Dynamic Analysis of a Piezoelectrically Layered Perforated Nonlocal Strain Gradient Nanobeam with Flexoelectricity

Abstract

This study presents a mathematical size-dependent model capable of investigating the dynamic behavior of a sandwich perforated nanobeam incorporating the flexoelectricity effect. The nonlocal strain gradient elasticity theory is developed for both continuum mechanics and flexoelectricity. Closed forms of the equivalent perforated geometrical variables are developed. The Hamiltonian principle is exploited to derive the governing equation of motion of the sandwich beam including the flexoelectric effect. Closed forms for the eigen values are derived for different boundary conditions. The accuracy of the developed model is verified by comparing the obtained results with the available published results. Parametric studies are conducted to explore the effects of the perforation parameters, geometric dimensions, nonclassical parameters, flexoelectric parameters, as well as the piezoelectric parameters on the vibration behavior of a piezoelectric perforated sandwich nanobeam. The obtained results demonstrate that both the flexoelectric and piezoelectric parameters increased the vibration frequency of the nanobeam. The nonlocal parameter reduced the natural vibration frequency due to a decrease in the stiffness of the structures. However, the strain gradient parameter increased the stiffness of the structures and hence increased the natural vibration frequency. The natural vibration frequency based on the NSGT can be increased or decreased, depending on the ration of the value of the nonlocal parameter to the strain gradient parameter. This model can be employed in the analysis and design of NEMS, nanosensors, and nanoactuators.

1. Introduction

In recent years, piezoelectric material, a special type of material where both the electrical and the elastic mechanical fields are coupled, has been used extensively for smart structures (sensor transistors, medical equipment, nano-electronic equipment, energy harvesters, energy absorbents, MEMS, NEMS, etc.) in the mechanical, chemistry, and aerospace industries, according to Ghobadi et al. [1]. Therefore, many scientists directed their research to the nanotechnology and nanoscience of piezoelectric materials [2]. The scale of nanostructures is close to their interatomic distances, which were not included in classical mechanics theory [3]. To study and investigate the mechanical responses of a nanostructure accurately, modified continuum model theories such as nonlocal elasticity, couple stress theory, strain gradient theory, and nonlocal strain gradient theory (NLSGT) have been applied [4].

Based on nonlocal strain gradient theory, Lim et al. [5] developed a higher-order model which couples strain gradients and nonlocal stresses to investigate the wave propagation of nanostructures. Li et al. [6] examined analytically the longitudinal vibration of size-dependent nanorods in the frame of nonlocal strain gradient theory. Corresponding to the elastodynamic problem, Apuzzo et al. [7] analyzed analytically the axial and flexural free vibrations of Bernoulli–Euler nanobeams by using NLSGT. Esfahani et al. [8] illustrated the nonlinear vibration of an electrostatic FG NLSGT nanoresonator with surface effects. Fakher et al. [9] developed the exact solution as well as a finite-element model to study the transverse vibrations of nanobeams within the framework of the two-phase local and nonlocal strain gradient theory. Abdelrahman et al. [10,11] studied the free and forced vibration of a nonlocal strain gradient nanobeam under a moving load by Timoshenko and higher-order shear deformation theories. Alazwari et al. [12] investigated the effect of multiphysics thermo-magnetio-mechanical loads on the dynamic response of functionally graded (FG) NLSGT nanobeams. Daikh et al. [13,14] illustrated the size and microstructure effects on the mechanical responses of multilayer FG carbon nanotube-reinforced composite NLSGT nanoplate by using the Galerkin approach. Shariati et al. [15] presented a size parameter calibration related to non-classical continuum theories by utilizing molecular dynamics simulations. Esen et al. [16] developed a mathematical model to study the vibration and buckling of FG-NLSGT nanobeams exposed to magnetic and thermal fields. Esen et al. [17,18] presented a comprehensive analysis of the dynamic response of NLSGT nanobeams under a moving load with sigmoid or symmetric FG gradation reinforced by carbon nanotubes. Wang and Zhang [19] examined the thermal buckling and postbuckling of temperature-dependent graphene platelet-reinforced porous nanocomposite beams. Hamidi et al. [20] exploited the NLSGT and surface energy to investigate the torsional vibration of nanobeams under the distributed external torque and moving external harmonic torque. Nuhu and Safaei [21] presented vibration analysis of small-sized structures by using nonclassical continuum theories of elasticity.

For piezoelectric structures, Anton and Sodano [22] presented a comprehensive review on power harvesting using piezoelectric materials. Ke et al. [23] examined the nonlinear vibration of piezoelectric nonlocal Timoshenko nanobeams under an applied voltage and thermal load. Arani et al. [24] developed an analytical solution to study the wave propagation of piezoelectric nanobeam systems using the nonlocal elasticity theory of Eringen. Liang et al. [25] proved the presence of strong effects of surface and flexoelectricity on the bending behavior of Euler–Bernoulli piezoelectric nanobeams. Arani et al. [26] investigated analytically the nonlinear vibration of a piezoelectric nanobeam via strain gradient theory. Li et al. [27] introduced a new formulation for flexoelectric theory by splitting the strain gradient tensor into two independent tensors with two modes. Mehralian and Beni [28] predicted the vibration bahvior of a size-dependent bimorph FG piezoelectric NLSGT shell structure. Eltaher et al. [3] studied the influence of the surface energy properties on the bending and vibrational response of a piezoelectric nonlocal nanobeam by using the Gurtin–Murdoch model. Eltaher et al. [29,30] investigated the influence of the size scale on the static deflection and natural frequency of a perforated nanobeam by using the nonlocal elasticity and surface energy theories. Kachapi et al. [31] studied the nonlinear vibration and stability response of a piezo-harmo-electrostatic nanoresonator cylindrical nanoshell in the frame of NLSGT and the Gurtin–Murdoch surface energy. Ghobadi et al. [1] developed a continuous size-dependent electromechanical model to study the nonlinear free vibration of a strain gradient Kirchhoff plate via a modified flexoelectric theory. Ali et al. [32] presented the impact of the viscoelastic properties of the bonding layer on the performance of piezoelectric actuators attached to elastic structures by using the finite element method. Mechkour [33] developed a two-scale homogenization technique of the elastic-electric coupling equation with rapidly oscillating coefficients in a periodically perforated piezoelectric body. Saeed [34] exploited the hybrid Laplace transform and finite element methods to study thermo-elastic interactions in a piezo-thermo-elastic medium.

For sandwich piezoelectric nanobeams, Qi et al. [35] presented the size-dependent and flexoelectricity effects on the bending of an electro-elastic bilayer nanobeam. Ray [36] and Sidhardh and Ray [37] derived exact solutions for the bending responses of nanobeams integrated with a flexoelectric layer. Shu et al. [38] presented a brief report about the recent discoveries in flexoelectricity, focusing on the flexoelectric materials, the theoretical developments, and their applications. Ebrahimi et al. [39] developed an analytical wave dispersion of advanced piezoelectric double-layered nanobeam systems. Wang et al. [40] presented analysis of an array of flexoelectric layered nanobeams in consideration of the surface effect used in vibration energy harvesting. Sobamowo [41] studied the nonlinear vibration of a piezoelectric nanobeam embedded in multiple layers of elastic media in a thermo-magnetic environment.

Based on the above literature, Investigation of the size-dependent vibration behavior of piezoelectrically layered perforated sandwich nanobeams with flexoelectricity has not been handled elsewhere. This work is devoted to filling this gap and developing a size-dependent model based on the strain gradient elasticity theory to study and analyze the vibration behavior of piezoelectrically layered perforated sandwich nanobeams with flexoelectricity for the first time. A regularly squared perforation pattern is assumed for the perforated core. An equivalent geometrical model is developed for the equivalent geometrical variables due to the perforation process. The dynamic equations of motion are derived using the Hamiltonian principle. Analytical solutions for the natural frequencies for different boundary conditions are derived. The developed methodology is verified and compared. Numerical results are obtained and discussed. Finally, the concluding remarks are summarized.

