Vibration of thermally postbuckled FG-GRC laminated plates resting on elastic foundations

Abstract

This paper investigates the small- and large-amplitude vibrations of thermally postbuckled graphene-reinforced composite (GRC) laminated plates resting on elastic foundations. The piecewise GRC layers are arranged in a functionally graded (FG) pattern along the thickness direction of the plate. The anisotropic and temperature-dependent material properties of the FG-GRC layers are estimated through the extended Halpin–Tsai micromechanical model. Based on the Reddy's higher order shear deformation plate theory and the von Kármán strain–displacement relationships, the motion equations of the plates are derived. The foundation support, the thermal effect, and the initial deflection caused by thermal postbuckling are also included in the derivation. A two-step perturbation approach is applied to determine the thermal postbuckling equilibrium paths as well as the nonlinear vibration solutions for the FG-GRC laminated plates. The numerical illustrations concern small- and large-amplitude vibration characteristics of thermally postbuckled FG-GRC laminated plates under a uniform temperature field. The effects of graphene reinforcement distributions and foundation stiffnesses on the vibration responses of FG-GRC laminated plates are examined in detail.

Keywords

Nonlinear vibration plates postbuckling functionally graded materials nanocomposites temperature-dependent properties

1. Introduction

Composite laminated plates are widely used in the aerospace industry owing to their specific higher strength-to-weight and stiffness-to-weight ratios. These plates may be subjected to mechanical and/or thermal loadings. This may lead to postbuckling of the plate where the plate deflection is in the order of the plate thickness. It is well known that the thermal postbuckling equilibrium path of a plate under a uniform temperature field is stable. Hence, the vibration response of a thermally postbuckled plate subjected to a uniform or nonuniform temperature rise is an important problem for engineering design. The geometrically nonlinear vibration characteristics of thermally postbuckled isotropic/anisotropic laminated plates were studied by Yang and Han (1983), Librescu et al. (1996a, 1996b), Librescu and Lin (1997), Lee and Lee (1997a, 1997b), Oh et al. (2000), Girish and Ramachandra (2005a, 2005b), Fazzolari and Carrera (2013) and Samadpour et al. (2015). Thermally postbuckled plates made of functionally graded (FG) metal/ceramic composite were studied by Park and Kim (2006), Xia and Shen (2008), and Taczala et al. (2016), whereas thermally postbuckled plates made of FG carbon nanotube-reinforced composite (CNTRC) were investigated recently by Fan and Wang (2016), Shen and Wang, (2017), and Fazzolari (2018). A comprehensive survey of these studies can be found in Shen and Wang (2017).

Many studies have been carried out on the thermal postbuckling of FG material (FGM) plates and FG-CNTRC plates subjected to thermal loading (Bouderba et al., 2016; El-Haina et al., 2017; Menasria et al., 2017). However, as in the case of simply supported asymmetric cross-ply laminated plates, due to the stretching–bending coupling effect the bifurcation buckling temperature does not exist for the simply supported FGM and FG-CNTRC plates subjected to uniform or nonuniform temperature rise unless the FGM or FG-CNTRC plate is geometrically mid-plane symmetric (Shen, 2007; Shen and Zhang, 2010); otherwise the results may be physically incorrect (Javaheri and Eslami, 2002; Wu, 2004; Ganapathi and Prakash, 2006; Bodaghi and Saidi, 2010; Talha and Singh, 2011).

As pointed out by Alijani et al. (2011), Huang and Shen (2004) were the pioneers in studying the nonlinear vibrations of thick FG plates. Gupta and Talha (2017) studied the nonlinear vibrations of geometrically imperfect FG plates. The main difference for the initial deflection caused by geometric imperfection and by thermal postbuckling is that the initial geometric imperfection is usually assumed to be a constant in the whole large deflection region, whereas the initial deflection caused by thermal postbuckling varies in the thermal postbuckling region as the temperature rises.

Graphene possesses exceptional mechanical, thermal, and electrical properties which enable graphene sheets to be considered as ideal reinforcement materials for creating high performance nanocomposites. Researchers have reported that the material properties of graphene sheets are anisotropic and temperature dependent (Reddy et al., 2006; Giannopoulos and Kallivokas, 2014; Ni et al., 2010; Shen et al., 2010). Ni et al. (2010) observed anisotropic mechanical properties for a graphene sheet along different load directions. Reddy et al. (2006) confirmed that graphene behaves like an orthotropic material; in particular the shear modulus is much lower than that of a graphene when it is treated as an isotropic material. Shen et al. (2010) found that the Young's modulus of a single layer graphene sheet decreases with increasing in temperature, whereas the shear modulus of the graphene sheet depends weakly on temperature variation. Graphene sheets can be aligned in polymer matrix to achieve better reinforcement as reported by Zhao et al. (2010) and Liang et al. (2011). Yousefi et al. (2014) obtained highly aligned graphene/polymer nanocomposites by an all aqueous casting method. It is well known that the physical interactions between graphene and polymer matrix are weak due to graphene's perfect bonding structure among its carbon atoms. Therefore, the load transfer efficiency between graphene and polymer matrix in nanocomposites is relatively low (Milani et al., 2013). When comparing with carbon fiber reinforced composites which may contain more than 60% volume fraction of carbon fibers, the volume fraction of graphene reinforcement in nanocomposites may only reach about 21% (Putz et al., 2010) and further increase of the volume fraction of graphene reinforcement may actually lead to the deterioration of the mechanical properties of the nanocomposites (Milani et al., 2013). Motivated by the idea of FGM, Shen et al. (2017a, 2017b) introduced the concept of FG graphene-reinforced composite (GRC) in which aligned graphene sheets are functionally distributed in a piecewise pattern to improve the mechanical properties and obtain the desired response of the structures. Shen et al. (2017a, 2017b) confirmed that FG-GRC thermal postbuckling equilibrium paths of FG-GRC laminated plates are stable, and the frequency-amplitude curves of FG-GRC laminated plates are hardening type. In their analysis, the graphene sheets are assumed to be aligned and oriented in the matrix layer-by-layer and the anisotropic and temperature-dependent material properties of the GRC are estimated by the extended Halpin–Tsai model which contains the graphene efficiency parameters. Song et al. (2017) and Wu et al. (2017) proposed a transverse isotropic multilayer model for the linear vibration and thermal buckling analyses of graphene platelet-reinforced composite (GPLRC) beams with each layer being isotropic but having different GPL weight fraction. In their analyses, the GPLs are assumed to be randomly oriented and uniformly dispersed in the matrix, and the material properties are assumed to be temperature independent. The equivalent isotropic Young's modulus of the GPLRC is obtained by using the modified Halpin–Tsai model, where two homogenization weight coefficients 3/8 and 5/8 are used. Note that the GPL should have aspect ratio l_GPL/w_GPL = 5/3 when this equivalent isotropic model is adopted, otherwise the results may be incorrect (Gholami and Ansari, 2018; Reddy et al., 2018; Sahmani et al., 2018). Following the works of Shen et al. (2017a, 2017b), Kiani (2018a, 2018b) studied the thermal postbuckling and nonlinear vibration of graphene reinforced laminated plates based on the first order shear deformation plate theory.

In the current study, small- and large-amplitude free vibration analyses are presented for a thermally postbuckled GRC laminated plate under a uniform temperature field and resting on an elastic foundation. We consider that the plate is of the graphene reinforcement either uniformly distributed (UD) or functionally graded (FG) in a piecewise pattern. The anisotropic and temperature-dependent material properties of GRCs are obtained by the extended Halpin–Tsai model which contains the graphene efficiency parameters estimated by matching the Halpin–Tsai model results against the elastic moduli of GRCs from the molecular dynamics (MD) simulations. The Reddy's higher order shear deformation plate theory is employed to derive the motion equations of the GRC laminated plate which take into consideration of the von Kármán-type nonlinear strain–displacement relationships, the initial thermal deflection caused by thermal postbuckling and the plate-foundation interaction. The motion equations can be separated into two sets of nonlinear equations which may be solved in sequence. From the first set of nonlinear equations, we will obtain the thermal postbuckling deflection that will serve as initial thermal deflection of the plate for the nonlinear vibration analysis. Thereafter, we need to solve a nonlinear vibration problem of an initially deflected plate to determine the nonlinear frequencies in the prebuckling and postbuckling states.

2. Large-amplitude vibration of postbuckled plates

We consider a rectangular plate of length a, width b, and thickness h under thermal environmental conditions. The plate consists of N plies with each ply being a mixture of graphene reinforcement and polymer matrix and having different graphene volume fraction. The graphene reinforcement is either zigzag (refer to as 0-ply) or armchair (refer to as 90-ply). The origin of the rectangular coordinate system is set at the corner of the plate on the middle plane. The plate rests on an elastic foundation. As is customary (Lue et al., 2009; Bakhadda et al., 2018), the foundation is assumed to be the Pasternak-type elastic foundation which provides not only a support in the normal direction, but also a shear layer support between the plate and the foundation. The interaction force p₀ between the plate and the foundation can be obtained by $p_{0} = {\bar{K}}_{1} \bar{W} - {\bar{K}}_{2} \nabla^{2} \bar{W}$ , where $\bar{W}$ is the plate displacement in the Z direction, ${\bar{K}}_{1}$ denotes the Winkler foundation stiffness, ${\bar{K}}_{2}$ denotes the shearing layer stiffness of the foundation, and $\nabla^{2}$ is the Laplace operator in X and Y, where X is the direction in the length of the plate and Y is the direction in the width of the plate.

