The Application of Fractional Derivative Viscoelastic Models in the Finite Element Method: Taking Several Common Models as Examples
Abstract
:1. Introduction
2. Finite Element Method
2.1. Fractional Derivative Viscoelastic Model
2.2. Classical Viscoelastic Model
2.3. Linear Elasticity Model
3. Finite Element Calculation under Static Load
3.1. Convergence Analysis
3.2. Numerical Solution Verification
3.3. Finite Element Analysis under Several Fractional Derivative and Classical Viscoelastic Models
3.3.1. Several Classical Viscoelastic Models
3.3.2. Several Fractional Derivative Viscoelastic Models
3.3.3. FTS and Three-Parameter Solid Model
3.3.4. FTS under Different Temperatures
3.3.5. FTS under Different Gradations
4. Finite Element Calculation under Dynamic Load
4.1. Convergence Analysis
4.2. Numerical Solution Verification
4.3. Finite Element Analysis under Several Fractional Derivative and Classical Viscoelastic Models
4.3.1. Several Classical Viscoelastic Models
4.3.2. Several Fractional Derivative Viscoelastic Models
4.3.3. FTS and Three-Parameter Solid Model
4.3.4. FTS under Different Temperatures
4.3.5. FTS under Different Gradations
5. Finite Element Calculation under Moving Load
5.1. Convergence Analysis
5.2. Numerical Solution Verification
5.3. Finite Element Analysis under Several Fractional Derivative and Classical Viscoelastic Models
5.3.1. Several Classical Viscoelastic Models
5.3.2. Several Fractional Derivative Viscoelastic Models
5.3.3. FTS and Three-Parameter Solid Model
5.3.4. FTS under Different Temperatures
5.3.5. FTS under Different Gradations
6. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Appendix A
References
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Parameter Types | FK | FM | FTS |
---|---|---|---|
Elastic modulus | |||
Poisson’s ratio |
Asphalt Mixture | Temperature (°C) | Fitting Parameters | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
Kelvin Model | Kelvin Model | Three-Parameter Solid Model | |||||||||
E | η | R2 | E | η | R2 | E1 | E2 | η | R2 | ||
ARHM-13 | 15 | 4929.89 | 2.69 | 0.4646 | 12,602.79 | 957.70 | 0.8462 | 3115.43 | 14,552.65 | 43.08 | 0.9262 |
30 | 1683.62 | 1.04 | 0.5375 | 4604.64 | 299.60 | 0.8195 | 1031.70 | 5526.33 | 13.03 | 0.9408 | |
45 | 589.37 | 0.32 | 0.5532 | 1461.25 | 108.33 | 0.7521 | 433.68 | 1755.65 | 4.24 | 0.9391 | |
60 | 341.00 | 0.16 | 0.5339 | 736.44 | 67.92 | 0.6305 | 305.63 | 867.06 | 2.45 | 0.9332 | |
ARHM-20 | 15 | 6659.82 | 3.21 | 0.4270 | 15,638.63 | 1395.49 | 0.8229 | 4740.86 | 17,698.89 | 61.16 | 0.9119 |
30 | 2345.27 | 1.31 | 0.5056 | 5999.82 | 440.16 | 0.8142 | 1553.55 | 7061.87 | 18.81 | 0.9335 | |
45 | 1169.86 | 0.64 | 0.5284 | 2906.24 | 218.61 | 0.7767 | 835.69 | 3451.10 | 8.86 | 0.9358 | |
60 | 522.61 | 0.24 | 0.5192 | 1123.76 | 105.48 | 0.6409 | 466.50 | 1314.37 | 3.86 | 0.9306 | |
