Bending and free vibration analyses of antisymmetrically laminated carbon nanotube-reinforced functionally graded plates

Abstract

In this paper, a simple four-variable first-order shear deformation theory is further applied to solve the bending and free vibration problems of antisymmetrically laminated functionally graded carbon nanotube (FG-CNT)-reinforced composite plates. The adopted four-variable theory contains only four unknowns in its displacement field which is less than the Reddy’s first-order theory. The equations of motion are derived from the Hamilton’s principle with the help of specific boundary conditions. Laminated FG-CNT-reinforced plates with different distribution types of carbon nanotube through the thickness are considered. The material properties of individual layer are estimated by using the extended rule of mixture. Analytical solutions of various simply supported antisymmetric cross-ply and angle-ply laminates are given for case study. The effects of carbon nanotube volume fraction, length to width ratio and thickness to width ratio on the non-dimensional fundamental frequency and the central deflection are investigated for antisymmetrically laminated FG-CNT-reinforced plates.

Keywords

Bending free vibration carbon nanotube functionally graded antisymmetric laminate

Introduction

Carbon nanotubes (CNTs) have received considerable attention in engineering fields owing to their outstanding mechanical, electrical and thermal properties, especially for the elastic moduli which are as high as 1 TPa for the shear modulus and 5 TPa for the Young’s modulus. Thus, the CNTs are nowadays considered as one of the most promising reinforcements for composite materials and have great potential in replacing traditional reinforcement materials for engineering applications. Thus, this topic has inspired great interest for researchers and numerous research works have been fulfilled related to the CNT-reinforced composite structures since last decades.

The static and dynamic characteristics of CNT-reinforced composite have been studied analytically, numerically and experimentally since the recent decades. There are a lot of literature works that introduce the current research state of CNT-reinforced composites with uniformly distributed or functionally distributed carbon nanotube (FG-CNT) alignment through the thickness. The modeling of FG-CNT has many similarities with conventional functionally graded plates and shells^1,2 for which the theories have been well established. Since the FG-CNT plates are reinforced by the CNT, the effective material properties of the CNT-reinforced nanocomposites are conventionally estimated by a micromechanics model, the Eshelby–Mori–Tanaka approach or the extended rule of mixture,^3,4 considering the small-scale effect of CNTs on the overall material properties. Aragh et al.⁵ studied the vibrational behavior of continuously graded CNT-reinforced cylindrical panels based on the Eshelby–Mori–Tanaka approach to estimate the effective constitutive law of the elastic isotropic medium with oriented, straight CNTs. Wattanasakulpong and Ungbhakorn⁶ provided the analytical solutions for bending, buckling and vibration responses of CNT-reinforced composite beams resting on the Pasternak elastic foundation using different shear deformation theories. Zhang et al.⁷ carried out the analysis of flexural strength of FG-CNT-reinforced cylindrical panels based on the first-order shear deformation shell theory considering four types of CNT distributions and conducted detail parametric studies to reveal the effects of volume fraction of CNTs, edge to radius ratio and thickness on the flexural strength and free vibration responses of cylindrical panels. Zhu et al.⁸ studied the static bending and free vibration analyses of CNT-reinforced composite plates by using the finite element approach based on the first-order shear deformation plate theory. The size-dependent vibration behavior of FG-CNT-reinforced polymer microcantilevers by introducing a material length scale parameter was studied by Rokni et al.⁹ In their work, the fundamental frequency was improved by the proposed optimum CNT dispersion pattern rather than the commonly used uniform CNT distribution pattern in their work. Mehrabadi and Aragh¹⁰ analyzed the stress of functionally graded open cylindrical shell reinforced by agglomerated CNTs subjected to mechanical loads. They discretized the governing equations by the two-dimensional differential quadrature method in the thickness and longitudinal directions and the trigonometric functions in tangential direction. The thermoelastic analysis of FG-CNT-reinforced composite plate was studied by Alibeigloo and Liew¹¹ using the Fourier series expansion along the in-plane directions and the state space technique across the thickness direction for the analytical solution. The buckling and postbuckling problems of FG-CNT-reinforced composite plates and shells have also been studied well. Lei et al.¹² presented the buckling results of functionally graded CNT-reinforced composite plates using the element-free kp-Ritz method under various in-plane mechanical loads. The postbuckling problem of CNT-reinforced functionally graded plates with edges elastically restrained against translation and rotation under axial compression was studied by Zhang et al.¹³ Yas and Samadi¹⁴ also studied the buckling problem of Timoshenko beams based on elastic foundation that is reinforced by single-walled CNTs and investigated the parameter effect on the buckling characteristics of the beam.

The free vibration and non-linear vibration of FG-CNT reinforced composites have also been comprehensively studied. For the free vibration, Yas et al.¹⁵ investigated the three-dimensional free vibration problems of FG-CNT-reinforced cylindrical panels. Zhang et al.¹⁶ investigated the free vibration problems of FG-CNT-reinforced triangular plates using the FSDT and element-free IMLS-Ritz method. Malekzadeh and Zarei¹⁷ investigated the free vibration of quadrilateral laminated plates with CNT-reinforced composite layers. The Rayleigh–Ritz method¹⁸ was also adopted to analyze the free vibration of rectangular nanoplates. All these works well revealed the important effect of parameters, such as volume fraction of CNT, CNT distribution type, aspect ratio, thickness to width ratio and boundary condition on the natural frequencies. There is also a large quantity of works regarding the non-linear analysis of FG-CNT-reinforced composites. He et al.¹⁹ conducted a comprehensive study on large amplitude free and forced vibration responses of CNT/fibers/polymer laminated multiscale composite beams, where the governing equations were derived by the Euler-Bernoulli beam theory and Von Kármán geometric nonlinearity. Rafiee et al.²⁰ investigated the large amplitude free vibration of FG-CNT-reinforced composite beams with piezoelectric layers subjected to temperature and electrical loadings based on Euler-Bernoulli beam theory. They²¹ also investigated the non-linear dynamic stability of piezoelectric FG-CNT-reinforced composite plates with initial geometric imperfection under thermal and electrical loadings, where the geometric imperfect shape can be of arbitrary types which take the form of the products of trigonometric functions and hyperbolic functions in the in-plane plane. Ke et al.²² also studied the non-linear free vibration of FG-CNT-reinforced composite beams, but with a Ritz method they used to derive the governing eigenvalue equations which were further solved by an iterative approach. Shen and Xiang³ conducted the non-linear analysis of CNT-reinforced composite beams resting on elastic foundations in thermal environments, including large amplitude vibration, non-linear bending and thermal postbuckling of nanocomposite beams. They⁴ also investigated the non-linear vibration of CNT-reinforced composite cylindrical shells in thermal environments. They studied two types of CNT distributions, uniformly distributed and functionally distributed cases. The equations of motion were obtained based on a higher order shear deformation theory and solved by a two-step perturbation technique. They²³ later extended their approach to the non-linear vibration of FG-CNT-reinforced composite cylindrical panels resting on elastic foundations in thermal environments. Guo and Zhang²⁴ investigated the non-linear vibrations of CNT-reinforced composite plates based on Reddy’s third-order shear deformation theory, and the governing equations were also solved by the perturbation method.

