**Figure 1.**
Eshelby inclusion model of randomly oriented carbon nanotubes.

**Figure 1.**
Eshelby inclusion model of randomly oriented carbon nanotubes.

**Figure 2.**
CNTs volume fractions distributions illustration.

**Figure 2.**
CNTs volume fractions distributions illustration.

**Figure 3.**
Representation of the quadrilateral plate.

**Figure 3.**
Representation of the quadrilateral plate.

**Figure 4.**
Young′s modulus for different levels of agglomeration and CNT volume fraction.

**Figure 4.**
Young′s modulus for different levels of agglomeration and CNT volume fraction.

**Figure 5.**
Mechanical properties evolution for partially agglomerated states. (**a.1**) Young’s modulus for Vf = 0.001, (**b.1**) Poisson’s ratio for Vf = 0.001, (**a.2**) Young’s modulus for Vf = 0.005, (**b.2**) Poisson’s ratio for Vf = 0.005, (**a.3**) Young’s modulus for Vf = 0.01, (**b.3**) Poisson’s ratio for Vf = 0.01.

**Figure 5.**
Mechanical properties evolution for partially agglomerated states. (**a.1**) Young’s modulus for Vf = 0.001, (**b.1**) Poisson’s ratio for Vf = 0.001, (**a.2**) Young’s modulus for Vf = 0.005, (**b.2**) Poisson’s ratio for Vf = 0.005, (**a.3**) Young’s modulus for Vf = 0.01, (**b.3**) Poisson’s ratio for Vf = 0.01.

**Figure 6.**
Beam schematic representation.

**Figure 6.**
Beam schematic representation.

**Figure 7.**
Vibrational modes of the beam: (**a**) 1st mode, (**b**) 2nd mode, (**c**) 3rd mode, (**d**) an additional mode related to the plate natural modes between the 2nd and the 3rd modes.

**Figure 7.**
Vibrational modes of the beam: (**a**) 1st mode, (**b**) 2nd mode, (**c**) 3rd mode, (**d**) an additional mode related to the plate natural modes between the 2nd and the 3rd modes.

**Figure 8.**
Vibrational modes for UD distribution without the agglomeration effect: (**a**) 1st mode, (**b**) 2nd mode, (**c**) 4th mode, (**d**) 5th mode.

**Figure 8.**
Vibrational modes for UD distribution without the agglomeration effect: (**a**) 1st mode, (**b**) 2nd mode, (**c**) 4th mode, (**d**) 5th mode.

**Figure 9.**
Vibrational modes for USFG distribution without the agglomeration effect: (**a**) 1st mode, (**b**) 2nd mode, (**c**) 3rd mode, (**d**) 4th mode, (**e**) 5th mode.

**Figure 9.**
Vibrational modes for USFG distribution without the agglomeration effect: (**a**) 1st mode, (**b**) 2nd mode, (**c**) 3rd mode, (**d**) 4th mode, (**e**) 5th mode.

**Figure 10.**
Vibrational modes for SFG distribution without the agglomeration effect: (**a**) 1st mode, (**b**) 2nd mode, (**c**) 4th mode, (**d**) 5th mode.

**Figure 10.**
Vibrational modes for SFG distribution without the agglomeration effect: (**a**) 1st mode, (**b**) 2nd mode, (**c**) 4th mode, (**d**) 5th mode.

**Figure 11.**
Vibrational modes for the SFG distribution with η = 1 and μ = 0.25: (**a**) 1st mode, (**b**) 2nd mode, (**c**) 4th mode, (**d**) 5th mode.

**Figure 11.**
Vibrational modes for the SFG distribution with η = 1 and μ = 0.25: (**a**) 1st mode, (**b**) 2nd mode, (**c**) 4th mode, (**d**) 5th mode.

**Figure 12.**
Vibrational modes for the SFG distribution with η = 1 and μ = 0.75: (**a**) 1st mode, (**b**) 2nd mode, (**c**) 4th mode, (**d**) 5th mode.

**Figure 12.**
Vibrational modes for the SFG distribution with η = 1 and μ = 0.75: (**a**) 1st mode, (**b**) 2nd mode, (**c**) 4th mode, (**d**) 5th mode.

**Figure 13.**
Vibrational modes for the UD distribution without the agglomeration effect: (**a**) 1st mode, (**b**) 2nd mode, (**c**) 3rd mode, (**d**) 4th mode, (**e**) 5th mode.

