Independent coordinate coupling method for vibration analysis of a functionally graded carbon nanotube–reinforced plate with central hole

Abstract

In this article, the free vibration of a functionally graded carbon nanotube–reinforced plate with central hole is investigated by means of the independent coordinates-based Rayleigh–Ritz method. For the proposed method, the kinematic and potential energies are substituted into Lagrange’s equation in order to obtain the equation of motion. However, the total energies are computed by the difference of energies between the hole domain and the plate domain. By applying the displacement matching condition at the hole domain, two coordinate systems are coupled. For the Rayleigh–Ritz method, the mode shape functions of uniform beams are assumed as admissible functions. By this method, convergent results can be obtained with certain number of terms of admissible functions. The present results clearly reflect the effects of the carbon nanotube distribution type, carbon nanotube volume fraction, hole size, and boundary condition on the nondimensional natural frequencies. The provided results show that the present method is efficient in studying the vibration problems of functionally graded carbon nanotube–reinforced plate with central hole.

Background introduction

Carbon nanotubes (CNTs) have received more and more attention since the last decade because of their excellent properties, especially for Young’s modulus and shear modulus. Because of their excellent properties, the CNTs are taken as one kind of promising reinforcement materials for manufacturing lightweight, but high-strength composite materials. In many practical applications, the composite plates are often required to open holes for specific design purposes. Since opening holes reduces both structural weight and stiffness, it is of vital importance to study the dynamic response of plates with open holes for the sake of design requirement as well as structural safety. Moreover, when applying the CNT reinforcements, the CNTs are usually distributed functionally through the thickness. The functionally graded (FG) properties make the analysis harder than the uniformly distributed case. Therefore, it is very meaningful to develop mathematical model to analyze the functionally graded carbon nanotube (FG-CNT)-reinforced plate with open holes.

The mechanical modeling of CNT-reinforced laminated composite structures has been widely studied since the recent decades. The modeling works of FG-CNT-reinforced plates are based on the conventional FG plates and shells.^1–5 The Eshelby–Mori–Tanaka approach^6–8 is normally used to estimate the effective material properties of composites reinforced by CNT. Sobhani Aragh et al. ⁸ also used the Eshelby–Mori–Tanaka approach to evaluate the effective properties of elastic isotropic medium with CNTs for vibration analysis of cylindrical panels. Zhang and colleagues^9–11 conducted the study of flexural strength of cylindrical panels with FG properties by means of the first-order shear deformation theory and conducted the detail parametric studies. Mehrabadi and Sobhani Aragh ¹² analyzed the stress of FG cylindrical shell reinforced by agglomerated CNTs which is subjected to mechanical loads. Zhang and colleagues^13–16 also investigated the postbuckling of FG-CNT-reinforced plates under axial compression, where the translation and rotation of edges are elastically restrained. For the free vibration, Yas et al. ¹⁷ investigated the three-dimensional vibration of CNT-reinforced cylindrical panels with FG properties. The free vibration of FG-CNT-reinforced triangular plates was also studied by Zhang and colleagues^18–21 using the first-order shear deformation theory and element-free, improved moving least-squares (IMLS)–Ritz method. The free vibration of quadrilateral laminates with CNT-reinforced layers was later investigated by Malekzadeh and Zarei. ²² The Rayleigh–Ritz method ²³ was also commonly adopted to study the vibration problems of rectangular nanoplates. The parameter effects, such as CNT volume fraction and distribution type, thickness to width ratio, aspect ratio, and boundary condition, on the natural frequencies have been well revealed in these works. The large-amplitude free vibration based on the Euler–Bernoulli beam theory and nonlinear geometric relation was studied by He et al. ²⁴ for CNT/fibers/polymer laminated beams. And for FG-CNT-reinforced beams with piezoelectric layers, Rafiee et al. ²⁵ also investigated the large-amplitude free vibration for the temperature load and electrical load. Ke et al. ²⁶ used an iterative approach to study the nonlinear vibration problem of FG-CNT-reinforced beams by means of Ritz method. Shen and Xiang ²⁷ studied the nonlinear vibration of FG-CNT-reinforced cylindrical panels resting on elastic foundations in thermal environments.

The aforementioned works are all about the CNT-reinforced plates and shells without open holes. When the structures have open holes, the modeling work becomes different and more complicated. The rectangular plates with rectangular cutouts were studied by Ali and Atwal ²⁸ with a Rayleigh–Ritz method–based approach. Using the partitioning method and orthogonal polynomials as admissible functions, Lam and Hung ²⁹ investigated the vibrations of plates with stiffened openings. Similar problems were also studied by Paramasivam, ³⁰ but with the finite difference method. The isotropic plates with circular holes have also been studied by many literature works. ³¹ However, the circular geometry of holes results in more complicated calculations for the static and dynamic analyses. The least-squares point-matching method was adopted by Hegarty and Ariman ³² and Lee and Lim ³³ to analyze a rectangular plate with circular hole under polar coordinate. The discrete Ritz method was advanced by Liew et al. ³⁴ for analysis of vibration of rectangular plates with cutouts. Kwak and Han ³⁵ studied the vibration of isotropic plates which have rectangular and circular holes by means of an independent coordinate coupling method.

Until now, many literature works have introduced the uniform plates and shells with holes, while the FG plates with open holes have not been comprehensively studied yet, especially for the FG-CNT-reinforced composite plates. Therefore, we study the vibration of FG-CNT-reinforced plate with central hole by means of a simple independent coordinate coupling method based Rayleigh–Ritz approach ³⁶ in this article. For the CNT-reinforced composite materials, the effective material properties are evaluated by the extended rule of mixture. The proposed approach adopts two coordinate systems for the plate and hole domains that are independent and later coupled by applying the displacement matching condition at the hole domain. By means of Lagrange’s equation, where the total energies are computed by the energy difference between the plate and hole domains, the equations of motion can be obtained, which further forms a standard eigenvalue problem. Different types of beam mode shape functions are taken as admissible functions in both in-plane directions. The numerical results are given for four types of CNT distributions. The effects of CNT volume fraction, hole size, and boundary condition on the nondimensional frequencies are investigated. With the proposed method, it is computationally accurate and efficient to solve the free vibration problem of FG-CNT-reinforced plate with central hole.

Statement of the problem

Consider a rectangular CNT-reinforced plate with rectangular hole shown in Figure 1; the geometric dimensions are denoted by the symbols a, b, and h for the length, width, and thickness, respectively. The length and width of the hole are represented by a_h and b_h, respectively. Two coordinates are shown in the figure as well, where (x, y, z) and (x_h, y_h, z_h) represent the coordinate system for the plate and hole, individually. Different CNT distributions are considered as shown in Figure 2, where the CNTs are assumed to be FG through the thickness only. As shown in Figure 2, UD defines the uniform distribution of CNT through the thickness; FG-V, FG-O, and FG-X define three different distributions of CNT through the thickness. Based on these distribution assumptions, the volume fraction of CNT through the thickness can be expressed as follows, where $V_{\mathrm{CNT}}$ is the function of the coordinate z

$$V_{\mathrm{CNT}}=V_{\mathrm{CNT}}^{*}\left(\mathrm{UD}\right)$$

(1a)

$$V_{\mathrm{CNT}}\left(z\right)=(1+2z/h)V_{\mathrm{CNT}}^{*}\left(\mathrm{FG}\mathrm{-}\mathrm{V}\right)$$

(1b)

$$V_{\mathrm{CNT}}\left(z\right)=2(1-2\left|z\right|/h)V_{\mathrm{CNT}}^{*}\left(\mathrm{FG}\mathrm{-}\mathrm{O}\right)$$

(1c)

$$V_{\mathrm{CNT}}\left(z\right)=2(2\left|z\right|/h)V_{\mathrm{CNT}}^{*}\left(\mathrm{FG}\mathrm{-}\mathrm{X}\right)$$

(1d)

where

$$V_{\mathrm{CNT}}^{*}=\frac{w_{\mathrm{CNT}}}{w_{\mathrm{CNT}}+({\rho }^{\mathrm{CNT}}/{\rho }^{\mathrm{m}})-({\rho }^{\mathrm{CNT}}/{\rho }^{\mathrm{m}})w_{\mathrm{CNT}}}$$

(2)

and mass fraction of CNT is denoted by $w_{\mathrm{CNT}}$ ; the densities of CNT and matrix are denoted by ${\rho }^{\mathrm{CNT}}$ and ${\rho }^{\mathrm{m}}$ , respectively.

Figure 1.

Coordinate systems of a rectangular plate with a central rectangular hole.

