Free vibration analysis of micro plate reinforced with functionally graded graphene nanoplatelets based on modified strain-gradient formulation

Abstract

In this paper, modified strain-gradient theory is developed for size-dependent formulation of micro plate reinforced with functionally graded graphene nanoplatelets. The reinforced micro plate is subjected to thermal and mechanical loads. The functionally graded graphene nanoplatelets are distributed along the thickness direction based on various patterns. The effective material properties of reinforced structure including modulus of elasticity and density or Poisson’s ratio are calculated based on Halpin–Tsai model and rule of mixture, respectively. The kinematic relations are developed based on third-order shear deformation theory. The solution procedure is proposed based on analytical work for custom boundary condition. Before presentation of numerical results, a comprehensive comparative study is performed for validation of present formulation. The numerical results are presented to investigate the influence of important parameters such as weight fraction of GNPs, various distribution of GNPs, three micro length scale parameters and some non-dimensional geometric parameters on the vibration responses.

Keywords

Modified strain gradient functionally graded graphene nanoplatelets third-order theory Halpin–Tsai model rule of mixture

Introduction

New materials and new methods of material production have been developed by various researchers to be used in new various conditions. These materials and production methods give a new opportunity for designer to produce materials and structures with novel properties applicable in various environments. In recent years, production of composite materials including nano reinforcements such as carbon nanotubes and graphene nanoplatelets offered some excellent options for designer to reach the structures with high stiffness and low density. Addition of carbon nanotubes and graphene nanoplatelets to matrixes such as polymer leads to increase of stiffness and decrease of density made of them. Some instances can be presented to show novel properties of composite structures reinforced with carbon nanotubes and graphene nanoplatelets. Some experiments were carried out to show that increase of 31% in Young’s modulus of elasticity of epoxy may be reached by addition of 0.1% weight fraction of graphene nanoplatelet reinforcement [1]. Another experiment was performed to show that increase of 76% in tensile strength and increase of 62% in Young’s modulus of elasticity are reached by the addition of 0.7% weight fraction of graphene platelet reinforcement [2]. One can conclude that although some experimental and theoretical works on the structures reinforced with carbon nanotube were published, however, the number of works about theoretical works of composite structures reinforced with graphene nanoplatelets is very limited. A literature review on the subject of this paper is presented to justify necessity of this paper.

The literature review is divided to three sections: first section is related to carbon nanotube reinforcements, second section is related to graphene nanoplatelets and third section is related to the application of non-classical theories such modified strain gradient and couple stress theories.

Wu et al. [3] investigated the nonlinear vibration analysis of imperfect composite beam reinforced with functionally graded carbon nanotubes based on first-order shear deformation theory (FSDT). The imperfection was assumed based on trigonometric and hyperbolic functions. The governing equations of motion were solved using Ritz method. Dynamic stability analysis of nanocomposite beam reinforced with single-walled carbon nanotubes was studied by Ke et al. [4] based on differential quadrature method. The numerical results including free vibration and buckling responses were provided in terms of nanotube volume fraction, slenderness ratio and end supports. Arefi and Zenkour [5,6] developed higher-order SDT (HSDT) to bending and free vibration analysis of sandwich plates including piezoelectric layers. Free vibration analysis of a composite elliptical plate reinforced with functionally graded carbon nanotube was studied by Ansari et al. [7] based on FSDT and Hamilton’s principle. Thermoelastic nonlinear frequency analysis of a sandwich structure reinforced with temperature-dependent functionally graded single-walled carbon nanotube was studied by Mehar et al. [8]. To include nonlinearity in derivation of governing equations of motion, Green–Lagrange nonlinear strain components were employed. The solution and numerical results were provided based on finite element method. Thang et al. [9] studied the nonlinear buckling analysis of imperfect functionally graded carbon nano-reinforced composite subjected to axial loading based on Kirchhoff model with von Karman-type of nonlinearity. The closed-form solution was derived based on Galerkin method and the Airy stress function.

Rafiee et al. [10] summarized some novel properties of graphene nanoribbons. They expressed that addition of 0.3% weight fraction of nanofillers leads to 30% increase in Young’s modulus of the epoxy composite while 0.3% weight fraction of multiwalled carbon nanotubes leads to 22% increase in Young’s modulus. Montazeri and Rafii-Tabar [11] presented a comprehensive work for the calculation of elastic constants of a polymeric nanocomposite embedded with graphene sheets, and carbon nanotubes based on molecular dynamics, molecular structural mechanics and finite element method. The reinforcement role of these nanofillers was investigated in transverse directions. In addition, the effect of sliding motion of graphene layers on the elastic constants of the nanocomposite was studied. Liu et al. [12] studied buckling and free vibration analysis of functionally graded cylindrical shell with initially stress reinforced with non-uniformly distributed graphene platelets (GPLs) based on three-dimensional elasticity theory and state-space formulation. The modified Halpin–Tsai model and rule of mixture are used for the calculation of effective material properties. They concluded on the influence of important parameters such as GPL weight fraction, dispersion pattern, geometry and size as well as the influence of initial stress on the buckling and free vibration responses. In addition, it was concluded that the addition of a small amount of GPLs leads to significant increase of buckling load and natural frequencies. Buckling analysis of composite cylindrical shell with opening reinforced with GPL was studied by Wang et al. [13] based on finite element method. The effective material properties of composite cylindrical shell were studied based on Halpin–Tsai micromechanics model and rule of mixture. The buckling load was evaluated in terms of important parameters such as location and geometry of opening, weight fraction and shape of GPL fillers. Critical buckling load and natural frequency analysis of nanocomposite plate reinforced with functionally graded graphene subjected to periodic uniaxial in-plane force and a uniform temperature rise was studied by Wu et al. [14] based on FSDT. The parametric analysis was performed to investigate the influence of various distributions, temperature change, static in-plane force, plate geometry and boundary condition on the parametric instability of functionally graded multilayer GPLRC plates. Thermoelastic bending analysis of nanocomposite rectangular plate reinforced with graphene nanoplatelets was studied by Yang et al. [15]. It was assumed that weight fraction of graphene nanoplatelets is changed gradually along the thickness direction for various distributions. The numerical results including stress and deformation distribution were presented in terms of GPL’s weight fraction, distribution pattern, geometry, temperature change and plate boundary conditions of the nanocomposite plates. Thermoelastic bending analysis of functionally graded polymer nanocomposite circular and annular plates reinforced with graphene nanoplatelets was studied by Yang et al. [16] based on three-dimensional elasticity theory. Three-dimensional thermoelastic analysis of a fucntionally graded elliptical plate with clamped edges was studied based in three-dimensional elasticity theory by Yang [17] for the case that weight fraction is changed gradually along the thickness direction.

Nonlinear size-dependent aanlysis of a fucntionally graded micro beam was studied by Arbind et al. [18] based on modified couple stress theory (MCST) and thrid-order SDT (TSDT). Nonlinearity was accounted based on von Karman strain-displacement relations. The bending results were derived based on an analytical methos in terms of micro length scale parameters. In addition, the nonlinear results were evaluated based on finite element method. Arefi et al. [19] studied free vibration analysis of three-layered nanoplate including functionally graded core and two piezoelectric layers based on nonlocal elasticity theory and concept of neutral surface. The numerical results were presented in terms of aspect ratio, non-dimensional geometric parameters and nonlocal parameter. Thai and Choi [20] investigated on the size-dependent bending, buckling and free vibration analysis of fucntionally graded plate based on Kirchhoff and Mindlin plate theories and MCST. They presented a model with one material length scale parameter instead of previous models that contain three material length scales. Modified couple stress formulation and a four variable-refined plate theory were employed by He et al. [21] for bending, buckling and free vibration responses of functionally graded rectangular microplate based on Hamilton’s principle. They performed a comparative work to show that employing the size dependent four variable model leads to more accurate results rather than size-dependent TSDT. Free vibration analysis of fucntionally graded Timoshenko micro beam was studied by Ansari et al. [22] based on strain-gradient theory (SGT). The governing equations of motion were derived based on Hamilton’s principle for a functionally graded beam obeying Mori–Tanaka scheme along the thickness direction. The effect of higher-order shear deformations and initial electric and magnetic potentials were studied on the sandwich nanoplate by Arefi and Zenkour [23]. To account size dependency and anisotropy, bending and free vibration analysis of fucntionally graded piezoelectric Timoshenko beam was studied by Li et al. [24] based on modified strain-gradient theory (MSGT). Size dependency was modelled using a model containing three material length scale parameters. Size dependency effect in micro scale and shear strains were included in derivation of governing equations of motion for bending, buckling and free vibration analyses of functionally graded micro-plates by Thai et al. [25]. Wang et al. [26] presented static bending and free vibration analysis of a micro scale Timoshenko beam based on strain-gradient elasticity theory. The governing equations of motion were derived based on Hamilton’s principle as well as initial conditions and boundary conditions. The problem was reduced to special case with removing two or all material length scale parameters. A refined SDT and strain-gradient elasticity theory were employed for bending, buckling and free vibration analysis of fucntionally graded microplate by Zhang et al. [27]. The advantage of this model was satisfying the stress-free boundary conditions on the top and bottom of plate without requiring the shear stress correction factor. The effects of material length scale parameter, material-gradient index, aspect ratio and transverse shear deformation were studied on the bending, buckling and vibration characteristics of microplate.

The aim of this work is to formulate a micro plate reinforced with functionally graded nanoplatelets based on TSDT and MSGT. To account size-dependency, MSGT is used. In addition, for accurate modelling of the micro plate with various ranges of dimensionless parameters, TSDT is employed. The micro plate is subjected to thermal loads while is resting on Pasternak’s foundation. Halpin–Tsai model and rule of mixture are employed to calculate the effective modulus of elasticity and Poisson’s ratio or density of composite-reinforced micro plate. The governing equations of motion are derived based on Hamilton’s principle. The natural frequencies of the reinforced micro plate are calculated in terms of weight fraction of GNPs, various distribution of GNPs, three micro length scale parameters and some non-dimensional geometric parameters such as side length to thickness ratio and thickness to micro length scale ratio. To verify the present formulation and corresponding numerical results, our formulations are reduced to modified couple stress and modified strain-gradient theories, and then the responses are compared with the literature results.

Modified strain-gradient formulation

The MSGT has been organized for size-dependent analysis of structures. This theory is organized using addition of the new terms to classical strain energy. The new terms are including the dilatation-gradient tensor $γ_{i}$ , deviatoric stretch-gradient tensor $η_{ijk}^{(1)}$ and symmetric part of rotation-gradient tensor $χ_{ij}^{(s)}$ .

One can conclude that the first term is dilatation gradient as a first-order tensor, which implies the dilatations in each direction. In addition, the deviatoric stretch gradient is third-order tensor, which represents the stretching effect. Rotation gradient is second-order tensor, which expresses the rotation effect. The corresponding terms are including a micro length scale parameter in each term.

Strain energy of a structure is defined based on MSGT as follows [25]

U = \frac{1}{2} \int_{V} (σ_{ij} ε_{ij} + p_{i} γ_{i} + τ_{ijk}^{(1)} η_{ijk}^{(1)} + m_{ij}^{(s)} χ_{ij}^{(s)}) d V

(1)

In which the components of strain tensor and corresponding terms of MSGT are defined as follows [25]

ε_{ij} = \frac{1}{2} (\frac{\partial u_{i}}{\partial x_{j}} + \frac{\partial u_{j}}{\partial x_{i}})

(2)

γ_{i} = \frac{\partial ε_{mm}}{\partial x_{i}}

(3)

\begin{array}{l} η_{ijk}^{(1)} = \frac{1}{3} (\frac{\partial ε_{jk}}{\partial x_{i}} + \frac{\partial ε_{ki}}{\partial x_{j}} + \frac{\partial ε_{ij}}{\partial x_{k}}) - \frac{1}{15} δ_{ij} (\frac{\partial ε_{mm}}{\partial x_{k}} + 2 \frac{\partial ε_{mk}}{\partial x_{m}}) \\ - \frac{1}{15} δ_{jk} (\frac{\partial ε_{mm}}{\partial x_{i}} + 2 \frac{\partial ε_{mi}}{\partial x_{m}}) - \frac{1}{15} δ_{ki} (\frac{\partial ε_{mm}}{\partial x_{j}} + 2 \frac{\partial ε_{mj}}{\partial x_{m}}) \end{array}

(4)

χ_{ij}^{(s)} = \frac{1}{4} (e_{imn} \frac{\partial^{2} u_{n}}{\partial x_{mj}^{2}} + e_{jmn} \frac{\partial^{2} u_{n}}{\partial x_{mi}^{2}})

(5)

where

u_{i}

are displacement components; in addition,

δ_{ij}

is Kronecker delta and

e_{ijk}

is permutation symbol. The components of classic stress tensor

σ_{ij}

in the presence of thermal strains are defined as

[\begin{matrix} σ_{xx} \\ σ_{yy} \\ σ_{yz} \\ σ_{xz} \\ σ_{xy} \end{matrix}] = [\begin{matrix} c_{11} & c_{12} & 0 & 0 & 0 \\ c_{12} & c_{22} & 0 & 0 & 0 \\ 0 & 0 & c_{44} & 0 & 0 \\ 0 & 0 & 0 & c_{55} & 0 \\ 0 & 0 & 0 & 0 & c_{66} \end{matrix}] [\begin{matrix} ε_{xx} - α_{C} Δ T \\ ε_{yy} - α_{C} Δ T \\ γ_{yz} \\ γ_{xz} \\ γ_{xy} \end{matrix}]

(6)

In equation (6), $α_{C}$ is heat expansion coefficients, $Δ T$ is temperature rising and $c_{ij}$ are stiffness coefficients that are defined as

c_{11} = c_{22} = \frac{E_{C} (z)}{1 - ν_{C}^{2} (z)}

(7a)

c_{12} = \frac{ν E_{C} (z)}{1 - ν_{C}^{2} (z)}

(7b)

c_{44} = c_{55} = c_{66} = \frac{E_{C} (z)}{2 (1 + ν_{C} (z))}

(7c)

In which $E_{C} (z)$ and $ν_{C} (z)$ are modulus of elasticity and Poisson’s ratio of composite micro plate. Figure 1 shows the used coordinates and geometry of plate.

