# Static and Dynamic Response of FG-CNT-Reinforced Rhombic Laminates

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

^{0}element consisting of seven nodal unknowns per node. The final material properties of FG-CNT-reinforced rhombic laminates are estimated using the rule of mixture. The obtained numerical are compared with the results available in the literature to verify the reliability of the present model. The present study investigates the effect of CNT distribution, loading pattern, volume fraction, and various combinations of boundary constraints by developing a finite element code in FORTRAN.

## 1. Introduction

^{0}nine-noded finite element. Since results for FG-CNT-reinforced rhombic laminates subjected to trigonometrical loading are not available in the literature, the present analysis results may serve as a benchmark for the researchers working in this field.

## 2. Carbon Nanotube-Reinforced Laminates

_{CNT}is the mass fraction of the CNTs in the CNT-reinforced rhombic plates, whereas ρ

^{m}and ρ

^{CNT}are the densities of the polymer matrix and carbon nanotubes, respectively.

_{11}

^{CNT}and E

_{22}

^{CNT}are Young’s moduli and G

_{12}

^{CNT}is the shear modulus of singly walled CNTs, respectively. E

^{m}and G

^{m}are known as Young’s modulus and shear modulus of the isotropic matrix. ν

_{12}

^{CNT}and ν

^{m}represent the Poisson’s ratio of CNTs and matrix respectively. V

_{m}and V

_{CNT}are the volume fractions of the matrix and carbon nanotubes, respectively, and the sum of both volume fractions equals to unity. η

_{1}, η

_{2}, η

_{3}are the scale-dependent material properties and they can be calculated by matching the effective properties of CNT-reinforced composite obtained from the MD simulations with those from the rule of mixture.

## 3. Mathematical Formulation

#### 3.1. Displacement Fields and Strains

_{0}, v

_{0}and w

_{0}) are displacements at the mid-plane and θ

_{x}, θ

_{y}are the bending rotations defined at the midplane about the y and x-axes respectively. ξ

_{x}, ξ

_{y}, ζ

_{x}and ζ

_{y}are higher order terms of Taylor’s series expansion. The function ξ

_{x}, ξ

_{y}, ζ

_{x}and ζ

_{y}will be calculated by vanishing shear stress at top and bottom of the plate. By applying the boundary conditions ${\gamma}_{\mathrm{xz}}\left(\mathrm{x},\mathrm{y},\pm h/2\right)={\gamma}_{\mathrm{yz}}\left(\mathrm{x},\mathrm{y},\pm h/2\right)=0$ at the upper layer and lower layer of the plate in Equation (7) and rearranging the terms that appear in the displacement field (u and v), we obtained

^{1}continuity requirement in the higher-order theory may arise due to the existence of first-order derivatives of transverse displacement. By applying C

^{0}continuity to the present problem, the out of plane derivatives are exchanged by the following relations in Equation (10):

^{0}continuity may be presented in the following manner:

_{0}, ν

_{0}, w

_{0}, θ

_{x}, θ

_{y}, ψ

_{x}, and ψ

_{y}for each node. Mathematically, the nodal displacement vector $\left\{\delta \right\}$ corresponding to displacement field in Equation (12) may be represented as

#### 3.2. Constitutive Relationship

## 4. Finite Element Formulation

#### 4.1. Element Description

^{0}isoparametric Lagrangian element was utilized in the present investigation. The element has a total of 63 degrees of freedom and each node has seven degrees of freedom. The element has inconsistent rectangular geometry in the x–y coordinate system. In order to ensure a consistent rectangular geometry, the element was plotted to the ξ–η plane. For the assumed nine-node element, the expressions for shape functions N

_{i}are described below.

#### 4.2. Governing Equation for Bending Analysis

#### 4.3. Governing Equation for Free Vibration Analysis

_{0}, v

_{0}and w

_{0}) as

_{i}) and global displacement vector {X}:

#### 4.4. Skew Boundary Transformation

## 5. Numerical Results and Discussion

^{m}= 2.1 GPa, ρ

^{m}= 1150 kg/m

^{3}and ν

^{m}= 0.34 at room temperature (300 K). The material properties of (10,10) SWCNTs at 300K are tabulated in Table 1. Three types volume fraction were used in present study. In the case of V*

_{CNT}= 0.11, η

_{1}= 0.934 and η

_{2}= 0.149, in the case of V*

_{CNT}= 0.14, η

_{1}= 0.150 and η

_{2}= 0.941, and for V*

_{CNT}= 0.17, η

_{1}= 0.149 and η

_{2}= 1.381. We assume that η

_{2}= η

_{3}and G

_{12}= G

_{13}= G

_{23}. The abovementioned values are used for the following numerical results.

- For the bending analysis$$\overline{w}=\frac{w}{h},{\overline{\sigma}}_{\mathrm{x}}={\sigma}_{\mathrm{x}}\left(\frac{a}{2},\frac{b}{2},z\right)\frac{{h}^{2}}{{q}_{0}{a}^{2}}.$$
- For the free vibration analysis$$\overline{\omega}=\omega \left({a}^{2}/h\right)\sqrt{{\rho}^{m}/{E}^{m}}.$$

- Simply supported (SSSS):$$v=w={\theta}_{\mathrm{y}}={\psi}_{\mathrm{y}}=0\mathrm{at}\mathrm{x}=0,a\phantom{\rule{0ex}{0ex}}u=w={\theta}_{\mathrm{x}}={\psi}_{\mathrm{x}}=0\mathrm{at}y=0,b.$$
- Clamped (CCCC):$$u=v=w={\theta}_{\mathrm{x}}={\theta}_{\mathrm{y}}={\psi}_{\mathrm{x}}={\psi}_{\mathrm{y}}=0\mathrm{at}\mathrm{x}=0,a\mathrm{and}y=0,b.$$
- Clamped and simply supported (CCSS):$$u=v=w={\theta}_{\mathrm{x}}={\theta}_{\mathrm{y}}={\psi}_{\mathrm{x}}={\psi}_{\mathrm{y}}=0\mathrm{at}\mathrm{x}=0,a\phantom{\rule{0ex}{0ex}}u=w={\theta}_{\mathrm{x}}={\psi}_{\mathrm{x}}=0\mathrm{at}y=0,b.$$

#### 5.1. Convergence and Validation of Present Formulation

#### 5.1.1. Free Vibration Analysis

**Example**

**1.**

_{CNT}= 0.11 and a skew angle of 15°. The convergence study indicated that 16 × 16 mesh is satisfactory for the free vibration analysis of functionally graded CNTRC rhombic plate using current nine-noded isoparametric elements. Hence, 16 × 16 mesh size was adopted for all the cases of free vibration analysis of a functionally graded CNT-reinforced rhombic plate.

**Example**

**2.**

**Example**

**3.**

**= 0.11 and 0.14) and three side-to-thickness ratios (a/h = 10, 20 and 50) were taken for comparison. The frequency parameter of simply supported and clamped boundary condition was found to be closer to Zhu et al. [5].**

_{CNT}#### 5.1.2. Bending Analysis

**Example**

**4.**

**= 0.11. The convergence study showed that 16 × 16 mesh size is acceptable for the present model using the discussed nine-noded isoparametric elements. Hence, 16 × 16 mesh size was chosen for all the parametric studies of the bending analysis of functionally graded CNTRC rhombic plate.**

_{CNT}**Example**

**5.**

**= 0.11, 0.14 and a/h = 10, 20, 50 were used for the comparison study. The dimensionless central deflection of different types of boundary condition was found to be in decent agreement with Zhu et al. [5].**

_{CNT}#### 5.2. Results and Discussion

_{CNT}), and different boundary conditions (SSSS, CCCC, CCSS, CSCS, CCFF, and CFCF) on the bending and free vibration behavior of functionally graded CNT-reinforced composite rhombic plate.

