Size-dependent free vibration analysis of three-layered exponentially graded nanoplate with piezomagnetic face-sheets resting on Pasternak’s foundation

Abstract

This work is devoted to the free vibration nonlocal analysis of an elastic three-layered nanoplate with exponentially graded graphene sheet core and piezomagnetic face-sheets. The rectangular elastic three-layered nanoplate is resting on Pasternak’s foundation. Material properties of the core are supposed to vary along the thickness direction based on the exponential function. The governing equations of motion are derived from Hamilton’s principle based on first-order shear deformation theory. In addition, Eringen’s nonlocal piezo-magneto-elasticity theory is used to consider size effects. The analytical solution is presented to solve seven governing equations of motion using Navier’s solution. Eventually, the natural frequency is scrutinized for different side length ratio, nonlocal parameter, inhomogeneity parameter, and parameters of foundation numerically. The comparison with various references is performed for validation of our analytical results.

Keywords

Free vibration three-layered nanoplate nonlocal piezoelasticity exponentially graded graphene sheet

Introduction

We know that materials are composed of molecules in a defined scale and in more small scale consist of atoms. Based on this description, the materials are not continuous. However, experimental analysis of structures due to various types of loading and conditions shows that the reaction of material due to applied loadings is independent of the molecular structure of materials. The above mentioned theory is known as Continuum theory. Nevertheless, the new theories have been developed to account small-scale effects in nanostructures or microstructures. These theories include nonlocal theory, strain gradient theory, and coupled stress theory. In this article, we employ the nonlocal elasticity theory to investigate vibration analysis of an exponentially graded three-layered nanoplate based on first-order shear deformation theory (FSDT). A comprehensive literature survey is presented as follows.

Nayak et al. (2002) used Reddy’s higher order shear deformation theory to derive governing equations of motion for anisotropic composite and sandwich plates based on finite element approach. Khorshidi (2011) presented vibro-acoustic analysis of Mindlin rectangular plates resting on an elastic foundation. The exact closed-form sound pressure equations were derived for six cases having two opposite sides simply supported. Amabili et al. (2011) studied nonlinear vibrations of rectangular laminated composite plates with different boundary conditions. In particular, numerical results of the classical Von Kárman theory, the FSDT, and the third-order shear deformation theory (TSDT) were compared. Khorshidi and Farhadi (2013) studied free vibration analysis of a laminated composite rectangular plate in contact with a bounded fluid. Obtained results showed that the accuracy of fundamental natural frequencies computed by classical laminated plate theory (CLPT) decreases as the thickness ratio increases. Sobhy (2013) investigated vibration and buckling analyses of exponentially graded sandwich plate resting on Pasternak’s foundation. The problem was solved for various boundary conditions using new functions for mid-plane stretching. The effect of some significant parameters of the problem such as inhomogeneity parameter, aspect ratio, thickness ratio, and the foundation parameters on the natural frequencies and critical buckling loads was investigated. Kim and Reddy (2013) employed general third-order plate theory to investigate bending, buckling, and free vibration responses of micro rectangular plate based on modified couple stress theory. The Navier solution was proposed to derive analytical solutions for simply supported rectangular plates. The numerical results have indicated that the effect of power-law function and small-scale parameter is very important. Khorshidi and Bakhsheshy (2014) investigated free natural frequency analysis of a functionally graded composite rectangular plate coupled with fluid using Rayleigh–Ritz method. The numerical results showed the effects of boundary conditions, aspect ratios, thickness ratios, gradient index, and material properties of the functionally graded plate on natural frequencies are very significant. Reddy et al. (2014) studied free vibration characteristics of a functionally graded thick plate based on higher order shear deformation plate theory. The governing equations of motion were derived using principle of virtual work. A comparison with literature was performed to show accuracy and correctness of the employed theory. A new first-order shear deformation plate theory was employed by Thai et al. (2014) to investigate bending, buckling, and free vibration analyses of a three-layered rectangular plate under various boundary conditions. The plate was composed of an isotropic core and two functionally graded face-sheets. They concluded that new proposed theory can present valid and correct results comparable with higher order shear deformation theory. Khorshidi and Bakhsheshy (2015) presented free vibration analysis of a functionally graded rectangular plate in contact with a bounded fluid. Zenkour et al. (2015) analyzed thermo-mechanical bending analysis of the exponentially graded thick rectangular plates resting on Pasternak’s foundations based on trigonometric shear and normal deformations plate theory. They mentioned that gradation of material properties using exponential model significantly changes thermo-mechanical behaviors of the plate. Khorshidi et al. (2015) studied free vibration analysis of functionally graded rectangular nanoplates based on nonlocal exponential shear deformation theory. It is shown that the frequency ratio decreases with an increase in the mode number and the value of the nonlocal parameter, and also, increase in the power-law index causes the non-dimensional frequencies to decrease. Khorshidi and Fallah (2016) studied buckling analysis of functionally graded rectangular nanoplate based on nonlocal exponential shear deformation theory. This showed that the critical buckling load decreases by increasing the value of the nonlocal parameter, and also, increase in the power-law index causes the non-dimensional critical buckling to decrease.

Influence of surface stress effects on the bending and vibration responses of simply supported laminated and isotropic plates based on nonlocal TSDT was performed by Raghu et al. (2016). Rahmat Talabi and Saidi (2013) studied free vibration analysis of a three-layered functionally graded circular/annular plate including a functionally graded core and two piezoelectric layers based on Reddy’s plate theory. They discussed on the influence of inhomogeneous index and various boundary conditions. Farajpour et al. (2016) studied large amplitude vibration of magneto-electro-elastic nanoplate based on Kirchhoff plate theory and nonlocal electro-magneto-elastic theory. The geometric nonlinearity was considered using Von Karman relation. The nanoplate was subjected to applied electric and magnetic potentials. The influence of applied electric and magnetic potentials as well as nonlocal parameter on the large amplitude vibration characteristics of the problem was studied. Arefi et al. (2016) formulated three-dimensional problem of a functionally graded three-layered cylindrical shell including an inhomogeneous core and two piezoelectric plates using FSDT. They mentioned that with an increase in the inhomogeneous index of a cylindrical shell, the natural frequencies decreased due to the decrease in bending stiffness of material. In addition, it was concluded that with an increase in the ratio of core thickness to cylinder length, the natural frequencies of the cylinder increased considerably. Application of normal and shear deformation theory to thermo-electro-magnetic analysis of a sandwich nanobeam was studied by Arefi and Zenkour (2016a). Application of sinusoidal shear deformation plate theory to transient analysis of piezomagnetic sandwich nanoplate was studied by Arefi and Zenkour (2016b). Zenkour and Arefi (2017a) studied wave propagation characteristics of a functionally graded piezoelectric Love nanorod model based on coupled stress components and surface elasticity theory. Influence of Pasternak’s foundation and nonlocal parameter on the thermo-electro-mechanical analysis of a functionally graded single-layer graphene sheet was studied by Zenkour and Arefi (2017b). Arefi and Zenkour (2017d) studied free vibration and bending analyses of a sandwich microbeam with two integrated piezomagnetic layers based on strain gradient theory and FSDT. The sandwich structure was subjected to applied electric and magnetic potentials resting on visco-Pasternak foundation. Arefi and Zenkour (2017a, 2017b, 2017c) studied nonlocal analysis of sandwich nanoplates and nanobeams using trigonometric plate theory.

The main purpose of this research is to present free vibration analysis of an exponentially graded three-layered nanoplate resting on Pasternak’s foundation based on FSDT and nonlocal piezo-magneto-elasticity relations. Seven governing equations of motion are derived using Hamilton’s principle and nonlocal elasticity. The analytical approach is proposed for a simply supported nanoplate to investigate influence of important parameters of the problem. The influence of important parameters of the problem such as nonlocal parameter, inhomogeneous index, ratio of side lengths, the ratio of thicknesses, and two parameters of foundation on the numerical results is considered.