Figure 2.

Four types of CNT distributions through the thickness with central hole: (a) UD, (b) FG-V, (c) FG-O, and (d) FG-X.

The extended rule of mixture is used to estimate the material properties of FG-CNT-reinforced composite

$$E_{11}={\eta }_1{V}_{\mathrm{CNT}}{E}_{11}^{\mathrm{CNT}}+V_{\mathrm{m}}{E}^{\mathrm{m}}$$

(3a)

$$\frac{{\eta }_2}{E_{22}}=\frac{V_{\mathrm{CNT}}}{E_{22}^{\mathrm{CNT}}}+\frac{V_{\mathrm{m}}}{E^{\mathrm{m}}}$$

(3b)

$$\frac{{\eta }_3}{G_{12}}=\frac{V_{\mathrm{CNT}}}{G_{12}^{\mathrm{CNT}}}+\frac{V_{\mathrm{m}}}{G^{\mathrm{m}}}$$

(3c)

where the superscripts CNT and m represent the material properties of CNT and matrix material, respectively. The efficiency parameters of CNT are defined by ${\eta }_i$ .

Poisson’s ratio $v_{12}$ and density $\rho$ are calculated by the following two equations

$$v_{12}=V_{\mathrm{CNT}}^{*}{v}_{12}^{\mathrm{CNT}}+V_{\mathrm{m}}{v}^{\mathrm{m}}$$

(4a)

$$\rho =V_{\mathrm{CNT}}^{*}{\rho }^{\mathrm{CNT}}+V_{\mathrm{m}}{\rho }^{\mathrm{m}}$$

(4b)

where the volume fraction of the matrix $V_{\mathrm{m}}$ has the following relation to $V_{\mathrm{CNT}}^{*}$

$$V_{\mathrm{CNT}}^{*}+V_{\mathrm{m}}=1$$

(5)

The constitutive relation for the CNT-reinforced plate is given by

$$\left\{\begin{array}{c}{\sigma }_1\hfill \\ \multicolumn{1}{c}{{\sigma }_2}\\ \multicolumn{1}{c}{{\sigma }_4}\\ \multicolumn{1}{c}{{\sigma }_5}\\ \multicolumn{1}{c}{{\sigma }_6}\end{array}\right\}=\left[\begin{array}{ccccc}{Q}_{11}\hfill & Q_{12}\hfill & 0\hfill & 0\hfill & Q_{16}\hfill \\ \multicolumn{1}{c}{}& Q_{22}\hfill & 0\hfill & 0\hfill & Q_{26}\hfill \\ \multicolumn{1}{c}{}& \hfill & Q_{44}\hfill & Q_{45}\hfill & 0\hfill \\ \multicolumn{1}{c}{}& \mathrm{symm}.\hfill & \hfill & Q_{55}\hfill & 0\hfill \\ \multicolumn{1}{c}{}& \hfill & \hfill & \hfill & Q_{66}\hfill \end{array}\right]\left\{\begin{array}{c}{\epsilon }_1\hfill \\ \multicolumn{1}{c}{{\epsilon }_2}\\ \multicolumn{1}{c}{{\epsilon }_4}\\ \multicolumn{1}{c}{{\epsilon }_5}\\ \multicolumn{1}{c}{{\epsilon }_6}\end{array}\right\}$$

(6)

where $Q_{\mathrm{ij}}$ are the material constants in the plate coordinate system which are dependent of the thickness coordinate z. The kinematic and potential energies are computed for the plate and hole domains in each individual coordinate system. For the plate domain

$$T=\frac{1}{2}\underset{0}{\overset{a}{\int }}\underset{0}{\overset{b}{\int }}\underset{-h/2}{\overset{h/2}{\int }}\rho \left(z\right)\stackrel{·}{w}{(x,y,t)}^2\mathrm{dxdydz}$$

(7a)

$$\begin{array}{c}\multicolumn{1}{c}{V=\frac{1}{2}\underset{0}{\overset{a}{\int }}\underset{0}{\overset{b}{\int }}(D_{11}{\left(\frac{{\partial }^{2}w}{\partial x^2}\right)}^2+2D_{12}\frac{{\partial }^{2}w}{\partial x^2}\frac{{\partial }^{2}w}{\partial y^2}}\\ \multicolumn{1}{c}{+4D_{66}{\left(\frac{{\partial }^{2}w}{\partial x\partial y}\right)}^2+D_{22}{\left(\frac{{\partial }^{2}w}{\partial y^2}\right)}^2)\mathrm{dxdy}}\end{array}$$

(7b)

and for the hole domain

$$T_h=\frac{1}{2}\underset{0}{\overset{a_h}{\int }}\underset{0}{\overset{b_h}{\int }}\underset{-h/2}{\overset{h/2}{\int }}\rho \left(z\right){\stackrel{·}{w}}_h{(x_h,y_h,t)}^{2}d{x}_{h}d{y}_{h}d{z}_h$$

(7c)

$$\begin{array}{c}\multicolumn{1}{c}{V_h=\frac{1}{2}\underset{0}{\overset{a_h}{\int }}\underset{0}{\overset{b_h}{\int }}(D_{11}{\left(\frac{{\partial }^2{w}_h}{\partial x_h^2}\right)}^2+2D_{12}\frac{{\partial }^2{w}_h}{\partial x_h^2}\frac{{\partial }^2{w}_h}{\partial y_h^2}}\\ \multicolumn{1}{c}{+4D_{66}{\left(\frac{{\partial }^2{w}_h}{\partial x_h\partial y_h}\right)}^2+D_{22}{\left(\frac{{\partial }^2{w}_h}{\partial y_h^2}\right)}^2){\mathrm{dx}}_h{\mathrm{dy}}_h}\end{array}$$

(7d)

where $w$ and $w_h$ describe the deflections of the plate and hole, respectively; ρ is the density; and D_ij represents the bending stiffness component which is calculated by the following equation

$$D_{\mathrm{ij}}=\underset{-h/2}{\overset{h/2}{\int }}{\overline{Q}}_{\mathrm{ij}}\left(z\right)z^2\mathrm{dz}$$

(8)

Introducing the nondimensional coordinates for both the plate coordinate and the hole coordinate as follows

$$\xi =x/a,\eta =y/b$$

(9a)

$${\xi }_h=x_h/a_h,{\eta }_h=y_h/b_h$$

(9b)

Therefore, the deflections of the plate and the hole can be expressed using the assumed mode method as follows

$$w(\xi ,\eta ,t)=\mathrm{\Phi }(\xi ,\eta )q\left(t\right)$$

(10a)

$$w_h({\xi }_h,{\eta }_h,t)={\mathrm{\Phi }}_h({\xi }_h,{\eta }_h)q_h\left(t\right)$$

(10b)

where Φ and Φ _h are admissible functions consisting of n and n_h terms, respectively. They are represented by the following forms

$$\mathrm{\Phi }(\xi ,\eta )={\left[\begin{array}{cccc}{\mathrm{\Phi }}_1\hfill & {\mathrm{\Phi }}_2\hfill & \cdots \hfill & {\mathrm{\Phi }}_n\hfill \end{array}\right]}_{1\times n}$$

(11a)

$$q\left(t\right)={\left[\begin{array}{cccc}{q}_1\hfill & q_2\hfill & \cdots \hfill & q_n\hfill \end{array}\right]}_{n\times 1}^T$$

(11b)

$${\mathrm{\Phi }}_h({\xi }_h,{\eta }_h)={\left[\begin{array}{cccc}{\mathrm{\Phi }}_{h1}\hfill & {\mathrm{\Phi }}_{h2}\hfill & \cdots \hfill & {\mathrm{\Phi }}_{h{n}_h}\hfill \end{array}\right]}_{1\times n_h}$$

(11c)

$$q_h\left(t\right)={\left[\begin{array}{cccc}{q}_{h1}\hfill & q_{h2}\hfill & \cdots \hfill & q_{h{n}_h}\hfill \end{array}\right]}_{n_h\times 1}^T$$

(11d)

The energy equations can be expressed as following using the expressions of equation (7). For the plate domain

$$T=\frac{1}{2}{\stackrel{·}{q}}^{T}M\stackrel{·}{q}$$

(12a)

$$V=\frac{1}{2}{q}^T\mathrm{Kq}$$

(12b)

where

$$M=\mathrm{ab}\underset{-h/2}{\overset{h/2}{\int }}\rho \left(z\right)\mathrm{dz}·\overline{M}$$