2. Mathematical Formulation

Due to the large areas of applicability of the piezoelectric smart nanobeam structure in modern control technology, it is necessary to study and analyze its electrostatic and electrodynamic behaviors. Consider a sandwich piezoelectric nanobeam composed of a perforated core layered with two piezoelectric layers, as shown in Figure 1. The core layer is made of the non-piezoelectric material having a Young’s modulus E_c and a height h_c. Both the upper and bottom layers are made of the same piezoelectric material with a height h_p, while the perforated core layer and the piezoelectric layers have the same length L and the same width W_b. The total thickness of the entire beam is denoted as h, and h = h_c + 2h_p. The polarization direction of both the piezoelectric upper and bottom layers is upward. The applied electric fields on the upper and bottom layers are in opposite directions. The x-axis is directed along the length direction, and the z-axis is along the thickness direction. Due to the symmetry of the nanobeam in the thickness direction, the neutral plane of the nanobeam is coincident with the longitudinal x-axis. The basic governing equations that describe the physical phenomena are presented in the following subsections.

2.1. Equivalent Geometrical Model

The equivalent geometrical variables are obtained based on a regular squared perforation pattern. As depicted in Figure 1, the spatial period is l_s, the hole side is l_s−t_s, and the number of holes throughout the beam in the section is N. The perforation filling ratio α can be expressed as [35,36].

$$\alpha =\frac{t_s}{l_s} 0\le \alpha \le 1, \alpha =\left\{\begin{array}{c}0 \mathrm{Fully} \mathrm{perforated}\\ 1 \mathrm{Fully} \mathrm{filled} \end{array} \right.$$

(1)

where t_s and l_s are the spatial period and the period length, respectively.

The equivalent bending and shear stiffness of the perforated beam are given by

$$\frac{{\left(EI\right)}_{eq}}{{\left(EI\right)}_s}=\left\{\frac{\alpha \left(N+1\right)\left(N^2+2N+{\alpha }^2\right)}{\left(1-{\alpha }^2+{\alpha }^3\right)N^3+3\alpha N^2+\left(3+2\alpha -3{\alpha }^2+{\alpha }^3\right){\alpha }^{2}N+{\alpha }^3}\right\}$$

(2a)

$$\frac{{\left(GA\right)}_{eq}}{{\left(EA\right)}_s}=\left(\frac{\left(1+N\right){\alpha }^3}{2N}\right)$$

(2b)

where (EI)_eq, (EI)_s are the equivalent bending stiffnesses of the perforated and solid beams, respectively, and (GA)_eq, (GA)_s are the equivalent shear stiffnesses of the perforated and solid beams, respectively. The equivalent mass per unit length (ρA)_eq and equivalent first moment of inertia (ρI)_eq are given by

$$\frac{{\left(\rho A\right)}_{eq}}{{\left(\rho A\right)}_s}=\left\{\frac{\left[1-N\left(\alpha -2\right)\right]\alpha }{N+\alpha }\right\}$$

(3a)

$$\frac{{\left(\rho I\right)}_{eq}}{{\left(\rho I\right)}_s}=\left\{\frac{\alpha \left[\left(2-\alpha \right)N^3+3N^2-2\left(\alpha -3\right)\left({\alpha }^2-\alpha +1\right)N+{\alpha }^2+1\right]}{{\left(N+\alpha \right)}^3}\right\}$$

(3b)

2.2. Kinematic Relations

In the context of the nonlocal strain gradient in the vibration analysis of piezoelectric materials incorporating the flexoelectricity effect, the vibration behavior is dependent on both the strain field and the electric field. Based on the Euler–Bernoulli beam theory, the displacement and the electric fields are given by [32,42,43,44]

$$\left\{\begin{array}{c}u\left(x,z,t\right)\\ w\left(x,z,t\right)\\ E_z\end{array}\right\}=\left\{\begin{array}{c}-z\frac{\partial w\left(x,t\right)}{\partial x}\\ w\left(x,t\right)\\ -\frac{\partial {\varphi }_z\left(z,t\right)}{\partial z}\end{array}\right\}$$

(4)

where the mid-plane subfunctions w and ϕ_z denote the transverse displacement and the electric potential field, respectively. The nonzero strain component is given by [44,45]

$${\epsilon }_{xx}\left(x,z,t\right)=-z\frac{{\partial }^{2}w\left(x,t\right)}{\partial x^2}$$

(5)

Incorporating the electric and flexoelectric effects, the electric enthalpy energy density function is expressed as [25]

$$H=-\frac{1}{2}{a}_{kl}{E}_k{E}_l+\frac{1}{2}{c}_{ijkl}{\epsilon }_{ij}{\epsilon }_{kl}-e_{ijk} E_k {\epsilon }_{ij}-{\mu }_{ijkl} E_i {\epsilon }_{jk,l}$$

(6)

where a_kl denotes the second-order permittivity tensor, c_ijkl refers to the fourth-order elasticity tensor, e_ijk is the piezoelectric coefficient tensor, and μ_ijkl is the electric field strain gradient coupling coefficient, which denotes the higher-order electromechanical coupling induced by the strain gradients.

The electric displacement is related to the electric enthalpy energy density function as follows:

$$D_i=-\frac{\partial H}{\partial E_i}=a_{ij}{E}_j+e_{ijk} {\epsilon }_{jk}+{\mu }_{ijkl} {\epsilon }_{jk,l}$$

(7)

According to the Gaussian theorem, the following condition is verified:

$$\frac{\partial D_z}{\partial z}=0\Rightarrow a_{33}\frac{\partial E_z}{\partial z}+e_{311} \frac{\partial {\epsilon }_{xx}}{\partial z}+{\mu }_{3111}\frac{ {\partial }^2{\epsilon }_{xx}}{\partial x\partial z}+{\mu }_{3113}\frac{ {\partial }^2{\epsilon }_{xx}}{\partial z^2}=0$$

(8)

Substituting Equation (5) into Equation (8) yields

$$a_{33}\frac{\partial E_z}{\partial z}-e_{311} \left(\frac{{\partial }^{2}w\left(x,t\right)}{\partial x^2}\right)-{\mu }_{3111}\left(\frac{{\partial }^{3}w\left(x,t\right)}{\partial x^3}\right)=0$$

(9)

By rearranging the terms in Equation (9), the first derivative of the electric field is given by [44]

$$\frac{\partial E_z}{\partial z}=\frac{1}{a_{33}}\left[e_{311} \left(\frac{{\partial }^{2}w\left(x,t\right)}{\partial x^2}\right)+{\mu }_{3111}\left(\frac{{\partial }^{3}w\left(x,t\right)}{\partial x^3}\right)\right]$$

(10)

By integrating Equation (10) with respect to z, one can write

$$E_z-E_{z0}=\frac{z}{a_{33}}\left[e_{311} \left(\frac{{\partial }^{2}w\left(x,t\right)}{\partial x^2}\right)+{\mu }_{3111}\left(\frac{{\partial }^{3}w\left(x,t\right)}{\partial x^3}\right)\right]$$

(11a)

Equation (11a) can be rewritten as

$$E_z=E_{z0}-\frac{1}{a_{33}}\left[e_{311} \left({\epsilon }_{xx}\right)+{\mu }_{3111}\left(\frac{\partial {\epsilon }_{xx}}{\partial x}\right)\right]$$

(11b)

in which E_z0 denotes the initial electric field in the z direction.