#### 5.2.1. Free Vibration Analysis

#### 5.2.2. Static Analysis

_{CNT}increases from 0.11 to 0.17 and approximately 6% decreases are noticed when V*

_{CNT}changes from 0.11 to 0.14. Maximum dimensionless deflection decreases with an increase in the skew angle because it reduces the length of the shorter diagonal leading to an enhancement in the stiffness of the rhombic plate. Thus, the deflection is reduced.

## 6. Conclusions

^{0}finite element model based on TSDT were presented. The actual material properties at any given section are calculated using the rule of mixture. The following conclusions written below were drawn from the obtained results for numerous values of side-to-thickness ratio, skew angle, and aspect ratio, and different types of end support.

- The FG-O and FG-X type distributions inside the CNT rhombic plates have lower and higher non-dimensional frequency parameter as well as higher and lower dimensionless deflection, respectively.
- The rise in the CNTs volume fraction results in a decrease in the deflection and an increase in the frequency parameter of the CNT-reinforced rhombic plate.
- The dimensionless frequency parameter increases along with the skew angle, irrespective of the CNT distribution and boundary condition.
- Maximum dimensionless deflection and dimensionless normal stresses decrease along with the skew angle.
- Higher values of non-dimensional fundamental frequencies and lower values of dimensionless deflection are found for greater constraints on boundaries.

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Element Stiffness Matrix

_{0}] is the penalty matrix added to the stiffness part to compensate for the replacement of the derivatives of transverse displacement $\left(\partial w/\partial \mathrm{x}\mathrm{and}\partial w/\partial \mathrm{y}\right)$ demanding C

^{1}continuity by new C

^{0}continuous variables (w

_{1}and w

_{2}) following penalty approach, which is a well-known procedure in the finite element analysis. The penalty [p

_{0}] matrix, is expressed as

^{5}.

#### Appendix A.2. Element Mass Matrix

_{i}is the mass density of the i-th layer and the matrix [L] is

_{e}] (which is 63 × 63 in the present formulation) and mass matrix [m

_{e}] are computed for all the elements and assembled to form the overall stiffness matrix, [K], and mass matrix, [M], for the total structure. The skyline storage technique is used to keep these large size matrices [K] and [M] in a single array; thus, a considerable amount of storage space in core memory is saved in an efficient manner. This has been implemented systematically in the computer code developed in the present study.