(12c)

$$K=b/a^3\overline{K}$$

(12d)

$$\overline{M}=\underset{0}{\overset{1}{\int }}\underset{0}{\overset{1}{\int }}{\mathrm{\Phi }}^T\mathrm{\Phi }d\xi d\eta$$

(12e)

$$\begin{array}{c}\multicolumn{1}{c}{\overline{K}=\underset{0}{\overset{1}{\int }}\underset{0}{\overset{1}{\int }}(D_{11}\frac{{\partial }^2{\mathrm{\Phi }}^T}{\partial {\xi }^2}\frac{{\partial }^2\mathrm{\Phi }}{\partial {\xi }^2}+2D_{12}{\alpha }^2\frac{{\partial }^2{\mathrm{\Phi }}^T}{\partial {\xi }^2}\frac{{\partial }^2\mathrm{\Phi }}{\partial {\eta }^2}+4D_{66}{\alpha }^2\frac{{\partial }^2{\mathrm{\Phi }}^T}{\partial \xi \partial \eta }\frac{{\partial }^2\mathrm{\Phi }}{\partial \xi \partial \eta }+D_{22}{\alpha }^4\frac{{\partial }^2{\mathrm{\Phi }}^T}{\partial {\eta }^2}\frac{{\partial }^2\mathrm{\Phi }}{\partial {\eta }^2})d\xi d\eta }\end{array}$$

(12f)

and for the hole domain

$$T_h=\frac{1}{2}{\stackrel{·}{q}}_h^T{M}_h{\stackrel{·}{q}}_h$$

(13a)

$$V_h=\frac{1}{2}{q}_h^T{K}_h{q}_h$$

(13b)

where

$$M_h=a_h{b}_h\underset{-h/2}{\overset{h/2}{\int }}\rho \left(z\right)\mathrm{dz}·{\overline{M}}_h$$

(13c)

$$K_h=b_h/b_h{a}_h^3{a}_h^3{\overline{K}}_h$$

(13d)

$${\overline{M}}_h=\underset{0}{\overset{1}{\int }}\underset{0}{\overset{1}{\int }}{\mathrm{\Phi }}_h^T{\mathrm{\Phi }}_{h}d{\xi }_{h}d{\eta }_h$$

(13e)

$${\overline{K}}_h=\underset{0}{\overset{1}{\int }}\underset{0}{\overset{1}{\int }}(D_{11}\frac{{\partial }^2{\mathrm{\Phi }}_h^T}{\partial {\xi }_h^2}\frac{{\partial }^2{\mathrm{\Phi }}_h}{\partial {\xi }_h^2}+2D_{12}{\alpha }_h^2\frac{{\partial }^2{\mathrm{\Phi }}_h^T}{\partial {\xi }_h^2}\frac{{\partial }^2{\mathrm{\Phi }}_h}{\partial {\eta }_h^2}+4D_{66}{\alpha }_h^2\frac{{\partial }^2{\mathrm{\Phi }}_h^T}{\partial {\xi }_h\partial {\eta }_h}\frac{{\partial }^2{\mathrm{\Phi }}_h}{\partial {\xi }_h\partial {\eta }_h}+D_{22}{\alpha }_h^4\frac{{\partial }^2{\mathrm{\Phi }}_h^T}{\partial {\eta }_h^2}\frac{{\partial }^2{\mathrm{\Phi }}_h}{\partial {\eta }_h^2})d{\xi }_{h}d{\eta }_h$$

(13f)

The coefficients α and α_h are denoted by α = a/b and α_h = a_h/b_h in the above two equations. To solve the free vibration problem, the nondimensional mass matrix and stiffness matrix given by equations (12) and (13) have to be obtained first, so that the eigen-problem can be solved. Thus, using the valuable separation, the admissible functions (equation (11a) and equation (11c)) are divided into two separate admissible functions and expressed as follows

$${\mathrm{\Phi }}_i(\xi ,\eta )={\varphi }_i\left(\xi \right){\psi }_i\left(\eta \right),(i=1,2,\dots ,n)$$

(14a)

$${\mathrm{\Phi }}_{\mathrm{hi}}({\xi }_h,{\eta }_h)={\varphi }_{\mathrm{hi}}\left({\xi }_h\right){\psi }_{\mathrm{hi}}\left({\eta }_h\right),(i=1,2,\dots ,n_h)$$

(14b)

First, we calculate the nondimensionalized mass and stiffness matrices for the plate domain. Substituting equation (14a) into equations (12e) and (12f), the following equations can be obtained

$${\overline{M}}_{\mathrm{ij}}={\mathrm{X}}_{1,\mathrm{ij}}{\mathrm{Y}}_{1,\mathrm{ij}}$$

(15a)

$$\begin{array}{c}\multicolumn{1}{c}{{\overline{K}}_{\mathrm{ij}}=D_{11}{\mathrm{X}}_{3,\mathrm{ij}}{\mathrm{Y}}_{1,\mathrm{ij}}+2D_{12}{\alpha }^2{\mathrm{X}}_{4,\mathrm{ij}}{\mathrm{Y}}_{4,\mathrm{ij}}}\\ \multicolumn{1}{c}{+4D_{66}{\alpha }^2{\mathrm{X}}_{2,\mathrm{ij}}{\mathrm{Y}}_{2,\mathrm{ij}}+D_{22}{\alpha }^4{\mathrm{X}}_{1,\mathrm{ij}}{\mathrm{Y}}_{3,\mathrm{ij}}}\end{array}$$

(15b)

where i, j = 1, 2, …, m and the coefficient matrices X _i and Y _i are denoted by the integrations of admissible functions and given by the following forms

$$\begin{array}{c}\multicolumn{1}{c}{{\mathrm{X}}_{1,\mathrm{ij}}=\underset{0}{\overset{1}{\int }}{\varphi }_i{\varphi }_{j}d\xi ,{\mathrm{X}}_{2,\mathrm{ij}}=\underset{0}{\overset{1}{\int }}\varphi {\prime }_i\varphi {\prime }_{j}d\xi ,}\\ \multicolumn{1}{c}{{\mathrm{X}}_{3,\mathrm{ij}}=\underset{0}{\overset{1}{\int }}{\varphi ″}_i{\varphi ″}_{j}d\xi ,{\mathrm{X}}_{4,\mathrm{ij}}=\underset{0}{\overset{1}{\int }}{\varphi }_i{\varphi ″}_{j}d\xi }\end{array}$$

(15c)

$$\begin{array}{c}\multicolumn{1}{c}{{\mathrm{Y}}_{1,\mathrm{ij}}=\underset{0}{\overset{1}{\int }}{\psi }_i{\psi }_{j}d\eta ,{\mathrm{Y}}_{2,\mathrm{ij}}=\underset{0}{\overset{1}{\int }}\psi {\prime }_i\psi {\prime }_{j}d\eta ,}\\ \multicolumn{1}{c}{{\mathrm{Y}}_{3,\mathrm{ij}}=\underset{0}{\overset{1}{\int }}{\psi ″}_i{\psi ″}_{j}d\eta ,{\mathrm{Y}}_{4,\mathrm{ij}}=\underset{0}{\overset{1}{\int }}{\psi }_i{\psi ″}_{j}d\eta }\end{array}$$

(15d)

where i, j = 1, 2, …, n. Assuming that N separated admissible functions are adopted in both the x and y directions, the total combination of admissible functions is N² that is n. By denoting the admissible functions for the x and y directions as ${\chi }_i(i=1,2,\dots ,N)$ and ${\gamma }_i(i=1,2,\dots ,N)$ , respectively; equation (14a) can be expressed as follows

$${\varphi }_k=\{\begin{array}{cc}{\chi }_1\hfill & 1\le k\le N\hfill \\ \multicolumn{1}{c}{{\chi }_2}& N+1\le k\le 2N\hfill \\ \multicolumn{1}{c}{{\chi }_3}& 2N+1\le k\le 3N\hfill \\ \multicolumn{1}{c}{⋮}& ⋮\hfill \\ \multicolumn{1}{c}{{\chi }_N}& (N-1)N+1\le k\le N^2\hfill \end{array}$$

(16a)

$${\psi }_k=\{\begin{array}{cc}{\gamma }_k\hfill & 1\le k\le N\hfill \\ \multicolumn{1}{c}{{\gamma }_{k-N}}& N+1\le k\le 2N\hfill \\ \multicolumn{1}{c}{{\gamma }_{k-2N}}& 2N+1\le k\le 3N\hfill \\ \multicolumn{1}{c}{⋮}& ⋮\hfill \\ \multicolumn{1}{c}{{\gamma }_{k-(N-1)N}}& (N-1)N+1\le k\le N^2\hfill \end{array}$$