2.3. The Nonclassical Constitutive Relations

Based on the nonlocal strain gradient theory (NSGT), the stress field ${\sigma }_{ij}^t$ accounts for not only the nonlocal elastic stress field but also the strain gradient stress field [30,33]. According to this theory, by incorporating the flexoelectricity effect, the following relations for the components of the stress, the higher-order nonlocal stress, and the nonlocal electric potential are obtained:

$$\left(1-{\left(e_{0}a\right)}^2\frac{{\partial }^2}{\partial x^2}\right){\sigma }_{xx}^t=\left\{E\left(1-l^2\frac{{\partial }^2}{\partial x^2}\right)+\frac{e_{311}^2}{a_{33}}\right\}{\epsilon }_{xx}+\frac{e_{311}}{a_{33}}{\mu }_{3111}\frac{\partial {\epsilon }_{xx}}{\partial x}-e_{311}{E}_{z0}$$

(12a)

$$\left(1-{\left(e_{0}a\right)}^2\frac{{\partial }^2}{\partial x^2}\right){\sigma }_{113}=\frac{{\mu }_{3113}}{a_{33}}\left(e_{311}{\epsilon }_{xx}+{\mu }_{3111}\frac{\partial {\epsilon }_{xx}}{\partial x}\right)-{\mu }_{3113}{E}_{z0}$$

(12b)

$$\left(1-{\left(e_{0}a\right)}^2\frac{{\partial }^2}{\partial x^2}\right){\sigma }_{111}=\frac{{\mu }_{3111}}{a_{33}}\left(e_{311}{\epsilon }_{xx}+{\mu }_{3111}\frac{\partial {\epsilon }_{xx}}{\partial x}\right)-{\mu }_{3111}{E}_{z0}$$

(12c)

$$\left(1-{\left(e_{0}a\right)}^2\frac{{\partial }^2}{\partial x^2}\right)D_z=a_{33}{E}_{z0}+{\mu }_{3113}\frac{\partial {\epsilon }_{xx}}{\partial z}$$

(12d)

3. Dynamic Equation of Motion

The dynamic equation of motion is obtained by performing the Hamiltonian principle as follows:

$$\delta {{\displaystyle \int }}_{t_1}^{t_2}\left[T-{{\displaystyle \int }}_{\mathsf{\Omega }}Hd\mathsf{\Omega }+W_{ex}\right]dt=0$$

(13)

where Ω refers to the volume of the sandwich beam and δT is the variation of the total kinetic energy, which can be expressed as

$$\delta T=\delta \left\{\frac{1}{2}{{\displaystyle \int }}_{\mathsf{\Omega }}\rho \left(z\right)\left({\dot{u}}^2+{\dot{w}}^2\right)d\mathsf{\Omega }\right\}={{\displaystyle \int }}_0^L\left\{\left({\left[{\rho }_c{I}_c\right]}_{eq}+{\rho }_p{I}_p\right)\frac{{\partial }^{4}w}{\partial x^2\partial t^2}-\left({\left[{\rho }_c{A}_c\right]}_{eq}+2{\rho }_p{A}_p\right)\frac{{\partial }^{2}w}{\partial t^2}\right\}\delta wdx$$

(14a)

with

$$I_c=\frac{h_c^3}{12}, I_p=\frac{h^3}{12}-I_c, A_c=h_c{w}_b, A_p=h_p{w}_b$$

(14b)

while δ ∫_Ω HdΩ denotes the variation of the electric enthalpy energy density function, which can be evaluated as follows:

$$\delta {{\displaystyle \int }}_{\mathsf{\Omega }}Hd\mathsf{\Omega }={{\displaystyle \int }}_{\mathsf{\Omega }}\left({\sigma }_{xx}^t\delta {\epsilon }_{xx}-D_z\delta E_z+{\sigma }_{111} \delta {\epsilon }_{xx,x}+{\sigma }_{113} \delta {\epsilon }_{xx,z}\right)d\mathsf{\Omega }={{\displaystyle \int }}_0^L\left(\overline{M}\delta {\epsilon }_{xx0}+{\overline{M}}^{\prime }\delta {\epsilon }_{xx0,x}\right)d\mathrm{x}$$

(15a)

where

$${\epsilon }_{xx0}=-\frac{{\partial }^{2}w}{\partial x^2}, \overline{M}=M+\frac{e_{311}}{a_{33}}{M}_D+N_{113 }, and {\overline{M}}^{\prime }=\frac{{\mu }_{3111}}{a_{33}}{M}_D+N_{111}$$

(15b)

In addition, according to the nonlocal strain gradient elasticity theory, the following relations can be written:

$$\left(1-{\left(e_{0}a\right)}^2\frac{{\partial }^2}{\partial x^2}\right)M=\left(1-l^2\frac{{\partial }^2}{\partial x^2}\right)\{{\left(E_c{I}_c\right)}_{eq}-E_p{I}_p\}{\epsilon }_{xx0}+\frac{e_{311}^2}{a_{33}}{I}_p{\epsilon }_{xx}+\frac{e_{311}{\mu }_{3111}{I}_p}{a_{33}}\frac{\partial {\epsilon }_{xx0}}{\partial x}-e_{311}{I}_1{E}_{z0}$$

(15c)

$$\left(1-{\left(e_{0}a\right)}^2\frac{{\partial }^2}{\partial x^2}\right)N_{113}=0$$

(15d)

$$\left(1-{\left(e_{0}a\right)}^2\frac{{\partial }^2}{\partial x^2}\right)M_{111}=w_b\left\{\frac{{\mu }_{3111}}{a_{33}}{I}_p\left(e_{311}{\epsilon }_{xx0}+{\mu }_{3111}\frac{\partial {\epsilon }_{xx0}}{\partial x}\right)-{\mu }_{3111}{I}_1{E}_{z0}\right\}$$

(15e)

$$\left(1-{\left(e_{0}a\right)}^2\frac{{\partial }^2}{\partial x^2}\right)M_D=w_b{a}_{33}{I}_1{E}_{z0}$$

(15f)

with

$$I_1=\frac{h^2}{4}-\frac{h_c^2}{4}$$

(15g)

The variation of the external work done W_ex is given by

$$\delta W_{ex}=-{{\displaystyle \int }}_0^{L}q\left(x\right)\delta wdx$$

(16)

where q(x) is the applied external uniform distributed load. By substituting Equations (14)–(16) into Equation (13), the governing equation of motion is given by

$$\frac{{\partial }^2\overline{M}}{\partial x^2}-\frac{{\partial }^3{\overline{M}}^{\prime }}{\partial x^3}-q\left(x\right)=\left[{\left({\rho }_c{A}_c\right)}_{eq}+2{\rho }_p{A}_p\right]\frac{{\partial }^{2}w}{\partial t^2}-\left[{\left({\rho }_c{I}_c\right)}_{eq}+{\rho }_p{I}_p\right]\frac{{\partial }^{4}w}{\partial x^2\partial t^2}$$

(17)

For convenience, the governing equations are normalized by introducing

$$W=\frac{w}{h} \mathrm{and} X=\frac{x}{L}$$

(18a)

$$dx=LdX and \frac{dW}{dX}=\left(\frac{L}{h}\right)\frac{dw}{dx}\to \frac{dw}{dx}=\left(\frac{h}{L}\right)\frac{dW}{dX}$$

(18b)

$$\frac{d^{4}w}{d{x}^4}=\frac{d^3}{d{x}^3}\left[\left(\frac{h}{L}\right)\frac{dW}{dX}\right]=\left(\frac{h}{L^4}\right)\frac{d^{2}W}{d{X}^2} \mathrm{and} \frac{d^{6}w}{d{x}^6}=\left(\frac{h}{L^6}\right)\frac{d^{2}W}{d{X}^2}$$