## References

- Iijima, S.; Ichihashi, T. Single-shell carbon nanotubes of 1-nm diameter. Nature
**1993**, 363, 603–605. [Google Scholar] [CrossRef] - Hone, J.; Llaguno, M.C.; Biercuk, M.J.; Johnson, A.T.; Batlogg, B.; Benes, Z.; Fischer, J.E. Thermal properties of carbon nanotubes and nanotube-based materials. Appl. Phys. A Mater. Sci. Process.
**2002**, 74, 339–343. [Google Scholar] [CrossRef] - Dresselhaus, M.S.; Dresselhaus, G.; Charlier, J.C.; Hernandez, E. Electronic, thermal and mechanical properties of carbon nanotubes. Philos. Trans. R. Soc. A Math. Phys. Eng. Sci.
**2004**, 362, 2065–2098. [Google Scholar] [CrossRef] [PubMed] - Reissner, E. On the theory of transverse bending of elastic plates. Int. J. Solids Struct.
**1976**, 12, 545–554. [Google Scholar] [CrossRef] - Zhu, P.; Lei, Z.X.; Liew, K.M. Static and free vibration analyses of carbon nanotube-reinforced composite plates using finite element method with first order shear deformation plate theory. Compos. Struct.
**2012**, 94, 1450–1460. [Google Scholar] [CrossRef] - Lei, Z.X.; Liew, K.M.; Yu, J.L. Free vibration analysis of functionally graded carbon nanotube-reinforced composite plates using the element-free kp-Ritz method in thermal environment. Compos. Struct.
**2013**, 106, 128–138. [Google Scholar] [CrossRef] - Moradi-dastjerdi, R.; Foroutan, M.; Pourasghar, A.; Sotoudeh-bahreini, R. Static analysis of functionally graded carbon nanotube-reinforced composite cylinders by a mesh-free method. J. Reinf. Plast. Compos.
**2013**, 32, 593–601. [Google Scholar] [CrossRef] - Yas, M.H.; Pourasghar, A.; Kamarian, S.; Heshmati, M. Three-dimensional free vibration analysis of functionally graded nanocomposite cylindrical panels reinforced by carbon nanotube. Mater. Des.
**2013**, 49, 583–590. [Google Scholar] [CrossRef] - Malekzadeh, P.; Zarei, A.R. Free vibration of quadrilateral laminated plates with carbon nanotube reinforced composite layers. Thin Walled Struct.
**2014**, 82, 221–232. [Google Scholar] [CrossRef] - Nami, M.R.; Janghorban, M. Thermal buckling analysis of functionally graded rectangular nanoplates based on nonlocal third-order shear deformation theory. Adv. Compos. Mater.
**2015**, 24, 439–450. [Google Scholar] [CrossRef] - Shahrbabaki, E.A.; Alibeigloo, A. Three-dimensional free vibration of carbon nanotube-reinforced composite plates with various boundary conditions using Ritz method. Compos. Struct.
**2014**, 111, 362–370. [Google Scholar] [CrossRef] - Sankar, A.; Natarajan, S.; Ganapathi, M. Dynamic instability analysis of sandwich plates with CNT reinforced facesheets. Compos. Struct.
**2016**, 146, 187–200. [Google Scholar] [CrossRef] - Mehar, K.; Panda, S.K. Vibration analysis of functionally graded carbon nanotube reinforced composite plate in thermal environment. J. Sandw. Struct. Mater.
**2015**, 18, 151–173. [Google Scholar] [CrossRef] - Mayandi, K.; Jeyaraj, P. Bending, buckling and free vibration characteristics of FG-CNT polymer composite beam under non-uniform thermal load. Proc. Inst. Mech. Eng. Part L J. Mater. Des. Appl.
**2015**, 229, 13–28. [Google Scholar] [CrossRef] - Zhang, L.W.; Cui, W.C.; Liew, K.M. Vibration analysis of functionally graded carbon nanotube reinforced composite thick plates with elastically restrained edges. Int. J. Mech. Sci.
**2015**, 103, 9–21. [Google Scholar] [CrossRef] - Huang, B.; Guo, Y.; Wang, J.; Du, J.; Qian, Z.; Ma, T.; Yi, L. Bending and free vibration analyses of antisymmetrically laminated carbon nanotube-reinforced functionally graded plates. J. Compos. Mater.
**2016**, 51, 3111–3125. [Google Scholar] [CrossRef] - García-macías, E.; Castro-triguero, R.; Saavedra, E.I.; Friswell, M.I.; Gallego, R. Static and free vibration analysis of functionally graded carbon nanotube reinforced skew plates. Compos. Struct.
**2016**, 140, 473–490. [Google Scholar] [CrossRef] - Mirzaei, M.; Kiani, Y. Free vibration of functionally graded carbon nanotube reinforced composite cylindrical panels. Compos. Struct.
**2016**, 142, 45–56. [Google Scholar] [CrossRef] - Song, Z.G.; Zhang, L.W.; Liew, K.M. Vibration analysis of CNT-reinforced functionally graded composite cylindrical shells in thermal environments. Int. J. Mech. Sci.
**2016**, 115–116, 339–347. [Google Scholar] [CrossRef] - Thomas, B.; Roy, T. Vibration analysis of functionally graded carbon nanotube-reinforced composite shell structures. Acta Mech.
**2016**, 599, 581–599. [Google Scholar] [CrossRef] - Duong, H.M.; Gong, F.; Liu, P.; Tran, T.Q. Advanced Fabrication and Properties of Aligned Carbon Nanotube Composites: Experiments and Modeling. In Carbon Nanotubes—Current Progress of Their Polymer Composites; Intech: Houston, TX, USA, 2016; pp. 47–72. ISBN 978-953-51-2470-2. [Google Scholar]
- Selim, B.A.; Zhang, L.W.; Liew, K.M. Vibration analysis of CNT reinforced functionally graded composite plates in a thermal environment based on Reddy’ s higher-order shear deformation theory. Compos. Struct.
**2016**, 156, 276–290. [Google Scholar] [CrossRef] - Zhang, L.W.; Selim, B.A. Vibration analysis of CNT-reinforced thick laminated composite plates based on Reddy’s higher-order shear deformation theory. Compos. Struct.
**2017**, 160, 689–705. [Google Scholar] [CrossRef] - Manevitch, L.I.; Smirnov, V.V.; Strozzi, M.; Pellicano, F. Nonlinear optical vibrations of single-walled carbon nanotubes. Procedia Eng.
**2017**, 199, 705–710. [Google Scholar] [CrossRef] - Fantuzzi, N.; Tornabene, F.; Bacciocchi, M.; Dimitri, R. Free vibration analysis of arbitrarily shaped Functionally Graded Carbon Nanotube-reinforced plates. Compos. Part B
**2017**, 115, 384–408. [Google Scholar] [CrossRef] - Tornabene, F.; Fantuzzi, N.; Bacciocchi, M. Linear static response of nanocomposite plates and shells reinforced by agglomerated carbon nanotubes. Compos. Part B
**2017**, 115, 449–476. [Google Scholar] [CrossRef] - Romano, G.; Barretta, R. Nonlocal elasticity in nanobeams: The stress-driven integral model. Int. J. Eng. Sci.
**2017**, 115, 14–27. [Google Scholar] [CrossRef] - Romano, G.; Barretta, R.; Diaco, M. On nonlocal integral models for elastic nano-beams. Int. J. Mech. Sci.
**2017**, 131–132, 490–499. [Google Scholar] [CrossRef] - Mahmoudpour, E.; Hosseini-Hashemi, S.H.; Faghidian, S.A. Nonlinear vibration analysis of FG nano-beams resting on elastic foundation in thermal environment using stress-driven nonlocal integral model. Appl. Math. Model.
**2018**, 57, 302–315. [Google Scholar] [CrossRef] - Barretta, R.; Ali Faghidian, S.; Luciano, R.; Medaglia, C.M.; Penna, R. Stress-driven two-phase integral elasticity for torsion of nano-beams. Compos. Part B Eng.
**2018**, 145, 62–69. [Google Scholar] [CrossRef] - Oskouie, M.F.; Ansari, R.; Rouhi, H. Bending of Euler–Bernoulli nanobeams based on the strain-driven and stress-driven nonlocal integral models: A numerical approach. Acta Mech. Sin.
**2018**. [Google Scholar] [CrossRef] - Barretta, R.; Faghidian, S.A.; Luciano, R. Longitudinal vibrations of nano-rods by stress-driven integral elasticity. Mech. Adv. Mater. Struct.
**2018**. [Google Scholar] [CrossRef] - Ansari, R.; Torabi, J.; Shakouri, A.H. Vibration analysis of functionally graded carbon nanotube-reinforced composite elliptical plates using a numerical strategy. Aerospace Sci. Technol.
**2017**, 60, 152–161. [Google Scholar] [CrossRef] - Tornabene, F.; Bacciocchi, M.; Fantuzzi, N.; Reddy, J.N. Multiscale Approach for Three-Phase CNT/Polymer/Fiber Laminated Nanocomposite Structures. Polym. Compos.
**2017**. [Google Scholar] [CrossRef] - Ardestani, M.M.; Zhang, L.W.; Liew, K.M. Isogeometric analysis of the effect of CNT orientation on the static and vibration behaviors of CNT-reinforced skew composite plates. Comput. Methods Appl. Mech. Eng.
**2017**, 317, 341–379. [Google Scholar] [CrossRef] - Esawi, A.M.K.; Farag, M.M. Carbon nanotube reinforced composites: Potential and current challenges. Mater. Des.
**2007**, 28, 2394–2401. [Google Scholar] [CrossRef] - Fidelus, J.D.; Wiesel, E.; Gojny, F.H.; Schulte, K.; Wagner, H.D. Thermo-mechanical properties of randomly oriented carbon/epoxy nanocomposites. Compos. Part A Appl. Sci. Manuf.
**2005**, 36, 1555–1561. [Google Scholar] [CrossRef] - Shen, H.S. Nonlinear bending of functionally graded carbon nanotube-reinforced composite plates in thermal environments. Compos. Struct.
**2009**, 91, 9–19. [Google Scholar] [CrossRef] - Reddy, J.N. A simple higher-order theory for laminated composite plates. J. Appl. Mech.
**1984**, 51, 745–752. [Google Scholar] [CrossRef] - Han, Y.; Elliott, J. Molecular dynamics simulations of the elastic properties of polymer/carbon nanotube composites. Comput. Mater. Sci.
**2007**, 39, 315–323. [Google Scholar] [CrossRef] - Reddy, J.N. An Introduction to the Finite Element Method; McGraw-Hill: New York, NY, USA, 1993. [Google Scholar]
- Srinivas, S.; Joga Rao, C.V.; Rao, A.K. An exact analysis for vibration of simply-supported homogeneous and laminated thick rectangular plates. J. Sound Vib.
**1970**, 12, 187–199. [Google Scholar] [CrossRef] - Mantari, J.L.; Oktem, A.S.; Soares, C.G. Bending and free vibration analysis of isotropic and multilayered plates and shells by using a new accurate higher-order shear deformation theory. Compos. Part B
**2012**, 43, 3348–3360. [Google Scholar] [CrossRef]

**Figure 3.**Variation of non-dimensional frequency parameter of FG-CNT-reinforced rhombic plate with aspect ratio; (

**A**) α = 15° and (

**B**) α = 30°.

**Figure 4.**The free vibration mode shapes of a SSSS square FG-V CNT-reinforced rhombic plate for skew angle 30° (

**A**) 1st Mode; (

**B**) 2nd Mode; (

**C**) 3rd Mode and (

**D**) 4th Mode.

**Figure 5.**Non-dimensional deflection of FG-V CNT-reinforced rhombic plate along the central line for (

**A**) 15°; (

**B**) 30°; (

**C**) 45° and (

**D**) 60° skew angle subjected to sin-sin loading.

**Figure 6.**Variation of non-dimensional deflection of FG-V CNT-reinforced rhombic plate with the skew angle for (

**A**) SSSS and (

**B**) CCCC boundary condition.

**Figure 7.**Variation of non-dimensional deflection of FG-CNT-reinforced rhombic plate with the skew angle subjected to sin-sin loading for (

**A**) CCSS, (

**B**) CSCS, (

**C**) CCFF and (

**D**) CFCF boundary condition.

**Figure 8.**Non-dimensional deflection of FG-CNTRC skew plate along the length (y/b = 0.5) subjected to sin-sin loading for (

**A**) a/h = 10; (

**B**) a/h = 20; (

**C**) a/h = 50 and (

**D**) a/h = 100.

**Figure 9.**Variation of non-dimensional axial stress for FG-CNT-reinforced rhombic plate subjected to sin-sin loading for (

**A**) α = 15°, (

**B**) α = 30°, (

**C**) α = 45° and (

**D**) α = 60°.

**Figure 10.**Variation of non-dimensional axial stress for FG-CNT-reinforced rhombic plate subjected to sin-sin loading (

**A**) α = 15°, (

**B**) α = 30°, (

**C**) α = 45° and (

**D**) α = 60°.

**Table 1.**Temperature-dependent material properties of (10, 10) SWCNT (L = 9.26 nm, R = 0.68 nm, h = 0.067 nm, ${\rho}_{12}^{\mathrm{CNT}}=0.175$).