(16b)

Thus, the coefficients in equation (15c) and (15d) can be found by solving N² integrations as follows

$${\Sigma }_{\mathrm{ij}}=\underset{0}{\overset{1}{\int }}{\chi }_i{\chi }_{j}d\xi$$

(17a)

$${\overline{\Sigma }}_{ij}=\underset{0}{\overset{1}{\int }}{{\chi }^{\prime }}_i{{\chi }^{\prime }}_{j}d\xi$$

(17b)

$${\widehat{\Sigma }}_{ij}=\underset{0}{\overset{1}{\int }}{{\chi }^{″}}_i{{\chi }^{″}}_{j}d\xi$$

(17c)

$${\tilde{\Sigma }}_{\mathrm{ij}}=\underset{0}{\overset{1}{\int }}{\chi }_i{\chi ″}_{j}d\xi$$

(17d)

$${\mathrm{\Gamma }}_{\mathrm{ij}}=\underset{0}{\overset{1}{\int }}{\gamma }_i{\gamma }_{j}d\eta$$

(17e)

$${\overline{\mathrm{\Gamma }}}_{ij}=\underset{0}{\overset{1}{\int }}{{\gamma }^{\prime }}_i{{\gamma }^{\prime }}_{j}d\eta$$

(17f)

$${\hat{\mathrm{\Gamma }}}_{\mathrm{ij}}=\underset{0}{\overset{1}{\int }}{\gamma ″}_i{\gamma ″}_{j}d\eta$$

(17g)

$${\tilde{\mathrm{\Gamma }}}_{ij}=\underset{0}{\overset{1}{\int }}{\gamma }_i{{\gamma }^{″}}_{j}d\eta$$

(17h)

where i, j = 1, 2, …, N. And, the coefficient matrices can be represented by the following expressions

$${\mathrm{X}}_1=\left[\begin{array}{cccc}{\Sigma }_{11}\overline{I}\hfill & {\Sigma }_{12}\overline{I}\hfill & \cdots \hfill & {\Sigma }_{1N}\overline{I}\hfill \\ \multicolumn{1}{c}{{\Sigma }_{21}\overline{I}}& {\Sigma }_{22}\overline{I}\hfill & \cdots \hfill & {\Sigma }_{2N}\overline{I}\hfill \\ \multicolumn{1}{c}{⋮}& ⋮\hfill & \ddots \hfill & ⋮\hfill \\ \multicolumn{1}{c}{{\Sigma }_{N1}\overline{I}}& {\Sigma }_{N2}\overline{I}\hfill & \cdots \hfill & {\Sigma }_{\mathrm{NN}}\overline{I}\hfill \end{array}\right]$$

(18a)

$${\mathrm{X}}_2=\left[\begin{array}{cccc}{\overline{\Sigma }}_{11}\overline{I}\hfill & {\overline{\Sigma }}_{12}\overline{I}\hfill & \cdots \hfill & {\overline{\Sigma }}_{1N}\overline{I}\hfill \\ \multicolumn{1}{c}{{\overline{\Sigma }}_{21}\overline{I}}& {\overline{\Sigma }}_{22}\overline{I}\hfill & \cdots \hfill & {\overline{\Sigma }}_{2N}\overline{I}\hfill \\ \multicolumn{1}{c}{⋮}& ⋮\hfill & \ddots \hfill & ⋮\hfill \\ \multicolumn{1}{c}{{\overline{\Sigma }}_{N1}\overline{I}}& {\overline{\Sigma }}_{N2}\overline{I}\hfill & \cdots \hfill & {\overline{\Sigma }}_{\mathrm{NN}}\overline{I}\hfill \end{array}\right]$$

(18b)

$${\mathrm{X}}_3=\left[\begin{array}{cccc}{\hat{\Sigma }}_{11}\overline{I}\hfill & {\hat{\Sigma }}_{12}\overline{I}\hfill & \cdots \hfill & {\hat{\Sigma }}_{1N}\overline{I}\hfill \\ \multicolumn{1}{c}{{\hat{\Sigma }}_{21}\overline{I}}& {\hat{\Sigma }}_{22}\overline{I}\hfill & \cdots \hfill & {\hat{\Sigma }}_{2N}\overline{I}\hfill \\ \multicolumn{1}{c}{⋮}& ⋮\hfill & \ddots \hfill & ⋮\hfill \\ \multicolumn{1}{c}{{\hat{\Sigma }}_{N1}\overline{I}}& {\hat{\Sigma }}_{N2}\overline{I}\hfill & \cdots \hfill & {\hat{\Sigma }}_{\mathrm{NN}}\overline{I}\hfill \end{array}\right]$$

(18c)

$${\mathrm{X}}_4=\left[\begin{array}{cccc}{\tilde{\Sigma }}_{11}\overline{I}\hfill & {\tilde{\Sigma }}_{12}\overline{I}\hfill & \cdots \hfill & {\tilde{\Sigma }}_{1N}\overline{I}\hfill \\ \multicolumn{1}{c}{{\tilde{\Sigma }}_{21}\overline{I}}& {\tilde{\Sigma }}_{22}\overline{I}\hfill & \cdots \hfill & {\tilde{\Sigma }}_{2N}\overline{I}\hfill \\ \multicolumn{1}{c}{⋮}& ⋮\hfill & \ddots \hfill & ⋮\hfill \\ \multicolumn{1}{c}{{\tilde{\Sigma }}_{N1}\overline{I}}& {\tilde{\Sigma }}_{N2}\overline{I}\hfill & \cdots \hfill & {\tilde{\Sigma }}_{\mathrm{NN}}\overline{I}\hfill \end{array}\right]$$

(18d)

$${\mathrm{Y}}_1=\left[\begin{array}{cccc}\mathrm{\Gamma }\hfill & \mathrm{\Gamma }\hfill & \cdots \hfill & \mathrm{\Gamma }\hfill \\ \multicolumn{1}{c}{\mathrm{\Gamma }}& \mathrm{\Gamma }\hfill & \cdots \hfill & \mathrm{\Gamma }\hfill \\ \multicolumn{1}{c}{⋮}& ⋮\hfill & \ddots \hfill & ⋮\hfill \\ \multicolumn{1}{c}{\mathrm{\Gamma }}& \mathrm{\Gamma }\hfill & \cdots \hfill & \mathrm{\Gamma }\hfill \end{array}\right]$$

(18e)

$${\mathrm{Y}}_2=\left[\begin{array}{cccc}\overline{\mathrm{\Gamma }}\hfill & \overline{\mathrm{\Gamma }}\hfill & \cdots \hfill & \overline{\mathrm{\Gamma }}\hfill \\ \multicolumn{1}{c}{\overline{\mathrm{\Gamma }}}& \overline{\mathrm{\Gamma }}\hfill & \cdots \hfill & \overline{\mathrm{\Gamma }}\hfill \\ \multicolumn{1}{c}{⋮}& ⋮\hfill & \ddots \hfill & ⋮\hfill \\ \multicolumn{1}{c}{\overline{\mathrm{\Gamma }}}& \overline{\mathrm{\Gamma }}\hfill & \cdots \hfill & \overline{\mathrm{\Gamma }}\hfill \end{array}\right]$$

(18f)

$${\mathrm{Y}}_3=\left[\begin{array}{cccc}\hat{\mathrm{\Gamma }}\hfill & \hat{\mathrm{\Gamma }}\hfill & \cdots \hfill & \hat{\mathrm{\Gamma }}\hfill \\ \multicolumn{1}{c}{\hat{\mathrm{\Gamma }}}& \hat{\mathrm{\Gamma }}\hfill & \cdots \hfill & \hat{\mathrm{\Gamma }}\hfill \\ \multicolumn{1}{c}{⋮}& ⋮\hfill & \ddots \hfill & ⋮\hfill \\ \multicolumn{1}{c}{\hat{\mathrm{\Gamma }}}& \hat{\mathrm{\Gamma }}\hfill & \cdots \hfill & \hat{\mathrm{\Gamma }}\hfill \end{array}\right]$$

(18g)

$${\mathrm{Y}}_4=\left[\begin{array}{cccc}\tilde{\mathrm{\Gamma }}\hfill & \tilde{\mathrm{\Gamma }}\hfill & \cdots \hfill & \tilde{\mathrm{\Gamma }}\hfill \\ \multicolumn{1}{c}{\tilde{\mathrm{\Gamma }}}& \tilde{\mathrm{\Gamma }}\hfill & \cdots \hfill & \tilde{\mathrm{\Gamma }}\hfill \\ \multicolumn{1}{c}{⋮}& ⋮\hfill & \ddots \hfill & ⋮\hfill \\ \multicolumn{1}{c}{\tilde{\mathrm{\Gamma }}}& \tilde{\mathrm{\Gamma }}\hfill & \cdots \hfill & \tilde{\mathrm{\Gamma }}\hfill \end{array}\right]$$

(18h)

where $\overline{I}$ is an n by n matrix full of ones.

