# Design Theory and Experimental Research of Ultrasonic Fatigue Test

^{*}

## Abstract

**:**

_{1}, L

_{2}, L

_{3}, R

_{1}and R

_{2}, on the resonant frequency and maximum stress of the sample. According to the optimized results, the sample was processed, and an ultrasonic fatigue test was carried out to ensure the sample fatigue fracture finally occurred. Finally, the S-N curve of the material was plotted based on the data recorded in the test and compared with the conventional fatigue life curve to verify the feasibility of the ultrasonic fatigue test device and test method. The fracture of the sample was observed using an optical microscope, and its macroscopic fracture morphology was analyzed. The fracture morphology of the sample can be divided into three typical zones: the fatigue crack source zone, the extension zone and the transient zone, where the fatigue cracks all originate from on the surface of the sample. The results demonstrate the validity of the ultrasonic fatigue test results and provide new ideas for the design and optimization of ultrasonic fatigue samples and shorter processing times, providing a reference for subsequent ultrasonic fatigue tests on typical materials.

## 1. Introduction

^{8}~10

^{11}cycles [1,2]. For example, aircraft engines generally operate at high speeds of tens of thousands of revolutions per minute and the blades in the engine are subjected to forces between 170 Hz and 2000 Hz; the axles of high-speed railway trains are subjected to 10

^{9}cycles of dynamic loads during an overhaul period (200 × 10

^{4}km); a gasoline engine operating at 3000 rpm is subjected to 8.5 × 10

^{9}cycles of dynamic loads during its 15-year service life. This kind of vibration load is characterized by small stress amplitude, but high vibration frequency [3], which can reach thousands of Hertz [4]. The compound damage caused by high-frequency and low-amplitude loads superimposed with other loads will greatly accelerate the damage of structural components, and seriously threaten the safety and reliability of structural components. In conventional fatigue testing, it is difficult to guarantee the requirements of key technologies [5], such as long life and high reliability for large aircraft projects and key projects such as high-speed railways. There is an urgent need for the development and manufacture of test methods and reliable test equipment for low-amplitude and ultra-high cycle fatigue life testing of mechanical structures [6].

^{7}cycles are time-consuming, costly and even impossible. In contrast, ultrasonic fatigue testing is fast, efficient and consistent with the real forces on the component. Ultrasonic frequencies can reach tens of thousands of Hertz per second [7], and using a 20 kHz ultrasonic vibration fatigue test machine can save time compared to conventional methods, and can study the fatigue life range from millions to billions of times. For example, it takes 10 h to perform a 10

^{6}-cycle fatigue test on a 30 Hz conventional fatigue testing machine, but less than 1 min on a 20 kHz ultrasonic vibration fatigue testing machine. It takes 11 years to perform a 10

^{10}cycle fatigue test with a conventional fatigue testing machine, but only 6 days with a 20 kHz ultrasonic vibration fatigue test machine [5]. Conventional fatigue testing machine can only extend the fatigue life to 10

^{6}cycles, whereas with a 20 kHz ultrasonic vibration fatigue testing machine, the fatigue life can be extended to 10

^{10}cycles. At the same time, the ultrasonic fatigue test method saves energy compared to conventional fatigue test techniques due to the shorter test time and the relatively low power requirement of the resonance system.

## 2. Test Equipment and Test Sample

#### 2.1. Test Principle

#### 2.2. Test Equipment

- (1)
- Ultrasonic generator

- (2)
- Ultrasonic transducer

- (3)
- Ultrasonic amplitude transformer

- (4)
- Ultrasonic sample

- (5)
- Electronic measurement and control system

#### 2.3. Test Sample Design

_{d}is the Young’s modulus.

_{1}(x) is the cross-sectional area of the intermediate equal section; S

_{2}(x) is the cross-sectional area of the variable section and S

_{3}(x) is the end cylindrical cross-sectional area, and

_{S}is proportional to the constant M, the magnitude of M reflects the magnitude of the stress amplification level, and the resonant length L

_{3}equation is obtained analytically by:

## 3. Simulation and Testing

#### 3.1. Initial Sizing and Simulation of Test Sample

_{b}= 412 MPa. For harmonic response vibration, the type of stress needed is a symmetrical cycle with a cyclic characteristic of r = −1, which belongs to the category of dynamic loading. For fatigue damage to occur in the middle section of the sample, the value of the stress acting on it must be greater than the material’s endurance limit. For the same material, the endurance limit σ

_{−1}under symmetrical cyclic alternating stresses is much lower than the strength limit σ

_{b}under static loading. For an aluminum alloy material, tension and compression tests are:

#### 3.2. Finite Element Optimization of Test Sample

_{1}, L

_{2}, L

_{3}, R

_{1}and R

_{2}of the sample simulation model as shown in Figure 7 are parameterized and set as the independent variables for optimization. The test resonant frequency and the maximum stress in the middle of the sample are set as the dependent variables for optimization, and the optimization conditions are set to a frequency close to 20 kHz and a maximum stress in the sample greater than 142.17 MPa.