(18c)

Substitution from Equations (15a–g) and (18a–c) into Equation (17) of the following dynamic equation of motion is obtained in terms of the normalized variables as

$$\begin{array}{ll}-[\{{\left(E_c{I}_c\right)}_{eq}+E_p{I}_p\}& +I_p\frac{e_{311}^2}{a_{33}}]\left(\frac{h}{L^4}\right)\left[\frac{{\partial }^{4}W}{\partial X^4}\right]+\left(l^2\left[{\left(E_c{I}_c\right)}_{eq}-E_p{I}_p\right]+I_p\frac{{\mu }_{3111}^2}{a_{33}}\right)\left(\frac{h}{L^6}\right)\frac{{\partial }^{6}W}{\partial X^6}+q\left(x\right)\\ & =h\left[{\left({\rho }_c{A}_c\right)}_{eq}+2{\rho }_p{A}_p\right]\left[\frac{{\partial }^{2}W}{\partial t^2}-\frac{{\left(e_{0}a\right)}^2}{L^2}\frac{{\partial }^{4}W}{\partial X^2\partial t^2}\right]\\ & -\left(\frac{h}{L^2}\right)\left[{\left({\rho }_c{I}_c\right)}_{eq}+{\rho }_p{I}_p\right]\left[\frac{{\partial }^{4}W}{\partial X^2\partial t^2}-\frac{{\left(e_{0}a\right)}^2}{L^2}\frac{{\partial }^{6}W}{\partial X^4\partial t^2}\right]\end{array}$$

(19a)

Neglecting the nonlocality effect, the equation of motion of the sandwich piezoelectric nanobeam, including the microstructure effect, is written as

$$\begin{gathered} -\left[\left\{{\left(E_c{I}_c\right)}_{eq}+E_p{I}_p\right\}+I_p\frac{e_{311}^2}{a_{33}}\right]\left(\frac{h}{L^4}\right)\left[\frac{{\partial }^{4}W}{\partial X^4}\right]+\left(l^2\left[{\left(E_c{I}_c\right)}_{eq}-E_p{I}_p\right]+I_p\frac{{\mu }_{3111}^2}{a_{33}}\right)\left(\frac{h}{L^6}\right)\frac{{\partial }^{6}W}{\partial X^6}+q\left(x\right) \\ =h\left[{\left({\rho }_c{A}_c\right)}_{eq}+2{\rho }_p{A}_p\right]\left[\frac{{\partial }^{2}W}{\partial t^2}\right]-\left(\frac{h}{L^2}\right)\left[{\left({\rho }_c{I}_c\right)}_{eq}+{\rho }_p{I}_p\right]\left[\frac{{\partial }^{4}W}{\partial X^2\partial t^2}\right] \end{gathered}$$

(19b)

Considering only the nonlocality effect, the equation of motion is given in the following form:

$$\begin{gathered} -\left[\left\{{\left(E_c{I}_c\right)}_{eq}+E_p{I}_p\right\}+I_p\frac{e_{311}^2}{a_{33}}\right]\left(\frac{h}{L^4}\right)\left[\frac{{\partial }^{4}W}{\partial X^4}\right]+\left(I_p\frac{{\mu }_{3111}^2}{a_{33}}\right)\left(\frac{h}{L^6}\right)\frac{{\partial }^{6}W}{\partial X^6}+q\left(x\right) \\ =h\left[{\left({\rho }_c{A}_c\right)}_{eq}+2{\rho }_p{A}_p\right]\left[\frac{{\partial }^{2}W}{\partial t^2}-\frac{{\left(e_{0}a\right)}^2}{L^2}\frac{{\partial }^{4}W}{\partial X^2\partial t^2}\right] \\ -\left(\frac{h}{L^2}\right)\left[{\left({\rho }_c{I}_c\right)}_{eq}+{\rho }_p{I}_p\right]\left[\frac{{\partial }^{4}W}{\partial X^2\partial t^2}-\frac{{\left(e_{0}a\right)}^2}{L^2}\frac{{\partial }^{6}W}{\partial X^4\partial t^2}\right] \end{gathered}$$

(19c)

Neglecting the nonclassical effects leads to the classical equation of motion of sandwich piezoelectric beam:

$$\begin{gathered} -\left[\left\{{\left(E_c{I}_c\right)}_{eq}+E_p{I}_p\right\}+I_p\frac{e_{311}^2}{a_{33}}\right]\left(\frac{h}{L^4}\right)\left[\frac{{\partial }^{4}W}{\partial X^4}\right]+\left(I_p\frac{{\mu }_{3111}^2}{a_{33}}\right)\left(\frac{h}{L^6}\right)\frac{{\partial }^{6}W}{\partial X^6}+q\left(x\right) \\ =h\left[{\left({\rho }_c{A}_c\right)}_{eq}+2{\rho }_p{A}_p\right]\left[\frac{{\partial }^{2}W}{\partial t^2}\right]-\left(\frac{h}{L^2}\right)\left[{\left({\rho }_c{I}_c\right)}_{eq}+{\rho }_p{I}_p\right]\left[\frac{{\partial }^{4}W}{\partial X^2\partial t^2}\right] \end{gathered}$$

(19d)

Neglecting the flexoelectric, piezoelectric, and nonclassical effects leads to the classical equation of motion of a perforated beam:

$$-{\left(E_c{I}_c\right)}_{eq}\left(\frac{h}{L^4}\right)\left[\frac{{\partial }^{4}W}{\partial X^4}\right]+q\left(x\right)=h\left[{\left({\rho }_c{A}_c\right)}_{eq}\right]\left[\frac{{\partial }^{2}W}{\partial t^2}\right]-\left(\frac{h}{L^2}\right)\left[{\left({\rho }_c{I}_c\right)}_{eq}\right]\left[\frac{{\partial }^{4}W}{\partial X^2\partial t^2}\right]$$

(19e)

4. Free Vibration Solution

Assume that the solution of Equation (19a–e) is given in the following form:

$$W={{\displaystyle \sum }}_{n=1}^{\infty }{W}_n{\mathsf{\Phi }}_n\left(X\right)e^{i{\omega }_{n}t}$$

(20)

with W_n and Φ_n(X) referring to the undetermined variable and the structural mode shape function, respectively, which must satisfy the boundary conditions i² = −1, ω_n denoting the vibration frequency for each mode. Substituting Equation (20) into Equation (19a) yields

$$\begin{array}{ll}-[\{{\left(E_c{I}_c\right)}_{eq}+E_p{I}_p\}& +I_p\frac{e_{311}^2}{a_{33}}]\left(\frac{h}{L^4}\right)\left[ {\displaystyle {\displaystyle \sum }_{n=1}^{\infty }}{W}_n{\mathsf{\Phi }}_n^{\left(4\right)}{e}^{i{\omega }_{n}t}\right]\\ & +\left(l^2\left[{\left(E_c{I}_c\right)}_{eq}-E_p{I}_p\right]+I_p\frac{{\mu }_{3111}^2}{a_{33}}\right)\left(\frac{h}{L^6}\right){\displaystyle {\displaystyle \sum }_{n=1}^{\infty }}{W}_n{\mathsf{\Phi }}_n^{\left(6\right)}{e}^{i{\omega }_{n}t}+q\left(x\right)\\ & =h\left[{\left({\rho }_c{A}_c\right)}_{eq}+2{\rho }_p{A}_p\right]\left[{\displaystyle {\displaystyle \sum }_{n=1}^{\infty }}-{\omega }_n^2{W}_n{\mathsf{\Phi }}_n{e}^{i{\omega }_{n}t}+\frac{{\left(e_{0}a\right)}^2}{L^2}\left({\omega }_n^2{\displaystyle {\displaystyle \sum }_{n=1}^{\infty }}{W}_n{\mathsf{\Phi }}_n^{\left(2\right)}{e}^{i{\omega }_{n}t}\right)\right]\\ & -\left(\frac{h}{L^2}\right)\left[{\left({\rho }_c{I}_c\right)}_{eq}+{\rho }_p{I}_p\right]\left[\left(-{\omega }_n^2{\displaystyle {\displaystyle \sum }_{n=1}^{\infty }}{W}_n{\mathsf{\Phi }}_n^{\left(2\right)}{e}^{i{\omega }_{n}t}\right)+\frac{{\left(e_{0}a\right)}^2}{L^2}\left({\omega }_n^2{\displaystyle {\displaystyle \sum }_{n=1}^{\infty }}{W}_n{\mathsf{\Phi }}_n^{\left(4\right)}{e}^{i{\omega }_{n}t}\right)\right]\end{array}$$