For different boundary conditions, the admissible functions can be selected from different mode shape functions of beam. The details are introduced as following three cases:

• 1. Simply supported condition

$${\chi }_i=\sqrt{2}\sin \left(i\pi \xi \right),(i=1,2,\dots ,N)$$

(19)

• 2. Clamped condition

$$\begin{array}{c}\multicolumn{1}{c}{{\chi }_i=\cosh \left({\lambda }_i\xi \right)-\cos \left({\lambda }_i\xi \right)-{\sigma }_i(\sinh \left({\lambda }_i\xi \right)-\sin \left({\lambda }_i\xi \right)),}\\ \multicolumn{1}{c}{(i=1,2,\dots ,N)}\end{array}$$

(20)

where λ_i = 4.730, 7.853, 10.996, 14.137, …, ${\sigma }_i=(\cosh ({\lambda }_i)-\cos ({\lambda }_i))/(\sinh ({\lambda }_i)-\sin ({\lambda }_i))$ .

• 3. Free-edge condition

$${\chi }_1=1$$

(21a)

$${\chi }_2=\sqrt{12}(\xi -1)$$

(21b)

$$\begin{array}{c}\multicolumn{1}{c}{{\chi }_{i+2}=\cosh \left({\lambda }_i\xi \right)+\cos \left({\lambda }_i\xi \right)-{\sigma }_i(\sinh \left({\lambda }_i\xi \right)+\sin \left({\lambda }_i\xi \right)),}\\ \multicolumn{1}{c}{(i=1,2,\dots ,N-2)}\end{array}$$

(21c)

By substituting equations (19)–(21) into equation (17a)–(17d), N² integrations can be computed for different boundary conditions. The coefficient matrices X _i can be consequently obtained from equation (18a)–(18d). The same method can be adopted for calculating the matrices Y _i . Thus, the the nondimensionalized matrices can be obtained from equation (15a) and (15b). It should be noted that using different admissible functions represents the different boundary conditions of the plate. Thus, different admissible functions can be combined, and the model is valid for various boundary conditions.

After that, it has to calculate the nondimensionalized matrices for the hole domain by repeating the above same method. While, for the hole domain, the free edge boundary type is used only since four edges of the hole are free. For the sake of brevity, they will not be introduced again. By applying the displacement matching condition, we can get

$$w_h({\xi }_h,{\eta }_h,t)=w(\xi ,\eta ,t)$$

(22)

Using equations (10) and (14), equation (22) can be expressed by

$$\begin{array}{c}\multicolumn{1}{c}{\sum _{j=1}^{n_h}{\mathrm{\Phi }}_{\mathrm{hj}}({\xi }_h,{\eta }_h)q_{\mathrm{hj}}\left(t\right)=\sum _{j=1}^{n_h}{\varphi }_{\mathrm{hj}}\left({\xi }_h\right){\psi }_{\mathrm{hj}}\left({\eta }_h\right)q_{\mathrm{hj}}\left(t\right)}\\ \multicolumn{1}{c}{=\sum _{k=1}^n{\mathrm{\Phi }}_k(\xi ,\eta )q_k\left(t\right)=\sum _{k=1}^n{\varphi }_k\left(\xi \right){\psi }_k\left(\eta \right)q_k\left(t\right)}\end{array}$$

(23)

Multiplying ${\varphi }_{\mathrm{hi}}\left({\xi }_h\right){\psi }_{\mathrm{hi}}\left({\eta }_h\right)$ and performing the integration, equation (23) can be expressed as follows

$$\begin{array}{c}\multicolumn{1}{c}{\sum _{j=1}^{n_h}\underset{1}{\overset{1}{\int }}\underset{0}{\overset{1}{\int }}{\varphi }_{\mathrm{hi}}\left({\xi }_h\right){\psi }_{\mathrm{hi}}\left({\eta }_h\right){\varphi }_{\mathrm{hj}}\left({\xi }_h\right){\psi }_{\mathrm{hj}}\left({\eta }_h\right)d{\xi }_{h}d{\eta }_h{q}_{\mathrm{hj}}\left(t\right)}\\ \multicolumn{1}{c}{=\sum _{k=1}^n\underset{0}{\overset{1}{\int }}\underset{0}{\overset{1}{\int }}{\varphi }_{\mathrm{hi}}\left({\xi }_h\right){\psi }_{\mathrm{hi}}\left({\eta }_h\right){\varphi }_k\left(\xi \right){\psi }_k\left(\eta \right)d{\xi }_{h}d{\eta }_h{q}_k\left(t\right),}\\ \multicolumn{1}{c}{(i=1,2,\dots ,n_h)}\end{array}$$

(24)

According to the orthogonal property of the eigenfunctions, equation (24) can be simplified as

$$\begin{array}{c}\multicolumn{1}{c}{q_{\mathrm{hi}}\left(t\right)=\sum _{k=1}^n\underset{0}{\overset{1}{\int }}{\varphi }_{\mathrm{hi}}\left({\xi }_h\right){\varphi }_k\left(\xi \right)d{\xi }_h\underset{0}{\overset{1}{\int }}{\psi }_{\mathrm{hi}}\left({\eta }_h\right){\psi }_k\left(\eta \right)d{\eta }_h{q}_k\left(t\right)}\\ \multicolumn{1}{c}{=\sum _{k=1}^n{\left(S_h\right)}_{\mathrm{ik}}{q}_k\left(t\right),(i=1,2,\dots ,n_h)}\end{array}$$

(25)

Equation (25) can be rewritten into the matrix form as follows

$$q_h=S_{h}q$$

(26)

where S_h is the coordinate transformation matrix which has the dimension of n_h by n. The above equation represents the relation between the matrices in two coordinate systems. Two coordinate systems have to satisfy the following geometric relation

$$\xi =\frac{d_x}{a}+\frac{a_h}{a}{\xi }_h,\eta =\frac{d_y}{b}+\frac{b_h}{b}{\eta }_h,z=z_h$$

(27)

Thus, substituting equation (27) into equation (25), the transformation matrix can be obtained using the numerical integration. The total kinetic and potential energies are defined by the energy difference between the plate domain and the hole domain. And, with the help of equation (26), they can be calculated as follows

$$T_{\mathrm{total}}=\frac{1}{2}{\stackrel{·}{q}}^T{M}_T\stackrel{·}{q}$$

(28a)

$$V_{\mathrm{total}}=\frac{1}{2}{q}^T{K}_{T}q$$

(28b)

where

$$M_T=M-S_h^T{M}_h{S}_h$$

(29a)

$$K_T=K-S_h^T{K}_h{S}_h$$

(29b)

Equation (29) can be rewritten as following equation by using the nondimensional variables

$$M_T=\mathrm{ab}\underset{-h/2}{\overset{h/2}{\int }}\rho \left(z\right)\mathrm{dz}·{\overline{M}}_T$$

(30a)

$$K_T=b/a^3{\overline{K}}_T$$

(30b)

where

$${\overline{M}}_T=\overline{M}-\left({\overline{a}}_h{\overline{b}}_h\right)S_h^T{\overline{M}}_h{S}_h$$

(31a)

$${\overline{K}}_T=\overline{K}-{\overline{b}}_h/{\overline{a}}_h^3{S}_h^T{\overline{K}}_h{S}_h$$

(31b)

$${\overline{a}}_h=a_h/a$$

(31c)

$${\overline{b}}_h=b_h/b$$

(31d)

The equation of motion can be calculated by substituting equation (28) into Lagrange’s equation that forms a standard eigenvalue problem as follows

$$[K_T-{\omega }^2{M}_T]A=0$$

(32)

The above eigenvalue problem can be solved with equations (30) and (31). The natural frequencies and the corresponding modal shapes can be obtained.

Numerical results and discussion

The numerical examples of FG-CNT-reinforced plates with central hole will be demonstrated in this section, and the numerical results will be discussed to reveal the effects of CNT distribution type, CNT volume fraction, hole size, and boundary condition on the nondimensional natural frequencies. The material properties ³⁷ of the matrix and CNT are shown in Table 1. The efficiency parameters of CNT are defined in Table 2. For numerical study, simple assumptions that ${\eta }_3={\eta }_2$ and $G_{23}=G_{12}=G_{13}$ are made.