The plate is assumed to be subjected to a transverse dynamic load q in a uniform temperature field. Based on the Reddy's higher order shear deformation plate theory (Reddy, 1984) with the von Kármán-type of kinematic nonlinearity, the motion equations for a GRC laminated plate can be expressed by

\begin{matrix} {\tilde{L}}_{11} (\bar{W}) - {\tilde{L}}_{12} ({\bar{Ψ}}_{x}) - {\tilde{L}}_{13} ({\bar{Ψ}}_{y}) + {\tilde{L}}_{14} (\bar{F}) - {\tilde{L}}_{15} ({\bar{N}}^{T}) \\ - {\tilde{L}}_{16} ({\bar{M}}^{T}) + ({\bar{K}}_{1} \bar{W} - {\bar{K}}_{2} \nabla^{2} \bar{W}) \\ = \tilde{L} (\bar{W} + {\bar{W}}^{*}, \bar{F} + {\bar{F}}^{*}) + {\tilde{L}}_{17} (\frac{\partial^{2} \bar{W}}{\partial {\bar{t}}^{2}}) \\ + I_{8} (\frac{\partial^{3} {\bar{Ψ}}_{x}}{\partial X \partial {\bar{t}}^{2}} + \frac{\partial^{3} {\bar{Ψ}}_{y}}{\partial Y \partial {\bar{t}}^{2}}) + q \end{matrix}

(1)

\begin{matrix} {\tilde{L}}_{21} (\bar{F}) + {\tilde{L}}_{22} ({\bar{Ψ}}_{x}) + {\tilde{L}}_{23} ({\bar{Ψ}}_{y}) - {\tilde{L}}_{24} (\bar{W}) - {\tilde{L}}_{25} ({\bar{N}}^{T}) \\ = - \frac{1}{2} \tilde{L} (\bar{W} + {\bar{W}}^{*}, \bar{W} + {\bar{W}}^{*}) \end{matrix}

(2)

\begin{matrix} {\tilde{L}}_{31} (\bar{W}) + {\tilde{L}}_{32} ({\bar{Ψ}}_{x}) - {\tilde{L}}_{33} ({\bar{Ψ}}_{y}) + {\tilde{L}}_{34} (\bar{F}) - {\tilde{L}}_{35} ({\bar{N}}^{T}) \\ - {\tilde{L}}_{36} ({\bar{S}}^{T}) = I_{9} \frac{\partial^{3} \bar{W}}{\partial X \partial {\bar{t}}^{2}} + I_{10} \frac{\partial^{2} {\bar{Ψ}}_{x}}{\partial {\bar{t}}^{2}} \end{matrix}

(3)

\begin{matrix} {\tilde{L}}_{41} (\bar{W}) - {\tilde{L}}_{42} ({\bar{Ψ}}_{x}) + {\tilde{L}}_{43} ({\bar{Ψ}}_{y}) + {\tilde{L}}_{44} (\bar{F}) - {\tilde{L}}_{45} ({\bar{N}}^{T}) \\ - {\tilde{L}}_{46} ({\bar{S}}^{T}) = I_{9} \frac{\partial^{3} \bar{W}}{\partial Y \partial {\bar{t}}^{2}} + I_{10} \frac{\partial^{2} {\bar{Ψ}}_{y}}{\partial {\bar{t}}^{2}} \end{matrix}

(4)

where

{\bar{W}}^{*}

is the initial deflection caused by thermal postbuckling,

{\bar{Ψ}}_{x}

and

{\bar{Ψ}}_{y}

are the rotations of the normals to the middle surface with respect to the Y- and X-axes, and

\bar{F}

is the stress function defined by

{\bar{N}}_{x} = \partial^{2} \bar{F} / \partial Y^{2}

{\bar{N}}_{y} = \partial^{2} \bar{F} / \partial X^{2}

, and

{\bar{N}}_{xy} = - \partial^{2} \bar{F} / \partial X \partial Y

. In equations (1)–(4), the linear operators

{\tilde{L}}_{ij}

( ) are defined as in Shen (2013) and the geometric nonlinearity in the von Kármán sense is given in the term

\tilde{L}

( ) which can be expressed by

\begin{matrix} \tilde{L} () = \frac{\partial^{2}}{\partial X^{2}} \frac{\partial^{2}}{\partial Y^{2}} - 2 \frac{\partial^{2}}{\partial X \partial Y} \frac{\partial^{2}}{\partial X \partial Y} + \frac{\partial^{2}}{\partial Y^{2}} \frac{\partial^{2}}{\partial X^{2}} \end{matrix}

(5)

In the above equations, $\bar{t}$ is time and $I_{j}$ are the inertias which are defined in Appendix A.

The plate–foundation interaction is included in the terms associated with ${\bar{K}}_{1}$ and ${\bar{K}}_{2}$ in equation (1), the temperature effect is considered in the material properties, and the terms denoted using superscript T, which are the thermal forces, moments, and higher order moments, ${\bar{N}}^{T}$ , ${\bar{M}}^{T}$ , and ${\bar{P}}^{T}$ , of the plate associated with the elevated temperature, are given by

\begin{matrix} [\begin{matrix} {\bar{N}}_{x}^{T} \\ {\bar{N}}_{y}^{T} \\ {\bar{N}}_{xy}^{T} \end{matrix} \begin{matrix} {\bar{M}}_{x}^{T} \\ {\bar{M}}_{y}^{T} \\ {\bar{M}}_{xy}^{T} \end{matrix} \begin{matrix} {\bar{P}}_{x}^{T} \\ {\bar{P}}_{y}^{T} \\ {\bar{P}}_{xy}^{T} \end{matrix}] = \sum_{k = 1}^{N} {\int_{t_{k - 1}}^{t_{k}} [\begin{matrix} A_{x} \\ A_{y} \\ A_{xy} \end{matrix}]}_{k} (1, Z, Z^{3}) Δ T d Z \end{matrix}

(6a)

and

{\bar{S}}^{T}

is given by

\begin{matrix} [\begin{matrix} {\bar{S}}_{x}^{T} \\ {\bar{S}}_{y}^{T} \\ {\bar{S}}_{xy}^{T} \end{matrix}] = [\begin{matrix} {\bar{M}}_{x}^{T} \\ {\bar{M}}_{y}^{T} \\ {\bar{M}}_{xy}^{T} \end{matrix}] - \frac{4}{3 h^{2}} [\begin{matrix} {\bar{P}}_{x}^{T} \\ {\bar{P}}_{y}^{T} \\ {\bar{P}}_{xy}^{T} \end{matrix}] \end{matrix}

(6b)

in which the temperature rise

Δ T

= T − T₀, where T₀ is the reference temperature at which there are no thermal strains in the plate, and

\begin{matrix} [\begin{matrix} A_{x} \\ A_{y} \\ A_{xy} \end{matrix}] = - [\begin{matrix} {\bar{Q}}_{11} & {\bar{Q}}_{12} & {\bar{Q}}_{16} \\ {\bar{Q}}_{12} & {\bar{Q}}_{22} & {\bar{Q}}_{26} \\ {\bar{Q}}_{16} & {\bar{Q}}_{26} & {\bar{Q}}_{66} \end{matrix}] [\begin{matrix} 1 & 0 \\ 0 & 1 \\ 0 & 0 \end{matrix}] [\begin{matrix} α_{11} \\ α_{22} \end{matrix}] \end{matrix}

(7)

where

α_{11}

and

α_{22}

are, respectively, the longitudinal and transverse thermal expansion coefficients of the GRC layer, and

{\bar{Q}}_{ij}

(i, j = 1, 2, 6) are the transformed elastic constants for a GRC layer. Since the graphene reinforcement in GRC layer is assumed to be zigzag or armchair type, we have

{\bar{Q}}_{ij} = Q_{ij}

, in which

\begin{matrix} Q_{11} = \frac{E_{11}}{1 - ν_{12} ν_{21}}, Q_{12} = \frac{ν_{21} E_{11}}{1 - ν_{12} ν_{21}}, \\ Q_{22} = \frac{E_{22}}{1 - ν_{12} ν_{21}}, \\ Q_{16} = Q_{26} = 0, Q_{66} = G_{12}, Q_{44} = G_{23}, Q_{55} = G_{13} \end{matrix}

(8)

where the effective Young's moduli E₁₁ and E₂₂, shear modulus G₁₂, and Poisson's ratios ν₁₂ and ν₂₁ of the GRC layer have their usual meanings.

