1, the experimental results can be found in the authors' previous study [6]. The fatigue life decreases when ξ 1 or ξ 2 increase. For the condition of ξ 2 = 1, fatigue life decreases significantly when ξ 1 increases from 1 to 2, while it decreases by 25% when ξ 1 increases from 2 to 4. For the condition of ξ 1 = 1, fatigue life also decreases significantly when ξ 2 increases from 1 to 2, while fatigue life has no obvious change with the increase of ξ 2 from 2 to 4.

Stress Analysis of Individual Loading Blocks
Under multiaxial asynchronous loading, Figure 4 shows the stress components on an arbitrary plane, and the plane orientation is defined as φ. The x axis is consistent with the specimen axis.
x,a n 1 x y , a 2 sin 2 sin cos 2 sin 2 where σn(t) and τn(t) are the cyclic normal and shear stress on an arbitrary plane. Under different frequency ratios, Table 4 shows the directions of the maximum shear stress amplitude (MSSA) and maximum normal (MN) planes, and the stresses on the two planes. In addition, the MN and MSSA on an arbitrary plane in a loading block are analyzed and given in Figure 5. There are four MSSA planes and two MN planes exist for each of the four frequency ratios in a loading block. In addition, it should be noted that there are two MSSA planes for AS-1. Two MN planes are axisymmetric about the axis of the specimen, and the values of τn,a on the two MN planes are equal.

Stress Analysis of Individual Loading Blocks
Under multiaxial asynchronous loading, Figure 4 shows the stress components on an arbitrary plane, and the plane orientation is defined as ϕ. The x axis is consistent with the specimen axis.

Stress Analysis of Individual Loading Blocks
Under multiaxial asynchronous loading, Figure 4 shows the stress components on an arbitrary plane, and the plane orientation is defined as φ. The x axis is consistent with the specimen axis. Under different frequency ratios, the stress components on a plane are calculated as x,a n 1 x y , a 2 sin 2 sin cos 2 sin 2 where σn(t) and τn(t) are the cyclic normal and shear stress on an arbitrary plane. Under different frequency ratios, Table 4 shows the directions of the maximum shear stress amplitude (MSSA) and maximum normal (MN) planes, and the stresses on the two planes. In addition, the MN and MSSA on an arbitrary plane in a loading block are analyzed and given in Figure 5. There are four MSSA planes and two MN planes exist for each of the four frequency ratios in a loading block. In addition, it should be noted that there are two MSSA planes for AS-1. Two MN planes are axisymmetric about the axis of the specimen, and the values of τn,a on the two MN planes are equal. Under different frequency ratios, the stress components on a plane are calculated as σ n (t) = σ x,a cos 2 ϕ sin(ξ 1 ωt) + τ xy,a sin 2ϕ sin(ξ 2 ωt), where σ n (t) and τ n (t) are the cyclic normal and shear stress on an arbitrary plane. Under different frequency ratios, Table 4 shows the directions of the maximum shear stress amplitude (MSSA) and maximum normal (MN) planes, and the stresses on the two planes. In addition, the MN and MSSA on an arbitrary plane in a loading block are analyzed and given in Figure 5. There are four MSSA planes and two MN planes exist for each of the four frequency ratios in a loading block. In addition, it should be noted that there are two MSSA planes for AS-1. Two MN planes are axisymmetric about the axis of the specimen, and the values of τ n,a on the two MN planes are equal.   Figure 5. Variation in the stress components on different planes.
According to previous research [6], the crack growth behavior is related to the stresses on the MSSA and MN planes. Figures 6 and A1-A3 show the stresses on the MSSA planes under four loading paths. In Figure 6a,b, there are two MSSA planes, which are perpendicular and parallel to the direction of the specimen axis, respectively, when ξ1 = 2 and ξ2 = 1. There is only one shear stress cycle on each MSSA plane in a loading block. For normal stress, there are two cycles on the 0° plane, and the value of σn is zero on the 90° plane. Under AS-2 with ξ1 = 4 and ξ2 = 1, the directions of the MSSA planes are ±13.3° and ±76.7°. Each MSSA plane contains three shear stress cycles in a loading block. There are four normal stress cycles on the planes of ±13.3°, and only one normal stress cycle with small stress on the planes of ±76.7°, as shown in Figure A1. When ξ1 = 1 and ξ2 = 2, there are two shear stress cycles on each plane in a loading block. However, it contains one larger normal stress cycle on the plane of ±10.0° and two smaller normal stress cycles on According to previous research [6], the crack growth behavior is related to the stresses on the MSSA and MN planes. Figures 6 and A1, Figures A2 and A3 show the stresses on the MSSA planes under four loading paths. In Figure 6a,b, there are two MSSA planes, which are perpendicular and parallel to the direction of the specimen axis, respectively, when ξ 1 = 2 and ξ 2 = 1. There is only one shear stress cycle on each MSSA plane in a loading block. For normal stress, there are two cycles on the 0 • plane, and the value of σ n is zero on the 90 • plane. Under AS-2 with ξ 1 = 4 and ξ 2 = 1, the directions of the MSSA planes are ±13.3 • and ±76.7 • . Each MSSA plane contains three shear stress cycles in a loading block. There are four normal stress cycles on the planes of ±13.3 • , and only one normal stress cycle with small stress on the planes of ±76.7 • , as shown in Figure A1. When ξ 1 = 1 and ξ 2 = 2, there are two shear stress cycles on each plane in a loading block. However, it contains one larger normal stress cycle on the plane of ±10.0 • and two smaller normal stress cycles on the plane of ±80.0 • , as shown in Figure  the plane of ±80.0°, as shown in Figure A2. For the condition of ξ1 = 1 and ξ2 = 4, each of the four MSSA planes contains four normal and shear stress cycles, respectively, in a loading block. The values of MN on the plane of ±12.4° are larger than those on the plane of ±77.6°, as shown in Figure A3.

