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

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^{2}

^{3}

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^{†}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Materials

#### 2.2. Experimental Methods

_{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. The tests were conducted under four different frequency ratios (ξ

_{1}:ξ

_{2}) 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.

## 3. Results and Discussion

#### 3.1. Multiaxial Fatigue Life

_{f}is the fatigue failure life (or blocks).

_{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.

#### 3.2. Stress Analysis of Individual Loading Blocks

_{n}(t) and τ

_{n}(t) are the cyclic normal and shear stress on an arbitrary plane.

_{n}

_{,a}on the two MN planes are equal.

_{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 A2. For the condition of ξ

_{1}= 1 and ξ

_{2}= 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.

#### 3.3. Crack Initiation and Propagation

#### 3.3.1. Crack Growth Paths of AS-1

_{1}= 2 and ξ

_{2}= 1, the crack morphologies of specimens DF-4 and DF-6 were observed and are shown in Figure 8. The main surface cracks of the two specimens initiate on the edge of defects; both the defects are spherical, and the diameters are approximately 34 μm for DF-4 and 21 μm for DF-6, respectively. The crack growth lengths of DF-4 on the left and right sides of the defect are approximately 30 and 55 μm at 35,000 blocks (66.5% of N

_{f}). The crack propagation direction is close to the MN planes on the right side of the defect, while it is close to the MSSA plane and several small cracks branch into the direction of MN planes on the left side. For specimen DF-6, Figure 8b shows that there are four initiation cracks along the MN planes at 35,000 blocks (81.4% of N

_{f}). A secondary crack in DF-6 was also observed, as shown in Figure 8c. The stage I crack initiation on the MSSA plane and stage II cracks propagation along the MN planes were both observed. The length of stage I crack is approximately 10 μm and shorter than the diameter of defect. Consequently, the cracks initiation from the defect of DF-6 are the stage II cracks along the MN planes. The crack morphologies of DF-4 and DF-6 are similar, and the cracks propagate along the MN planes. There are two MN planes where the normal stress and shear stress amplitude are both equal, thus, the cracks branch along the MN planes of both DF-4 and DF-6 on the left and right sides of the defects as the main cracks propagate.

#### 3.3.2. Crack Growth Paths of AS-2

_{1}= 4 and ξ

_{2}= 1, at 32,000 blocks (89.6% of N

_{f}), six secondary cracks in specimen DF-9 were observed and the morphologies are shown in Figure 9a–f. For these secondary cracks, the initiation directions are all close to the MSSA planes, which are ±13.3° and bear larger normal stresses. No initiation cracks were observed on the MSSA planes of ±76.7°, this is because the normal stress on ±13.3° planes are both 476.59 MPa and larger than that on the ±76.7° plane with the value of 167.43 MPa. The lengths of stage I cracks are approximately 40–50 μm, and then the cracks propagate along the MN planes. As 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.

#### 3.3.3. Crack Growth Paths of AS-3

_{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.

#### 3.3.4. Crack Growth Paths of AS-4

_{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.

_{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.

_{1}= 2 or ξ

_{1}= 4 and ξ

_{2}= 1, once the crack initiates, it will propagate rapidly.

#### 3.4. Crack Length versus Loading Blocks

_{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}.

#### 3.5. Fatigue Life Prediction

#### 3.5.1. Existing Multiaxial Cycle Counting Method

_{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.

_{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.

_{n}

_{,m}is the mean normal stress, S is the material parameter and S = 1.5–2.0 for steel materials.

#### 3.5.2. Fatigue Life Prediction Results

## 4. Conclusions

- (a)
- The experimental results show that fatigue failure life under asynchronous loadings decreases when the value of ξ
_{1}or ξ_{2}increase from 1 to 2, and there is no significant change when the value of ξ_{1}or ξ_{2}increase from 2 to 4. - (b)
- Based on the observation of the surface crack path, the crack initiates on the maximum shear stress amplitude plane with larger normal stress and propagates along the maximum normal stress planes. The proportion of the stage II crack propagation life is more than 50% of the fatigue failure life.
- (c)
- Under asynchronous loadings, the increasing of ξ
_{1}or ξ_{2}results in more shear stress cycles and larger shear stress amplitude on the MSSA planes in a loading block, and this may be the reason for the increase of secondary cracks. Furthermore, the difference between the load cycle amplitudes on the MN plane causes the crack retardation and leads to the crack growth length being longer for the constant amplitude loading than that for the asynchronous loading under the same fatigue life ratio. - (d)
- The applicability of Bannantine-Socie and Wang-Brown counting method is verified under multiaxial asynchronous fatigue loading with different fatigue failure criteria and Palmgren-Miner’s cumulative damage rule. The results indicate that the accuracy of the Bannantine-Socie model with the section critical plane method is higher than that of the others.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

