An isogeometric approach to static and transient analysis of fluid-infiltrated porous metal foam piezoelectric nanoplates with flexoelectric effects and variable nonlocal parameters

Abstract

In this work, a novel refined higher-order shear deformation plate theory is integrated with nonlocal elasticity theory for analyzing the free vibration, bending, and transient behaviors of fluid-infiltrated porous metal foam piezoelectric nanoplates resting on Pasternak elastic foundation with flexoelectric effects. Isogeometric analysis (IGA) and the Navier solution are applied to the problem. The innovation in the present study is that the influence of the in-plane variation of the nonlocal parameter on the free and forced vibration of the piezoelectric nanoplates is investigated for the first time. The nonlocal parameter and material characteristics are assumed to be material-dependent and vary gradually over the thickness of structures. Based on Hamilton’s principle, equations of motion are built, then the IGA approach combined with the Navier solution is used to analyze the static and dynamic response of the nanoplate. Lastly, we investigate the effects of the porosity coefficients, flexoelectric parameters, elastic stiffness, thickness, and variation of the nonlocal parameters on the mechanical behaviors of the rectangular and elliptical piezoelectric nanoplates.

Graphical abstract

Keywords

isogeometric analysis / fluid-infiltrated porous / piezoelectric nanoplates / static and transient analysis / flexoelectric effect / variable nonlocal parameters

1 Introduction

The reaction of electric polarization to a change in the mechanical strain of a material is known as the flexoelectric phenomenon. It is possible to think of it as a higher-order influence in terms of piezoelectricity [1–4]. The flexoelectric phenomenon becomes apparent at the nanoscale, where large strain gradients are anticipated to exist. It is noted that, in contrast to the piezoelectric effects, flexoelectric effects may occur in any material since these are dictated by symmetry. Because of these attributes, flexoelectricity has garnered an increasing amount of attention by scientists over the last decade. At the moment, its function in the physics of dielectrics and semiconductors is generally acknowledged, and the effect is considered to have great potential for use in practice. In addition, the obtained results of these structures theoretically and experimentally are rather inconsistent, which is evidence that our knowledge of the topic is still lacking [5]. In nanoelectromechanical systems, strong electromechanical coupling leads to piezoelectric and flexoelectric nanostructures that have been impressively used in nanosensors, nanoresonators, and nanogenerators. It has been observed in several investigations that the elastic and piezoelectric constants of materials depend on their characteristic size. To correctly explain the electromechanical interaction, it is crucial to investigate the underlying processes of this size dependence in nanoscale structures.

The most recent and typical research on nanostructures with the flexoelectric effect may be presented as follows. Bagheri and Tadi [6] researched the size-dependent nonlinear forced oscillation of viscoelastic/flexoelectric nanobeam structures. In their study, the hypotheses of a non-classical continuous medium and the classical beam model have been employed to compute nanobeams. Ray [7] evaluated the influence of material length scale on the static behaviors of simply supported flexoelectric nanobeam structures using a few strain theories. In his study, mechanical fields and electric potential parts of the beam were determined in both closed and open circuit circumstances. Tho et al. [8] studied the static and free vibration behaviors of nanoscale piezoelectric beams under the effects of flexoelectricity and geometrical imperfection using the finite element method (FEM) and a third-order shear deformation theory. Yu et al. [9] explored dynamic flexoelectricity in the static bending and vibration of FG piezoelectric nanobeams, and they modified the electric field equilibrium equation from the conventional electric Gauss equations. Numerical research was carried out by Ansari and colleagues [10] to examine the static bending manners of piezoelectric nanobeam structures under electrical stress. Baroudi and Najar [11] used the Euler-Bernoulli theory to examine the behavior of a flexoelectric piezoelectric nanobeam under various actuation situations. Baroudi et al. [12] conducted an analytical investigation on the mechanical manners of nanobeams under a variety of boundary conditions and electrical loads. This investigation took into account gradient elasticity, piezoelectricity, and flexoelectricity. Chu et al. [13] explored thermally induced dynamic manners of FG flexoelectric nanobeams while taking into account the neutral surface idea and thermally induced von Karman nonlinearity. Zhao et al. [14] expanded a size-dependent classical nanobeam model with porous AFG flexoelectric material.