(21)

The considered boundary conditions must be satisfied by the assumed solution function (Khabaz et al. [46], Kaghazian et al. [47], and Chu et al. [48]) and the corresponding mode shapes function are listed in

Table 1. Boundary conditions and the associated mode shape function.

BCs.	at X = 0	at X = 1	${\mathsf{\Phi }}_{\mathit{n}}\left(\mathit{X}\right)$
SS	$W=\frac{{\partial }^{2}W}{\partial X^2}=0$	$W=\frac{{\partial }^{2}W}{\partial X^2}=0$	${\mathsf{\Phi }}_n\left(X\right)=\sin \left(n\pi X\right)$
CF	$W=\frac{\partial W}{\partial X}=0$	$\frac{{\partial }^{3}W}{\partial X^3}=\frac{{\partial }^{2}W}{\partial X^2}=0$	${\mathsf{\Phi }}_n\left(X\right)=\cosh \left(k_{n}X\right)-\cos \left(k_{n}X\right)-{\beta }_n\left[\sinh \left(k_{n}X\right)-\sin \left(k_{n}X\right)\right]$ ${\beta }_n=\frac{\cosh \left(k_n\right)+\cos \left(k_n\right)}{\sinh \left(k_n\right)+\sin \left(k_n\right)}, \cos \left(k_n\right)\cosh \left(k_n\right)=-1$
CC	$W=\frac{\partial W}{\partial X}=0$	$W=\frac{\partial W}{\partial X}=0$	${\mathsf{\Phi }}_n\left(X\right)=\cosh \left(k_{n}X\right)-\cos \left(k_{n}X\right)-{\beta }_n\left[\sinh \left(k_{n}X\right)-\sin \left(k_{n}X\right)\right]$ ${\beta }_n=\frac{\cosh \left(k_n\right)-\cos \left(k_n\right)}{\sinh \left(k_n\right)-\sin \left(k_n\right)}, \cos \left(k_n\right)\cosh \left(k_n\right)=1$
CS	$W=\frac{\partial W}{\partial X}=0$	$W=\frac{{\partial }^{2}W}{\partial X^2}=0$	${\mathsf{\Phi }}_n\left(X\right)=\cosh \left(k_{n}X\right)-\cos \left(k_{n}X\right)-{\beta }_n\left[\sinh \left(k_{n}X\right)-\sin \left(k_{n}X\right)\right]$ ${\beta }_n=\frac{\cosh \left(k_n\right)-\cos \left(k_n\right)}{\sinh \left(k_n\right)-\sin \left(k_n\right)}, \tan \left(k_n\right)=\tanh \left(k_n\right)$

.

For a simply supported beam SS, the natural frequencies are obtained by direct substitution from Table 1 into Equation (21), and when neglecting the applied external load, one can write

$$\begin{gathered} \left\{-\left[\left\{{\left(E_c{I}_c\right)}_{eq}+E_p{I}_p\right\}+I_p\frac{e_{311}^2}{a_{33}}\right]\left(\frac{h}{L^4}\right){\left(n\pi \right)}^4+\left(l^2\left[{\left(E_c{I}_c\right)}_{eq}-E_p{I}_p\right]-I_p\frac{{\mu }_{3111}^2}{a_{33}}\right)\left(\frac{h}{L^6}\right){\left(n\pi \right)}^4\right\}W \\ =\left\{h\left[{\left({\rho }_c{A}_c\right)}_{eq}-2{\rho }_p{A}_p\right]\left[1+\frac{{\left(e_{0}a\right)}^2}{L^2}\left({\left(n\pi \right)}^2\right)\right] \\ -\left(\frac{h}{L^2}\right)\left[{\left({\rho }_c{I}_c\right)}_{eq}+{\rho }_p{I}_p\right]\left[\left({\left(n\pi \right)}^2\right)+\frac{{\left(e_{0}a\right)}^2}{L^2}\left({\left(n\pi \right)}^4\right)\right]\right\}{\omega }_n^{2}W \end{gathered}$$

(22)

The natural frequencies for the SS are given in the following form:

$${\omega }_n=\sqrt{\frac{\left\{-\left[\left\{{\left(E_c{I}_c\right)}_{eq}+E_p{I}_p\right\}+I_p\frac{e_{311}^2}{a_{33}}\right]\left(\frac{h}{L^4}\right){\left(n\pi \right)}^4+\left(l^2\left[{\left(E_c{I}_c\right)}_{eq}-E_p{I}_p\right]-I_p\frac{{\mu }_{3111}^2}{a_{33}}\right)\left(\frac{h}{L^6}\right){\left(n\pi \right)}^4\right\}}{h\left[{\left({\rho }_c{A}_c\right)}_{eq}-2{\rho }_p{A}_p\right]\left[1+\frac{{\left(e_{0}a\right)}^2}{L^2}\left({\left(n\pi \right)}^2\right)\right]-\left(\frac{h}{L^2}\right)\left[{\left({\rho }_c{I}_c\right)}_{eq}+{\rho }_p{I}_p\right]\left[\left({\left(n\pi \right)}^2\right)+\frac{{\left(e_{0}a\right)}^2}{L^2}\left({\left(n\pi \right)}^4\right)\right]}}$$

(23)

Simplifying Equation (23) yields

$${\omega }_n=\sqrt{\frac{\left({\left(E_c{I}_c\right)}_{eq}+\left(E_p{I}_p\right)+\frac{I_p{e}_{311}^2}{a_{33}}\right)\frac{{\left(n\pi \right)}^4}{L^4}+\left(l^2\left\{{\left(E_c{I}_c\right)}_{eq}+\left(E_p{I}_p\right)\right\}+\frac{I_p{\mu }_{3111}^2}{a_{33}}\right)\frac{{\left(n\pi \right)}^6}{L^6}}{\left\{\left[{\left({\rho }_c{A}_c\right)}_{eq}+2{\rho }_p{A}_p\right]+\left[{\left({\rho }_c{I}_c\right)}_{eq}+{\rho }_p{I}_p\right]{\left(\frac{n\pi }{L}\right)}^2\right\}\left(1+{\left(e_{0}a\right)}^2{\left(\frac{n\pi }{L}\right)}^2\right)}}$$

(24)