E | Young’s modulus |

G | shear modulus |

N_{f} | fatigue failure life |

N_{50} | logarithmic mean fatigue failure life |

S | material parameter of Smith-Watson-Topper model |

k | material parameter of Fatemi-Socie model |

b | axial fatigue strength exponent |

b_{0} | shear fatigue strength exponent |

c | axial fatigue ductility exponent |

c_{0} | shear fatigue ductility exponent |

ξ_{1} | axial stress frequency |

ξ_{2} | shear stress frequency |

φ | direction of an arbitrary plane |

ν’ | effective Poisson Ratio |

σ_{y} | tensile yield strength |

σ_{u} | tensile ultimate strength |

τ_{y} | torsional yield strength |

τ_{u} | torsional ultimate strength |

σ_{x}(t) | axial stress |

σ_{x,a} | axial stress amplitude |

τ_{xy}(t) | shear stress |

τ_{xy,a} | torsion stress amplitude |

σ_{n}(t) | cyclic normal stress on the plane of φ |

τ_{n}(t) | cyclic shear stress on the plane of φ |

σ_{n,max} | maximum normal stress |

σ_{n} | normal stress on the plane of φ |

σ_{n,a} | normal stress amplitude on the plane of φ |

τ_{n,a} | shear stress amplitude on the plane of φ |

σ_{n,m} | mean normal stress |

τ_{na}_{,max} | maximum shear stress amplitude |

γ_{na}_{,max} | maximum shear strain amplitude |

ε_{n,a} | principal strain amplitude |

σ’_{f} | axial fatigue strength coefficient |

τ’_{f} | shear fatigue strength coefficient |

ε’_{f} | axial fatigue ductility coefficient |

γ’_{f} | shear fatigue ductility coefficient |

## Abbreviations

AS | asynchronous loading path |

MN | maximum normal |

MSSA | maximum shear stress amplitude |

## Appendix A

Spec. ID | Contents | N_{1} | N_{2} | N_{3} | N_{4} | N_{5} | N_{6} |
---|---|---|---|---|---|---|---|

DF-4 | blocks | 35,000 | 40,000 | 43,000 | 46,000 | 48,500 | 49,500 |

2a (μm) | 122.5 | 186 | 276 | 403 | 669 | 993 | |

DF-6 | blocks | 35,000 | 38,000 | 39,300 | 40,100 | — | — |

2a (μm) | 261 | 577 | 800 | 1054 | — | — | |

DF-2 | blocks | 25,000 | 35,000 | 40,000 | 44,000 | 46,000 | — |

2a (μm) | 81 | 146 | 270 | 495 | 1050 | — | |

DF-5 | blocks | 25,000 | 35,000 | 39,000 | 42,000 | 44,000 | 45,500 |

2a (μm) | 87 | 163 | 242 | 364 | 567 | 848 | |

DF-9 | blocks | 32,000 | — | — | — | — | — |

2a (μm) | 920 | — | — | — | — | — | |

DF-7 | blocks | 37,000 | 50,000 | 57,000 | 62,000 | 65,000 | 66,800 |

2a (μm) | 75 | 150 | 207 | 318 | 593 | 964 | |

DF-8 | blocks | 20,000 | 23,000 | 27,000 | 29,000 | 31,500 | 32,800 |

2a (μm) | 136 | 200 | 280 | 400 | 683 | 987 |

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**Table 1.**Chemical composition and weight percent of 30CrMnSiA steel (wt %). (Reproduced with permission from Liu, Crack initiation and propagation of 30CrMnSiA steel under uniaxial and multiaxial cyclic loading, published by Elsevier, 2019).

