Two Fatigue Life Prediction Models Based on the Critical Plane Theory and Artificial Neural Networks
Abstract
:1. Introduction
2. Stress–Strain Analysis under Multiaxial Loading
2.1. Thin-Walled Tube under Multiaxial Loading
2.2. Notched Specimen under Multiaxial Loading
3. Fatigue Life Prediction Model
3.1. A New Multiaxial Fatigue Model Based on the Critical Plane Theory
3.2. Establishment of a Neural Network Model
3.2.1. Structure of a BPNN
3.2.2. Establishment of a BPNN Model
3.3. Existing Multiaxial Fatigue Theoretical Models
4. Experimental Data Verification
4.1. Life Prediction of the Two Material Specimens under Constant-Amplitude Loading
4.1.1. Life Prediction
4.1.2. Prediction Result Analysis
4.2. Life Prediction of the Two Material Specimens under Variable-Amplitude Loading
4.2.1. Life Prediction of the S45C Steel Specimen under Variable-Amplitude Loading in the Time Domain
4.2.2. Life Prediction of the 7075-T651 Aluminum Alloy Specimen under Random-Vibration Loading in the Frequency Domain
- (1)
- Finite Element Simulation
- (2)
- Life Prediction
5. Discussion
6. Conclusions
- (1)
- In Section 3.1, the WYT model not only considers the influences of shear strain and normal strain on the critical plane but also those of normal stress and shear stress. By predicting the lives of two material specimens under constant-amplitude and variable-amplitude loading, respectively, in Section 4, the WYT model has better prediction effects for the two material specimens than the SWT model and SHD model based on Table 4, Table 5, Table 6 and Table 9.
- (2)
- In Section 3.2, a BPNN model with four input parameters (, , , ) for multiaxial fatigue life prediction is established by using the shear strain amplitude, normal strain amplitude, mean shear stress, and mean normal stress on the critical plane as input variables of the BPNN. In Section 4, the prediction effect of the trained BPNN model with four input parameters (, , , ) is better than that of the WYT model for the same material specimen based on Table 4, Table 5 and Table 6 and Table 9. This is because the BPNN model can effectively and autonomously learn the nonlinear mapping relationships between fatigue life and multiple variables from existing experimental data, which is its advantage.
- (3)
- In Section 5, the influences of different variable groups as input parameters of the neural network on the prediction effect are discussed. Two variables (, ), three variables (, , ), and four variables (, , , ) are used as the input parameters of the BPNN model. Their effects for the two material specimens are better than those of the three theoretical models (SWT, SHD, and WYT) based on Table 4, Table 5, Table 10 and Table 11; the prediction results of the BPNN model with four input parameters (, , , ) and the BPNN model with four input parameters (, , , ) for S45C are both within the two-time error band based on Table 4 and Table 10; the prediction results of the BPNN model with four input parameters (, , , ) and the BPNN model with four input parameters (, , , ) for 7075-T651 are both within the three-time error band based on Table 5 and Table 11.
- (4)
- Although the prediction effect of the WYT model is not better than that of the BPNN model for the same material specimen, it is better than those reported earlier. On the contrary, the construction of neural network models is not easy for engineers and technicians who are not familiar with neural networks. The purpose of this study is to provide more options for engineers when predicting a structural life under different loadings.
- (5)
- This study uses small-sample data to train the proposed BPNN model. If more test data are used to train the BPNN model, it is believed that its prediction accuracy may be further improved. If the WYT model is to be used to simulate and calculate life data under different loading conditions, and these are used to train the BPNN model, the effect of the trained BPNN model on the prediction of the life of a specimen needs further study.