Figure 1.

The schematic figure of composite micro plate reinforced with functionally graded graphene nanoplatelets resting on Pasternak’s foundation.

The non-classical and higher-order terms of stress ( $p_{i}$ , $η_{ijk}^{(1)}$ , $m_{ij}^{(s)}$ ) are defined as [25]

p_{i} = 2 μ l_{0}^{2} γ_{i}

(8)

τ_{ijk}^{(1)} = 2 μ l_{1}^{2} η_{ijk}^{(1)}

(9)

m_{ij}^{(s)} = 2 μ l_{2}^{2} χ_{ij}^{(s)}

(10)

where

l_{0}, l_{1}, l_{2}

are the material length scale parameters corresponding to the dilatation-gradient tensor, deviatoric stretch-gradient tensor and symmetric rotation-gradient tensor, respectively. The shear modulus

μ

is defined as follows

μ = \frac{E_{C} (z)}{2 (1 + ν_{C} (z))}

(11)

Our problem in this paper is a composite rectangular micro-plate made of polymer matrix reinforced with graphene nanoplatelets. The coordinate system is defined as x,y,z located at the middle of plate. The length, width and thickness of micro-plate is assumed as a, b and h, respectively. The micro-plate is resting on Pasternak’s foundation. The effective modulus of elasticity based on extended Halpin–Tsai model is defined as follows [14,28,30]

E_{C} = \frac{3}{8} \frac{1 + ξ_{L} η_{L} V_{GNP}}{1 - η_{L} V_{GNP}} \times E_{M} + \frac{5}{8} \frac{1 + ξ_{w} η_{w} V_{GNP}}{1 - η_{w} V_{GNP}} \times E_{M}

(12)

where

V_{GNP}

is the volume fraction of the GNPs and

E_{M}

and

E_{GNP}

refer to the Young’s modulus of the polymer matrix and GNPs, respectively. Two additional parameters

η_{L}, η_{w}

are introduced as follows

η_{L} = \frac{(E_{GNP} / E_{M}) - 1}{(E_{GNP} / E_{M}) + ξ_{L}}

(13)

η_{w} = \frac{(E_{GNP} / E_{M}) - 1}{(E_{GNP} / E_{M}) + ξ_{W}}

(14)

where

ξ_{L}

and

ξ_{w}

depend on the geometry and size of the GNP nanofillers as follows

ξ_{L} = 2 (\frac{l_{GNP}}{h_{GNP}})

(15)

ξ_{w} = 2 (\frac{w_{GNP}}{h_{GNP}})

(16)

and

l_{GNP}

w_{GNP}

and

h_{GNP}

refer to the average length, width and thickness of the GNPs, respectively. In addition, the volume fraction

V_{GNP}

is defined as follows [14,30]

V_{GNP} = \frac{g_{GNP}}{g_{GNP} + (ρ_{GNP} / ρ_{M}) (1 - g_{GNP})}

(17)

where

g_{GNP}

is the weight fraction of GNPs. Moreover,

ρ_{GNP}

and

ρ_{M}

refer to the density of GNPs and polymer, respectively. The effective density, Poisson’s ratio and thermal expansion coefficient are defined, according to the rule of mixtures, as follows [14]

ρ_{C} = ρ_{GNP} V_{GNP} + ρ_{M} V_{M}

(18)

ν_{C} = ν_{GNP} V_{GNP} + ν_{M} V_{M}

(19)

α_{C} = α_{GNP} V_{GNP} + α_{M} V_{M}

(20)

where

ν_{GNP}

and

ν_{M}

are Poisson’s ratios of GNPs and polymer matrix,

α_{GNP}

and

α_{M}

are thermal expansion coefficient of GNPs and the matrix, respectively. The relation between volume fractions of epoxy and reinforcement is defined as follows

V_{GNP} + V_{M} = 1

(21)

In this work, three models are employed for the distribution of graphene nanoplatelets. These models that are presented in Figure 2 are described as follows [17].

Linear distribution with GNP weight fraction changing from the highest value on the top surface to zero at the bottom surface

g_{GNP} (z) = λ_{1} W_{GNP}^{0} (\frac{1}{2} + \frac{z}{h})

(22)

Parabolic distribution where the GNP weight fraction is the maximum at the top and bottom surfaces and zero at the mid-plane

g_{GNP} (z) = \frac{4}{h^{2}} λ_{2} W_{GNP}^{0} z^{2}

(23)

Uniform GNP distribution with constant weight fraction along the plate thickness

g_{GNP} (z) = λ_{3} W_{GNP}^{0}

(24)

Figure 2.

Various distributions of functionally graded graphene nanoplatelets.

Kinematic and constitutive relations

The displacement field is assumed based on TSDT, (Figure 1) as follows

u (x, y, z, t) = u_{0} (x, y, t) - z \frac{\partial w_{0} (x, y, t)}{\partial x} + f (z) ψ_{1} (x, y, t)

(25a)

v (x, y, z, t) = v_{0} (x, y, t) - z \frac{\partial w_{0} (x, y, t)}{\partial y} + f (z) ψ_{2} (x, y, t)

(25b)

w (x, y, z, t) = w_{0} (x, y, t)

(25c)

In which $u, v, w$ are displacement fields along the x,y,z directions, respectively. In addition, $u_{0}$ , $v_{0}$ , $w_{0}$ are displacements of middle surface, and $ψ_{1}$ , $ψ_{2}$ are higher-order rotation components.

The shape function is assumed based on TSDT as follows

f (z) = z (1 - \frac{4 z^{2}}{3 h^{2}})

(26)

It is noticeable that this theory does not need to shear stress correction factor. The strain components are defined as follows

ε_{xx} = \frac{\partial u_{0}}{\partial x} - z \frac{\partial^{2} w_{0}}{\partial x^{2}} + f (z) \frac{\partial ψ_{1}}{\partial x}

(27a)

ε_{yy} = \frac{\partial v_{0}}{\partial y} - z \frac{\partial^{2} w_{0}}{\partial y^{2}} + f (z) \frac{\partial ψ_{2}}{\partial y}

(27b)

γ_{yz} = 2 ε_{yz} = f^{'} (z) ψ_{2}

(27c)

γ_{xz} = 2 ε_{xz} = f^{'} (z) ψ_{1}

(27d)

γ_{xy} = 2 ε_{xy} = \frac{\partial u_{0}}{\partial y} + \frac{\partial v_{0}}{\partial x} - 2 z \frac{\partial^{2} w_{0}}{\partial x \partial y} + f (z) (\frac{\partial ψ_{1}}{\partial y} + \frac{\partial ψ_{2}}{\partial x})

(27e)

Using the displacement field and strain components defined in equations (25) and (27), the components of dilatation-gradient tensor $γ_{i}$ are defined as

γ_{x} = \frac{\partial^{2} u_{0}}{\partial x^{2}} + \frac{\partial^{2} v_{0}}{\partial x \partial y} - z (\frac{\partial^{3} w_{0}}{\partial x^{3}} + \frac{\partial^{3} w_{0}}{\partial x \partial y^{2}}) + f (z) (\frac{\partial^{2} ψ_{1}}{\partial x^{2}} + \frac{\partial^{2} ψ_{2}}{\partial x \partial y})

(28a)

γ_{y} = \frac{\partial^{2} u_{0}}{\partial x \partial y} + \frac{\partial^{2} v_{0}}{\partial y^{2}} - z (\frac{\partial^{3} w_{0}}{\partial x^{2} \partial y} + \frac{\partial^{3} w_{0}}{\partial y^{3}}) + f (z) (\frac{\partial^{2} ψ_{1}}{\partial x \partial y} + \frac{\partial^{2} ψ_{2}}{\partial y^{2}})

(28b)

γ_{z} = - \frac{\partial^{2} w_{0}}{\partial x^{2}} + f' (z) (\frac{\partial ψ_{1}}{\partial x} + \frac{\partial ψ_{2}}{\partial y}) - \frac{\partial^{2} w_{0}}{\partial y^{2}}

(28c)

The components of symmetric part of rotation-gradient tensor are defined as

χ_{xx}^{(s)} = \frac{\partial^{2} w_{0}}{\partial x \partial y} - \frac{1}{2} f^{'} (z) \frac{\partial ψ_{2}}{\partial x}

(29a)

χ_{yy}^{(s)} = - \frac{\partial^{2} w_{0}}{\partial x \partial y} + \frac{1}{2} f^{'} (z) \frac{\partial ψ_{1}}{\partial y}

(29b)

χ_{zz}^{(s)} = \frac{1}{2} f^{'} (z) (\frac{\partial ψ_{2}}{\partial x} - \frac{\partial ψ_{1}}{\partial y})

(29c)

χ_{xy}^{(s)} = \frac{1}{2} (\frac{\partial^{2} w_{0}}{\partial y^{2}} - \frac{\partial^{2} w_{0}}{\partial x^{2}}) + \frac{1}{4} f^{'} (z) (\frac{\partial ψ_{1}}{\partial x} - \frac{\partial ψ_{2}}{\partial y})

(29d)

χ_{yz}^{(s)} = \frac{1}{4} (\frac{\partial^{2} v_{0}}{\partial x \partial y} - \frac{\partial^{2} u_{0}}{\partial y^{2}}) + \frac{1}{4} f (z) (\frac{\partial^{2} ψ_{2}}{\partial x \partial y} - \frac{\partial^{2} ψ_{1}}{\partial y^{2}}) + \frac{1}{4} f^{″} (z) ψ_{1}

(29e)

χ_{xz}^{(s)} = \frac{1}{4} (\frac{\partial^{2} v_{0}}{\partial x^{2}} - \frac{\partial^{2} u_{0}}{\partial x \partial y}) + \frac{1}{4} f (z) (\frac{\partial^{2} ψ_{2}}{\partial x^{2}} - \frac{\partial^{2} ψ_{1}}{\partial x \partial y}) - \frac{1}{4} f^{″} (z) ψ_{2}

(29f)

And the components of deviatoric stretch-gradient tensor are defined as

\begin{array}{l} η_{xxx}^{(1)} = \frac{1}{5} (2 \frac{\partial^{2} u_{0}}{\partial x^{2}} - 2 \frac{\partial^{2} v_{0}}{\partial x \partial y} - \frac{\partial^{2} u_{0}}{\partial y^{2}}) + \frac{1}{5} z (3 \frac{\partial^{3} w_{0}}{\partial x \partial y^{2}} - 2 \frac{\partial^{3} w_{0}}{\partial x^{3}}) \\ + \frac{1}{5} f (z) (2 \frac{\partial^{2} ψ_{1}}{\partial x^{2}} - 2 \frac{\partial^{2} ψ_{2}}{\partial x \partial y} - \frac{\partial^{2} ψ_{1}}{\partial y^{2}}) - \frac{1}{5} f^{″} (z) ψ_{1} \end{array}

(30a)

\begin{array}{l} η_{xxy}^{(1)} = η_{xyx}^{(1)} = η_{yxx}^{(1)} = (\frac{8}{15} \frac{\partial^{2} u_{0}}{\partial x \partial y} + \frac{4}{15} \frac{\partial^{2} v_{0}}{\partial x^{2}} - \frac{1}{5} \frac{\partial^{2} v_{0}}{\partial y^{2}}) + z (- \frac{4}{5} \frac{\partial^{3} w_{0}}{\partial x^{2} \partial y} + \frac{1}{5} \frac{\partial^{3} w_{0}}{\partial y^{3}}) \\ + f (z) (\frac{8}{15} \frac{\partial^{2} ψ_{1}}{\partial x \partial y} + \frac{4}{15} \frac{\partial^{2} ψ_{2}}{\partial x^{2}} - \frac{1}{5} \frac{\partial^{2} ψ_{2}}{\partial y^{2}}) - \frac{1}{15} f^{″} (z) ψ_{2} \end{array}