Recent researches found that laminated structures of beams, plates and shells present higher mechanical performance than single layer structures. The merits of laminated structures have also inspired the researchers to study the laminated FG-CNT-reinforced composite which are capable of sustaining mechanical loads from various directions due to different CNT orientations. However, these works have not been comprehensively introduced until now, only a few studies have been introduced. Lei et al.²⁵ studied the free vibration problems of laminated FG-CNT-reinforced rectangular plates, but only for the symmetric laminations. They later²⁶ investigated the bending problem of laminated CNT-reinforced functionally graded plates using the element-free kp-Ritz method based on the first-order shear deformation theory. These works are limited to the symmetrically laminated FG-CNT-reinforced composites. Until now, the static and dynamic characteristics of antisymmetrically laminated CNT-reinforced composites have not been comprehensively studied yet. However, the antisymmetrically laminated structures also have significant applications for engineering structures. Thus, in this work, we investigate the bending and free vibration problems of antisymmetrically laminated FG-CNT-reinforced composite plates by using a simple four-variable theory. The effective properties are also estimated by the extended rule of mixture. The Navier solutions are given for the simply supported boundary conditions. Four types of CNT distributions, various volume fractions of CNTs, thickness to width ratios, length to width ratios and laminations are examined to investigate their effects on the static bending and free vibration responses.

Theoretical formulations

Laminated FG-CNT-reinforced plate

Consider a perfectly bonded rectangular CNT-reinforced composite laminate with layer thickness t and N layers which is shown in Figure 1, the length, width and total thickness are denoted by a, b and h, respectively. The coordinate system is shown in the figure as well. Four types of CNT distribution through the layer thickness are shown in Figure 2 that are conventional used and given by other existed works,^26–28 where UD represents the uniform distribution of CNT through the thickness of the matrix and FG-V, FG-O and FG-X denote the functionally graded distributions of CNT through the thickness. Therefore, according to their distributions, the CNT volume fraction $V CNT$ for an individual layer can be expressed by the following equations, which is the function of the thickness coordinate z. It should be noted that the total volume of CNT are same for all four types.

V CNT = V_{CNT}^{*} (UD)

(1a)

V CNT (z) = (1 + 2 z / h) V_{CNT}^{*} (FG - V)

(1b)

V CNT (z) = 2 (1 - 2 | z | / h) V_{CNT}^{*} (FG - O)

(1c)

V CNT (z) = 2 (2 | z | / h) V_{CNT}^{*} (FG - X)

(1d)

where

V_{CNT}^{*} = \frac{w CNT}{w CNT + (ρ CNT / ρ m) - (ρ CNT / ρ m) w CNT}

(2)

and

w CNT

is the mass fraction of CNT,

ρ CNT

and

ρ m

are the densities of the CNT and the matrix material, respectively.

Figure 1.

Coordinate system and geometry of a laminated CNT-reinforced plate.

Figure 2.

Four types of CNT distributions through the thickness: (a) UD; (b) FG-V; (c) FG-O; (d) FG-X.

The effective material properties of FG-CNT-reinforced materials can be simply expressed by the extended rule of mixture as follows.

E 11 = η 1 V CNT E_{11}^{CNT} + V m E m

(3a)

\frac{η 2}{E 22} = \frac{V CNT}{E_{22}^{CNT}} + \frac{V m}{E m}

(3b)

\frac{η 3}{G 12} = \frac{V CNT}{G_{12}^{CNT}} + \frac{V m}{G m}

(3c)

where

E_{11}^{CNT}

E_{22}^{CNT}

and

G_{12}^{CNT}

are the Young’s moduli of CNT,

E m

and

G m

are the Young’s modulus and shear modulus of the isotropic matrix. The CNT efficiency parameters are denoted by

η i (i = 1, 2, 3)

which account for the scale-dependent parameters calculated from the simulation of molecular dynamics.

For the Poisson’s ratio, $v 12$ and density ρ which are also dependent of the thickness and can be calculated by

v 12 = V_{CNT}^{*} v_{12}^{CNT} + V m v m

(4a)

ρ = V CNT ρ CNT + V m ρ m

(4c)

where

v_{12}^{CNT}

and

v m

are the Poisson’s ratios of the CNT and the matrix,

V m

is the volume fraction of the matrix which satisfies the following relation with

V CNT

V CNT + V m = 1

(5)

Kinematics

The displacement field of the first-order shear deformation theory²⁹ is given by the following equations which have five structural unknowns

u 1 (x, y, z) = u (x, y) + z φ x (x, y)

(6a)

u 2 (x, y, z) = v (x, y) + z φ y (x, y)

(6b)

u 3 (x, y, z) = w (x, y)

(6c)

where

(u, v, w, φ x, φ y)

are the structural unknowns, representing three displacements in x, y, z directions, respectively, and two rotations of the normal to the midplane about x and y axes, respectively.

The adopted first order shear deformation theory given in the literature^30–33 can be expressed by dividing the transverse displacement w into the bending and shear parts (i.e. $w = w b + w s$ ) and making the assumptions of the following equations

φ x = - \frac{\partial w b}{\partial x}, φ y = - \frac{\partial w b}{\partial y}

(7)

Thus, the displacements of the adopted first-order shear deformation theory can be given by the following equations which contain only four structural unknowns (u, v, w_b, w_s)

u 1 (x, y, z) = u (x, y) - z \frac{\partial w b}{\partial x}

(8a)

u 2 (x, y, z) = v (x, y) - z \frac{\partial w b}{\partial y}

(8b)

u 3 (x, y, z) = w b (x, y) + w s (x, y)

(8c)

Then, the non-zero strain components can be calculated by the following equations based on the small deformation theory

ɛ x = \frac{\partial u}{\partial x} - z \frac{\partial 2 w b}{\partial x 2}

(9a)

ɛ y = \frac{\partial v}{\partial y} - z \frac{\partial 2 w b}{\partial y 2}

(9b)

γ xy = \frac{\partial u}{\partial y} + \frac{\partial v}{\partial x} - 2 z \frac{\partial 2 w b}{\partial x \partial y}

(9c)

γ xz = \frac{\partial w s}{\partial x}

(9d)

γ yz = \frac{\partial w s}{\partial y}

(9e)

Constitutive equations

For a rectangular plate, the linear constitutive relation for an individual layer can be expressed by the following form

{σ x σ y σ xy σ yz σ xz} = [Q 11 Q 12 000 Q 12 Q 22 00000 Q 66 00000 Q 44 00000 Q 55] {ɛ x ɛ y γ xy γ yz γ xz}

(10)

where Q_ij are the material constants in the material axes of the layer. Since the Young’s modulus, shear modulus and Poisson’s ratio are function of z for the FG-CNT-reinforced materials, the material constants Q_ij(z) are also function of z which are given by the following equations

Q 11 = \frac{E 1}{1 - v 12 v 21}, Q 22 = \frac{E 2}{1 - v 12 v 21}, Q 12 = \frac{v 21 E 1}{1 - v 12 v 21}

Q 66 = G 12, Q 44 = G 23, Q 55 = G 13

(11)

Since the laminate consists of several individual layers with different material orientations with respect to the laminate coordinates, the general form of the constitutive relation has to be transformed into the laminate coordinate through the coordinate transformation and can be further expressed by the following equation for the kth layer.