Several micromechanical models have been developed to predict the effective material properties of GRCs, e.g. the Mori–Tanaka model (García-Macías et al., 2018), the Voigt model (rule of mixture) (Sperling, 2006), and the Halpin–Tsai model (Halpin and Kardos, 1976). The Mori–Tanaka model is applicable to microparticles and the rule of mixture model is suitable for the fiber fillers. The Halpin–Tsai model was developed for two-dimensionally aligned anisotropic fillers. It has been reported that in nanoscale these three models cannot predict the effective material properties of GRCs accurately and should be modified (Hu et al., 2014). Following Shen et al. (2017a, 2017b), the effective material properties of the GRCs can be predicted by the extended Halpin–Tsai model:

\begin{matrix} E_{11} = η_{1} \frac{1 + 2 (a_{G} / h_{G}) γ_{11} V_{G}}{1 - γ_{11}^{G} V_{G}} E^{m} \end{matrix}

(9a)

\begin{matrix} E_{22} = η_{2} \frac{1 + (2 b_{G} / h_{G}) γ_{22} V_{G}}{1 - γ_{22}^{G} V_{G}} E^{m} \end{matrix}

(9b)

\begin{matrix} G_{12} = η_{3} \frac{1}{1 - γ_{12}^{G} V_{G}} G^{m} \end{matrix}

(9c)

in which

\begin{matrix} γ_{11}^{G} = \frac{E_{11}^{G} / E^{m} - 1}{E_{11}^{G} / E^{m} + 2 a_{G} / h_{G}} \end{matrix}

(10a)

γ_{22}^{G} = \frac{E_{22}^{G} / E^{m} - 1}{E_{22}^{G} / E^{m} + 2 b_{G} / h_{G}}

(10b)

γ_{12}^{G} = \frac{G_{12}^{G} / G^{m} - 1}{G_{12}^{G} / G^{m}}

(10c)

where a_G is the length, b_G is the width, and h_G is the effective thickness of the graphene sheet, E^m and G^m are the Young's modulus and shear modulus of the matrix, and

E_{11}^{G}

E_{22}^{G}

, and

G_{12}^{G}

are the Young's moduli and shear modulus of the graphene sheet. The volume fractions of the graphene reinforcement and the matrix are denoted by V_G and V_m, respectively, and we have V_G + V_m = 1.

Due to the surface effects, strain gradients effect and intermolecular effect, in nanoscale the GRC mechanical properties cannot be directly estimated by the conventional Halpin–Tsai model. Therefore, we introduce the efficiency parameters $η_{j}$ (j = 1, 2, 3) to the Halpin–Tsai model to account for these size-dependent effects. The values of $η_{1}$ , $η_{2}$ , and $η_{3}$ may be determined by matching the elastic modulus of GRCs obtained from the MD simulations with the numerical results predicted by the extended Halpin–Tsai model.

In the current study, the material properties of both the graphene sheet and the matrix will be considered to be temperature dependent (Lin et al., 2017). The thermal expansion coefficients in the longitudinal and transverse directions of the GRC layer are defined as

\begin{matrix} α_{11} = \frac{V_{G} E_{11}^{G} α_{11}^{G} + V_{m} E^{m} α^{m}}{V_{G} E_{11}^{G} + V_{m} E^{m}} \end{matrix}

(11a)

α_{22} = (1 + ν_{12}^{G}) V_{G} α_{22}^{G} + (1 + ν^{m}) V_{m} α^{m} - ν_{12} α_{11}

(11b)

where

α_{11}^{G}

α_{22}^{G}

, and

α^{m}

are thermal expansion coefficients of the graphene and the matrix. The Poisson's ratios

ν^{m}

and

ν_{12}^{G}

are assumed to be weakly dependent on temperature variation and the Poisson's ratio

ν_{12}

of the GRC layer is defined by

ν_{12} = V_{G} ν_{12}^{G} + V_{m} ν^{m}

(12)

Similarly, the mass density ρ of the GRC layer is given by

ρ = V_{G} ρ^{G} + V_{m} ρ^{m}

(13)

in which

ρ^{G}

and

ρ^{m}

are, respectively, the densities of the graphene and the matrix.

All four edges of the GRC laminated plates are assumed to be simply supported with in-plane immovable, i.e. no in-plane displacements in the X- and Y-directions. For such a plate, the boundary conditions can be given by

X = 0, a:

\bar{W} = {\bar{Ψ}}_{y} = 0

(14a)

{\bar{M}}_{x} = {\bar{P}}_{x} = 0

(14b)

\bar{U} = 0

(14c)

Y = 0, b:

\bar{W} = {\bar{Ψ}}_{x} = 0

(14d)

{\bar{M}}_{y} = {\bar{P}}_{y} = 0

(14e)

\bar{V} = 0

(14f)

where

\bar{U}

and

\bar{V}

are displacements in the X- and Y-directions, and the bending moments

{\bar{M}}_{x}

and

{\bar{M}}_{y}

and the higher order moments

{\bar{P}}_{x}

and

{\bar{P}}_{y}

are defined as in Reddy (1984).

The conditions expressing the in-plane immovability conditions (14c) and (14f) are fulfilled on the average sense as

\int_{0}^{b} \int_{0}^{a} \frac{\partial \bar{U}}{\partial X} d X d Y = 0

(15a)

\int_{0}^{a} \int_{0}^{b} \frac{\partial \bar{V}}{\partial Y} d Y d X = 0

(15b)

\begin{array}{l} \int_{0}^{b} \int_{0}^{a} {[(A_{11}^{*} \frac{\partial^{2} \bar{F}}{\partial Y^{2}} + A_{12}^{*} \frac{\partial^{2} \bar{F}}{\partial X^{2}}) + (B_{11}^{*} - \frac{4}{3 h^{2}} E_{11}^{*}) \frac{\partial {\bar{Ψ}}_{x}}{\partial X} + (B_{12}^{*} - \frac{4}{3 h^{2}} E_{12}^{*}) \frac{\partial {\bar{Ψ}}_{y}}{\partial Y} \\ - \frac{4}{3 h^{2}} (E_{11}^{*} \frac{\partial^{2} \bar{W}}{\partial X^{2}} + E_{12}^{*} \frac{\partial^{2} \bar{W}}{\partial Y^{2}}) - \frac{1}{2} {(\frac{\partial (\bar{W} + {\bar{W}}^{*})}{\partial X})}^{2} - (A_{11}^{*} {\bar{N}}_{x}^{T} + A_{12}^{*} {\bar{N}}_{y}^{T})] d X d Y \end{array}

(16a)

\begin{matrix} \int_{0}^{a} \int_{0}^{b} {[(A_{22}^{*} \frac{\partial^{2} \bar{F}}{\partial X^{2}} + A_{12}^{*} \frac{\partial^{2} \bar{F}}{\partial Y^{2}}) + (B_{21}^{*} - \frac{4}{3 h^{2}} E_{21}^{*}) \frac{\partial {\bar{Ψ}}_{x}}{\partial X} \\ + (B_{22}^{*} - \frac{4}{3 h^{2}} E_{22}^{*}) \frac{\partial {\bar{Ψ}}_{y}}{\partial Y}] - \frac{4}{3 h^{2}} (E_{21}^{*} \frac{\partial^{2} \bar{W}}{\partial X^{2}} + E_{22}^{*} \frac{\partial^{2} \bar{W}}{\partial Y^{2}}) \\ - \frac{1}{2} (\frac{\partial (\bar{W} + {\frac{}{W}}^{*})}{\partial Y})^{2} - (A_{12}^{*} {\bar{N}}_{x}^{T} + A_{22}^{*} {\bar{N}}_{y}^{T})] d Y d X \end{matrix}

(16b)

In the above equations, the reduced stiffness matrices [ $A_{ij}^{*}$ ], [ $B_{ij}^{*}$ ], [ $D_{ij}^{*}$ ], [ $E_{ij}^{*}$ ], [ $F_{ij}^{*}$ ], and [ $H_{ij}^{*}$ ] are defined in Appendix A.