Table 1.

Material properties of CNT and matrix materials.

CNT	Matrix
$E_{11}^{\mathrm{CNT}}=5.6466\mathrm{TPa}$	E m = 2.1 GPa
$E_{22}^{\mathrm{CNT}}=7.0800\mathrm{TPa}$	v m = 0.34
$G_{12}^{\mathrm{CNT}}=1.9445\mathrm{TPa}$	ρ m = 1150 kg/m3
$v_{12}^{\mathrm{CNT}}=0.175$
ρ CNT = 1400 kg/m3

CNT: carbon nanotubes.

Table 2.

CNT efficiency parameters with respect to different volume fractions.

$V_{\mathrm{CNT}}^{*}$	${\eta }_1$	${\eta }_2$
0.11	0.149	0.934
0.14	0.150	0.941
0.17	0.149	1.381

CNT: carbon nanotubes.

First, we present the results for the case of four-edge simply supported (SSSS) square plate without hole. The results are obtained with the volume fraction of CNT equivalent to 0.11 and for four types of CNT distributions. The natural frequencies are nondimensionalized by $\overline{\omega }=\omega (b^2/h)\sqrt{{\rho }^{\mathrm{m}}/E^{\mathrm{m}}}$ , and the first nine natural frequencies are given in Table 3 for different CNT distribution types. The present results are convergent, which are obtained using 49 terms of admissible functions. The results are also compared with those given by Zhu et al. ³⁷ It is found that the results agree well with the solutions provided by Zhu et al. except a small difference for the FG-V case. For the plate without central hole, it can be seen that the FG-X case presents the largest frequencies, but the FG-O case presents the smallest frequencies among four types. It can also be found that the UD case and the FG-V case possess similar frequencies. The four-edge clamped (CCCC) boundary condition is also examined, and the corresponding frequency results are shown in Table 4. The natural frequencies are also nondimensionalized by $\overline{\omega }=\omega (b^2/h)\sqrt{{\rho }^{\mathrm{m}}/E^{\mathrm{m}}}$ and validated by Zhu et al. ³⁷ It is found that the frequencies under the CCCC boundary condition are much larger than the frequencies under the SSSS boundary condition. The FG-X case also possesses the largest frequencies among four cases, and FG-O case also possesses the smallest frequencies. These results are also compared with the reference work. Although some difference exists between two methods, it is still tolerable to predict the free vibration using the present Rayleigh–Ritz method.

Table 3.

Nondimensionalized natural frequencies of four-edge simply supported (SSSS) CNT-reinforced functionally graded square plate without hole, $V_{\mathrm{CNT}}^{*}=0.11(\overline{\omega }=\omega (b^2/h)\sqrt{{\rho }_m/E_m})$ .

	Modes	1	2	3	4	5	6	7	8	9
UD	Present	19.581	23.704	34.538	52.695	76.134	77.551	78.325	83.861	94.816
	Zhu et al. 37	19.223	23.408	34.669	54.043	70.811	72.900	–	–	–
FG-V	Present	19.551	23.638	34.524	52.821	76.101	77.879	78.205	83.643	94.552
	Zhu et al. 37	16.252	21.142	33.350	53.430	60.188	62.780	–	–	–
FG-O	Present	14.373	19.299	31.089	49.652	54.764	57.492	64.290	74.403	77.196
	Zhu et al. 37	14.302	19.373	31.615	51.370	53.035	55.825	–	–	–
FG-X	Present	23.616	27.257	37.523	55.561	80.797	92.644	94.463	99.230	109.030
	Zhu et al. 37	22.984	26.784	37.591	56.946	83.150	84.896	–	–	–

CNT: carbon nanotubes.

Table 4.

Nondimensionalized natural frequencies of four-edge clamped (CCCC) CNT-reinforced functionally graded square plate without hole, $V_{\mathrm{CNT}}^{*}=0.11(\overline{\omega }=\omega (b^2/h)\sqrt{{\rho }_m/E_m})$ .

	Modes	1	2	3	4	5	6	7	8	9
UD	Present	43.660	47.587	57.523	75.096	100.279	118.717	121.071	126.612	132.512
	Zhu et al. 37	39.730	43.876	54.768	74.488	98.291	100.537	–	–	–
FG-V	Present	43.645	47.563	57.556	75.273	100.675	118.692	120.979	126.452	133.182
	Zhu et al. 37	34.165	39.043	51.204	72.202	86.291	89.054	–	–	–
FG-O	Present	31.707	36.616	48.227	67.319	85.297	88.271	93.383	95.206	108.003
	Zhu et al. 37	30.303	35.163	46.811	65.631	76.338	80.460	–	–	–
FG-X	Present	52.951	56.382	65.441	82.192	107.015	139.433	144.562	146.537	151.310
	Zhu et al. 37	46.166	49.934	60.225	79.534	108.694	110.921	–	–	–

CNT: carbon nanotubes.

Next is to examine the effects of CNT volume fraction, distribution type, and varying hole size on the nondimensionalized frequencies. The SSSS boundary condition is examined first as well, and the corresponding frequency results are given in Tables 5–8 for the cases of UD, FG-V, FG-O, and FG-X, respectively. In Table 5, the frequencies of the UD case are demonstrated. Two factors, CNT volume fraction and varying hole size, are investigated. It can be found that increasing the CNT volume fraction, the nondimensionalized frequencies increase significantly due to the factor that the strength of the CNT is much larger than the matrix. The square hole is predefined to be located at the geometric center of the plate. Therefore, the natural frequencies also increase significantly if increasing the hole size. When the hole size is 0.5, which represents the ratio of a_h/a, the nondimensionalized fundamental frequencies are 29.174, 31.747, and 36.074 for the volume fractions of 0.11, 0.14, and 0.17, respectively, that are 49%, 45%, and 50% larger than the corresponding cases without hole. Table 6 gives the results of FG-V plate with central hole. The CNT volume fraction and hole size are also investigated. The values of the nondimensional frequencies are very close to the previous UD case. Similar comments on the effects of volume fraction and hole size can be made according to the previous case. Increasing the CNT volume fraction also enhances the natural frequencies. Increasing the hole size, the dimenisonalized natural frequencies also increase significantly, but not linearly. Tables 7 and 8 give the frequency results for the FG-O and FG-X cases, respectively. It is also found that with the existence of the central hole, the frequencies of the FG-X case also present largest values among four types, while FG-O case present smallest values. This is due to the factor that if the CNTs distributed close to the surface, it enhances the stiffness of the plate, which will increase the frequencies in turn. However, when the CNTs distributed close to the midplane, it reduces the stiffness and decreases the frequencies. Therefore, the effect of CNT distribution types has been well demonstrated by the given results.

Table 5.

Nondimensionalized frequencies of simply supported (SSSS) FG-CNT square plate with various dimensions of hole for UD case.

$V_{\mathrm{CNT}}^{*}$	ac/a	1	2	3	4	5	6	7	8	9
0.11	0	19.581	23.704	34.538	52.695	76.134	77.551	78.325	83.861	94.816
	0.1	19.964	23.719	35.214	52.820	76.181	78.277	78.879	83.910	94.670
	0.2	21.085	23.922	37.180	54.265	76.783	78.240	80.851	84.503	94.709
	0.3	22.929	24.670	40.192	58.477	78.550	78.814	79.763	86.610	96.607
	0.4	25.568	26.364	43.059	65.623	77.646	80.834	80.869	91.791	101.963
	0.5	29.174	29.529	43.500	70.028	82.693	84.032	84.702	100.002	109.151
0.14	0	21.896	25.768	36.290	54.375	79.425	85.565	87.584	92.731	103.072
	0.1	22.322	25.784	37.003	54.503	80.794	85.618	87.540	92.786	102.934
	0.2	23.550	26.004	39.106	55.997	82.884	86.282	87.523	93.453	103.017
	0.3	25.513	26.810	42.436	60.367	81.786	88.098	88.172	95.955	105.106
	0.4	28.224	28.617	45.843	67.866	79.525	90.184	90.281	102.277	111.093
	0.5	31.747	31.967	46.856	72.647	86.332	91.726	94.069	111.251	111.485
0.17	0	24.116	29.328	42.972	65.759	93.693	96.463	96.901	103.467	117.313
	0.1	24.587	29.347	43.812	65.914	93.751	96.402	98.558	103.527	117.128
	0.2	25.973	29.598	46.252	67.717	94.493	96.351	101.006	104.258	117.168
	0.3	28.259	30.525	49.972	72.969	96.692	97.056	99.645	106.832	119.511
	0.4	31.548	32.627	53.471	81.861	97.018	99.566	99.632	113.125	126.110
	0.5	36.074	36.558	53.924	87.314	101.971	104.419	104.946	123.232	135.891

FG-CNT: functionally graded carbon nanotube.