For plate and shell structures, Naskar et al. [15] examined the electro-mechanical manners of graphene-reinforced piezoelectric FG nanocomposite plates using a semi-analytical approach based on the extended Kantorovich method and Ritz solution. Lan et al. [16] investigated the vibration modes of the flexoelectric circular plate. Using the finite element modeling and shear deformation hypothesis functions, Duc et al. [17] analyzed the oscillation behavior and the static buckling of flexoelectric nanostructures with different thicknesses, in which, the thickness varied according to linear and nonlinear criteria. Ji [18] considered the flexoelectric manners of circular plate-type intelligent components. By using von Kármán plate theory with the flexoelectric hypothesis based on Hamilton’s law, a nonlinear flexoelectric circular plate model was presented in their work. van Minh and van Ke [19] investigated the flexoelectric impact on the behaviors of piezoelectric nanoplates considering non-uniform thickness in both the longitudinal and transverse directions. Thai et al. [20] considered the effect of the flexoelectric phenomenon of piezoelectric nanoplates supported by elastic foundations under static load. Most recently, Qu et al. [21] proposed the couple stress theory and a curvature-based flexoelectricity theory to compute the static bending, free vibration, and dynamic response of circular cylindrical shell structures. Zhang et al. [22] presented the static bending and free vibration of simply supported parabolic-cylindrical shells subjected to mechanical load and the effect of flexoelectric actuators. Fattaheian and Tadi [23] developed a continuous electromechanical model based on the FSDT shell model and on a novel flexoelectric theory, to investigate the static and vibration behavior of the FG magneto-electro-elastic conical nanoshells. Fan et al. [24] developed a neural network to investigate the effect of flexoelectric actuators on a rectangular plate. In addition, several recent plate and shell studies suggest promising new theories for future flexoelectric research [25–27].

The isogeometric analysis approach (IGA), is a relatively new technique in the field of computational mechanics. It allows the combination of numerical methods with Computer-Aided Design, resulting in a single, streamlined procedure. The time it takes to get from design to analysis is cut down and therefore this significantly reduces actual engineering design costs [28–30]. Hughes and the members of his group at the University of Texas at Austin are considered the first developers of this approach [31,32]. Isogeometric techniques have since then found significant usage in the domain of computational methods, notably in the contents of computational mechanics of complex and intelligent structures. Further works [33–48] illustrate the wide use of the IGA approach in engineering computation.

Exploring the mechanical characteristics of nanobeam, nanoplate, and nanoshell structures, and in particular the phenomenon of flexoelectricity and the size-dependent effects, appears to have a huge appeal based on reports by scientists from all over the world. However, studies using IGA for investigation of the mechanical behavior of nanobeams and nanoplates, considering the influence of flexoelectricity, are limited in number. Therefore, this study examines the static bending, free vibration, and dynamic behavior of rectangular and elliptical fluid-infiltrated porous metal foam (FPF) piezoelectric nanoplates on a Pasternak elastic foundation, accounting for flexoelectric effects and varying nonlocal parameters through the plate thickness. Hamilton’s principle, the nonlocal elasticity theory, and a novel refined higher-order shear deformation theory are used to establish governing equations of motion for the FPF piezoelectric nanoplate. In addition, both the IGA approach and Navier’s solution are employed to yield results that validate the accuracy of solutions to certain previously unsolved problems. The parametric study is carried out to demonstrate the effects of the porosity parameter, flexoelectric parameter, elastic stiffness parameter, thickness, and the variation of the nonlocal parameter on the static and dynamic responses of FPF nanoplates. The results of this study can serve as a basis for design calculations of microelectromechanical devices, as well as a reference for studies on the behavior of nanostructures with flexoelectric effects.