{σ x σ y σ xy σ yz σ xz} (k) = [\bar{Q} 11 \bar{Q} 12 \bar{Q} 16 00 \bar{Q} 12 \bar{Q} 22 \bar{Q} 26 00 \bar{Q} 16 \bar{Q} 26 \bar{Q} 66 00000 \bar{Q} 44 \bar{Q} 45 000 \bar{Q} 45 \bar{Q} 55] (k) {ɛ x ɛ y γ xy γ yz γ xz}

(12)

where

\bar{Q} ij

are the transformed material constants expressed in terms of material constants and θ is the angle between the global laminate coordinate and the material coordinate of individual layer

\bar{Q} 11 = Q 11 \cos 4 θ + 2 (Q 12 + 2 Q 66) \sin 2 θ \cos 2 θ + Q 22 \sin 4 θ

\bar{Q} 12 = (Q 11 + Q 22 - 4 Q 66) \sin 2 θ \cos 2 θ + Q 12 (\sin 4 θ + \cos 4 θ)

\bar{Q} 22 = Q 11 \sin 4 θ + 2 (Q 12 + 2 Q 66) \sin 2 θ \cos 2 θ + Q 22 \cos 4 θ

\bar{Q} 16 = (Q 11 - Q 12 - 2 Q 66) \sin θ \cos 3 θ + (Q 12 - Q 22 + 2 Q 66) \sin 3 θ \cos θ

\bar{Q} 26 = (Q 11 - Q 12 - 2 Q 66) \sin 3 θ \cos θ + (Q 12 - Q 22 + 2 Q 66) \sin θ \cos 3 θ

(13)

\bar{Q} 66 = (Q 11 + Q 22 - 2 Q 12 - 2 Q 66) \sin 2 θ \cos 2 θ + Q 66 (\sin 4 θ + \cos 4 θ)

\bar{Q} 44 = Q 44 \cos 2 θ + Q 55 \sin 2 θ

\bar{Q} 45 = (Q 55 - Q 44) \cos θ \sin θ

\bar{Q} 55 = Q 55 \cos 2 θ + Q 44 \sin 2 θ

Equations of motion

The equations of motion are derived by applying the Hamilton’s principle, which can be stated by the following form

0 = \int_{0}^{T} (δ U + δ V - δ K) dt

(14)

where δU, δV and δK are the variations of the strain energy, the work done the external force and the kinetic energy, respectively. Firstly, the variation of strain energy can be calculated by

δ U = \int Ω 0 {\int_{- h / 2}^{h / 2} [σ x δ ɛ x + σ y δ ɛ y + σ xy δ γ xy + σ xz δ γ xz + σ yz δ γ yz] dz} dA = \int Ω 0 [N x \frac{\partial δ u}{\partial x} - M x \frac{\partial 2 δ w b}{\partial x 2} + N y \frac{\partial v}{\partial y} - M y \frac{\partial 2 δ w b}{\partial y 2} + N xy (\frac{\partial δ u}{\partial y} + \frac{\partial δ v}{\partial x}) - 2 M xy \frac{\partial 2 δ w b}{\partial x \partial y} + Q x \frac{\partial δ w s}{\partial x} + Q y \frac{\partial δ w s}{\partial y}] dA

(15)

where N, M and Q are the stress resultants defined by

(N x, N y, N xy) = \int_{- h / 2}^{h / 2} (σ x, σ y, σ xy) dz

(16a)

(M x, M y, M xy) = \int_{- h / 2}^{h / 2} (σ x, σ y, σ xy) zdz

(16b)

(Q x, Q y) = \int_{- h / 2}^{h / 2} (σ xz, σ yz) dz

(16c)

Substituting the strains, equation (9), and the constitutive equation, equation (12) into equation (16), the stress resultants can be defined in terms of displacements

{N x N y N xy} = [A 11 A 12 A 16 A 12 A 22 A 26 A 16 A 26 A 66] {\partial u / \partial x \partial v / \partial y \partial u / \partial y + \partial v / \partial x} - [B 11 B 12 B 16 B 12 B 22 B 26 B 16 B 26 B 66] {\partial 2 w b / \partial x 2 \partial 2 w b / \partial y 2 2 \partial 2 w b / \partial x \partial y}

(17a)

{M x M y M xy} = [B 11 B 12 B 16 B 12 B 22 B 26 B 16 B 26 B 66] {\partial u / \partial x \partial v / \partial y \partial u / \partial y + \partial v / \partial x} - [D 11 D 12 D 16 D 12 D 22 D 26 D 16 D 26 D 66] {\partial 2 w b / \partial x 2 \partial 2 w b / \partial y 2 2 \partial 2 w b / \partial x \partial y}

(17b)

{Q x Q y} = κ [A 55 A 45 A 45 A 44] {\partial w s / \partial x \partial w s / \partial y}

(17c)

where κ is the shear correction factor and

(A ij, B ij, D ij)

are the stiffness coefficients defined by

(A ij, B ij, D ij) = \int_{- h / 2}^{h / 2} (1, z, z 2) \bar{Q} ij (z) dz (i, j = 1, 2, 6)

(18a)

A_{ij}^{s} = \int_{- h / 2}^{h / 2} \bar{Q} ij (z) dz (i, j = 4, 5)

(18b)

The shear correction factor κ is taken to be the following form³⁴ for the functionally graded materials

κ = \frac{5}{6 - (v_{12}^{CNT} V CNT + v m V m)}

(19)

Consider a plate composed of N orthotropic layers, the stiffness coefficients can be further defined by

(A ij, B ij, D ij) = \sum_{k = 1}^{N} \int_{z k}^{z k + 1} (1, z, z 2) \bar{Q} ij (z) dz

(20a)

A_{ij}^{s} = \sum_{k = 1}^{N} \int_{z k}^{z k + 1} \bar{Q} ij (z) dz

(20b)

The variation of work done by the external transverse load q can be expressed by

δ V = - \int Ω 0 q δ (w b + w s) dA

(21)

The variation of kinetic energy can be written as

δ K = \int Ω 0 \int_{- h / 2}^{h / 2} ρ (z) (\overset{\cdot}{u} 1 δ \overset{\cdot}{u} 1 + \overset{\cdot}{u} 2 δ \overset{\cdot}{u} 2 + \overset{\cdot}{u} 3 δ \overset{\cdot}{u} 3) dzdA = \int Ω 0 [I 0 (\overset{\cdot}{u} δ \overset{\cdot}{u} + \overset{\cdot}{v} δ \overset{\cdot}{v} + (\overset{\cdot}{w} b + \overset{\cdot}{w} s) δ (\overset{\cdot}{w} b + \overset{\cdot}{w} s)) - I 1 (\overset{\cdot}{u} \frac{\partial δ \overset{\cdot}{w} b}{\partial x} + \frac{\partial \overset{\cdot}{w} b}{\partial x} δ \overset{\cdot}{u} + \frac{\partial \overset{\cdot}{w} b}{\partial y} δ \overset{\cdot}{v} + \overset{\cdot}{v} \frac{\partial δ \overset{\cdot}{w} b}{\partial y}) + I 2 (\frac{\partial \overset{\cdot}{w} b}{\partial x} \frac{\partial δ \overset{\cdot}{w} b}{\partial x} + \frac{\partial \overset{\cdot}{w} b}{\partial y} \frac{\partial δ \overset{\cdot}{w} b}{\partial y})] dA

(22)

where the dot-superscript convention denotes the differentiation with respect to time t; ρ(z) is the mass density; and (I₀, I₁, I₂) are the mass inertias defined by

(I 0, I 1, I 2) = \int_{- h / 2}^{h / 2} (1, z, z 2) ρ (z) dz

(23)

By substituting equation (15), equation (21) and equation (22) into equation (14), integration by parts, the following equations of motion can be obtained

δ u 0 : \frac{\partial N x}{\partial x} + \frac{\partial N xy}{\partial y} = I 0 \overset{··}{u} - I 1 \frac{\partial \overset{··}{w} b}{\partial x}

(24a)

δ v 0 : \frac{\partial N y}{\partial y} + \frac{\partial N xy}{\partial x} = I 0 \overset{··}{v} - I 1 \frac{\partial \overset{··}{w} b}{\partial y}

(24b)

δ w b : \frac{\partial 2 M x}{\partial x 2} + \frac{\partial 2 M y}{\partial y 2} + 2 \frac{\partial 2 M xy}{\partial x \partial y} + q = I 0 (\overset{··}{w} b + \overset{··}{w} s) + I 1 (\frac{\partial \overset{··}{u}}{\partial x} + \frac{\partial \overset{··}{v}}{\partial y}) - I 2 (\frac{\partial 2 \overset{··}{w} b}{\partial x 2} + \frac{\partial 2 \overset{··}{w} b}{\partial y 2})