3. Solution procedure

Many different methods are used to solve the nonlinear problems of FG plate structures (Lai et al., 2009; Civalek, 2013; Zhu et al., 2014; Kiani, 2018a, 2018b). A two-step perturbation technique has been developed for nonlinear analysis of beam, plate and shell structures (Shen, 2013). This approach gives explicit analytical expressions of all the variables in the large deflection region. The advantage of this method is that it is unnecessary to guess the forms of solutions which can be obtained step by step, and such solutions satisfy both the governing equations and the boundary conditions accurately in the asymptotic sense. To use this two-step perturbation approach, the motion equations (1)–(4) are first rewritten in the dimensionless forms as

\begin{matrix} L_{11} (W) - L_{12} (Ψ_{x}) - L_{13} (Ψ_{y}) + γ_{14} L_{14} (F) - L_{16} (M^{T}) \\ + (K_{1} W - K_{2} \nabla^{2} W) = γ_{14} β^{2} L (W + W^{*}, F + F^{*}) \\ + L_{17} (\frac{\partial^{2} W}{\partial t^{2}}) + γ_{80} (\frac{\partial^{3} Ψ_{x}}{\partial x \partial t^{2}} + β \frac{\partial^{3} Ψ_{y}}{\partial y \partial t^{2}}) + λ_{q} \end{matrix}

(17)

\begin{matrix} L_{21} (F) + γ_{24} L_{22} (Ψ_{x}) + γ_{24} L_{23} (Ψ_{y}) - γ_{24} L_{24} (W) \\ = - \frac{1}{2} γ_{24} β^{2} L (W + W^{*}, W + W^{*}) \end{matrix}

(18)

\begin{matrix} L_{31} (W) + L_{32} (Ψ_{x}) - L_{33} (Ψ_{y}) + γ_{14} L_{34} (F) \\ - L_{36} (S^{T}) = γ_{90} \frac{\partial^{3} W}{\partial x \partial t^{2}} + γ_{10} \frac{\partial^{2} Ψ_{x}}{\partial t^{2}} \end{matrix}

(19)

\begin{matrix} L_{41} (W) - L_{42} (Ψ_{x}) + L_{43} (Ψ_{y}) + γ_{14} L_{44} (F) \\ - L_{46} (S^{T}) = γ_{90} β \frac{\partial^{3} W}{\partial y \partial t^{2}} + γ_{10} \frac{\partial^{2} Ψ_{y}}{\partial t^{2}} \end{matrix}

(20)

where the nondimensional operators L_ij( ) and L( ) are defined as in Shen (2013). In equations (17)–(20), the dimensionless parameters are defined as

\begin{matrix} x = π \frac{X}{a}, y = π \frac{Y}{b}, β = \frac{a}{b}, \\ (W, W^{*}) = \frac{(\bar{W}, {\bar{W}}^{*})}{{[D_{11}^{*} D_{22}^{*} A_{11}^{*} A_{22}^{*}]}^{1 / 4}}, \\ (Ψ_{x}, Ψ_{y}) = \frac{a}{π} \frac{({\bar{Ψ}}_{x}, {\bar{Ψ}}_{y})}{{[D_{11}^{*} D_{22}^{*} A_{11}^{*} A_{22}^{*}]}^{1 / 4}}, (F, F^{*}) = \frac{(\bar{F}, {\bar{F}}^{*})}{{[D_{11}^{*} D_{22}^{*}]}^{1 / 2}}, \\ ( \\ M_{x}, P_{x}) = \frac{a^{2}}{π^{2}} \frac{1}{D_{11}^{*} {[D_{11}^{*} D_{22}^{*} A_{11}^{*} A_{22}^{*}]}^{1 / 4}} ({\bar{M}}_{x}, \frac{4}{3 h^{2}} {\bar{P}}_{x}), \\ τ = \frac{π t}{a} \sqrt{\frac{E_{0}}{ρ_{0}}}, ω_{L} = Ω_{L} \frac{a}{π} \sqrt{\frac{ρ_{0}}{E_{0}}}, \\ γ_{14} = [\frac{D_{22}^{*}}{D_{11}^{*}}]^{1 / 2}, γ_{24} = [\frac{A_{11}^{*}}{A_{22}^{*}}]^{1 / 2}, γ_{5} = - \frac{A_{12}^{*}}{A_{22}^{*}}, γ_{170} = - \frac{I_{1} E_{0} a^{2}}{π^{2} ρ_{0} D_{11}^{*}}, \\ γ_{171} & = \frac{4 E_{0} (I_{5} I_{1} - I_{4} I_{2})}{3 ρ_{0} h^{2} I_{1} D_{11}^{*}}, \\ (γ_{80}, γ_{90}, γ_{10}) = (I_{8}, I_{9}, I_{10}) \frac{E_{0}}{ρ_{0} D_{11}^{*}}, \\ (γ_{T 1}, γ_{T 2}) = \frac{a^{2}}{π^{2}} \frac{(A_{x}^{T}, A_{y}^{T})}{{[D_{11}^{*} D_{22}^{*}]}^{1 / 2}}, \\ (γ_{T 3}, γ_{T 4}, γ_{T 6}, γ_{T 7}) \\ = \frac{a^{2}}{π^{2} {hD}_{11}^{*}} (D_{x}^{T}, D_{y}^{T}, \frac{4}{3 h^{2}} F_{x}^{T}, \frac{4}{3 h^{2}} F_{y}^{T}), \\ (K_{1}, k_{1}) = {\bar{K}}_{1} (\frac{a^{4}}{π^{4} D_{11}^{*}}, \frac{b^{4}}{E_{0} h^{3}}), \\ (K_{2}, k_{2}) = {\bar{K}}_{2} (\frac{a^{2}}{π^{2} D_{11}^{*}}, \frac{b^{2}}{E_{0} h^{3}}), \\ λ_{q} = \frac{q a^{4}}{π^{4} D_{11}^{*} {[D_{11}^{*} D_{22}^{*} A_{11}^{*} A_{22}^{*}]}^{1 / 4}}, λ_{T} = Δ T \end{matrix}

(21)

where k₁ and k₂ are the alternative dimensionless forms of foundation stiffnesses which are only used for the numerical examples, E₀ and

ρ_{0}

are the reference values of E^m and

ρ^{m}

at T₀ = 300 K (room temperature), and

A_{x}^{T}

D_{x}^{T}

F_{x}^{T}

, etc., are defined by

[\begin{matrix} \begin{matrix} A_{x}^{T} \\ A_{y}^{T} \end{matrix} & \begin{matrix} D_{x}^{T} \\ D_{y}^{T} \end{matrix} & \begin{matrix} F_{x}^{T} \\ F_{y}^{T} \end{matrix} \end{matrix}] = - \sum_{k = 1} \int_{t_{k - 1}}^{t_{k}} [\begin{matrix} A_{x} \\ A_{y} \end{matrix}] (1, Z, Z^{3}) d Z

(22)

where A_x and A_y are given in equation (7).

The boundary conditions of equation (14) become

x = 0, π:

W = Ψ_{y} = 0

(23a)

M_{x} = P_{x} = 0

(23b)

δ_{x} = 0

(23c)

y = 0, π:

W = Ψ_{x} = 0

(23d)

M_{y} = P_{y} = 0

(23e)

δ_{y} = 0

(23f)

where

\begin{matrix} δ_{x} = - \frac{1}{4 π^{2} β^{2} γ_{24}} \int_{0}^{π} \int_{0}^{π} [(γ_{24}^{2} β^{2} \frac{\partial^{2} F}{\partial y^{2}} - γ_{5} \frac{\partial^{2} F}{\partial x^{2}}) + γ_{24} (γ_{511} \frac{\partial Ψ_{x}}{\partial x} + γ_{233} β \frac{\partial Ψ_{y}}{\partial y}) \\ - γ_{24} (γ_{611} \frac{\partial^{2} W}{\partial x^{2}} + γ_{244} β^{2} \frac{\partial^{2} W}{\partial y^{2}}) \\ - \frac{1}{2} γ_{24} (\frac{\partial (W + W^{*})}{\partial x})^{2} + (γ_{24}^{2} γ_{T 1} - γ_{5} γ_{T 2}) λ_{T}] d x d y \end{matrix}

(24a)

\begin{matrix} δ_{y} = - \frac{1}{4 π^{2} β^{2} γ_{24}} \int_{0}^{π} \int_{0}^{π} [(\frac{\partial^{2} F}{\partial x^{2}} - γ_{5} β^{2} \frac{\partial^{2} F}{\partial y^{2}}) \\ + γ_{24} (γ_{220} \frac{\partial Ψ_{x}}{\partial x} + γ_{522} β \frac{\partial Ψ_{y}}{\partial y}) \\ - γ_{24} (γ_{240} \frac{\partial^{2} W}{\partial x^{2}} + γ_{622} β^{2} \frac{\partial^{2} W}{\partial y^{2}}) \\ - \frac{1}{2} γ_{24} β^{2} (\frac{\partial (W + W^{*})}{\partial y})^{2} + (γ_{T 2} - γ_{5} γ_{T 1}) λ_{T}] d y d x \end{matrix}

(24b)

In equations (17), (18), (24a), and (24b), $W^{*} (x, y)$ is the thermal deflection in a thermal postbuckling state of the GRC laminated plate and $F^{*} (x, y)$ is the stress function corresponding to $W^{*} (x, y)$ . To this end, the thermal postbuckling problem of the GRC laminated plate in a uniform temperature field should be firstly solved. In such a case, W is only a function of x and all terms with respect to time in equations (17)–(20) are neglected. It is worth noting that, for the thermal postbuckling problem, $Δ T$ is an unknown; hence, an iterative scheme is necessary when the material properties are assumed to be temperature dependent. Let $λ_{q} = 0$ , the solution is obtained easily as previously reported in Shen et al. (2017a).