(30b)

η_{xxz}^{(1)} = η_{xzx}^{(1)} = η_{zxx}^{(1)} = \frac{1}{15} (- 4 \frac{\partial^{2} w_{0}}{\partial x^{2}} + \frac{\partial^{2} w_{0}}{\partial y^{2}}) + \frac{2}{15} f^{'} (z) (4 \frac{\partial ψ_{1}}{\partial x} - \frac{\partial ψ_{2}}{\partial y})

(30c)

\begin{array}{l} η_{xyy}^{(1)} = η_{yxy}^{(1)} = η_{yyx}^{(1)} = (\frac{8}{15} \frac{\partial^{2} v_{0}}{\partial x \partial y} + \frac{4}{15} \frac{\partial^{2} u_{0}}{\partial y^{2}} - \frac{1}{5} \frac{\partial^{2} u_{0}}{\partial x^{2}}) + z (- \frac{4}{5} \frac{\partial^{3} w_{0}}{\partial x \partial y^{2}} + \frac{1}{5} \frac{\partial^{3} w_{0}}{\partial x^{3}}) \\ + f (z) (\frac{8}{15} \frac{\partial^{2} ψ_{2}}{\partial x \partial y} + \frac{4}{15} \frac{\partial^{2} ψ_{1}}{\partial y^{2}} - \frac{1}{5} \frac{\partial^{2} ψ_{1}}{\partial x^{2}}) - \frac{1}{15} f^{″} (z) ψ_{1} \end{array}

(30d)

\begin{array}{l} η_{xzz}^{(1)} = η_{zxz}^{(1)} = η_{zzx}^{(1)} = (- \frac{2}{15} \frac{\partial^{2} v_{0}}{\partial x \partial y} - \frac{1}{15} \frac{\partial^{2} u_{0}}{\partial y^{2}} - \frac{1}{5} \frac{\partial^{2} u_{0}}{\partial x^{2}}) + z (\frac{1}{5} \frac{\partial^{3} w_{0}}{\partial x \partial y^{2}} + \frac{1}{5} \frac{\partial^{3} w_{0}}{\partial x^{3}}) \\ + f (z) (- \frac{2}{15} \frac{\partial^{2} ψ_{2}}{\partial x \partial y} - \frac{1}{15} \frac{\partial^{2} ψ_{1}}{\partial y^{2}} - \frac{1}{5} \frac{\partial^{2} ψ_{1}}{\partial x^{2}}) + \frac{4}{15} f^{″} (z) ψ_{1} \end{array}

(30e)

\begin{array}{l} η_{yyy}^{(1)} = \frac{1}{5} (2 \frac{\partial^{2} v_{0}}{\partial y^{2}} - 2 \frac{\partial^{2} u_{0}}{\partial x \partial y} - \frac{\partial^{2} v_{0}}{\partial x^{2}}) + \frac{1}{5} z (3 \frac{\partial^{3} w_{0}}{\partial x^{2} \partial y} - 2 \frac{\partial^{3} w_{0}}{\partial y^{3}}) \\ + \frac{1}{5} f (z) (2 \frac{\partial^{2} ψ_{2}}{\partial y^{2}} - 2 \frac{\partial^{2} ψ_{1}}{\partial x \partial y} - \frac{\partial^{2} ψ_{2}}{\partial x^{2}}) - \frac{1}{5} f^{″} (z) ψ_{2} \end{array}

(30f)

η_{yyz}^{(1)} = η_{yzy}^{(1)} = η_{zyy}^{(1)} = \frac{1}{15} (- 4 \frac{\partial^{2} w_{0}}{\partial y^{2}} + \frac{\partial^{2} w_{0}}{\partial x^{2}}) + \frac{2}{15} f^{'} (z) (4 \frac{\partial ψ_{2}}{\partial y} - \frac{\partial ψ_{1}}{\partial x})

(30g)

\begin{array}{l} η_{yzz}^{(1)} = η_{zyz}^{(1)} = η_{zzy}^{(1)} = (- \frac{2}{15} \frac{\partial^{2} u_{0}}{\partial x \partial y} - \frac{1}{15} \frac{\partial^{2} v_{0}}{\partial x^{2}} - \frac{1}{5} \frac{\partial^{2} v_{0}}{\partial y^{2}}) + z (\frac{1}{5} \frac{\partial^{3} w_{0}}{\partial x^{2} \partial y} + \frac{1}{5} \frac{\partial^{3} w_{0}}{\partial y^{3}}) \\ + f (z) (- \frac{2}{15} \frac{\partial^{2} ψ_{1}}{\partial x \partial y} - \frac{1}{15} \frac{\partial^{2} ψ_{2}}{\partial x^{2}} - \frac{1}{5} \frac{\partial^{2} ψ_{2}}{\partial y^{2}}) + \frac{4}{15} f^{″} (z) ψ_{2} \end{array}

(30h)

η_{zzz}^{(1)} = \frac{1}{5} (\frac{\partial^{2} w_{0}}{\partial x^{2}} + \frac{\partial^{2} w_{0}}{\partial y^{2}}) - \frac{2}{5} f^{'} (z) (\frac{\partial ψ_{1}}{\partial x} + \frac{\partial ψ_{2}}{\partial y})

(30i)

η_{xyz}^{(1)} = η_{yxz}^{(1)} = η_{yzx}^{(1)} = η_{zxy}^{(1)} = η_{zyx}^{(1)} = η_{xzy}^{(1)} = - \frac{1}{3} \frac{\partial^{2} w_{0}}{\partial x \partial y} + \frac{1}{3} f^{'} (z) (\frac{\partial ψ_{1}}{\partial y} + \frac{\partial ψ_{2}}{\partial x})

(30j)

The non-classical components of stress are calculated using the above equations.

The Hamilton’s principle is used to derive governing equations of motion. This theory is defined as

δ \int_{t_{1}}^{t_{2}} (U - T - W) d t = 0

(31)

in which U is strain energy, T is kinetic energy and W is the external works. The total strain energy is introduced by indexes 1, 2, 3 and 4 as follows

δ U = δ U_{1} + δ U_{2} + δ U_{3} + δ U_{4}

(32)

Substitution of variation of strain components into variational form of strain energy leads to resultant form of strain energy as

\begin{array}{l} δ U_{1} = \int_{V} σ_{ij} δ ε_{ij} d V = \int_{V} [σ_{xx} δ ε_{xx} + σ_{yy} δ ε_{yy} + 2 σ_{xy} δ ε_{xy} + 2 σ_{yz} δ ε_{yz} + 2 σ_{xz} δ ε_{xz}] d V \\ = \int_{A} [(- \frac{\partial N_{xx}}{\partial x} - \frac{\partial N_{xy}}{\partial y}) δ u_{0} + (- \frac{\partial N_{yy}}{\partial y} - \frac{\partial N_{xy}}{\partial x}) δ v_{0} \\ + (- \frac{\partial^{2} M_{xx}}{\partial x^{2}} - \frac{\partial^{2} M_{yy}}{\partial y^{2}} - 2 \frac{\partial^{2} M_{xy}}{\partial x \partial y}) δ w_{0} + (- \frac{\partial Q_{xx}^{1}}{\partial x} - \frac{\partial Q_{xy}^{1}}{\partial y} + Q_{xz}^{2}) δ ψ_{1} \\ + (- \frac{\partial Q_{yy}^{1}}{\partial y} - \frac{\partial Q_{xy}^{1}}{\partial x} + Q_{yz}^{2}) δ ψ_{2}] d A \end{array}

(33)

The variation form of strain energy of dilatation-gradient tensor is defined as

\begin{array}{l} δ U_{2} = \int_{V} p_{i} δ γ_{i} d V = \int_{V} [p_{1} δ γ_{1} + p_{2} δ γ_{2} + p_{3} δ γ_{3}] d V \\ = \int_{A} [(\frac{\partial^{2} P_{1}^{1}}{\partial x^{2}} + \frac{\partial^{2} P_{2}^{1}}{\partial x \partial y}) δ u_{0} + (\frac{\partial^{2} P_{1}^{1}}{\partial x \partial y} + \frac{\partial^{2} P_{2}^{1}}{\partial y^{2}}) δ v_{0} \\ + (\frac{\partial^{3} P_{1}^{2}}{\partial x^{3}} + \frac{\partial^{3} P_{1}^{2}}{\partial x \partial y^{2}} + \frac{\partial^{3} P_{2}^{2}}{\partial x^{2} \partial y} + \frac{\partial^{3} P_{2}^{2}}{\partial y^{3}} - \frac{\partial^{2} P_{3}^{1}}{\partial x^{2}} - \frac{\partial^{2} P_{3}^{1}}{\partial y^{2}}) δ w_{0} \\ + (\frac{\partial^{2} P_{1}^{3}}{\partial x^{2}} + \frac{\partial^{2} P_{2}^{3}}{\partial x \partial y} - \frac{\partial^{2} P_{3}^{4}}{\partial x}) δ ψ_{1} + (\frac{\partial^{2} P_{1}^{3}}{\partial x \partial y} + \frac{\partial^{2} P_{2}^{3}}{\partial y^{2}} - \frac{\partial^{2} P_{3}^{4}}{\partial y}) δ ψ_{2}] d A \end{array}

(34)

The variation of strain energy of deviatoric stretch-gradient tensor is including very long terms. Due to this situation, the variation of strain energy of deviatoric stretch-gradient tensor is divided to 10 terms as follows

δ U_{3} = δ U_{3, 1} + δ U_{3, 2} + δ U_{3, 3} + δ U_{3, 4} + δ U_{3, 5} + δ U_{3, 6} + δ U_{3, 7} + δ U_{3, 8} + δ U_{3, 9} + δ U_{3, 10}

(35)

In which the sub-terms are defined as follows

\begin{array}{l} δ U_{3, 1} = \int_{V} [τ_{xxx}^{(1)} δ η_{xxx}^{(1)}] d V \\ = \int_{A} [(\frac{2}{5} \frac{\partial^{2} N_{xxx}}{\partial x^{2}} - \frac{1}{5} \frac{\partial^{2} N_{xxx}}{\partial y^{2}}) δ u_{0} + (- \frac{2}{5} \frac{\partial^{2} N_{xxx}}{\partial x \partial y}) δ v_{0} + (\frac{2}{5} \frac{\partial^{3} M_{xxx}^{1}}{\partial x^{3}} - \frac{3}{5} \frac{\partial^{3} M_{xxx}^{1}}{\partial x \partial y^{2}}) δ w_{0} \\ + (\frac{2}{5} \frac{\partial^{2} M_{xxx}^{2}}{\partial x^{2}} - \frac{1}{5} \frac{\partial^{2} M_{xxx}^{2}}{\partial y^{2}} - \frac{1}{5} M_{xxx}^{4}) δ ψ_{1} + (- \frac{2}{5} \frac{\partial^{2} M_{xxx}^{2}}{\partial x \partial y}) δ ψ_{2}] d A \end{array}

(36)

\begin{array}{l} δ U_{3, 2} = \int_{V} [3 τ_{xxy}^{(1)} δ η_{xxy}^{(1)}] d V \\ = \int_{A} 3 [(\frac{8}{15} \frac{\partial^{2} N_{xxy}}{\partial x \partial y}) δ u_{0} + (\frac{4}{15} \frac{\partial^{2} N_{xxy}}{\partial x^{2}} - \frac{1}{5} \frac{\partial^{2} N_{xxy}}{\partial y^{2}}) δ v_{0} + (\frac{4}{5} \frac{\partial^{3} M_{xxy}^{1}}{\partial x^{2} \partial y} - \frac{1}{5} \frac{\partial^{3} M_{xxy}^{1}}{\partial y^{3}}) δ w_{0} \\ + (\frac{8}{15} \frac{\partial^{2} M_{xxy}^{2}}{\partial x \partial y}) δ ψ_{1} + (\frac{4}{15} \frac{\partial^{2} M_{xxy}^{2}}{\partial x^{2}} - \frac{1}{5} \frac{\partial^{2} M_{xxy}^{2}}{\partial y^{2}} - \frac{1}{15} M_{xxy}^{4}) δ ψ_{2}] d A \end{array}