Formulation

In this section, the formulation of the problem is presented. Our model is a three-layered nanoplate including an exponentially graded core and two piezomagnetic face-sheets with lengths a and b (Figure 1). The thickness of core and face-sheets are considered as h and $h_{p}$ . The material properties of the functionally graded core are changed based on exponential function along the thickness direction of nanoplate. The material properties of the core are assumed as (Zenkour et al., 2015)

\bar{P} (z) = P_{0} e^{k (\frac{z}{h} + \frac{1}{2})}

(1)

where $\bar{P} (z)$ described material properties such as Young’s modulus and density through the thickness direction and $P_{0}$ described the homogeneous material properties corresponding to material properties at $z = - h / 2$ . The inhomogeneous index k denotes the variation of material properties along the thickness direction.

After definition of gradation of material properties, the basic governing equations can be implemented. In this article, nonlocal piezo-magneto-elasticity theory is used. Our assumption for the current formulation is expressed as follows:

There is a perfect connection between layers so that there is no slippage at the interfaces among the three layers of the nanoplate.

The top and bottom piezomagnetic layers have the same and homogeneous properties.

Eringen’s nonlocal theory is a well-known theory that considers small effects. Nonlocal piezo-magneto-elasticity relations indicate that the stress, electric displacement, and magnetic induction at any point not only depend on strain, electric, and magnetic fields at same point but also on strain, electric, and magnetic fields at all other neighborhood points. For nonlocal piezomagnetic face-sheets, the basic equations can be expressed as (Eringen, 1983); Ghorbanpour Arani et al. (2015), Li et al. (2015) and Zhang et al. (2014).

\begin{matrix} σ_{i j} = \underset{v}{∭} α (| x^{'} - x |, τ) [c_{i j k l} ε_{k l} (x^{'}) - e_{k i j} E_{k} (x^{'}) - f_{k i j} H_{k} (x^{'})] d x^{'} \\ D_{i} = \underset{v}{∭} α (| x^{'} - x |, τ) [e_{i j k l} ε_{k l} (x^{'}) + κ_{i k} E_{k} (x^{'}) + g_{i k} H_{k} (x^{'})] d x^{'} \end{matrix}

(2)

\begin{array}{l} B_{i} = \underset{v}{∭} α (| x^{'} - x |, τ) [f_{i k l} ε_{k l} (x^{'}) + g_{i k} E_{k} (x^{'}) + μ_{i k} H_{k} (x^{'})] d x^{'} \\ σ_{i j, j} = ρ {\overset{\cdot\cdot}{u}}_{i}, D_{i, i} = 0, B_{i, i} = 0 \end{array}

(3)

ε_{ij} = \frac{1}{2} (u_{i, j} + u_{j, i}), E_{i} = - Φ_{, i}, H_{i} = - Ψ_{, i}

(4)

where $σ_{ij}$ , $D_{i}$ , and $B_{i}$ are the nonlocal stress, electric displacement, and magnetic induction, respectively. Also $c_{ijkl}$ , $e_{kij}$ , $f_{kij}$ , $κ_{ik}$ , and $μ_{ik}$ represent elastic constant, piezoelectric, piezomagnetic, dielectric, magneto-electric and magnetic constants correspondingly. $ε_{ij}$ , $u_{i}$ , $E_{i}$ , and $H_{i}$ are strain, displacement, electric field, and magnetic field, respectively. Electric and magnetic potentials are prescribe by $Φ$ and $Ψ$ , respectively. $α$ is the nonlocal modulus and $| x' - x |$ is the Euclidean distance. Small-scale coefficient is defined by $τ = e_{0} a / l$ . e₀ is the nonlocal material constant and determined from experimental or dynamics simulation results. a and l define internal characteristic scale and external characteristic scale, respectively. Based on nonlocal electro-magneto-elastic relations, the stress, electric displacement, and magnetic induction are defined as (Arefi, 2015; Arefi and Zenkour, 2016b, 2017b, 2017d); Ke et al. (2015) and Murmu and Pradhan (2009).

\begin{matrix} σ_{ij} - η \nabla^{2} σ_{ij} = c_{ijkl} ε_{kl} - e_{kij} E_{k} - f_{kij} H_{k} \\ D_{i} - η \nabla^{2} D_{i} = e_{ijkl} ε_{kl} + κ_{ik} E_{k} + g_{ik} H_{k} \\ B_{i} - η \nabla^{2} B_{i} = f_{ikl} ε_{kl} + g_{ik} E_{k} + μ_{ik} H_{k} \end{matrix}

(5)

where $\nabla^{2}$ represents the Laplacian operator and $η$ is equal to $(e_{0} a)^{2}$ related to the impact of small scale on the mechanical model in nano-measurement.

Based on nonlocal electro-magneto-elastic relations, the constitutive equations are derived by Pan (2001) as

\begin{matrix} {\begin{matrix} σ_{xx} \\ σ_{yy} \\ τ_{yz} \\ τ_{xz} \\ τ_{xy} \end{matrix}} - η \nabla^{2} {\begin{matrix} σ_{xx} \\ σ_{yy} \\ τ_{yz} \\ τ_{xz} \\ τ_{xy} \end{matrix}} = [\begin{matrix} c_{11} & c_{12} & c_{13} & 0 & 0 & 0 \\ c_{12} & c_{22} & c_{13} & 0 & 0 & 0 \\ 0 & 0 & 0 & c_{44} & 0 & 0 \\ 0 & 0 & 0 & 0 & c_{55} & 0 \\ 0 & 0 & 0 & 0 & 0 & c_{66} \end{matrix}] {\begin{matrix} ε_{xx} \\ ε_{yy} \\ 2 ε_{yz} \\ 2 ε_{xz} \\ 2 ε_{xy} \end{matrix}} - [\begin{matrix} 0 & 0 & e_{31} \\ 0 & 0 & e_{32} \\ 0 & e_{24} & 0 \\ e_{15} & 0 & 0 \\ 0 & 0 & 0 \end{matrix}] {\begin{matrix} E_{x} \\ E_{y} \\ E_{z} \end{matrix}} \\ - [\begin{matrix} 0 & 0 & f_{31} \\ 0 & 0 & f_{32} \\ 0 & f_{24} & 0 \\ f_{15} & 0 & 0 \\ 0 & 0 & 0 \end{matrix}] {\begin{matrix} H_{x} \\ H_{y} \\ H_{z} \end{matrix}} \end{matrix}

(6)

{\begin{matrix} D_{x} \\ D_{y} \\ D_{z} \end{matrix}} - η \nabla^{2} {\begin{matrix} D_{x} \\ D_{y} \\ D_{z} \end{matrix}} = [\begin{matrix} 0 & 0 & e_{15} & 0 & 0 \\ 0 & 0 & e_{24} & 0 & 0 \\ e_{31} & e_{32} & e_{33} & 0 & 0 \end{matrix}] {\begin{matrix} ε_{xx} \\ ε_{yy} \\ 2 ε_{yz} \\ 2 ε_{xz} \\ 2 ε_{xy} \end{matrix}} + [\begin{matrix} κ_{11} & 0 & 0 \\ 0 & κ_{22} & 0 \\ 0 & 0 & κ_{33} \end{matrix}] {\begin{matrix} E_{x} \\ E_{y} \\ E_{z} \end{matrix}} + [\begin{matrix} g_{11} & 0 & 0 \\ 0 & g_{22} & 0 \\ 0 & 0 & g_{33} \end{matrix}] {\begin{matrix} H_{x} \\ H_{y} \\ H_{z} \end{matrix}}

