Analytical Solutions Based on Fourier Cosine Series for the Free Vibrations of Functionally Graded Material Rectangular Mindlin Plates
Abstract
1. Introduction
2. Methodology
2.1. Material Models
2.2. Governing Equations and Boundary Conditions
- Simply supported: ;
- Clamped: = = = = = 0, and
- Free: .
2.3. Series Solutions
3. Convergence Studies and Comparisons
4. Numerical Results
- The constraint increases when a free boundary condition changes to a simply supported boundary condition. The constraint further increases in a clamped boundary condition. Higher constraint results in higher plate stiffness and larger natural frequencies. Therefore, > > > and > > > > > (where the subscripts indicate the boundary conditions) if the first six rigid body modes with zero frequencies are considered for plates with FFFF boundary conditions.
- The Mori–Tanaka material model provides a larger Young’s modulus than the power-law material model does; however, both models yield the same density distribution (Figure 2). Consequently, FGM plates following the Mori-Tanaka material model have larger natural frequencies than those following the power-law material model.
- No in-plane displacement-dominated mode exists in the first six modes for thin square plates with h/a = 0.02; however, such a mode may exist for moderately thick plates with h/a = 0.1.
- The nondimensional frequencies () of plates with h/a = 0.1 are less than those of plates with h/a = 0.02 because h/a is involved in the definition of . When converting to , one finds that the trend is opposite for because the plate rigidity increases with h/a.
5. Concluding Remarks
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
Appendix A
References
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Material | Properties | ||
---|---|---|---|
E (GPa) | Poisson’s Ratio (ν) | (kg/m3) | |
Aluminum (Al) | 70.0 | 0.3 | 2702 |
Alumina (Al2O3) | 380 | 0.3 | 3800 |
Zirconia (ZrO2) | 200 | 0.3 | 5700 |
Material Model | Material Ingredient | Mode | Exact Closed-Form Sol. | Published | ||||||
---|---|---|---|---|---|---|---|---|---|---|
5 | 10 | 15 | 25 | 30 | 35 | |||||
Power-Law | Al/Al2O3 | 1 | 4.510 | 4.433 | 4.422 | 4.419 | 4.418 | 4.418 | 4.419 (1,1) | <4.420> (4.347) |
2 | 11.03 | 10.63 | 10.60 | 10.60 | 10.58 | 10.58 | 10.59 (1,2) | <10.59> (10.42) | ||
3 | 11.03 | 10.63 | 10.60 | 10.56 | 10.58 | 10.58 | 10.59 (2,1) | </> (10.42) | ||
4 * | 16.22 | 16.20 | 16.20 | 16.20 | 16.20 | 16.20 | 16.20 (1,0) | <×> (15.94) | ||
5 * | 16.22 | 16.20 | 16.20 | 16.20 | 16.20 | 16.20 | 16.20 (0,1) | <×> (/) | ||
6 | 16.90 | 16.34 | 16.31 | 16.30 | 16.30 | 16.30 | 16.31 (2,2) | <16.31> (/) | ||
M-T | Al/ZrO2 | 1 | 5.288 | 5.205 | 5.193 | 5.190 | 5.190 | 5.190 | 5.192 (1,1) | {5.096} |
2 | 12.90 | 12.45 | 12.42 | 12.41 | 12.41 | 12.41 | 12.41 (1,2) | {12.30} | ||
3 | 12.90 | 12.45 | 12.42 | 12.41 | 12.41 | 12.41 | 12.41 (2,1) | {12.30} | ||