(37)

\begin{array}{l} δ U_{3, 3} = \int_{V} [3 τ_{xxz}^{(1)} δ η_{xxz}^{(1)}] d V \\ = \int_{A} 3 [(- \frac{4}{15} \frac{\partial^{2} N_{xxz}}{\partial x^{2}} + \frac{1}{15} \frac{\partial^{2} N_{xxz}}{\partial y^{2}}) δ w_{0} + (- \frac{8}{15} \frac{\partial M_{xxz}^{3}}{\partial x}) δ ψ_{1} + (\frac{2}{15} \frac{\partial M_{xxz}^{3}}{\partial y}) δ ψ_{2}] d A \end{array}

(38)

\begin{array}{l} δ U_{3, 4} = \int_{V} [3 τ_{xyy}^{(1)} δ η_{xyy}^{(1)}] d V = \int_{A} 3 [(\frac{4}{15} \frac{\partial^{2} N_{xyy}}{\partial y^{2}} - \frac{1}{5} \frac{\partial^{2} N_{xyy}}{\partial x^{2}}) δ u_{0} + (\frac{8}{15} \frac{\partial^{2} N_{xyy}}{\partial x \partial y}) δ v_{0} \\ + (\frac{4}{5} \frac{\partial^{3} M_{xyy}^{1}}{\partial x \partial y^{2}} - \frac{1}{5} \frac{\partial^{3} M_{xyy}^{1}}{\partial x^{3}}) δ w_{0} + (\frac{4}{15} \frac{\partial^{2} M_{xyy}^{2}}{\partial y^{2}} - \frac{1}{5} \frac{\partial^{2} M_{xyy}^{2}}{\partial x^{2}} - \frac{1}{15} M_{xyy}^{4}) δ ψ_{1} \\ + (\frac{8}{15} \frac{\partial^{2} M_{xyy}^{2}}{\partial x \partial y}) δ ψ_{2}] d A \end{array}

(39)

\begin{array}{l} δ U_{3, 5} = \int_{V} [3 τ_{xzz}^{(1)} δ η_{xzz}^{(1)}] d V = \int_{A} 3 [(- \frac{1}{5} \frac{\partial^{2} N_{xzz}}{\partial x^{2}} - \frac{1}{15} \frac{\partial^{2} N_{xzz}}{\partial y^{2}}) δ u_{0} + (- \frac{2}{15} \frac{\partial^{2} N_{xzz}}{\partial x \partial y}) δ v_{0} \\ + (- \frac{1}{5} \frac{\partial^{3} M_{xzz}^{1}}{\partial x^{3}} - \frac{1}{5} \frac{\partial^{3} M_{xzz}^{1}}{\partial x \partial y^{2}}) δ w_{0} + (- \frac{1}{5} \frac{\partial^{2} M_{xzz}^{2}}{\partial x^{2}} - \frac{1}{15} \frac{\partial^{2} M_{xzz}^{2}}{\partial y^{2}} + \frac{4}{15} M_{xzz}^{4}) δ ψ_{1} \\ + (- \frac{2}{15} \frac{\partial^{2} M_{xzz}^{2}}{\partial x \partial y}) δ ψ_{2}] d A \end{array}

(40)

\begin{array}{l} δ U_{3, 6} = \int_{V} [τ_{yyy}^{(1)} δ η_{yyy}^{(1)}] d V = \int_{A} [(- \frac{2}{5} \frac{\partial^{2} N_{yyy}}{\partial x \partial y}) δ u_{0} + (\frac{2}{5} \frac{\partial^{2} N_{yyy}}{\partial y^{2}} - \frac{1}{5} \frac{\partial^{2} N_{yyy}}{\partial x^{2}}) δ v_{0} \\ + (\frac{2}{5} \frac{\partial^{3} M_{yyy}^{1}}{\partial y^{3}} - \frac{3}{5} \frac{\partial^{3} M_{yyy}^{1}}{\partial x^{2} \partial y}) δ w_{0} + (- \frac{2}{5} \frac{\partial^{2} M_{yyy}^{2}}{\partial x \partial y}) δ ψ_{1} \\ + (\frac{2}{5} \frac{\partial^{2} M_{yyy}^{2}}{\partial y^{2}} - \frac{1}{5} \frac{\partial^{2} M_{yyy}^{2}}{\partial x^{2}} - \frac{1}{5} M_{yyy}^{4}) δ ψ_{2}] d A \end{array}

(41)

\begin{array}{l} δ U_{3, 7} = \int_{V} [3 τ_{yyz}^{(1)} δ η_{yyz}^{(1)}] d V \\ = \int_{A} 3 [(\frac{1}{15} \frac{\partial^{2} N_{yyz}}{\partial x^{2}} - \frac{4}{15} \frac{\partial^{2} N_{yyz}}{\partial y^{2}}) δ w_{0} + (\frac{2}{15} \frac{\partial M_{yyz}^{3}}{\partial x}) δ ψ_{1} + (- \frac{8}{15} \frac{\partial M_{yyz}^{3}}{\partial y}) δ ψ_{2}] d A \end{array}

(42)

\begin{array}{l} δ U_{3, 8} = \int_{V} [3 τ_{yzz}^{(1)} δ η_{yzz}^{(1)}] d V = \int_{A} 3 [(- \frac{2}{15} \frac{\partial^{2} N_{yzz}}{\partial x \partial y}) δ u_{0} + (- \frac{1}{15} \frac{\partial^{2} N_{yzz}}{\partial x^{2}} - \frac{1}{5} \frac{\partial^{2} N_{yzz}}{\partial y^{2}}) δ v_{0} \\ + (- \frac{1}{5} \frac{\partial^{3} M_{yzz}^{1}}{\partial x^{2} \partial y} - \frac{1}{5} \frac{\partial^{3} M_{yzz}^{1}}{\partial y^{3}}) δ w_{0} + (- \frac{2}{15} \frac{\partial^{2} M_{yzz}^{2}}{\partial x \partial y}) δ ψ_{1} \\ + (- \frac{1}{15} \frac{\partial^{2} M_{yzz}^{2}}{\partial x^{2}} - \frac{1}{5} \frac{\partial^{2} M_{yzz}^{2}}{\partial y^{2}} + \frac{4}{15} M_{yzz}^{4}) δ ψ_{2}] d A \end{array}

(43)

\begin{array}{l} δ U_{3, 9} = \int_{V} [τ_{zzz}^{(1)} δ η_{zzz}^{(1)}] d V \\ = \int_{A} [(\frac{1}{5} \frac{\partial^{2} N_{zzz}}{\partial x^{2}} + \frac{1}{5} \frac{\partial^{2} N_{zzz}}{\partial y^{2}}) δ w_{0} + (\frac{2}{5} \frac{\partial M_{zzz}^{3}}{\partial x}) δ ψ_{1} + (\frac{2}{5} \frac{\partial M_{zzz}^{3}}{\partial y}) δ ψ_{2}] d A \end{array}

(44)

\begin{array}{l} δ U_{3, 10} = \int_{V} [6 τ_{xyz}^{(1)} δ η_{xyz}^{(1)}] d V \\ = \int_{A} 6 [(- \frac{1}{3} \frac{\partial^{2} N_{xyz}}{\partial x \partial y}) δ w_{0} + (- \frac{1}{3} \frac{\partial M_{xyz}^{3}}{\partial y}) δ ψ_{1} + (- \frac{1}{3} \frac{\partial M_{xyz}^{3}}{\partial x}) δ ψ_{2}] d A \end{array}

(45)

The variation form of strain energy of symmetric part of rotation-gradient tensor is defined as

\begin{array}{l} δ U_{4} = \int_{V} m_{ij} δ χ_{ij} d V = \int_{V} [m_{xx} δ χ_{xx} + m_{yy} δ χ_{yy} + 2 m_{xy} δ χ_{xy} + 2 m_{yz} δ χ_{yz} + 2 m_{xz} δ χ_{xz}] d V \\ = \int_{A} [(- \frac{1}{2} \frac{\partial^{2} S_{yz}^{1}}{\partial y^{2}} - \frac{1}{2} \frac{\partial^{2} S_{xz}^{1}}{\partial x \partial y}) δ u_{0} + (\frac{1}{2} \frac{\partial^{2} S_{xz}^{1}}{\partial x^{2}} + \frac{1}{2} \frac{\partial^{2} S_{yz}^{1}}{\partial x \partial y}) δ v_{0} \\ + (\frac{\partial^{2} S_{xy}^{1}}{\partial y^{2}} - \frac{\partial^{2} S_{xy}^{1}}{\partial x^{2}} - \frac{\partial^{2} S_{yy}^{1}}{\partial x \partial y} + \frac{\partial^{2} S_{xx}^{1}}{\partial x \partial y}) δ w_{0} \\ + (- \frac{1}{2} \frac{\partial^{2} S_{yz}^{2}}{\partial y^{2}} - \frac{1}{2} \frac{\partial^{2} S_{xz}^{2}}{\partial x \partial y} + \frac{1}{2} S_{yz}^{4} - \frac{1}{2} \frac{\partial S_{xy}^{3}}{\partial x} - \frac{1}{2} \frac{\partial S_{yy}^{3}}{\partial y} + \frac{1}{2} \frac{\partial S_{zz}^{3}}{\partial y}) δ ψ_{1} \\ + (\frac{1}{2} \frac{\partial^{2} S_{xz}^{2}}{\partial x^{2}} - \frac{1}{2} S_{xz}^{4} + \frac{1}{2} \frac{\partial^{2} S_{yz}^{2}}{\partial x \partial y} - \frac{1}{2} \frac{\partial S_{zz}^{3}}{\partial x} + \frac{1}{2} \frac{\partial S_{xy}^{3}}{\partial y} + \frac{1}{2} \frac{\partial S_{xx}^{3}}{\partial x}) δ ψ_{2}] d A \end{array}

(46)

In which A is area of plate. The resultant components are defined as

[N_{ij}, M_{ij}, Q_{ij}^{1}, Q_{ij}^{2}] = \int_{- h / 2}^{h / 2} σ_{ij} [1, z, f (z), f^{'} (z)] d z

(47a)

[P_{i}^{1}, P_{i}^{2}, P_{i}^{3}, P_{i}^{4}] = \int_{- h / 2}^{h / 2} p_{i} [1, z, f (z), f^{'} (z)] d z

(47b)

[N_{ijk}, M_{ijk}^{1}, M_{ijk}^{2}, M_{ijk}^{3}, M_{ijk}^{4}] = \int_{- h / 2}^{h / 2} τ_{ijk} [1, z, f (z), f^{'} (z), f^{″} (z)] d z

(47c)

[S_{ij}^{1}, S_{ij}^{2}, S_{ij}^{3}, S_{ij}^{4}] = \int_{- h / 2}^{h / 2} m_{ij} [1, f (z), f^{'} (z), f^{″} (z)] d z

(47d)

The variation of kinetic energy is defined as

\begin{array}{l} δ T = \int_{V} (ρ \vec{V} . δ V) d V \\ = \int_{A} [(- I_{0} \frac{\partial^{2} u_{0}}{\partial t^{2}} + I_{1} \frac{\partial^{3} w_{0}}{\partial x \partial t^{2}} - I_{2} \frac{\partial^{2} ψ_{1}}{\partial t^{2}}) δ u_{0} + (- I_{0} \frac{\partial^{2} v_{0}}{\partial t^{2}} + I_{1} \frac{\partial^{3} w_{0}}{\partial y \partial t^{2}} - I_{2} \frac{\partial^{2} ψ_{2}}{\partial t^{2}}) δ v_{0} \\ + (- I_{1} \frac{\partial^{3} u_{0}}{\partial x \partial t^{2}} + I_{3} \frac{\partial^{4} w_{0}}{\partial x^{2} \partial t^{2}} - I_{4} \frac{\partial^{3} ψ_{1}}{\partial x \partial t^{2}} - I_{1} \frac{\partial^{3} v_{0}}{\partial y \partial t^{2}} + I_{3} \frac{\partial^{4} w_{0}}{\partial y^{2} \partial t^{2}} - I_{4} \frac{\partial^{3} ψ_{2}}{\partial y \partial t^{2}} - I_{0} \frac{\partial^{2} w_{0}}{\partial t^{2}}) δ w_{0} \\ + (- I_{2} \frac{\partial^{2} u_{0}}{\partial t^{2}} + I_{4} \frac{\partial^{3} w_{0}}{\partial x \partial t^{2}} - I_{5} \frac{\partial^{2} ψ_{1}}{\partial t^{2}}) δ ψ_{1} + (- I_{2} \frac{\partial^{2} v_{0}}{\partial t^{2}} + I_{4} \frac{\partial^{3} w_{0}}{\partial y \partial t^{2}} - I_{5} \frac{\partial^{2} ψ_{2}}{\partial t^{2}}) δ ψ_{2}] d A \end{array}

(48)

In which the integration constants are defined as

[I_{0}, I_{1}, I_{2}, I_{3}, I_{4}, I_{5}] = \int_{- h / 2}^{h / 2} ρ [1, z, f (z), z^{2}, zf (z), f^{2} (z)] d z