Thereafter, a large-amplitude vibration problem for a thermally postbuckled plate needs to be solved. We assume the solution of equations (17)–(20) to be such that

\begin{matrix} W (x, y, τ, ɛ) = \sum_{j = 1} ɛ^{j} w_{j} (x, y, τ), \\ Ψ_{x} (x, y, τ, ɛ) = \sum_{j = 1} ɛ^{j} Ψ_{xj} (x, y, τ), \\ Ψ_{y} (x, y, τ, ɛ) & = \sum_{j = 1} ɛ^{j} Ψ_{yj} (x, y, τ), \\ F (x, y, τ, ɛ) = \sum_{j = 0} ɛ^{j} f_{j} (x, y, τ), \\ λ_{q} (x, y, τ, ɛ) & = \sum_{j = 1} ɛ^{j} λ_{j} (x, y, τ) \end{matrix}

(25)

where

τ = ɛ t

and ɛ is a small perturbation parameter that has no physical meaning in the first step. The simply supported boundary conditions in the space domain are considered in this study, hence, we assume the first term of w_j (x, y, τ) to have

\begin{matrix} w_{1} (x, y, τ) = A_{11}^{(1)} (τ) sin mx sin ny \end{matrix}

(26)

in which the vibration mode (m, n) denotes the half-waves along the X- and Y-directions of the plate, respectively. The initial conditions are considered to be

\begin{matrix} W |_{τ = 0} = \frac{\partial W}{\partial τ} |_{τ = 0} = 0, Ψ_{x} |_{τ = 0} = \frac{\partial Ψ_{x}}{\partial τ} |_{τ = 0} = 0, \\ Ψ_{y} |_{τ = 0} = \frac{\partial Ψ_{y}}{\partial τ} |_{τ = 0} = 0 \end{matrix}

(27)

Substituting equation (25) into equations (17)–(20), collecting the terms of the same order of ɛ, we obtain a set of perturbation equations. Applying equation (26) as the first step solution to the perturbation equations, the fourth order asymptotic solutions can be obtained step by step:

\begin{matrix} W (x, y, t) = ɛ A_{11}^{(1)} (t) sin mx sin ny + (ɛ A_{11}^{(1)} (t))^{3} \\ \times [a_{331} sin 3 mx sin ny + a_{313} sin mx sin 3 ny] + O (ɛ^{4}) \end{matrix}

(28)

\begin{matrix} Ψ_{x} (x, y, t) = [(ɛ A_{11}^{(1)} (t)) c_{111} + (ɛ \frac{\partial^{2} A_{11}^{(1)} (t)}{\partial t^{2}}) c_{311}] \\ \times cos mx sin ny + (ɛ A_{11}^{(1)} (t))^{2} c_{202} sin 2 mx \\ + (ɛ A_{11}^{(1)} (t))^{3} [c_{331} cos 3 mx sin ny \\ + c_{313} cos mx sin 3 ny] + O (ɛ^{4}) \end{matrix}

(29)

\begin{matrix} Ψ_{y} (x, y, t) = [(ɛ A_{11}^{(1)} (t)) d_{111} + (ɛ \frac{\partial^{2} A_{11}^{(1)} (t)}{\partial t^{2}}) d_{311}] \\ \times sin mx cos ny + (ɛ A_{11}^{(1)} (t))^{2} d_{220} sin 2 ny \\ + (ɛ A_{11}^{(1)} (t))^{3} [d_{331} sin 3 mx cos ny \\ + d_{313} sin mx cos 3 ny] + O (ɛ^{4}) \end{matrix}

(30)

\begin{matrix} F (x, y, t) = - B_{00}^{(0)} y^{2} / 2 - b_{00}^{(0)} x^{2} / 2 \\ + [(ɛ A_{11}^{(1)} (t)) \\ b_{111} + (ɛ \frac{\partial^{2} A_{11}^{(1)} (t)}{\partial t^{2}}) b_{311}] \\ \times sin mx sin ny + ɛ^{2} (- B_{00}^{(2)} y^{2} / 2 - b_{00}^{(2)} x^{2} / 2) \\ + (ɛ A_{11}^{(1)} (t))^{2} [b_{220} cos 2 mx + b_{202} cos 2 ny] \\ + (ɛ A_{11}^{(1)} (t))^{3} [b_{331} sin 3 mx sin ny \\ + b_{313} sin mx sin 3 ny] + O (ɛ^{4}) \end{matrix}

(31)

\begin{matrix} λ_{q} (x, y, t) = ɛ [g_{30} A_{11}^{(1)} (t) + g_{31} (\frac{\partial^{2} A_{11}^{(1)} (t)}{\partial t^{2}})] sin mx sin ny \\ + [(ɛ A_{11}^{(1)} (t))^{2} g_{201} + (ɛ A_{11}^{(1)} (t)) (ɛ A_{11}^{*}) g_{202}] cos 2 mx \\ + [(ɛ A_{11}^{(1)} (t))^{2} g_{021} + (ɛ A_{11}^{(1)} (t)) (ɛ A_{11}^{*}) g_{022}] cos 2 ny \\ + [(ɛ A_{11}^{(1)} (t))^{3} g_{331} + (ɛ A_{11}^{(1)} (t)) (ɛ A_{11}^{*})^{2} g_{332}] \\ \times sin mx sin ny + O (ɛ^{4}) \end{matrix}

(32)

Note that equation (32) contains ( $ɛ A_{11}^{*}$ ) which comes from thermal postbuckling. In equations (28)–(32), τ is replaced back by t. Then in the second step, we take ( $ɛ A_{11}^{(1)}$ ) as the second perturbation parameter which relates to the dimensionless vibration amplitude. For free vibration we have $λ_{q} = 0$ . Applying the Galerkin procedure to equation (32) yields

\begin{matrix} g_{30} \frac{d^{2} (ɛ A_{11}^{(1)})}{d t^{2}} + g_{31} (ɛ A_{11}^{(1)}) + g_{32} (ɛ A_{11}^{(1)})^{2} + g_{33} (ɛ A_{11}^{(1)})^{3} = 0 \end{matrix}

(33)

In equation (33), the terms g₃₀, g₃₁, g₃₂, and g₃₄ are described in detail in Appendix B, and the solution of which may be expressed by

ω_{NL} = [\frac{g_{31}}{g_{30}} (1 + \frac{9 g_{31} g_{33} - 10 g_{32}^{2}}{12 g_{31}^{2}} A^{2})]^{1 / 2} (Δ T / Δ T_{cr} < 1)

(34a)

\begin{matrix} ω_{NL} = [\frac{3}{4} \frac{g_{33}}{g_{31}} A^{2}]^{1 / 122} (Δ T / Δ T_{cr} = 1) \end{matrix}

(34b)

\begin{matrix} ω_{NL} = [\frac{G_{31}}{g_{30}} (1 + \frac{9 g_{31} g_{33} - 10 g_{32}^{2}}{12 g_{31}^{2}} A^{2})]^{1 / 2} \\ (Δ T / Δ T_{cr} > 1) \end{matrix}

(34c)

where

A = {\bar{W}}_{max} / [D_{11}^{*} D_{22}^{*} A_{11}^{*} A_{22}^{*}]^{1 / 4}

is the dimensionless maximum amplitude of the plate. From equation (21), the corresponding linear frequency can be expressed as

Ω_{L} = ω_{L} (π / a) (E_{0} / ρ_{0})^{1 / 2}

, where

ω_{L} = [g_{31} / g_{30}]^{1 / 2}

4. Numerical results and discussions

The thermal postbuckling and large-amplitude vibration behaviors are presented in this section for GRC laminated plates resting on elastic foundations under a uniform temperature field. Poly(methyl methacrylate), referred to as PMMA, is selected for the matrix, and the material properties of PMMA are assumed to be ρ^m = 1150 kg/m³, ν^m = 0.34, α^m = 45(1 + 0.0005ΔT) × 10⁻⁶/K, and E^m = (3.52−0.0034T) GPa, where T = T₀ + ΔT and T₀ = 300 K (room temperature). Hence, we have α^m = 45.0 × 10⁻⁶/K and E^m = 2.5 GPa when T = 300 K. The zigzag (refer to as 0-ply) graphene sheets with ρ_G = 4118 kg/m³ and effective thickness h_G = 0.188 nm are selected as reinforcements. The temperature-dependent material properties of zigzag graphene sheets are adopted as previously reported in Lin et al. (2017). The variation of thermomechanical properties of zigzag graphene sheets with respect to temperature within 300 ≤ T ≤ 1000 are as follows:

\begin{matrix} E_{11}^{G} = (2.16637 - 0.00193 T + 2 . 9370110^{- 6} T^{2} \\ - 1 . 5177510^{- 9} T^{3}) TPa, \\ E_{22}^{G} & = (2.16868 - 0.00193 T + 2 . 8595410^{- 6} T^{2} \\ - 1 . 4514510^{- 9} T^{3}) TPa, \\ G_{12}^{G} & = (0.53514 + 8 . 2443610^{- 4} T - 1 . 293210^{- 6} T^{2} \\ + 5 . 7850710^{- 10} T^{3}) TPa, \\ α_{11}^{G} & = (- 3.83788 + 0.01416 T - 1 . 6335510^{- 5} T^{2} \\ + 6 . 3358910^{- 9} T^{3}) \times 10^{- 6} / K, \\ α_{22}^{G} = (- 3.73997 + 0.01296 T - 1 . 3503310^{- 5} T^{2} \\ + 4 . 6039210^{- 9} T^{3}) \times 10^{- 6} / K, \\ ν_{12}^{G} = 0.177 \end{matrix}

(35)

As mentioned previously, the conventional Halpin–Tsai model cannot predict the effective material properties of GRCs accurately due to the small scale effect and other effects on the GRC material properties. We need to extend the conventional Halpin–Tsai model with the graphene efficiency parameters

η_{j}

(j = 1, 2, 3) to evaluate the GRC material properties. We obtain the graphene efficiency parameters

η_{j}

(j = 1, 2, 3) by calibrating the Halpin–Tsai model against the Young's moduli E₁₁ and E₂₂ and shear modulus G₁₂ of GRCs obtained from the MD simulations, as previously reported in Shen et al. (2017a, 2017b). Table 1 shows the obtained graphene efficiency parameters and in this study we take G₁₃ = G₂₃ = 0.5G₁₂.