_{1}, L

_{2}, L

_{3}and R

_{2}are negatively related to it, and R

_{1}is positively related to it. Additionally, for the maximum stress, L

_{1}, L

_{2}, L

_{3}and R

_{1}are negatively related to it, and R

_{2}is positively related to it. In addition, in terms of the rate at which the curve rises or falls, L

_{1}, L

_{2}and L

_{3}affect frequency and stress to a lesser extent, while R

_{1}and R

_{2}affect them to a greater extent. The next step is to combine the local sensitivity diagrams and set the optimization conditions to arrive at the optimal results.

_{1}and R

_{2}. In order to avoid the influence due to this situation, in the actual processing of the sample, R

_{1}and R

_{2}should be paid more attention, while the deviations of L

_{1}, L

_{2}and L

_{3}and the relative surface roughness requirements can be appropriately relaxed.

_{max}> 142.17 MPa. After comparing the optimization results of different algorithms, the “Screening” method was finally selected. Based on the above two constraints, the optimum design simulation gives three sets of optimal solutions, as shown in Table 4. According to the calculation results, the inherent frequencies and stress levels corresponding to the three groups of sample sizes can meet the resonance conditions and fatigue test requirements.

_{1}= 5.1 mm, L

_{2}= 10.5 mm, L

_{3}= 12.7 mm, R

_{1}= 2.1 mm, R

_{2}= 5.6 mm. The model was remodeled with the optimized dimensions and imported into Workbench for verification. The resonant frequency of the sample in axial tension and compression was found to be 20,087 Hz, as shown in Figure 11a, which meets the required frequency for the test. The maximum stress of 195.7 MPa > 142.17 MPa obtained from the harmonic response analysis under the applied 30 μm displacement condition is shown in Figure 11b, which meets the stress requirement for fatigue fracture of the sample in the ultrasonic fatigue tensile test.

#### 3.3. Test Procedure

## 4. Analysis of Results

#### 4.1. Fatigue Life Analysis of Sample

#### 4.2. Fracture Morphology Analysis of Sample

## 5. Conclusions

_{1}, L

_{2}, L

_{3}and R

_{2}and decreases with R

_{1}; maximum stress decreases with L

_{1}, L

_{2}, L

_{3}and R

_{1}and decreases with R

_{2}. Based on the trend and sensitivity of each dimensional parameter on the intrinsic frequency and maximum stress of the sample, the deviation range was determined to reduce the processing difficulty.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Ultrasonic fatigue test system (

**a**) Schematic diagram of ultrasonic fatigue system; (

**b**) actual test machine.

**Figure 3.**The variable-section sample: (

**a**) The hourglass-shaped sample; (

**b**) the dog-bone-shaped sample.

**Figure 5.**Finite element analysis results of an actual machined dog-bone-shaped sample: (

**a**) Modal analysis; (

**b**) harmonious response analysis.

**Figure 11.**Finite element analysis results of the optimized sample: (

**a**) Modal analysis; (

**b**) harmonious response analysis.

**Figure 19.**Fracture morphology of 6063-aluminum alloy samples with different powers. (

**a**) The third set of test pieces, test power = 45%; (

**b**) the eighth set of test pieces, test power = 60%.

**Figure 20.**Section morphology of 6063-aluminum alloy samples with different powers (

**a**) The second set of test pieces, test power = 40%; (

**b**) the seventh set of test pieces, test power = 55%.

**Figure 21.**Direction of crack expansion in aluminum alloy samples with different powers (

**a**) The first set of test pieces, test power = 30%; (

**b**) the third set of test pieces, test power = 45%.

Material | σ_{b}[MPa] | E_{d}[Gpa] | ρ [kg/m ^{3}] | μ |
---|---|---|---|---|

6063Al | 412 | 71 | 2770 | 0.33 |

Material | L_{1}[mm] | L_{2}[mm] | L_{3}[mm] | R_{1}[mm] | R_{2}[mm] |
---|---|---|---|---|---|

6063Al | 5 | 10 | 14 | 2 | 6 |

No. | L_{1}[mm] | L_{2}[mm] | L_{3}[mm] | R_{1}[mm] | R_{2}[mm] | Frequency [Hz] | Stress [MPa] |
---|---|---|---|---|---|---|---|