Temperature (K) | ${\mathit{E}}_{11}^{\mathbf{CNT}}$ (TPa) | ${\mathit{E}}_{22}^{CNT}$ (TPa) | ${\mathit{G}}_{12}^{CNT}$ (TPa) | ${\mathit{\alpha}}_{11}^{\mathbf{CNT}}$ (10^{−6}/K) | ${\mathit{\alpha}}_{22}^{\mathbf{CNT}}$ (10^{−6}/K) |
---|---|---|---|---|---|

300 | 5.6466 | 7.0800 | 1.9445 | 3.4584 | 5.1682 |

500 | 5.5308 | 6.9348 | 1.9643 | 4.5361 | 5.0189 |

700 | 5.4744 | 6.8641 | 1.9644 | 4.6677 | 4.8943 |

Mesh | UD | FG-V | FG-O | FG-X |
---|---|---|---|---|

8 × 8 | 18.5932 | 18.0731 | 16.6424 | 19.2637 |

10 × 10 | 18.5915 | 18.0719 | 16.4112 | 19.2623 |

12 × 12 | 18.5906 | 18.0709 | 16.4093 | 19.2612 |

14 × 14 | 18.5894 | 18.0697 | 16.4085 | 19.2606 |

16 × 16 | 18.5892 | 18.0694 | 16.4084 | 19.2604 |

**Table 3.**Comparison of non-dimensional maximum deflection and normal stress of square isotropic plate subjected to uniform load.

a/h | Source | $\overline{\mathit{w}}$ | ${\overline{\mathit{\sigma}}}_{\mathbf{x}}$ |
---|---|---|---|

10 | Present | 4.666 | 0.289 |

Reddy [41] | 4.770 | 0.289 | |

20 | Present | 4.491 | 0.287 |

Reddy [41] | 4.570 | 0.268 | |

50 | Present | 4.408 | 0.284 |

Reddy [41] | 4.496 | 0.266 |

**Table 4.**Comparison of non-dimensional frequency parameter of the simply supported square isotropic plate.

Source | Mode | ||
---|---|---|---|

(1, 1) | (1, 2) | (1, 3) | |

Present | 0.093 | 0.221 | 0.415 |

Mantari et al. [43] | 0.093 | 0.222 | 0.415 |

Srinivas et al. [42] | 0.093 | 0.223 | 0.417 |

**Table 5.**Comparison study of first six non-dimensional frequency parameter of UD CNT-reinforced composite plate.

BC | V*_{CNT} | Mode | a/h = 10 | a/h = 20 | a/h = 50 | |||
---|---|---|---|---|---|---|---|---|

Zhu et al. [5] | Present | Zhu et al. [5] | Present | Zhu et al. [5] | Present | |||

CCCC | 0.11 | 1 | 17.625 | 18.284 | 28.400 | 29.232 | 39.730 | 41.246 |

2 | 23.041 | 23.793 | 33.114 | 34.108 | 43.876 | 45.501 | ||

3 | 33.592 | 34.188 | 44.559 | 45.456 | 54.768 | 56.313 | ||

4 | 33.729 | 35.188 | 59.198 | 60.708 | 74.488 | 75.080 | ||

5 | 37.011 | 38.536 | 61.851 | 63.003 | 98.291 | 100.577 | ||

6 | 37.317 | 38.738 | 63.043 | 63.553 | 100.537 | 101.437 | ||

0.14 | 1 | 18.127 | 18.854 | 29.911 | 30.795 | 43.583 | 45.216 | |

2 | 23.572 | 24.374 | 34.516 | 35.558 | 47.479 | 49.218 | ||

3 | 34.252 | 34.874 | 45.898 | 46.830 | 57.968 | 59.617 | ||

4 | 34.650 | 36.267 | 61.628 | 63.337 | 77.395 | 78.064 | ||

5 | 37.921 | 39.384 | 64.199 | 64.457 | 106.371 | 104.359 | ||

6 | 37.972 | 39.592 | 64.496 | 66.100 | 106.487 | 108.807 | ||

SSSS | 0.11 | 1 | 13.532 | 13.885 | 17.355 | 18.014 | 19.223 | 20.124 |

2 | 17.700 | 18.199 | 21.511 | 22.278 | 23.408 | 24.396 | ||

3 | 19.449 | 19.422 | 32.399 | 33.231 | 34.669 | 35.734 | ||

4 | 19.449 | 19.427 | 38.898 | 38.844 | 54.043 | 54.658 | ||

5 | 27.569 | 28.121 | 38.898 | 38.854 | 70.811 | 73.189 | ||

6 | 32.563 | 33.291 | 50.199 | 50.524 | 72.900 | 75.313 | ||

0.14 | 1 | 14.306 | 14.668 | 18.921 | 19.618 | 21.354 | 22.359 | |

2 | 18.362 | 18.870 | 22.867 | 23.666 | 25.295 | 26.373 | ||

3 | 19.791 | 19.769 | 33.570 | 34.419 | 36.267 | 37.393 | ||

4 | 19.791 | 19.774 | 39.583 | 39.538 | 55.608 | 56.238 | ||

5 | 28.230 | 28.784 | 39.583 | 39.548 | 78.110 | 80.675 | ||

6 | 33.646 | 34.492 | 51.422 | 51.737 | 80.015 | 82.137 |

**Table 6.**Comparison study of first six non-dimensional frequency parameter of FG-V CNT-reinforced composite plate.

BC | V*_{CNT} | Mode | a/h = 10 | a/h = 20 | a/h = 50 | |||
---|---|---|---|---|---|---|---|---|

Zhu et al. [5] | Present | Zhu et al. [5] | Present | Zhu et al. [5] | Present | |||

CCCC | 0.11 | 1 | 17.211 | 17.753 | 26.304 | 26.693 | 34.165 | 34.480 |

2 | 22.812 | 23.462 | 31.496 | 32.099 | 39.043 | 39.584 | ||

3 | 33.070 | 34.035 | 43.589 | 44.133 | 51.204 | 51.815 | ||

4 | 33.552 | 34.355 | 56.249 | 57.061 | 72.202 | 71.954 | ||

5 | 36.528 | 37.889 | 59.249 | 60.253 | 86.291 | 86.133 | ||

6 | 37.437 | 38.841 | 62.608 | 62.218 | 89.054 | 89.105 | ||

0.14 | 1 | 17.791 | 18.405 | 27.926 | 28.371 | 37.568 | 37.909 | |

2 | 23.413 | 24.113 | 32.976 | 33.629 | 42.175 | 42.733 | ||

3 | 34.101 | 34.792 | 44.989 | 45.573 | 53.963 | 54.590 | ||

4 | 34.275 | 35.553 | 58.951 | 59.968 | 74.785 | 74.546 | ||

5 | 37.538 | 39.053 | 61.816 | 63.051 | 94.022 | 93.911 | ||

6 | 38.159 | 39.574 | 64.135 | 63.758 | 96.573 | 96.680 | ||

SSSS | 0.11 | 1 | 12.452 | 12.601 | 15.110 | 15.291 | 16.252 | 16.465 |

2 | 17.060 | 17.409 | 19.903 | 20.297 | 21.142 | 21.573 | ||

3 | 19.499 | 19.479 | 31.561 | 32.106 | 33.350 | 33.993 | ||

4 | 19.499 | 19.484 | 38.998 | 38.959 | 53.430 | 53.670 | ||

5 | 27.340 | 27.762 | 38.998 | 38.969 | 60.188 | 60.337 | ||

6 | 31.417 | 31.903 | 47.739 | 47.899 | 62.198 | 63.042 | ||

0.14 | 1 | 13.256 | 13.415 | 16.510 | 16.701 | 17.995 | 18.228 | |

2 | 17.734 | 18.090 | 21.087 | 21.483 | 22.643 | 23.082 | ||

3 | 19.879 | 19.871 | 32.617 | 33.163 | 34.660 | 35.306 | ||

4 | 19.879 | 19.876 | 39.759 | 39.742 | 54.833 | 55.062 | ||

5 | 28.021 | 28.449 | 39.759 | 39.752 | 66.552 | 66.712 | ||

6 | 32.678 | 33.284 | 51.078 | 51.122 | 68.940 | 69.212 |

**Table 7.**Convergence of non-dimensional maximum deflection of CNT-reinforced functionally graded clamped rhombic plate.