Table 1.

Temperature-dependent efficiency parameters of graphene/PMMA nanocomposites.

T (K)	V _G	$η_{1}$	$η_{2}$	$η_{3}$
300	0.03	2.929	2.855	11.842
	0.05	3.068	2.962	15.944
	0.07	3.013	2.966	23.575
	0.09	2.647	2.609	32.816
	0.11	2.311	2.260	33.125
400	0.03	2.977	2.896	13.928
	0.05	3.128	3.023	15.229
	0.07	3.060	3.027	22.588
	0.09	2.701	2.603	28.869
	0.11	2.405	2.337	29.527
500	0.03	3.388	3.382	16.712
	0.05	3.544	3.414	16.018
	0.07	3.462	3.339	23.428
	0.09	3.058	2.936	29.754
	0.11	2.736	2.665	30.773

As part of the validation of the present method, the buckling temperature T_cr (in K) for (0/90/0/90/0)_S GRC laminated plates under a uniform temperature field are calculated and compared in Table 2 with the isogeometric finite element method (FEM) results of Mirzaei and Kiani (2017) based on the first-order shear deformation theory (FSDT). The geometric parameters of the plate are taken to be a/b = 1, h = 2 mm, and b/h = 15, 20, and 50. The material properties are the same as used in Mirzaei and Kiani (2017). As a second example, the dimensionless fundamental frequencies of (0/90/0/90/0)_S GRC laminated plates under different thermal environmental conditions T = 300, 400, and 500 K are compared in Table 3 with the isogeometric FEM results of Kiani (2018b) based on the FSDT. The plate has a/b = 1, b/h = 10, and h = 2 mm. The dimensionless frequencies are defined by

\tilde{Ω} = Ω (a^{2} / h) \sqrt{ρ_{0} / ρ_{0} E_{0} E_{0}}

. These two comparison studies reveal that the present solutions agree well with existing results.

Table 2.

Comparisons of buckling temperatures T_cr (in K) for perfect, (0/90/0/90/0)_S GRC plates under a uniform temperature field [a/b = 1, h = 2 mm, (m, n) = (1, 1)].

b/h	Source	UD	FG-X	FG-O
15	Mirzaei and Kiani (2017)	454.189	490.860	422.298
	Present and Shen et al. (2017a)	453.858	487.778	423.264
20	Mirzaei and Kiani (2017)	392.930	417.302	372.654
	Present and Shen et al. (2017a)	392.830	416.202	372.783
50	Mirzaei and Kiani (2017)	316.255	321.200	312.294
	Present and Shen et al. (2017a)	316.262	321.178	312.298

Table 3.

Comparisons of natural frequency $\tilde{Ω} = Ω (a^{2} / h) \sqrt{ρ_{0} / ρ_{0} E_{0} E_{0}}$ of (0/90/0/90/0)_S GRC plates in thermal environments (a/b = 1, b/h = 10, h = 2 mm).

	Source	${\tilde{Ω}}_{11}$	${\tilde{Ω}}_{12}$	${\tilde{Ω}}_{21}$	${\tilde{Ω}}_{22}$	${\tilde{Ω}}_{13}$	${\tilde{Ω}}_{31}$
T = 300 K
UD	Kiani (2018b)	28.0794	64.8676	64.9378	95.4811	116.8946	117.0263
	Present and Shen et al. (2017b)	28.0982	64.9584	65.0364	95.6904	116.1739	116.3356
FG-V&Λ	Kiani (2018b)	26.2182	59.4712	59.5815	88.8070	107.4205	107.8222
	Present and Shen et al. (2017b)	25.3900	59.2889	59.4084	88.6182	107.5928	107.8485
FG-X	Kiani (2018b)	30.0881	68.2699	68.3945	100.5284	120.2327	120.4635
	Present and Shen et al. (2017b)	29.5212	65.7134	65.8394	95.6296	113.5548	113.8022
FG-O	Kiani (2018b)	23.4260	55.4163	55.4212	82.9171	101.7395	101.7495
	Present and Shen et al. (2017b)	23.5769	56.2111	56.2103	84.5669	104.1813	104.1911
T = 400 K
UD	Kiani (2018b)	21.9430	55.8754	56.4669	83.8599	102.5441	103.4163
	Present and Shen et al. (2017b)	21.9591	55.9631	56.5605	84.0641	102.8508	103.7435
FG-V&Λ	Kiani (2018b)	20.4385	51.0325	51.7530	78.0404	94.9630	96.0897
	Present and Shen et al. (2017b)	19.7323	51.2779	52.0123	78.3923	95.9517	97.0796
FG-X	Kiani (2018b)	24.3027	59.3343	59.8177	88.2192	106.6484	107.3818
	Present and Shen et al. (2017b)	23.8138	57.2699	57.7903	84.4584	101.4343	102.2760
FG-O	Kiani (2018b)	17.4008	47.1036	47.5887	72.5463	89.5566	90.2131
	Present and Shen et al. (2017b)	17.5581	47.8107	48.2951	73.9946	91.5963	92.2668
T = 500 K
UD	Kiani (2018b)	15.0636	47.9939	49.4060	74.4269	91.9083	93.9191
	Present and Shen et al. (2017b)	15.0776	48.0719	49.5070	74.6317	92.2023	94.2890
FG-V&Λ	Kiani (2018b)	14.1465	44.2592	50.7232	74.3433	90.3340	98.4017
	Present and Shen et al. (2017b)	14.8439	48.1197	49.5077	75.4086	95.5734	97.5242
FG-X	Kiani (2018b)	18.7757	52.6440	53.6839	79.9368	97.5167	99.0306
	Present and Shen et al. (2017b)	18.2245	50.6451	51.7409	76.4767	92.7234	94.3692
FG-O	Kiani (2018b)	9.5522	39.9600	41.1008	64.6199	80.8077	82.2606
	Present and Shen et al. (2017b)	9.8082	40.7210	41.8332	66.0909	82.8460	84.2405

After validating the present formulation and the method of solution, parametric studies have been carried out. Typical results are shown in Figures 1 –6. For the following examples, the geometric parameters of the plate are taken to be a/b = 1, h = 2 mm, and b/h = 20, 30, and 40. In the present analysis, only two types of FG-GRC laminated plates, namely FG-O and FG-X, are considered. It has been reported that GRCs may contain the volume fraction of graphene reinforcement by up to 21% (Putz et al., 2010). The maximum volume fraction of graphene reinforcement in a GRC layer is 11% in the current study. The FG-O GRC laminated plate has a mid-plane symmetric graded distribution of graphene reinforcements [0.03/0.05/0.07/0.09/0.11]_S, while the FG-X GRC laminated plate has the reversed graphene distributions [0.11/0.09/0.07/0.05/0.03]_S. The UD GRC laminated plate has ten GRC layers with the same graphene volume fraction V_G = 0.07. In such a way, the UD and the two FG GRC laminated plates will have the same value of total volume fraction of graphene reinforcement.

Figure 1.

Thermal postbuckling load–deflection curves of (0/90/0/90/0)_S GRC laminated plates of three types of graphene reinforcement distributions.

Figure 2.

Thermal postbuckling load–deflection curves of FG-X (0/90/0/90/0)_S GRC laminated plates with different values of b/h ratio.

Figures 1 –3 present the nondimensional thermal postbuckling equilibrium paths of (0/90/0/90/0)_S GRC laminated plates resting on or without elastic foundations and in a uniform temperature field. It is worth noting that the material properties are assumed to be functions of temperature, and the temperature is also assumed to be an unknown in the thermal postbuckling analysis, hence, an iterative scheme is necessary to obtain the numerical results as previously shown in Shen (2001). It can be observed that the thermal postbuckling load–deflection curves for all cases in Figures 1 –3 show stable equilibrium paths with buckling mode of m = 1 which enables us to determine the temperatures and the deflections of the postbuckled plates that will be used as inputs for the nonlinear vibration analysis of thermally postbuckled GRC laminated plates. It is worth noting that, unlike in Shen (2017a), three dimensionless thermal postbuckling load-deflection curves in each figure have opposite orders as those shown in Shen (2017a).

Figure 3.

Thermal postbuckling load–deflection curves of FG-X (0/90/0/90/0)_S GRC laminated plates supported by elastic foundations.