Table 6.

Nondimensionalized frequencies of simply supported (SSSS) FG-CNT square plate with various dimensions of hole for FG-V case.

$V_{\mathrm{CNT}}^{*}$	ac/a	1	2	3	4	5	6	7	8	9
0.11	0	19.551	23.638	34.524	52.821	76.101	77.879	78.205	83.643	94.552
	0.1	19.933	23.653	35.199	52.946	76.148	78.163	79.211	83.692	94.423
	0.2	21.054	23.856	37.164	54.394	76.748	78.140	81.186	84.285	94.494
	0.3	22.896	24.602	40.173	58.616	78.497	78.722	80.093	86.406	96.404
	0.4	25.534	26.294	43.033	65.771	77.971	80.731	80.755	91.643	101.760
	0.5	29.143	29.457	43.469	70.165	82.515	84.369	84.564	99.906	109.726
0.14	0	21.871	25.727	36.342	54.631	79.965	85.537	87.485	92.565	102.908
	0.1	22.297	25.743	37.056	54.760	81.343	85.590	87.446	92.619	102.786
	0.2	23.525	25.963	39.161	56.260	83.441	86.252	87.441	93.287	102.898
	0.3	25.490	26.768	42.489	60.651	82.333	88.050	88.097	95.804	105.001
	0.4	28.209	28.577	45.882	68.174	80.064	90.087	90.196	102.171	110.992
	0.5	31.749	31.933	46.878	72.946	86.894	91.587	93.963	111.194	112.374
0.17	0	24.089	29.329	43.200	66.391	93.660	96.357	98.071	103.336	117.315
	0.1	24.560	29.347	44.044	66.547	93.718	96.302	99.746	103.397	117.149
	0.2	25.947	29.599	46.493	68.367	94.460	96.267	102.209	104.127	117.227
	0.3	28.240	30.527	50.216	73.666	96.649	96.983	100.829	106.708	119.590
	0.4	31.549	32.636	53.693	82.621	98.186	99.493	99.558	113.030	126.193
	0.5	36.123	36.583	54.094	88.078	101.863	104.349	106.166	123.206	135.943

FG-CNT: functionally graded carbon nanotube.

Table 7.

Nondimensionalized frequencies of simply supported (SSSS) FG-CNT square plate with various dimensions of hole for FG-O case.

$V_{\mathrm{CNT}}^{*}$	ac/a	1	2	3	4	5	6	7	8	9
0.11	0	14.373	19.299	31.089	49.652	54.764	57.492	64.290	74.403	77.196
	0.1	14.656	19.311	31.692	49.769	54.798	57.436	64.328	75.635	77.045
	0.2	15.515	19.478	33.385	51.121	55.248	57.356	64.767	77.015	77.241
	0.3	17.007	20.099	35.777	55.002	56.748	57.744	66.127	76.183	78.518
	0.4	19.294	21.534	37.633	59.423	59.431	61.349	68.898	74.484	82.574
	0.5	22.727	24.256	37.041	62.330	63.186	64.989	73.851	79.865	87.724
0.14	0	15.953	20.569	32.048	50.507	61.327	63.812	70.095	75.319	82.277
	0.1	16.267	20.582	32.671	50.626	61.365	63.762	70.136	76.584	82.133
	0.2	17.209	20.759	34.441	52.005	61.863	63.702	70.621	78.331	82.135
	0.3	18.820	21.418	37.008	55.988	63.464	64.155	72.190	77.259	83.757
	0.4	21.245	22.931	39.139	62.592	65.962	66.058	75.403	75.672	88.188
	0.5	24.799	25.789	38.805	66.454	68.514	69.796	81.129	81.854	94.151
0.17	0	17.621	23.689	38.210	61.061	67.127	70.483	78.856	91.523	94.758
	0.1	17.968	23.704	38.951	61.204	67.169	70.414	78.903	93.037	94.572
	0.2	19.022	23.909	41.031	62.867	67.720	70.316	79.442	94.533	95.006
	0.3	20.851	24.672	43.967	67.637	69.561	70.790	81.107	93.706	96.377
	0.4	23.660	26.434	46.240	72.850	72.860	75.436	84.495	91.622	101.352
	0.5	27.877	29.777	45.503	76.436	77.473	79.906	90.548	98.228	107.659

FG-CNT: functionally graded carbon nanotube.

Table 8.

Nondimensionalized frequencies of simply supported (SSSS) FG-CNT square plate with various dimensions of hole for FG-X case.

$V_{\mathrm{CNT}}^{*}$	ac/a	1	2	3	4	5	6	7	8	9
0.11	0	23.616	27.257	37.523	55.561	80.797	92.644	94.463	99.230	109.030
	0.1	24.073	27.275	38.261	55.692	82.196	92.701	94.426	99.288	108.913
	0.2	25.374	27.507	40.469	57.219	84.366	93.406	94.436	100.014	109.058
	0.3	27.405	28.352	44.050	61.702	83.261	95.141	95.200	102.866	111.305
	0.4	30.125	30.236	47.902	69.450	80.901	97.036	97.271	110.118	117.787
	0.5	33.519	33.709	49.389	74.511	88.028	98.355	100.907	113.261	119.263
0.14	0	26.493	29.962	40.022	58.139	83.820	104.265	105.971	110.478	116.438
	0.1	27.003	29.981	40.813	58.276	85.278	104.329	105.937	110.543	117.002
	0.2	28.425	30.236	43.216	59.876	87.587	105.102	105.966	111.368	119.931
	0.3	30.563	31.154	47.243	64.586	86.466	106.746	106.899	114.761	122.426
	0.4	33.176	33.306	51.842	72.816	83.932	108.530	108.937	123.340	123.407
	0.5	36.548	36.871	54.011	78.406	91.647	109.693	112.456	117.103	132.171
0.17	0	29.143	33.948	47.363	70.713	103.216	114.175	116.574	122.872	135.790
	0.1	29.709	33.969	48.294	70.880	104.999	114.245	116.525	122.945	135.636
	0.2	31.331	34.259	51.058	72.823	107.738	115.121	116.528	123.837	135.799
	0.3	33.896	35.317	55.483	78.517	106.316	117.400	117.425	127.277	138.586
	0.4	37.391	37.685	60.115	88.314	103.345	119.898	120.114	136.012	146.575
	0.5	41.850	42.066	61.696	94.612	112.296	121.692	124.864	144.905	147.739

FG-CNT: functionally graded carbon nanotube.

The CCCC boundary condition case is examined as well, and the corresponding results are given in Tables 9–12 for the cases of UD, FG-V, FG-O, and FG-X, respectively. The square hole is also predefined at the middle of the square plate with varying dimensions. First, the present nondimensionalized frequencies are much larger than the frequencies of the simply supported boundary conditions. Second, with the same amount of CNTs, the FG-X plate possesses larger stiffness than the other three cases. Thus, the FG-X case presents the largest frequencies as well for the CCCC boundary condition. Third, the frequencies change significantly along with the varying hole size. Significant increment can be observed when increasing the hole size. These conclusions are very close to the previous SSSS boundary condition case. Therefore, by these results, the effects of CNT distribution type, CNT volume fraction, and hole size on the first nine natural frequencies are well demonstrated for the CCCC boundary condition. The conclusions are also clearly given according to the present results.

Table 9.

Nondimensionalized frequencies of clamped (CCCC) FG-CNT square plate with various dimensions of hole for UD case.