(24c)

δ w s : \frac{\partial Q x}{\partial x} + \frac{\partial Q y}{\partial y} + q = I 0 (\overset{··}{w} b + \overset{··}{w} s)

(24d)

The boundary conditions can be written in the explicit form as:

Clamped edge

u = v = w b = w s = \frac{\partial w b}{\partial x} = 0, at x = 0, a

(25a)

u = v = w b = w s = \frac{\partial w b}{\partial y} = 0, at y = 0, b

(25b)

Simply supported edge (cross-ply laminate)

N x = v = w b = w s = M x = 0, at x = 0, a

(26a)

u = N y = w b = w s = M y = 0, at y = 0, b

(26b)

Simply supported edge (angle-ply laminate)

u = N xy = w b = w s = M x = 0, at x = 0, a

(27a)

N xy = v = w b = w s = M y = 0, at y = 0, b

(27b)

Free edge

N x = N xy = \frac{\partial M x}{\partial x} + 2 \frac{\partial M xy}{\partial y} + I 2 \frac{\partial \overset{··}{w} b}{\partial x} = Q x = M x = 0, at x = 0, a

(28a)

N xy = N y = 2 \frac{\partial M xy}{\partial x} + \frac{\partial M y}{\partial y} + I 2 \frac{\partial \overset{··}{w} b}{\partial y} = Q y = M y = 0, at y = 0, b

(28b)

By using the stress resultants from equation (17), the equations of motion can be expressed in terms of the displacement unknowns

A 11 \frac{\partial 2 u}{\partial x 2} + 2 A 16 \frac{\partial 2 u}{\partial x \partial y} + A 66 \frac{\partial 2 u}{\partial y 2} + A 16 \frac{\partial 2 v}{\partial x 2} + (A 12 + A 66) \frac{\partial 2 v}{\partial x \partial y} + A 26 \frac{\partial 2 v}{\partial y 2} - B 11 \frac{\partial 3 w b}{\partial x 3} - 3 B 16 \frac{\partial 3 w b}{\partial x 2 \partial y} - (B 12 + 2 B 66) \frac{\partial 3 w b}{\partial x \partial y 2} - B 26 \frac{\partial 3 w b}{\partial y 3} = I 0 \overset{··}{u} - I 1 \frac{\partial \overset{··}{w} b}{\partial x}

(29a)

A 16 \frac{\partial 2 u}{\partial x 2} + (A 12 + A 66) \frac{\partial 2 u}{\partial x \partial y} + A 26 \frac{\partial 2 u}{\partial y 2} + A 66 \frac{\partial 2 v}{\partial x 2} + 2 A 26 \frac{\partial 2 v}{\partial x \partial y} + A 22 \frac{\partial 2 v}{\partial y 2} - B 16 \frac{\partial 3 w b}{\partial x 3} - (B 12 + 2 B 66) \frac{\partial 3 w b}{\partial x 2 \partial y} - 3 B 26 \frac{\partial 3 w b}{\partial x \partial y 2} - B 22 \frac{\partial 3 w b}{\partial y 3} = I 0 \overset{··}{v} - I 1 \frac{\partial \overset{··}{w} b}{\partial y}

(29b)

B 11 \frac{\partial 3 u}{\partial x 3} + 3 B 16 \frac{\partial 3 u}{\partial x 2 \partial y} + (B 12 + 2 B 66) \frac{\partial 3 u}{\partial x \partial y 2} + B 26 \frac{\partial 3 u}{\partial y 3} + B 16 \frac{\partial 3 v}{\partial x 3} + (B 12 + 2 B 66) \frac{\partial 3 v}{\partial x 2 \partial y} + 3 B 26 \frac{\partial 3 v}{\partial x \partial y 2} + B 22 \frac{\partial 3 v}{\partial y 3} - D 11 \frac{\partial 4 w b}{\partial x 4} - 4 D 16 \frac{\partial 4 w b}{\partial x 3 \partial y} - 2 (D 12 + 2 D 66) \frac{\partial 4 w b}{\partial x 2 \partial y 2} - 4 D 26 \frac{\partial 4 w b}{\partial x \partial y 3} - D 22 \frac{\partial 4 w b}{\partial y 4} + q = I 0 (\overset{··}{w} b + \overset{··}{w} s) + I 1 (\frac{\partial \overset{··}{u}}{\partial x} + \frac{\partial \overset{··}{v}}{\partial y}) - I 2 (\frac{\partial 2 \overset{··}{w} b}{\partial x 2} + \frac{\partial 2 \overset{··}{w} b}{\partial y 2})

(29c)

κ (A 55 \frac{\partial 2 w s}{\partial x 2} + 2 A 45 \frac{\partial 2 w s}{\partial x \partial y} + A 44 \frac{\partial 2 w s}{\partial y 2}) + q = I 0 (\overset{··}{w} b + \overset{··}{w} s)

(29d)

Analytical solutions for antisymmetric laminates

The antisymmetrically layered FG-CNT laminates are investigated in this work. Consider a simply supported rectangular laminate, the Navier solutions are given by the following equations for cross-ply and angle-ply laminates, respectively.

u (x, y, t) = \sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} U mn e iwt \cos α x \sin β y v (x, y, t) = \sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} V mn e iwt \sin α x \cos β y (antisymmetriccross-ply)

(30a)

u (x, y, t) = \sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} U mn e iwt \sin α x \cos β y v (x, y, t) = \sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} V mn e iwt \cos α x \sin β y (antisymmetricangle-ply)

(30b)

w b (x, y, t) = \sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} W bmn e iwt \sin α x \sin β y w s (x, y, t) = \sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} W smn e iwt \sin α x \sin β y

(30c)

where α = mπ/a, β = nπ/b are the coefficients, ω is the natural frequency. The transverse load q is also defined in the double-Fourier sine series as

q (x, y) = \sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} Q mn \sin α x \sin β y

(31)

The coefficients Q_mn can be calculated for the uniform load and sinusoidal load as

It should be noted that the Navier solutions for simply supported boundary condition exist only when the laminate stacking sequences are that

A 16 = A 26 = B 16 = B 26 = D 16 = D 26 = 0 (forantisymmetriccross-ply)

(33a)

A 16 = A 26 = B 11 = B 12 = B 22 = B 66 = D 16 = D 26 = 0 (forantisymmetricangle-ply)

(33b)

Therefore, substituting equation (30) and equation (31) into the equations of motion, equation (29), the analytical solutions can be obtained from the following equation

([S 11 S 12 S 13 0 S 12 S 22 S 23 0 S 13 S 23 S 33 0000 S 44] - ω 2 [I 0 0 - I 1 α 00 I 0 - I 1 β 0 - I 1 α - I 1 β I 0 + I 2 (α 2 + β 2) I 0 00 I 0 I 0]) {U mn V mn W bmn W smn} = {00 Q mn Q mn}

(34)

where

S 11 = A 11 α 2 + A 66 β 2, S 12 = (A 12 + A 66) α β, S 22 = A 66 α 2 + A 22 β 2

S 33 = D 11 α 4 + 2 (D 12 + 2 D 66) α 2 β 2 + D 22 β 4

S 44 = κ (A 55 α 2 - 2 A 45 α β + A 44 β 2)

(35a)

Numerical results

In this section, various numerical examples are presented and discussed to reveal the effects of volume fractions of CNT, length to width ratio, thickness to length ratio and CNT distribution type on the central deflection and fundamental natural frequency of antisymmetrically laminated FG-CNT-reinforced composite plates. Various antisymmetric layup stacking sequences and number of layers are examined as well. The material properties for the matrix and CNT are given in Table 1. The properties of armchair (10,10) SWCNTs are used in this paper which are obtained at the temperature of 300 K. The CNT efficiency parameters are defined following the previous works by Zhu et al., as shown in Table 2.⁸ For numerical study, the assumptions that

η 3 = η 2

and

G 23 = G 12 = G 13

are also made according to the Zhu et al.⁸

Table 1.