As for the remaining parts of this work they are laid out as follows. In Section 2, the theoretical modeling and material of nanoplates are presented. Further development of the governing equations is shown in Section 3. In Section 4, we give the numerical findings, where the verification instances are carried out to validate the precision of the presented theory and model. Moreover, a substantial amount of discussion is included here as well. Section 5 concludes with some of the most crucial findings from this research.

2 Modeling and material of nanoplates

2.1 Fluid-infiltrated porous metal foam piezoelectric nanoplate

A rectangular FPF piezoelectric nanoplate with dimensions a\times b\times h, subjected to a force q(x,y), can be placed on Pasternak’s elastic medium with two stiffness parameters: k_{\mathrm{w}}, k_{\mathrm{s}} as Fig.1. Material characteristics, including Young’s modulus, shear modulus, and mass density, vary according to the symmetric and asymmetric porosity functions, which can be expressed by the formulas below [49,50]:

Symmetric porosity distribution (Type 1)

(1)\{\begin{array}{l}\begin{array}{l}E\left(\mathit{\text{z}}\right)=E_1\left(1-{\lambda }_0\mathrm{c}\mathrm{o}\mathrm{s}\left({\displaystyle \frac{\text{π}\mathit{\text{z}}}{h}}\right)\right),\\ G\left(\mathit{\text{z}}\right)=G_1\left(1-{\lambda }_0\mathrm{c}\mathrm{o}\mathrm{s}\left({\displaystyle \frac{\text{π}\mathit{\text{z}}}{h}}\right)\right),\\ \rho \left(\mathit{\text{z}}\right)={\rho }_1\left(1-\sqrt{1-{\lambda }_0}\mathrm{c}\mathrm{o}\mathrm{s}\left({\displaystyle \frac{\text{π}\mathit{\text{z}}}{h}}\right)\right),\end{array}\end{array}

where {\lambda }_0 is a porosity parameter represented by

(2)

where E_1, G_1, and {\rho }_1 are the maxima of each of the Young’s modulus, shear modulus, and mass density of the porous metal foam, respectively; and E_2 and G_2 are the minima of the aforementioned variables, respectively.

Asymmetric porosity distribution (Type 2)

(3)\{\begin{array}{rl}& E\left(\mathit{\text{z}}\right)=E_1\left(1-{\lambda }_0\mathrm{c}\mathrm{o}\mathrm{s}\left(\frac{\text{π}\mathit{\text{z}}}{2h}+\frac{\text{π}}{4}\right)\right),\\ & G\left(\mathit{\text{z}}\right)=G_1\left(1-{\lambda }_0\mathrm{c}\mathrm{o}\mathrm{s}\left(\frac{\text{π}\mathit{\text{z}}}{2h}+\frac{\text{π}}{4}\right)\right),\\ & \rho \left(\mathit{\text{z}}\right)={\rho }_1\left(1-\sqrt{1-{\lambda }_0}\mathrm{c}\mathrm{o}\mathrm{s}\left(\frac{\text{π}\mathit{\text{z}}}{2h}+\frac{\text{π}}{4}\right)\right).\end{array}

In the present study, the variation of the Poisson ratio is small enough for it to be assumed to be constant [51].

2.2 The nonlocal elasticity hypothesis for piezoelectric and flexoelectric effects

Based on the nonlocal elasticity theory [52,53] and the linear poroelasticity theory of Biot [54]. When the flexoelectric effect is taken into consideration, the stress components and electric displacement vector for a nanoscale dielectric material are expressed as follows:

(4)\begin{array}{rl}{{\sigma }_{ij}-{\mu }^2{\mathrm{\nabla }}^2\sigma }_{ij}=& 2G{\epsilon }_{ij}+\frac{2G{v}_{\mathrm{u}}}{1-v_{\mathrm{u}}}ϵ{\delta }_{ij}-\tilde{M}\left(\vartheta -\alpha ϵ\right)\alpha {\delta }_{ij}\\ & -e_{kij}{\overline{E}}_k+f_{klij}{\overline{E}}_{k,x_l},\end{array}