**Figure 13.**
Vibrational modes for the UD distribution without the agglomeration effect: (**a**) 1st mode, (**b**) 2nd mode, (**c**) 3rd mode, (**d**) 4th mode, (**e**) 5th mode.

**Figure 14.**
Vibrational modes for the USFG distribution without the agglomeration effect: (**a**) 1st mode, (**b**) 2nd mode, (**c**) 3rd mode, (**d**) 4th mode, (**e**) 5th mode.

**Figure 14.**
Vibrational modes for the USFG distribution without the agglomeration effect: (**a**) 1st mode, (**b**) 2nd mode, (**c**) 3rd mode, (**d**) 4th mode, (**e**) 5th mode.

**Figure 15.**
Vibrational modes for the SFG distribution without the agglomeration effect: (**a**) 1st mode, (**b**) 2nd mode, (**c**) 3rd mode, (**d**) 4th mode, (**e**) 5th mode.

**Figure 15.**
Vibrational modes for the SFG distribution without the agglomeration effect: (**a**) 1st mode, (**b**) 2nd mode, (**c**) 3rd mode, (**d**) 4th mode, (**e**) 5th mode.

**Figure 16.**
Vibrational modes for the USFG distribution with η = 1 and μ = 0.25: (**a**) 1st mode, (**b**) 2nd mode, (**c**) 3rd mode, (**d**) 4th mode, (**e**) 5th mode.

**Figure 16.**
Vibrational modes for the USFG distribution with η = 1 and μ = 0.25: (**a**) 1st mode, (**b**) 2nd mode, (**c**) 3rd mode, (**d**) 4th mode, (**e**) 5th mode.

**Figure 17.**
Vibrational modes for the USFG distribution with η = 1 and μ = 0.5: (**a**) 1st mode, (**b**) 2nd mode, (**c**) 3rd mode, (**d**) 4th mode, (**e**) 5th mode.

**Figure 17.**
Vibrational modes for the USFG distribution with η = 1 and μ = 0.5: (**a**) 1st mode, (**b**) 2nd mode, (**c**) 3rd mode, (**d**) 4th mode, (**e**) 5th mode.

**Table 1.**
Properties of the equivalent fibre for SWCNT (10,10) [

10].

**Table 1.**
Properties of the equivalent fibre for SWCNT (10,10) [

10].

Equivalent Fibre |
---|

Longitudinal elastic modulus (GPa) | 649.12 |

Transverse elastic modulus (GPa) | 11.27 |

Transverse shear modulus (GPa) | 5.13 |

Poisson′s ratio | 0.284 |

Density (kg/m^{3}) | 1400 |

**Table 2.**
Hill’s moduli of the SWCNT [

28].

**Table 2.**
Hill’s moduli of the SWCNT [

28].

Hill’s Elastic Moduli (GPa) |
---|

${k}_{r}$ | 271 |

${l}_{r}$ | 88 |

${m}_{r}$ | 17 |

${n}_{r}$ | 1089 |

${p}_{r}$ | 442 |

Density (kg/m^{3}) | 1400 |

**Table 3.**
Convergence study for the first natural frequency of a square plate.

**Table 3.**
Convergence study for the first natural frequency of a square plate.

Mesh | a/h = 10 | a/h = 100 |
---|

Q4 (DOF) | Q9 (DOF) | Analytic Solution | Q4 (DOF) | Q9 (DOF) | Analytic Solution |
---|

5 × 5 | - | 0.9305 (605) | 0.9300 | - | 0.0963 (605) | 0.0963 |

10 × 10 | 0.9399 (605) | 0.9303 (2205) | 0.0973 (605) | 0.0963 (2205) |

15 × 15 | 0.9346 (1280) | 0.9303 (4805) | 0.0968 (1280) | 0.0963 (4805) |

20 × 20 | 0.9327 (2205) | - | 0.0965 (2205) | - |

25 × 25 | 0.9318 (3380) | - | 0.0965 (3380) | - |

**Table 4.**
Convergence analysis using the bi-linear element Q4 and the bi-quadratic element Q9.

**Table 4.**
Convergence analysis using the bi-linear element Q4 and the bi-quadratic element Q9.

| Number of Elements | |
---|

**λ** | **15** | **25** | **50 (dev)** | **75** | **100 (dev)** | **Timoshenko beam [5]** |

**Q4** |

1 | 5.24450 | 5.21481 | 5.20024 (1.994) | 5.19698 | 5.19571 (1.905) | 5.098585 |

2 | 8.71002 | 8.59864 | 8.54931 (7.757) | 8.53935 | 8.53566 (7.585) | 7.93386 |

3 | 12.21564 | 11.93952 | 11.82283 (13.41) | 11.80034 | 11.79223 (13.11) | 10.42527 |