Crack Growth Paths of AS-1
For the condition of ξ1 = 2 and ξ2 = 1, the crack morphologies of specimens DF-4 and DF-6 were observed and are shown in  the plane of ±80.0°, as shown in Figure A2. For the condition of ξ1 = 1 and ξ2 = 4, each of the four MSSA planes contains four normal and shear stress cycles, respectively, in a loading block. The values of MN on the plane of ±12.4° are larger than those on the plane of ±77.6°, as shown in Figure A3.

Crack Growth Paths of AS-1
For the condition of ξ1 = 2 and ξ2 = 1, the crack morphologies of specimens DF-4 and DF-6 were observed and are shown in    Figure 9g, the stage Ι crack with length of 65 μm is observed for DF-9, and then the direction of crack propagation changes and branches along the two MN planes. After that, the direction of crack propagation is perpendicular to the specimen axis.  Figure 9g, the stage I crack with length of 65 µm is observed for DF-9, and then the direction of crack propagation changes and branches along the two MN planes. After that, the direction of crack propagation is perpendicular to the specimen axis.  Figure 9. Secondary cracks (a-f) and main crack (g) in specimen DF-9.

Crack Growth Paths of AS-3
For the condition of ξ1 = 1 and ξ2 = 2, Figure 10 shows the crack morphologies of DF-2 and DF-5. For DF-2 at 25,000 blocks (50.1% of Nf), the surface crack initiates on the MSSA planes with larger normal stress and propagates along the MN planes; however, the length of the stage I crack is short and only approximately 10 μm. For DF-5 at 25,000 blocks (49.5% of Nf), the main crack initiates on the edge of the defect and propagates along the MN planes. Otherwise, one secondary crack of DF-5 with a length of 150 μm was also observed at 45,500 blocks (90.1% of Nf). The transformation from stage I to stage II crack was observed. The secondary crack initiates on the MSSA plane with a length of approximately 35 μm and propagates along the MN planes. Figure 10d,e show the main crack propagation morphologies of specimens DF-2 at 46,000 blocks (92.2% of Nf) and DF-5 at 45,500 blocks (90.1% of Nf), respectively, and they propagate along the MN planes. For DF-5, the defect has little effect on the fatigue life. In [52], the traditional theories based on stress concentration factors are not applicable to the small defect, and the small defect problem should be treated as the small-crack problem. Thus, the stress intensity factors rather than stress concentration factors are suggested to deal with the small defect problem. Figure 9. Secondary cracks (a-f) and main crack (g) in specimen DF-9.