ARHM-25 | 15 | 8771.20 | 4.18 | 0.4126 | 20,531.81 | 1861.68 | 0.8319 | 6191.55 | 23,092.45 | 83.16 | 0.9087 |
30 | 2933.94 | 1.70 | 0.5124 | 7734.54 | 537.87 | 0.8239 | 1856.92 | 9148.62 | 23.51 | 0.9369 | |
45 | 1634.33 | 0.95 | 0.5394 | 4249.58 | 295.72 | 0.7941 | 1095.27 | 5088.36 | 12.33 | 0.9398 | |
60 | 693.00 | 0.33 | 0.5386 | 1557.14 | 134.24 | 0.6765 | 586.32 | 1843.44 | 4.97 | 0.9355 |
Asphalt Mixture | Temperature (°C) | Fitting Parameters | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
FK | FM | FTS | ||||||||||||
E | η | α | R2 | E | η | α | R2 | E1 | E2 | η | α | R2 | ||
ARHM-13 | 15 | 0.00 | 3725.60 | 0.1648 | 0.9979 | 4.83 × 104 | 3893.66 | 0.2001 | 1.0000 | 123.53 | 4.37 × 104 | 3782.63 | 0.2081 | 1.0000 |
30 | 78.77 | 1113.10 | 0.1898 | 0.9999 | 1.09 × 1026 | 1189.09 | 0.1834 | 0.9997 | 165.58 | 3.96 × 104 | 1042.49 | 0.2133 | 1.0000 | |
45 | 132.81 | 304.55 | 0.2000 | 1.0000 | 5.06 × 1034 | 438.42 | 0.1641 | 0.9950 | 133.84 | 4.15 × 1022 | 303.35 | 0.2004 | 0.9997 | |
60 | 116.76 | 153.58 | 0.1902 | 0.9999 | 2.57 × 1020 | 279.65 | 0.1328 | 0.9877 | 115.29 | 3.11 × 1029 | 155.46 | 0.1886 | 0.9999 | |
ARHM-20 | 15 | 0.00 | 5262.83 | 0.1475 | 0.9972 | 5.12 × 104 | 5754.83 | 0.1853 | 1.0000 | 109.51 | 4.85 × 104 | 5665.14 | 0.1901 | 1.0000 |
30 | 53.81 | 1686.00 | 0.1708 | 0.9999 | 2.76 × 106 | 1741.71 | 0.1680 | 0.9998 | 219.34 | 4.06 × 104 | 1566.38 | 0.1994 | 1.0000 | |
45 | 171.29 | 702.53 | 0.1845 | 1.0000 | 1.40 × 1020 | 876.18 | 0.1631 | 0.9982 | 184.66 | 9.78 × 104 | 694.44 | 0.1898 | 1.0000 | |
60 | 161.27 | 255.52 | 0.1809 | 1.0000 | 1.04 × 1019 | 429.57 | 0.1317 | 0.9908 | 154.04 | 5.09 × 1020 | 263.83 | 0.1776 | 0.9999 | |
ARHM-25 | 15 | 0.00 | 6911.07 | 0.1473 | 0.9954 | 5.51 × 104 | 7768.56 | 0.1952 | 1.0000 | 77.09 | 5.39 × 104 | 7703.05 | 0.1978 | 1.0000 |
30 | 30.18 | 2105.14 | 0.1757 | 0.9998 | 2.25 × 105 | 2147.92 | 0.1783 | 0.9998 | 281.09 | 4.36 × 104 | 1909.56 | 0.2117 | 1.0000 | |
45 | 204.71 | 983.34 | 0.1917 | 1.0000 | 3.76 × 1044 | 1189.39 | 0.1730 | 0.9986 | 248.75 | 6.20 × 105 | 952.45 | 0.2047 | 1.0000 | |
60 | 214.14 | 325.49 | 0.1930 | 1.0000 | 1.51 × 1016 | 553.09 | 0.1417 | 0.9901 | 215.54 | 1.05 × 1034 | 323.85 | 0.1935 | 0.9999 |
Frequency (Hz) | α | |||||
---|---|---|---|---|---|---|
0 | 0.00001 | 0.0001 | 0.001 | 0.01 | 0.1 | |
1/32 | 1.41523 | 1.41524 | 1.41528 | 1.41573 | 1.42019 | 1.46471 |
1/16 | 1.41523 | 1.41524 | 1.41527 | 1.41557 | 1.41855 | 1.44843 |
1/8 | 1.41523 | 1.41524 | 1.41525 | 1.41540 | 1.41692 | 1.43207 |
1/4 | 1.41523 | 1.41523 | 1.41523 | 1.41524 | 1.41528 | 1.41568 |
1/2 | 1.41523 | 1.41523 | 1.41522 | 1.41508 | 1.41365 | 1.39928 |
1 | 1.41523 | 1.41523 | 1.41520 | 1.41491 | 1.41201 | 1.38292 |
2 | 1.41524 | 1.41523 | 1.41519 | 1.41475 | 1.41038 | 1.36664 |