Figure 4(a) presents the linear fundamental frequencies ${\bar{ω}}_{L}$ (Hz) of the pre- and postbuckled (0/90/0/90/0)_S GRC plates of three types of graphene reinforcement distributions in a uniform temperature field. The plate has b/h = 30. The results show that the fundamental frequencies of (0/90/0/90/0)_S GRC plates decrease as temperature rises in the prebuckling states. Evidently, the fundamental frequency approaches zero at the bifurcation point, whereas in the postbuckling states, the fundamental frequency will further increase due to the thermal deflection W* obtained from the thermal postbuckling equilibrium path of the same plate. It can be seen that the FG-X (0/90/0/90/0)_S GRC plate has a higher frequency than the UD plate has, whereas the FG-O (0/90/0/90/0)_S GRC plate has a lower frequency than the UD plate has in the prebuckling region. On the contrary, the results are reversed in the postbuckling region. Figure 4(b) shows the nonlinear to linear frequency ratio $ω_{NL} / ω_{0}$ of the pre- and postbuckled (0/90/0/90/0)_S GRC plates of three types of graphene reinforcement distributions under a uniform temperature field, where $ω_{0}$ represents $ω_{L}$ at $Δ T = 0$ . Three cases, i.e. $Δ T / Δ T_{cr} = 0$ , 1, and 2, are considered. $Δ T / Δ T_{cr} = 0$ represents no temperature rise, $Δ T / Δ T_{cr} = 1$ represents bifurcation buckling case, and $Δ T / Δ T_{cr} = 2$ represents thermal postbuckled equilibrium case. For the case $Δ T / Δ T_{cr} = 0$ , the nonlinear frequency-amplitude curves of the (0/90/0/90/0)_S GRC plate are the same as those of Shen et al. (2017b). For the case $Δ T / Δ T_{cr} = 1$ , the frequency-amplitude curves of the (0/90/0/90/0)_S GRC plate appear to be linear. For the case $Δ T / Δ T_{cr} = 2$ , a large-amplitude free vibration about a thermally postbuckled equilibrium state of the (0/90/0/90/0)_S GRC plate is observed. In all three cases, the FG-X (0/90/0/90/0)_S GRC laminated plate with a given postbuckled displacement has lower nonlinear to linear frequency ratio than the UD (0/90/0/90/0)_S GRC plate has, while the FG-O (0/90/0/90/0)_S GRC laminated plate has higher nonlinear to linear frequency ratio than the UD (0/90/0/90/0)_S GRC plate has. Hence, only FG-X (0/90/0/90/0)_S GRC plates are considered in the following examples.

Figure 4.

Vibration of thermally buckled (0/90/0/90/0)_S GRC laminated plates of three types of graphene reinforcement distributions: (a) linear frequencies; (b) nonlinear to linear frequency ratio ω_NL/ω₀.

Figure 5 depicts the effect of the plate width-to-thickness ratio b/h ( = 20, 30, and 40) on the linear fundamental frequencies ${\bar{ω}}_{L}$ (Hz) and the nonlinear to linear frequency ratio $ω_{NL} / / ω_{0}$ of the FG-X (0/90/0/90/0)_S GRC plates in the pre- and post-buckling states. It is observed that in both prebuckling and postbuckling regions, the fundamental frequency is decreased when the plate width-to-thickness ratio b/h increases, as shown in Figure 5(a). Figure 5(b) also shows that for a given postbuckled displacement, the nonlinear to linear frequency ratio of FG-X (0/90/0/90/0)_S GRC plates becomes lower when the plate width-to-thickness ratio b/h increases.

Figure 5.

The effect of b/h ratio on the vibration characteristics of thermally buckled FG-X (0/90/0/90/0)_S GRC laminated plates: (a) linear frequencies; (b) nonlinear to linear frequency ratio ω_NL/ω₀.

Figure 6.

The effect of foundation stiffness on the vibration characteristics of thermally buckled FG-X (0/90/0/90/0)_S GRC laminated plates: (a) linear frequencies; (b) nonlinear to linear frequency ratio ω_NL/ω₀.

The influence of foundation stiffness on the linear fundamental frequencies ${\bar{ω}}_{L}$ (Hz) and the nonlinear to linear frequency ratio $ω_{NL} / / ω_{0}$ of the pre- and postbuckled FG-X (0/90/0/90/0)_S GRC plates resting on elastic foundations is illustrated in Figure 6. The plate has b/h = 30. Two foundation models are considered. The stiffnesses are (k₁, k₂) = (100, 10) for the Pasternak elastic foundation, (k₁, k₂) = (100, 0) for the Winkler elastic foundation and (k₁, k₂) = (0, 0) for the plate without any elastic foundation. Figure 6(a) shows that the fundamental frequencies are increased in the prebuckled states, but are decreased in the postbuckled states as foundation stiffness increases. Figure 6(b) also shows that for a given postbuckled displacement, the nonlinear to linear frequency ratio decreases as foundation stiffness increases.

5. Concluding remarks

Small- and large-amplitude vibration characteristics for thermally postbuckled GRC laminated plates have been investigated using a multi-scale approach. The volume fraction of graphene in each ply may be different, which results in a piecewise FG-GRC plate. The extended Halpin–Tsai model is adopted to estimate the anisotropic and temperature-dependent material properties of GRCs. The novelty of this study is that the temperature-dependent material properties and initial deflections caused by the thermal postbuckling are both taken into account, and such an initial deflection is not a constant in the large-amplitude vibration region. Numerical illustrations have been carried out for UD, FG-O, and FG-X GRC laminated plates with low graphene volume fractions. The buckling temperature, thermal postbuckling equilibrium path and temperature-dependent frequency for FG GRC laminated plates are obtained. We observe that the fundamental frequencies of the GRC laminated plates are increased but the nonlinear to linear frequency ratios of the GRC laminated plates are decreased with increase in foundation stiffness. The results show that the plate width-to-thickness ratio has a significant effect on the fundamental frequencies, but has a weak effect on the frequency-amplitude curves in both prebuckling and postbuckling states. The results confirm that a piecewise FG reinforcement distribution has a significant effect on the nonlinear vibration characteristics of thermally postbuckled GRC laminated plates.

Footnotes

Acknowledgement

The authors are very grateful for financial support.

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 (grant number 51779138) and the Australian Research Council (grant number DP140104156).

References

Alijani

Amabili

Bakhtiari-Nejad

(2011) Thermal effects on nonlinear vibrations of functionally graded doubly curved shells using higher order shear deformation theory. Composite Structures 93: 2541–2553.

Bakhadda

Bouiadjra

Bourada

et al. (2018) Dynamic and bending analysis of carbon nanotube- reinforced composite plates with elastic foundation. Wind and Structures 27: 311–324.

Bodaghi

Saidi

(2011) Thermoelastic buckling behavior of thick functionally graded rectangular plates. Archive of Applied Mechanics 81: 1555–1572.

Bouderba

Houari

MSA

Tounsi

et al. (2016) Thermal stability of functionally graded sandwich plates using a simple shear deformation theory. Structural Engineering and Mechanics 58: 397–422.

Civalek

(2013) Nonlinear dynamic response of laminated plates resting on nonlinear elastic foundations by the discrete singular convolution-differential quadrature coupled approaches. Composites Part B: Engineering 50: 171–179.

El-Haina

Bakora

Bousahla

et al. (2017) A simple analytical approach for thermal buckling of thick functionally graded sandwich plates. Structural Engineering and Mechanics 63: 585–595.

Fan

Wang

(2016) Thermal postbuckling and vibration of postbuckled matrix cracked hybrid laminated plates containing carbon nanotube reinforced composite layers on elastic foundation. Composite Structures 157: 386–397.

Fazzolari

(2018) Thermoelastic vibration and stability of temperature-dependent carbon nanotube-reinforced composite plates. Composite Structures 196: 199–214.

Fazzolari

Carrera

(2013) Accurate free vibration analysis of thermo-mechanically pre/post-buckled anisotropic multilayered plates based on a refined hierarchical trigonometric Ritz formulation. Composite Structures 95: 381–402. .

10.

Ganapathi

Prakash

(2006) Thermal buckling of simply supported functionally graded skew plates. Composite Structures 74: 247–250.

11.

García-Macías

Rodríguez-Tembleque

Sáez

(2018) Bending and free vibration analysis of functionally graded graphene vs. carbon nanotube reinforced composite plates. Composite Structures 186: 123–138.

12.

Gholami

Ansari

(2018) Nonlinear harmonically excited vibration of third-order shear deformable functionally graded graphene platelet-reinforced composite rectangular plates. Engineering Structures 156: 197–209.

13.

Giannopoulos

Kallivokas

(2014) Mechanical properties of graphene based nanocomposites incorporating a hybrid interphase. Finite Elements Analysis and Design 90: 31–40. .

14.

Girish

Ramachandra

(2005a) Thermal postbuckled vibrations of symmetrically laminated composite plates with initial geometric imperfections. Journal of Sound and Vibration 282: 1137–1153.

15.