Total DOF | 160 | 260 | 510 | 760 | 1010 | |

**λ** | **Q9** | |

1 | 5.19881 | 5.19589 | 5.19410 (1.873) | 5.19371 | 5.19358 (1.863) | 5.098585 |

2 | 8.53882 | 8.53377 | 8.53089 (7.525) | 8.53028 | 8.53008 (7.515) | 7.93386 |

3 | 11.79418 | 11.78565 | 11.78162 (13.01) | 11.78080 | 11.78055 (13.00) | 10.42527 |

Total DOF | 465 | 765 | 1515 | 2265 | 3015 | |

**Table 5.**
Natural frequencies in order of the material distribution and boundary conditions.

**Table 5.**
Natural frequencies in order of the material distribution and boundary conditions.

λ | B. C. | Euler-Bernoulli Beam Element [5] | Timoshenko Beam Element [5] |
---|

UD | USFG | SFG | UD | USFG | SFG |
---|

1 | C-C | 5.4647 | 5.3708 | 5.7495 | 5.098585 | 5.031699 | 5.294657 |

2 | 9.0571 | 8.9005 | 9.5291 | 7.93386 | 7.85222 | 8.167861 |

3 | 12.6469 | 12.4262 | 13.3056 | 10.42527 | 10.3392 | 10.66974 |

1 | H-H | 3.63 | 3.6182 | 3.8192 | 3.574603 | 3.563695 | 3.748668 |

2 | 7.2489 | 7.1216 | 7.6266 | 6.854168 | 6.757675 | 7.133685 |

3 | 10.8457 | 10.6722 | 11.4107 | 9.71082 | 9.610907 | 10.02423 |

1 | C-F | 2.1672 | 2.13 | 2.2802 | 2.151246 | 2.115347 | 2.259759 |

2 | 5.4175 | 5.324 | 5.6998 | 5.167211 | 5.09299 | 5.385917 |

3 | 9.0443 | 8.887 | 9.5155 | 8.194459 | 8.096988 | 8.475179 |

1 | C-H | 4.5367 | 4.4696 | 4.7731 | 4.356794 | 4.302608 | 4.546786 |

2 | 8.1535 | 8.0173 | 8.5784 | 7.426815 | 7.341066 | 7.685675 |

3 | 11.7464 | 11.5427 | 12.3583 | 10.08701 | 9.991091 | 10.3645 |

**Table 6.**
Natural frequencies in order of the material distribution and boundary conditions. Present model with Q9 element.

**Table 6.**
Natural frequencies in order of the material distribution and boundary conditions. Present model with Q9 element.

λ | B. C. | Present Model [FSDT] |
---|

UD (dev) | USFG (dev) | SFG (dev) |
---|

1 | C-C | 5.19410 (1.873) | 5.23446 (4.030) | 5.41031 (2.184) |

2 | 8.53089 (7.525) | 8.59440 (9.452) | 8.87200 (8.621) |

3 | 11.78162 (13.01) | 11.86476 (14.75) | 12.23028 (14.63) |

1 | H-H | 3.46974 (2.934) | 3.52617 (1.053) | 3.61832 (3.477) |

2 | 6.89656 (0.618) | 6.95265 (2.885) | 7.18576 (0.730) |

3 | 10.24292 (5.479) | 10.33215 (7.504) | 10.65856 (6.328) |

1 | C-F | 2.07402 (3.590) | 2.09119 (1.142) | 2.16316 (4.275) |

2 | 5.16153 (0.110) | 5.20323 (2.165) | 5.37898 (0.129) |

3 | 8.55821 (4.439) | 8.62481 (6.519) | 8.90793 (5.106) |

1 | C-H | 4.32579 (0.712) | 4.36675 (1.491) | 4.50872 (0.837) |

2 | 7.72167 (3.970) | 7.78485 (6.045) | 8.03845 (4.590) |

3 | 11.02165 (9.266) | 11.10498 (11.15) | 11.45557 (10.53) |

**Table 7.**
First five natural frequencies for the FG-CNTRC square plate with different distributions without the agglomeration effect.

**Table 7.**
First five natural frequencies for the FG-CNTRC square plate with different distributions without the agglomeration effect.