C | Mn | Si | P | Ni | Cr | W | Mo | V | Cu | Ti |
---|---|---|---|---|---|---|---|---|---|---|

0.31 | 0.85 | 0.99 | 0.01 | 0.05 | 0.87 | 0.01 | 0.02 | 0.01 | 0.18 | 0.003 |

**Table 2.**Mechanical properties of 30CrMnSiA steel. (Reproduced with permission from Liu, Crack initiation and propagation of 30CrMnSiA steel under uniaxial and multiaxial cyclic loading, published by Elsevier, 2019).

E (GPa) | σ_{y} (MPa) | σ_{u} (MPa) | G (GPa) | τ_{y} (MPa) | τ_{u} (MPa) |
---|---|---|---|---|---|

207 | 1196 | 1334 | 77.2 | 825 | 1040 |

Load Path | σ_{x}_{,a} (MPa) | τ_{xy}_{,a} (MPa) | ξ_{1} | ξ_{2} | Spec. ID | N_{f} (blocks) | N_{50} (blocks) |
---|---|---|---|---|---|---|---|

— | 350 | 350 | 1 | 1 | G-10 | 185,261 | 141,984 |

G-11 | 175,013 | ||||||

G-12 | 119,687 | ||||||

G-100 | 136,694 | ||||||

G-104 | 108,778 | ||||||

AS-1 | 350 | 350 | 2 | 1 | DF-4 | 52,632 | 47,560 |

DF-6 | 42,976 | ||||||

AS-2 | 350 | 350 | 4 | 1 | DF-9 | 35,702 | 35,710 |

DF-10 | 35,717 | ||||||

AS-3 | 350 | 350 | 1 | 2 | DF-2 | 49,892 | 50,195 |

DF-5 | 50,500 | ||||||

AS-4 | 350 | 350 | 1 | 4 | DF-7 | 70,330 | 50,033 |

DF-8 | 35,593 |

Load Path | ξ_{1} | ξ_{2} | MSSA | σ_{n}/MPa | τ_{na}_{,max} (MPa) | MN | σ_{n}_{,max} (MPa) | τ_{n}_{,a} (MPa) |
---|---|---|---|---|---|---|---|---|

AS-1 | 2 | 1 | 0°/90° | 350/0 | 350.00 | ±30.0° | 494.98 | 285.77 |

AS-2 | 4 | 1 | ±13.3°/±76.7° | 476.59/167.43 | 372.16 | ±31.0° | 544.26 | 307.08 |

AS-3 | 1 | 2 | ±10.0°/±80.0° | 299.42/127.23 | 371.88 | ±34.0° | 504.46 | 256.13 |

AS-4 | 1 | 4 | ±12.4°/±77.6° | 385.62/161.73 | 385.62 | ±32.5° | 548.07 | 295.18 |

BS-SWT | BS-FS | BS-SCPM | WB-S = 1.5 | WB-S = 2.0 | WB-SCPM | |
---|---|---|---|---|---|---|

Ei ≤ 2 | 25.0 | 75.0 | 100.0 | 50.0 | 37.5 | 50.0 |

Ei ≤ 3 | 37.5 | 75.0 | 100.0 | 75.0 | 75.0 | 75.0 |

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## Share and Cite

**MDPI and ACS Style**

Liu, T.; Qi, X.; Shi, X.; Gao, L.; Zhang, T.; Zhang, J. Effect of Loading Frequency Ratio on Multiaxial Asynchronous Fatigue Failure of 30CrMnSiA Steel. *Materials* **2021**, *14*, 3968.
https://doi.org/10.3390/ma14143968

**AMA Style**

Liu T, Qi X, Shi X, Gao L, Zhang T, Zhang J. Effect of Loading Frequency Ratio on Multiaxial Asynchronous Fatigue Failure of 30CrMnSiA Steel. *Materials*. 2021; 14(14):3968.
https://doi.org/10.3390/ma14143968

**Chicago/Turabian Style**

Liu, Tianqi, Xinxin Qi, Xinhong Shi, Limin Gao, Tian Zhang, and Jianyu Zhang. 2021. "Effect of Loading Frequency Ratio on Multiaxial Asynchronous Fatigue Failure of 30CrMnSiA Steel" *Materials* 14, no. 14: 3968.
https://doi.org/10.3390/ma14143968