Girish

Ramachandra

(2005b) Postbuckling and vibration analysis of antisymmetric angle-ply composite plates. Journal of Thermal Stresses 28: 1145–1159.

16.

Gupta

Talha

(2017) Nonlinear flexural and vibration response of geometrically imperfect gradient plates using hyperbolic higher-order shear and normal deformation theory. Composites Part B: Engineering 123: 241–261.

17.

Halpin

Kardos

(1976) The Halpin–Tsai equations: a review. Polymer Engineering Science 16: 344–352.

18.

Kulkarni

Choi

et al. (2014) Graphene–polymer nanocomposites for structural and functional applications. Progress in Polymer Science 39: 1934–1972.

19.

Huang

X-L

Shen

H-S

(2004) Nonlinear vibration and dynamic response of functionally graded plates in thermal environments. International Journal of Solids and Structures 41: 2403–2427.

20.

Javaheri

Eslami

(2002) Thermal buckling of functionally graded plates. AIAA Journal 40: 162–169.

21.

Kiani

(2018a) NURBS-based isogeometric thermal postbuckling analysis of temperature dependent graphene-reinforced composite laminated plates. Thin-Walled Structures 125: 211–219.

22.

Kiani

(2018b) Isogeometric large amplitude free vibration of graphene reinforced laminated plates in thermal environment using NURBS formulation. Computer Methods in Applied Mechanics and Engineering 332: 86–101.

23.

Lai

Lim

Xiang

et al. (2009) On asymptotic analysis for large amplitude nonlinear free vibration of simply supported laminated plates. Journal of Vibration and Acoustics 131: 051010.

24.

Lee

D-M

Lee

(1997a) Vibration behaviors of thermally postbuckled anisotropic plates using first-order shear deformable plate theory. Computers and Structures 63: 371–378.

25.

Lee

D-M

Lee

(1997b) Vibration analysis of stiffened laminated plates including thermally postbuckled deflection effect. Journal of Reinforced Plastics and Composites 16: 1138–1154.

26.

Liang

Yao

Wang

et al. (2011) A three-dimensional vertically aligned functionalized multilayer graphene architecture: an approach for graphene-based thermal interfacial materials. ACS Nano 5: 2392–2401.

27.

Librescu

Lin

(1997) Vibration of thermomechanically loaded flat and curved panel taking into account geometric imperfections and tangential edge restraints. International Journal of Solids and Structures 34: 2161–2181.

28.

Librescu

Lin

Nemeth

et al. (1996a) Vibration of geometrically imperfect panels subjected to thermal and mechanical loads. Journal of Spacecraft Rockets 33: 285–291.

29.

Librescu

Lin

Nemeth

et al. (1996b) Frequency–load interaction of geometrically imperfect curved panels subjected to heating. AIAA Journal 34: 166–177.

30.

Lin

Xiang

Shen

H-S

(2017) Temperature dependent mechanical properties of graphene reinforced polymer nanocomposites: a molecular dynamics simulation. Composites Part B: Engineering 111: 261–269.

31.

Lue

Lim

Chen

(2009) Exact solutions for free vibrations of functionally graded thick plates on elastic foundations. Mechanics of Advanced Materials and Structures 16: 576–584.

32.

Menasria

Bouhadra

Tounsi

et al. (2017) A new and simple HSDT for thermal stability analysis of FG sandwich plates. Steel and Composite Structures 25: 157–175.

33.

Milani

González

Quijada

et al. (2013) Polypropylene/graphene nanosheet nanocomposites by in situ polymerization: synthesis, characterization and fundamental properties. Composites Science and Technology 84: 1–7.

34.

Mirzaei

Kiani

(2017) Isogeometric thermal buckling analysis of temperature dependent FG graphene reinforced laminated plates using NURBS formulation. Composite Structures 180: 606–616.

35.

Zou

et al. (2010) Anisotropic mechanical properties of graphene sheets from molecular dynamics. Physica B 405: 1301–1306.

36.

Han

Lee

(2000) Postbuckling and vibration characteristics of piezolaminated composite plate subjected to thermo-piezoelectric loads. Journal of Sound and Vibration 233: 19–40.

37.

Park

Kim

(2006) Thermal postbuckling and vibration analyses of functionally graded plates. Journal of Sound and Vibration 289: 77–93.

38.

Putz

Compton

Palmeri

et al. (2010) High-nanofiller-content graphene oxide-polymer nanocomposites via vacuum-assisted self-assembly. Advanced Functional Materials 20: 3322–3329.

39.

Reddy

Rajendran

Liew

(2006) Equilibrium configuration and elastic properties of finite grapheme. Nanotechnology 17: 864–870.

40.

Reddy

(1984) A refined nonlinear theory of plates with transverse shear deformation. International Journal of Solids and Structures 20: 881–896.

41.

Reddy

RMR

Karunasena

Lokuge

(2018) Free vibration of functionally graded-GPL reinforced composite plates with different boundary conditions. Aerospace Science and Technology 78: 147–156.

42.

Sahmani

Aghdam

Rabczuk

(2018) Nonlocal strain gradient plate model for nonlinear large-amplitude vibrations of functionally graded porous micro/nano-plates reinforced with GPLs. Composite Structures 198: 51–62.

43.

Samadpour

Sadighi

Shakeri

et al. (2015) Vibration analysis of thermally buckled SMA hybrid composite sandwich plate. Composite Structures 119: 251–263.

44.

Shen

H-S

(1997) Kármán-type equations for a higher-order shear deformation plate theory and its use in the thermal postbuckling analysis. Applied Mathematics and Mechanics 18: 1137–1152.

45.

Shen

H-S

(2001) Thermal postbuckling behavior of imperfect shear deformable laminated plates with temperature-dependent properties. Computer Methods in Applied Mechanics and Engineering 190: 5377–5390.

46.

Shen

H-S

(2007) Thermal postbuckling behavior of shear deformable FGM plates with temperature-dependent properties. International Journal of Mechanical Sciences 49: 466–478.

47.

Shen

H-S

(2013) A Two-Step Perturbation Method in Nonlinear Analysis of Beams, Plates and Shells, Hoboken, NJ: John Wiley & Sons.

48.

Shen

H-S

Wang

(2017) Nonlinear vibration of compressed and thermally postbuckled nanotube-reinforced composite plates resting on elastic foundations. Aerospace Science and Technology 64: 63–74.

49.

Shen

H-S

Zhang

C-L

(2010) Thermal buckling and postbuckling behavior of functionally graded carbon nanotube-reinforced composite plates. Materials & Design 31: 3403–3411.

50.

Shen

H-S

Xiang

Lin

(2017a) Thermal buckling and postbuckling of functionally graded graphene-reinforced composite laminated plates resting on elastic foundations. Thin-Walled Structures 118: 229–237.

51.

Shen

H-S

Xiang

Lin

(2017b) Nonlinear vibration of functionally graded graphene-reinforced composite laminated plates in thermal environments. Computer Methods in Applied Mechanics and Engineering 319: 175–193.

52.

Shen

H-S

Zhang

C-L

(2010) Temperature-dependent elastic properties of single layer graphene sheets. Materials & Design 31: 4445–4449.

53.

Song

Kitipornchai

Yang

(2017) Free and forced vibrations of functionally graded polymer composite plates reinforced with graphene nanoplatelets. Composite Structures 159: 579–588.

54.

Sperling

(2006) Introduction to Physical Polymer Science, 4th ed. Hoboken, NJ: John Wiley & Sons.

55.

Taczala

Buczkowski

Kleiber

(2016) Nonlinear free vibration of pre- and post-buckled FGM plates on two-parameter foundation in the thermal environment. Composite Structures 137: 85–92.

56.

Talha

Singh

(2011) Thermo-mechanical buckling analysis of finite element modeled functionally graded ceramic-metal plates. International Journal of Applied Mechanics 3: 867–880.

57.

Kitipornchai

Yang

(2017) Thermal buckling and postbuckling of functionally graded graphene nanocomposite plates. Materials & Design 132: 430–41.

58.

(2004) Thermal buckling of simply supported moderately thick rectangular FGM plate. Composite Structures 64: 211–218.

59.

Xia

X-K

Shen

H-S

(2008) Vibration of postbuckled FGM hybrid laminated plates in thermal environment. Engineering Structures 30: 2420–2435.

60.

Yang

Han

(1983) Buckled plate vibrations and large amplitude vibrations using high-order triangular elements. AIAA Journal 21: 758–766.

61.

Yousefi

Sun

Lin

et al. (2014) Highly aligned graphene/polymer nanocomposites with excellent dielectric properties for high-performance electromagnetic interference shielding. Advanced Materials 26: 5480–5487.

62.

Zhao

Zhang

Hao

et al. (2010) Alternate multilayer films of poly(vinyl alcohol) and exfoliated graphene oxide fabricated via a facial layer-by-layer assembly. Macromolecules 43: 9411–9416.

63.

Zhu

Zhang

Liew

(2014) Geometrically nonlinear thermomechanical analysis of moderately thick functionally graded plates using a local Petrov–Galerkin approach with moving Kriging interpolation. Composite Structures 107: 298–314.