Crack Growth Paths of AS-3
For the condition of ξ 1 = 1 and ξ 2 = 2, Figure 10 shows the crack morphologies of DF-2 and DF-5. For DF-2 at 25,000 blocks (50.1% of N f ), the surface crack initiates on the MSSA planes with larger normal stress and propagates along the MN planes; however, the length of the stage I crack is short and only approximately 10 µm. For DF-5 at 25,000 blocks (49.5% of N f ), the main crack initiates on the edge of the defect and propagates along the MN planes. Otherwise, one secondary crack of DF-5 with a length of 150 µm was also observed at 45,500 blocks (90.1% of N f ). The transformation from stage I to stage II crack was observed. The secondary crack initiates on the MSSA plane with a length of approximately 35 µm and propagates along the MN planes. Figure 10d,e show the main crack propagation morphologies of specimens DF-2 at 46,000 blocks (92.2% of N f ) and DF-5 at 45,500 blocks (90.1% of N f ), respectively, and they propagate along the MN planes. For DF-5, the defect has little effect on the fatigue life. In [52], the traditional theories based on stress concentration factors are not applicable to the small defect, and the small defect problem should be treated as the small-crack problem. Thus, the stress intensity factors rather than stress concentration factors are suggested to deal with the small defect problem.

Crack Growth Paths of AS-4
For the condition of ξ 1 = 1 and ξ 2 = 4, the crack morphologies of DF-7 and DF-8 are shown in Figure 11. For specimen DF-7 at 37,000 blocks (52.6% of N f ), only a stage I crack initiation along the MSSA plane was observed. For specimen DF-8 at 20,000 blocks (56.2% of N f ), crack initiation along the MSSA plane with a length of 35 µm and propagation along the MN planes were both observed. Figure 11c,d show the main crack propagation morphologies of specimens DF-7 and DF-8, respectively. The cracks in specimens DF-7 and DF-8 propagate along the MN planes of 32.5 • and −32.5 • with the equal normal and shear stresses, respectively. For the specimen DF-7, the crack gradually changes direction at 5000 blocks (71.1% of N f ), and the length of the stage I crack is approximately 100 µm, which is obviously longer than that in DF-8. In addition, the branch cracks propagation along MN planes are also observed in specimens DF-7 and DF-8.

Crack Growth Paths of AS-4
For the condition of ξ1 = 1 and ξ2 = 4, the crack morphologies of DF-7 and DF-8 are shown in Figure 11. For specimen DF-7 at 37,000 blocks (52.6% of Nf), only a stage I crack initiation along the MSSA plane was observed. For specimen DF-8 at 20,000 blocks (56.2% of Nf), crack initiation along the MSSA plane with a length of 35 μm and propagation along the MN planes were both observed. Figure 11c,d show the main crack propagation morphologies of specimens DF-7 and DF-8, respectively. The cracks in specimens DF-7 and DF-8 propagate along the MN planes of 32.5° and-32.5° with the equal normal and shear stresses, respectively. For the specimen DF-7, the crack gradually changes direction at 5000 blocks (71.1% of Nf), and the length of the stage I crack is approximately 100 μm, which is obviously longer than that in DF-8. In addition, the branch cracks propagation along MN planes are also observed in specimens DF-7 and DF-8.