4 * | 18.10 | 18.09 | 18.08 | 18.08 | 18.08 | 18.08 | 18.08 (1,0) | {17.49} | ||
5 * | 18.10 | 18.09 | 18.08 | 18.08 | 18.08 | 18.08 | 18.08 (0,1) | {17.49} | ||
6 | 19.74 | 19.12 | 19.08 | 19.07 | 19.06 | 19.06 | 19.09 (2,2) | {18.87} |
Mode | Published | |||||||
---|---|---|---|---|---|---|---|---|
5 | 10 | 15 | 25 | 30 | 35 | |||
0 | 1 | 8.183 | 8.079 | 8.073 | 8.071 | 8.071 | 8.070 | {8.070} <8.070> |
2 | 15.37 | 14.91 | 14.88 | 14.87 | 14.86 | 14.86 | {14.86} <14.86> | |
3 | 18.25 | 17.95 | 17.93 | 17.92 | 17.92 | 17.92 | {17.92} <17.92> | |
4 * | 19.50 | 19.49 | 19.48 | 19.48 | 19.48 | 19.48 | [19.48] < × > | |
5 | 24.49 | 23.91 | 23.87 | 23.85 | 23.85 | 23.85 | {23.85} <23.85> | |
6 | 27.72 | 26.44 | 26.32 | 26.29 | 26.29 | 26.28 | {26.28} </> | |
1 | 1 | 6.320 | 6.228 | 6.223 | 6.221 | 6.221 | 6.221 | <6.220> |
2 | 11.89 | 11.51 | 11.48 | 11.47 | 11.47 | 11.47 | <11.47> | |
3 | 14.17 | 13.94 | 13.92 | 13.91 | 13.91 | 13.91 | <13.92> | |
4 * | 16.22 | 16.20 | 16.20 | 16.20 | 16.20 | 16.20 | <×> | |
5 | 19.05 | 18.57 | 18.54 | 18.53 | 18.53 | 18.53 | <18.54> | |
6 | 21.62 | 20.48 | 20.38 | 20.35 | 20.35 | 20.35 | </> |
Mode | Published | |||||||
---|---|---|---|---|---|---|---|---|
5 | 10 | 15 | 25 | 30 | 35 | |||
0 | 1 | 1.039 | 1.038 | 1.038 | 1.038 | 1.038 | 1.038 | {1.038} (1.030) |
2 | 2.399 | 2.428 | 2.435 | 2.438 | 2.439 | 2.439 | {2.440} (2.391) | |
3 | 6.134 | 6.082 | 6.079 | 6.079 | 6.079 | 6.079 | {6.080} (6.005) | |
4 * | 6.548 | 6.576 | 6.578 | 6.580 | 6.581 | 6.581 | {/} (7.636) | |
5 | 7.742 | 7.702 | 7.712 | 7.715 | 7.716 | 7.716 | {7.716} (/) | |
6 | 8.417 | 8.518 | 8.533 | 8.544 | 8.545 | 8.546 | {8.548} (/) | |
5 | 1 | 0.6833 | 0.6826 | 0.6827 | 0.6828 | 0.6828 | 0.6828 | (0.6768) |
2 | 1.575 | 1.594 | 1.599 | 1.601 | 1.601 | 1.601 | (1.568) | |
3 | 4.017 | 3.983 | 3.981 | 3.981 | 3.981 | 3.981 | (3.927) | |
4 * | 4.253 | 4.272 | 4.273 | 4.274 | 4.274 | 4.275 | (4.263) | |
5 | 5.065 | 5.039 | 5.045 | 5.047 | 5.047 | 5.047 | (/) | |
6 | 5.510 | 5.577 | 5.586 | 5.593 | 5.594 | 5.594 | (/) |
Mode | Published | |||||||
---|---|---|---|---|---|---|---|---|
5 | 15 | 25 | 35 | 40 | 45 | |||
0 | 1 | 3.823 | 3.842 | 3.846 | 3.847 | 3.849 | 3.849 | {3.849} |
2 | 6.921 | 5.794 | 5.745 | 5.737 | 5.736 | 5.736 | {5.733} | |
3 | 7.821 | 7.091 | 7.064 | 7.059 | 7.060 | 7.060 | {7.058} | |
4 | 10.08 | 9.665 | 9.656 | 9.655 | 9.660 | 9.660 | {9.660} | |
5 | 10.08 | 9.665 | 9.656 | 9.655 | 9.660 | 9.660 | {9.660} | |
6 | 16.93 | 16.76 | 16.74 | 16.74 | 16.75 | 16.75 | {16.75} | |
5 | 1 | 2.508 | 2.521 | 2.523 | 2.524 | 2.524 | 2.524 | (2.512) |
2 | 4.516 | 3.790 | 3.759 | 3.753 | 3.752 | 3.752 | (3.746) | |
3 | 5.111 | 4.640 | 4.623 | 4.620 | 4.620 | 4.619 | (4.608) | |
4 | 6.579 | 6.314 | 6.309 | 6.308 | 6.308 | 6.308 | (6.270) | |
5 | 6.579 | 6.314 | 6.309 | 6.308 | 6.308 | 6.308 | (6.270) | |
6 | 11.03 | 10.92 | 10.91 | 10.91 | 10.91 | 10.91 | (/) |