λ | UD | USFG | SFG |
---|

Q4 | Q9 | Q4 | Q9 | Q4 | Q9 |
---|

1 | 11.1259 | 11.0975 | 10.2156 | 10.1876 | 13.0043 | 12.9717 |

2 | 26.6616 | 26.4300 | 24.0251 | 23.8584 | 30.7728 | 30.5147 |

3 | 26.6616 | 26.4300 | 24.3895 | 24.1794 | 30.7728 | 30.5147 |

4 | 40.8513 | 40.5000 | 26.0625 | 26.0013 | 46.6761 | 46.2997 |

5 | 50.3098 | 49.3144 | 29.7480 | 29.7034 | 57.1369 | 56.0610 |

**Table 8.**
First five natural frequencies for the UD distribution for three different states of complete agglomeration.

**Table 8.**
First five natural frequencies for the UD distribution for three different states of complete agglomeration.

λ | η = 1 μ = 0.25 | η = 1 μ = 0.5 | η = 1 μ = 0.75 |
---|

1 | 7.2119 | 9.8010 | 8.5212 |

2 | 17.1866 | 23.3547 | 20.3099 |

3 | 17.1866 | 23.3547 | 20.3099 |

4 | 26.3491 | 35.8032 | 31.1414 |

5 | 32.0929 | 43.6063 | 37.9327 |

**Table 9.**
First five natural frequencies for the USFG distribution for three different states of complete agglomeration.

**Table 9.**
First five natural frequencies for the USFG distribution for three different states of complete agglomeration.

λ | η = 1 μ = 0.25 | η = 1 μ = 0.5 | η = 1 μ = 0.75 |
---|

1 | 6.9518 | 7.9533 | 8.9882 |

2 | 16.5096 | 18.8644 | 21.2173 |

3 | 16.5822 | 18.9727 | 21.4074 |

4 | 16.7036 | 19.5689 | 22.5596 |

5 | 19.0766 | 22.3055 | 25.7035 |

**Table 10.**
First five natural frequencies for the SFG distribution for three different states of complete agglomeration.

**Table 10.**
First five natural frequencies for the SFG distribution for three different states of complete agglomeration.

λ | η = 1 μ = 0.25 | η = 1 μ = 0.5 | η = 1 μ = 0.75 |
---|

1 | 7.2801 | 10.6382 | 8.8357 |

2 | 17.3136 | 25.1455 | 20.9576 |

3 | 17.3136 | 25.1455 | 20.9576 |

4 | 26.4997 | 38.2979 | 32.0087 |

5 | 32.2457 | 46.4714 | 38.9016 |

**Table 11.**
First five natural frequencies for the UD distribution for two different states of partial agglomeration.

**Table 11.**
First five natural frequencies for the UD distribution for two different states of partial agglomeration.

λ | η = 0.25 μ = 0.5 | η = 0.75 μ = 0.5 |
---|

1 | 10.7993 | 10.6423 |

2 | 25.7258 | 25.3509 |

3 | 25.7258 | 25.3509 |

4 | 39.4283 | 38.8528 |

5 | 48.0147 | 47.3131 |

**Table 12.**
First five natural frequencies for the USFG distribution for two different states of partial agglomeration.

**Table 12.**
First five natural frequencies for the USFG distribution for two different states of partial agglomeration.

λ | η = 0.25 μ = 0.5 | η = 0.75 μ = 0.5 |
---|

1 | 9.9679 | 9.8321 |

2 | 23.3603 | 23.0535 |

3 | 23.6652 | 23.3503 |

4 | 25.3939 | 24.9567 |

5 | 28.9605 | 28.4688 |

**Table 13.**
First five natural frequencies for the SFG distribution for two different states of partial agglomeration.

**Table 13.**
First five natural frequencies for the SFG distribution for two different states of partial agglomeration.

λ | η = 0.25 μ = 0.5 | η = 0.75 μ = 0.5 |
---|

1 | 12.5694 | 12.2965 |

2 | 29.5889 | 28.9567 |

3 | 29.5889 | 28.9567 |

4 | 44.9194 | 43.9722 |

5 | 54.4064 | 53.2675 |

**Table 14.**
First five natural frequencies for the FG-CNTRC rectangular plate with different distributions without the agglomeration effect.

**Table 14.**
First five natural frequencies for the FG-CNTRC rectangular plate with different distributions without the agglomeration effect.