Mesh | UD | FG-V | FG-O | FG-X |
---|---|---|---|---|

8 × 8 | 0.00372 | 0.00422 | 0.00588 | 0.00309 |

10 × 10 | 0.00361 | 0.00411 | 0.00576 | 0.00297 |

12 × 12 | 0.00355 | 0.00406 | 0.00568 | 0.00292 |

14 × 14 | 0.00349 | 0.00402 | 0.00561 | 0.00289 |

16 × 16 | 0.00345 | 0.00401 | 0.00557 | 0.00287 |

**Table 8.**Comparison study of non-dimensional maximum deflection of various volume fraction of UD CNT-reinforced composite plate.

BC | V*_{CNT} | a/h = 10 | a/h = 20 | a/h = 50 | |||
---|---|---|---|---|---|---|---|

Zhu et al. [5] | Present | Zhu et al. [5] | Present | Zhu et al. [5] | Present | ||

CCCC | 0.11 | 0.00222 | 0.00207 | 0.01339 | 0.01257 | 0.2618 | 0.24056 |

0.14 | 0.00208 | 0.00192 | 0.01188 | 0.01115 | 0.2131 | 0.19644 | |

SSSS | 0.11 | 0.00373 | 0.00354 | 0.03628 | 0.03352 | 1.1550 | 1.04729 |

0.14 | 0.00330 | 0.00314 | 0.03001 | 0.02779 | 0.9175 | 0.83205 | |

SCSC | 0.11 | 0.00332 | 0.00313 | 0.03393 | 0.03127 | 1.0990 | 0.99624 |

0.14 | 0.00297 | 0.00281 | 0.02852 | 0.02634 | 0.8890 | 0.80555 | |

SFSF | 0.11 | 0.00344 | 0.00339 | 0.03341 | 0.03223 | 1.0680 | 1.01428 |

0.14 | 0.00302 | 0.00297 | 0.02760 | 0.02654 | 0.8505 | 0.80295 |

**Table 9.**Comparison study of non-dimensional maximum deflection of various volume fraction of UD CNT-reinforced composite plate.

BC | V*_{CNT} | a/h = 10 | a/h = 20 | a/h = 50 | |||
---|---|---|---|---|---|---|---|

Zhu et al. [5] | Present | Zhu et al. [5] | Present | Zhu et al. [5] | Present | ||

CCCC | 0.11 | 0.00211 | 0.00191 | 0.01150 | 0.01052 | 0.18940 | 0.16721 |

0.14 | 0.00198 | 0.00179 | 0.01036 | 0.00954 | 0.15600 | 0.13941 | |

SSSS | 0.11 | 0.00318 | 0.00294 | 0.02701 | 0.02398 | 0.79000 | 0.67655 |

0.14 | 0.00284 | 0.00266 | 0.02256 | 0.02021 | 0.62710 | 0.53777 | |

SCSC | 0.11 | 0.00286 | 0.00264 | 0.02587 | 0.02297 | 0.77280 | 0.66351 |

0.14 | 0.00258 | 0.00240 | 0.02184 | 0.01955 | 0.62060 | 0.53313 | |

SFSF | 0.11 | 0.00290 | 0.00276 | 0.02484 | 0.02281 | 0.73380 | 0.70308 |

0.14 | 0.00259 | 0.00248 | 0.02078 | 0.01916 | 0.58540 | 0.54605 |

**Table 10.**Variation of first six natural non-dimensional frequency parameters of simply supported FG-CNT-reinforced rhombic plate.

Types | V*_{CNT} | Skew Angle | Mode | |||||
---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | |||