(49)

Finally, the variation of external works including the Pasternak’s foundation and pre-loads are defined as

δ W = \int_{A} (- F_{f} - N_{x_{0}} \frac{\partial^{2} w_{0}}{\partial x^{2}} - N_{y_{0}} \frac{\partial^{2} w_{0}}{\partial y^{2}}) δ w_{0} d A

(50)

In which the reaction of Pasternak’s foundation $F_{f}$ is defined as [29,30]

F_{f} = k_{1} w_{0} - k_{2} (\frac{\partial^{2} w_{0}}{\partial x^{2}} + \frac{\partial^{2} w_{0}}{\partial y^{2}})

(51)

where

k_{1}

k_{2}

are spring and shear parameters of Pasternak’s foundation. In addition,

N_{x_{0}}

and

N_{y_{0}}

are thermal pre-loads along the x and y directions, respectively

N_{x_{0}} = \int_{- h / 2}^{h / 2} [α Δ T (c_{11} + c_{12})] d z

(52a)

N_{y_{0}} = \int_{- h / 2}^{h / 2} [α Δ T (c_{12} + c_{22})] d z

(52b)

Substitution of variation of strain energy, kinetic energy and energy due to external works into Hamilton’s principle leads to governing equations of motion as follows

\begin{array}{l} δ u_{0} : - \frac{\partial N_{xx}}{\partial x} - \frac{\partial N_{xy}}{\partial y} + \frac{\partial^{2} P_{1}^{1}}{\partial x^{2}} + \frac{\partial^{2} P_{2}^{1}}{\partial x \partial y} + \frac{2}{5} \frac{\partial^{2} N_{xxx}}{\partial x^{2}} - \frac{1}{5} \frac{\partial^{2} N_{xxx}}{\partial y^{2}} + \frac{4}{5} \frac{\partial^{2} N_{xyy}}{\partial y^{2}} - \frac{3}{5} \frac{\partial^{2} N_{xyy}}{\partial x^{2}} \\ - \frac{3}{5} \frac{\partial^{2} N_{xzz}}{\partial x^{2}} - \frac{1}{5} \frac{\partial^{2} N_{xzz}}{\partial y^{2}} - \frac{2}{5} \frac{\partial^{2} N_{yyy}}{\partial x \partial y} + \frac{8}{5} \frac{\partial^{2} N_{xxy}}{\partial x \partial y} - \frac{2}{5} \frac{\partial^{2} N_{yzz}}{\partial x \partial y} - \frac{1}{2} \frac{\partial^{2} S_{yz}^{1}}{\partial y^{2}} - \frac{1}{2} \frac{\partial^{2} S_{xz}^{1}}{\partial x \partial y} \\ + I_{0} \frac{\partial^{2} u_{0}}{\partial t^{2}} - I_{1} \frac{\partial^{3} w_{0}}{\partial x \partial t^{2}} + I_{2} \frac{\partial^{2} ψ_{1}}{\partial t^{2}} = 0 \end{array}

(53)

\begin{array}{l} δ v_{0} : - \frac{\partial N_{yy}}{\partial y} - \frac{\partial N_{xy}}{\partial x} + \frac{\partial^{2} P_{1}^{1}}{\partial x \partial y} + \frac{\partial^{2} P_{2}^{1}}{\partial y^{2}} - \frac{2}{5} \frac{\partial^{2} N_{xxx}}{\partial x \partial y} + \frac{8}{5} \frac{\partial^{2} N_{xyy}}{\partial x \partial y} - \frac{2}{5} \frac{\partial^{2} N_{xzz}}{\partial x \partial y} \\ + \frac{2}{5} \frac{\partial^{2} N_{yyy}}{\partial y^{2}} - \frac{1}{5} \frac{\partial^{2} N_{yyy}}{\partial x^{2}} + \frac{4}{5} \frac{\partial^{2} N_{xxy}}{\partial x^{2}} - \frac{3}{5} \frac{\partial^{2} N_{xxy}}{\partial y^{2}} - \frac{1}{5} \frac{\partial^{2} N_{yzz}}{\partial x^{2}} - \frac{3}{5} \frac{\partial^{2} N_{yzz}}{\partial y^{2}} \\ + \frac{1}{2} \frac{\partial^{2} S_{yz}^{1}}{\partial x \partial y} + \frac{1}{2} \frac{\partial^{2} S_{xz}^{1}}{\partial x^{2}} + I_{0} \frac{\partial^{2} v_{0}}{\partial t^{2}} - I_{1} \frac{\partial^{3} w_{0}}{\partial y \partial t^{2}} + I_{2} \frac{\partial^{2} ψ_{2}}{\partial t^{2}} = 0 \end{array}

(54)

\begin{array}{l} δ w_{0} : - \frac{\partial^{2} M_{xx}}{\partial x^{2}} - \frac{\partial^{2} M_{yy}}{\partial y^{2}} - 2 \frac{\partial^{2} M_{xy}}{\partial x \partial y} + \frac{\partial^{3} P_{1}^{2}}{\partial x \partial y^{2}} + \frac{\partial^{3} P_{1}^{2}}{\partial x^{3}} - \frac{\partial^{2} P_{3}^{1}}{\partial y^{2}} + \frac{\partial^{3} P_{2}^{2}}{\partial y^{3}} - \frac{\partial^{2} P_{3}^{1}}{\partial x^{2}} \\ + \frac{\partial^{3} P_{2}^{2}}{\partial x^{2} \partial y} - 2 \frac{\partial^{2} N_{xyz}}{\partial x \partial y} + \frac{1}{5} \frac{\partial^{2} N_{zzz}}{\partial x^{2}} + \frac{1}{5} \frac{\partial^{2} N_{zzz}}{\partial y^{2}} - \frac{4}{5} \frac{\partial^{2} N_{yyz}}{\partial y^{2}} + \frac{1}{5} \frac{\partial^{2} N_{yyz}}{\partial x^{2}} - \frac{4}{5} \frac{\partial^{2} N_{xxz}}{\partial x^{2}} \\ + \frac{1}{5} \frac{\partial^{2} N_{xxz}}{\partial y^{2}} - \frac{3}{5} \frac{\partial^{3} M_{xxx}^{1}}{\partial x \partial y^{2}} + \frac{2}{5} \frac{\partial^{3} M_{xxx}^{1}}{\partial x^{3}} - \frac{3}{5} \frac{\partial^{3} M_{yzz}^{1}}{\partial y^{3}} - \frac{3}{5} \frac{\partial^{3} M_{yzz}^{1}}{\partial x^{2} \partial y} - \frac{3}{5} \frac{\partial^{3} M_{yyy}^{1}}{\partial x^{2} \partial y} \\ + \frac{2}{5} \frac{\partial^{3} M_{yyy}^{1}}{\partial y^{3}} - \frac{3}{5} \frac{\partial^{3} M_{xzz}^{1}}{\partial x \partial y^{2}} - \frac{3}{5} \frac{\partial^{3} M_{xzz}^{1}}{\partial x^{3}} - \frac{3}{5} \frac{\partial^{3} M_{xyy}^{1}}{\partial x^{3}} + \frac{12}{5} \frac{\partial^{3} M_{xyy}^{1}}{\partial x \partial y^{2}} - \frac{3}{5} \frac{\partial^{3} M_{xxy}^{1}}{\partial y^{3}} \\ + \frac{12}{5} \frac{\partial^{3} M_{xxy}^{1}}{\partial x^{2} \partial y} + \frac{\partial^{2} S_{xx}^{1}}{\partial x \partial y} - \frac{\partial^{2} S_{xy}^{1}}{\partial x^{2}} - \frac{\partial^{2} S_{yy}^{1}}{\partial x \partial y} + \frac{\partial^{2} S_{xy}^{1}}{\partial y^{2}} + N_{x_{0}} \frac{\partial^{2} w_{0}}{\partial x^{2}} + N_{y_{0}} \frac{\partial^{2} w_{0}}{\partial y^{2}} + F_{f} I_{0} \frac{\partial^{2} w_{0}}{\partial t^{2}} \\ + I_{1} \frac{\partial^{3} u_{0}}{\partial x \partial t^{2}} - I_{3} \frac{\partial^{4} w_{0}}{\partial x^{2} \partial t^{2}} + I_{4} \frac{\partial^{3} ψ_{1}}{\partial x \partial t^{2}} + I_{1} \frac{\partial^{3} v_{0}}{\partial y \partial t^{2}} - I_{3} \frac{\partial^{4} w_{0}}{\partial y^{2} \partial t^{2}} + I_{4} \frac{\partial^{3} ψ_{2}}{\partial y \partial t^{2}} = 0 \end{array}

(55)

\begin{array}{l} δ ψ_{1} : Q_{xz}^{2} - \frac{\partial Q_{xx}^{1}}{\partial x} - \frac{\partial Q_{xy}^{1}}{\partial y} + \frac{\partial^{2} P_{2}^{3}}{\partial x \partial y} + \frac{\partial^{2} P_{1}^{3}}{\partial x^{2}} - \frac{\partial P_{3}^{4}}{\partial x} + \frac{4}{5} M_{xzz}^{4} - \frac{1}{5} M_{xyy}^{4} + \frac{2}{5} \frac{\partial^{2} M_{xxx}^{2}}{\partial x^{2}} \\ - \frac{1}{5} M_{xxx}^{4} + \frac{2}{5} \frac{\partial M_{zzz}^{3}}{\partial x} + \frac{2}{5} \frac{\partial M_{yyz}^{3}}{\partial x} - \frac{8}{5} \frac{\partial M_{xxz}^{3}}{\partial x} - 2 \frac{\partial M_{xyz}^{3}}{\partial y} - \frac{2}{5} \frac{\partial^{2} M_{yzz}^{2}}{\partial x \partial y} - \frac{2}{5} \frac{\partial^{2} M_{yyy}^{2}}{\partial x \partial y} \end{array} \begin{array}{l} - \frac{3}{5} \frac{\partial^{2} M_{xzz}^{2}}{\partial x^{2}} - \frac{1}{5} \frac{\partial^{2} M_{xzz}^{2}}{\partial y^{2}} - \frac{3}{5} \frac{\partial^{2} M_{xyy}^{2}}{\partial x^{2}} + \frac{4}{5} \frac{\partial^{2} M_{xyy}^{2}}{\partial y^{2}} + \frac{8}{5} \frac{\partial^{2} M_{xxy}^{2}}{\partial x \partial y} - \frac{1}{5} \frac{\partial^{2} M_{xxx}^{2}}{\partial y^{2}} \\ - \frac{1}{2} \frac{\partial S_{xy}^{3}}{\partial x} + \frac{1}{2} S_{yz}^{4} - \frac{1}{2} \frac{\partial^{2} S_{yz}^{2}}{\partial y^{2}} - \frac{1}{2} \frac{\partial^{2} S_{xz}^{2}}{\partial x \partial y} + \frac{1}{2} \frac{\partial S_{zz}^{3}}{\partial y} - \frac{1}{2} \frac{\partial S_{yy}^{3}}{\partial y} + I_{2} \frac{\partial^{2} u_{0}}{\partial t^{2}} \\ - I_{4} \frac{\partial^{3} w_{0}}{\partial x \partial t^{2}} + I_{5} \frac{\partial^{2} ψ_{1}}{\partial t^{2}} = 0 \end{array}

(56)

\begin{array}{l} δ ψ_{2} : Q_{yz}^{2} - \frac{\partial Q_{yy}^{1}}{\partial y} - \frac{\partial Q_{xy}^{1}}{\partial x} + \frac{\partial^{2} P_{1}^{3}}{\partial x \partial y} + \frac{\partial^{2} P_{2}^{3}}{\partial y^{2}} - \frac{\partial P_{3}^{4}}{\partial y} - \frac{2}{5} \frac{\partial^{2} M_{xxx}^{2}}{\partial x \partial y} - \frac{1}{5} M_{xxy}^{4} - \frac{1}{5} M_{yyy}^{4} \\ - 2 \frac{\partial M_{xyz}^{3}}{\partial x} + \frac{2}{5} \frac{\partial M_{zzz}^{3}}{\partial y} - \frac{8}{5} \frac{\partial M_{yyz}^{3}}{\partial y} - \frac{3}{5} \frac{\partial^{2} M_{yzz}^{2}}{\partial y^{2}} - \frac{1}{5} \frac{\partial^{2} M_{yzz}^{2}}{\partial x^{2}} - \frac{1}{5} \frac{\partial^{2} M_{yyy}^{2}}{\partial x^{2}} + \frac{2}{5} \frac{\partial^{2} M_{yyy}^{2}}{\partial y^{2}} \\ - \frac{2}{5} \frac{\partial^{2} M_{xzz}^{2}}{\partial x \partial y} + \frac{8}{5} \frac{\partial^{2} M_{xyy}^{2}}{\partial x \partial y} + \frac{4}{5} \frac{\partial^{2} M_{xxy}^{2}}{\partial x^{2}} + \frac{2}{5} \frac{\partial M_{xxz}^{3}}{\partial y} + \frac{4}{5} M_{yzz}^{4} - \frac{3}{5} \frac{\partial^{2} M_{xxy}^{2}}{\partial y^{2}} + \frac{1}{2} \frac{\partial S_{xx}^{3}}{\partial x} \\ + \frac{1}{2} \frac{\partial S_{xy}^{3}}{\partial y} - \frac{1}{2} \frac{\partial S_{zz}^{3}}{\partial x} - \frac{1}{2} S_{xz}^{4} + \frac{1}{2} \frac{\partial^{2} S_{yz}^{2}}{\partial x \partial y} + \frac{1}{2} \frac{\partial^{2} S_{xz}^{2}}{\partial x^{2}} + I_{2} \frac{\partial^{2} v_{0}}{\partial t^{2}} - I_{4} \frac{\partial^{3} w_{0}}{\partial y \partial t^{2}} \\ + I_{5} \frac{\partial^{2} ψ_{2}}{\partial t^{2}} = 0 \end{array}