Regarding the other boundary conditions, following the same procedure and utilizing Galerkin’s method obtains the solutions for the natural frequencies. Solutions for the natural frequencies are obtained in the following form:

$${\omega }_n=\sqrt{\frac{\left({\left(E_c{I}_c\right)}_{eq}+\left(E_p{I}_p\right)+\frac{I_p{e}_{311}^2}{a_{33}}\right){\left(\frac{k_n}{L}\right)}^4+\left(l^2\left\{{\left(E_c{I}_c\right)}_{eq}+\left(E_p{I}_p\right)\right\}+\frac{I_p{\mu }_{3111}^2}{a_{33}}\right){\xi }_n{\left(\frac{k_n}{L}\right)}^6}{\left\{\left[{\left({\rho }_c{A}_c\right)}_{eq}+2{\rho }_p{A}_p\right]\left(1+{\left(e_{0}a\right)}^2{\xi }_n{\left(\frac{k_n}{L}\right)}^2\right)+\left[{\left({\rho }_c{I}_c\right)}_{eq}+{\rho }_p{I}_p\right]{\xi }_n{\left(\frac{k_n}{L}\right)}^2\right\}\left({\xi }_n+{\left(e_{0}a\right)}^2{\left(\frac{k_n}{L}\right)}^2\right)}}$$

(25a)

with ${\xi }_n$ expressed as

$$\begin{gathered} {\xi }_n={{\displaystyle \int }}_0^1\left\{\left(\cosh \left(k_n\right)+\cos \left(k_n\right)\right)-{\beta }_n\left(\sinh \left(k_{n}X\right)+\sin \left(k_{n}X\right)\right)\right\}\left\{\left(\cosh \left(k_n\right)-\cos \left(k_n\right)\right) \\ -{\beta }_n\left(\sinh \left(k_{n}X\right)-\sin \left(k_{n}X\right)\right)\right\}dX \end{gathered}$$

(25b)

The values of $k_n, {\beta }_n, \mathrm{and} {\xi }_n$ are found for the different boundary conditions and structural mode shapes as indicated in

Table 2. Numerical values for $k_n, {\beta }_n, \mathrm{and} {\xi }_n$ for different boundary conditions at different structural mode shapes.

n	CF ${\mathit{k}}_{\mathit{n}}\mathbf{\cong }\frac{\left(\mathbf{2}\mathit{n}\mathbf{-}\mathbf{1}\right)\mathit{\pi }}{\mathbf{2}}$			CC ${\mathit{k}}_{\mathit{n}}\mathbf{\cong }\frac{\left(\mathbf{2}\mathit{n}\mathbf{+}\mathbf{1}\right)\mathit{\pi }}{\mathbf{2}}$			CS ${\mathit{k}}_{\mathit{n}}\mathbf{\cong }\frac{\left(\mathbf{4}\mathit{n}\mathbf{+}\mathbf{1}\right)\mathit{\pi }}{\mathbf{4}}$
	${\mathit{k}}_{\mathit{n}}$	${\mathit{\beta }}_{\mathit{n}}$	${\mathit{\xi }}_{\mathit{n}}$	${\mathit{k}}_{\mathit{n}}$	${\mathit{\beta }}_{\mathit{n}}$	${\mathit{\xi }}_{\mathit{n}}$	${\mathit{k}}_{\mathit{n}}$	${\mathit{\beta }}_{\mathit{n}}$	${\mathit{\xi }}_{\mathit{n}}$
1	1.8751	0.7341	0.2441	4.730	0.9825	0.5499	3.9266	1.0008	0.7467
2	4.6941	1.0185	0.6033	7.8532	1.0008	0.7467	7.0686	1.0000	0.8585
3	7.8548	0.9992	0.7440	10.9956	0.9999	0.8181	10.2102	1.0000	0.9021
4	10.9955	1.0000	0.8181	14.1372	1.0000	0.8585	13.3518	1.0000	0.9251
5	14.1372	0.9999	0.8585	17.2788	1.0000	0.8843	16.4934	1.0000	0.9394
6	17.2788	1.0000	0.8843	20.4204	1.0000	0.9021	19.6350	1.0000	0.9491
7	20.4204	1.0000	0.9021	23.5620	1.0000	0.9151	22.7765	1.0000	0.9561
8	23.5620	1.0000	0.9151	26.7035	1.0000	0.9251	25.9181	1.0000	0.9614
9	26.7035	1.0000	0.9251	29.8451	1.0000	0.9330	29.0597	1.0000	0.9656
10	29.8451	1.0000	0.9330	32.9867	1.0000	0.9706	32.2013	1.0000	0.9688

.

5. Numerical Results and Discussion

5.1. Validation of the Developed Methodology

The analytical solutions for the natural frequency of the developed methodology were verified by comparing the obtained natural frequencies with the corresponding available results in the literature. Three verification cases were considered. The first case concerned verifying the obtained natural frequencies of a simply supported SS sandwich beam in the absence of piezoelectricity and size-dependent effects. The second case considered the verification of the nonclassical natural frequencies of a single-layer nanobeam without the piezoelectric effect. The third case handled the verification case of the natural frequencies of a single-layer piezoelectric nanobeam for different boundary conditions.

Consider an SS sandwich beam with the following geometrical and material characteristics: beam length L = 1.2 m, core thickness h_c = 10 mm, piezoelectric layer thickness h_p = 0.5 mm, and total thickness h_t = 11 mm. The elastic moduli of the elastic core and the piezoelectric layer are E_c = 130 MPa and E_p = 32 GPa, respectively. The mass densities of the elastic core and the piezoelectric layer are ρ_c = 126 kg/m³ and ρ_p = 1380 kg/m³, respectively. The nonclassical as well as the piezoelectric parameters are neglected. The presented results in

Table 3. Comparison of the estimated classical natural frequencies in Hz for the lowest eight vibration modes of SS sandwich beam in the absence of piezoelectricity effect for L = 1.2 m, h_c = 10 mm, h_p = 0.5 mm, h_t = 11 mm, E_c = 130 MPa, E_p = 32 GPa, ρ_c = 126 kg/m³, ρ_p = 1380 kg/m³, l = ea = 0 nm, a₃₃ = 0, e₃₁₁ = 0 C/m², and μ₃₁₁₁ = 0 C/m.

Comparison of the estimated classical natural frequencies in Hz for the lowest eight vibration modes of SS sandwich beam in the absence of piezoelectricity effect for L = 1.2 m, h_c = 10 mm, h_p = 0.5 mm, h_t = 11 mm, E_c = 130 MPa, E_p = 32 GPa, ρ_c = 126 kg/m³, ρ_p = 1380 kg/m³, l = ea = 0 nm, a₃₃ = 0, e₃₁₁ = 0 C/m², and μ₃₁₁₁ = 0 C/m.

Mode	Present Model	[44] Analytical	[48] FE	[48] Analytical	% Error with [44]
1	20.0667	19.9436	19.946	19.902	0.6172
2	80.2513	79.7746	79.784	79.083	0.5976
3	180.5081	179.4928	176.023	179.513	0.5656
4	320.7608	319.0984	308.349	319.135	0.5210
5	500.9028	498.5912	473.037	498.648	0.4636
6	720.7976	717.9713	666.657	718.053	0.3937
7	980.2793	977.2387	885.597	977.349	0.3111
8	1279.1529	1276.3934	1126.269	1276.538	0.2162

show good agreement between the obtained results and the corresponding results reported by Zeng et al. [44] and Chanthanumataporn and Watanabe [48], verifying the effectiveness of the developed procedure in effectively investigating the natural frequency of a sandwich beam structure without the piezoelectric or size effects.