Case | SFSF | SSSF | SCSF | SCSS | SFFF | SSFF | CSFF | CSSF |
Ave. Differences (%) | 0.080 | 0.045 | 0.040 | 0.054 | 0.088 | 0.045 | 0.056 | 0.030 |
Case | CFSF | CFCF | CSCF | CCFF | CCSF | CCSS | CCCF | CCCS |
Ave. Differences (%) | 0.056 | 0.044 | 0.020 | 0.055 | 0.028 | 0.045 | 0.024 | 0.049 |
b/a | h/a | Mode | ||||||
---|---|---|---|---|---|---|---|---|
1 | 2 | 3 | 4 | 5 | 6 | |||
1 | 0.02 | 0 | 10.84 | 22.03 | 22.03 | 32.36 | 39.29 | 39.49 |
0.5 | 9.184 | 18.67 | 18.67 | 27.44 | 33.32 | 33.48 | ||
2 | 7.527 | 15.30 | 15.30 | 22.49 | 27.30 | 27.44 | ||
5 | 7.133 | 14.49 | 14.49 | 21.29 | 25.84 | 25.97 | ||
0.1 | 0 | 9.842 | 18.77 | 18.77 | 26.31 | 31.00 | 31.30 | |
0.5 | 8.409 | 16.11 | 16.11 | 22.64 | 26.73 | 26.98 | ||
2 | 6.902 | 13.23 | 13.23 | 18.58 | 21.94 | 22.15 | ||
5 | 6.451 | 12.27 | 12.27 | 17.15 | 20.18 | 20.38 | ||
2 | 0.02 | 0 | 7.413 | 9.593 | 13.48 | 19.05 | 19.23 | 21.34 |
0.5 | 6.281 | 8.128 | 11.43 | 16.14 | 16.29 | 18.09 | ||
2 | 5.148 | 6.662 | 9.363 | 13.23 | 13.35 | 14.83 | ||
5 | 4.879 | 6.313 | 8.872 | 12.53 | 12.65 | 14.04 | ||
0.1 | 0 | 6.897 | 8.815 | 12.16 | 16.64 | 16.75 | 18.30 | |
0.5 | 5.882 | 7.523 | 10.39 | 14.27 | 14.33 | 15.70 | ||
2 | 4.827 | 6.173 | 8.521 | 11.71 | 11.75 | 12.88 | ||
5 | 4.526 | 5.779 | 7.960 | 10.88 | 10.95 | 11.95 |
b/a | h/a | Mode | ||||||
---|---|---|---|---|---|---|---|---|
1 | 2 | 3 | 4 | 5 | 6 | |||
1 | 0.02 | 0 | 4.038 | 5.976 | 7.361 | 10.44 | 10.44 | 18.41 |
0.5 | 3.421 | 5.067 | 6.238 | 8.849 | 8.849 | 15.59 | ||
2 | 2.804 | 4.153 | 5.112 | 7.252 | 7.252 | 12.78 | ||
5 | 2.657 | 3.932 | 4.843 | 6.869 | 6.869 | 12.11 | ||
0.1 | 0 | 3.849 | 5.735 | 7.060 | 9.660 | 9.660 | 16.75 | |
0.5 | 3.269 | 4.861 | 5.988 | 8.213 | 8.213 | 14.25 | ||
2 | 2.677 | 3.969 | 4.893 | 6.711 | 6.711 | 11.63 | ||
5 | 2.525 | 3.752 | 4.621 | 6.312 | 6.312 | 10.91 | ||
2 | 0.02 | 0 | 1.656 | 1.997 | 4.389 | 4.511 | 6.690 | 7.604 |
0.5 | 1.405 | 1.693 | 3.719 | 3.821 | 5.670 | 6.442 | ||
2 | 1.152 | 1.387 | 3.048 | 3.132 | 4.647 | 5.280 | ||
5 | 1.090 | 1.315 | 2.888 | 2.969 | 4.401 | 5.004 | ||
0.1 | 0 | 1.610 | 1.927 | 4.196 | 4.382 | 6.419 | 7.176 | |
0.5 | 1.364 | 1.636 | 3.563 | 3.717 | 5.443 | 6.099 | ||
2 | 1.117 | 1.341 | 2.918 | 3.042 | 4.446 | 4.991 | ||
5 | 1.058 | 1.267 | 2.753 | 2.875 | 4.201 | 4.700 |
BC | h/a | Material Model | Mode | ||||||
---|---|---|---|---|---|---|---|---|---|
1 | 2 | 3 | 4 | 5 | 6 | ||||
CFFF | 0.02 | 0 | Power-Law | 1.049 | 2.552 | 6.418 | 8.180 | 9.273 | 16.17 |
0.5 | 0.8888 | 2.162 | 5.437 | 6.930 | 7.857 | 13.70 | |||
2 | 0.7284 | 1.772 | 4.456 | 5.678 | 6.439 | 11.23 | |||
5 | 0.6907 | 1.680 | 4.224 | 5.383 | 6.102 | 10.63 | |||
0.1 | 0 | Power-Law or M-T | 1.038 | 2.439 | 6.079 | 6.581 * | 7.716 | 8.546 | |
0.5 | Power-Law | 0.8000 | 2.072 | 5.168 | 5.907 * | 6.555 | 7.280 | ||
M-T | 0.8089 | 1.960 | 4.879 | 5.606 * | 6.186 | 6.877 | |||
2 | Power-Law | 0.7211 | 1.698 | 4.230 | 4.946 * | 5.359 | 5.962 | ||