(57)

Solution procedure

The solution procedure is developed in this section for functionally graded rectangular micro plate reinforced with graphene nanoplatelets. The harmonic displacement field for a simply supported rectangular micro plate is defined as

{\begin{array}{l} u_{0} \\ v_{0} \\ w_{0} \\ ψ_{1} \\ ψ_{2} \end{array}} = \sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} {\begin{array}{l} U_{0} \cos (\frac{m π x}{a}) \sin (\frac{n π y}{b}) \\ V_{0} \sin (\frac{m π x}{a}) \cos (\frac{n π y}{b}) \\ W_{0} \sin (\frac{m π x}{a}) \sin (\frac{n π y}{b}) \\ Ψ_{1} \cos (\frac{m π x}{a}) \sin (\frac{n π y}{b}) \\ Ψ_{2} \sin (\frac{m π x}{a}) \cos (\frac{n π y}{b}) \end{array}} e^{i ω t}

(58)

in which

U_{0}

V_{0}

W_{0}

Ψ_{1}

and

Ψ_{2}

are amplitudes of displacements and rotations and

ω

is natural frequency of composite micro plate. Substitution of proposed solutions into governing equations of motion leads to known format of characteristics equations as follows

([K] - ω^{2} [M]) {X} = {0}

(59)

The natural frequencies are obtained using solution of characteristic equation as

Det ([K] - ω^{2} [M]) = 0

(60)

Results and discussion

Numerical results and corresponding discussions on them are presented in this section. The numerical results are presented in terms of significant parameters of the problem such as weight fraction of GNPs, various distribution of GNPs, three micro length scale parameters and some non-dimensional geometric parameters such as side length to thickness ratio and thickness to micro length scale ratio.

Before presentation of full numerical results, a validation should be provided. The validation section is included in comparison with the results based on MCST and MSGT.

It is noticeable that if all micro length scale parameters $l_{0}$ , $l_{1}$ and $l_{2}$ are fixed to zero, the MSGT is reduced to classical elasticity theory (CET). In addition, if $l_{0}$ , $l_{1}$ are fixed to zero and only $l_{2} = l$ is assumed non zero, the original theory is reduced to the MCST.

Validation

The validation of present numerical results is performed with the results of previous works based on MCST. The numerical results are calculated for FGM plate with simply supported boundary conditions. The material properties of FGM plate are considered based on the following relations

\begin{matrix} E (z) = E_{2} + (E_{1} - E_{2}) {(\frac{1}{2} + \frac{z}{h})}^{n} \\ ρ (z) = ρ_{2} + (ρ_{1} - ρ_{2}) {(\frac{1}{2} + \frac{z}{h})}^{n} \\ ν (z) = ν_{2} + (ν_{1} - ν_{2}) {(\frac{1}{2} + \frac{z}{h})}^{n} \end{matrix}

in which

(E_{1}, ρ_{1}, ν_{1})

and

(E_{2}, ρ_{2}, ν_{2})

are modulus of elasticity, density and Poisson ratio of top and bottom of FGM plate. The material properties of FGM are assumed as follows [20]

\begin{array}{l} E_{1} = 14.4 GPa, ρ_{1} = 12200 kg / m^{3}, E_{2} = 1.44 GPa, ρ_{2} = 1220 kg / m^{3}, \\ ν_{1} = ν_{2} = 0.38, h = 17.6 μm \end{array}

The above material properties and functionality are related to the results of Table 1. For this case, the non-dimensional natural frequencies are assumed as follows

\bar{ω} = ω \frac{a^{2}}{h} \sqrt{\frac{ρ_{2}}{E_{2}}}

Table 1.

Comparison of non-dimensional natural frequencies of FGM square plate with simply supported boundary condition based on modified couple stress theory.

$a / h$	$h / l$	References	n
$a / h$	$h / l$	References	0	1	10
5	$\infty$	KPT [20]	5.9671	5.2960	6.2322
		FSDT [20]	5.3871	4.8744	5.5818
		HSDT [21]	5.3885	4.8755	5.3793
		Present, KPT	5.9671	5.2960	6.2322
		Present, TSDT	5.3885	4.8755	5.3793
	5	KPT [20]	6.3957	5.7810	6.6413
		FSDT [20]	5.7797	5.3239	5.9551
		HSDT [21]	5.8439	5.3749	5.9200
		Present, KPT	6.3957	5.7810	6.6413
		Present, TSDT	5.8439	5.3749	5.9200
	2.5	KPT [20]	7.5366	7.0383	7.7399
		FSDT [20]	6.7996	6.4600	6.9333
		HSDT [21]	7.0322	6.6512	7.2275
		Present, KPT	7.5366	7.0383	7.7399
		Present, TSDT	7.0322	6.6512	7.2275
	$5 / 3$	KPT [20]	9.1264	8.7406	9.2865
		FSDT [20]	8.1595	7.9298	8.2517
		HSDT [21]	8.6525	8.3550	8.9116
		Present, KPT	9.1264	8.7406	9.2865
		Present, TSDT	8.6525	8.3550	8.9116
	1.25	KPT [20]	10.9718	10.6773	11.0953
		FSDT [20]	9.6451	9.4998	9.7045
		HSDT [21]	10.5052	10.2755	10.7941
		Present, KPT	10.9718	10.6773	11.0953
		Present, TSDT	10.5052	10.2755	10.7941
	1	KPT [20]	12.9640	12.7419	13.0578
		FSDT [20]	11.1311	11.0451	11.1666
		HSDT [21]	12.4874	12.3118	12.7942
		Present, KPT	12.9640	12.7419	13.0578
		Present, TSDT	12.4874	12.3118	12.7942
10	$\infty$	KPT [20]	6.1103	5.3953	6.3958
		FSDT [20]	5.9301	5.2697	6.1903
		HSDT [21]	5.9302	5.2698	6.1157
		Present, KPT	6.1103	5.3953	6.3958
		Present, TSDT	5.9302	5.2698	6.1157
	5	KPT [20]	6.5491	5.8894	6.8156
		FSDT [20]	6.3559	5.7518	6.5967
		HSDT [21]	6.3770	5.7676	6.5814
		Present, KPT	6.5491	5.8894	6.8156
		Present, TSDT	6.3770	5.7676	6.5814
	2.5	KPT [20]	7.7174	7.1702	7.9431
		FSDT [20]	7.4807	6.9920	7.6797
		HSDT [21]	7.5590	7.0532	7.7786
		Present, KPT	7.7174	7.1702	7.9431
		Present, TSDT	7.5590	7.0532	7.7786
	$5 / 3$	KPT [20]	9.3453	8.9045	9.5303
		FSDT [20]	9.0261	8.6477	9.1829
		HSDT [21]	9.1954	8.7870	9.4095
		Present, KPT	9.3453	8.9045	9.5303
		Present, TSDT	9.1954	8.7870	9.4095
	1.25	KPT [20]	11.2349	10.8775	11.3866
		FSDT [20]	10.7848	10.4942	10.9066
		HSDT [21]	11.0863	10.7543	11.2882
		Present, KPT	11.2349	10.8775	11.3866
		Present, TSDT	11.0863	10.7543	11.2882
	1	KPT [20]	13.2749	12.9808	13.4006
		FSDT [20]	12.6360	12.4128	12.7303
		HSDT [21]	13.1220	12.8483	13.3131
		Present, KPT	13.2749	12.9808	13.4006
		Present, TSDT	13.1220	12.8483	13.3131
20	∞	KPT [20]	6.1477	5.4210	6.4387
		FSDT [20]	6.0997	5.3880	6.3837
		HSDT [21]	6.0997	5.3880	6.3627
		Present, KPT	6.1477	5.4210	6.4387
		Present, TSDT	6.0997	5.3880	6.3627
	5	KPT [20]	6.5892	5.9175	6.8614
		FSDT [20]	6.5376	5.8812	6.8026
		HSDT [21]	6.5434	5.8854	6.7981
		Present, KPT	6.5892	5.9175	6.8614
		Present, TSDT	6.5434	5.8854	6.7981
	2.5	KPT [20]	7.7646	7.2044	7.9965
		FSDT [20]	7.7009	7.1571	7.9251
		HSDT [21]	7.7224	7.1736	7.9523
		Present, KPT	7.7646	7.2044	7.9965
		Present, TSDT	7.7224	7.1736	7.9523
	$5 / 3$	KPT [20]	9.4026	8.9470	9.5943
		FSDT [20]	9.3158	8.8781	9.4993
		HSDT [21]	9.3625	8.9159	9.5619
		Present, KPT	9.4026	8.9470	9.5943
		Present, TSDT	9.3625	8.9159	9.5619
	1.25	KPT [20]	11.3037	10.9293	11.4631
		FSDT [20]	11.1801	10.8255	11.3303
		HSDT [21]	11.2640	10.8968	11.4366
		Present, KPT	11.3037	10.9293	11.4631
		Present, TSDT	11.2640	10.8968	11.4366
	1	KPT [20]	13.3562	13.0426	13.4906
		FSDT [20]	13.1786	12.8871	13.3030
		HSDT [21]	13.3153	13.0076	13.4670
		Present, KPT	13.3562	13.0426	13.4906
		Present, TSDT	13.3153	13.0076	13.4670

Table 1 lists the non-dimensional natural frequencies of FGM plate in terms of various values of side length to thickness ratio a/h for various values of thickness to micro length scale parameter ratio h/l and various non-homogeneous indexes. The numerical results are compared with the results of Thai and Choi [20] based on Kirchhoff plate theory (KPT) and FSDT and in another case with the results of He et al. [21] based on HSDT.

The numerical results indicate that the present numerical results are in good agreement with the previous results. In addition, for a plate with higher values of a/h, smaller values of non-homogeneous index and infinite values of h/l, the results obtained from various theories mentioned by Thai et al. [20] and He et al. [21] converge to a same value.

With decrease of $a / h$ ratio, the difference between results is increased due to some simplification and limitation of KPT for thick plates. Furthermore, with increase of h/l ratio and non-homogeneous index, the difference between theories is increased significantly.

Table 2 lists the variation of non-dimensional natural frequencies of FGM square plate with simply supported boundary condition based MSGT. The numerical results obtained from present formulation are compared with the results of Thai et al. [25] and Zhang et al. [27]. The reference material properties for FGM plate is assumed as [27]

E_{1} = 380 GPa, ρ_{1} = 3800 kg / m^{3}, E_{2} = 70 GPa, ρ_{2} = 2702 kg / m^{3}, ν_{1} = ν_{2} = 0.3

Table 2.

Comparison of non-dimensional natural frequencies of FGM square plate with simply supported boundary condition based on modified strain gradient theory.