λ | UD | USFG | SFG |
---|

1 | 49.3032 | 29.4880 | 56.0488 |

2 | 61.7868 | 45.0712 | 69.7297 |

3 | 81.0005 | 56.8115 | 90.5044 |

4 | 105.2993 | 74.2900 | 116.3783 |

5 | 133.2387 | 76.6857 | 145.6953 |

**Table 15.**
First five natural frequencies for the UD distribution for three different states of complete agglomeration.

**Table 15.**
First five natural frequencies for the UD distribution for three different states of complete agglomeration.

λ | η = 1 μ = 0.25 | η = 1 μ = 0.5 | η = 1 μ = 0.75 |
---|

1 | 32.0856 | 37.9241 | 43.5964 |

2 | 40.2250 | 47.5490 | 54.6530 |

3 | 52.7614 | 62.3764 | 71.6812 |

4 | 68.6291 | 81.1477 | 93.2317 |

5 | 86.8894 | 102.7540 | 118.0290 |

**Table 16.**
First five natural frequencies for the USFG distribution for three different states of complete agglomeration.

**Table 16.**
First five natural frequencies for the USFG distribution for three different states of complete agglomeration.

λ | η = 1 μ = 0.25 | η = 1 μ = 0.5 | η = 1 μ = 0.75 |
---|

1 | 19.0417 | 22.2483 | 25.5956 |

2 | 31.0008 | 35.5023 | 40.0258 |

3 | 38.8980 | 44.6043 | 50.3604 |

4 | 49.8835 | 58.2905 | 65.8665 |

5 | 50.4338 | 58.3212 | 66.9380 |

**Table 17.**
First five natural frequencies for the SFG distribution for three different states of complete agglomeration.

**Table 17.**
First five natural frequencies for the SFG distribution for three different states of complete agglomeration.

λ | η = 1 μ = 0.25 | η = 1 μ = 0.5 | η = 1 μ = 0.75 |
---|

1 | 32.2384 | 38.8929 | 46.4611 |

2 | 40.3660 | 48.6204 | 57.9616 |

3 | 52.8553 | 63.5232 | 75.5133 |

4 | 68.6205 | 82.2699 | 97.4962 |

5 | 86.7151 | 103.7140 | 122.5369 |

**Table 18.**
First five natural frequencies for the UD distribution for two different states of partial agglomeration.

**Table 18.**
First five natural frequencies for the UD distribution for two different states of partial agglomeration.

λ | η = 0.25 μ = 0.5 | η = 0.75 μ = 0.5 |
---|

1 | 48.0037 | 47.3023 |

2 | 60.1669 | 59.2865 |

3 | 78.8925 | 77.7359 |

4 | 102.5815 | 101.0744 |

5 | 129.8282 | 127.9167 |

**Table 19.**
First five natural frequencies for the USFG distribution for two different states of partial agglomeration.

**Table 19.**
First five natural frequencies for the USFG distribution for two different states of partial agglomeration.

λ | η = 0.25 μ = 0.5 | η = 0.75 μ = 0.5 |
---|

1 | 28.7626 | 28.2805 |

2 | 44.1295 | 43.5517 |

3 | 55.6138 | 54.8666 |

4 | 72.7059 | 71.7045 |

5 | 74.9295 | 73.6747 |

**Table 20.**
First five natural frequencies for the SFG distribution for two different states of partial agglomeration.

**Table 20.**
First five natural frequencies for the SFG distribution for two different states of partial agglomeration.

λ | η = 0.25 μ = 0.5 | η = 0.75 μ = 0.5 |
---|

1 | 54.3945 | 53.2559 |

2 | 67.6982 | 66.2947 |

3 | 87.9146 | 86.1159 |

4 | 113.1130 | 110.8318 |

5 | 141.6856 | 138.8681 |

**Table 21.**
Summary of the agglomeration effect in the natural modes′ shapes of the quadrilateral plates.

**Table 21.**
Summary of the agglomeration effect in the natural modes′ shapes of the quadrilateral plates.

**Square Plate** |

CNTs′ distribution | η = 1 | μ = 0.5 |

μ = 0.25 | μ = 0.5 | μ = 0.75 | η = 0.25 | η = 0.75 |

UD | - | - | - | - | - |

USFG | - | - | - | - | - |

SFG | 5th | 5th | 5th | - | - |

**Rectangular Plate** |

CNTs′ distribution | η = 1 | μ = 0.5 |

μ = 0.25 | μ = 0.5 | μ = 0.75 | η = 0.25 | η = 0.75 |

UD | - | - | - | - | - |

USFG | 4th, 5th | 5th | - | - | - |

SFG | - | - | - | - | - |