To verify the effectiveness of the developed methodology and accurately investigate the nonclassical dynamic behavior of a nanosized beam structure, an SS nanobeam having a length L = 10 nm and thickness h_t = 10 nm were used. The piezoelectric effect was neglected. The nondimensional frequency parameter ${\lambda }_i=\frac{{\omega }_{ni}{L}^2}{h_t}\sqrt{\frac{12\rho }{E}}$ was used for this verification case. The same problem was previously analyzed by Lu et al. [48] and Zeng et al. [44] with the same geometry and boundary conditions. A comparison of the estimated nonclassical nondimensional natural frequency parameters ${\lambda }_i$ for the first and second vibration modes for the SS homogenous nanobeam without the piezoelectricity effect is depicted in

Table 4. Comparison of the estimated nonclassical nondimensional natural frequency parameters ${\lambda }_i=\frac{{\omega }_{ni}{L}^2}{h_t}\sqrt{\frac{12\rho }{E}}$ for the first and the second vibration modes for an SS homogenous nanobeam without the piezoelectricity effect, where L = 10 nm and h_t = 1 nm.

$\left({\mathit{e}}_0\mathit{a}\right) \left(\mathbf{nm}\right)$	Method of Analysis	First Vibration Mode (λ1)			Second Vibration Mode (λ2)
		Strain Gradient Parameter (l)			Strain Gradient Parameter (l)
		0	0.5 ht	ht	0	0.5 ht	ht
0	[40]	9.8293	9.9498	10.3029	38.8446	40.7164	45.8759
	Present	9.8291	9.9496	10.3027	38.8415	40.7132	45.8722
	[37]	9.8293	9.9498	10.3029	38.8446	40.7164	45.8759
	% Error	0.0020	0.0020	0.0019	0.0080	0.0079	0.0081
0.5	[40]	9.7102	9.8293	10.1781	37.0589	38.844 6	43.7669
	Present	9.710	9.8291	10.1779	37.0559	38.8415	43.7634
	[37]	9.7102	9.8293	10.1781	37.0589	38.8446	43.7669
	% Error	0.0021	0.0020	0.0020	0.0081	0.0080	0.0080
1.0	[40]	9.3774	9.4924	9.8293	32.8910	34.4759	38.8446
	Present	9.3772	9.4922	9.8291	32.8884	34.4732	38.8415
	[37]	9.3774	9.4924	9.8293	32.8910	34.4759	38.8446
	% Error	0.0021	0.0021	0.0020	0.0079	0.0078	0.0080
1.5	[40]	8.8915	9.000 5	9.319 9	28.2683	29.6305	33.3851
	Present	8.8913	9.0003	9.3197	28.266	29.6281	33.3825
	[37]	8.8915	9.0005	9.3199	28.2683	29.6305	33.3851
	% Error	0.0022	0.0022	0.0021	0.0081	0.0081	0.0078
2.0	[40]	8.3228	8.4248	8.7238	24.1877	25.3532	28.5659
	Present	8.3226	8.4246	8.7236	24.1857	25.3512	28.5636
	[37]	8.3228	8.4248	8.7238	24.187 7	25.3532	28.5659
	% Error	0.0024	0.0024	0.0023	0.0083	0.0079	0.0081

. The obtained results showed good agreement with the corresponding results obtained by Lu et al. [49] and Zeng et al. [44].

Seeking for deeper verification of the developed procedure to effectively investigate the nonclassical electromechanical dynamic behavior of sandwich beam structures for different boundary conditions, we considered a sandwich beam composed of an elastic core and two piezoelectric layers with the following geometrical parameters: core thickness h_c = 0, total beam thickness h_t = 2 nm, and piezoelectric layer thickness h_p = 1 nm. The modulus of elasticity of the piezoelectric layer E_p and the mass density ρ_p were E_p = 132 GPa and ρ_p = 7500 kg/m³, respectively. The piezoelectric parameters were given as follows: e₃₁₁ = −4.1 C/m² and a₃₃ = 7.124 × 10⁻⁹ N/(m²·K). The nonlocal parameter was given as ea = 0.1 × L nm, and the strain gradient parameter was l = 0 nm. The same problem was previously handled by Ke et al. [23] and Zeng et al. [44]. The nondimensional frequency parameter for this verification case was given as ${\lambda }_1={\omega }_{n1}L\sqrt{\frac{{\rho }_p}{E_p}}.$ A comparison of the estimated nonclassical fundamental natural frequency parameters ${\lambda }_1$ for a piezoelectric nanobeam at different beam aspect ratios L/h_t for different boundary conditions is shown in

Table 5. Comparison of the estimated nonclassical fundamental natural frequency parameters ${\lambda }_1={\omega }_{n1}L\sqrt{\frac{{\rho }_p}{E_p}}$ for piezoelectric nanobeam at different beam aspect ratios L/h_t for different boundary conditions for h_c = 0, h_t = 2 nm, h_p = 1 nm, E_p = 132 GPa, ρ_p = 7500 kg/m³ e₃₁₁ = −4.1 C/m², a₃₃ = 7.124 × 10⁻⁹ N/(m²·K), and ea = 0.1 × L nm.

L/ht	SS				CC			CS				CF
	Present	[23]	[44]	% Error	Present	[44]	% Error	Present	[23]	[44]	% Error	Present	[44]	% Error
6	0.4519	0.4570	0.4571	1.1376	1.0149	1.0245	0.9370	0.7012	0.7077	0.7087	1.0583	0.1699	0.1714	0.8751
8	0.3406	0.3428	0.3428	0.6418	0.7644	0.7684	0.5206	0.5284	0.5310	0.5315	0.5833	0.1274	0.1285	0.8560
10	0.2731	0.2742	0.2742	0.4012	0.6127	0.6167	0.6486	0.4236	0.4250	0.4252	0.3763	0.1020	0.1028	0.7782
16	0.1711	0.1714	0.1714	0.1750	0.3838	0.3842	0.1041	0.2654	0.2658	0.2658	0.1505	0.0637	0.0643	0.9331
20	0.1370	0.1371	0.1371	0.0729	0.3072	0.3073	0.0325	0.2124	0.2127	0.2126	0.0941	0.0510	0.0514	0.7782
30	0.0914	0.0914	0.0914	0	0.2049	0.2049	0	0.1417	0.1420	0.1417	0	0.0340	0.0343	0.8746

. It is noticed that there was a good agreement between the obtained results reported by Ke et al. [23] and Zeng et al. [44].

5.2. Parametric Studies

To demonstrate the applicability of the developed procedure to analyze the nonclassical electromechanical dynamic behavior of sandwich nanobeams, consider a sandwich beam structure composed of an elastic core and two piezoelectric layers. The composite beam has the following characteristics proposed by Zeng et al. [44]: total beam thickness h_t = Wp = 2.5 nm, piezoelectric layer thickness h_p = 0.1 × h_t nm, core thickness h_c = 0.8 × h_t, and beam length L = 20 × h_t. The modulus of elasticity of the elastic core and the piezoelectric layer are E_c = 130 MPa and E_p = 132 GPa, respectively. The mass densities are ρ_c = 1380 kg/m³ and ρ_p = 7500 kg/m³. The piezoelectric parameters of the piezoelectric layers are given as a₃₃ = 7.124 × 10⁻⁹ N/(m²·K), μ₃₁₁₁ = 5 × 10⁻⁸ C/m, and e₃₁₁ = −4.1 C/m². The dimensionless frequency parameter ${\mathsf{\Omega }}_{ni}$ is given by ${\mathsf{\Omega }}_{ni}={\omega }_{ni}{L}^2\sqrt{\frac{{\left(\rho A\right)}_c+2{\left(\rho A\right)}_p}{{\left(EI\right)}_c+{\left(EI\right)}_p}}$.

Depending on the fundamental frequency parameter, λ₁ on the perforation filling ratio for nonclassical and classical electromechanical and mechanical analyses at different values of the strain gradient and nonlocal parameters for different boundary conditions are depicted in Figure 2. It was observed that for the considered material and geometrical parameters, due to the larger increase in the overall system mass than that of the overall system stiffness, increasing the perforation filling ratio increased the fundamental natural frequencies for all values of the strain gradient and nonlocal parameters for all boundary conditions. When comparing the considered boundary conditions, the CF boundary conditions resulted in a flexible structure that produced smaller values for the fundamental frequency parameters compared with other boundary conditions. In addition, an insignificant effect was observed between the classical and nonclassical cases or the electromechanical and mechanical cases of the CF boundary condition.