(7)

\begin{matrix} {\begin{matrix} B_{x} \\ B_{y} \\ B_{z} \end{matrix}} - η \nabla^{2} {\begin{matrix} B_{x} \\ B_{y} \\ B_{z} \end{matrix}} = [\begin{matrix} 0 & 0 & f_{15} & 0 & 0 \\ 0 & 0 & f_{24} & 0 & 0 \\ f_{31} & f_{32} & f_{33} & 0 & 0 \end{matrix}] {\begin{matrix} ε_{xx} \\ ε_{yy} \\ 2 ε_{yz} \\ 2 ε_{xz} \\ 2 ε_{xy} \end{matrix}} + [\begin{matrix} g_{11} & 0 & 0 \\ 0 & g_{22} & 0 \\ 0 & 0 & g_{33} \end{matrix}] {\begin{matrix} E_{x} \\ E_{y} \\ E_{z} \end{matrix}} \\ + [\begin{matrix} μ_{11} & 0 & 0 \\ 0 & μ_{22} & 0 \\ 0 & 0 & μ_{33} \end{matrix}] {\begin{matrix} H_{x} \\ H_{y} \\ H_{z} \end{matrix}} \end{matrix}

(8)

FSDT is used in this study for description of displacement components. Based on FSDT, three displacement components are expressed as Sayyad and Ghugal (2015) and Ghorbanpour Arani and Jalaei (2016)

\begin{matrix} u (x, y, z, t) = u_{0} (x, y, t) + z ϕ_{x} (x, y, t) \\ v (x, y, z, t) = v_{0} (x, y, t) + z ϕ_{y} (x, y, t) \\ w (x, y, z, t) = w_{0} (x, y, t) \end{matrix}

(9)

where $(u, v, w)$ are displacements along the $(x, y, z)$ directions and $(u_{0}, v_{0}, w_{0})$ are mid-surfaces. $ϕ_{x}$ and $ϕ_{y}$ represent rotation around the y- and x-axes, respectively. Based on displacement field defined in equation (9) and using linear strain–displacement relations, five in-plane strain components are derived as

\begin{matrix} ε_{xx} = \frac{\partial u_{0}}{\partial x} + z \frac{\partial ϕ_{x}}{\partial x} \\ ε_{yy} = \frac{\partial v_{0}}{\partial y} + z \frac{\partial ϕ_{y}}{\partial y} \\ ε_{yz} = \frac{1}{2} (\frac{\partial w_{0}}{\partial y} + ϕ_{y}) \\ ε_{xz} = \frac{1}{2} (\frac{\partial w_{0}}{\partial x} + ϕ_{x}) \\ ε_{xy} = \frac{1}{2} (\frac{\partial u_{0}}{\partial y} + \frac{\partial v_{0}}{\partial x} + z \frac{\partial ϕ_{x}}{\partial y} + z \frac{\partial ϕ_{y}}{\partial x}) \end{matrix}

(10)

For completion of basic relations, the electric and magnetic fields and potentials are required. Electric and magnetic potentials are assumed as a combination of a cosine function along the planar direction and a linear function along the thickness direction. The second term implies applied electric and magnetic potentials and the first term imposes homogeneous boundary conditions. The distributions of electric and magnetic potentials are assumed as (Arefi and Zenkour, 2017a, 2017b, 2017c, 2017d; Ke et al., 2014)

Φ (x, y, z, t) = - \cos (\frac{π \tilde{z}}{h_{p}}) ϕ (x, y, t) + \frac{2 \tilde{z} ϕ_{0}}{h_{p}}

(11)

Ψ (x, y, z, t) = - \cos (\frac{π \tilde{z}}{h_{p}}) ψ (x, y, t) + \frac{2 \tilde{z} ψ_{0}}{h_{p}}

(12)

where $ϕ (x, y, t)$ and $ψ (x, y, t)$ express planar distribution of electric and magnetic potentials and $ψ_{0}$ and $ϕ_{0}$ are applied electric and magnetic potentials. The new variable $\tilde{z} = z \pm h / 2 \pm h_{p} / 2$ is defined for piezomagnetic face-sheets. For the top layer, $\tilde{z} = z^{t} = z - h / 2 - h_{p} / 2$ , and for the bottom layer, $\tilde{z} = z^{b} = z + h / 2 + h_{p} / 2$ . Based on electric and magnetic potentials defined in equations (11) and (12), the electric and magnetic fields are derived as Asemi et al. (2014)

\begin{matrix} E_{x} = \cos (\frac{π \tilde{z}}{h}) \frac{\partial ϕ}{\partial x}, E_{y} = \cos (\frac{π \tilde{z}}{h}) \frac{\partial ϕ}{\partial y}, \\ E_{z} = - \frac{π}{h} \sin (\frac{π \tilde{z}}{h}) ϕ - \frac{2 ϕ_{0}}{h} \end{matrix}

(13)

\begin{matrix} H_{x} = \cos (\frac{π \tilde{z}}{h}) \frac{\partial ψ}{\partial x}, H_{y} = \cos (\frac{π \tilde{z}}{h}) \frac{\partial ψ}{\partial y,} \\ H_{z} = - \frac{π}{h} \sin (\frac{π \tilde{z}}{h}) ψ - \frac{2 ψ_{0}}{h} \end{matrix}

(14)

The governing equations of motion are derived by employing Hamilton’s principle as follow Ke et al. (2012)

δ Π = \int (δ K_{e} - δ U_{s} - δ W_{l})

(15)

where $(K_{e}, U_{s}, W_{l})$ are kinetic energy, strain energy, and energy due to foundation reaction which can be expressed as

K_{e} = \frac{1}{2} \underset{v}{\int \int \int} ρ [{(\frac{\partial u_{x}}{\partial t})}^{2} + {(\frac{\partial u_{y}}{\partial t})}^{2} + {(\frac{\partial u_{z}}{\partial t})}^{2}] dV

(16)

\begin{matrix} U_{s} = \\ \frac{1}{2} \int \int \int ((σ_{xx} ε_{xx} + σ_{yy} ε_{yy} + 2 σ_{xy} ε_{xy} + 2 σ_{xz} ε_{xz} + 2 σ_{yz} ε_{yz}) \\ - (D_{x} E_{x} + D_{y} E_{y} + D_{z} E_{z}) - (B_{x} H_{x} + B_{y} H_{y} + B_{z} H_{z})) dV \end{matrix}

(17)

W_{l} = - \frac{1}{2} \int \int F (x, y) w dA

(18)

where $F (x, y)$ is the transverse loading that can be calculated as Mohammadimehr et al. (2016)

F (x, y) = k_{w} w - k_{G} (\frac{\partial^{2} w}{\partial x^{2}} + \frac{\partial^{2} w}{\partial y^{2}})

(19)

where $k_{w}$ and $k_{G}$ are Pasternak foundation parameters related to Winkler and shear constants, respectively.