UD | 0.11 | 15° | 14.0884 | 18.3200 | 18.8946 | 22.6183 | 29.3328 | 33.6414 |

30° | 14.8977 | 18.8299 | 21.3717 | 28.4322 | 33.1078 | 35.1178 | ||

45° | 17.1862 | 20.5115 | 26.9900 | 37.0144 | 39.6322 | 39.6568 | ||

60° | 22.7998 | 24.0366 | 38.2235 | 46.2487 | 47.8936 | 50.9246 | ||

0.14 | 15° | 14.8687 | 18.6487 | 19.5642 | 23.0233 | 30.0198 | 34.8422 | |

30° | 15.6755 | 19.1758 | 22.0498 | 28.9455 | 33.9035 | 36.3247 | ||

45° | 17.9727 | 20.9102 | 27.7332 | 37.7063 | 40.6588 | 40.8949 | ||

60° | 23.2808 | 24.8769 | 39.2472 | 47.5170 | 49.0156 | 52.2774 | ||

0.17 | 15° | 17.4811 | 22.9180 | 23.5104 | 28.2964 | 36.5676 | 41.9224 | |

30° | 18.4931 | 23.5450 | 26.6092 | 35.5615 | 41.2831 | 43.7657 | ||

45° | 21.3530 | 25.6345 | 33.6340 | 46.2801 | 49.4285 | 49.4321 | ||

60° | 28.4860 | 29.9161 | 47.6618 | 57.6946 | 59.7955 | 63.5234 | ||

FG-V | 0.11 | 15° | 12.8212 | 18.1269 | 18.3759 | 22.6786 | 28.9595 | 32.2712 |

30° | 13.6868 | 18.8528 | 20.6993 | 28.4857 | 32.6175 | 33.8100 | ||

45° | 16.0825 | 20.5023 | 26.3727 | 36.9725 | 38.5706 | 38.8945 | ||

60° | 22.7651 | 23.1363 | 37.4158 | 45.9406 | 47.6860 | 50.0097 | ||

0.14 | 15° | 13.6306 | 18.7223 | 18.8344 | 23.1366 | 29.6827 | 33.6486 | |

30° | 14.4859 | 19.2484 | 21.3970 | 29.0713 | 33.4956 | 35.1799 | ||

45° | 16.8766 | 20.9596 | 27.1726 | 37.8010 | 39.9021 | 40.0565 | ||

60° | 23.3107 | 23.9748 | 38.5628 | 47.3845 | 48.9510 | 51.4979 | ||

0.17 | 15° | 15.8726 | 22.5869 | 23.1041 | 28.5207 | 36.1984 | 40.2043 | |

30° | 16.9639 | 23.7009 | 25.8142 | 35.8176 | 40.7628 | 42.1365 | ||

45° | 19.9783 | 25.7658 | 32.9286 | 46.4775 | 48.1150 | 48.5974 | ||

60° | 28.6027 | 28.8423 | 46.7338 | 57.6186 | 59.8787 | 62.5277 | ||

FG-O | 0.11 | 15° | 11.2246 | 16.8568 | 18.2727 | 22.5605 | 27.7024 | 28.6900 |

30° | 12.1133 | 18.7752 | 19.3640 | 28.3558 | 30.3139 | 30.7798 | ||

45° | 14.5197 | 20.4388 | 24.7329 | 35.2696 | 36.2118 | 36.9006 | ||

60° | 21.5223 | 22.7001 | 34.8852 | 45.8698 | 46.6015 | 47.6229 | ||

0.14 | 15° | 11.9547 | 17.3832 | 18.6042 | 22.9689 | 28.2193 | 30.0605 | |

30° | 12.8181 | 19.1248 | 19.9022 | 28.8740 | 31.5416 | 31.6441 | ||

45° | 15.1892 | 20.8433 | 25.3918 | 36.4990 | 37.2763 | 37.6006 | ||

60° | 22.1785 | 23.1888 | 35.8993 | 47.2005 | 47.9486 | 48.7680 | ||

0.17 | 15° | 13.8793 | 20.7785 | 22.8791 | 22.8791 | 34.1824 | 35.9124 | |

30° | 14.9671 | 23.4971 | 23.8865 | 35.4973 | 37.8814 | 38.1425 | ||

45° | 17.9248 | 25.5652 | 30.5918 | 43.9152 | 45.0463 | 46.1779 | ||

60° | 26.5756 | 28.3845 | 43.3295 | 57.2497 | 58.0859 | 59.5035 | ||

FG-X | 0.11 | 15° | 15.3579 | 18.4618 | 19.9956 | 22.7929 | 30.4082 | 35.0962 |

30° | 16.1745 | 18.9803 | 22.4702 | 28.6545 | 34.2453 | 36.6028 | ||

45° | 18.4949 | 20.6851 | 28.1124 | 37.3147 | 40.9182 | 41.2241 | ||

60° | 23.0073 | 25.4117 | 39.5510 | 46.7919 | 48.3756 | 52.4841 | ||

0.14 | 15° | 16.0673 | 18.8614 | 20.7220 | 23.2855 | 31.2995 | 36.1529 | |

30° | 16.9039 | 19.3980 | 23.2367 | 29.2774 | 35.2156 | 37.6988 | ||

45° | 19.2847 | 21.1602 | 28.9762 | 38.1470 | 42.0428 | 42.4396 | ||

60° | 23.5712 | 26.3605 | 40.6662 | 48.1906 | 49.6636 | 53.9309 | ||

0.17 | 15° | 19.0583 | 23.2864 | 25.1259 | 28.7507 | 38.4760 | 43.6385 | |

30° | 20.1180 | 23.9280 | 28.2892 | 36.1352 | 43.2193 | 45.5938 | ||

45° | 23.1055 | 26.0614 | 35.4283 | 47.0376 | 51.4225 | 51.5545 | ||

60° | 28.9750 | 31.9346 | 49.7188 | 58.8146 | 60.8682 | 65.8386 |

**Table 11.**Variation of first six natural non-dimensional frequency parameters of clamped FG-CNT-reinforced rhombic plate.

Types | V*_{CNT} | Skew Angle | Mode | |||||
---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | |||

UD | 0.11 | 15° | 18.5892 | 24.6235 | 35.3293 | 35.6482 | 39.8059 | 40.1566 |

30° | 19.7907 | 27.5192 | 37.5758 | 38.7334 | 44.7699 | 44.8668 | ||

45° | 23.1240 | 33.7718 | 43.5280 | 45.2188 | 54.8007 | 56.3342 | ||

60° | 32.8768 | 46.6645 | 58.8579 | 62.5042 | 70.8589 | 76.4394 | ||

0.14 | 15° | 19.1594 | 25.2162 | 36.0641 | 36.7261 | 40.8256 | 40.8465 | |

30° | 20.3677 | 28.1613 | 38.6503 | 39.6187 | 45.6147 | 45.7882 | ||

45° | 23.7315 | 34.5566 | 44.6059 | 46.2941 | 55.7146 | 57.5412 | ||

60° | 33.6101 | 47.7344 | 60.1998 | 63.7591 | 72.4446 | 77.7145 | ||

0.17 | 15° | 23.1763 | 30.7160 | 44.0894 | 44.4545 | 49.6574 | 50.1117 | |

30° | 24.6779 | 34.3325 | 46.8674 | 48.3332 | 55.8713 | 55.9847 | ||

45° | 28.8430 | 42.1399 | 54.3158 | 56.4308 | 68.3713 | 70.3186 | ||

60° | 41.0318 | 58.2465 | 73.4787 | 78.0518 | 88.4742 | 95.3523 | ||

FG-V | 0.11 | 15° | 18.0694 | 24.2993 | 34.8395 | 35.1140 | 39.2444 | 40.2623 |

30° | 19.3073 | 27.1997 | 36.8576 | 38.3514 | 44.4246 | 44.9823 | ||

45° | 22.7117 | 33.3930 | 43.0375 | 44.7306 | 54.9298 | 55.8898 | ||

60° | 32.5888 | 46.2233 | 58.3954 | 62.3472 | 70.4436 | 76.6148 | ||

0.14 | 15° | 18.7213 | 24.9627 | 35.9325 | 36.0340 | 40.3826 | 41.0223 | |

30° | 19.9628 | 27.9173 | 38.0427 | 39.3412 | 45.5331 | 45.8315 | ||

45° | 23.3936 | 34.2698 | 44.2154 | 45.9247 | 55.9691 | 57.3257 | ||

60° | 33.3992 | 47.4078 | 59.8831 | 63.7136 | 72.2043 | 78.0564 | ||

0.17 | 15° | 22.5493 | 30.3816 | 43.5223 | 43.9471 | 49.0751 | 50.5020 | |

30° | 24.1057 | 34.0165 | 46.0643 | 47.9801 | 55.5987 | 56.4159 | ||

45° | 28.3819 | 41.7701 | 53.8425 | 55.9614 | 68.8770 | 69.9530 | ||

60° | 40.7818 | 57.8482 | 73.1009 | 78.1001 | 88.2112 | 96.0396 | ||

FG-O | 0.11 | 15° | 16.4084 | 22.9046 | 31.6159 | 33.6029 | 36.6562 | 40.0537 |

30° | 17.7077 | 25.7508 | 33.8501 | 36.2830 | 42.3527 | 44.7506 | ||

45° | 21.2099 | 31.6586 | 40.5674 | 42.2311 | 52.8152 | 53.9899 | ||

60° | 31.2046 | 44.1065 | 55.5461 | 60.6186 | 66.8835 | 76.2352 | ||

0.14 | 15° | 17.0452 | 23.4601 | 32.8241 | 34.3062 | 37.6525 | 40.7285 | |

30° | 18.3275 | 26.3574 | 34.9993 | 37.1966 | 43.2631 | 45.5052 | ||

45° | 21.8175 | 32.4388 | 41.6106 | 43.3410 | 54.1879 | 55.1259 | ||

60° | 31.8862 | 45.1773 | 56.9465 | 61.7865 | 68.6013 | 77.5228 | ||

0.17 | 15° | 20.4935 | 28.4369 | 39.6386 | 41.7340 | 45.7071 | 50.0278 | |

30° | 22.0844 | 31.9798 | 42.3651 | 45.1869 | 52.6957 | 55.8895 | ||

45° | 26.3966 | 39.3908 | 50.6104 | 52.6850 | 65.9841 | 67.2724 | ||

60° | 38.8088 | 55.0062 | 69.4149 | 75.5700 | 83.7054 | 95.1822 | ||

FG-X | 0.11 | 15° | 19.2604 | 25.3608 | 36.1914 | 36.8063 | 40.4675 | 40.9340 |

30° | 20.4754 | 28.2998 | 38.7220 | 39.7120 | 45.2151 | 45.8357 | ||

45° | 23.8434 | 34.6543 | 44.6261 | 46.3103 | 55.2276 | 57.4279 | ||

60° | 33.6486 | 47.6758 | 59.9923 | 63.4780 | 72.0668 | 77.0374 | ||

0.14 | 15° | 19.8084 | 26.0430 | 37.1106 | 37.8532 | 41.2910 | 42.0442 | |

30° | 21.0506 | 29.0493 | 39.8008 | 40.7351 | 46.1353 | 47.0096 | ||

45° | 24.4920 | 35.5521 | 45.7961 | 47.4919 | 56.3516 | 58.8017 | ||

60° | 34.4934 | 48.8548 | 61.4386 | 64.9465 | 73.7658 | 78.6049 | ||

0.17 | 15° | 24.0671 | 31.9338 | 45.6821 | 45.9560 | 50.9173 | 51.3375 | |

30° | 25.6288 | 35.6476 | 48.4261 | 49.9805 | 56.8856 | 57.6593 | ||

45° | 29.9300 | 43.6056 | 55.9981 | 58.1580 | 69.4729 | 72.2037 | ||

60° | 42.3389 | 59.8760 | 75.2616 | 79.8475 | 90.3762 | 96.8913 |

**Table 12.**Variation of fundamental natural non-dimensional frequency parameters of FG-CNT-reinforced rhombic plate.

Types | V*_{CNT} | Skew Angle | Boundary Condition | |||
---|---|---|---|---|---|---|