$V_{\mathrm{CNT}}^{*}$	ac/a	1	2	3	4	5	6	7	8	9
0.11	0	43.660	47.587	57.523	75.096	100.279	118.717	121.071	126.612	132.512
	0.1	44.910	47.652	59.031	102.412	118.858	121.007	126.758	133.386	154.386
	0.2	47.853	48.460	64.097	105.598	120.386	121.157	128.402	139.535	155.148
	0.3	51.220	51.450	73.957	103.016	123.167	123.622	135.593	143.468	155.107
	0.4	54.470	58.640	84.482	102.720	125.156	133.106	134.204	150.806	156.866
	0.5	56.997	71.130	85.737	126.436	129.111	137.625	152.337	154.784	184.803
0.14	0	48.965	52.615	62.043	79.133	104.092	133.469	135.637	136.399	140.772
	0.1	50.342	52.686	63.693	106.335	133.626	135.579	137.302	140.931	167.025
	0.2	53.362	53.577	69.427	109.873	135.257	135.785	142.832	143.699	167.870
	0.3	56.506	56.866	80.868	107.326	137.801	138.554	147.883	151.366	167.938
	0.4	59.400	64.820	92.969	107.273	138.187	139.492	149.156	166.024	172.645
	0.5	61.608	78.472	93.106	137.564	140.638	141.905	168.405	169.797	198.130
0.17	0	53.750	58.727	71.288	93.422	125.053	146.092	149.072	156.091	165.470
	0.1	55.293	58.807	73.152	127.707	146.265	148.990	156.271	166.561	191.206
	0.2	58.964	59.805	79.383	131.638	148.161	149.167	158.285	174.225	192.147
	0.3	63.223	63.497	91.446	128.400	151.681	152.198	167.054	179.107	192.080
	0.4	67.372	72.370	104.309	128.010	154.233	163.879	167.574	185.962	193.993
	0.5	70.610	87.810	106.052	155.862	160.395	171.812	187.763	191.376	229.270

FG-CNT: functionally graded carbon nanotube.

Table 10.

Nondimensionalized frequencies of clamped (CCCC) FG-CNT square plate with various dimensions of hole for FG-V case.

$V_{\mathrm{CNT}}^{*}$	ac/a	1	2	3	4	5	6	7	8	9
0.11	0	43.645	47.563	57.556	75.273	100.675	118.692	120.979	126.452	133.182
	0.1	44.899	47.627	59.061	102.818	118.832	120.923	126.595	134.062	154.254
	0.2	47.854	48.435	64.121	106.008	120.362	121.094	128.235	140.249	155.017
	0.3	51.238	51.425	73.968	103.402	123.105	123.575	135.458	144.193	154.974
	0.4	54.503	58.612	84.489	103.104	125.042	133.070	134.878	150.706	156.765
	0.5	57.035	71.106	85.830	126.288	129.486	138.308	152.304	154.695	184.844
0.14	0	48.956	52.615	62.143	79.449	104.718	133.449	135.566	137.413	140.663
	0.1	50.337	52.686	63.791	106.976	133.606	135.514	138.325	140.819	167.067
	0.2	53.379	53.577	69.518	110.518	135.239	135.739	142.717	144.775	167.913
	0.3	56.557	56.869	80.943	107.931	137.757	138.523	148.979	151.275	167.976
	0.4	59.483	64.823	93.062	107.850	139.208	139.416	149.137	166.041	172.635
	0.5	61.709	78.494	93.339	138.111	140.539	142.937	168.497	169.793	198.418
0.17	0	53.747	58.788	71.594	94.174	126.408	146.071	149.009	156.045	167.556
	0.1	55.297	58.867	73.460	129.090	146.244	148.936	156.222	168.662	191.685
	0.2	59.007	59.867	79.680	133.031	148.148	149.136	158.227	176.424	192.628
	0.3	63.348	63.565	91.691	129.735	151.675	152.189	166.993	181.341	192.552
	0.4	67.593	72.448	104.497	129.332	154.213	163.888	169.676	186.028	194.391
	0.5	70.905	87.928	106.438	155.827	161.669	173.941	187.877	191.745	230.300

FG-CNT: functionally graded carbon nanotube.

Table 11.

Nondimensionalized frequencies of clamped (CCCC) FG-CNT square plate with various dimensions of hole for FG-O case.

$V_{\mathrm{CNT}}^{*}$	ac/a	1	2	3	4	5	6	7	8	9
0.11	0	31.707	36.616	48.227	67.319	85.297	88.271	93.383	95.206	108.003
	0.1	32.651	36.667	49.445	85.400	88.196	95.278	95.322	126.691	127.983
	0.2	35.216	37.292	53.155	86.622	88.223	96.447	97.661	128.585	132.223
	0.3	38.916	39.611	59.438	89.770	89.976	95.238	100.640	128.425	135.288
	0.4	43.285	45.131	65.327	92.963	94.965	96.898	110.976	127.326	128.836
	0.5	47.259	54.951	66.794	94.872	112.301	113.812	121.124	130.204	146.910
0.14	0	35.366	39.896	50.926	69.547	95.374	95.575	98.280	104.653	116.602
	0.1	36.408	39.951	52.226	95.689	97.347	98.212	104.777	128.667	135.617
	0.2	39.128	40.632	56.321	97.018	98.287	99.993	106.049	134.431	136.264
	0.3	42.785	43.154	63.662	97.478	100.089	100.272	111.136	136.128	137.871
	0.4	46.785	49.175	71.067	97.153	102.777	107.989	123.411	129.371	136.691
	0.5	50.158	59.815	72.967	104.390	118.078	124.766	132.034	132.405	159.469
0.17	0	38.871	44.924	59.235	82.752	104.554	108.216	114.841	116.763	132.535
	0.1	40.029	44.986	60.731	104.681	108.123	116.905	117.170	155.838	157.156
	0.2	43.178	45.754	65.280	106.179	108.154	118.283	120.091	157.895	162.636
	0.3	47.732	48.598	72.970	110.052	110.302	117.113	123.410	157.699	166.387
	0.4	53.121	55.371	80.157	113.994	116.780	118.788	136.057	156.615	158.200
	0.5	58.036	67.422	81.939	116.349	137.683	139.895	148.579	160.152	180.284

FG-CNT: functionally graded carbon nanotube.

Table 12.

Nondimensionalized frequencies of clamped (CCCC) FG-CNT square plate with various dimensions of hole for FG-X case.

$V_{\mathrm{CNT}}^{*}$	ac/a	1	2	3	4	5	6	7	8	9
0.11	0	52.951	56.382	65.441	82.192	107.015	139.433	144.562	146.537	151.310
	0.1	54.421	56.458	67.198	109.345	140.361	144.732	146.490	151.477	176.353
	0.2	57.406	57.466	73.451	113.159	146.417	146.756	146.948	153.587	177.266
	0.3	60.415	60.922	86.043	110.695	148.716	149.760	151.311	163.140	177.467
	0.4	63.064	69.447	98.992	111.097	141.292	150.180	161.196	176.089	185.620
	0.5	65.068	83.941	98.221	144.305	145.267	151.214	178.729	181.972	208.370
0.14	0	59.510	62.757	71.464	87.902	112.707	145.493	162.728	164.583	169.086
	0.1	61.125	62.841	73.414	115.190	146.464	162.918	164.540	169.271	188.926
	0.2	63.894	64.194	80.527	119.482	153.382	164.688	164.864	171.729	188.210
	0.3	66.924	67.781	94.821	117.265	158.050	166.763	168.220	182.705	187.809
	0.4	69.329	77.270	108.223	119.120	147.484	168.057	180.992	193.242	207.708
	0.5	71.148	93.147	106.472	151.899	156.453	169.020	196.346	201.885	227.346
0.17	0	65.301	69.850	81.793	103.688	135.894	177.751	178.149	180.761	187.079
	0.1	67.131	69.943	83.974	138.840	178.359	178.933	180.697	187.286	220.106
	0.2	71.041	71.126	91.646	143.551	180.488	181.010	187.305	189.851	221.232
	0.3	74.986	75.486	107.042	140.288	183.574	184.720	192.804	201.470	221.390
	0.4	78.565	86.047	123.193	140.417	180.095	185.565	198.852	219.373	229.488
	0.5	81.276	104.109	122.913	181.111	185.027	186.942	222.622	225.456	260.724

FG-CNT: functionally graded carbon nanotube.

Concluding remarks

In this work, we presented an approximate approach for predicting the free vibration problem of FG-CNT-reinforced plate with central hole. The present approach is based on two independent coordinates and the Rayleigh–Ritz procedure. The beam mode shape functions are adopted as the assumption of admissible functions. Various boundary conditions are considered, and the total energies are defined by the energy difference between the plate domain and the hole domain. The equation of motion is derived by applying Lagrange’s equation and further forms a standard eigenvalue problem. The numerical results well demonstrate the effects of CNT distribution types, CNT volume fraction, and hole size on the nondimensional natural frequencies. Some of the results are validated by the existed literature works to verify the correctness of the present methodology. This work is easy to implement and computationally efficient in calculating the FG-CNT-reinforced plate with varying size of central hole. The provided results can be used as references in engineering design of CNT-reinforced plate when required to open holes.