Material properties of CNT and matrix materials.

CNT	Matrix
$E_{11}^{CNT} = 5.6466 TPa$ $E_{22}^{CNT} = 7.0800 TPa$ $G_{12}^{CNT} = 1.9445 TPa$ $v_{12}^{CNT} = 0.175$ $ρ CNT = 1400 kg / m 3$	$E m = 2.1 GPa$ $v m = 0.34$ $ρ m = 1150 kg / m 3$

Table 2.

CNT efficiency parameters with respect to different volume fractions.

$V_{CNT}^{*}$	$η 1$	$η 2$
0.11	0.149	0.934
0.14	0.150	0.941
0.17	0.149	1.381

Firstly, the free vibration of four-edge simply-supported (SSSS) single-layer CNT-reinforced composite (CNTRC) plates is investigated. The fundamental frequencies are non-dimensionalized by

\bar{ω} = ω (a 2 / h) \sqrt{ρ m / E m}

and given in Table 3 for different CNT volume fractions, thickness to width ratios and CNT distribution types. The results are also compared with the solutions given by Zhu et al.⁸ It can be seen that the present results agree well with the solutions of the given references. For the case that h/b = 0.02, the fundamental frequencies have small differences compared with the reference results for difference volume fractions. When the h/b ratio increases, the difference also increases a bit. This is due to the choice of shear correction factor that is used in the first-order shear deformation theory. It is still difficult to determine the exact value of shear correction factor for the first-order shear deformation theories. However, the difference is not significant and it is tolerable to use the present method for studying the vibration and bending problems of FG-CNT-reinforced composite plates.

Table 3.

Non-dimensional fundamental natural frequency $\bar{ω} = ω (a 2 / h) \sqrt{ρ m / E m}$ for single layer CNTRC square plate (SSSS).

$V_{CNT}^{*}$	h/b	UD		FG-V		FG-O		FG-X
$V_{CNT}^{*}$	h/b	Present	Zhu et al.⁸	Present	Zhu et al.⁸	Present	Zhu et al.⁸	Present	Zhu et al.⁸
0.11	0.02	19.354	19.223	16.236	16.252	14.300	14.302	23.204	22.984
	0.05	18.280	17.355	15.574	15.110	13.840	13.523	21.425	19.939
	0.10	15.550	13.532	13.753	12.452	12.512	11.550	17.362	14.616
0.14	0.02	21.591	21.354	18.043	17.995	15.856	15.801	25.937	25.555
	0.05	20.182	18.921	17.181	16.510	15.261	14.784	23.598	21.643
	0.10	16.773	14.306	14.900	13.256	13.591	12.338	18.591	15.638
0.17	0.02	23.844	23.697	19.938	19.982	17.563	17.544	28.619	28.413
	0.05	22.557	21.456	19.151	18.638	17.017	16.628	26.475	24.764
	0.10	19.258	16.815	16.970	15.461	15.429	14.282	21.536	18.278

Tables 4 and 5 present the non-dimensional fundamental frequencies

\bar{ω} = ω (b 2 / h) \sqrt{ρ m / E m}

of antisymmetric cross-ply laminate (0°/90°)_n considering the effects of length to width ratio and thickness to width ratio, respectively. The simply supported FG-CNT-reinforced plate consists of 2n layers and has the geometric parameters of a/b = 1, 1.5, 2 in Table 4 and h/b = 0.02, 0.05, 0.10 in Table 5 for parameter study. From both tables, it is found that the fundamental frequencies increase with increase of the volume fraction of CNTs due to the fact that the elastic properties of CNT are higher than the matrix material. It is also found that the fundamental frequencies increase with increase of the number of layers for all four types of CNT distribution. By increasing the length to width ratio (a/b) and thickness to width ratio (h/b), we found that the fundamental frequencies decrease for both cases shown in Tables 4 and 5. Among four types of CNT distribution, it is found that the FG-O case presents the smallest fundamental frequencies compared with other UD, FG-V and FG-X cases, while the FG-X case presents the largest fundamental frequencies compared with other three functionally graded plates. Thus, we can conclude that the CNT distributed close to the surfaces enhances the stiffness, while the CNT distributed close to the midplane lowers the stiffness of antisymmetrically laminated FG-CNT-reinforced composite plates. Therefore, the effects of the length to width ratio, the thickness to width ratio and the layer number on the non-dimensional fundamental frequencies of the cross-ply antisymmetrically laminated FG-CNT reinforced composite plates have been well demonstrated by these two tables.

Table 4.

Effects of length to width ratio and CNT volume fraction on the non-dimensional fundamental frequency $\bar{ω} = ω (b 2 / h) \sqrt{ρ m / E m}$ for antisymmetric cross-ply (0°/90°)_n laminates (SSSS, h/b = 0.02).

$V_{CNT}^{*}$	a/b	UD			FG-V			FG-O			FG-X
$V_{CNT}^{*}$	a/b	n = 1	n = 2	n = 4	n = 1	n = 2	n = 4	n = 1	n = 2	n = 4	n = 1	n = 2	n = 4
0.11	1.0	11.348	17.714	18.958	10.056	17.495	18.883	9.182	17.378	18.856	13.064	17.975	18.995
	1.5	8.665	13.639	14.609	7.830	13.499	14.563	6.974	13.383	14.536	10.022	13.850	14.645
	2.0	8.038	12.752	13.669	7.295	12.634	13.634	6.431	12.518	13.607	9.342	12.961	13.710
0.14	1.0	12.395	19.726	21.142	10.876	19.484	21.065	9.874	19.354	21.036	14.396	20.032	21.193
	1.5	9.484	15.210	16.316	8.459	15.049	16.267	7.516	14.926	16.239	11.062	15.456	16.362
	2.0	8.817	14.240	15.283	7.890	14.099	15.244	6.949	13.978	15.217	10.328	14.479	15.333
0.17	1.0	14.035	21.831	23.358	12.435	21.565	23.271	11.367	21.421	23.238	16.180	22.165	23.411
	1.5	10.713	16.805	17.996	9.666	16.633	17.942	8.631	16.493	17.910	12.408	17.075	18.045
	2.0	9.934	15.711	16.836	8.999	15.564	16.795	7.957	15.424	16.763	11.561	15.975	16.891

Table 5.

Effects of thickness to width ratio and CNT volume fraction on the non-dimensional fundamental frequency $\bar{ω} = ω (b 2 / h) \sqrt{ρ m / E m}$ for antisymmetric cross-ply (0°/90°)_n square laminates (SSSS, a/b = 1).