CCSS | CSCS | CCFF | CFCF | |||

UD | 0.11 | 15° | 17.5656 | 16.1325 | 17.1211 | 5.8025 |

30° | 18.3524 | 17.1106 | 17.4540 | 5.8490 | ||

45° | 20.7007 | 19.8447 | 18.4603 | 5.9919 | ||

60° | 28.2486 | 27.7948 | 21.7009 | 6.3069 | ||

0.14 | 15° | 18.1342 | 16.7688 | 17.7012 | 6.2424 | |

30° | 18.9229 | 17.7494 | 18.0355 | 6.2905 | ||

45° | 21.2864 | 20.5072 | 19.0480 | 6.4353 | ||

60° | 28.9146 | 28.5757 | 22.3161 | 6.7554 | ||

0.17 | 15° | 21.8982 | 20.0801 | 21.3412 | 7.1743 | |

30° | 22.8820 | 21.3025 | 21.7584 | 7.2326 | ||

45° | 25.8166 | 24.7178 | 23.0172 | 7.4116 | ||

60° | 35.2478 | 34.6467 | 27.0680 | 7.8059 | ||

FG-V | 0.11 | 15° | 17.0086 | 15.3184 | 16.5114 | 5.0487 |

30° | 17.8273 | 16.3356 | 16.8584 | 5.0936 | ||

45° | 20.2441 | 19.1336 | 17.8998 | 5.2408 | ||

60° | 27.9303 | 27.1387 | 21.2303 | 5.5617 | ||

0.14 | 15° | 17.6618 | 16.0166 | 17.1825 | 5.4678 | |

30° | 18.4787 | 17.0329 | 17.5285 | 5.5141 | ||

45° | 20.9045 | 19.8505 | 18.5709 | 5.6624 | ||

60° | 28.6673 | 27.9750 | 21.9207 | 5.9884 | ||

0.17 | 15° | 21.2161 | 19.0620 | 20.5857 | 6.2207 | |

30° | 22.2464 | 20.3420 | 21.0234 | 6.2774 | ||

45° | 25.2838 | 23.8570 | 22.3339 | 6.4629 | ||

60° | 34.9349 | 33.9038 | 26.5208 | 6.8657 | ||

FG-O | 0.11 | 15° | 15.2966 | 13.7187 | 14.7110 | 4.3016 |

30° | 16.1749 | 14.7745 | 15.0980 | 4.3467 | ||

45° | 18.6942 | 17.6072 | 16.2095 | 4.4929 | ||

60° | 26.5093 | 25.5699 | 19.6311 | 4.7975 | ||

0.14 | 15° | 15.9573 | 14.3811 | 15.4087 | 4.6680 | |

30° | 16.8163 | 15.4187 | 15.7844 | 4.7147 | ||

45° | 19.3078 | 18.2378 | 16.8762 | 4.8622 | ||

60° | 27.1313 | 26.2562 | 20.2771 | 5.1739 | ||

0.17 | 15° | 19.1484 | 17.0768 | 18.4438 | 5.2956 | |

30° | 20.2204 | 18.3681 | 18.9177 | 5.3519 | ||

45° | 23.3092 | 21.8497 | 20.2813 | 5.5337 | ||

60° | 32.9609 | 31.7016 | 24.4972 | 5.9132 | ||

FG-X | 0.11 | 15° | 18.2140 | 17.0254 | 17.7792 | 6.5860 |

30° | 19.0078 | 18.0146 | 18.1123 | 6.6352 | ||

45° | 21.3832 | 20.7913 | 19.1261 | 6.7811 | ||

60° | 29.0061 | 28.8746 | 22.3997 | 7.1022 | ||

0.14 | 15° | 18.7355 | 17.6246 | 18.2942 | 7.0150 | |

30° | 19.5470 | 18.6364 | 18.6340 | 7.0670 | ||

45° | 21.9758 | 21.4797 | 19.6696 | 7.2180 | ||

60° | 29.7641 | 29.7602 | 23.0163 | 7.5489 | ||

0.17 | 15° | 22.7038 | 21.2134 | 22.1242 | 8.1485 | |

30° | 23.7285 | 22.4908 | 22.5506 | 8.2111 | ||

45° | 26.7784 | 26.0495 | 23.8494 | 8.3974 | ||

60° | 36.4902 | 36.3121 | 28.0370 | 8.8055 |

**Table 13.**Variation of fundamental natural non-dimensional frequency parameters of FG-CNT-reinforced rhombic plate.

Types | V*_{CNT} | Skew Angle | a/h | |||
---|---|---|---|---|---|---|

5 | 20 | 50 | 100 | |||

UD | 0.11 | 15° | 9.0743 | 18.2012 | 20.3073 | 20.6787 |

30° | 9.4150 | 18.9711 | 21.0669 | 21.4408 | ||

45° | 10.2558 | 21.3334 | 23.4743 | 23.8687 | ||

60° | 11.3999 | 29.4063 | 32.2438 | 32.8085 | ||

0.14 | 15° | 9.3243 | 19.7973 | 22.5305 | 23.0303 | |

30° | 9.5879 | 20.5395 | 23.2468 | 23.7472 | ||

45° | 10.4551 | 22.8478 | 25.5594 | 26.0749 | ||

60° | 11.6404 | 30.8885 | 34.2574 | 34.9408 | ||

0.17 | 15° | 11.3132 | 22.4497 | 24.9546 | 25.3933 | |

30° | 11.7725 | 23.4173 | 25.9121 | 26.3544 | ||

45° | 12.8172 | 26.3808 | 28.9389 | 29.4075 | ||

60° | 14.2430 | 36.4827 | 39.9125 | 40.5928 | ||

FG-V | 0.11 | 15° | 8.7986 | 15.5123 | 16.6902 | 16.8853 |

30° | 9.4269 | 16.4011 | 17.5995 | 17.8014 | ||

45° | 10.2475 | 19.0147 | 20.3313 | 20.5637 | ||

60° | 11.3736 | 27.4772 | 29.5441 | 29.9513 | ||

0.14 | 15° | 9.1228 | 16.9116 | 18.4397 | 18.6988 | |

30° | 9.6238 | 17.7666 | 19.3058 | 19.5706 | ||

45° | 10.4761 | 20.3260 | 21.9636 | 22.2570 | ||

60° | 11.6465 | 28.7997 | 31.1984 | 31.6713 | ||

0.17 | 15° | 10.9766 | 19.0705 | 20.4470 | 20.6737 | |

30° | 11.8498 | 20.1959 | 21.5996 | 21.8348 | ||

45° | 12.8769 | 23.4938 | 25.0480 | 25.3214 | ||

60° | 14.2886 | 34.1237 | 36.6062 | 37.0957 | ||

FG-O | 0.11 | 15° | 8.0040 | 13.1635 | 13.9508 | 14.0776 |

30° | 8.7975 | 14.0945 | 14.9077 | 15.0418 | ||

45° | 10.2194 | 16.7496 | 17.6897 | 17.8561 | ||

60° | 11.3501 | 25.0679 | 26.6826 | 27.0149 | ||

0.14 | 15° | 8.3081 | 14.3293 | 15.3437 | 15.5098 | |

30° | 9.1008 | 15.2139 | 16.2464 | 16.4188 | ||

45° | 10.4216 | 17.7910 | 18.9359 | 19.1387 | ||

60° | 11.5944 | 26.0648 | 27.8938 | 28.2650 | ||

0.17 | 15° | 9.9887 | 16.1395 | 17.0377 | 17.1815 | |

30° | 10.9641 | 17.2789 | 18.2081 | 18.3607 | ||

45° | 12.7826 | 20.5334 | 21.6118 | 21.8026 | ||

60° | 14.1922 | 30.7514 | 32.6180 | 33.0048 | ||

FG-X | 0.11 | 15° | 9.2309 | 21.2410 | 24.8589 | 25.5557 |

30° | 9.4901 | 21.9714 | 25.5444 | 26.2392 | ||