1 | 5.350 | 9.620 | 13.972 | 1.820 | 5.786 | 17,989 | 200.54 |

2 | 4.890 | 9.740 | 13.860 | 1.892 | 5.412 | 19,957 | 204.69 |

3 | 4.770 | 10.220 | 12.628 | 2.012 | 5.460 | 21,403 | 209.31 |

4 | 5.490 | 9.940 | 13.636 | 2.148 | 5.988 | 19,940 | 188.96 |

5 | 5.130 | 9.180 | 13.076 | 2.132 | 6.204 | 20,388 | 207.54 |

6 | 4.850 | 9.420 | 14.532 | 1.836 | 6.540 | 16,704 | 222.73 |

7 | 5.470 | 10.460 | 15.092 | 2.020 | 6.060 | 17,806 | 190.11 |

8 | 4.550 | 10.50 | 14.868 | 1.956 | 5.652 | 19,176 | 199.12 |

9 | 4.670 | 9.980 | 12.796 | 2.140 | 6.108 | 20,837 | 207.51 |

10 | 4.790 | 10.940 | 13.188 | 2.004 | 6.314 | 18,420 | 201.01 |

11 | 4.590 | 9.140 | 13.356 | 2.076 | 5.700 | 21,684 | 218.96 |

12 | 5.070 | 10.380 | 13.020 | 1.812 | 6.252 | 17,258 | 201.95 |

13 | 4.750 | 9.540 | 12.852 | 1.932 | 6.468 | 18,509 | 215.46 |

14 | 4.99 | 10.980 | 14.644 | 1.964 | 6.036 | 17,818 | 192.5 |

15 | 5.290 | 9.060 | 13.524 | 1.996 | 5.628 | 20,453 | 205.92 |

16 | 5.170 | 10.420 | 15.204 | 1.884 | 5.556 | 18,192 | 190.23 |

17 | 4.570 | 9.260 | 14.756 | 1.868 | 5.748 | 18,970 | 217.04 |

18 | 5.270 | 10.860 | 14.252 | 2.060 | 6.564 | 17,382 | 195.14 |

19 | 5.390 | 10.020 | 12.964 | 1.940 | 5.580 | 19,814 | 194.31 |

20 | 5.230 | 10.820 | 12.740 | 2.100 | 5.844 | 20,215 | 189.08 |

21 | 5.090 | 9.780 | 13.412 | 2.180 | 5.532 | 22,004 | 198.9 |

22 | 5.450 | 9.660 | 14.924 | 2.044 | 5.508 | 19,849 | 187.86 |

23 | 5.310 | 10.70 | 14.028 | 2.052 | 5.484 | 20,010 | 183.45 |

24 | 4.930 | 9.30 | 12.684 | 1.876 | 5.772 | 19,799 | 215.75 |

25 | 4.910 | 9.020 | 13.916 | 1.948 | 6.132 | 18,936 | 212.33 |

No. | L_{1}[mm] | L_{2}[mm] | L_{3}[mm] | R_{1}[mm] | R_{2}[mm] | Frequency [Hz] | Stress [MPa] |
---|---|---|---|---|---|---|---|

1 | 4.7758 | 10.735 | 12.669 | 2.1168 | 6.1663 | 20,048 | 199 |

2 | 5.4919 | 10.039 | 12.69 | 1.9159 | 5.4596 | 19,998 | 194.81 |

3 | 5.0447 | 10.973 | 12.779 | 2.1757 | 6.1803 | 19,998 | 190.5 |

No. | Temperature [°C] | Power [%] | End Amplitudes [μm] | Cross-Sectional Stress [MPa] | Load Time [s] | Number of Cycles |
---|---|---|---|---|---|---|

1 | 15–50 | 30 | 20.37 | 123.84 | 5657.28 | 1.13 × 10^{8} |

2 | 35 | 21.79 | 126.98 | 2343.52 | 4.69 × 10^{7} | |

3 | 40 | 23.45 | 132.84 | 569.39 | 1.29 × 10^{7} | |

4 | 45 | 24.39 | 139.65 | 330.59 | 6.62 × 10^{6} | |

5 | 50 | 26.21 | 147.63 | 123.77 | 2.93 × 10^{6} | |

6 | 55 | 27.46 | 156.52 | 65.43 | 1.33 × 10^{6} | |

7 | 60 | 28.49 | 165.85 | 36.58 | 7.40 × 10^{5} | |

8 | 65 | 30.28 | 171.52 | 30.44 | 6.06 × 10^{5} | |

9 | 70 | 31.27 | 178.75 | 13.47 | 2.77 × 10^{5} |

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**MDPI and ACS Style**

Feng, N.; Wang, X.; Guo, J.; Li, Q.; Yu, J.; Zhang, X.
Design Theory and Experimental Research of Ultrasonic Fatigue Test. *Machines* **2022**, *10*, 635.
https://doi.org/10.3390/machines10080635

**AMA Style**

Feng N, Wang X, Guo J, Li Q, Yu J, Zhang X.
Design Theory and Experimental Research of Ultrasonic Fatigue Test. *Machines*. 2022; 10(8):635.
https://doi.org/10.3390/machines10080635

**Chicago/Turabian Style**

Feng, Ning, Xin Wang, Jiazheng Guo, Qun Li, Jiangtao Yu, and Xuecheng Zhang.
2022. "Design Theory and Experimental Research of Ultrasonic Fatigue Test" *Machines* 10, no. 8: 635.
https://doi.org/10.3390/machines10080635