Crack Growth Paths of AS-4
For the condition of ξ1 = 1 and ξ2 = 4, the crack morphologies of DF-7 and DF-8 are shown in Figure 11. For specimen DF-7 at 37,000 blocks (52.6% of Nf), only a stage I crack initiation along the MSSA plane was observed. For specimen DF-8 at 20,000 blocks (56.2% of Nf), crack initiation along the MSSA plane with a length of 35 μm and propagation along the MN planes were both observed. Figure 11c,d show the main crack propagation morphologies of specimens DF-7 and DF-8, respectively. The cracks in specimens DF-7 and DF-8 propagate along the MN planes of 32.5° and-32.5° with the equal normal and shear stresses, respectively. For the specimen DF-7, the crack gradually changes direction at 5000 blocks (71.1% of Nf), and the length of the stage I crack is approximately 100 μm, which is obviously longer than that in DF-8. In addition, the branch cracks propagation along MN planes are also observed in specimens DF-7 and DF-8. Under AS-4, Figure 12 shows that several secondary cracks were observed in DF-7 at 66,800 blocks (95.0% of Nf) and in DF-8 at 32,800 blocks (92.2% of Nf). The directions of secondary crack initiation and propagation are almost consistent with those of the main cracks. Otherwise, the branch cracks are also observed, and propagate along the MN planes.  Under AS-4, Figure 12 shows that several secondary cracks were observed in DF-7 at 66,800 blocks (95.0% of N f ) and in DF-8 at 32,800 blocks (92.2% of N f ). The directions of secondary crack initiation and propagation are almost consistent with those of the main cracks. Otherwise, the branch cracks are also observed, and propagate along the MN planes. Under AS-4, Figure 12 shows that several secondary cracks were observed in DF-7 at 66,800 blocks (95.0% of Nf) and in DF-8 at 32,800 blocks (92.2% of Nf). The directions of secondary crack initiation and propagation are almost consistent with those of the main cracks. Otherwise, the branch cracks are also observed, and propagate along the MN planes.  As discussed above, under multiaxial asynchronous fatigue loadings with different frequency ratios, the surface stage I crack initiates on the MSSA plane with a large normal stress and the stage II cracks mainly propagate along the MN planes. For some of the loading paths, since the normal and shear stresses are both equal on the two MN planes, the cracks branching along the MN planes were observed during the main crack propagation. As the tension or torsion frequency ratio increases, more secondary cracks can be found on surface of the specimen. This may be due to the increase of shear stress amplitude and shear stress cycles in a loading block. In a previous study [6], the authors found that for pure torsional loading, high stress level will lead to more initiation of small cracks and the fatigue failure under high stress level is caused by the connection of small cracks. Under the same loading path, the morphologies of the main crack and secondary cracks are similar. For the condition where ξ 1 = 2 or ξ 1 = 4 and ξ 2 = 1, once the crack initiates, it will propagate rapidly.

Crack Length versus Loading Blocks
The main surface crack lengths versus loading blocks of different specimens were recorded and are given in Table A1 and Figure 13. The test results for G-100 under constant amplitude loading path can be found in a previous study by the authors [6]. From Figure 13, it can be seen that the crack growth length is longer for the constant amplitude loading than that for the asynchronous loading under the same fatigue life ratio. That is because that the crack length shown in the figure is a stage II-mode I crack, and the crack propagation is determined by the stress state on the MN planes. According to the stress analysis in Section 3.2, there are two different normal and shear stress cycles on each MN plane in a loading block for the conditions of ξ 1 :ξ 2 = 2:1 and ξ 1 :ξ 2 = 1:2 and four different normal and shear stress cycles for the conditions of ξ 1 :ξ 2 = 4:1 and ξ 1 :ξ 2 = 1:4. Difference between the load cycle amplitudes may lead to the crack retardation effect. However, the stage II crack propagation is not affected by the above factors for the constant amplitude loading. In addition, it can also be seen from Figure 13 that the crack growth life of stage II accounts for more than 50% of N f . When the crack length is approximately 500 µm, the crack propagation life accounts for more than 85% of N f . As discussed above, under multiaxial asynchronous fatigue loadings wi frequency ratios, the surface stage I crack initiates on the MSSA plane with a la stress and the stage II cracks mainly propagate along the MN planes. For s loading paths, since the normal and shear stresses are both equal on the two M the cracks branching along the MN planes were observed during the main cr gation. As the tension or torsion frequency ratio increases, more secondary cr found on surface of the specimen. This may be due to the increase of shear st tude and shear stress cycles in a loading block. In a previous study [6], the aut that for pure torsional loading, high stress level will lead to more initiation of s and the fatigue failure under high stress level is caused by the connection of sm Under the same loading path, the morphologies of the main crack and secon are similar. For the condition where ξ1 = 2 or ξ1 = 4 and ξ2 = 1, once the crack will propagate rapidly.