By substituting variation forms of kinetic energy, strain energy, and energy due to external works into Hamilton’s principle and equating the coefficients of $δ u_{0}, δ v_{0}, δ w_{0}, δ ϕ_{x}, δ ϕ_{y}, δ ϕ$ , and $δ ψ$ to zero, the following equations of motion can be obtained

\begin{array}{l} δ u : \frac{\partial N_{x x}}{\partial x} + \frac{\partial N_{x y}}{\partial y} = I_{0} \frac{\partial^{2} u}{\partial t^{2}} + I_{1} \frac{\partial^{2} ϕ_{x}}{\partial t^{2}} \\ δ v : \frac{\partial N_{y y}}{\partial y} + \frac{\partial N_{x y}}{\partial x} = I_{0} \frac{\partial^{2} v}{\partial t^{2}} + I_{1} \frac{\partial^{2} ϕ_{y}}{\partial t^{2}} \\ δ w : \frac{\partial N_{x z}}{\partial x} + \frac{\partial N_{y z}}{\partial y} - k_{w} w + k_{G} (\frac{\partial^{2} w}{\partial x^{2}} + \frac{\partial^{2} w}{\partial y^{2}}) = I_{0} \frac{\partial^{2} w}{\partial t^{2}} \\ δ ϕ_{x} : - N_{x z} + \frac{\partial M_{x x}}{\partial x} + \frac{\partial M_{x y}}{\partial y} = I_{1} (\frac{\partial^{2} u}{\partial t^{2}}) + I_{2} (\frac{\partial^{2} ϕ_{x}}{\partial t^{2}}) \\ δ ϕ_{y} : - N_{y z} + \frac{\partial M_{y y}}{\partial y} + \frac{\partial M_{x y}}{\partial x} = I_{1} (\frac{\partial^{2} v}{\partial t^{2}}) + I_{2} (\frac{\partial^{2} ϕ_{y}}{\partial t^{2}}) \\ δ ϕ : \int_{- h / 2}^{h / 2} [\cos (\frac{π z}{h}) \frac{\partial D_{x}}{\partial x} + \cos (\frac{π z}{h}) \frac{\partial D_{y}}{\partial y} + \frac{π}{h} \sin (\frac{π z}{h}) D_{z}] d z \\ δ ψ : \int_{- h / 2}^{h / 2} [\cos (\frac{π z}{h}) \frac{\partial B_{x}}{\partial x} + \cos (\frac{π z}{h}) \frac{\partial B_{y}}{\partial y} + \frac{π}{h} \sin (\frac{π z}{h}) B_{z}] d z \end{array}

(20)

where $(I_{0}, I_{1}, I_{2})$ are defined in Appendix 1.

$N_{ij}$ and $M_{ij}$ indicate the in-plane stress resultants and bending moments, respectively. They are described by the following

\begin{array}{l} N_{x x} = \int_{- h / 2}^{h / 2} σ_{x x} d z, N_{y y} = \int_{- h / 2}^{h / 2} σ_{y y} d z, N_{x y} = \int_{- h / 2}^{h / 2} σ_{x y} d z \\ N_{x z} = k_{s} \int_{- h / 2}^{h / 2} σ_{x z} d z, N_{y z} = k_{s} \int_{- h / 2}^{h / 2} σ_{y z} d z \\ M_{x x} = \int_{- h / 2}^{h / 2} σ_{x x} z d z, M_{y y} = \int_{- h / 2}^{h / 2} σ_{y y} z d z, M_{x y} = \int_{- h / 2}^{h / 2} σ_{x y} z d z \end{array}

(21)

where $k_{s}$ denotes the shear stress correction factor and consider equal to 5/6 for the plate.

By substitution of equations (10), (13), and (14) into equations (6) to (8) and then using equation (21), the seven equations of motions, equation (20), can be derived as

\begin{matrix} δ u : (u_{0, yy} + v_{0, xy}) A_{66} + (ϕ_{x, yy} + ϕ_{y, xy}) B_{66} \\ + B_{11} ϕ_{x, xx} + u_{0, xx} A_{11} + B_{12} ϕ_{y, xy} \\ + v_{0, xy} A_{12} + ϕ_{, x} Ξ_{31} + ψ_{, x} β_{31} \\ - (1 - η \nabla^{2}) (I_{0} {\overset{\cdot\cdot}{u}}_{0} + I_{1} {\overset{\cdot\cdot}{ϕ}}_{x}) = 0 \end{matrix}

(22)

\begin{matrix} δ v : A_{66} (v_{0, xx} + u_{0, xy}) + B_{66} (ϕ_{y, xx} + ϕ_{x, xy}) + A_{22} v_{0, yy} \\ + B_{22} ϕ_{y, yy} + A_{12} u_{0, xy} \\ + B_{12} ϕ_{x, xy} + ϕ_{, y} Ξ_{32} + ψ_{, y} β_{32} - (1 - η \nabla^{2}) (I_{0} {\overset{\cdot\cdot}{v}}_{0} + I_{1} {\overset{\cdot\cdot}{ϕ}}_{y}) = 0 \end{matrix}

(23)

\begin{matrix} δ w : - k_{s} (k_{15} ψ_{, xx} - M_{15} ϕ_{, xx} - M_{24} ϕ_{, yy} - k_{24} . ψ_{, yy}) + k_{s} A_{44} (w_{0, yy} + ϕ_{y, y}) \\ + k_{s} A_{55} (w_{0, xx} + ϕ_{x, x}) + (1 - η \nabla^{2}) (k_{w} w_{0} - I_{0} {\overset{\cdot\cdot}{w}}_{0} - k_{G} w_{0}) = 0 \end{matrix}

(24)

\begin{matrix} δ ϕ_{x} : k_{s} (M_{15} ϕ_{, x} + O_{15} ψ_{, x}) - k_{s} A_{55} (w_{0, x} + ϕ_{x}) + Q_{66} (ϕ_{x, yy} + ϕ_{y, xy}) + B_{66} (u_{0, yy} + v_{0, xy}) \\ + E_{31} ϕ_{, x} + F_{31} ψ_{, x} + Q_{11} ϕ_{x, xx} + B_{11} u_{0, xx} + Q_{12} ϕ_{y, xy} + B_{12} v_{0, xy} - (1 - η \nabla^{2}) (I_{1} {\overset{\cdot\cdot}{u}}_{0} + I_{2} {\overset{\cdot\cdot}{ϕ}}_{x}) = 0 \end{matrix}

(25)

\begin{matrix} δ ϕ_{y} : k_{s} (M_{24} ϕ_{, y} + O_{24} . ψ_{, y}) - k_{s} A_{44} (w_{0, y} + ϕ_{y}) + Q_{66} (ϕ_{y, xx} + ϕ_{x, xy}) + B_{22} v_{0, yy} \\ + B_{66} (v_{0, xx} + u_{0, xy}) + E_{32} ϕ_{, y} + F_{32} ψ_{, y} + Q_{12} ϕ_{x, xy} + Q_{22} ϕ_{y, yy} + Q_{22} ϕ_{y, yy} + B_{12} u_{0, xy} \\ - (1 - η \nabla^{2}) (I_{1} {\overset{\cdot\cdot}{v}}_{0} + I_{2} {\overset{\cdot\cdot}{ϕ}}_{y}) = 0 \end{matrix}

(26)

\begin{matrix} δ ϕ : k_{s} M_{15} (w_{0, xx} + ϕ_{x, x}) + k_{s} M_{24} (w_{0, yy} + ϕ_{y, y}) + X_{11} ϕ_{, xx} + X_{22} ϕ_{, yy} \\ + Y_{11} ψ_{, xx} + Y_{22} ψ_{, yy} + E_{31} ϕ_{x, x} + E_{32} ϕ_{y, y} - X_{33} ϕ - Y_{33} ψ + Ξ_{31} u_{0, x} + Ξ_{32} v_{0, y} = 0 \end{matrix}

(27)

\begin{matrix} δ ψ : Y_{11} ϕ_{, xx} + Y_{22} ϕ_{, yy} + Z_{11} ψ_{, xx} + Z_{22} ψ_{, yy} + F_{31} ϕ_{x, x} + F_{32} ϕ_{y, y} + β_{31} . u_{0, x} \\ + β_{32} v_{0, y} - Y_{33} ϕ - Z_{33} ψ + k_{s} O_{15} (w_{0, xx} + ϕ_{x, x}) + k_{s} O_{24} (w_{0, yy} + ϕ_{y, y}) = 0 \end{matrix}

(28)

where $(A_{ij}, B_{ij}, E_{ij}, F_{ij}, O_{ij}, Q_{ij}, X_{ij}, Y_{ij}, Z_{ij}, β_{ij}, M_{ij}, Ξ_{ij})$ are defined in Appendix 1.