$V_{CNT}^{*}$	h/b	UD			FG-V			FG-O			FG-X
$V_{CNT}^{*}$	h/b	n = 1	n = 2	n = 4	n = 1	n = 2	n = 4	n = 1	n = 2	n = 4	n = 1	n = 2	n = 4
0.11	0.02	11.348	17.714	18.958	10.056	17.495	18.883	9.182	17.378	18.856	13.064	17.975	18.995
	0.05	11.095	16.872	17.944	9.872	16.683	17.881	9.038	16.581	17.859	12.694	17.099	17.977
	0.10	10.326	14.640	15.338	9.300	14.518	15.302	8.582	14.450	15.288	11.609	14.794	15.363
0.14	0.02	12.395	19.726	21.142	10.876	19.484	21.065	9.874	19.354	21.036	14.396	20.032	21.193
	0.05	12.083	18.624	19.811	10.656	18.424	19.754	9.705	18.314	19.729	13.930	18.888	19.859
	0.10	11.152	15.832	16.555	9.981	15.717	16.534	9.174	15.649	16.519	12.592	16.007	16.596
0.17	0.02	14.035	21.831	23.358	12.435	21.565	23.271	11.367	21.421	23.238	16.180	22.165	23.411
	0.05	13.728	20.820	22.142	12.213	20.596	22.078	11.193	20.470	22.049	15.733	21.120	22.197
	0.10	12.793	18.124	18.993	11.520	17.993	18.975	10.640	17.909	18.957	14.415	18.344	19.052

Tables 6 and 7 show the non-dimensional central deflection

\bar{w} = wE m h 3 \times 102 / (q 0 b 4)

for antisymmetric cross-ply (0°/90°)_n laminate subjected to a uniform transverse load q₀ = 0.1 MPa. Different geometric parameters are investigated as well, and the deflection values are obtained at the position of (a/2, b/2). Table 6 gives the central deflections for the cases of various a/b ratios, volume fractions and numbers of layers. It can be observed that by increasing the a/b ratio, the central deflection increases. While increase the volume fraction and the number of layers, the central deflection decreases for all four types. This can also be explained by the effect to the stiffness of the laminate. Table 7 lists the central deflections affected by the thickness to width ratio (h/b). We found that the thickness to width ratio has significant influence on the central deflection of antisymmetrically laminated FG-CNT-reinforced composite plates. As we known, a thick plate has smaller deflection than a thinner plate under the same load condition. However, since the deflections are normalized by

\bar{w} = wE m h 3 \times 102 / (q 0 b 4)

in this work, it is found that the central deflection increases as the h/b value increases.

Table 6.

Effects of length to width ratio and CNT volume fraction on non-dimensional central deflection $\bar{w} = wE m h 3 \times 103 / (q 0 b 4)$ for antisymmetric cross-ply (0°/90°)_n laminates under uniform load q₀ = 0.1 MPa (SSSS, h/b = 0.02).

$V_{CNT}^{*}$	a/b	UD			FG-V			FG-O			FG-X
$V_{CNT}^{*}$	a/b	n = 1	n = 2	n = 4	n = 1	n = 2	n = 4	n = 1	n = 2	n = 4	n = 1	n = 2	n = 4
0.11	1.0	12.083	4.959	4.329	15.380	5.084	4.364	18.449	5.154	4.377	9.120	4.816	4.313
	1.5	20.502	8.275	7.211	25.036	8.446	7.258	31.648	8.595	7.284	15.329	8.024	7.177
	2.0	23.121	9.171	7.981	27.904	9.340	8.021	36.144	9.518	8.053	17.102	8.878	7.933
0.14	1.0	10.065	3.973	3.458	13.069	4.073	3.484	15.857	4.128	3.493	7.462	3.853	3.442
	1.5	17.006	6.610	5.744	21.328	6.752	5.778	27.075	6.864	5.799	12.502	6.401	5.712
	2.0	19.087	7.306	6.341	23.719	7.450	6.373	30.752	7.582	6.396	13.901	7.066	6.300
0.17	1.0	7.800	3.224	2.816	9.933	3.305	2.838	11.886	3.349	2.846	5.870	3.128	2.804
	1.5	13.243	5.382	4.693	16.223	5.493	4.721	20.402	5.588	4.738	9.875	5.214	4.667
	2.0	14.946	5.967	5.195	18.114	6.077	5.22	23.317	6.191	5.24	11.027	5.771	5.161

Table 7.

Effects of thickness to width ratio and CNT volume fraction on non-dimensional central deflection $\bar{w} = wE m h 3 \times 103 / (q 0 b 4)$ for antisymmetric cross-ply (0°/90°)_n square laminates under uniform load q₀ = 0.1 MPa (SSSS, a/b = 1).

$V_{CNT}^{*}$	h/b	UD			FG-V			FG-O			FG-X
$V_{CNT}^{*}$	h/b	n = 1	n = 2	n = 4	n = 1	n = 2	n = 4	n = 1	n = 2	n = 4	n = 1	n = 2	n = 4
0.11	0.02	12.083	4.959	4.329	15.380	5.084	4.364	18.449	5.154	4.377	9.120	4.816	4.313
	0.05	12.530	5.407	4.777	15.827	5.531	4.811	18.895	5.600	4.823	9.566	5.263	4.759
	0.10	14.128	7.004	6.374	17.421	7.125	6.405	20.489	7.194	6.417	11.160	6.857	6.353
0.14	0.02	10.065	3.973	3.458	13.069	4.073	3.484	15.857	4.128	3.493	7.462	3.853	3.442
	0.05	10.494	4.402	3.887	13.496	4.500	3.910	16.284	4.555	3.920	7.889	4.279	3.869
	0.10	12.027	5.935	5.420	15.021	6.024	5.435	17.808	6.079	5.445	9.413	5.804	5.393
0.17	0.02	7.800	3.224	2.816	9.933	3.305	2.838	11.886	3.349	2.846	5.870	3.128	2.804
	0.05	8.082	3.506	3.098	10.212	3.584	3.117	12.166	3.629	3.125	6.150	3.407	3.083
	0.10	9.090	4.515	4.107	11.211	4.582	4.115	13.164	4.627	4.124	7.148	4.406	4.082

Tables 8 and 9 present the non-dimensional fundamental frequencies

\bar{ω} = ω (b 2 / h) \sqrt{ρ m / E m}

for antisymmetric angle-ply laminate (45°/−45°)_n. For angle-ply laminate, the effects of geometric parameters, such as a/b ratio and h/b ratio, on the fundamental frequencies are also investigated. Similar observations for volume fractions, a/b ratios, h/b ratios and layer numbers are found for the simply supported boundary conditions. It is also found that the FG-O case possesses the smallest fundamental frequencies, while the FG-X case possesses the largest fundamental frequencies for the antisymmetrically laminated angle-ply plate. This phenomenon is obviously same with the previous cross-ply case. However, we found that the fundamental frequencies of the FG-V case are smaller than the UD case for the antisymmetric angle-ply laminate. The central deflections of antisymmetric angle-ply laminate (45°/−45°)_n are listed in Tables 10 and 11 for different values of length to width ratios and thickness to width ratios, respectively. By increasing the length to width ratio (a/b) and thickness to width ratio (h/b), it is found that the central deflections increase. The FG-O case presents the largest deflections than other cases in both tables due to its lowest stiffness and the FG-X presents the smallest deflections due to its highest stiffness. Thus, the effects of CNT volume fraction, length to width ratio, thickness to width ratio, number of layers and CNT distribution types on the central deflections are well demonstrated for the antisymmetrically laminated angle-ply (45°/-45°)_n FG-CNT reinforced composite plates.

Table 8.

Effects of length to width ratio and CNT volume fraction on the non-dimensional fundamental frequency $\bar{ω} = ω (b 2 / h) \sqrt{ρ m / E m}$ for antisymmetric angle-ply (45°/-45°)_n laminates (SSSS, h/b = 0.02).