45° | 10.3425 | 24.2577 | 27.7886 | 28.4923 | ||

60° | 11.5036 | 32.2825 | 36.4515 | 37.3272 | ||

0.14 | 15° | 9.4307 | 22.9391 | 27.5913 | 28.5358 | |

30° | 9.6990 | 23.6674 | 28.2521 | 29.1913 | ||

45° | 10.5801 | 25.9622 | 30.4437 | 31.3820 | ||

60° | 11.7856 | 34.0782 | 39.1212 | 40.2237 | ||

0.17 | 15° | 11.6432 | 26.2338 | 30.6098 | 31.4489 | |

30° | 11.9640 | 27.1943 | 31.5188 | 32.3564 | ||

45° | 13.0307 | 30.1785 | 34.4733 | 35.3262 | ||

60° | 14.4875 | 40.5269 | 45.7146 | 46.8021 |

**Table 14.**Variation of non-dimensional maximum deflection of FG-CNT-reinforced simply supported rhombic plate under uniform loading.

Types | V*_{CNT} | Skew Angle | |||
---|---|---|---|---|---|

15° | 30° | 45° | 60° | ||

UD | 0.11 | 0.00345 | 0.00311 | 0.00234 | 0.00119 |

0.14 | 0.00306 | 0.00278 | 0.00212 | 0.00110 | |

0.17 | 0.00222 | 0.00199 | 0.00150 | 0.00076 | |

FG-V | 0.11 | 0.00401 | 0.00365 | 0.00267 | 0.00128 |

0.14 | 0.00354 | 0.00323 | 0.00240 | 0.00118 | |

0.17 | 0.00257 | 0.00234 | 0.00170 | 0.00081 | |

FG-O | 0.11 | 0.00557 | 0.00479 | 0.00333 | 0.00149 |

0.14 | 0.00486 | 0.00424 | 0.00302 | 0.00140 | |

0.17 | 0.00360 | 0.00310 | 0.00216 | 0.00097 | |

FG-X | 0.11 | 0.00287 | 0.00261 | 0.00201 | 0.00106 |

0.14 | 0.00260 | 0.00237 | 0.00183 | 0.00098 | |

0.17 | 0.00185 | 0.00167 | 0.00127 | 0.00066 |

**Table 15.**Variation of non-dimensional maximum deflection of FG-CNT-reinforced clamped rhombic plate under uniform loading.

Types | V*_{CNT} | Skew Angle | |||
---|---|---|---|---|---|

15° | 30° | 45° | 60° | ||

UD | 0.11 | 0.00201 | 0.00178 | 0.00131 | 0.00065 |

0.14 | 0.00187 | 0.00167 | 0.00124 | 0.00061 | |

0.17 | 0.00127 | 0.00113 | 0.00083 | 0.00041 | |

FG-V | 0.11 | 0.00214 | 0.00189 | 0.00137 | 0.00066 |

0.14 | 0.00197 | 0.00175 | 0.00128 | 0.00062 | |

0.17 | 0.00136 | 0.00120 | 0.00087 | 0.00042 | |

FG-O | 0.11 | 0.00263 | 0.00226 | 0.00158 | 0.00072 |

0.14 | 0.00241 | 0.00210 | 0.00148 | 0.00069 | |

0.17 | 0.00166 | 0.00144 | 0.00101 | 0.00046 | |

FG-X | 0.11 | 0.00186 | 0.00166 | 0.00123 | 0.00061 |

0.14 | 0.00174 | 0.00155 | 0.00116 | 0.00058 | |

0.17 | 0.00117 | 0.00104 | 0.00077 | 0.00038 |

**Table 16.**Variation of non-dimensional maximum deflection of FG-CNT-reinforced simply supported rhombic plate under sin-sin loading.

Types | V*_{CNT} | Skew Angle | |||
---|---|---|---|---|---|

15° | 30° | 45° | 60° | ||

UD | 0.11 | 0.00215 | 0.00158 | 0.00092 | 0.00038 |

0.14 | 0.00192 | 0.00142 | 0.00084 | 0.00036 | |

0.17 | 0.00138 | 0.00101 | 0.00059 | 0.00024 | |

FG-V | 0.11 | 0.00248 | 0.00183 | 0.00102 | 0.00040 |

0.14 | 0.00220 | 0.00163 | 0.00093 | 0.00037 | |

0.17 | 0.00159 | 0.00117 | 0.00065 | 0.00026 | |

FG-O | 0.11 | 0.00339 | 0.00237 | 0.00126 | 0.00047 |

0.14 | 0.00297 | 0.00211 | 0.00115 | 0.00044 | |

0.17 | 0.00219 | 0.00153 | 0.00081 | 0.00030 | |

FG-X | 0.11 | 0.00181 | 0.00135 | 0.00081 | 0.00035 |

0.14 | 0.00165 | 0.00123 | 0.00074 | 0.00032 | |

0.17 | 0.00116 | 0.00086 | 0.00051 | 0.00022 |

**Table 17.**Variation of non-dimensional maximum deflection of FG-CNT-reinforced clamped rhombic plate under sin-sin loading.

Types | V*_{CNT} | Skew Angle | |||
---|---|---|---|---|---|

15° | 30° | 45° | 60° | ||

UD | 0.11 | 0.00136 | 0.00101 | 0.00059 | 0.00025 |

0.14 | 0.00127 | 0.00095 | 0.00056 | 0.00023 | |

0.17 | 0.00086 | 0.00064 | 0.00038 | 0.00016 | |

FG-V | 0.11 | 0.00145 | 0.00106 | 0.00062 | 0.00025 |

0.14 | 0.00134 | 0.00099 | 0.00058 | 0.00024 | |

0.17 | 0.00092 | 0.00067 | 0.00039 | 0.00016 | |

FG-O | 0.11 | 0.00177 | 0.00128 | 0.00071 | 0.00027 |

0.14 | 0.00163 | 0.00118 | 0.00067 | 0.00026 | |

0.17 | 0.00112 | 0.00081 | 0.00045 | 0.00017 | |

FG-X | 0.11 | 0.00126 | 0.00094 | 0.00056 | 0.00023 |

0.14 | 0.00118 | 0.00088 | 0.00052 | 0.00022 | |

0.17 | 0.00080 | 0.00059 | 0.00035 | 0.00015 |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Ansari, M.I.; Kumar, A.; Barnat-Hunek, D.; Suchorab, Z.; Andrzejuk, W.; Majerek, D.
Static and Dynamic Response of FG-CNT-Reinforced Rhombic Laminates. *Appl. Sci.* **2018**, *8*, 834.
https://doi.org/10.3390/app8050834

**AMA Style**

Ansari MI, Kumar A, Barnat-Hunek D, Suchorab Z, Andrzejuk W, Majerek D.
Static and Dynamic Response of FG-CNT-Reinforced Rhombic Laminates. *Applied Sciences*. 2018; 8(5):834.
https://doi.org/10.3390/app8050834

**Chicago/Turabian Style**

Ansari, Md Irfan, Ajay Kumar, Danuta Barnat-Hunek, Zbigniew Suchorab, Wojciech Andrzejuk, and Dariusz Majerek.
2018. "Static and Dynamic Response of FG-CNT-Reinforced Rhombic Laminates" *Applied Sciences* 8, no. 5: 834.
https://doi.org/10.3390/app8050834