Solution procedure

An analytical solution for a simply supported rectangular exponentially graded graphene sheet (EGGS) plate is obtained using Navier solution technique. The boundary conditions of a simply supported rectangular plate can be expressed as

\begin{matrix} x = 0, l and y = 0, l \\ w = ϕ = ψ = ϕ_{x} = ϕ_{y} = 0 \\ w ″ = ϕ ″ = ψ ″ = {ϕ ″}_{x} = {ϕ ″}_{y} = 0 \end{matrix}

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The displacements are assumed as the series of double trigonometric functions that satisfy boundary conditions

\begin{array}{l} u (x, y, t) = \sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} U_{m n} \cos (\frac{m π}{a} x) \sin (\frac{n π}{b} y) e^{i ω t} \\ v (x, y, t) = \sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} V_{m n} \sin (\frac{m π}{a} x) \cos (\frac{n π}{b} y) e^{i ω t} \\ w (x, y, t) = \sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} W_{m n} \sin (\frac{m π}{a} x) \sin (\frac{n π}{b} y) e^{i ω t} \\ ϕ_{x} (x, y, t) = \sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} ϕ_{x m n} \cos (\frac{m π}{a} x) \sin (\frac{n π}{b} y) e^{i ω t} \\ ϕ_{y} (x, y, t) = \sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} ϕ_{y m n} \sin (\frac{m π}{a} x) \cos (\frac{n π}{b} y) e^{i ω t} \\ ϕ (x, y, t) = \sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} ϕ_{m n} \sin (\frac{m π}{a} x) \sin (\frac{n π}{b} y) e^{i ω t} \\ ψ (x, y, t) = \sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} ψ_{m n} \sin (\frac{m π}{a} x) \sin (\frac{n π}{b} y) e^{i ω t} \end{array}

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By substituting equation (30) into governing equations (22) to (28), the stiffness and mass matrices for finding the eigenvalues can be defined as

{[\begin{matrix} K_{11} & K_{12} & 0 & K_{14} & K_{15} & K_{16} & K_{17} \\ K_{21} & K_{22} & 0 & K_{24} & K_{25} & K_{26} & K_{27} \\ 0 & 0 & K_{33} & K_{34} & K_{35} & K_{36} & K_{37} \\ K_{41} & K_{42} & K_{43} & K_{44} & K_{45} & K_{46} & K_{47} \\ K_{51} & K_{52} & K_{53} & K_{54} & K_{55} & K_{56} & K_{57} \\ K_{61} & K_{62} & K_{63} & K_{64} & K_{65} & K_{66} & K_{67} \\ K_{71} & K_{72} & K_{73} & K_{74} & K_{75} & K_{76} & K_{77} \end{matrix}] - [\begin{matrix} I_{0} & 0 & 0 & I_{1} & 0 & 0 & 0 \\ 0 & I_{0} & 0 & 0 & I_{1} & 0 & 0 \\ 0 & 0 & I_{0} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & I_{2} & 0 & 0 & 0 \\ I_{1} & I_{1} & 0 & 0 & I_{2} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}] \times ω^{2}} {\begin{matrix} U_{mn} \\ V_{mn} \\ W_{mn} \\ ϕ_{xmn} \\ ϕ_{ymn} \\ ϕ_{mn} \\ ψ_{mn} \end{matrix}} = {0}

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and $K_{ij}$ are defined in Appendix 1.

Numerical investigation

The numerical results of the problem are presented in this section. Before the presentation of numerical results, a comparison with literature for verification and validation is required.

Verification

In order to verify this study, the current results are compared with corresponding literature. For validation of results, the integrated piezomagnetic face-sheets are removed and the material properties are assumed based on Table 1 according to Pradhan and Kumar (2011).

Table 1.

The material properties of core.

E₀ (TPa)	ρ₀ (kg/m³)	ν
1.06	2250	0.25

For comparison, the frequency ratio is considered as

frequency ratio = \frac{frequency using of nonlocal theory}{frequency using of local theory}

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Table 2 shows the comparison between our numerical results with Pradhan and Kumar (2011) and Pradhan and Phadikar (2009) for various nonlocal parameters. This comparison indicates that the present numerical results are in good agreement with literatures.

Table 2.

Comparison of numerical results of single-layer nanoplate with literatures.

Nonlocal parameter (nm²)	0	1	2	3
Present	1	0.913865	0.846733	0.792509
Pradhan and Kumar (2011)	1	0.9139	0.8467	0.7925
Pradhan and Phadikar (2009)	1	0.9139	0.8468	0.7926

Numerical results of three-layered nanoplate

In this section, the numerical results of three-layered nanoplate made of exponentially graded materials are presented. The material properties of the core are assumed based on Table 1 Xiao et al. (2005), Shokrieh and Rafiee (2010) and material properties of piezomagnetic face-sheets are assumed by Farajpour et al. (2016) in Table 3.

Table 3.

The material properties of piezomagnetic face-sheets.

c₁₁ (GPa)	c₁₂ (GPa)	c₁₃ (GPa)	c₂₂ (GPa)	c₃₃ (GPa)	c₄₄ (GPa)
226	125	124	226	216	44.2
c₅₅ (GPa)	c₆₆ (GPa)	e₃₁ (C/m²)	e₁₅ (C/m²)	e₃₂ (C/m²)	e₂₄ (C/m²)
44.2	50.5	−2.2	5.8	−2.2	5.8
e₃₃ (C/m²)	f₃₁ (N/A m)	f₁₅ (N/A m)	f₃₂ (N/A m)	f₂₄ (N/A m)	f₃₃ (N/A m)
9.3	290.1	275	290.1	275	349.9
κ₁₁ (C/V m)	κ₂₂ (C/V m)	κ₃₃ (C/V m)	g₁₁ (N s/CV)	g₂₂ (N s/CV)	g₃₃ (N s/CV)
5.64 × 10⁻⁹	5.64 × 10⁻⁰⁹	6.35 × 10⁻⁹	5.367 × 10⁻¹²	5.367 × 10⁻¹²	2737.5 × 10⁻¹²
	μ₁₁ (N s²/CV)	μ₂₂ (N s²/CV)	μ₃₃ (N s²/CV)	ρ (kg/m³)
	−0.000297	−0.000297	83.5 × 10⁻⁶	5.55 × 10³

Figure 2 shows fundamental and second natural frequencies of nanoplate in terms of nonlocal parameter in THz. One can conclude that with an increase in nonlocal parameter, all natural frequencies decreased significantly. We can mention that with an increase in nonlocal parameter, the stiffness of nanostructures decreased and consequently the natural frequencies decreased.

The influence of inhomogeneous index of exponentially graded core on the vibration responses of nanoplate is studied in Figure 3. Figure 3 shows fundamental and second natural frequencies of nanoplate in terms of inhomogeneous index k, in THz. The numerical results indicate that with an increase in the inhomogeneous index k, all natural frequencies increased. The reason for this increase is that with an increase in inhomogeneous index k, based on equation (1), all material properties such as modulus of elasticity and density increased and this increase leads to a stiffer nanoplate.

Figure 4 shows fundamental and second natural frequencies of nanoplate in terms of ratio $h / h_{p}$ for cases in which the total thickness $h + 2 h_{p}$ of nanoplate is assumed fixed. We see that with an increase in ratio $h / h_{p}$ , all natural frequencies increased. It can be discussed that with an increase in ratio $h / h_{p}$ , the portion of core rather than piezomagnetic face-sheets increased. Since the stiffness of core is more than the stiffness of face-sheets, the increase in ratio $h / h_{p}$ leads to a stiffer plate and finally reaches to greater natural frequencies.