4 | 1.41526 | 1.41525 | 1.41519 | 1.41461 | 1.40876 | 1.35049 |
Frequency (Hz) | α | |||||||
---|---|---|---|---|---|---|---|---|
0 | 1 × 10−10 | 1 × 10−9 | 1 × 10−8 | 1 × 10−7 | 1 × 10−6 | 1 × 10−5 | 1 × 10−4 | |
1/32 | 2.1431 × 10−7 | 2.1435 × 10−7 | 2.1474 × 10−7 | 2.1864 × 10−7 | 2.5762 × 10−7 | 6.4742 × 10−7 | 4.5454 × 10−6 | 4.3525 × 10−5 |
1/16 | 2.1593 × 10−7 | 2.1597 × 10−7 | 2.1636 × 10−7 | 2.2026 × 10−7 | 2.5924 × 10−7 | 6.4904 × 10−7 | 4.5470 × 10−6 | 4.3537 × 10−5 |
1/8 | 2.2019 × 10−7 | 2.2023 × 10−7 | 2.2062 × 10−7 | 2.2452 × 10−7 | 2.6350 × 10−7 | 6.5329 × 10−7 | 4.5513 × 10−6 | 4.3531 × 10−5 |
1/4 | 2.1555 × 10−7 | 2.1560 × 10−7 | 2.1599 × 10−7 | 2.1988 × 10−7 | 2.5886 × 10−7 | 6.4866 × 10−7 | 4.5467 × 10−6 | 4.3527 × 10−5 |
1/2 | 2.0931 × 10−7 | 2.0935 × 10−7 | 2.0974 × 10−7 | 2.1364 × 10−7 | 2.5262 × 10−7 | 6.4242 × 10−7 | 4.5404 × 10−6 | 4.3520 × 10−5 |
1 | 1.8919 × 10−7 | 1.8924 × 10−7 | 1.8963 × 10−7 | 1.9353 × 10−7 | 2.3250 × 10−7 | 6.2230 × 10−7 | 4.5203 × 10−6 | 4.3500 × 10−5 |
2 | 9.7809 × 10−7 | 9.7852 × 10−7 | 9.8242 × 10−8 | 1.0214 × 10−7 | 1.4112 × 10−7 | 5.3092 × 10−7 | 4.4289 × 10−6 | 4.3409 × 10−5 |
4 | −1.7937 × 10−8 | −1.7893 × 10−8 | −1.7504 × 10−8 | −1.3605 × 10−8 | 2.5377 × 10−8 | 4.1520 × 10−7 | 4.3134 × 10−6 | 4.3296 × 10−5 |
Working Conditions | α | |||||
---|---|---|---|---|---|---|
0 | 0.00001 | 0.0001 | 0.001 | 0.01 | 0.1 | |
v = 25,000 mm/s, t = 0.01 s | 0.317347 | 0.317346 | 0.317333 | 0.3172 | 0.315874 | 0.302417 |
v = 2500 mm/s, t = 0.1 s | 0.317347 | 0.317347 | 0.317345 | 0.317322 | 0.317092 | 0.31466 |
v = 250 mm/s, t = 1 s | 0.317347 | 0.317348 | 0.317357 | 0.317444 | 0.31831 | 0.326984 |
v = 25 mm/s, t = 10 s | 0.317347 | 0.31735 | 0.317369 | 0.31757 | 0.31953 | 0.33905 |
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Zheng, G.; Zhang, N.; Lv, S. The Application of Fractional Derivative Viscoelastic Models in the Finite Element Method: Taking Several Common Models as Examples. Fractal Fract. 2024, 8, 103. https://doi.org/10.3390/fractalfract8020103
Zheng G, Zhang N, Lv S. The Application of Fractional Derivative Viscoelastic Models in the Finite Element Method: Taking Several Common Models as Examples. Fractal and Fractional. 2024; 8(2):103. https://doi.org/10.3390/fractalfract8020103
Chicago/Turabian StyleZheng, Guozhi, Naitian Zhang, and Songtao Lv. 2024. "The Application of Fractional Derivative Viscoelastic Models in the Finite Element Method: Taking Several Common Models as Examples" Fractal and Fractional 8, no. 2: 103. https://doi.org/10.3390/fractalfract8020103
APA StyleZheng, G., Zhang, N., & Lv, S. (2024). The Application of Fractional Derivative Viscoelastic Models in the Finite Element Method: Taking Several Common Models as Examples. Fractal and Fractional, 8(2), 103. https://doi.org/10.3390/fractalfract8020103