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Appendix A. Experimental Results for S45C Steel [44]
Loading Paths | Phase Angle (°) | (MPa) | (MPa) | (Cycles) | ||
a | - | 0 | 0.025 | 0 | 595.19 | 110 |
a | - | 0 | 0.01 | 0 | 480.15 | 852 |
a | - | 0 | 0.005 | 0 | 389.1 | 3383 |
a | - | 0 | 0.004 | 0 | 361.9 | 5514 |
a | - | 0 | 0.015 | 0 | 509.02 | 421 |
a | - | 0 | 0.004 | 0 | 365.46 | 8933 |
a | - | 0 | 0.003 | 0 | 325.9 | 22071 |
a | - | 0 | 0.015 | 0 | 500.16 | 407 |
b | - | 0.015 | 0 | 287.14 | 0 | 1151 |
b | - | 0.015 | 0 | 283.81 | 0 | 1761 |
b | - | 0.015 | 0 | 286.3 | 0 | 1771 |
b | - | 0.009 | 0 | 244.25 | 0 | 5644 |
b | - | 0.008 | 0 | 229.38 | 0 | 14930 |
c | 0 | 0.0052 | 0.006 | 103.55 | 370.33 | 2278 |
c | 0 | 0.0082 | 0.009 | 111.4 | 395.16 | 568 |
c | 0 | 0.0065 | 0.0072 | 108.88 | 391.11 | 1366 |
c | 0 | 0.0065 | 0.0036 | 164.4 | 285.14 | 4647 |
c | 0 | 0.0041 | 0.009 | 57.28 | 430.58 | 1181 |
d | 22.5 | 0.0082 | 0.018 | 184.9 | 554.4 | 215 |
e | 45 | 0.0055 | 0.006 | 214.82 | 419.13 | 1631 |
e | 45 | 0.0082 | 0.018 | 189 | 542.51 | 191 |
f | 90 | 0.0041 | 0.009 | 156.27 | 456.77 | 678 |
f | 90 | 0.0055 | 0.006 | 204.27 | 432.96 | 1617 |
f | 90 | 0.0066 | 0.0077 | 267.5 | 514.23 | 435 |
Appendix B. Experimental Results for 7075-T651 Aluminum Alloy [45]
Loading Paths | Phase Angle (°) | (MPa) | (MPa) | (MPa) | (MPa) | (Cycles) |
a | - | 0 | 315 | 0 | 0 | 35,094 |
a | - | 0 | 260 | 0 | 0 | 181,817 |
a | - | 0 | 260 | 0 | 0 | 159,980 |
a | - | 0 | 235 | 0 | 0 | 848,760 |
a | - | 0 | 215 | 0 | 0 | 1,187,357 |
g | - | 0 | 203.04 | 0 | 228.95 | 87,622 |
g | - | 0 | 183.54 | 0 | 206.96 | 198,247 |
g | - | 0 | 183.3 | 0 | 206.69 | 175,509 |
g | - | 0 | 183.3 | 0 | 206.69 | 139,329 |
g | - | 0 | 183.3 | 0 | 206.69 | 391,636 |
g | - | 0 | 181.42 | 0 | 204.56 | 660,226 |
g | - | 0 | 178.6 | 0 | 201.38 | 373,265 |
g | - | 0 | 169.2 | 0 | 190.79 | 1,134,075 |
h | - | 105.83 | 0 | 119.34 | 0 | 730,491 |
h | - | 117.16 | 0 | 132.11 | 0 | 243,363 |
b | - | 117 | 0 | 0 | 0 | 2,159,208 |
b | - | 117 | 0 | 0 | 0 | 1,217,964 |
h | - | 151.33 | 0 | 170.64 | 0 | 19,500 |
h | - | 151.33 | 0 | 170.64 | 0 | 46,893 |
b | - | 151.33 | 0 | 0 | 0 | 207,445 |
b | - | 151.33 | 0 | 0 | 0 | 187,227 |
h | - | 139.12 | 0 | 156.88 | 0 | 209,904 |
h | - | 134.24 | 0 | 151.38 | 0 | 210,668 |
h | - | 134.24 | 0 | 151.38 | 0 | 310,472 |
h | - | 134.24 | 0 | 151.38 | 0 | 246,343 |
b | - | 134.24 | 0 | 0 | 0 | 1,179,056 |
b | - | 180 | 0 | 0 | 0 | 56,421 |
b | - | 210 | 0 | 0 | 0 | 13,630 |
b | - | 210 | 0 | 0 | 0 | 23,898 |
i | 0 | 83.47 | 144.57 | 94.12 | 163.02 | 65,046 |
i | 0 | 83.47 | 144.57 | 94.12 | 163.02 | 53,058 |
i | 0 | 75.02 | 129.94 | 84.6 | 146.53 | 270,597 |
i | 0 | 75.02 | 129.94 | 84.6 | 146.53 | 126,438 |
j | 30 | 79.63 | 137.92 | 89.79 | 155.53 | 178,920 |
j | 30 | 75.02 | 129.94 | 84.6 | 146.53 | 747,389 |
j | 30 | 88.26 | 152.87 | 99.53 | 172.39 | 36,299 |
j | 30 | 88.26 | 152.87 | 99.53 | 172.39 | 37,525 |
j | 30 | 83.47 | 144.57 | 94.12 | 163.02 | 64,231 |
j | 30 | 83.47 | 144.57 | 94.12 | 163.02 | 82,445 |
k | 45 | 75.02 | 129.94 | 84.6 | 146.53 | 934,710 |
k | 45 | 75.02 | 129.94 | 84.6 | 146.53 | 1,527,482 |
k | 45 | 79.63 | 137.92 | 89.79 | 155.53 | 89,852 |
k | 45 | 83.47 | 144.57 | 94.12 | 163.02 | 167,161 |
k | 45 | 83.47 | 144.57 | 94.12 | 163.02 | 56,775 |
k | 45 | 85.39 | 147.89 | 96.29 | 166.77 | 105,315 |
l | 90 | 83.47 | 144.57 | 94.12 | 163.02 | 94,718 |
l | 90 | 83.47 | 144.57 | 94.12 | 163.02 | 70,333 |
l | 90 | 88.26 | 152.87 | 99.53 | 172.39 | 51,324 |
l | 90 | 88.26 | 152.87 | 99.53 | 172.39 | 68,455 |