Incorporating the piezoelectric and flexoelectricity effects resulted in a stiffening effect which produced larger values for the electromechanical fundamental frequency parameters compared with the corresponding mechanical cases.

Introducing the nonlocality effect led to a softening effect, resulting in smaller values of the fundamental frequency parameters for both the electromechanical and mechanical cases. On the other hand, introduction of the strain gradient parameter produced a stiffening effect, and thus larger values of the fundamental frequency parameters were produced compared with the corresponding classical cases.

The variation of the fundamental frequency parameter with the number of hole rows throughout the beam’s cross-sectional area for the nonclassical and classical electromechanical and mechanical analyses for different beam boundary conditions is shown in Figure 3.

It can be seen that increasing the number of hole rows throughout the beam’s cross-sectional area produced larger values for the fundamental frequency parameter for all cases due to a larger decreasing rate for the system equivalent mass than that in the system’s equivalent stiffness. In a comparison between the different boundary conditions, the CF boundary conditions produced a flexible structure that resulted in smaller values for the fundamental frequency parameters compared with the other boundary conditions.

Figure 4 demonstrates the size dependence due to the strain gradient effect of the piezoelectricity and flexoelectricity on the fundamental frequency parameter at different beam aspect ratios for different boundary conditions. It can be seen that larger values for the fundamental frequency parameter were detected with the increasing strain gradient parameter for all beam aspect ratios and boundary conditions. In the absence of the piezoelectric and flexoelectricity effects, smaller values for the mechanical fundamental frequency parameter while considering the larger effect values of the fundamental frequency parameter were produced. In addition, increasing the beam aspect ratio decreased the fundamental frequency parameter. The strain gradient effect resulted in a stiffening effect, which produced larger values for the fundamental frequency parameter. It was also noticed that smaller values for the fundamental frequency parameters were produced in the CF boundary condition compared with the other considered boundary conditions.

Neglecting the strain gradient effect, introduction of the nonlocality effect led to a softening effect which produced smaller values for the fundamental frequency parameter, as demonstrated in Figure 5. The rate of the fundamental frequency parameter decrease increased at smaller values of the beam aspect ratio. Additionally, incorporating the piezoelectric and flexoelectricity effects led to larger values for the fundamental frequency parameter compared with the corresponding mechanical cases. Regarding the boundary condition effect, the CF boundary condition produced a flexible system which resulted in smaller values for the fundamental frequency parameters compared with the corresponding cases of the other boundary conditions.

Figure 6 demonstrates the variation of the fundamental frequency parameter λ₁ with the piezoelectric coefficient e₃₁₁ for the nonclassical and classical electromechanical and mechanical analyses of piezoelectric sandwich nanobeams at different values of the strain gradient and nonlocal parameters for different beam boundary conditions. It can be seen that as the absolute values of the piezoelectric coefficient e₃₁₁ increased as the fundamental frequency parameter increased. Larger values for the fundamental frequency parameter were detected with increases in the strain gradient parameter, while smaller values were observed with increases in the nonlocal parameter compared with the corresponding electromechanical and mechanical classical cases. It can also be observed that, when compared with the other boundary conditions, smaller values for the fundamental frequency parameter were observed for the CF boundary conditions, with a slight difference between the classical and nonclassical electromechanical and mechanical behavior.

The piezoelectric layer’s thickness significantly affected the natural frequency parameter. Figure 7 shows the variation of the fundamental frequency parameter λ₁ with the piezoelectric coefficient e₃₁₁ for the nonclassical and classical electromechanical and mechanical analyses of piezoelectric sandwich nanobeams at different ratios of the piezoelectric layer thickness to the total beam thickness h_p/h_t for different beam boundary conditions. It can be noticed that the effects of the piezoelectricity on the fundamental natural frequency parameter were more noticeable for a smaller-scale nanobeam. Smaller values for the fundamental frequency parameter were produced at larger values of the piezoelectric thickness ratio h_p/h_t. It is also depicted that the CF boundary condition produced smaller values for the fundamental frequency parameter compared with the other considered boundary conditions.

The electric field–strain gradient coupling coefficient had a significant effect on the dynamic behavior of the sandwich nanobeam. Figure 8 shows the variation of the fundamental frequency parameter λ₁ with the electric field–strain gradient coupling coefficient μ₃₁₁₁ for the nonclassical and classical electromechanical and mechanical analyses of piezoelectric sandwich nanobeams at different values of the strain gradient and nonlocal parameters for different beam boundary conditions. It can be observed that increasing the absolute values of the electric field–strain gradient coupling coefficient produced larger values for the fundamental frequency parameter for all values of the nonclassical parameters for all considered boundary conditions. Regarding the effect of the boundary conditions, due to more flexibility, the CF resulted in smaller values for the fundamental frequency parameters.

The piezoelectric layer thickness hp significantly affected the electromechanical dynamic behavior of the sandwich nanobeam. Variation of the fundamental frequency parameter λ₁ with the electric field–strain gradient coupling coefficient μ₃₁₁₁ for the nonclassical and classical electromechanical and mechanical analyses of piezoelectric sandwich nanobeams at different ratios of the piezoelectric layer thickness to the total beam thickness hp/ht for different beam boundary conditions is demonstrated in Figure 9. It is noticeable that increasing the piezoelectric layer thickness ratio hp/ht produced smaller values of the fundamental frequency parameter for both the classical and nonclassical cases for all boundary conditions. This was because of the fundamental frequency parameter being controlled by both the system mass and stiffness. Increasing the piezoelectric layer’s thickness produced a greater increase in the overall system mass than that produced in the overall system stiffness.

6. Conclusions

The electromechanical size-dependent dynamic behavior of a sandwich nanobeam with a perforated core and two piezoelectric face sheets incorporating the flexoelectricity effect was developed. Size dependency was adopted using the nonlocal strain gradient elasticity theory. The coupled electromechanical equation of motion was developed using the Hamilton principle. Analytical forms for the natural frequencies were derived for different boundary conditions. The accuracy of the developed procedure was verified and compared. Numerical results were obtained and discussed. Based on the obtained numerical results, the following concluding remarks are revealed:

➢

The perforation parameters significantly affected the electromechanical and mechanical dynamic behavior of the sandwich nanobeams. Based on the selected geometrical and material parameters, increasing the perforation filling ratio decreased the fundamental frequency parameter for all considered beam boundary conditions. On the other hand, increasing the number of hole rows through the beam cross-section increased the fundamental frequency parameters. Thus, both the electromechanical and mechanical dynamic behavior of the perforated sandwich nanobeam could be controlled by careful selection of the perforation parameters.

➢

The electromechanical and mechanical dynamic behavior of the sandwich perforated nanobeam could also be controlled by the size-dependent parameters. The nonlocal strain gradient theory could be used to incorporate either softening or stiffening effects. Increasing the strain gradient parameter resulted in a stiffening effect, while increasing the nonlocal parameter led to a softening effect.

➢

Incorporating the piezoelectricity and flexoelectricity resulted in increasing the frequency parameter. Furthermore, the effects of the flexoelectricity on the dimensionless natural frequency were more noticeable for a smaller nanobeam aspect ratio.

➢

Increasing the piezoelectric layer thickness resulted in decreasing the natural frequency parameter, and thus both the piezoelectric and flexoelectricity effects were more significant for smaller sizes.