$V_{CNT}^{*}$	a/b	UD			FG-V			FG-O			FG-X
$V_{CNT}^{*}$	a/b	n = 1	n = 2	n = 4	n = 1	n = 2	n = 4	n = 1	n = 2	n = 4	n = 1	n = 2	n = 4
0.11	1.0	15.109	24.180	25.912	14.116	23.946	25.871	11.974	23.761	25.818	17.697	24.601	26.012
	1.5	10.575	16.942	18.167	10.083	16.827	18.145	8.390	16.642	18.098	12.377	17.236	18.235
	2.0	8.693	13.888	14.892	8.364	13.810	14.874	6.917	13.638	14.832	10.150	14.124	14.944
0.14	1.0	16.631	27.005	28.961	15.388	26.694	28.908	12.996	26.536	28.859	19.600	27.481	29.077
	1.5	11.637	18.936	20.326	11.021	18.785	20.299	9.100	18.601	20.250	13.710	19.270	20.404
	2.0	9.556	15.521	16.663	9.147	15.421	16.642	7.493	15.241	16.597	11.236	15.791	16.724
0.17	1.0	18.665	29.789	31.917	17.489	29.521	31.878	14.806	29.277	31.810	21.889	30.323	32.051
	1.5	13.065	20.869	22.374	12.487	20.740	22.353	10.375	20.503	22.293	15.310	21.241	22.464
	2.0	10.741	17.107	18.341	10.357	17.020	18.324	8.555	16.802	18.269	12.557	17.406	18.410

Table 9.

Effects of thickness to width ratio and CNT volume fraction on the non-dimensional fundamental frequency $\bar{ω} = ω (b 2 / h) \sqrt{ρ m / E m}$ for antisymmetric angle-ply (45°/-45°)_n square laminates (SSSS, a/b = 1).

$V_{CNT}^{*}$	h/b	UD			FG-V			FG-O			FG-X
$V_{CNT}^{*}$	h/b	n = 1	n = 2	n = 4	n = 1	n = 2	n = 4	n = 1	n = 2	n = 4	n = 1	n = 2	n = 4
0.11	0.02	15.109	24.180	25.912	14.116	23.946	25.871	11.974	23.761	25.818	17.697	24.601	26.012
	0.05	14.558	22.192	23.516	12.916	21.950	23.489	11.683	21.868	23.450	16.847	22.521	23.596
	0.10	13.013	17.772	18.441	11.526	17.703	18.422	10.807	17.610	18.416	14.608	17.949	18.487
0.14	0.02	16.631	27.005	28.961	15.388	26.694	28.908	12.996	26.536	28.859	19.600	27.481	29.077
	0.05	15.935	24.402	25.825	13.955	24.220	25.804	12.644	24.064	25.766	18.511	24.764	25.921
	0.10	14.040	18.991	19.648	12.562	18.918	19.645	11.602	18.847	19.643	15.754	19.182	19.712
0.17	0.02	18.665	29.789	31.917	17.489	29.521	31.878	14.806	29.277	31.810	21.889	30.323	32.051
	0.05	17.999	27.403	29.043	15.841	27.236	29.007	14.455	27.021	28.985	20.865	27.839	29.168
	0.10	16.124	22.045	22.885	14.668	21.910	22.944	13.397	21.879	22.898	18.151	22.311	22.989

Table 10.

Effects of length to width ratio and CNT volume fraction on non-dimensional central deflection $\bar{w} = wE m h 3 \times 103 / (q 0 b 4)$ for antisymmetric angle-ply (45°/-45°)_n laminates under uniform load q₀ = 0.1 MPa (SSSS, h/b = 0.02).

$V_{CNT}^{*}$	a/b	UD			FG-V			FG-O			FG-X
$V_{CNT}^{*}$	a/b	n = 1	n = 2	n = 4	n = 1	n = 2	n = 4	n = 1	n = 2	n = 4	n = 1	n = 2	n = 4
0.11	1.0	6.693	2.608	2.270	8.138	2.654	2.276	10.669	2.701	2.286	4.874	2.519	2.252
	1.5	13.608	5.295	4.603	14.769	5.360	4.613	21.638	5.487	4.638	9.928	5.115	4.569
	2.0	19.961	7.816	6.796	21.303	7.895	6.811	31.531	8.105	6.852	14.639	7.556	6.749
0.14	1.0	5.487	2.076	1.804	6.295	2.120	1.810	8.996	2.150	1.817	3.947	2.005	1.790
	1.5	11.165	4.209	3.652	12.271	4.271	3.661	18.270	4.363	3.680	8.039	4.064	3.624
	2.0	16.415	6.216	5.392	17.683	6.289	5.404	26.703	6.447	5.435	11.869	6.005	5.353
0.17	1.0	4.331	1.697	1.477	5.550	1.725	1.481	6.890	1.757	1.487	3.146	1.637	1.465
	1.5	8.805	3.446	2.997	11.011	3.484	3.002	13.973	3.570	3.019	6.408	3.326	2.973
	2.0	12.910	5.086	4.424	15.720	5.133	4.432	20.353	5.273	4.459	9.445	4.913	4.391

Table 11.

Effects of thickness to width ratio and CNT volume fraction on non-dimensional central deflection $\bar{w} = wE m h 3 \times 103 / (q 0 b 4)$ for antisymmetric angle-ply (45°/-45°)_n square laminates under uniform load q₀ = 0.1 MPa (SSSS, a/b = 1).

$V_{CNT}^{*}$	h/b	UD			FG-V			FG-O			FG-X
$V_{CNT}^{*}$	h/b	n = 1	n = 2	n = 4	n = 1	n = 2	n = 4	n = 1	n = 2	n = 4	n = 1	n = 2	n = 4
0.11	0.02	6.693	2.608	2.270	8.138	2.654	2.276	10.669	2.701	2.286	4.874	2.519	2.252
	0.05	7.140	3.055	2.717	9.125	3.103	2.726	11.115	3.147	2.733	5.321	2.965	2.699
	0.10	8.738	4.653	4.315	10.719	4.697	4.320	12.709	4.741	4.327	6.915	4.559	4.293
0.14	0.02	5.487	2.076	1.804	6.295	2.120	1.810	8.996	2.150	1.817	3.947	2.005	1.790
	0.05	5.916	2.505	2.233	7.600	2.552	2.443	9.422	2.577	2.244	4.374	2.431	2.217
	0.10	7.449	4.038	3.766	9.125	4.066	3.768	10.947	4.102	3.769	5.898	3.956	3.741
0.17	0.02	4.331	1.697	1.477	5.550	1.725	1.481	6.890	1.757	1.487	3.146	1.637	1.465
	0.05	4.613	1.979	1.760	5.891	2.014	1.764	7.170	2.036	1.767	3.426	1.917	1.745
	0.10	5.622	2.987	2.768	6.989	3.003	2.763	8.168	3.035	2.765	4.424	2.915	2.743

Figures 3 and 4 depict the fundamental non-dimensional frequencies of angle-ply (θ/-θ) laminate to investigate the effect of angle θ on the fundamental frequencies. The volume fraction is selected as $V_{CNT}^{*}$ = 0.11 for these figures. Figure 3 gives the curves for a square laminate with different values of thickness to width ratios. We found that the curves are symmetric to the line of θ = 45° and it presents the largest values of fundamental frequencies for UD, FG-V and FG-X cases at θ = 45°. The effect of thickness to width ratio shows that a thin plate possesses higher fundamental frequency than a thick plate in this work. Figure 4 gives the variation of the fundamental frequency due to the effect caused by different values of length to width ratio. The results are obtained for the case that h/b = 0.02. When a/b = 1, the plate is square and the frequency curve is symmetric to the line of θ = 45°. But for non-squared plate, the deflection increases with the increase of θ. When increase the a/b ratio, the fundamental frequency decreases for all four types due to the fact that a slender plate possess smaller fundamental frequency. Therefore, the effect of angle θ is well demonstrated in these two figures.

Figure 3.

Variation of the fundamental frequency $\bar{ω} = ω (b 2 / h) \sqrt{ρ m / E m}$ for angle-ply (θ/-θ) square laminate with different values of thickness to width ratio for four CNT distribution types.

Figure 4.