The influence of ratio of side lengths $a / b$ on the free vibration characteristics of nanoplate is presented in Figure 5. It is observed that with an increase in ratio of side lengths $a / b$ , the natural frequencies decreased where $a / b \leq 1$ . For the increase in $a / b$ , the natural frequencies decreased where $a / b > 1$ . Figure 6 shows the variation of natural frequencies in terms of spring parameter of foundation ( $k_{w}$ ). One can conclude that increase in spring parameter of foundation leads to increase in stiffness of foundation and consequently leads to increase in natural frequencies of nanoplate. Effect of shear parameter of foundation ( $k_{G}$ ) on the natural frequencies of nanoplate can be considered in Figure 7. We can conclude that with an increase in the shear parameter, the stiffness of foundation increased and consequently the natural frequencies increased.

Conclusion

Free vibration analysis of a shear deformable three-layered nanoplate resting on Pasternak’s foundation was studied in this article. The surrounding medium was described by Pasternak’s model including both direct and shear effects. Eringen’s nonlocal piezo-magneto-elasticity theory was used to consider size effects. The nonlocal constitutive relations were developed for exponentially graded core and two piezomagnetic face-sheets. The governing equations of motion were derived using Hamilton’s principle based on FSDT. The analytical method has been proposed for the solution of the governing equations of motion to study the influence of parameters of the nanostructure such as inhomogeneous index, nonlocal parameter, two parameters of foundation, and non-dimensional geometric parameters on the free vibration responses. The main results of our analysis are expressed as follows:

Increase in inhomogeneous index of exponentially graded core leads to a stiffer core and consequently increases the natural frequencies of nanoplate.

Discussion on the influence of nonlocal parameter indicates that with an increase in this parameter, all natural frequencies decreased. One can conclude that this decrease is due to decrease in stiffness of nanomaterial with an increase in nonlocal parameter.

The numerical results indicate that with an increase in the non-dimensional thickness ratio $h / h_{p}$ , the natural frequencies increased. This is due to increase in bending stiffness of the three-layered structure. The material properties indicate that the core is very stiffer than face-sheets and consequently leads to a stiffer structure.

The non-dimensional side length ratio (a/b) has a significant influence on the vibration characteristics of the three-layered nanoplate. The numerical results indicate that vibration behavior of plate is depending on the value of a/b. For $a / b < 1$ , the natural frequencies decreased with an increase in $a / b$ , and for $a / b > 1$ , the natural frequencies increased with an increase in $a / b$ .

Two parameters of foundation can significantly change the natural frequencies of nanoplate. One can conclude that increase in both parameters of foundation increases the natural frequencies significantly.

Footnotes

Appendix 1

\begin{array}{l} I_{0} = \int_{- \frac{h}{2}}^{\frac{h}{2}} ρ^{c} d z + \int_{\frac{h}{2}}^{\frac{h}{2} + h_{p}} ρ^{p} d z + \int_{- \frac{h}{2} - h_{p}}^{- \frac{h}{2}} ρ^{p} d z \\ I_{1} = \int_{- \frac{h}{2}}^{\frac{h}{2}} ρ^{c} z d z + \int_{\frac{h}{2}}^{\frac{h}{2} + h_{p}} ρ^{p} z d z + \int_{- \frac{h}{2} - h_{p}}^{- \frac{h}{2}} ρ^{p} z d z \\ I_{2} = \int_{- \frac{h}{2}}^{\frac{h}{2}} ρ^{c} z^{2} d z + \int_{\frac{h}{2}}^{\frac{h}{2} + h_{p}} ρ^{p} z^{2} d z + \int_{- \frac{h}{2} - h_{p}}^{- \frac{h}{2}} ρ^{p} z^{2} d z \end{array}

\begin{matrix} K_{11} & = - \frac{π^{2} A_{66}}{b^{2}} - \frac{π^{2} A_{11}}{a^{2}} \\ K_{12} & = - \frac{π^{2} A_{66}}{ab} - \frac{π^{2} A_{12}}{ab} \\ K_{14} & = - \frac{π^{2} B_{66}}{b^{2}} - \frac{π^{2} B_{11}}{a^{2}} \\ K_{15} & = - \frac{π^{2} B_{66}}{ab} - \frac{π^{2} B_{12}}{ab} \\ K_{16} & = \frac{π Ξ_{31}}{a} \\ K_{17} & = \frac{π β_{31}}{a} \end{matrix}

\begin{matrix} K_{21} & = - \frac{π^{2} A_{66}}{ab} - \frac{π^{2} A_{12}}{ab} \\ K_{22} & = - \frac{π^{2} A_{22}}{b^{2}} - \frac{π^{2} A_{66}}{a^{2}} \\ K_{24} & = - \frac{π^{2} B_{66}}{ab} - \frac{π^{2} B_{12}}{ab} \\ K_{25} & = - \frac{π^{2} B_{22}}{b^{2}} - \frac{π^{2} B_{66}}{a^{2}} \\ K_{26} & = \frac{π Ξ_{32}}{b} \\ K_{27} & = \frac{π β_{32}}{b} \end{matrix}

\begin{array}{l} K_{33} = - \frac{k_{G} π^{2}}{a^{2}} - \frac{k_{G} π^{2}}{b^{2}} - k_{W} - \frac{k_{s} π^{2} A_{55}}{a^{2}} - \frac{k_{s} π^{2} A_{44}}{b^{2}} \\ K_{34} = - \frac{π k_{s}}{a} A_{55} \end{array}

\begin{array}{l} K_{35} = - \frac{π k_{s}}{b} A_{44} \\ K_{36} = \frac{π^{2} k_{s} M_{15}}{a^{2}} + \frac{π^{2} k_{s} M_{24}}{b^{2}} \\ K_{37} = \frac{π^{2} k_{s} O_{24}}{b^{2}} + \frac{π^{2} k_{s} O_{15}}{a^{2}} \end{array}

\begin{matrix} K_{41} = - \frac{π^{2} B_{66}}{b^{2}} - \frac{π^{2} B_{11}}{a^{2}} \\ K_{42} = \frac{π^{2} B_{66}}{ab} - \frac{π^{2} B_{12}}{ab} \\ K_{43} = - \frac{π k_{s}}{a} A_{55} \\ K_{44} = - k_{s} A_{55} - \frac{π^{2} Q_{66}}{b^{2}} - \frac{π^{2} Q_{11}}{a^{2}} \\ K_{45} = - \frac{π^{2} Q_{66}}{ab} - \frac{π^{2} Q_{12}}{ab} \\ K_{46} = \frac{π k_{s} M_{15}}{a} + \frac{π E_{31}}{a} \\ K_{47} = \frac{π k_{s} O_{15}}{a} + \frac{π F_{31}}{a} \end{matrix}

\begin{matrix} K_{51} = - \frac{π^{2} B_{66}}{ab} - \frac{π^{2} B_{12}}{ab} \\ K_{52} = - \frac{π^{2} B_{22}}{b^{2}} - \frac{π^{2} B_{66}}{a^{2}} \\ K_{53} = - \frac{π k_{s}}{b} A_{44} \\ K_{54} = - \frac{π^{2} Q_{66}}{ab} - \frac{π^{2} Q_{12}}{ab} \\ K_{55} = - k_{s} A_{44} - \frac{π^{2} Q_{66}}{a^{2}} - \frac{π^{2} Q_{22}}{b^{2}} \\ K_{56} = \frac{k_{s} π M_{24}}{b} + \frac{π E_{32}}{b} \\ K_{57} = \frac{k_{s} π O_{24}}{b} + \frac{π 1 mu F_{32}}{b} \end{matrix}

\begin{matrix} K_{61} = \frac{π Ξ_{31}}{a} \\ K_{62} = \frac{π Ξ_{32}}{b} \\ K_{63} = \frac{π^{2} k_{s} M_{15}}{a^{2}} + \frac{π^{2} k_{s} M_{24}}{b^{2}} \\ K_{64} = \frac{k_{s} π M_{15}}{a} + \frac{π 1 mu E_{31}}{a} \\ K_{65} = \frac{k_{s} π M_{24}}{b} + \frac{π 1 mu E_{32}}{b} \\ K_{66} = \frac{π^{2} X_{22}}{b^{2}} + \frac{π^{2} X_{11}}{a^{2}} + X_{33} \\ K_{67} = \frac{π^{2} Y_{22}}{b^{2}} + \frac{π^{2} Y_{11}}{a^{2}} + Y_{33} \end{matrix}

\begin{array}{l} K_{71} = \frac{π β_{31}}{a} \\ K_{72} = \frac{π β_{32}}{b} \end{array}

\begin{array}{l} K_{73} = \frac{π^{2} k_{s} O_{24}}{a^{2}} + \frac{π^{2} k_{s} O_{15}}{a^{2}} \\ K_{74} = \frac{k_{s} π O_{15}}{a} + \frac{π F_{31}}{a} \\ K_{75} = \frac{k_{s} π O_{24}}{b} + \frac{π F_{32}}{b} \\ K_{76} = \frac{π^{2} Y_{22}}{b^{2}} + \frac{π^{2} Y_{11}}{a^{2}} + Y_{33} \\ K_{77} = \frac{π^{2} Z_{11}}{a^{2}} + \frac{π^{2} Z_{22}}{b^{2}} + Z_{33} \end{array}

\begin{array}{l} A_{11} = \int_{- \frac{h}{2}}^{\frac{h}{2}} c_{11}^{c} d z + \int_{\frac{h}{2}}^{\frac{h}{2} + h_{p}} c_{11}^{p} d z + \int_{- \frac{h}{2} - h_{p}}^{- \frac{h}{2}} c_{11}^{p} d z \\ A_{22} = \int_{- \frac{h}{2}}^{\frac{h}{2}} c_{22}^{c} d z + \int_{\frac{h}{2}}^{\frac{h}{2} + h_{p}} c_{22}^{p} d z + \int_{- \frac{h}{2} - h_{p}}^{- \frac{h}{2}} c_{22}^{p} d z \\ A_{44} = \int_{- \frac{h}{2}}^{\frac{h}{2}} c_{44}^{c} d z + \int_{\frac{h}{2}}^{\frac{h}{2} + h_{p}} c_{44}^{p} d z + \int_{- \frac{h}{2} - h_{p}}^{- \frac{h}{2}} c_{44}^{p} d z \\ A_{55} = \int_{- \frac{h}{2}}^{\frac{h}{2}} c_{55}^{c} d z + \int_{\frac{h}{2}}^{\frac{h}{2} + h_{p}} c_{55}^{p} d z + \int_{- \frac{h}{2} - h_{p}}^{- \frac{h}{2}} c_{55}^{p} d z \\ A_{12} = \int_{- \frac{h}{2}}^{\frac{h}{2}} c_{12}^{c} d z + \int_{\frac{h}{2}}^{\frac{h}{2} + h_{p}} c_{12}^{p} d z + \int_{- \frac{h}{2} - h_{p}}^{- \frac{h}{2}} c_{12}^{p} d z \\ A_{66} = \int_{- \frac{h}{2}}^{\frac{h}{2}} c_{66}^{c} d z + \int_{\frac{h}{2}}^{\frac{h}{2} + h_{p}} c_{66}^{p} d z + \int_{- \frac{h}{2} - h_{p}}^{- \frac{h}{2}} c_{66}^{p} d z \end{array}

\begin{array}{l} B_{11} = \int_{- \frac{h}{2}}^{\frac{h}{2}} c_{11}^{c} z d z + \int_{\frac{h}{2}}^{\frac{h}{2} + h_{p}} c_{11}^{p} z d z + \int_{- \frac{h}{2} - h_{p}}^{- \frac{h}{2}} c_{12}^{p} z d z \\ B_{22} = \int_{- \frac{h}{2}}^{\frac{h}{2}} c_{22}^{c} z d z + \int_{\frac{h}{2}}^{\frac{h}{2} + h_{p}} c_{22}^{p} z d z + \int_{- \frac{h}{2} - h_{p}}^{- \frac{h}{2}} c_{22}^{p} z d z \\ B_{12} = \int_{- \frac{h}{2}}^{\frac{h}{2}} c_{12}^{c} z d z + \int_{\frac{h}{2}}^{\frac{h}{2} + h_{p}} c_{12}^{p} z d z + \int_{- \frac{h}{2} - h_{p}}^{- \frac{h}{2}} c_{12}^{p} z d z \\ B_{66} = \int_{- \frac{h}{2}}^{\frac{h}{2}} c_{66}^{c} z d z + \int_{\frac{h}{2}}^{\frac{h}{2} + h_{p}} c_{66}^{p} z d z + \int_{- \frac{h}{2} - h_{p}}^{- \frac{h}{2}} c_{66}^{p} z d z \end{array}

Q_{11} = \int_{- \frac{h}{2}}^{\frac{h}{2}} c_{11}^{c} z^{2} d z + \int_{\frac{h}{2}}^{\frac{h}{2} + h_{p}} c_{11}^{p} z^{2} d z + \int_{- \frac{h}{2} - h_{p}}^{- \frac{h}{2}} c_{11}^{p} z^{2} d z

\begin{array}{l} Q_{22} = \int_{- \frac{h}{2}}^{\frac{h}{2}} c_{22}^{c} z^{2} d z + \int_{\frac{h}{2}}^{\frac{h}{2} + h_{p}} c_{22}^{p} z^{2} d z + \int_{- \frac{h}{2} - h_{p}}^{- \frac{h}{2}} c_{22}^{p} z^{2} d z \\ Q_{12} = \int_{- \frac{h}{2}}^{\frac{h}{2}} c_{12}^{c} z^{2} d z + \int_{\frac{h}{2}}^{\frac{h}{2} + h_{p}} c_{12}^{p} z^{2} d z + \int_{- \frac{h}{2} - h_{p}}^{- \frac{h}{2}} c_{12}^{p} z^{2} d z \\ Q_{66} = \int_{- \frac{h}{2}}^{\frac{h}{2}} c_{66}^{c} z^{2} d z + \int_{\frac{h}{2}}^{\frac{h}{2} + h_{p}} c_{66}^{p} z^{2} d z + \int_{- \frac{h}{2} - h_{p}}^{- \frac{h}{2}} c_{66}^{p} z^{2} d z \end{array}

\overset{b}{z} = z - \frac{h}{2} - \frac{h_{p}}{2}, z^{t} = z + \frac{h}{2} + \frac{h_{p}}{2}

\begin{array}{l} E_{31} = \int_{\frac{h}{2}}^{\frac{h}{2} + h_{p}} \frac{π}{h} z e_{31} \sin (\frac{π \overset{t}{z}}{h_{p}}) d z + \int_{- \frac{h}{2} - h_{p}}^{- \frac{h}{2}} \frac{π}{h} z e_{31} \sin (\frac{π \overset{b}{z}}{h_{p}}) d z \\ E_{32} = \int_{\frac{h}{2}}^{\frac{h}{2} + h_{p}} \frac{π}{h} z e_{32} \sin (\frac{π \overset{t}{z}}{h_{p}}) d z + \int_{- \frac{h}{2} - h_{p}}^{- \frac{h}{2}} \frac{π}{h} z e_{32} \sin (\frac{π \overset{b}{z}}{h_{p}}) d z \end{array}

\begin{array}{l} F_{31} = \int_{\frac{h}{2}}^{\frac{h}{2} + h_{p}} \frac{π}{h} z f_{31} \sin (\frac{π \overset{t}{z}}{h_{p}}) d z + \int_{- \frac{h}{2} - h_{p}}^{- \frac{h}{2}} \frac{π}{h} z f_{31} \sin (\frac{π \overset{b}{z}}{h_{p}}) d z \\ F_{32} = \int_{\frac{h}{2}}^{\frac{h}{2} + h_{p}} \frac{π}{h} z f_{32} \sin (\frac{π \overset{t}{z}}{h_{p}}) d z + \int_{- \frac{h}{2} - h_{p}}^{- \frac{h}{2}} \frac{π}{h} z f_{32} \sin (\frac{π \overset{b}{z}}{h_{p}}) d z \end{array}

\begin{array}{l} X_{11} = \int_{\frac{h}{2}}^{\frac{h}{2} + h_{p}} κ_{11} \sin^{2} (\frac{π \overset{t}{z}}{h_{p}}) d z + \int_{- \frac{h}{2} - h_{p}}^{- \frac{h}{2}} κ_{11} \sin^{2} (\frac{π \overset{b}{z}}{h_{p}}) d z \\ X_{22} = \int_{\frac{h}{2}}^{\frac{h}{2} + h_{p}} κ_{22} \sin^{2} (\frac{π \overset{t}{z}}{h_{p}}) d z + \int_{- \frac{h}{2} - h_{p}}^{- \frac{h}{2}} κ_{22} \sin^{2} (\frac{π \overset{b}{z}}{h_{p}}) d z \\ X_{33} = \int_{\frac{h}{2}}^{\frac{h}{2} + h_{p}} \frac{π^{2}}{h^{2}} κ_{33} \sin^{2} (\frac{π \overset{t}{z}}{h_{p}}) d z + \int_{- \frac{h}{2} - h_{p}}^{- \frac{h}{2}} \frac{π^{2}}{h^{2}} κ_{33} \sin^{2} (\frac{π \overset{b}{z}}{h_{p}}) d z \end{array}

\begin{array}{l} Y_{11} = \int_{\frac{h}{2}}^{\frac{h}{2} + h_{p}} g_{11} \sin^{2} (\frac{π \overset{t}{z}}{h_{p}}) d z + \int_{- \frac{h}{2} - h_{p}}^{- \frac{h}{2}} g_{11} \sin^{2} (\frac{π \overset{b}{z}}{h_{p}}) d z \\ Y_{22} = \int_{\frac{h}{2}}^{\frac{h}{2} + h_{p}} g_{22} \sin^{2} (\frac{π \overset{t}{z}}{h_{p}}) d z + \int_{- \frac{h}{2} - h_{p}}^{- \frac{h}{2}} g_{22} \sin^{2} (\frac{π \overset{b}{z}}{h_{p}}) d z \\ Y_{33} = \int_{\frac{h}{2}}^{\frac{h}{2} + h_{p}} \frac{π^{2}}{h^{2}} g_{33} \sin^{2} (\frac{π \overset{t}{z}}{h_{p}}) d z + \int_{- \frac{h}{2} - h_{p}}^{- \frac{h}{2}} \frac{π^{2}}{h^{2}} g_{33} \sin^{2} (\frac{π \overset{b}{z}}{h_{p}}) d z \end{array}

\begin{array}{l} Z_{11} = \int_{\frac{h}{2}}^{\frac{h}{2} + h_{p}} μ_{11} \sin^{2} (\frac{π \overset{t}{z}}{h_{p}}) d z + \int_{- \frac{h}{2} - h_{p}}^{- \frac{h}{2}} μ_{11} \sin^{2} (\frac{π \overset{b}{z}}{h_{p}}) d z \\ Z_{22} = \int_{\frac{h}{2}}^{\frac{h}{2} + h_{p}} μ_{22} \sin^{2} (\frac{π \overset{t}{z}}{h_{p}}) d z + \int_{- \frac{h}{2} - h_{p}}^{- \frac{h}{2}} μ_{22} \sin^{2} (\frac{π \overset{b}{z}}{h_{p}}) d z \\ Z_{33} = \int_{\frac{h}{2}}^{\frac{h}{2} + h_{p}} \frac{π^{2}}{h^{2}} μ_{33} \sin^{2} (\frac{π \overset{t}{z}}{h_{p}}) d z + \int_{- \frac{h}{2} - h_{p}}^{- \frac{h}{2}} \frac{π^{2}}{h^{2}} μ_{33} \sin^{2} (\frac{π \overset{b}{z}}{h_{p}}) d z \end{array}

\begin{array}{l} Ξ_{31} = \int_{\frac{h}{2}}^{\frac{h}{2} + h_{p}} \frac{π}{h} e_{31} \sin (\frac{π \overset{t}{z}}{h_{p}}) d z + \int_{- \frac{h}{2} - h_{p}}^{- \frac{h}{2}} \frac{π}{h} e_{31} \sin (\frac{π \overset{b}{z}}{h_{p}}) d z \\ Ξ_{32} = \int_{\frac{h}{2}}^{\frac{h}{2} + h_{p}} \frac{π}{h} e_{32} \sin (\frac{π \overset{t}{z}}{h_{p}}) d z + \int_{- \frac{h}{2} - h_{p}}^{- \frac{h}{2}} \frac{π}{h} e_{32} \sin (\frac{π \overset{b}{z}}{h_{p}}) d z \end{array}

\begin{array}{l} β_{31} = \int_{\frac{h}{2}}^{\frac{h}{2} + h_{p}} \frac{π}{h_{p}} f_{31} \sin (\frac{π \overset{t}{z}}{h_{p}}) d z + \int_{- \frac{h}{2} - h_{p}}^{- \frac{h}{2}} \frac{π}{h_{p}} f_{31} \sin (\frac{π \overset{b}{z}}{h_{p}}) d z \\ β_{32} = \int_{\frac{h}{2}}^{\frac{h}{2} + h_{p}} \frac{π}{h_{p}} f_{32} \sin (\frac{π \overset{t}{z}}{h_{p}}) d z + \int_{- \frac{h}{2} - h_{p}}^{- \frac{h}{2}} \frac{π}{h_{p}} f_{32} \sin (\frac{π \overset{b}{z}}{h_{p}}) d z \end{array}

\begin{array}{l} M_{15} = \int_{\frac{h}{2}}^{\frac{h}{2} + h_{p}} e_{15} \sin (\frac{π \overset{t}{z}}{h_{p}}) d z + \int_{- \frac{h}{2} - h_{p}}^{- \frac{h}{2}} e_{15} \sin (\frac{π \overset{b}{z}}{h_{p}}) d z \\ M_{24} = \int_{\frac{h}{2}}^{\frac{h}{2} + h_{p}} e_{24} \sin (\frac{π \overset{t}{z}}{h_{p}}) d z + \int_{- \frac{h}{2} - h_{p}}^{- \frac{h}{2}} e_{24} \sin (\frac{π \overset{b}{z}}{h_{p}}) d z \end{array}

\begin{array}{l} O_{15} = \int_{\frac{h}{2}}^{\frac{h}{2} + h_{p}} f_{15} \sin (\frac{π \overset{t}{z}}{h_{p}}) d z + \int_{- \frac{h}{2} - h_{p}}^{- \frac{h}{2}} f_{15} \sin (\frac{π \overset{b}{z}}{h_{p}}) d z \\ O_{24} = \int_{\frac{h}{2}}^{\frac{h}{2} + h_{p}} f_{24} \sin (\frac{π \overset{t}{z}}{h_{p}}) d z + \int_{- \frac{h}{2} - h_{p}}^{- \frac{h}{2}} f_{24} \sin (\frac{π \overset{b}{z}}{h_{p}}) d z \end{array}

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 financially supported by the University of Kashan (Grant Number: 574613/026). M. Arefi would like to thank the Iranian Nanotechnology Development Committee for their financial support.

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