M-T | 0.6973 | 1.643 | 4.087 | 4.650 * | 5.176 | 5.759 | |||
5 | Power-Law | 0.6828 | 1.601 | 3.981 | 4.275 * | 5.047 | 5.594 | ||
M-T | 0.6666 | 1.564 | 3.886 | 4.073 * | 4.926 | 5.457 | |||
CFSF | 0.02 | 0 | Power-Law | 4.591 | 6.191 | 11.91 | 14.90 | 16.90 | 23.15 |
0.5 | 3.664 | 4.950 | 9.511 | 11.89 | 13.50 | 18.48 | |||
2 | 3.084 | 4.162 | 8.000 | 10.00 | 11.36 | 15.55 | |||
5 | 2.950 | 3.981 | 7.652 | 9.569 | 10.86 | 14.87 | |||
0.1 | 0 | 4.401 | 5.820 | 10.89 | 13.45 | 15.05 | 15.57 * | ||
0.5 | 3.526 | 4.678 | 8.742 | 10.83 | 12.14 | 13.27 * | |||
2 | 2.963 | 3.922 | 7.316 | 9.071 | 10.16 | 10.99 * | |||
5 | 2.818 | 3.724 | 6.935 | 8.563 | 9.569 | 9.641 * |
BC | Mode | ||||||
---|---|---|---|---|---|---|---|
1 | 2 | 3 | 4 | 5 | 6 | ||
SSFF | 0 | 0.9943 | 5.011 | 5.600 | 10.63 | 12.16 * | 14.10 |
0.5 | 0.8432 | 4.254 | 4.754 | 9.044 | 10.92 * | 12.00 | |
2 | 0.6910 | 3.481 | 3.891 | 7.398 | 9.134 * | 9.806 | |
5 | 0.6538 | 3.286 | 3.672 | 6.951 | 7.896 * | 9.204 | |
CSFF | 0 | 1.571 | 5.472 | 6.977 | 8.176 * | 11.71 | 14.51 |
0.5 | 1.333 | 4.648 | 5.936 | 7.340 | 9.984 | 12.35 | |
2 | 1.093 | 3.804 | 4.860 | 6.148 * | 8.171 | 10.10 | |
5 | 1.033 | 3.585 | 4.569 | 5.313 * | 7.653 | 9.470 | |
CSSF | 0 | 4.837 | 8.710 | 13.92 | 16.36 * | 17.23 | 17.83 |
0.5 | 4.113 | 7.416 | 11.89 | 14.63 * | 14.74 | 15.25 | |
2 | 3.372 | 6.072 | 9.747 | 11.98 | 12.31 * | 12.49 | |
5 | 3.175 | 5.703 | 9.106 | 10.60 * | 11.23 | 11.64 | |
CCFF | 0 | 2.019 | 6.700 | 7.481 | 12.71 | 15.41 * | 16.62 |
0.5 | 1.714 | 5.703 | 6.369 | 10.85 | 13.80 * | 14.22 | |
2 | 1.405 | 4.668 | 5.217 | 8.883 | 11.42 * | 11.78 | |
5 | 1.327 | 4.385 | 4.900 | 8.300 | 9.983 * | 10.87 | |
CCCF | 0 | 6.685 | 10.78 | 16.41 | 19.75 | 20.31 | 23.79 * |
0.5 | 5.701 | 9.207 | 14.01 | 16.90 | 17.43 | 21.36 * | |
2 | 4.678 | 7.546 | 11.55 | 13.83 | 14.29 | 17.88 * | |
5 | 4.385 | 7.050 | 10.72 | 12.85 | 13.25 | 15.46 * |
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Huang, C.-S.; Huang, S.H. Analytical Solutions Based on Fourier Cosine Series for the Free Vibrations of Functionally Graded Material Rectangular Mindlin Plates. Materials 2020, 13, 3820. https://doi.org/10.3390/ma13173820
Huang C-S, Huang SH. Analytical Solutions Based on Fourier Cosine Series for the Free Vibrations of Functionally Graded Material Rectangular Mindlin Plates. Materials. 2020; 13(17):3820. https://doi.org/10.3390/ma13173820
Chicago/Turabian StyleHuang, Chiung-Shiann, and S. H. Huang. 2020. "Analytical Solutions Based on Fourier Cosine Series for the Free Vibrations of Functionally Graded Material Rectangular Mindlin Plates" Materials 13, no. 17: 3820. https://doi.org/10.3390/ma13173820
APA StyleHuang, C.-S., & Huang, S. H. (2020). Analytical Solutions Based on Fourier Cosine Series for the Free Vibrations of Functionally Graded Material Rectangular Mindlin Plates. Materials, 13(17), 3820. https://doi.org/10.3390/ma13173820