Variation of the fundamental frequency $\bar{ω} = ω (b 2 / h) \sqrt{ρ m / E m}$ for angle-ply (θ/-θ) laminate with different values of length to width ratio for four CNT distribution types.

Conclusions

In this paper, the bending and free vibration problems of antisymmetrically laminated FG-CNT-reinforced composite plates are investigated by using the four-variable first-order shear deformation theory. The antisymmetrically laminated FG-CNT-reinforced composite plates are assumed to be perfectly bonded and the CNTs are assumed to be uniformly distributed or functionally distributed through the thickness. The effective properties are estimated by the extended rule of mixture with different CNT efficiency parameters counting for the size-dependent effect. The effects of CNT volume fractions, CNT distribution types, length to width ratio, thickness to width ratio and numbers of layers on both non-dimensional fundamental frequencies and deflections of simply supported antisymmetrically laminated plates are studied in detail. The solutions showed that these parameters have significant influence on the static and dynamic characteristics of antisymmetric FG-CNT-reinforced composite plates.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China (nos. 11272161, 11372145 and 11372146), the State Key Laboratory of Mechanics and Control of Mechanical Structures (Nanjing University of Aeronautics and astronautics) (Grant no. MCMS-0516Y01), Ningbo Natural Science Foundation (Grant no. 2016A610056), Zhejiang Provincial Top Key Discipline of Mechanics Open Foundation (Grant no. xklx1601), the research fund from Ningbo University (Grant no. XYL16008) and the K. C. Wong Magna Fund through Ningbo University.

References

Sofiyev

. Nonlinear free vibration of shear deformable orthotropic functionally graded cylindrical shells. Compos Struct 2016; 142: 35–44.

Akavci

. Mechanical behavior of functionally graded sandwich plates on elastic foundation. Compos B Eng 2016; 96: 136–152.

Shen

Xiang

. Nonlinear analysis of nanotube-reinforced composite beams resting on elastic foundations in thermal environments. Eng Struct 2013; 56: 698–708.

Shen

Xiang

. Nonlinear vibration of nanotube-reinforced composite cylindrical shells in thermal environments. Comput Meth Appl Mech Eng 2012; 213–216: 196–205.

Aragh

Barati

AHN

Hedayati

. Eshelby–Mori–Tanaka approach for vibrational behavior of continuously graded carbon nanotube-reinforced cylindrical panels. Compos B Eng 2012; 43: 1943–1954.

Wattanasakulpong

Ungbhakorn

. Analytical solutions for bending, buckling and vibration responses of carbon nanotube-reinforced composite beams resting on elastic foundation. Comput Mater Sci 2013; 71: 201–208.

Zhang

Lei

Liew

et al.

Static and dynamic of carbon nanotube reinforced functionally graded cylindrical panels. Compos Struct 2014; 111: 205–212.

Zhu

Lei

Liew

. 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.

Rokni

Milani

Seethaler

. Size-dependent vibration behavior of functionally graded CNT-reinforced polymer microcantilevers: modeling and optimization. Eur J Mech – A/Solids 2015; 49: 26–34.

10.

Mehrabadi

Aragh

. Stress analysis of functionally graded open cylindrical shell reinforced by agglomerated carbon nanotubes. Thin-Walled Struct 2014; 80: 130–141.

11.

Alibeigloo

Liew

. Thermoelastic analysis of functionally graded carbon nanotube-reinforced composite plate using theory of elasticity. Compos Struct 2013; 106: 873–881.

12.

Lei

Liew

. Buckling analysis of functionally graded carbon nanotube-reinforced composite plates using the element-free kp-Ritz method. Compos Struct 2013; 98: 160–168.

13.

Zhang

Liew

Reddy

. Postbuckling of carbon nanotube reinforced functionally graded plates with edges elastically restrained against translation and rotation under axial compression. Comput Meth Appl Mech Eng 2016; 298: 1–28.

14.

Yas

Samadi

. Free vibrations and buckling analysis of carbon nanotube-reinforced composite Timoshenko beams on elastic foundation. Int J Pressure Vessels Pip 2012; 98: 119–128.

15.

Yas

Pourasghar

Kamarian

et al.

Three-dimensional free vibration analysis of functionally graded nanocomposite cylindrical panels reinforced by carbon nanotube. Mater Des 2013; 49: 583–590.

16.

Zhang

Lei

Liew

. Free vibration analysis of functionally graded carbon nanotube-reinforced composite triangular plates using the FSDT and element-free IMLS-Ritz method. Compos Struct 2015; 120: 189–199.

17.

Malekzadeh

Zarei

. Free vibration of quadrilateral laminated plates with carbon nanotube reinforced composite layers. Thin-Walled Struct 2014; 82: 221–232.

18.

Chakraverty

Behera

. Free vibration of rectangular nanoplates using Rayleigh–Ritz method. Phys E: Low-dimensional Systems Nanostruct 2014; 56: 357–363.

19.

Rafiee

Mareishi

et al.

Large amplitude vibration of fractionally damped viscoelastic CNTs/fiber/polymer multiscale composite beams. Compos Struct 2015; 131: 1111–1123.

20.

Rafiee

Yang

Kitipornchai

. Large amplitude vibration of carbon nanotube reinforced functionally graded composite beams with piezoelectric layers. Compos Struct 2013; 96: 716–725.

21.

Rafiee

Liew

. Non-linear dynamic stability of piezoelectric functionally graded carbon nanotube-reinforced composite plates with initial geometric imperfection. Int J Non-Linear Mech 2014; 59: 37–51.

22.

Yang

Kitipornchai

. Nonlinear free vibration of functionally graded carbon nanotube-reinforced composite beams. Compos Struct 2010; 92: 676–683.

23.

Shen

Xiang

. Nonlinear vibration of nanotube-reinforced composite cylindrical panels resting on elastic foundations in thermal environments. Compos Struct 2014; 111: 291–300.

24.

Guo

Zhang

. Nonlinear vibrations of a reinforced composite plate with carbon nanotubes. Compos Struct 2016; 135: 96–108.

25.

Lei

Zhang

Liew

. Free vibration analysis of laminated FG-CNT reinforced composite rectangular plates using the kp-Ritz method. Compos Struct 2015; 127: 245–259.

26.

Lei

Zhang

Liew

. Analysis of laminated CNT reinforced functionally graded plates using the element-free kp-Ritz method. Compos B Eng 2016; 84: 211–221.

27.

Lei

Liew

. Large deflection analysis of functionally graded carbon nanotube-reinforced composite plates by the element-free kp-Ritz method. Comput Meth Appl Mech Eng 2013; 256: 189–199.

28.

Heydarpour

Aghdam

Malekzadeh

. Free vibration analysis of rotating functionally graded carbon nanotube-reinforced composite truncated conical shells. Compos Struct 2014; 117: 187–200.

29.

Reddy

. Mechanics of laminated composite plates and shells: theory and analysis, 2nd ed. Boca Raton, FL: CRC Press, 2004.

30.

Thai

Choi

. A simple first-order shear deformation theory for the bending and free vibration analysis of functionally graded plates. Compos Struct 2013; 101: 332–340.

31.

Shimpi

Patel

. A two variable refined plate theory for orthotropic plate analysis. Int J Solids Struct 2006; 43: 6783–6799.

32.

Thai

Kim

. A simple quasi-3D sinusoidal shear deformation theory for functionally graded plates. Compos Struct 2013; 99: 172–180.

33.

Thai

Choi

. A simple first-order shear deformation theory for laminated composite plates. Compos Struct 2013; 106: 754–763.

34.

Efraim

Eisenberger

. Exact vibration analysis of variable thickness thick annular isotropic and FGM plates. J Sound Vibr 2007; 299: 720–738.