Crack Length versus Loading Blocks
The main surface crack lengths versus loading blocks of different speci recorded and are given in Table A1 and Figure 13. The test results for G constant amplitude loading path can be found in a previous study by the autho Figure 13, it can be seen that the crack growth length is longer for the constant loading than that for the asynchronous loading under the same fatigue life ra because that the crack length shown in the figure is a stage II-mode I crack, an propagation is determined by the stress state on the MN planes. According t analysis in Section 3.2, there are two different normal and shear stress cycles o plane in a loading block for the conditions of ξ1:ξ2 = 2:1 and ξ1:ξ2 = 1:2 and fo normal and shear stress cycles for the conditions of ξ1:ξ2 = 4:1 and ξ1:ξ2 = 1:4. between the load cycle amplitudes may lead to the crack retardation effect. Ho stage II crack propagation is not affected by the above factors for the constant loading. In addition, it can also be seen from Figure 13 that the crack growth II accounts for more than 50% of Nf. When the crack length is approximately 5 crack propagation life accounts for more than 85% of Nf.

Fatigue Life Prediction
From the crack growth morphologies, the stage I crack initiates along on plane with larger normal stress, and stage II crack propagates along one of the Therefore, the stress components on these two planes are the main factors a

Fatigue Life Prediction
From the crack growth morphologies, the stage I crack initiates along on the MSSA plane with larger normal stress, and stage II crack propagates along one of the MN planes. Therefore, the stress components on these two planes are the main factors affecting the fatigue failure life. The relationship between shear stress, normal stress, the combination of the shear stress and normal stress and the fatigue life are given in Figure 14, it shows no obvious relationship. In fact, for the multiaxial asynchronous loading paths, it should be noted that one load block may contain several different fatigue cycles. Therefore, a cycle counting method must be used to predict fatigue life under multiaxial asynchronous loads.  Figure 14, it show no obvious relationship. In fact, for the multiaxial asynchronous loading paths, it should be noted that one load block may contain several different fatigue cycles. Therefore, a cy cle counting method must be used to predict fatigue life under multiaxial asynchronou loads.

Existing Multiaxial Cycle Counting Method
The counting process is more complex due to the coupling effect of tensile and shea stresses under multiaxial variable amplitude loading. Efforts have been made to apply th uniaxial cycle counting method to predict the multiaxial fatigue life. BS and WB method are commonly used in dealing with multiaxial fatigue loadings [46,47].
To identify multiaxial fatigue load reversals, Bannantine and Socie [35,36] used th rainflow cycle counting method and critical plane criterion. Whether the tensile or shea critical plane damage parameter is chosen depends on the failure mode. When the tensil stress is selected for cycle counting, the Smith-Watson-Topper [51] damage model (SWT is used and expressed as where k is the material parameter, γna,max is the maximum shear strain amplitude, σn,max i the maximum normal stress, τ'f are the shear fatigue strength and ductility coefficient, b