$a / h$	Mode	$h / l$	References	n
$a / h$	Mode	$h / l$	References	0	0.5	1	2	5	10
5	1	$\infty$	[25]	0.2113	0.1807	0.1631	0.1472	0.1358	0.1301
			[27]	0.2113	0.1804	0.1631	0.1472	0.1378	0.1358
			Present	0.2113	0.1807	0.1631	0.1472	0.1358	0.1301
		10	[25]	0.2264	0.1948	0.1766	0.1596	0.1463	0.1394
			[27]	0.2273	0.1953	0.1774	0.1603	0.1470	0.1402
			Present	0.2273	0.1957	0.1774	0.1603	0.1470	0.1401
		5	[25]	0.2666	0.2318	0.2118	0.1920	0.1740	0.1642
			[27]	0.2698	0.2344	0.2144	0.1944	0.1764	0.1666
			Present	0.2698	0.2347	0.2144	0.1944	0.1764	0.1666
		2	[25]	0.4566	0.4032	0.3729	0.3402	0.3029	0.2807
			[27]	0.4688	0.4131	0.3823	0.3490	0.3117	0.2896
			Present	0.4688	0.4136	0.3823	0.3490	0.3117	0.2895
		1	[25]	0.8344	0.7406	0.6867	0.6266	0.5540	0.5111
			[27]	0.7011	0.6290	0.5832	0.5270	0.4553	0.4150
			Present	0.7011	0.6285	0.5817	0.5249	0.4536	0.4141
	2	$\infty$	[25]	0.4623	0.3989	0.3607	0.3236	0.2918	0.2771
			[27]	0.4623	0.3982	0.3607	0.3236	0.2918	0.2772
			Present	0.4623	0.3989	0.3607	0.3236	0.2918	0.2771
		10	[25]	0.4978	0.4313	0.3919	0.3529	0.3181	0.3011
			[27]	0.5010	0.4335	0.3945	0.3553	0.3205	0.3035
			Present	0.5010	0.4342	0.3945	0.3553	0.3205	0.3034
		5	[25]	0.5909	0.5158	0.4724	0.4283	0.3857	0.3628
			[27]	0.6019	0.5246	0.4811	0.4364	0.3939	0.3709
			Present	0.6019	0.5254	0.4811	0.4363	0.3938	0.3708
		2	[25]	1.0255	0.9036	0.8374	0.7684	0.6914	0.6428
			[27]	1.0260	0.9206	0.8536	0.7712	0.6663	0.6074
			Present	1.0260	0.9209	0.8535	0.7711	0.6662	0.6073
		1	[25]	1.8858	1.6631	1.5460	1.4242	1.2806	1.1868
			[27]	1.3904	1.2475	1.1567	1.0451	0.9029	0.8231
			Present	1.3904	1.2468	1.1545	1.0422	0.9007	0.8218
10	1	$\infty$	[25]	0.0577	0.0490	0.0442	0.0401	0.0377	0.0364
			[27]	0.0577	0.0489	0.0442	0.0401	0.0377	0.0364
			Present	0.0577	0.0490	0.0442	0.0401	0.0377	0.0364
		10	[25]	0.0617	0.0528	0.0478	0.0434	0.0403	0.0387
			[27]	0.0619	0.0529	0.0480	0.0435	0.0405	0.0388
			Present	0.0619	0.0530	0.0480	0.0435	0.0404	0.0388
		5	[25]	0.0725	0.0629	0.0573	0.0520	0.0474	0.0449
			[27]	0.0730	0.0632	0.0578	0.0524	0.0478	0.0453
			Present	0.0730	0.0633	0.0578	0.0524	0.0478	0.0453
		2	[25]	0.1240	0.1098	0.1013	0.0918	0.0810	0.0750
			[27]	0.1258	0.1113	0.1028	0.0933	0.0825	0.0764
			Present	0.1258	0.1114	0.1028	0.0933	0.0825	0.0764
		1	[25]	0.2268	0.2023	0.1873	0.1698	0.1482	0.1359
			[27]	0.2309	0.2057	0.1907	0.1731	0.1514	0.1390
			Present	0.2309	0.2058	0.1907	0.1731	0.1514	0.1390
	2	$\infty$	[25]	0.1376	0.1174	0.1059	0.0958	0.0891	0.0856
			[27]	0.1377	0.1171	0.1059	0.0958	0.0891	0.0857
			Present	0.1377	0.1174	0.1059	0.0961	0.0891	0.0856
		10	[25]	0.1475	0.1266	0.1147	0.1038	0.0957	0.0915
			[27]	0.1479	0.1267	0.1150	0.1041	0.0961	0.0918
			Present	0.1479	0.1270	0.1150	0.1041	0.0961	0.0918
		5	[25]	0.1736	0.1508	0.1377	0.1247	0.1133	0.1071
			[27]	0.1750	0.1518	0.1388	0.1258	0.1144	0.1082
			Present	0.1750	0.1520	0.1388	0.1258	0.1144	0.1082
		2	[25]	0.2977	0.2632	0.2430	0.2209	0.1961	0.1818
			[27]	0.3027	0.2673	0.2471	0.2250	0.2001	0.1856
			Present	0.3027	0.2675	0.2471	0.2250	0.2001	0.1855
		1	[25]	0.5448	0.4837	0.4483	0.4090	0.3603	0.3312
			[27]	0.5130	0.4603	0.4268	0.3856	0.3331	0.3037
			Present	0.5130	0.4600	0.4258	0.3843	0.3321	0.3031

In addition, it is assumed that three material length scales are the same and are equal to $l = 15 μm$ . The non-dimensional natural frequencies of FGM plate in this case are defined as

\bar{ω} = ω h \sqrt{\frac{ρ_{1}}{E_{1}}}

Table 2 lists the variation of non-dimensional natural frequencies of FGM plate in terms of various values of side length to thickness ratio a/h, thickness to material length scale parameter ratio and non-homogeneous index. The results are compared with the results of Thai et al. [25] and Zhang et al. [27]. The numerical results show that our numerical results are in excellent agreement with results of Zhang et al. [27] while have a small deviation with respect to the results of Thai et al. [25].

Free vibration characteristics of composite micro plate reinforced with grapheme nanoplatelets

In this section, the full numerical results of composite micro plate reinforced with graphene nanoplatelets are presented in terms of some significant parameters of the problem. The material properties of polymer matrix and graphene nanoplatelets are assumed as follows

\begin{array}{l} E_{GNP} = 1.01 TPa, ρ_{GNP} = 1062.5 kg / m^{3}, ν_{GNP} = 0.186, α_{GNP} = 5 \times 10^{- 6} / K \\ E_{M} = 3 GPa, ρ_{M} = 1200 kg / m^{3}, ν_{M} = 0.34, α_{M} = 60 \times 10^{- 6} / K [14] \\ l_{GNP} = 2.5 μm, w_{GNP} = 1.5 μm, h_{GNP} = 1.5 nm, W_{GNP}^{0} = 0.01 [17] \\ a = b = 10 h \end{array}

In addition, $Δ T$ is temperature rising with respect to original temperature. The three material length scales are assumed the same. The micro length scale parameters of an isotropic homogeneous material are derived experimentally about $l = 17.6 μm$ by Lam et al. [31].

It is noticeable that the material length scale parameters should be derived from experimental results. However, there is no experimental work to predict the material length scale parameters for composite materials reinforced with graphene nanoplatelets. In this work, we use $l = 15 μm$ .

In this work, the influence of important parameters such as the type of distribution, the weight fraction of GNPs, the thickness to length ratio $h / l$ , temperature rising, side length ratio and two parameters of Pasternak’s foundation are studied on the responses based on modified strain gradient theory (MSGT) and MCST.

The non-dimensional natural frequencies are assumed as

\bar{ω} = ω h \sqrt{\frac{ρ_{M}}{E_{M}}}

Table 3 lists the relationship between GNP-gradient index $λ_{i}$ and the total GNP content $g_{GNP}^{*}$ for three GNP distribution models in which $λ_{1}$ , $λ_{2}$ and $λ_{3}$ are corresponding to linear, parabolic and uniform GPL distributions, respectively.

Table 3.

The relationship between GNP gradient index $λ_{i}$ and the total GNP content $g_{GNP}^{*}$ [17].

$λ_{1}$ (Linear)	$λ_{2}$ Parabolic	$λ_{3}$ Uniform	$g_{GNP}^{*}$ (%)
0 (Epoxy)	0	0	0
1/3	1/3	1	2/3
1	1	3	2

Table 4 lists the non-dimensional fundamental natural frequencies of composite micro plate reinforced with GNPs in terms of $h / l$ in terms of various temperature rising based on two aforementioned theories (MSGT, MCST). The results are calculated for $k_{1} = k_{2} = 0$ and $λ_{1} = 2$ . It is concluded that with increase of temperature rising, the natural frequencies are decreased slightly. In addition, the obtained results indicate that with increase of $h / l,$ the natural frequencies are decreased significantly. One can conclude that the results of MSGT and MCST are converged to the same values for higher values of $h / l$ .

Table 4.

Non-dimensional fundamental natural frequencies of composite micro plate reinforced with GNPs in terms of $h / l$ in terms of various temperature rising based on two aforementioned theories (MSGT, MCST).

$h / l$	MSGT			MCST
	$Δ T (K)$
	0	30	60	0	30	60
1	0.4693	0.4669	0.4645	0.2699	0.2656	0.2614
2	0.2533	0.2488	0.2443	0.1648	0.1579	0.1506
3	0.1876	0.1815	0.1752	0.1368	0.1283	0.1192
4	0.1583	0.1510	0.1433	0.1255	0.1162	0.1061
5	0.1426	0.1345	0.1258	0.1199	0.1101	0.0994
6	0.1333	0.1246	0.1152	0.1168	0.1067	0.0956
7	0.1274	0.1182	0.1083	0.1148	0.1046	0.0932
8	0.1234	0.1139	0.1036	0.1136	0.1032	0.0916
9	0.1206	0.1108	0.1002	0.1127	0.1022	0.0905
10	0.1185	0.1086	0.0977	0.1120	0.1015	0.0897
11	0.1170	0.1069	0.0958	0.1116	0.101	0.0891
12	0.1158	0.1056	0.0943	0.1112	0.1006	0.0887
13	0.1148	0.1046	0.0932	0.1109	0.1003	0.0883
14	0.1141	0.1038	0.0923	0.1107	0.1000	0.0880
15	0.1135	0.1031	0.0915	0.1105	0.0998	0.0878

Table 5 lists the first six natural frequencies of composite micro plate in terms of $h / l$ based on two aforementioned theories (MSGT, MCST). The results are obtained with these conditions: $k_{1} = k_{2} = 0, Δ T = 0 (K), λ_{1} = 2$ . In this table, the natural frequencies for various modes are presented.

Table 5.

The first six natural frequencies of composite micro plate in terms of in terms of $h / l$ based on two aforementioned theories (MSGT, MCST).

Theory	$h / l$	m, n
Theory	$h / l$	1,1	2,1	2,2	3,1	3,2	3,3
MSGT	1	0.4693	1.0496	1.4342	1.6784	2.0357	2.6177
	2	0.2533	0.6078	0.9390	1.1497	1.4536	1.9348
	3	0.1876	0.4503	0.6957	0.8516	1.0762	1.4306
	4	0.1583	0.3797	0.5864	0.7175	0.9059	1.2025
	5	0.1426	0.3421	0.5279	0.6456	0.8146	1.0797
	6	0.1333	0.3197	0.4931	0.6028	0.7601	1.0063
	7	0.1274	0.3054	0.4708	0.5754	0.7252	0.9592
	8	0.1234	0.2957	0.4558	0.5569	0.7016	0.9273
	9	0.1206	0.2889	0.4451	0.5438	0.6849	0.9047
	10	0.1185	0.2839	0.4374	0.5342	0.6726	0.8882
MCST	1	0.2699	0.6569	1.0262	1.2639	1.6093	2.0306
	2	0.1648	0.3991	0.6209	0.7629	0.9686	1.2950
	3	0.1368	0.3299	0.5113	0.6268	0.7932	1.0555
	4	0.1255	0.3019	0.4668	0.5715	0.7217	0.9573
	5	0.1199	0.2880	0.4447	0.5439	0.6861	0.9083
	6	0.1168	0.2802	0.4323	0.5284	0.6659	0.8804
	7	0.1148	0.2754	0.4246	0.5188	0.6535	0.8632
	8	0.1136	0.2722	0.4195	0.5125	0.6453	0.8519
	9	0.1127	0.2700	0.4160	0.5081	0.6396	0.8440
	10	0.1120	0.2684	0.4134	0.5049	0.6355	0.8383

Figure 3 shows the variation of non-dimensional fundamental natural frequencies of composite micro plate reinforced with GNPs in terms of $h / l$ based on three theories MSGT, MCST, CET. The results are obtained for these characteristics: $k_{1} = k_{2} = 0, Δ T = 0 (K), λ_{2} = 3$ .

Figure 3.

Variation of non-dimensional natural frequencies in terms of $h / l$ based on various theories (MSGT, MCST, CET).

The numerical results indicate that the highest values of natural frequencies are predicted by MSGT while the lowest by CET. In addition, it is concluded that the natural frequencies obtained from various theories (MSGT, MCST, CET) coincide with them for large values of $h / l$ .

Variation of the non-dimensional natural frequencies of composite micro plate in terms of $h / l$ for various distributions and various weight fractions of GNPs are presented in Figures 4 and 5 based on MCST and MSGT, respectively. These results are obtained for $k_{1} = k_{2} = 0$ and $Δ T = 0 (K)$ . The numerical results indicate that with increase of volume fraction of GNPs, the natural frequencies are increased significantly. One can conclude that with increase of weight fraction of GNPs, the stiffness of constituent materials is increased, and consequently, the natural frequencies are increased significantly. In addition, it is observed that the highest natural frequencies are obtained for distribution 2 (parabolic distribution) while the lowest one for distribution 1 (linear distribution).