l | 90 | 88.26 | 152.87 | 99.53 | 172.39 | 147180 |
m | 180 | 83.47 | 144.57 | 94.12 | 163.02 | 78,147 |
m | 180 | 81.55 | 141.25 | 91.96 | 159.28 | 102,512 |
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Parameters | Settings |
---|---|
Neurons at the input layer | 4 |
Neurons at the hidden layer | 9 |
Neurons at the output layer | 1 |
Activation function of the hidden layer | logsig |
Activation function of the output layer | tansig |
Training function | trainlm |
Learning rate | 0.01 |
Properties | Symbols | S45C [44] | 7075-T651 [45,48] |
---|---|---|---|
Modulus of elasticity | E (GPa) | 186 | 71.7 |
Shear modulus | G (GPa) | 73 | 27.5 |
Yield strength | (MPa) | 496 | 501 |
Ultimate tensile strength | (MPa) | 770 | 561 |
Fatigue strength coefficient | (MPa) | 1206 * | 1235 |
Fatigue ductility coefficient | 0.29 * | 0.243 | |
Fatigue strength exponent | b | −0.09 * | −0.138 |
Fatigue ductility exponent | c | −0.56 * | −0.71 |
Shear fatigue strength coefficient | (MPa) | 696 * | 797 |
Shear fatigue ductility coefficient | 0.5 * | 5.42 | |
Shear fatigue strength exponent | b0 | −0.09 * | −0.126 |
Shear fatigue ductility exponent | c0 | −0.56 * | −1.173 |
Scatter Band | SWT | SHD | WYT | BPNN |
---|---|---|---|---|
±3 | 66.67% | 95.83% | 100% | 100% |
±2 | 50% | 62.5% | 91.67% | 100% |
Scatter Band | SWT | SHD | WYT | BPNN |
---|---|---|---|---|
±3 | 53.85% | 57.69% | 75% | 100% |
±2 | 32.69% | 36.54% | 48.08% | 86.54% |
Scatter Band | SWT | SHD | WYT | BPNN |
---|---|---|---|---|
±3 | 42% | 94.74% | 94.74% | 100% |
±2 | 10.53% | 68.42% | 89.47% | 73.68 |
Specimen No. | (g2/Hz) | (g2/Hz) | fmin (Hz) | fmid (Hz) | fmax (Hz) | Excitation Direction | Experimental Lives (s) |
---|---|---|---|---|---|---|---|
1 | 0.025 | 0.025 | 10 | - | 400 | Z | 13,032 |
2 | 0.05 | 0.05 | 10 | - | 400 | Z | 2196 |
3 | 0.075 | 0.075 | 10 | - | 400 | Z | 1326 |
4 | 0.02 | 0.15 | 10 | 150 | 350 | Z | 15,210 |
5 | 0.05 | 0.15 | 10 | 150 | 350 | Z | 1380 |
6 | 0.075 | 0.15 | 10 | 150 | 350 | Z | 1032 |
7 | 0.03 | 0.025 | 10 | 150 | 350 | Z | 7974 |
8 | 0.06 | 0.025 | 10 | 150 | 350 | Z | 1710 |
9 | 0.08 | 0.025 | 10 | 150 | 350 | Z | 534 |
First-Order Natural Frequency | Second-Order Natural Frequency | ||
---|---|---|---|
Experiment Frequency | Simulation Frequency | Experiment Frequency | Simulation Frequency |
30.7 Hz | 30.4 Hz | 262.2 Hz | 269.2 Hz |
Scatter Band | SWT | SHD | WYT | BPNN |
---|---|---|---|---|
±3 | 55.56% | 0% | 66.67% | 100% |
±2 | 0% | 0% | 44.44% | 66.67% |
Scatter Band | BPNN Model with Two Input Parameters () | BPNN Model with Three Input Parameters () | BPNN Model with Four Input Parameters () |
---|---|---|---|
±3 | 100% | 100% | 100% |
±2 | 100% | 100% | 100% |
Scatter band | BPNN Model with Two Input Parameters () | BPNN Model with Three Input Parameters () | BPNN Model with Four Input Parameters () |
---|---|---|---|
±3 | 94.23% | 88.46% | 100% |
±2 | 69.23% | 75% | 82.69 |
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Wang, Y.; Qiu, Y.; Li, J.; Bai, J. Two Fatigue Life Prediction Models Based on the Critical Plane Theory and Artificial Neural Networks. Metals 2024, 14, 938. https://doi.org/10.3390/met14080938
Wang Y, Qiu Y, Li J, Bai J. Two Fatigue Life Prediction Models Based on the Critical Plane Theory and Artificial Neural Networks. Metals. 2024; 14(8):938. https://doi.org/10.3390/met14080938
Chicago/Turabian StyleWang, Yantian, Yuanying Qiu, Jing Li, and Jin Bai. 2024. "Two Fatigue Life Prediction Models Based on the Critical Plane Theory and Artificial Neural Networks" Metals 14, no. 8: 938. https://doi.org/10.3390/met14080938