Existing Multiaxial Cycle Counting Method
The counting process is more complex due to the coupling effect of tensile and shear stresses under multiaxial variable amplitude loading. Efforts have been made to apply the uniaxial cycle counting method to predict the multiaxial fatigue life. BS and WB methods are commonly used in dealing with multiaxial fatigue loadings [46,47].
To identify multiaxial fatigue load reversals, Bannantine and Socie [35,36] used the rainflow cycle counting method and critical plane criterion. Whether the tensile or shear critical plane damage parameter is chosen depends on the failure mode. When the tensile stress is selected for cycle counting, the Smith-Watson-Topper [51] damage model (SWT) is used and expressed as where ε n,a is the principal strain amplitude, σ' f and ε' f are the axial and shear fatigue strength coefficient, b and c are the axial fatigue strength and ductility exponent. When the shear stress is selected to count the cycle, the Fatemi-Socie [53] damage model (FS) is usually used and expressed as max ϕ γ na,max 1 + k σ n,max σ y where k is the material parameter, γ na,max is the maximum shear strain amplitude, σ n,max is the maximum normal stress, τ' f are the shear fatigue strength and ductility coefficient, b 0 and c 0 are the shear fatigue strength and ductility exponent.
An example of the BS with the SWT method to extract the reversal counting cycles under AS-2 is shown in Figure A7.
The WB method [37][38][39] is based on rainflow cycle counting and the von Mises equivalent strain/stress to identify all reversals. Then, the damage can be calculated by the critical plane criterion or a modified strain-based multiaxial fatigue criterion due to Wang and Brown [54], as shown in Equation (8), to obtain the total fatigue life. The Wang and Brown criterion can be expressed as where ν' is the effective Poisson ratio, σ n,m is the mean normal stress, S is the material parameter and S = 1.5-2.0 for steel materials. An example of using the WB method to extract the reversal counting cycles under AS-2 is given in Figure A8.

Fatigue Life Prediction Results
The authors proposed a section critical plane method (SCPM) [55] for multiaxial high-cycle fatigue life prediction. Here it is also used for fatigue life prediction under asynchronous loadings.
To estimate the fatigue life under asynchronous loadings, the BS and WB cycle counting methods are used. Then, the fatigue life under simple cycles is predicted by the FS, SWT, WB, and SPCM criteria with the S-N curves of fully reversed uniaxial loading and fully reversed torsional loading. The S-N curves can be expressed by the following equations: Fully reversed uniaxial tension-compression loading: log N f = 6.9577 − 1.2294 log(σ x,a − 565.25).
Finally, the fatigue life is obtained by the Palmgren-Miner's linear cumulative damage rule. The comparison between the experimental and prediction results of 30CrMnSiA steel under multiaxial asynchronous loadings is given in Figure 15. The black solid line indicates that the prediction result is equal to the experimental result, the black dashed lines are used to represent the ±2 times scatter band of fatigue life, and the red dotted lines denote for the ±3 times scatter band of fatigue life. For multiaxial fatigue under stress loadings, there is no obvious engineering plastic strain. Hence only the elastic part is considered when the SWT, FS, and WB criteria are used for fatigue life prediction.
The prediction results pertaining to BS with the SWT method (mainly based on the tension stress and strain) are non-conservative and mostly exceed the 3 times range, and the discrepancies between the experimental and prediction results are large. Regarding the BS with the FS method, the prediction results lie outside of the 3 times range under AS-1, and others fall into the ±2 times range. Figure 15d,e show the results for S = 1.5 and S = 2.0, respectively. When S = 1.5, the prediction life is conservative and lies in the −3 times fatigue life scatter band except for AS-2. When S = 2.0, the prediction results all lie in the ±3 times range, and they are conservative for the condition of AS-2, while the others are non-conservative. For the two shear-based models (FS model and WB model), both the shear strain and normal strain (or stress) are taken into account, and the prediction results are much better than the SWT method. The prediction results obtained using the BS with the SCPM method all fall into the ±2 times range. Regarding the WB with the SCPM method, the prediction results lie in the ±3 times range except the condition of AS-3. The SCPM method considered the crack propagation direction under different loading paths and the prediction results are also good. For these four loading paths, when using SCPM model for life prediction, the stress components used are mainly the shear stress amplitude and normal stress on the MSSA plane.    Table 5 gives the summary of percentage in error index Ei. The error index Ei ≤ 2 and Ei ≤ 3 indicate the prediction results with ±2 and ±3 times scatter band of fatigue life, respectively. Compared with other models, the BS method with the SCPM criterion is more applicable for 30CrMnSiA steel under asynchronous loadings. Table 5. Percentage in two error indexes of each model (unit: %).

Conclusions
In this paper, multiaxial fatigue tests were performed on 30CrMnSiA steel under four asynchronous loadings. Subsequently, the effect of the frequency ratio on the fatigue failure life and crack behaviors were studied. As discussed above, the following conclusions can be drawn: