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Microsystem Technologies

, Volume 24, Issue 5, pp 2295–2316 | Cite as

Nanoscale mass nanosensor based on the vibration analysis of embedded magneto-electro-elastic nanoplate made of FGMs via nonlocal Mindlin plate theory

  • M. Hosseini
  • M. R. Mofidi
  • A. Jamalpoor
  • M. Safi Jahanshahi
Open Access
Technical Paper
  • 516 Downloads

Abstract

In the current paper, the sensitivity performance of functionally graded magneto-electro-elastic (FG-MEE) nanoplate with attached nanoparticles as a nanosensor is analyzed based on nonlocal Mindlin plate assumption. Power law distribution model is employed to display how the material properties of FG-MEE nanoplate vary across the thickness direction. It is supposed that FG-MEE nanoplate is under initial external electric and magnetic potentials. Boundary condition of each edge of FG-MEE nanoplate is assumed to be simply supported. Furthermore, a Pasternak substrate is applied for modelling the total reaction pressure between nanoplate and foundation. Partial differential equations and corresponding boundary conditions are first achieved using Hamilton’s variational principle and then analytically solved to determine the frequency shift utilizing Navier’s approach. Numerical examples are performed to elucidate the dependency of the sensitivity performance of FG-MEE nanosensor on the volume fraction exponent, nonlocal parameter, total attached mass and location of the nanoparticle, aspect ratio, mode number, initial external electric voltage, initial external magnetic potential, and Pasternak medium coefficients. It is clearly indicated that these factors have highly significant impacts on the variations of frequency shift.

1 Introduction

In the recent years, with the rapid growth in the use of the composite materials in a wide range of engineering structures such as aircraft structures (Baker and Baker 2004) and sensor and actuators (Tressler et al. 1999; Akdogan et al. 2005; Li et al. 2008) because of their excellent mechanical properties, the importance of recognizing the composite material behavior increases day to day. Magneto-electro-elastic (MEE) composite materials are an important class of smart composite materials combining piezoelectric and piezomagnetic phases. The capacity of converting energy among magnetism, electricity, or elasticity into another form has caused that the MEE materials become suitable for smart applications. Up to now, various analytical or numerical works have been carried out to investigate MEE plate, beam, and shell which contain studies on the static behavior (Wu and Tsai 2007; Wu et al. 2010), linear and nonlinear vibration (Ramirez et al. 2006; Razavi and Shooshtari 2015; Shirbani et al. 2017) and on the post-buckling behavior (Zhou et al. 2003). For the first time, the excellent behavior of MEE composite materials experimentally was reported by Van Den Boomgaard et al. (1974). A discrete layer model is applied to predict the natural frequencies of orthotropic functionally graded (FG) MEE plates combining various percentages of barium titanate (BaTiO3) and cobalt iron oxide (CoFe2O4) by Ramirez et al. (2006). Nonlinear forced vibration of a MEE plate is investigated via first-order shear deformation theory by Shooshtari and Razavi (2016). Liu et al. (2016) developed a MEE 3D model for studying the components of the MEE field. Surface effects include surface piezoelectricity, surface elasticity and surface piezomagneticity investigated on the bending, buckling and free vibration of the MEE Euler–Bernoulli beams by Xu et al. (2016).

During the recent years, the investigation of the mechanical behavior of MEE structures at nano- and micro- scales have attracted the interest of some researchers. It have proven that the classical approaches cannot predict satisfactorily the mechanical behavior of nano- and micro- structures (Pradhan and Phadikar 2009; Ansari et al. 2010; Hosseini et al. 2016a, b, c). Experiments and atomistic simulations (Chen et al. 2006; Stan et al. 2007) demonstrated that the influence of small size becomes significant when the dimensions of the structure become small and reach to the micro/nano length scale. Since the traditional classical elasticity and plasticity theories have no material internal length scale, these theories are not capable of taking into account the influence of small size scale. Therefore, these theories modified and included one or several independent internal length scale parameters. Some of the most popular modified theories include nonlocal elasticity theory suggested by Eringen (1972, 1983), modified couple stress theory presented (MCST) by Yang et al. (2002) and modified strain gradient theory (MSGT) suggested by Lam et al. (2003).

Our literature review here will focus on the analysis of MEE nanostructures based on the nonlocal elasticity of Eringen. According to the suggested model of Eringen, the elastic strain is described by a Fredholm type integral equations in which the stress is result of a convolution between the local response to an elastic strain and a smoothing kernel dependent on a nonlocal parameter. But some researchers showed the constitutive boundary conditions and paradoxes of nonlocal integral model (Romano and Barretta 2016, 2017a, b; Romano et al. 2017a, b; Apuzzo et al. 2017). They demonstrated that Eringen’s nonlocal integral model admits no solution. On the basis of the nonlocal elasticity theory of Eringen, Ansari et al. (2015c) proposed a third-order shear deformable beam model to study forced vibration behavior of magneto-electro-thermo-elastic nanobeams by taking into account the effects of small scale. Recently, by considering the von Kármán’s assumption, the influences of the external electric and magnetic potential on the nonlinear natural frequencies of the MEE nanoplates are shown by Farajpour et al. (2016). Wang et al. (2016) presented a novel two-dimensional linear elastic theory for demonstrating surface, nonlocal effects, poling directions, piezoelectric phase materials on the MEE nanoplate. Numerical results illustrated that driving frequency and output power density of magnetic energy harvesters have obvious size-dependent behavior because of surface effects. On the basis of the nonlocal Mindlin plate approach, Li et al. (2014) analytically carried out elastic instability and free vibration behavior of MEE nanoplate resting on Pasternak medium. It was shown that buckling load and natural frequency decrease (increase) linearly with electric potential (magnetic potential). Recently, the generalized differential quadrature method in conjunction with pseudo arc-length continuation are applied to calculate the nonlinear buckling and postbuckling of MEE Mindlin nanoplate under thermal load by Ansari and Gholami (2017). A mathematical model based on Euler–Bernoulli beam assumption is proposed by Vaezi et al. (2016) for investigating natural frequencies and buckling loads of MEE microbeam. Moreover, there are a number of literatures regarding the influences of small scale on the MEE nanostructures such as MEE nanobeams (Ke and Wang 2014; Ansari et al. 2015b; Li et al. 2016; Ma et al. 2017) and MEE nanoplates (Ke et al. 2014; Wu et al. 2015; Jamalpoor et al. 2017; Kiani et al. 2017) by taking into account the nonlocal parameter.

Functionally graded materials (FGMs) are a special subgroup of novel composite materials which contain the heterogeneous property and have a continuous variation of material properties from one surface to another. Due to the numerous profits containing high mechanical and thermal resistance, FGMs are vastly used in sensors and thermogenerators (Müller et al. 2003; Rahimi et al. 2011), biomedical fields (Pompe et al. 2003; Parthasarathy et al. 2011), aerospace (Kumar et al. 2013). With respect to the development of science and wide application of FGM nanostructures, the study of the size effect on the mechanical behavior of FGM nanostructures is of considerable practical interest. On the basis of nonlocal elasticity theory in conjunction with Gurtin–Murdoch elasticity theory, Hosseini and Jamalpoor (2015) reported the important role of the surface effects, i.e., surface elasticity, residual stresses and surface density on the free vibration of a double-FGM viscoelastic nanoplates-system under thermal load. They considered that the material traits of the plate follow power law repartition in the thickness direction. Numerical results expressed that an increase in the amount of the power law index makes a reduction of the natural frequencies. A modified power-law model in conjunction with the refined four-variable plate assumption are implemented by Barati et al. (2017) to study the influence of FGMs on the vibration frequency of smart piezoelectric plates under various boundary conditions. Wave propagation phenomena of a FG-MEE nanorod is investigated via nonlocal continuum mechanics by Narendar (2016). Furthermore, there are a number of literatures regarding the effects of FGMs on the nanostructures (Natarajan et al. 2012; Ansari et al. 2015a; Barretta et al. 2016; Thang et al. 2017; Mechab et al. 2016; Jamalpoor and Kiani 2017; Hosseini et al. 2017).

It should be pointed that sometimes one or more concentrated particles are attached to the structures. Thus, we need the mass detection technology for detecting the position of attached masses with high accuracy and sensitivity. In order to find the exact point of attached masses, sensors play a significant role. The main idea for determining the location of added masses such as bacterium/virus, biomolecules, and buckyballs is to measure the resonant frequency shift of the nanosensor caused by variations in total mass of the system (Zhou et al. 2014; Asemi et al. 2015; Karličić et al. 2015; Shi et al. 2015; Jalali et al. 2015; Sadeghzadeh 2016). It is that piezoelectric (ZnO-BaTiO3) (Guo et al. 2012; Alluri et al. 2015), magnetic/piezoelectric (Yeh et al. 2016), and magnetic (CoFe2O4) (Kim et al. 2009; Taei et al. 2016) can be suitable candidates for sensor devices because of having excellent properties like dimensional stability, abrasion, and corrosion resistance.

The above-reviewed papers explicitly illustrated that while there exist a considerable number of studies that deal with the static and dynamic responses of MEE nanoplate based on different (Kirchhoff or Mindlin) plate assumptions, nanoscale mass detection based on vibration FG-MEE Mindlin nanoplate seem to be nonexistent. Accordingly, in this work, a first-order shear deformable (Mindlin) plate theory in conjunction with the nonlocal elasticity theory is applied to take into account the size effect on the sensitivity of the sensor. In this regard, the key novelties of the presented study are summarized as follows:
  • Development of a nonlocal Mindlin plate theory for free vibration of FG-MEE nanoplate resting on the Pasternak medium.

  • FG-MEE Mindlin nanoplate with attached concentrated nanoparticles is presented as a mass nanosensor according to the vibration analysis.

Partial differential equations of motion of the system with multiple added masses are derived by employing Hamilton’s principle, then by applying the analytical Navier type solution, the frequency shift of the system is presented in the explicit closed-form.

2 Modeling of problem and formulation

2.1 Geometrical configuration

Consider a simply supported thick rectangular (\(l_{x} \times l_{y}\)) nanoplate with uniform thickness \(h\) consisting of BaTiO3-CoFe2O4 MEE, which is exposed to electric \({ \varPhi }\left( {x,y,z,t} \right)\) and magnetic potentials \({\varPsi }\left( {x,y,z,t} \right)\), as shown in Fig. 1. As can be seen from Fig. 1, it is considered that nanoplate with attached nanoparticles is resting on the Pasternak-type foundation with the transverse and shear coefficients \(k_{w}\) and \(k_{G}\), respectively. Also, the location of any rigid nanoparticle is assumed to be at (\(x_{e} , y_{e} ), e = 1,2, \ldots\). Furthermore, the material properties of MEE nanoplate are supposed to be graded in the thickness direction based on the power-law model in terms of the volume fractions of the constituents. It should be noted that in FG-MEE nanoplate, the materials properties vary continuously from a fully BaTiO3 phase at the top surface (\(z = \frac{h}{2}\)) to a fully CoFe2O4 phase at the bottom surface (\(z = - \frac{h}{2}\)).
Fig. 1

The geometry of FG-MEE nanoplate with multiple attached nanoparticle resting on Pasternak medium (color figure online)

2.2 Constitutive relations for nonlocal FG-MEE Mindlin nanoplate

2.2.1 FGMs

On the basis of the power law distribution, the effective material characteristics of MEE nanoplate vary continuously across the plate thickness and can be proposed as below:
$$PM\left( z \right) = PM_{2} V_{2} + PM_{1} V_{1}$$
(1)
where, \(PM_{1}\) and \(PM_{2}\) are the properties of CoFe2O4 phase and BaTiO3 phase, respectively. Also, \(V_{1}\) and \(V_{2}\) refer to the volume fractions of CoFe2O4 rich and BaTiO3 rich, respectively, which can be written as
$$V_{2} = \left( {\frac{1}{2} + \frac{z}{h}} \right)^{K} ,\;\; V_{1} = 1 - V_{2}$$
(2)
in which \(K\) is the volume fraction exponent and always has a positive value.

2.2.2 Displacement field

In order to obtain more accurate estimation of mechanical behavior of thick plate, we apply Mindlin plate hypothesis (first-order shear deformation theory) which incorporates the shear deformations through-the-thickness of a plate into classical plate theory. On the basis of Mindlin plate theory, the displacement fields of the plate can be proposed in the five unknown parameters in Cartesian coordinate as below
$$\begin{aligned} u_{x} \left( {x,y,z,t} \right) = &\, u_{0} \left( {x,y,t} \right) + z\theta_{x} (x,y,t) \\ u_{y} \left( {x,y,z,t} \right) = &\, v_{0} \left( {x,y,t} \right) + z\theta_{y} (x,y,t) \\ u_{z} \left( {x,y,z,t} \right) = &\, w_{0} \left( {x,y,t} \right) \\ \end{aligned}$$
(3)
in which \(u_{0}\) and \(v_{0}\) denote the in plane deflections of the mid-surface; \(\theta_{x}\) and \(\theta_{y}\) refer to the rotation of the middle plane in the \(x\) and \(y\) directions, respectively; and \(w_{0}\) is the lateral displacement.

2.2.3 Strains

The nonzero components of the linear strain–displacement relationships can be expressed as
$$\begin{aligned} \varepsilon_{xx} &= \frac{{\partial u_{0} }}{\partial x} + z\frac{{\partial \theta_{x} }}{\partial x} \hfill \\ \varepsilon_{yy} &= \frac{{\partial v_{0} }}{\partial y} + z\frac{{\partial \theta_{y} }}{\partial y} \hfill \\ \gamma_{xz} &= \theta_{x} + \frac{{\partial w_{0} }}{\partial x} \hfill \\ \gamma_{yz} &= \theta_{y} + \frac{{\partial w_{0} }}{\partial y} \hfill \\ \gamma_{xy} &= \frac{{\partial u_{0} }}{\partial y} + \frac{{\partial v_{0} }}{\partial x} + z\left( {\frac{{\partial \theta_{x} }}{\partial y} + \frac{{\partial \theta_{y} }}{\partial x}} \right) \hfill \\ \end{aligned}$$
(4)

2.3 Nonlocal elasticity theory

It is considered via Eringen’s nonlocal elastic theory (Eringen 1972, 1983) that the stress, electric displacement, and magnetic induction at a reference spot of MEE nanostructures are supposed to be a function of the strain, electric and magnetic components at each spot in material domain of body. According to the main idea of this nonlocal theory, the basic relations for FG-MEE substance without considering body force can be written as follows (Ansari et al. 2015b; Farajpour et al. 2016)
$$\begin{aligned} \sigma_{ij}^{nl} &= \mathop \int \nolimits {\mathcal{F}}\left( {\left| {x - x^{\prime}} \right|,\frac{{e_{0} a}}{b} } \right)\sigma_{ij}^{l} dV\left( {x^{\prime}} \right), \forall x\int V, \hfill \\ D_{i}^{nl} &= \mathop \int \nolimits {\mathcal{F}}\left( {\left| {x - x^{\prime}} \right|,\frac{{e_{0} a}}{b} } \right)D_{i}^{l} dV\left( {x^{\prime}} \right), \forall x\smallint V, \hfill \\ B_{i}^{nl} &= \mathop \int \nolimits {\mathcal{F}}\left( {\left| {x - x^{\prime}} \right|,\frac{{e_{0} a}}{b}} \right)B_{i}^{l} dV\left( {x^{\prime}} \right), \forall x\smallint V, \hfill \\ \sigma_{ij}^{l} &= \bar{C}_{iikl} \left( z \right)\varepsilon_{kl} \left( {x^{\prime}} \right) - \bar{e}_{mij} \left( z \right)E_{m} \left( {x^{\prime}} \right) - \bar{q}_{nij} \left( z \right)H_{n} \left( {x^{\prime}} \right), \hfill \\ D_{i}^{l} &= \bar{e}_{ikl} \left( z \right)\varepsilon_{kl} \left( {x^{\prime}} \right) + \bar{\eta }_{im} \left( z \right)E_{m} \left( {x^{\prime}} \right) + \bar{d}_{in} \left( z \right)H_{n} \left( {x^{\prime}} \right), \hfill \\ B_{i}^{l} &= \bar{q}_{ikl} \left( z \right)\varepsilon_{kl} \left( {x^{\prime}} \right) + \bar{d}_{im} \left( z \right)E_{m} \left( {x^{\prime}} \right) + \bar{\mu }_{in} \left( z \right)H_{n} \left( {x^{\prime}} \right), \hfill \\ \end{aligned}$$
(5)
where the terms \(\sigma_{ij}^{nl} = \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\sigma_{xx}^{nl} } & {\sigma_{yy}^{nl} } & {\sigma_{yz}^{nl} } \\ \end{array} } & {\sigma_{xz}^{nl} } & {\sigma_{xy}^{nl} } \\ \end{array} } \right]^{T}\), \(D_{i}^{nl} = \left[ {\begin{array}{*{20}c} {D_{x}^{nl} } & {D_{y}^{nl} } & {D_{z}^{nl} } \\ \end{array} } \right]^{T}\) and \(B_{i}^{nl} = \left[ {\begin{array}{*{20}c} {B_{x}^{nl} } & {B_{y}^{nl} } & {B_{z}^{nl} } \\ \end{array} } \right]^{T}\) (the superscript T signifies transpose) represent, respectively, the nonlocal forms of the stress tensor, electric displacement and magnetic induction. Also, \(\sigma_{ij}^{l}\), \(D_{i}^{l}\) and \(B_{i}^{l}\) demonstrate, respectively, the local forms of the stress tensor, electric displacement and magnetic induction. \({\mathcal{F}}\left( {\left| {x - x^{\prime}} \right|,\frac{{e_{0} a}}{\text{b}} } \right)\) indicates the Kernel function, and the Euclidean distance between reference point \(x\) and any points \(x^{\prime}\) in the body is illustrated with the symbol \(\left| {x - x^{\prime}} \right|\). Furthermore, in Eq. (5), \(E_{m}\) and \(H_{n}\) are electric field and magnetic field vectors, respectively. Also, symbol \(V\) is the volume of MEE substance. Nonlocal parameter that takes into account the small scale effects into the constitutive equations is demonstrated by \(\frac{{e_{0} a}}{b}\), which e 0 is a constant parameter determined experimentally for matching theoretical model with empirical results; \(a\) is the internal characteristic length (e.g. lattice parameter, granular size); and \(b\) refers to an external property of length (e.g. crack length, wavelength). However, the strain-driven integral constitutive nonlocal relation (5) proposed by Eringen (1983) is inapplicable to nonlocal elastic problems on bounded structural domains (Romano and Barretta 2016, 2017a, b; Romano et al. 2017a, b; Apuzzo et al. 2017). In this paper we assume the following phenomenological differential constitutive law (consequent but not equivalent to Eq. 5) to investigate size-dependent behavior of Mindlin plates.
$$\begin{aligned} \left[ {1 - \left( {e_{0} a} \right)^{2} \nabla^{2} } \right]\left[ {\begin{array}{ll} {\begin{array}{ll} {\sigma_{xx}^{nl} } \\ {\sigma_{yy}^{nl} } \\ {\sigma_{yz}^{nl} } \\ \end{array} } \\ {\sigma_{xz}^{nl} } \\ {\sigma_{xy}^{nl} } \\ \end{array} } \right] = \left[ {\bar{C}\left( z \right)} \right]\left[ {\begin{array}{ll} {\begin{array}{ll} {\varepsilon_{xx} } \\ {\varepsilon_{yy} } \\ {\gamma_{yz} } \\ \end{array} } \\ {\gamma_{xz} } \\ {\gamma_{xy} } \\ \end{array} } \right] - [\bar{e}\left( z \right)]\left[ {\begin{array}{ll} {E_{x} } \\ {E_{y} } \\ {E_{z} } \\ \end{array} } \right] - [\bar{q}\left( z \right)]\left[ {\begin{array}{ll} {H_{x} } \\ {H_{y} } \\ {H_{z} } \\ \end{array} } \right], \hfill \\ \left[ {1 - \left( {e_{0} a} \right)^{2} \nabla^{2} } \right]\left[ {\begin{array}{ll} {D_{x}^{nl} } \\ {D_{y}^{nl} } \\ {D_{z}^{nl} } \\ \end{array} } \right] = \left[ {\bar{e}\left( z \right)} \right]^{T} \left[ {\begin{array}{ll} {\begin{array}{ll} {\varepsilon_{xx} } \\ {\varepsilon_{yy} } \\ {\gamma_{yz} } \\ \end{array} } \\ {\gamma_{xz} } \\ {\gamma_{xy} } \\ \end{array} } \right] + \left[ {\bar{\eta }\left( z \right)} \right]\left[ {\begin{array}{ll} {E_{x} } \\ {E_{y} } \\ {E_{z} } \\ \end{array} } \right] + [\bar{d}\left( z \right)]\left[ {\begin{array}{ll} {H_{x} } \\ {H_{y} } \\ {H_{z} } \\ \end{array} } \right], \hfill \\ \left[ {1 - \left( {e_{0} a} \right)^{2} \nabla^{2} } \right]\left[ {\begin{array}{ll} {B_{x}^{nl} } \\ {B_{y}^{nl} } \\ {B_{z}^{nl} } \\ \end{array} } \right] = [\bar{q}\left( z \right)]^{T} \left[ {\begin{array}{ll} {\begin{array}{ll} {\varepsilon_{xx} } \\ {\varepsilon_{yy} } \\ {\gamma_{yz} } \\ \end{array} } \\ {\gamma_{xz} } \\ {\gamma_{xy} } \\ \end{array} } \right] + [\bar{d}\left( z \right)]\left[ {\begin{array}{ll} {E_{x} } \\ {E_{y} } \\ {E_{z} } \\ \end{array} } \right] + [\bar{\mu }\left( z \right)]\left[ {\begin{array}{ll} {H_{x} } \\ {H_{y} } \\ {H_{z} } \\ \end{array} } \right], \hfill \\ \end{aligned}$$
(6)
in which \(\nabla^{2} = \frac{{\partial^{2} }}{{\partial x^{2} }} + \frac{{\partial^{2} }}{{\partial y^{2} }}\) denotes the two-dimensional Laplacian operator. By not considering the size effect (\(e_{0} a = 0\)), the traditional classical constitutive relations of FG-MEE structures can be acquired from nonlocal basic relations. \([\bar{C}]\), \(\bar{e}\), \([\bar{q}]\), \(\bar{\eta }\), \([\bar{d}]\) and \([\bar{\mu }]\) denote, respectively, the elastic, piezoelectric, piezomagnetic, dielectric, magnetoelectric and magnetic constant matrices, which here are supposed to be varied continuously as a power law function of the plate thickness and are illustrated as below
$$ \begin{aligned} \left[ {\bar{C}\left( z \right)} \right] &= \left[ {\begin{array}{*{20}c} {\bar{C}_{11} \left( z \right)} & {\bar{C}_{12} \left( z \right)} & 0 & 0 & 0 \\ {\bar{C}_{12} \left( z \right)} & {\bar{C}_{22} \left( z \right)} & 0 & 0 & 0 \\ 0 & 0 & {\bar{C}_{44} \left( z \right)} & 0 & 0 \\ 0 & 0 & 0 & {\bar{C}_{55} \left( z \right)} & 0 \\ 0 & 0 & 0 & 0 & {\bar{C}_{66} \left( z \right)} \\ \end{array} } \right], \hfill \\ \left[ {\bar{e}\left( z \right)} \right] &= \left[ {\begin{array}{*{20}c} 0 & 0 & {\bar{e}_{31} \left( z \right)} \\ 0 & 0 & {\bar{e}_{32} \left( z \right)} \\ 0 & {\bar{e}_{24} \left( z \right)} & 0 \\ {\bar{e}_{15} \left( z \right)} & 0 & 0 \\ 0 & 0 & 0 \\ \end{array} } \right], \hfill \\ \left[ {\bar{q}\left( z \right)} \right] &= \left[ {\begin{array}{*{20}c} 0 & 0 & {\bar{q}_{31} \left( z \right)} \\ 0 & 0 & {\bar{q}_{32} \left( z \right)} \\ 0 & {\bar{q}_{24} \left( z \right)} & 0 \\ {\bar{q}_{15} \left( z \right)} & 0 & 0 \\ 0 & 0 & 0 \\ \end{array} } \right], \hfill \\ \left[ {\bar{\eta }\left( z \right)} \right] &= \left[ {\begin{array}{*{20}c} {\bar{\eta }_{11} \left( z \right)} & 0 & 0 \\ 0 & {\bar{\eta }_{22} \left( z \right)} & 0 \\ 0 & 0 & {\bar{\eta }_{33} \left( z \right)} \\ \end{array} } \right], \hfill \\ \left[ {\bar{d}\left( z \right)} \right] &= \left[ {\begin{array}{*{20}c} {\bar{d}_{11} \left( z \right)} & 0 & 0 \\ 0 & {\bar{d}_{22} \left( z \right)} & 0 \\ 0 & 0 & {\bar{d}_{33} \left( z \right)} \\ \end{array} } \right], \hfill \\ [\bar{\mu }\left( z \right)] &= \left[ {\begin{array}{*{20}c} {\bar{\mu }_{11} \left( z \right)} & 0 & 0 \\ 0 & {\bar{\mu }_{22} \left( z \right)} & 0 \\ 0 & 0 & {\bar{\mu }_{33} \left( z \right)} \\ \end{array} } \right]. \hfill \\ \end{aligned} $$
(7)
By supposing the plane stress state, the reduced material constants \( \bar{C}_{ij} \left( z \right), \bar{e}_{ij} \left( z \right), \bar{q}_{ij} \left( z \right), \bar{\eta }_{ij} \left( z \right), \bar{d}_{ij} \left( z \right) \) and \(\bar{\mu }_{ij} \left( z \right)\) can be denoted as follows
$$\begin{aligned} \bar{C}_{11} \left( z \right) & = C_{11} \left( z \right) - \frac{{C_{13} \left( z \right)^{2} }}{{C_{33} \left( z \right)}}, \bar{C}_{12} \left( z \right) = C_{12} \left( z \right) - \frac{{C_{13} \left( z \right)C_{23} \left( z \right)}}{{C_{33} \left( z \right)}}, \\ \bar{C}_{22} \left( z \right) & = C_{22} \left( z \right) - \frac{{C_{23} \left( z \right)^{2} }}{{C_{33} \left( z \right)}}, \\ \bar{C}_{44} \left( z \right) & = C_{44} \left( z \right), \; \bar{C}_{55} \left( z \right) = C_{55} \left( z \right), \;\bar{C}_{66} \left( z \right) = C_{66} \left( z \right), \\ \bar{e}_{31} \left( z \right) & = e_{31} \left( z \right) - \frac{{C_{13} \left( z \right)e_{33} \left( z \right)}}{{C_{33} \left( z \right)}}, \\ \bar{e}_{32} \left( z \right) & = e_{32} \left( z \right) - \frac{{C_{23} \left( z \right)e_{33} \left( z \right)}}{{C_{33} \left( z \right)}}, \\ \bar{e}_{24} \left( z \right) = & e_{24} \left( z \right), \bar{e}_{15} \left( z \right) = e_{15} \left( z \right), \\ \bar{q}_{31} \left( z \right) & = q_{31} \left( z \right) - \frac{{C_{13} \left( z \right)q_{33} \left( z \right)}}{{C_{33} \left( z \right)}}, \\ \bar{q}_{32} \left( z \right) & = q_{32} \left( z \right) - \frac{{C_{23} \left( z \right)q_{33} \left( z \right)}}{{C_{33} \left( z \right)}}, \\ \bar{q}_{24} \left( z \right) & = q_{24} \left( z \right),\; \bar{q}_{15} \left( z \right) = q_{15} \left( z \right),\; \\ \bar{\eta }_{11} \left( z \right) & = \eta_{11} \left( z \right), \;\bar{\eta }_{22} \left( z \right) = \eta_{11} \left( z \right), \\ \bar{\eta }_{33} \left( z \right) & = \eta_{33} \left( z \right) + \frac{{e_{33} \left( z \right)^{2} }}{{C_{33} \left( z \right)}}, \;\bar{d}_{11} \left( z \right) = d_{11} \left( z \right), \\ \bar{d}_{22} \left( z \right) & = d_{22} \left( z \right), \;\bar{d}_{33} \left( z \right) = d_{33} (z) + \frac{{q_{33} (z)e_{33} (z)}}{{C_{33} (z)}}, \\ \bar{\mu }_{11} \left( z \right) & = \mu_{11} (z),\;\bar{\mu }_{22} (z) = \mu_{22} (z),\; \\ \bar{\mu }_{33} (z) & = \mu_{33} (z) + \frac{{q_{33} (z)^{2} }}{{C_{33} (z)}}. \\ \end{aligned}$$
(8)
It should be noted that if the electric and magnetic fields are supposed to be the negative gradients of scalar electric \(\varPhi \left( {x,y,z,t} \right)\) and magnetic potentials \(\varPsi (x,y,z,t)\), respectively, Maxwell equations in the quasi-static approximation are satisfied as
$$\begin{aligned} E_{i} & = - \frac{{\partial \varPhi \left( {x,y,z,t} \right)}}{\partial i} \\ H_{i} & = - \frac{{\partial \varPsi \left( {x,y,z,t} \right)}}{\partial i}, (i = x,y,z) \\ \end{aligned}$$
(9)
However, in this work, the boundary conditions of magnetic and electric fields at the bottom and top surfaces of the FG-MEE nanoplate are taken to be as
$$\begin{aligned} \varPhi \left( {x,y, - \frac{h}{2},t} \right) = - \phi_{0} , \varPhi \left( {x,y,\frac{h}{2},t} \right) = \phi_{0} \hfill \\ \varPsi \left( {x,y, - \frac{h}{2},t} \right) = - \varPsi_{0} , \varPsi \left( {x,y,\frac{h}{2},t} \right) = \varPsi_{0} \hfill \\ \end{aligned}$$
(10)
where the terms \(\phi_{0}\) and \(\varPsi_{0}\) (\(- \phi_{0}\) and \(- \varPsi_{0} )\) indicate, respectively, the external electric potential and magnetic potential exerting to the top (bottom) surface of FG-MEE nanoplate. With respect to the electric-magnetic boundary conditions, the explicit distribution of electric and magnetic potentials are expressed as a combination of cosine and linear variation in the following form (Farajpour et al. 2016; Ansari and Gholami 2017)
$$\begin{aligned} \varPhi \left( {x,y,z,t} \right) = - \cos \left( {\frac{\pi z}{h}} \right)\phi \left( {x,y,t} \right) + \frac{{2z\phi_{0} }}{h} \hfill \\ \varPsi \left( {x,y,z,t} \right) = - \cos \left( {\frac{\pi z}{h}} \right)\psi \left( {x,y,t} \right) + \frac{{2z\psi_{0} }}{h} \hfill \\ \end{aligned}$$
(11)
in which \(\phi \left( {x,y,t} \right)\) and \(\psi \left( {x,y,t} \right)\) demonstrate the spatial variations related to the electric potential and magnetic potential along the \(x\) and \(y\) directions, respectively.

2.4 Governing equations and boundary conditions

In order to determine the governing differential equations of the motion of present model and related boundary conditions, in this section, Hamilton’s principle can be applied in the below form as
$$\mathop \int \limits_{{t_{1} }}^{{t_{2} }} \left( {\delta T - \delta U + \delta V} \right)dt = 0$$
(12)
where \(\delta T, \delta U\) and \(\delta V\), respectively, denote the virtual kinetic energy, the virtual strain energy and the virtual work exerted by external force \(q\). Here, \(q\) refers to the transverse load per unit area of FG-MEE nanoplate due to the added nanoparticle with mass \(m_{e}\) for \(e{\text{th}}\) mass in position (\(x_{e} , y_{e}\)), Pasternak elastic medium, and external loads which can be expressed as (Shen et al. 2012; Ke et al. 2014; Hosseini et al. 2016a)
$$\begin{aligned} q & = - k_{w} w_{0} + k_{G} \left( {\frac{{\partial^{2} w_{0} }}{{\partial x^{2} }} + \frac{{\partial^{2} w_{0} }}{{\partial y^{2} }}} \right) \\ & \;\; - \mathop \sum \limits_{e = 1}^{p} m_{e} \delta \left( {x - x_{e} } \right)\delta \left( {y - y_{e} } \right)\frac{{\partial^{2} w_{0} }}{{\partial t^{2} }} \\ & \;\; + \left( {N_{Ex} + N_{Mx} } \right)\frac{{\partial^{2} w_{0} }}{{\partial x^{2} }} + \left( {N_{Ey} + N_{My} } \right)\frac{{\partial^{2} w_{0} }}{{\partial y^{2} }} \\ \end{aligned}$$
(13)
in which \(\left( {N_{Ex} = \mathop \int \nolimits_{{ - \frac{h}{2}}}^{{\frac{h}{2}}} 2\frac{{\bar{e}_{31} (z)\phi_{0} }}{h},N_{Ey} = \mathop \int \nolimits_{{ - \frac{h}{2}}}^{{\frac{h}{2}}} 2\frac{{\bar{e}_{32} (z)\phi_{0} }}{h}} \right)\) and \(\left( {N_{Mx} = \mathop \int \nolimits_{{ - \frac{h}{2}}}^{{\frac{h}{2}}} 2\frac{{\bar{q}_{31} (z)\psi_{0} }}{h},N_{My} = \mathop \int \nolimits_{{ - \frac{h}{2}}}^{{\frac{h}{2}}} 2\frac{{\bar{q}_{32} (z)\psi_{0} }}{h}} \right)\) illustrate the normal loads induced by initial external electric potential and initial external magnetic potential along \(x\) and \(y\) coordinate directions, respectively. Also, the Dirac delta function in the two two-dimensional state is shown with symbol \(\delta \left( {x - x_{e} } \right)\delta \left( {y - y_{e} } \right)\), which define as
$$\begin{aligned} \delta \left( {x - x_{e} } \right)\delta \left( {y - y_{e} } \right) = 0\quad for (x, y) \ne (x_{e} , y_{e} ) \hfill \\ \mathop \int \limits_{ - \infty }^{\infty } \mathop \int \limits_{ - \infty }^{\infty } \delta \left( {x - x_{e} } \right)\delta \left( {y - y_{e} } \right)dxdy = 1. \hfill \\ \end{aligned}$$
(14)

2.4.1 Virtual kinetic energy

The virtual kinetic energy on the basis of the first shear order deformation theory can be written as follows
$$\delta T = \mathop \int \nolimits \left[ {I_{0} \left( {\dot{u}_{0} \delta \dot{u}_{0} + \dot{v}_{0} \delta \dot{v}_{0} + \dot{w}_{0} \delta \dot{w}_{0} } \right) + I_{1} \left( {\dot{u}_{0} \delta \dot{\theta }_{x} + \dot{\theta }_{x} \delta \dot{u}_{0} + \dot{v}_{0} \delta \dot{\theta }_{y} + \dot{\theta }_{y} \delta \dot{v}_{0} } \right) + I_{2} \left( {\dot{\theta }_{x} \delta \dot{\theta }_{x} + \dot{\theta }_{y} \delta \dot{\theta }_{y} } \right)} \right]dA$$
(15)
where \(I_{0} , I_{1}\) and \(I_{2}\) denote the mass inertia terms which can be expressed as \(\left\{ {I_{0} , I_{1} ,I_{2} } \right\} = \mathop \int \nolimits_{{ - \frac{h}{2}}}^{{\frac{h}{2}}} \{ \rho (z),\rho (z)z, \rho (z)z^{2} \} dz\).

2.4.2 Virtual strain energy

The variation of strain energy using the first shear order deformation theory can be expressed as
$$\begin{aligned} \delta U = \mathop \int \nolimits \left[ {N_{xx} \frac{{\partial \delta u_{0} }}{\partial x} + N_{yy} \frac{{\partial \delta v_{0} }}{\partial y} + M_{xx} \frac{{\partial \delta \theta_{x} }}{\partial x} + M_{yy} \frac{{\partial \delta \theta_{y} }}{\partial y} + N_{xy} \left( {\frac{{\partial \delta u_{0} }}{\partial y} + \frac{{\partial \delta v_{0} }}{\partial x}} \right) + M_{xy} \left( {\frac{{\partial \delta \theta_{x} }}{\partial y} + \frac{{\partial \delta \theta_{y} }}{\partial x}} \right) + Q_{xx} \left( {\delta \theta_{x} + \frac{{\partial \delta w_{0} }}{\partial x}} \right) + Q_{yy} \left( {\delta \theta_{y} + \frac{{\partial \delta w_{0} }}{\partial y}} \right)} \right]dA \hfill \\ \;\; - \mathop {\iint }\limits_{{ - \frac{h}{2}}}^{{\frac{h}{2}}} \left( {D_{x} \delta E_{x} + D_{y} \delta E_{y} + D_{z} \delta E_{z} + B_{x} \delta H_{x} + B_{y} \delta H_{y} + B_{z} \delta H_{z} } \right)dzdA \hfill \\ \end{aligned}$$
(16)
in which parameters \(N_{ij} , M_{ij}\) and \(Q_{ii}\) can be defined by the following integral expressions
$$\begin{aligned} \left\{ {N_{xx} , N_{yy} ,N_{xy} } \right\} & = \mathop \int \limits_{{ - \frac{h}{2}}}^{{\frac{h}{2}}} \left\{ {\sigma_{xx} ,\sigma_{yy} , \sigma_{xy} } \right\}dz , \left\{ {M_{xx} , M_{yy} ,M_{xy} } \right\} \\ & = \mathop \int \limits_{{ - \frac{h}{2}}}^{{\frac{h}{2}}} \left\{ {\sigma_{xx} ,\sigma_{yy} , \sigma_{xy} } \right\}zdz , \left\{ {Q_{xx} , Q_{yy} } \right\} = k_{s} \mathop \int \limits_{{ - \frac{h}{2}}}^{{\frac{h}{2}}} \left\{ {\sigma_{xz} ,\sigma_{yz} } \right\}dz. \\ \end{aligned}$$
(17)

The term \(k_{s}\) is the shear correction factor and here equals to \(5/6\).

2.4.3 Virtual work

Variation of work applied by external loads denotes by
$$\delta V = \mathop \int \nolimits q\delta w_{0} dA$$
(18)

2.4.4 The equilibrium equations in terms of the displacements

Finally, by implementing relations (1518) into Hamilton’s variational principle and integrating by parts, and factoring the coefficients of \(\delta u_{0} , \delta v_{0} , \delta w_{0} , \delta \theta_{x} , \delta \theta_{y} , \delta \phi\) and \(\delta \psi\), the following equilibrium equations of motion are derived as
$$\delta u_{0}: \frac{{\partial N_{xx} }}{\partial x} + \frac{{\partial N_{xy} }}{\partial y} = I_{0} \frac{{\partial^{2} u_{0} }}{{\partial t^{2} }} + I_{1} \frac{{\partial^{2} \theta_{x} }}{{\partial t^{2} }}$$
(19-1)
$$\delta v_{0}: \frac{{\partial N_{xy} }}{\partial x} + \frac{{\partial N_{yy} }}{\partial y} = I_{0} \frac{{\partial^{2} v_{0} }}{{\partial t^{2} }} + I_{1} \frac{{\partial^{2} \theta_{y} }}{{\partial t^{2} }}$$
(19-2)
$$\delta w_{0}: \frac{{\partial Q_{xx} }}{\partial x} + \frac{{\partial Q_{yy} }}{\partial y} + q = I_{0} \frac{{\partial^{2} w_{0} }}{{\partial t^{2} }}$$
(19-3)
$$\delta \theta_{x}: \frac{{\partial M_{xx} }}{\partial x} + \frac{{\partial M_{xy} }}{\partial y} - Q_{xx} = I_{1} \frac{{\partial^{2} u_{0} }}{{\partial t^{2} }} + I_{2} \frac{{\partial^{2} \theta_{x} }}{{\partial t^{2} }}$$
(19-4)
$$\delta \theta_{y}: \frac{{\partial M_{yy} }}{\partial y} + \frac{{\partial M_{xy} }}{\partial x} - Q_{yy} = I_{1} \frac{{\partial^{2} v_{0} }}{{\partial t^{2} }} + I_{2} \frac{{\partial^{2} \theta_{y} }}{{\partial t^{2} }}$$
(19-5)
$$\delta \phi: \mathop \int \limits_{{ - \frac{h}{2}}}^{{\frac{h}{2}}} \left( {\frac{{\partial D_{x} }}{\partial x}\cos \left( {\frac{\pi z}{h}} \right) + \frac{{\partial D_{y} }}{\partial y}\cos \left( {\frac{\pi z}{h}} \right) + \frac{\pi }{h}D_{z} \sin \left( {\frac{\pi z}{h}} \right)} \right)dz = 0$$
(19-6)
$$\delta \psi: \mathop \int \limits_{{ - \frac{h}{2}}}^{{\frac{h}{2}}} \left( {\frac{{\partial B_{x} }}{\partial x}\cos \left( {\frac{\pi z}{h}} \right) + \frac{{\partial B_{y} }}{\partial y}\cos \left( {\frac{\pi z}{h}} \right) + \frac{\pi }{h}B_{z} \sin \left( {\frac{\pi z}{h}} \right)} \right)dz = 0$$
(19-7)
and the related natural and necessary boundary conditions with normal vectors \(n_{x}\) and \(n_{y}\) are obtained along borders \(x\) and \(y\) as below
$$\delta u_{0} = 0 \quad {\text{or }}N_{xx} n_{x} + N_{xy} n_{y} = 0$$
(20-1)
$$\delta v_{0} = 0\quad {\text{or }}\quad N_{xy} n_{x} + N_{yy} n_{y} = 0$$
(20-2)
$$\delta w_{0} = 0\quad {\text{or}}\quad \left( {Q_{xx} + k_{G} \frac{\partial w}{\partial x}} \right)n_{x} + \left( {Q_{yy} + k_{G} \frac{\partial w}{\partial y}} \right)n_{y} = 0$$
(20-3)
$$\delta \theta_{x} = 0\quad {\text{or}}\quad M_{xx} n_{x} + M_{xy} n_{y} = 0$$
(20-4)
$$\delta \theta_{y} = 0\quad {\text{or}}\quad M_{xy} n_{x} + M_{yy} n_{y} = 0$$
(20-5)
$$\delta \phi = 0\quad {\text{or }}\quad\mathop \int \limits_{{ - \frac{h}{2}}}^{{\frac{h}{2}}} \left[ {n_{x} D_{x} \cos \left( {\frac{\pi z}{h}} \right) + n_{y} D_{y} \cos \left( {\frac{\pi z}{h}} \right)} \right]dz = 0$$
(20-6)
$$\delta \psi = 0\quad {\text{or }}\quad\mathop \int \limits_{{ - \frac{h}{2}}}^{{\frac{h}{2}}} \left[ {n_{x} B_{x} \cos \left( {\frac{\pi z}{h}} \right) + n_{y} B_{y} \cos \left( {\frac{\pi z}{h}} \right)} \right]dz = 0$$
(20-7)
The following nonlocal expressions in terms of displacements can be written by integrating Eq. (6) over the area of nanoplate cross-section
$$\left[ {1 - \left( {e_{0} a} \right)^{2} \nabla^{2} } \right]N_{xx} = A_{11} \frac{{\partial u_{0} }}{\partial x} + A_{12} \frac{{\partial v_{0} }}{\partial y} + B_{11} \frac{{\partial \theta_{x} }}{\partial x} + B_{12} \frac{{\partial \theta_{y} }}{\partial y} + E_{11} \phi + F_{11} \psi + N_{xx1}^{E} + N_{xx1}^{M}$$
(21-1)
$$\left[ {1 - \left( {e_{0} a} \right)^{2} \nabla^{2} } \right]N_{yy} = A_{12} \frac{{\partial u_{0} }}{\partial x} + A_{22} \frac{{\partial v_{0} }}{\partial y} + B_{12} \frac{{\partial \theta_{x} }}{\partial x} + B_{22} \frac{{\partial \theta_{y} }}{\partial y} + G_{11} \phi + H_{11} \psi + N_{yy1}^{E} + N_{yy1}^{M}$$
(21-2)
$$\left[ {1 - \left( {e_{0} a} \right)^{2} \nabla^{2} } \right]N_{xy} = A_{66} \left( {\frac{{\partial u_{0} }}{\partial y} + \frac{{\partial v_{0} }}{\partial x}} \right) + B_{66} \left( {\frac{{\partial \theta_{x} }}{\partial y} + \frac{{\partial \theta_{y} }}{\partial x}} \right)$$
(21-3)
$$\left[ {1 - \left( {e_{0} a} \right)^{2} \nabla^{2} } \right]M_{xx} = B_{11} \frac{{\partial u_{0} }}{\partial x} + B_{12} \frac{{\partial v_{0} }}{\partial y} + D_{11} \frac{{\partial \theta_{x} }}{\partial x} + D_{12} \frac{{\partial \theta_{y} }}{\partial y} + E_{22} \phi + F_{22} \psi + N_{xx2}^{E} + N_{xx2}^{M}$$
(21-4)
$$\left[ {1 - \left( {e_{0} a} \right)^{2} \nabla^{2} } \right]M_{yy} = B_{12} \frac{{\partial u_{0} }}{\partial x} + B_{22} \frac{{\partial v_{0} }}{\partial y} + D_{12} \frac{{\partial \theta_{x} }}{\partial x} + D_{22} \frac{{\partial \theta_{y} }}{\partial y} + G_{22} \phi + H_{22} \psi + N_{yy2}^{E} + N_{yy2}^{M}$$
(21-5)
$$\left[ {1 - \left( {e_{0} a} \right)^{2} \nabla^{2} } \right]M_{xy} = B_{66} \left( {\frac{{\partial u_{0} }}{\partial y} + \frac{{\partial v_{0} }}{\partial x}} \right) + D_{66} \left( {\frac{{\partial \theta_{x} }}{\partial y} + \frac{{\partial \theta_{y} }}{\partial x}} \right)$$
(21-6)
$$\left[ {1 - \left( {e_{0} a} \right)^{2} \nabla^{2} } \right]Q_{xx} = k_{s} A_{55} \left( {\theta_{x} + \frac{{\partial w_{0} }}{\partial x}} \right) - k_{s} J_{2} \frac{\partial \phi }{\partial x} - k_{s} J_{3} \frac{\partial \psi }{\partial x}$$
(21-7)
$$\left[ {1 - \left( {e_{0} a} \right)^{2} \nabla^{2} } \right]Q_{yy} = k_{s} A_{44} \left( {\theta_{y} + \frac{{\partial w_{0} }}{\partial y}} \right) - k_{s} L_{2} \frac{\partial \phi }{\partial y} - k_{s} L_{3} \frac{\partial \psi }{\partial y}$$
(21-8)
$$\mathop \int \limits_{{ - \frac{h}{2}}}^{{\frac{h}{2}}} \left[ {1 - \left( {e_{0} a} \right)^{2} \nabla^{2} } \right]D_{x} cos\left( {\frac{\pi z}{h}} \right) = J_{2} \left( {\theta_{x} + \frac{{\partial w_{0} }}{\partial x}} \right) + Q_{1} \frac{\partial \phi }{\partial x} + Q_{2} \frac{\partial \psi }{\partial x}$$
(21-9)
$$\mathop \int \limits_{{ - \frac{h}{2}}}^{{\frac{h}{2}}} \left[ {1 - \left( {e_{0} a} \right)^{2} \nabla^{2} } \right]D_{y} cos\left( {\frac{\pi z}{h}} \right) = L_{2} \left( {\theta_{y} + \frac{{\partial w_{0} }}{\partial y}} \right) + X_{1} \frac{\partial \phi }{\partial y} + X_{2} \frac{\partial \psi }{\partial y}$$
(21-10)
$$\mathop \int \limits_{{ - \frac{h}{2}}}^{{\frac{h}{2}}} \left[ {1 - \left( {e_{0} a} \right)^{2} \nabla^{2} } \right]D_{z} \left( {\frac{\pi }{h}} \right)sin\left( {\frac{\pi z}{h}} \right) = E_{11} \frac{{\partial u_{0} }}{\partial x} + G_{11} \frac{{\partial v_{0} }}{\partial y} + E_{22} \frac{{\partial \theta_{x} }}{\partial x} + G_{22} \frac{{\partial \theta_{y} }}{\partial y} - P_{1} \phi - P_{2} \psi - N_{xx3}^{E} + N_{xx3}^{M}$$
(21-11)
$$\mathop \int \limits_{{ - \frac{h}{2}}}^{{\frac{h}{2}}} \left[ {1 - \left( {e_{0} a} \right)^{2} \nabla^{2} } \right]B_{x} cos\left( {\frac{\pi z}{h}} \right) = J_{3} \left( {\theta_{x} + \frac{{\partial w_{0} }}{\partial x}} \right) + Q_{2} \frac{\partial \phi }{\partial x} + Q_{3} \frac{\partial \psi }{\partial x}$$
(21-12)
$$\mathop \int \limits_{{ - \frac{h}{2}}}^{{\frac{h}{2}}} \left[ {1 - \left( {e_{0} a} \right)^{2} \nabla^{2} } \right]B_{y} cos\left( {\frac{\pi z}{h}} \right) = L_{3} \left( {\theta_{y} + \frac{{\partial w_{0} }}{\partial y}} \right) + X_{2} \frac{\partial \phi }{\partial y} + X_{3} \frac{\partial \psi }{\partial y}$$
(21-13)
$$\mathop \int \limits_{{ - \frac{h}{2}}}^{{\frac{h}{2}}} \left[ {1 - \left( {e_{0} a} \right)^{2} \nabla^{2} } \right]B_{z} \left( {\frac{\pi }{h}} \right)sin\left( {\frac{\pi z}{h}} \right) = F_{11} \frac{{\partial u_{0} }}{\partial x} + H_{11} \frac{{\partial v_{0} }}{\partial y} + F_{22} \frac{{\partial \theta_{x} }}{\partial x} + H_{22} \frac{{\partial \theta_{y} }}{\partial y} - P_{2} \phi - P_{3} \psi - N_{yy3}^{E} + N_{yy3}^{M}$$
(21-14)
where parameters \(N_{xxi}^{E} , N_{xxi}^{M} , N_{yyi}^{E} , N_{yyi}^{M} , A_{ij} ,B_{ij} , D_{ij} , E_{ij} ,F_{ij} ,G_{ij} ,H_{ij} , J_{i} , {\text{L}}_{i} , Q_{i} , X_{i}\) and \(P_{i}\) are represented in integral forms, which are provided in appendix A. By substituting Eqs. (21-121-14) into (19-119-7), the neoclassical governing equations of FG-METE Mindlin nanoplate in terms of displacements can be obtained as:
$$\begin{aligned} &\delta u_{0}: A_{11} \frac{{\partial^{2} u_{0} }}{{\partial x^{2} }} + A_{12} \frac{{\partial^{2} v_{0} }}{\partial x\partial y} + B_{13} \frac{{\partial^{2} \theta_{x} }}{{\partial x^{2} }} + B_{12} \frac{{\partial^{2} \theta_{y} }}{\partial x\partial y} + E_{11} \frac{\partial \phi }{\partial x} + F_{11} \frac{\partial \psi }{\partial x} \hfill \\ &\quad+ A_{66} \left( {\frac{{\partial^{2} u_{0} }}{{\partial y^{2} }} + \frac{{\partial^{2} v_{0} }}{\partial x\partial y}} \right) + B_{66} \left( {\frac{{\partial^{2} \theta_{x} }}{{\partial y^{2} }} + \frac{{\partial^{2} \theta_{y} }}{\partial x\partial y}} \right) = \left[ {1 - \left( {e_{0} a} \right)^{2} \nabla^{2} } \right]\left( {I_{0} \frac{{\partial^{2} u_{0} }}{{\partial t^{2} }} + I_{1} \frac{{\partial^{2} \theta_{x} }}{{\partial t^{2} }}} \right) \hfill \\ \end{aligned}$$
(22)
$$\begin{aligned}& \delta v_{0}: A_{66} \left( {\frac{{\partial^{2} v_{0} }}{{\partial x^{2} }} + \frac{{\partial^{2} u_{0} }}{\partial x\partial y}} \right) + B_{66} \left( {\frac{{\partial^{2} \theta_{x} }}{\partial x\partial y} + \frac{{\partial^{2} \theta_{y} }}{{\partial x^{2} }}} \right) + A_{12} \frac{{\partial^{2} u_{0} }}{\partial x\partial y} + A_{22} \frac{{\partial^{2} v_{0} }}{{\partial y^{2} }} + B_{12} \frac{{\partial^{2} \theta_{x} }}{\partial x\partial y} \hfill \\ &\quad+ B_{22} \frac{{\partial^{2} \theta_{y} }}{{\partial y^{2} }} + G_{11} \frac{\partial \phi }{\partial y} + H_{11} \frac{\partial \psi }{\partial y} = \left[ {1 - \left( {e_{0} a} \right)^{2} \nabla^{2} } \right]\left( {I_{0} \frac{{\partial^{2} v_{0} }}{{\partial t^{2} }} + I_{1} \frac{{\partial^{2} \theta_{y} }}{{\partial t^{2} }}} \right) \hfill \\ \end{aligned}$$
(23)
$$\begin{aligned} &\delta w_{0}: k_{s} A_{55} \left( {\frac{{\partial \theta_{x} }}{\partial x} + \frac{{\partial^{2} w_{0} }}{{\partial x^{2} }}} \right) + k_{s} A_{44} \left( {\frac{{\partial \theta_{y} }}{\partial y} + \frac{{\partial^{2} w_{0} }}{{\partial y^{2} }}} \right) - k_{s} J_{2} \frac{{\partial^{2} \phi }}{{\partial x^{2} }} - k_{s} J_{3} \frac{{\partial^{2} \psi }}{{\partial x^{2} }} - k_{s} L_{2} \frac{{\partial^{2} \phi }}{{\partial y^{2} }} - k_{s} L_{3} \frac{{\partial^{2} \psi }}{{\partial y^{2} }} \hfill \\ &\quad\quad+ \left[ {1 - \left( {e_{0} a} \right)^{2} \nabla^{2} } \right]\left[ {\left( {N_{Ex} + N_{Mx} } \right)\frac{{\partial^{2} w_{0} }}{{\partial x^{2} }} + \left( {N_{Ey} + N_{My} } \right)\frac{{\partial^{2} w_{0} }}{{\partial y^{2} }} - k_{w} w_{0} + k_{G} \left( {\frac{{\partial^{2} w_{0} }}{{\partial x^{2} }} + \frac{{\partial^{2} w_{0} }}{{\partial y^{2} }}} \right)} \right] \hfill \\ &\quad= \left[ {1 - \left( {e_{0} a} \right)^{2} \nabla^{2} } \right]\left[ {\left( {I_{0} + \mathop \sum \limits_{e = 1}^{p} m_{e} \delta \left( {x - x_{e} } \right)\delta \left( {y - y_{e} } \right)} \right)\frac{{\partial^{2} w_{0} }}{{\partial t^{2} }}} \right] \hfill \\ \end{aligned}$$
(24)
$$\begin{aligned}& \delta \theta_{x}: B_{11} \frac{{\partial^{2} u_{0} }}{{\partial x^{2} }} + B_{12} \frac{{\partial^{2} v_{0} }}{\partial x\partial y} + D_{11} \frac{{\partial^{2} \theta_{x} }}{{\partial x^{2} }} + D_{12} \frac{{\partial^{2} \theta_{y} }}{\partial x\partial y} \hfill \\ &\quad+ \left( {E_{22} + k_{s} J_{2} } \right)\frac{\partial \phi }{\partial x} + \left( {F_{22} + k_{s} J_{3} } \right)\frac{\partial \psi }{\partial x} + B_{66} \left( {\frac{{\partial^{2} u_{0} }}{{\partial y^{2} }} + \frac{{\partial^{2} v_{0} }}{\partial x\partial y}} \right) \hfill \\&\quad + D_{66} \left( {\frac{{\partial^{2} \theta_{x} }}{{\partial y^{2} }} + \frac{{\partial^{2} \theta_{y} }}{\partial x\partial y}} \right) - k_{s} A_{55} \left( {\theta_{x} + \frac{{\partial w_{0} }}{\partial x}} \right) = \left[ {1 - \left( {e_{0} a} \right)^{2} \nabla^{2} } \right]\left( {I_{1} \frac{{\partial^{2} u_{0} }}{{\partial t^{2} }} + I_{2} \frac{{\partial^{2} \theta_{x} }}{{\partial t^{2} }}} \right) \hfill \\ \end{aligned}$$
(25)
$$\begin{aligned} &\delta \theta_{y}: B_{12} \frac{{\partial^{2} u_{0} }}{\partial x\partial y} + B_{22} \frac{{\partial^{2} v_{0} }}{{\partial y^{2} }} + D_{12} \frac{{\partial^{2} \theta_{x} }}{\partial x\partial y} + D_{22} \frac{{\partial^{2} \theta_{y} }}{{\partial y^{2} }} + \left( {G_{22} + k_{s} L_{2} } \right)\frac{\partial \phi }{\partial y} \hfill \\ &\quad+ \left( {H_{22} + k_{s} L_{3} } \right)\frac{\partial \psi }{\partial y} + B_{66} \left( {\frac{{\partial^{2} u_{0} }}{\partial x\partial y} + \frac{{\partial^{2} v_{0} }}{{\partial x^{2} }}} \right) \hfill \\ &\quad+ D_{66} \left( {\frac{{\partial^{2} \theta_{x} }}{\partial x\partial y} + \frac{{\partial^{2} \theta_{y} }}{{\partial x^{2} }}} \right) - k_{s} A_{44} \left( {\theta_{y} + \frac{{\partial w_{0} }}{\partial y}} \right) = \left[ {1 - \left( {e_{0} a} \right)^{2} \nabla^{2} } \right]\left( {I_{1} \frac{{\partial^{2} v_{0} }}{{\partial t^{2} }} + I_{2} \frac{{\partial^{2} \theta_{y} }}{{\partial t^{2} }}} \right) \hfill \\ \end{aligned}$$
(26)
$$\begin{aligned}& \delta \phi: J_{2} \left( {\frac{{\partial \theta_{x} }}{\partial x} + \frac{{\partial^{2} w_{0} }}{{\partial x^{2} }}} \right) + L_{2} \left( {\frac{{\partial \theta_{y} }}{\partial y} + \frac{{\partial^{2} w_{0} }}{{\partial y^{2} }}} \right) + Q_{1} \frac{{\partial^{2} \phi }}{{\partial x^{2} }} + Q_{2} \frac{{\partial^{2} \psi }}{{\partial x^{2} }} \hfill \\ &\quad+ X_{1} \frac{{\partial^{2} \phi }}{{\partial y^{2} }} + X_{2} \frac{{\partial^{2} \psi }}{{\partial y^{2} }} + E_{11} \frac{{\partial u_{0} }}{\partial x} + G_{11} \frac{{\partial v_{0} }}{\partial y} + E_{22} \frac{{\partial \theta_{x} }}{\partial x} + G_{22} \frac{{\partial \theta_{y} }}{\partial y} - P_{1} \phi - P_{2} \psi = 0 \hfill \\ \end{aligned}$$
(27)
$$\begin{aligned} &\delta \psi: J_{3} \left( {\frac{{\partial \theta_{x} }}{\partial x} + \frac{{\partial^{2} w_{0} }}{{\partial x^{2} }}} \right) + L_{3} \left( {\frac{{\partial \theta_{y} }}{\partial y} + \frac{{\partial^{2} w_{0} }}{{\partial y^{2} }}} \right) + Q_{2} \frac{{\partial^{2} \phi }}{{\partial x^{2} }} + Q_{3} \frac{{\partial^{2} \psi }}{{\partial x^{2} }} \hfill \\ &\quad+ X_{2} \frac{{\partial^{2} \phi }}{{\partial y^{2} }} + X_{3} \frac{{\partial^{2} \psi }}{{\partial y^{2} }} + F_{11} \frac{{\partial u_{0} }}{\partial x} + H_{11} \frac{{\partial v_{0} }}{\partial y} + F_{22} \frac{{\partial \theta_{x} }}{\partial x} + H_{22} \frac{{\partial \theta_{y} }}{\partial y} - P_{2} \phi - P_{3} \psi = 0 \hfill \\ \end{aligned}$$
(28)

2.5 Dimensionless expressions of equilibrium equations

By suggesting the following non-dimensional phrases, the linear equations (2228) can be rewritten in a non-dimensional form
$$\begin{aligned} X = \frac{x}{{l_{x} }}, \;\;Y = \frac{y}{{l_{y} }}, \;\;\bar{w} = \frac{{w_{0} }}{h}, \;\bar{\phi } = \frac{\phi }{{\bar{\phi }_{0} }},\;\; \bar{\psi } = \frac{\psi }{{\bar{\psi }_{0} }},\;\; \bar{u} = \frac{{u_{0} }}{h},\;\; \bar{v} = \frac{{v_{0} }}{h},\;\; \bar{\phi }_{0} = \sqrt {\frac{{A_{11}^{0} }}{{P_{1} }}} , \hfill \\ \bar{\psi }_{0} = \sqrt {\frac{{A_{11}^{0} }}{{P_{3} }}} , \;\;r = \frac{{l_{x} }}{h}, \;\;R = \frac{{l_{x} }}{{l_{y} }},\;\; \tau^{2} = \frac{{t^{2} A_{11}^{0} }}{{l_{x}^{2} I_{0}^{0} }},\;\; \bar{K}_{w} = \frac{{k_{w} l_{x}^{2} }}{{A_{11}^{0} }}, \;\;\bar{K}_{G} = \frac{{k_{G} }}{{A_{11}^{0} }}, \;\;\mu = \frac{{e_{0} a}}{{l_{x} }} \hfill \\ \end{aligned}$$
(29)
where \(A_{11}^{0} = \left( {\bar{C}_{11} } \right)_{{{\text{BaTiO}}_{3} }} h\) and \(I_{0}^{0} = \left( \rho \right)_{{{\text{BaTiO}}_{3} }} h\). After applying the above dimensionless phrases into Eqs. (2228), the non-dimensional form of the constitutive equations of FG-MEE nanoplate can be introduced as
$$\begin{aligned}& \delta\bar u: \bar{A}_{11} \frac{{\partial^{2} \bar{u}}}{{\partial X^{2} }} + \bar{A}_{12} R\frac{{\partial^{2} \bar{v}}}{\partial X\partial Y} + \bar{B}_{11} \frac{{\partial^{2} \theta_{x} }}{{\partial X^{2} }} + \bar{B}_{12} R\frac{{\partial^{2} \theta_{y} }}{\partial X\partial Y} \hfill \\ &\quad + \bar{E}_{11} r\frac{{\partial \bar{\phi }}}{\partial X} + \bar{F}_{11} r\frac{{\partial \bar{\psi }}}{\partial X} + \bar{A}_{66} \left( {R^{2} \frac{{\partial^{2} \bar{u}}}{{\partial Y^{2} }} + R\frac{{\partial^{2} \bar{v}}}{\partial X\partial Y}} \right) \hfill \\ &\quad+ \bar{B}_{66} \left( {R^{2} \frac{{\partial^{2} \theta_{x} }}{{\partial Y^{2} }} + R\frac{{\partial^{2} \theta_{y} }}{\partial X\partial Y}} \right) = \left[ {1 - \mu^{2} \left( {\frac{{\partial^{2} }}{{\partial X^{2} }} + R^{2} \frac{{\partial^{2} }}{{\partial Y^{2} }}} \right)} \right]\left( {\frac{{I_{0} }}{{I_{0}^{0} }}\frac{{\partial^{2} {\bar{\text{u}}}}}{{\partial \tau^{2} }} + \frac{{I_{1} }}{{I_{0}^{0} h}}\frac{{\partial^{2} \theta_{x} }}{{\partial \tau^{2} }}} \right) \hfill \\ \end{aligned}$$
(30)
$$\begin{aligned}& \delta\bar v: \bar{A}_{66} \left( {\frac{{\partial^{2} \bar{v}}}{{\partial X^{2} }} + R\frac{{\partial^{2} \bar{u}}}{\partial X\partial Y}} \right) + \bar{B}_{66} \left( {R\frac{{\partial^{2} \theta_{x} }}{\partial X\partial Y} + \frac{{\partial^{2} \theta_{y} }}{{\partial X^{2} }}} \right) + \bar{A}_{12} R\frac{{\partial^{2} \bar{u}}}{\partial X\partial Y} + \bar{A}_{22} R^{2} \frac{{\partial^{2} \bar{v}}}{{\partial Y^{2} }} \hfill \\ &\quad+ \bar{B}_{12} R\frac{{\partial^{2} \theta_{x} }}{\partial X\partial Y} + \bar{B}_{22} R^{2} \frac{{\partial^{2} \theta_{y} }}{{\partial Y^{2} }} + \bar{G}_{11} Rr\frac{{\partial \bar{\phi }}}{\partial Y} \hfill \\ &\quad+ \bar{H}_{11} Rr\frac{{\partial \bar{\psi }}}{\partial Y} = \left[ {1 - \mu^{2} \left( {\frac{{\partial^{2} }}{{\partial X^{2} }} + R^{2} \frac{{\partial^{2} }}{{\partial Y^{2} }}} \right)} \right]\left( {\frac{{I_{0} }}{{I_{0}^{0} }}\frac{{\partial^{2} \bar{v}}}{{\partial \tau^{2} }} + \frac{{I_{1} }}{{I_{0}^{0} h}}\frac{{\partial^{2} \theta_{y} }}{{\partial \tau^{2} }}} \right) \hfill \\ \end{aligned}$$
(31)
$$\begin{aligned}& \delta \bar w: k_{s} \bar{A}_{55} \left( {\frac{{\partial \theta_{x} }}{\partial X} + \frac{1}{r}\frac{{\partial^{2} \bar{w}}}{{\partial X^{2} }}} \right) + k_{s} \bar{A}_{44} \left( {R\frac{{\partial \theta_{y} }}{\partial Y} + \frac{{R^{2} }}{r}\frac{{\partial^{2} \bar{w}}}{{\partial Y^{2} }}} \right) - k_{s} \frac{{\bar{J}_{2} }}{r}\frac{{\partial^{2} \bar{\phi }}}{{\partial X^{2} }} - k_{s} \frac{{\bar{J}_{3} }}{r}\frac{{\partial^{2} \bar{\psi }}}{{\partial X^{2} }} \hfill \\ &\quad\quad- k_{s} \frac{{\bar{L}_{2} R^{2} }}{r}\frac{{\partial^{2} \bar{\phi }}}{{\partial Y^{2} }} - k_{s} \frac{{\bar{L}_{3} R^{2} }}{r}\frac{{\partial^{2} \bar{\psi }}}{{\partial Y^{2} }} + \frac{1}{r}\left[ {1 - \mu^{2} \left( {\frac{{\partial^{2} }}{{\partial X^{2} }} + R^{2} \frac{{\partial^{2} }}{{\partial Y^{2} }}} \right)} \right] \hfill \\ &\quad \quad\left[ {\frac{{\left( {N_{Ex} + N_{Mx} } \right)}}{{A_{11}^{0} }}\frac{{\partial^{2} \bar{w}}}{{\partial X^{2} }} + \frac{{\left( {N_{Ey} + N_{My} } \right)R^{2} }}{{A_{11}^{0} }}\frac{{\partial^{2} w_{0} }}{{\partial y^{2} }} - \bar{K}_{w} \bar{w} + \bar{K}_{G} \left( {\frac{{\partial^{2} \bar{w}}}{{\partial X^{2} }} + R^{2} \frac{{\partial^{2} \bar{w}}}{{\partial Y^{2} }}} \right)} \right] \hfill \\ &\quad= \frac{1}{r}\left[ {1 - \mu^{2} \left( {\frac{{\partial^{2} }}{{\partial X^{2} }} + R^{2} \frac{{\partial^{2} }}{{\partial Y^{2} }}} \right)} \right]\left[ {\left( {\frac{{I_{0} }}{{I_{0}^{0} }} + \mathop \sum \limits_{e = 1}^{p} \frac{{m_{e} }}{{I_{0}^{0} l_{x} l_{y} }}\delta \left( {X - X_{e} } \right)\delta \left( {Y - Y_{e} } \right)} \right)\frac{{\partial^{2} \bar{w}}}{{\partial \tau^{2} }}} \right] \hfill \\ \end{aligned}$$
(32)
$$\begin{aligned} &\delta \theta_{x}: \bar{B}_{11} \frac{{\partial^{2} \bar{u}}}{{\partial X^{2} }} + \bar{B}_{12} R\frac{{\partial^{2} \bar{v}}}{\partial X\partial Y} + \bar{D}_{11} \frac{{\partial^{2} \theta_{x} }}{{\partial X^{2} }} + \bar{D}_{12} R\frac{{\partial^{2} \theta_{y} }}{\partial X\partial Y} + \left( {\bar{E}_{22} + k_{s} \bar{J}_{2} } \right)r\frac{{\partial \bar{\phi }}}{\partial X} \hfill \\ &\quad+ \left( {\bar{F}_{22} + k_{s} \bar{J}_{3} } \right)r\frac{{\partial \bar{\psi }}}{\partial X} + \bar{B}_{66} \left( {R^{2} \frac{{\partial^{2} \bar{u}}}{{\partial Y^{2} }} + R\frac{{\partial^{2} \bar{v}}}{\partial X\partial Y}} \right) + \bar{D}_{66} \left( {R^{2} \frac{{\partial^{2} \theta_{x} }}{{\partial Y^{2} }} + R\frac{{\partial^{2} \theta_{y} }}{\partial X\partial Y}} \right) \hfill \\ &\quad- k_{s} \bar{A}_{55} \left( {r^{2} \theta_{x} + r\frac{{\partial \bar{w}}}{\partial X}} \right) = \left[ {1 - \mu^{2} \left( {\frac{{\partial^{2} }}{{\partial X^{2} }} + R^{2} \frac{{\partial^{2} }}{{\partial Y^{2} }}} \right)} \right]\left( {\frac{{I_{1} }}{{I_{0}^{0} h}}\frac{{\partial^{2} \bar{v}}}{{\partial \tau^{2} }} + \frac{{I_{2} }}{{I_{0}^{0} h^{2} }}\frac{{\partial^{2} \theta_{x} }}{{\partial \tau^{2} }}} \right) \hfill \\ \end{aligned}$$
(33)
$$\begin{aligned} &\delta \theta_{y}: \bar{B}_{12} R\frac{{\partial^{2}\bar u}}{\partial X\partial Y} + \bar{B}_{22} R^{2} \frac{{\partial^{2} \bar{v}}}{{\partial Y^{2} }} + \bar{D}_{12} R\frac{{\partial^{2} \theta_{x} }}{\partial X\partial Y} \hfill \\ &\quad\quad+ \bar{D}_{22} R^{2} \frac{{\partial^{2} \theta_{y} }}{{\partial Y^{2} }} + \left( {\bar{G}_{22} + k_{s} \bar{L}_{2} } \right)Rr\frac{{\partial \bar{\phi }}}{\partial Y} + \left( {\bar{H}_{22} + k_{s} \bar{L}_{3} } \right)Rr\frac{{\partial \bar{\psi }}}{\partial Y} + \bar{B}_{66} \left( {R\frac{{\partial^{2} \bar{u}}}{\partial X\partial Y} + \frac{{\partial^{2} \bar{v}}}{{\partial X^{2} }}} \right) \hfill \\ &\quad\quad+ \bar{D}_{66} \left( {R\frac{{\partial^{2} \theta_{x} }}{\partial X\partial Y} + \frac{{\partial^{2} \theta_{y} }}{{\partial X^{2} }}} \right) - k_{s} \bar{A}_{44} \left( {r^{2} \theta_{y} + rR\frac{{\partial \bar{w}}}{\partial Y}} \right) \hfill \\ &\quad= \left[ {1 - \mu^{2} \left( {\frac{{\partial^{2} }}{{\partial X^{2} }} + R^{2} \frac{{\partial^{2} }}{{\partial Y^{2} }}} \right)} \right]\left( {\frac{{I_{1} }}{{I_{0}^{0} h}}\frac{{\partial^{2} \bar{v}}}{{\partial \tau^{2} }} + \frac{{I_{2} }}{{I_{0}^{0} h^{2} }}\frac{{\partial^{2} \theta_{y} }}{{\partial \tau^{2} }}} \right) \hfill \\ \end{aligned}$$
(34)
$$\begin{aligned} &\delta \bar\phi: \bar{J}_{2} \left( {\frac{{\partial \theta_{x} }}{\partial X} + \frac{1}{r}\frac{{\partial^{2} \bar{w}}}{{\partial X^{2} }}} \right) + \bar{L}_{2} \left( {R\frac{{\partial \theta_{y} }}{\partial Y} + \frac{{R^{2} }}{r}\frac{{\partial^{2} \bar{w}}}{{\partial Y^{2} }}} \right) + \frac{1}{r}\bar{Q}_{1} \frac{{\partial^{2} \bar{\phi }}}{{\partial X^{2} }} + \frac{1}{r}\bar{Q}_{2} \frac{{\partial^{2} \bar{\psi }}}{{\partial X^{2} }} \hfill \\ &\quad+ \frac{{R^{2} }}{r}\bar{X}_{1} \frac{{\partial^{2} \bar{\phi }}}{{\partial Y^{2} }} + \frac{{R^{2} }}{r}\bar{X}_{2} \frac{{\partial^{2} \bar{\psi }}}{{\partial Y^{2} }} + \bar{E}_{11} \frac{{\partial \bar{u}}}{\partial X} \hfill \\ &\quad+ \bar{G}_{11} R\frac{{\partial \bar{v}}}{\partial Y} + \bar{E}_{22} \frac{{\partial \theta_{x} }}{\partial X} + \bar{G}_{22} R\frac{{\partial \theta_{y} }}{\partial Y} - \bar{P}_{1} r\bar{\phi } - \bar{P}_{2} r\bar{\psi } = 0 \hfill \\ \end{aligned}$$
(35)
$$\begin{aligned} &\delta \bar\psi: \bar{J}_{3} \left( {\frac{{\partial \theta_{x} }}{\partial X} + \frac{1}{r}\frac{{\partial^{2} \bar{w}}}{{\partial X^{2} }}} \right) + \bar{L}_{3} \left( {R\frac{{\partial \theta_{y} }}{\partial Y} + \frac{{R^{2} }}{r}\frac{{\partial^{2} \bar{w}}}{{\partial Y^{2} }}} \right) + \frac{1}{r}\bar{Q}_{2} \frac{{\partial^{2} \bar{\phi }}}{{\partial X^{2} }} \hfill \\ &\quad+ \frac{1}{r}\bar{Q}_{3} \frac{{\partial^{2} \bar{\psi }}}{{\partial X^{2} }} + \frac{{R^{2} }}{r}\bar{X}_{2} \frac{{\partial^{2} \bar{\phi }}}{{\partial Y^{2} }} + \frac{{R^{2} }}{r}\bar{X}_{3} \frac{{\partial^{2} \bar{\psi }}}{{\partial Y^{2} }} \hfill \\ &\quad+ \bar{F}_{11} \frac{{\partial \bar{u}}}{\partial X} + \bar{H}_{11} R\frac{{\partial \bar{v}}}{\partial Y} + \bar{F}_{22} \frac{{\partial \theta_{x} }}{\partial X} + \bar{H}_{22} R\frac{{\partial \theta_{y} }}{\partial Y} - \bar{P}_{2} r\bar{\phi } - \bar{P}_{3} r\bar\psi = 0 \hfill \\ \end{aligned}$$
(36)
in which
$$\begin{aligned} \left\{ {\bar{A}_{11} ,\bar{A}_{12} ,\bar{A}_{22} ,\bar{A}_{44} ,\bar{A}_{55} ,\bar{A}_{66} } \right\} &= \left\{ {\frac{{A_{11} }}{{A_{11}^{0} }}, \frac{{A_{12} }}{{A_{11}^{0} }}, \frac{{A_{22} }}{{A_{11}^{0} }}, \frac{{A_{44} }}{{A_{11}^{0} }}, \frac{{A_{55} }}{{A_{11}^{0} }}, \frac{{A_{66} }}{{A_{11}^{0} }}} \right\}, \hfill \\ \left\{ {\bar{B}_{11} ,\bar{B}_{12} ,\bar{B}_{22} ,\bar{B}_{66} } \right\} &= \left\{ {\frac{{B_{11} }}{{A_{11}^{0} h}}, \frac{{B_{12} }}{{A_{11}^{0} h}}, \frac{{B_{22} }}{{A_{11}^{0} h}}, \frac{{B_{66} }}{{A_{11}^{0} h}}} \right\}, \hfill \\ \left\{ {\bar{D}_{11} ,\bar{D}_{12} ,\bar{D}_{22} ,\bar{D}_{66} } \right\} &= \left\{ {\frac{{D_{11} }}{{A_{11}^{0} h^{2} }}, \frac{{D_{12} }}{{A_{11}^{0} h^{2} }}, \frac{{D_{22} }}{{A_{11}^{0} h^{2} }}, \frac{{D_{66} }}{{A_{11}^{0} h^{2} }}} \right\}, \hfill \\ \left\{ {\bar{E}_{11} ,\bar{E}_{22} } \right\} &= \left\{ {\frac{{E_{11} \bar{\phi }_{0} }}{{A_{11}^{0} }}, \frac{{E_{22} \bar{\phi }_{0} }}{{A_{11}^{0} h}}} \right\},\;\;\; \left\{ {\bar{F}_{11} ,\bar{F}_{22} } \right\} = \left\{ {\frac{{F_{11} \bar{\psi }_{0} }}{{A_{11}^{0} }}, \frac{{F_{22} \bar{\psi }_{0} }}{{A_{11}^{0} h}}} \right\}, \hfill \\ \left\{ {\bar{G}_{11} ,\bar{G}_{22} } \right\} &= \left\{ {\frac{{G_{11} \bar{\phi }_{0} }}{{A_{11}^{0} }}, \frac{{G_{22} \bar{\phi }_{0} }}{{A_{11}^{0} h}}} \right\},\;\; \left\{ {\bar{H}_{11} ,\bar{H}_{22} } \right\} = \left\{ {\frac{{H_{11} \bar{\psi }_{0} }}{{A_{11}^{0} }}, \frac{{H_{22} \bar{\psi }_{0} }}{{A_{11}^{0} h}}} \right\}, \hfill \\ \left\{ {\bar{J}_{2} ,\bar{J}_{3} } \right\} &= \left\{ {\frac{{J_{2} \bar{\phi }_{0} }}{{A_{11}^{0} h}}, \frac{{J_{3} \bar{\psi }_{0} }}{{A_{11}^{0} h}}} \right\},\;\;\; \left\{ {\bar{L}_{2} ,\bar{L}_{3} } \right\} = \left\{ {\frac{{L_{2} \bar{\phi }_{0} }}{{A_{11}^{0} h}}, \frac{{L_{3} \bar{\psi }_{0} }}{{A_{11}^{0} h}}} \right\}, \hfill \\ \left\{ {\bar{Q}_{1} ,\bar{X}_{1} ,\bar{P}_{1} } \right\} &= \left\{ {\frac{{Q_{1} \bar{\phi }_{0}^{2} }}{{A_{11}^{0} h^{2} }},\frac{{X_{1} \bar{\phi }_{0}^{2} }}{{A_{11}^{0} h^{2} }},\frac{{P_{1} \bar{\phi }_{0}^{2} }}{{A_{11}^{0} }}} \right\}, \hfill \\ \left\{ {\bar{Q}_{2} ,\bar{X}_{2} ,\bar{P}_{2} } \right\} &= \left\{ {\frac{{Q_{2} \bar{\phi }_{0} \bar{\psi }_{0} }}{{A_{11}^{0} h^{2} }},\frac{{X_{2} \bar{\phi }_{0} \bar{\psi }_{0} }}{{A_{11}^{0} h^{2} }},\frac{{P_{2} \bar{\phi }_{0} \bar{\psi }_{0} }}{{A_{11}^{0} }}} \right\}, \hfill \\ \left\{ {\bar{Q}_{3} ,\bar{X}_{3} ,\bar{P}_{3} } \right\} &= \left\{ {\frac{{Q_{3} \bar{\psi }_{0}^{2} }}{{A_{11}^{0} h^{2} }},\frac{{X_{3} \bar{\psi }_{0}^{2} }}{{A_{11}^{0} h^{2} }},\frac{{P_{3} \bar{\psi }_{0}^{2} }}{{A_{11}^{0} }}} \right\} \hfill \\ \end{aligned}$$
(37)

3 Analytical solution for vibration behavior of FG-MEE nanoplates

In the following section, the simply supported boundary conditions are considered on each the four ends of rectangular nanoplate. Thus, the deflections and moments at each end of FG-MEE nanoplate are assumed to be zero. Also, the short circuit type of edge supports which is used for the actuator case is selected for the electric and magnetic potentials (Ansari and Gholami 2016). On the basis of the short circuit type, the electric potential and magnetic potential are supposed to be zero at the ends of FG-MEE nanoplate. Overall, the related boundary conditions can be expressed as
$$\bar{u}\left( {X,0,\tau } \right) = \bar{u}\left( {X,1,\tau } \right) = \bar{v}\left( {0,Y,\tau } \right) = \bar{v}\left( {1,Y,\tau } \right) = 0$$
(38-1)
$$\bar{w}\left( {X,0,\tau } \right) = \bar{w}\left( {X,1,\tau } \right) = \bar{w}\left( {0,Y,\tau } \right) = \bar{w}\left( {1,Y,\tau } \right) = 0$$
(38-2)
$$\bar{\theta }_{x} \left( {X,0,\tau } \right) = \bar{\theta }_{x} \left( {X,1,\tau } \right) = \bar{\theta }_{y} \left( {0,Y,\tau } \right) = \bar{\theta }_{y} \left( {1,Y,\tau } \right) = 0$$
(38-3)
$$\bar{\phi }\left( {X,0,\tau } \right) = \bar{\phi }\left( {X,1,\tau } \right) = \bar{\psi }\left( {0,Y,\tau } \right) = \bar{\psi }\left( {1,Y,\tau } \right) = 0$$
(38-4)
$$\bar{\phi }\left( {0,Y,\tau } \right) = \bar{\phi }\left( {1,Y,\tau } \right) = \bar{\psi }\left( {X,0,\tau } \right) = \bar{\psi }\left( {X,1,\tau } \right) = 0$$
(38-5)
$$M_{xx} \left( {0,Y,\tau } \right) = M_{xx} \left( {1,Y,\tau } \right) = M_{yy} \left( {X,0,\tau } \right) = M_{yy} \left( {X,1,\tau } \right) = 0.$$
(38-6)
However, the other boundary conditions of engineering interest such as clamped and free boundary conditions can be expressed as
$$\begin{aligned} &\text{Clamped:}\\& \bar{u} = \bar{v} = \bar{\theta }_{x} = \bar{\theta }_{y} = \bar{\phi } = \bar{\psi } = 0 \quad {\text{at}}\quad X = 0,1; Y = 0,1\end{aligned}$$
(39)
Free:
$$M_{xx} = M_{xy} = Q_{xx} = \bar{\phi } = \bar{\psi } = 0\quad {\text{at}}\;X = 0,1$$
(40-1)
$$M_{yy} = M_{xy} = Q_{yy} = \bar{\phi } = \bar{\psi } = 0\quad {\text{at}}\;Y = 0,1.$$
(40-2)

To find more detail about analytically solving governing equations for clamped and free boundary conditions, the reader is referred to (Sobhy 2013; Ebrahimi et al. 2017).

According to Eqs. (38-138-6), an analytical approach, so-called Navier solution, is considered to solve dimensionless governing Eqs. (3036) and obtain the natural frequency and shift frequency. Based on Navier solution and with regards to relation (38-138-6), the solutions are presented in the double Fourier series as follows
$$\bar{u}\left( {X,Y,\tau } \right) = e^{i\varOmega \tau } \mathop \sum \limits_{m = 1}^{\infty } \mathop \sum \limits_{n = 1}^{\infty } \overline{\overline{U}} { \cos }(m\pi X){ \sin }\left( {n\pi Y} \right)$$
(41-1)
$$\bar{v}\left( {X,Y,\tau } \right) = e^{i\varOmega \tau } \mathop \sum \limits_{m = 1}^{\infty } \mathop \sum \limits_{n = 1}^{\infty } \overline{\overline{V}} { \sin }(m\pi X){ \cos }\left( {n\pi Y} \right)$$
(41-2)
$$\bar{w}\left( {X,Y,\tau } \right) = e^{i\varOmega \tau } \mathop \sum \limits_{m = 1}^{\infty } \mathop \sum \limits_{n = 1}^{\infty } \overline{\overline{W}} { \sin }(m\pi X){ \sin }\left( {n\pi Y} \right)$$
(41-3)
$$\bar{\theta }_{x} \left( {X,Y,\tau } \right) = e^{i\varOmega \tau } \mathop \sum \limits_{m = 1}^{\infty } \mathop \sum \limits_{n = 1}^{\infty } \overline{\overline{\theta }}_{x} { \cos }(m\pi X){ \sin }\left( {n\pi Y} \right)$$
(41-4)
$$\bar{\theta }_{y} \left( {X,Y,\tau } \right) = e^{i\varOmega \tau } \mathop \sum \limits_{m = 1}^{\infty } \mathop \sum \limits_{n = 1}^{\infty } \overline{\overline{\theta }}_{y} { \sin }(m\pi X){ \cos }\left( {n\pi Y} \right)$$
(41-5)
$$\bar{\phi }\left( {X,Y,\tau } \right) = e^{i\varOmega \tau } \mathop \sum \limits_{m = 1}^{\infty } \mathop \sum \limits_{n = 1}^{\infty } \overline{\overline{\phi }} { \sin }(m\pi X){ \sin }\left( {n\pi Y} \right)$$
(41-6)
$$\bar{\psi }\left( {X,Y,\tau } \right) = e^{i\varOmega \tau } \mathop \sum \limits_{m = 1}^{\infty } \mathop \sum \limits_{n = 1}^{\infty } \overline{\overline{\psi }} { \sin }(m\pi X){ \sin }\left( {n\pi Y} \right)$$
(41-7)
in which \(i = \sqrt { - 1}\) and \(\overline{\overline{U}} , \overline{\overline{V}} , \overline{\overline{W}} , \overline{\overline{\theta }}_{x} , \overline{\overline{\theta }}_{y} , \overline{\overline{\phi }} , \overline{\overline{\psi }}\) are the unknown coefficients. Also, the half wave numbers through \(X\) and \(Y\) directions are denoted by symbols \(m\) and \(n\), respectively. Furthermore, term \(\varOmega\) refers to the non-dimensional natural frequency of FG-MEE nanoplate. After applying assumed solutions into Eqs. (3036), and next multiplying both sides of the resulted error equations by \(\sin \left( {\frac{m\pi }{{l_{x} }}X} \right)\sin \left( {\frac{n\pi }{{l_{y} }}Y} \right)\) and integrating over the FG-MEE nanoplate area, yields the following constitutive matrix equation
$$\left\{ {\left[ M \right]\varOmega^{2} - \left[ K \right]} \right\}\left[ \varLambda \right] = 0$$
(42)
in which \(\left[ M \right]\) and \(\left[ K \right]\) demonstrate the mass and stiffness matrices, respectively, and \(\left[ \varLambda \right]\) shows the field variables vector, which are as
$$\begin{aligned} \left[ M \right] & = \left[ {\begin{array}{*{20}c} {M_{11} } & 0 & 0 & {M_{14} } & 0 & 0 & 0 \\ 0 & {M_{22} } & 0 & 0 & {M_{25} } & 0 & 0 \\ 0 & 0 & {M_{33} } & 0 & 0 & 0 & 0 \\ {M_{41} } & 0 & 0 & {M_{44} } & 0 & 0 & 0 \\ 0 & {M_{52} } & 0 & 0 & {M_{55} } & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right], \\ \left[ K \right] & = \left[ {\begin{array}{*{20}c} {K_{11} } & {K_{12} } & 0 & {K_{14} } & {K_{15} } & {K_{16} } & {K_{17} } \\ {K_{21} } & {K_{22} } & 0 & {K_{24} } & {K_{25} } & {K_{26} } & {K_{27} } \\ 0 & 0 & {K_{33} } & {K_{34} } & {K_{35} } & {K_{36} } & {K_{37} } \\ {K_{41} } & {K_{42} } & {K_{43} } & {K_{44} } & {K_{45} } & {K_{46} } & {K_{47} } \\ {K_{51} } & {K_{52} } & {K_{53} } & {K_{54} } & {K_{55} } & {K_{56} } & {K_{57} } \\ {K_{61} } & {K_{62} } & {K_{63} } & {K_{64} } & {K_{65} } & {K_{66} } & {K_{67} } \\ {K_{71} } & {K_{72} } & {K_{73} } & {K_{74} } & {K_{75} } & {K_{76} } & {K_{77} } \\ \end{array} } \right], \\ \left[ \varLambda \right] & = \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\overline{\overline{U}} } \\ {\overline{\overline{V}} } \\ {\overline{\overline{W}} } \\ \end{array} } \\ {\overline{\overline{\theta }}_{x} } \\ {\overline{\overline{\theta }}_{y} } \\ \end{array} } \\ {\overline{\overline{\phi }} } \\ {\overline{\overline{\psi }} } \\ \end{array} } \right] \\ \end{aligned}$$
(43)
where the elements coefficients of the matrices \(\left[ M \right]\) and \(\left[ K \right]\) are expressed in Appendix B. After calculating the non-trivial solution of equation (42), the natural frequencies can be obtained. For a non-trivial solution, the determinant of the coefficient matrix \(\left\{ {\left[ M \right]\varOmega^{2} - \left[ K \right]} \right\}\) must be equal to zero, which yields the eigenfrequencies of the system. Moreover, without loss of generality, the resonance frequency with and without considering the effect of attached nanomasses can be respectively presented as \(f_{mn} = \frac{{\varOmega_{mn} }}{2\pi }\) and \(f_{0,mn} = \frac{{\varOmega_{0,mn} }}{2\pi }\). Now, the natural frequency shift can be defined as follows
$$\Delta f_{mn} = f_{0,mn} - f_{mn} .$$
(44)

4 Numerical examples and discussions

4.1 Convergence and comparison study

Before embarking on the investigation of the variations of the suggested system frequency shift, some convergence analyses to determine the accuracy of the developed model are performed. Firstly, after eliminating Pasternak substrate and FGMs, the calculated the first two non-dimensional natural frequencies of MEE nanoplate for different nonlocal parameters based on current study are illustrated in Table 1 along with predictions according to the nonlocal Mindlin plate theory (MPT) (Ansari and Gholami 2016) and nonlocal Kirchhoff plate theory (KPT) (Ke et al. 2014). It is clear from Table 1 that proposed model results are in great agreement with those reported by (Ke et al. 2014; Ansari and Gholami 2016).
Table 1

Comparison study of the first two dimensionless natural frequencies of MEE nanoplate

  

\(\mu = 0\)

\(\mu = 0.2\)

\(\mu = 0.4\)

\(\omega_{11}\)

KPT (Ke et al. 2014)

0.3698

0.2764

0.1813

MPT(DQM) (Ansari and Gholami 2016)

0.3684

0.2756

0.1808

Present

0.3683

0.2755

0.1806

\(\omega_{12}\)

KPT (Ke et al. 2014)

0.9247

0.5362

0.3100

MPT(DQM) (Ansari and Gholami 2016)

0.9108

0.5284

0.3055

Present

0.9107

0.5283

0.3053

Another comparison study is also carried out through Fig. 2 in which the fundamental frequency shift calculated according to the present model for different values of nonlocal parameter and different added particles at location \(x_{1} = 0.5l_{x} , y_{1} = 0.5l_{y}\) without considering FGMs, Pasternak medium, and piezomagnetic effect, are compared to those of piezoelectric Kirchhoff nanoplate model of Asemi et al. (2015). Figure 2 denotes an excellent agreement between the outcomes of current paper and the results of Ref. (Asemi et al. 2015).
Fig. 2

Comparison study between the fundamental frequency shift predicted by present model and those reported in Ref. Asemi et al. (2015) (color figure online)

4.2 Benchmark results

In this subsection, the benchmark results are presented in the tabular and graphical form to better understanding and explaining the significant parameters such as the position of single or multiple attached masses on the natural frequency shift changes. To achieve this goal, we consider the three different configurations (single mass, two masses, and four masses) for the number of added particles. It is assumed that the total mass of added masses is identical for each state. However, the details of these three cases are given in Table 2. Furthermore, it is supposed that FG-MEE nanoplate to be composed of barium titanate (BaTiO3) and cobalt iron oxide (CoFe2O4) with effective material characteristics given in Table 3 (Aboudi 2001). Moreover, the utilized geometrical characteristics of nanoplate are assumed to be as \(h = 1\;{\text{nm}}, l_{x} = l_{y} = 10\;{\text{nm}}\), and half wave numbers are assumed to be \(m = n = 1\).
Table 2

The details of the three different configurations of attached nanoparticles

Pictorial representation

\(m_{Total}\)

Position \((X_{e} , Y_{e} )\)

Open image in new window

\(m_{1} = m_{Total}\)

\((0.5, 0.5)\)

Open image in new window

\(m_{1} = \frac{{m_{Total} }}{2}\)

\(m_{2} = \frac{{m_{Total} }}{2}\)

\((0.25, 0.5)\)

\((0.75, 0.5)\)

Open image in new window

\(m_{1} = \frac{{m_{Total} }}{4}\)

\(m_{2} = \frac{{m_{Total} }}{4}\)

\(m_{3} = \frac{{m_{Total} }}{4}\)

\(m_{4} = \frac{{m_{Total} }}{4}\)

\((0.25, 0.25)\)

\((0.25, 0.75)\)

\((0.75, 0.25)\)

\((0.75, 0.75)\)

Table 3

Material properties of MEE (physical constants are set in italic type and their units are given in parentheses)

Properties

BaTiO3

CoFe2O4

C 11 (\({\text{GPa}}\))

166

286

C 12

77

173

C 13

78

170.5

C 22

166

286

C 23

78

170.5

C 33

162

269.5

C 44

43

45.3

C 55

43

45.3

C 66

44.5

56.5

e 31 (C/m2)

− 4.4

0

e 32

− 4.4

0

e 24

11.6

0

e 15

11.6

0

e 33

18.6

0

q 31 (N/Am)

0

580.3

q 32

0

580.3

q 24

0

550

q 15

0

550

q 33

0

699.7

η 11 (10−9 C2/Nm2)

11.2

0.08

η 22

11.2

0.08

η 33

12.6

0.093

d 11 (s/m)

0

0

d 22

0

0

d 33

0

0

μ 11 (10−6 Ns2/C2)

5

− 590

μ 22

5

− 590

μ 33

10

157

ρ (kg/m3)

5800

5300

Initially, to study the influences of length to thickness ratio and material property gradient index on the changes of frequency shift, Table 4 is provided for the three figurations of the attached nanoparticles. Here, we took \(\bar{K}_{w} = \bar{K}_{G} = \phi_{0} =\psi_{0} = 0, \mu = 0.2, m_{Total} = 10^{ - 21} \;{\text{kg}}, {\text{R}} = 1\). It is clear that an increase in the volume fraction of barium titanate part (BaTiO3)/cobalt iron oxide component (CoFe2O4) leads to decreasing/increasing the bending stiffness of FG-MEE nanoplate, and consequently, frequency shift or sensibility efficiency of nanosensor decrease/increases. Furthermore, another remarkable point derived from this table is that the sensitivity property of the nanosensor reduces by increasing the number of added masses. Also, it can be found that frequency shift significantly decreases as the length to thickness ratio increases.
Table 4

The effect of length to thickness ratio and material property gradient index on the changes of frequency shift

\(\frac{{l_{x} }}{h}\)

Number of mass

Power law index, K

Fully BaTiO3

0.5

1

10

Fully CoFe2O4

5

Single mass

0.5600

0.5937

0.6020

0.6341

0.6494

Two masses

0.5090

0.5401

0.5482

0.5785

0.5905

Four masses

0.4413

0.4697

0.4773

0.5052

0.5162

10

Single mass

0.2428

0.2604

0.2656

0.2828

0.2896

Two masses

0.1979

0.2129

0.2175

0.2326

0.2384

Four masses

0.1480

0.1599

0.1637

0.1760

0.1805

15

Single mass

0.1292

0.1393

0.1425

0.1528

0.1567

Two masses

0.0952

0.1031

0.1056

0.1138

0.1169

Four masses

0.0633

0.0689

0.0708

0.0767

0.0788

Figure 3 gives a comparison of variations of frequency shift against to the total attached mass for different values of the nonlocal parameter, \(\bar{K}_{w} = \bar{K}_{G} = \phi_{0} = \psi_{0} = 0, {\text{R}} = {\text{K}} = 1\). The most important distinctions are as follows. It is observed that by taking into account the large values of the nonlocal parameter, the sensitivity performance of nanosensor reduces, particularly for large values of total attached mass (mainly due to the softening effect of the nonlocal parameter on the nanostructures). Furthermore, it is demonstrated that increasing the total attached mass value from \(10^{ - 21}\) to \(10^{ - 19} {\text{kg}}\) leads to a notable increase in the frequency shift of FG-MEE nanoplate. Moreover, it can be resulted from this figure that the influence of various figurations of attached nanoparticles on the frequency shift become lower for large values of the total attached mass and the nonlocal parameter. It is also found that for large values of nonlocal parameter, FG-MEE nanosensor is insensitive to the variation of the total attached mass.
Fig. 3

Coupled effects of the nonlocal parameter and number of added nanoparticles on the fundamental frequency shift versus different values of the total attached mass (color figure online)

Depicted in Fig. 4 is the influence of the aspect ratio on the sensing property of FG-MEE nanosensor for different values of the total attached mass, \(\bar{K}_{w} = \bar{K}_{G} = \phi_{0} = \psi_{0} = 0, \mu = 0.2, {\text{K}} = 1\). As can be seen, an increase in the aspect ratio causes to monotonically decreasing the frequency shift of FG-MEE nanoplate and the ability of nanosensor to sense attached particles reduces. Also, it is clear that the case of single mass has the maximum value of frequency shift.
Fig. 4

Changes of the fundamental frequency shift of nanosensor as a function of total attached mass for various values of aspect ratio (color figure online)

To study the effects of the transverse and shear stiffness parameters of Pasternak elastic substrate on the sensitivity performance of FG-MEE nanosensor, Figure 5 is plotted. Here, we did not consider the external electric and external magnetic potentials, and the nonlocal parameter is supposed to be 0.2. In particular, it is easily deduced that by taking into account stiffer polymer matrix, sensitivity property of nanosensor increases. Furthermore, it is noticed that under the maximum value of the total attached mass, the influence of Pasternak foundation on the frequency shift increases.
Fig. 5

The effect of Pasternak medium coefficients on the variation of the sensitivity performance of FG-MEE nanosensor (color figure online)

Figure 6 reveals the influence of the applied external electric voltage on the frequency shift curves versus the total attached mass for the three cases of mass, \(\bar{K}_{w} = \bar{K}_{G} = \psi_{0} = 0, \mu = 0.2, {\text{K}} = {\text{R}} = 1\). The most interesting result is perhaps the fact that a rise in the value of the electric voltage form \(- 0.2\) to \(0.05\;{\text{V}}\) causes a decrease in the sensitivity behavior of FG-MEE nanosensor. This may be explained by the fact that imposing the positive (negative) applied external electric potential, the compressive (tensile) in-plane loads tend to generate in FG-MEE nanosensor, and consequently, frequency shift reduces (increases). Furthermore, it is found from figure that when FG-MEE nanosensor is subjected to the positive external electric potential, the influence of the total attached mass on the sensitivity property remarkably become low.
Fig. 6

Coupled effects of external electric voltage and number of attached nanoparticles and the total attached mass of the nanoparticles on the fundamental frequency shift (color figure online)

The frequency shift curves of FG-MEE nanosensor with respect to the total attached mass for different values of external magnetic potential are displayed in Fig. 7. Here, we took \(\bar{K}_{w} = \bar{K}_{G} = \psi_{0} = 0, \mu = 0.2, K = {\text{R}} = 1\). It is illustrated that contrary to the influence of external electric voltage on the variation of frequency shift, by increasing external magnetic potential from \(- 0.01\) to \(0.02\;{\text{A}}\), the sensitivity property of FG-MEE nanoplate according to the mass sensor considerably increases. Also, it should be noted that when FG-MEE nanosensor is subjected to the negative external magnetic potential, the effect of the total attached mass on the sensitivity property significantly become low.
Fig. 7

Coupled effects of external magnetic potential and number of attached nanoparticles and the total attached mass of the nanoparticles on the fundamental frequency shift (color figure online)

Figure 8a–d show the effect of the higher modes on the difference between the natural frequency of FG-MEE nanoplate with and without added nanoparticle corresponding to the different positions of single attached nanoparticle. The results are calculated for \(\bar{K}_{w} = \bar{K}_{G} =\psi_{0} = \phi_{0} = 0, \mu = 0.2, K = {\text{R}} = 1, m_{Total} = 10^{ - 20} {\text{kg}}\). It can be seen that the sensitivity property of nanosensor increases when the attached nanoparticle is closer to the center of FG-MEE nanoplate. This implies that the sensitivity property of nanosensor vanishes when added nanoparticle is located at the edges of FG-MEE nanoplate. Moreover, it is denoted that by increasing mode number the frequency shift remarkably increases.
Fig. 8

The influence of attached single nanoparticle location on the sensitivity performance of FG-MEE nanosensor; (a) \(m = n = 1\), (b) \(m = 1, n = 2\), (c) \(m = n = 2\), (d) \(m = n = 3\) (color figure online)

5 Concluding remarks

The main contribution of the current study was to provide a nanosize mass detection model basis on the vibrating FG-MEE Mindlin nanoplate resting on Pasternak foundation. Also, external electric voltage and external magnetic potential were exerted to the FG-MEE nanosensor. Eringen’s differential constitutive law was employed to take into account small-scale effect into the constitutive equations. By using the power law distribution, the effective material properties of rectangular FG-MEE nanoplate change gradually in the direction of nanoplate thickness. Hamilton’s principle was utilized in conjunction with nonlocal Mindlin plate theory to derive the size-dependent governing partial differential equations of motion. Afterward, outlined discussions were depicted to demonstrate how to change the sensitivity property of FG-MEE nanosensor by varying the position of attached nanoparticle, the number of nanoparticles, the values of the nonlocal parameter, initial external electric and magnetic potentials, material property gradient index, the dimension of the FG-MEE nanoplate, and Pasternak substrate coefficients. Some noteworthy points derived from numerical results can be listed as
  • The difference between the natural frequency of FG-MEE nanoplate with and without attached nanoparticle notable increases by increasing external magnetic potential.

  • The sensitivity performance of FG-MEE nanosensor reduces under an increase of external electric voltage.

  • An increase in the value of the volume fraction exponent causes higher frequency shift.

  • It is demonstrated that nonlocal component has a considerable reduction on the sensibility of FG-MEE nanosensor.

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Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  • M. Hosseini
    • 1
  • M. R. Mofidi
    • 1
  • A. Jamalpoor
    • 2
  • M. Safi Jahanshahi
    • 1
  1. 1.Department of Mechanical EngineeringSirjan University of TechnologySirjanIran
  2. 2.Department of Mechanical EngineeringIran University of Science and TechnologyTehranIran

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