Figure 4.

The non-dimensional natural frequencies of composite micro plate in terms of $h / l$ for various distributions and various weight fractions of GNPs based on modified couple stress theory.

Figure 5.

The non-dimensional natural frequencies of composite micro plate in terms of $h / l$ for various distributions and various weight fractions of GNPs based on modified strain gradient theory.

The numerical results presented in Figure 5 indicate that the same conclusion can be expressed for results of MSGT. In addition, it is concluded that the obtained results based on MSGT is very larger than the corresponding results of MCST. One can conclude that employing the MCST leads to more flexible structure rather than modelling the structure using MSGT.

Shown in Figures 6 and 7 are variation of the non-dimensional natural frequencies of composite micro plate in terms of $h_{GNP} / l_{GNP}$ for various values of distributions of GNPs and two values of $l_{GNP} / w_{GNP} = 1, 5 / 3$ based on MCST and MSGT, respectively. These results are obtained for $k_{1} = k_{2} = 0, h / l = 5$ and $Δ T = 0 (K)$ . In addition, $l_{GNP}$ is assumed constant. It is observed that with increase of $h_{GNP} / l_{GNP}$ ratio, the natural frequencies are increased significantly. It is concluded that with increase of $h_{GNP} / l_{GNP}$ , the stiffness of composite micro plate is increased, and consequently, the natural frequencies are increased.

Figure 6.

The non-dimensional natural frequencies of composite micro plate in terms of $h_{GNP} / l_{GNP}$ for various distributions of GNPs and two values of $h_{GNP} / l_{GNP} = 1, 5 / 3$ based on modified couple stress theory.

Figure 7.

Investigation on the influence of weight fraction and various distributions of GNPs leads to interesting conclusions. It is concluded that the maximum stiffness and maximum natural frequencies are obtained for $λ_{2} = 3$ . Furthermore, increase of $l_{GNP} / w_{GNP}$ leads to the decrease of natural frequencies. One can conclude that with increase of $l_{GNP} / w_{GNP,}$ the stiffness of reinforced micro plate is decreased, and consequently, the natural frequencies are decreased significantly. It is noticeable that GNPs with larger area give larger stiffness and larger natural frequencies.

Figures 8 and 9 show variation of the non-dimensional natural frequencies of composite micro plate in terms of temperature rising $Δ T (K)$ for various distributions of GNPs based on MCST and MSGT for $k_{1} = k_{2} = 0$ and $h / l = 5$ . The numerical results show that with increase of temperature, the natural frequencies are decreased significantly. One can conclude that with increase of temperature, the stiffness of structure is decreased, and then the natural frequencies are decreased. These results are in accordance with the results of literature [14].

Figure 8.

The non-dimensional natural frequencies of composite micro plate in terms of temperature rising $Δ T (K)$ for various distributions of GNPs based on modified couple stress theory.

Figure 9.

The non-dimensional natural frequencies of composite micro plate in terms of temperature rising $Δ T (K)$ for various distributions of GNPs based on MSGT.

The influence of side length ratio $a / b$ is studied on the responses of composite micro plate. To investigate the influence of this parameter, the area of plate is assumed fixed ( $a \times b = Const$ ). Figures 10 and 11 show the variation of the non-dimensional natural frequencies of composite micro plate in terms of side length ratio $a / b$ for various distributions and various weight fractions of GNPs based on MCST and MSGT, respectively. The numerical results are obtained for $k_{1} = k_{2} = 0$ , $h / l = 3, Δ T = 0 (K)$ . The numerical results show that the minimum natural frequencies are obtained for $\frac{a}{b} = 1$ . For the side length ratio $a / b$ greater and lower than 1, the natural frequencies are increased significantly. One can conclude that the square plate $\frac{a}{b} = 1$ has minimum stiffness that leads to minimum natural frequencies.

Figure 10.

The non-dimensional natural frequencies of composite micro plate in terms of side length ratio $a / b$ for various distributions and various weight fractions of GNPs based on modified couple stress theory.

Figure 11.

Figure 12.

The non-dimensional natural frequencies of composite micro plate in terms of two parameters of Pasternak’s foundation.

The influence of two parameters of Pasternak’s foundation is studied on the natural frequencies of composite micro plate. Figure 12 shows two-dimensional distribution of natural frequencies in terms of spring and shear parameters of Pasternak’s foundation. The results are presented for $Δ T = 0 (K), h / l = 3$ and $λ_{1} = 2$ and based on MSGT.

Conclusion

Free vibration analysis of composite micro plate reinforced with functionally graded nanoplatelets was studied in this paper based on MSGT and TSDT. The composite micro plate was reinforced with functionally graded graphene nanoplatelets with various distribution along the thickness direction. Size dependency was accounted in governing equations of motion through accounting the three material length scale parameters. The numerical results were presented based the analytical method for a simply supported micro plate. The numerical results were verified through comparison with previous works using reduction of governing equations of motion for more simple case with setting some material length scale to zero. Full numerical results were presented in terms of weight fraction of GNPs, various distribution of GNPs, three micro length scale parameters and some non-dimensional geometric parameters such as side length ratio, side length to thickness ratio and thickness to micro length scale ratio. The main conclusions of this study are presented as follows:

Comparisons with previous works were provided using results of modified couple stress and modified strain-gradient formulations. One can conclude that the present formulation and corresponding numerical results are in good agreement with the aforementioned literature.

The comparison between results of MSGT and CET in terms of various $\frac{h}{l}$ ratio indicate that for $\frac{h}{l} < 10$ , employing the MSGT leads to more accurate results rather than classical plate theory while for $\frac{h}{l} > 10$ , the difference between various theories is vanished.

The numerical results were presented in terms of various distribution of graphene nanoplatelets. It is concluded that the maximum stiffness is obtained for parabolic distribution while the minimum stiffness is obtained for linear distribution.

The amount and geometry of graphene nanoplatelets have significant influence on the responses. It is concluded that increase of parameters $\frac{h_{GNP}}{l_{GNP}}$ , $\frac{l_{G N P}}{w_{GNP}}$ leads to the decrease of natural frequencies. In addition, an increase in weight fraction of nano filler leads to the increase of natural frequencies.

The effect of thermal environment was studied on the natural frequencies of composite microplate. It is concluded that an increase in thermal loads decreases natural frequencies significantly.

Footnotes

Acknowledgements

The first author would also like to thank the Iranian Nanotechnology Development Committee for their 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: The author(s) received financial support from the University of Kashan (Grant Number: 467893/0655) for the research, authorship, and/or publication of this article.

References

Rafiee

Wang

, et al. Enhanced mechanical properties of nanocomposites at low graphene content. ACS Nano 2009; 3: 3884–3890.

Liang

Wang

Huang

, et al. Electromagnetic interference shielding of graphene/epoxy composites. Carbon N Y 2008; 47: 922–925.

Yang

Kitipornchai

Nonlinear vibration of functionally graded carbon nanotube-reinforced composite beams with geometric imperfections.

Compos Part B Eng 2016; 90: 86–96.

Yang

Kitipornchai

Dynamic stability of functionally graded carbon nanotube-reinforced composite beams.

Mech Adv Mater Struct 2013; 20: 28–37.

Arefi

Zenkour

AM.

Effect of thermo-magneto-electro-mechanical fields on the bending behaviors of a three-layered nanoplate based on sinusoidal shear-deformation plate theory.

J Sandw Struct Mater 2019; 21: 639–669.

Arefi

Zenkour

AM.

Size-dependent free vibration and dynamic analyses of piezo-electro-magnetic sandwich nanoplates resting on viscoelastic foundation.

Phys B: Condensed Matter 2017; 521: 188–197.

Ansari

Torabi

Shakouri

AH.

Vibration analysis of functionally graded carbon nanotube-reinforced composite elliptical plates using a numerical strategy.

Aerosp Sci Technol 2017; 60: 152–161.

Mehar

Panda

Mahapatra

TR.

Thermoelastic nonlinear frequency analysis of CNT reinforced functionally graded sandwich structure.

Eur J Mech A/Solids 2017; 65: 384–396.

Thang

Nguyen

T-T

Lee

A new approach for nonlinear buckling analysis of imperfect functionally graded carbon nanotube-reinforced composite plates.

Compos Part B Eng 2017; 127: 166–174.

10.

Rafiee

Thomas

, et al. Graphene nanoribbon composites. ACS Nano 2010; 4: 7415–7420.

11.

Montazeri

Rafii-Tabar

Multiscale modeling of graphene- and nanotube-based reinforced polymer nanocomposites.

Phys Lett Sect A Gen Solid State Phys 2011; 375: 4034–4040.

12.

Liu

Kitipornchai

Chen

, et al. Three-dimensional buckling and free vibration analyses of initially stressed functionally graded graphene reinforced composite cylindrical shell. Compos Struct 2018; 189: 560–569.

13.

Wang

Feng

Zhao

, et al. Buckling of graphene platelet reinforced composite cylindrical shell with cutout. Int J Str Stab Dyn 2018; 18: 1850040

14.

Yang

Kitipornchai

Parametric instability of thermo-mechanically loaded functionally graded graphene reinforced nanocomposite plates.

Int J Mech Sci 2018; 135: 431–440.

15.

Yang

Kitipornchai

Thermoelastic analysis of functionally graded graphene reinforced rectangular plates based on 3D elasticity.

Meccanica 2017; 52: 2275–2292.

16.

Yang

Kitipornchai

Yang

, et al. 3D thermo-mechanical bending solution of functionally graded graphene reinforced circular and annular plates. Appl Math Model 2017; 49: 69–86.

17.

Yang

Mei

Chen

, et al. 3D thermo-mechanical solution of transversely isotropic and functionally graded graphene reinforced elliptical plates. Compos Struct 2018; 184: 1040–1048.

18.

Arbind

Reddy

Srinivasa

AR.

Modified couple stress-based third-order theory for nonlinear analysis of functionally graded beams.

Latin Am J Solids Struct 2014; 11: 459–487.

19.

Arefi

Mohammad-Rezaei Bidgoli

Zenkour

AM.

Free vibration analysis of a sandwich nano-plate including FG core and piezoelectric face-sheets by considering neutral surface. Mech Adv Mater Struct. Epub ahead of print 13 March 2018. DOI: 10.1080/15376494.2018.1455939.

20.

Thai

Choi

DH.

Size-dependent functionally graded Kirchhoff and Mindlin plate models based on a modified couple stress theory.

Compos Struct 2013; 95: 142–153.

21.

Lou

Zhang

, et al. A size-dependent four variable refined plate model for functionally graded microplates based on modified couple stress theory. Compos Struct 2015; 130: 107–115.

22.

Ansari

Gholami

Sahmani

Free vibration analysis of size-dependent functionally graded microbeams based on the strain gradient Timoshenko beam theory.

Compos Struct 2011; 94: 221–228.

23.

Arefi

Zenkour

AM.

Nonlocal electro-thermo-mechanical analysis of a sandwich nanoplate containing a Kelvin–Voigt viscoelastic nanoplate and two piezoelectric layers.

Acta Mech 2017; 228: 475–493.

24.

Feng

Cai

ZY.

Bending and free vibration of functionally graded piezoelectric beam based on modified strain gradient theory.

Compos Struct 2014; 115: 41–50.

25.

Thai

, et al. Size-dependant behaviour of functionally graded microplates based on the modified strain gradient elasticity theory and isogeometric analysis. Comput Struct 2017; 190: 219–241.

26.

Wang

Zhao

Zhou

A micro scale Timoshenko beam model based on strain gradient elasticity theory.

Eur J Mech A/Solids 2010; 29: 591–599.

27.

Zhang

Liu

, et al. An efficient size-dependent plate theory for bending, buckling and free vibration analyses of functionally graded microplates resting on elastic foundation. Appl Math Model 2015; 39: 3814–3845.

28.

Arefi

Mohammad-Rezaei Bidgoli E, Dimitri

, et al. Nonlocal bending analysis of curved nanobeams reinforced by grapheme nanoplatelets. Compos Part B Eng 2019; 166: 1–12.

29.

Arefi

Mohammad-Rezaei Bidgoli E Dimitri

, et al. Application of sinusoidal shear deformation theory and physical neutral surface to analysis of functionally graded piezoelectric plate. Compos Part B Eng 2018; 151: 35–50.

30.

Arefi

Mohammad-Rezaei Bidgoli E Dimitri

, et al. Free vibrations of functionally graded polymer composite nanoplates reinforced with graphene nanoplatelets. Aero Sci Tech 2018; 81: 108–117.

31.

Lam

DCC

Yang

Chong

ACM

, et al. Experiments and theory in strain gradient elasticity. J Mech Phys Solids 2003; 51: 1477–1508.