(5)\tilde{M}=\frac{2G\left(v_{\mathrm{u}}-v\right)}{{\alpha }^2\left(1-2v_{\mathrm{u}}\right)\left(1-2v\right)},v_{\mathrm{u}}=\frac{v+{\displaystyle \frac{\alpha \beta \left(1-2v\right)}{3}}}{1-{\displaystyle \frac{\alpha \beta \left(1-2v\right)}{3}}},

where e_{kij},c_{ijkl}, and f_{kijm} are the characteristics of the material for the piezoelectric, elastic, flexoelectric, and permittivity constant tensors, respectively, and {\overline{E}}_k is the electric field component. G represents the shear modulus, {\epsilon }_{ij} and {\sigma }_{ij} are the strain and stress. The parameter \tilde{M} denotes the Biot modulus, defined as the rise in the amount of fluid ; v_{\mathrm{u}} is the undrained Poisson ratio ; ϵ represents the volumetric strain; {\delta }_{ij} is the Kronecker delta factor; \vartheta is the variation in the liquid volume contained inside the porosities; \alpha expresses the Biot parameter of effective stress ; \beta is the Skempton parameter; \mu \left(\mathrm{n}\mathrm{m}\right) is a nonlocal parameter; {\mathrm{\nabla }}^2={\mathrm{\partial }}_{,xx}^2+{\mathrm{\partial }}_{,yy}^2 is the Laplacian operator.

In previous works on nanostructures, the nonlocal parameters were often assumed to be constant throughout the material. This is not true in the case of materials with variable mechanical properties. Therefore, to calculate more accurately the deflection and natural frequency of piezoelectric nanostructures, the nonlocal parameters are assumed to vary following the same law as the elastic modulus E\left(\mathit{\text{z}}\right).

Symmetric porosity distribution (Type 1):

(6)\mu \left(\mathit{\text{z}}\right)={\mu }_1\left(1-{\lambda }_0\mathrm{c}\mathrm{o}\mathrm{s}\left(\frac{\text{π}\mathit{\text{z}}}{h}\right)\right).

Asymmetric porosity distribution (Type 2):

(7)\mu \left(\mathit{\text{z}}\right)={\mu }_1\left(1-{\lambda }_0\mathrm{c}\mathrm{o}\mathrm{s}\left(\frac{\text{π}\mathit{\text{z}}}{2h}+\frac{\text{π}}{4}\right)\right).

The nonlocal parameter \mu \left(\mathit{\text{z}}\right) in two cases of porous distribution laws are shown in Fig.2 with {\mu }_1=2\mathrm{n}\mathrm{m} and various values of porosity parameter {\lambda }_0.

3 Governing equations

3.1 Novel refined high-order shear deformation theory

The displacement field is defined according to the novel refined high-order shear deformation hypothesis as follows [55,56]:

(8)\{\begin{array}{l}{u}_x\left(x,y,\mathit{\text{z}},t\right)=u_0\left(x,y,t\right)-\mathit{\text{z}}{\beta }_{,x}^{\mathrm{b}}-F\left(\mathit{\text{z}}\right){\beta }_{,x}^{\mathrm{s}},\\ u_y\left(x,y,\mathit{\text{z}},t\right)=v_0\left(x,y,t\right)-\mathit{\text{z}}{\beta }_{,y}^{\mathrm{b}}-F\left(\mathit{\text{z}}\right){\beta }_{,y}^{\mathrm{s}},\\ u_{\mathit{\text{z}}}\left(x,y,\mathit{\text{z}},t\right)={\beta }^{\mathrm{b}}+{\beta }^{\mathrm{s}},\end{array}

where u_0 and v_0 are the neutral surface’s displacements, respectively; {\beta }^{\mathrm{b}} and {\beta }^{\mathrm{s}} are the transverse displacement’s bending and shear parts, respectively. A novel inverse hyperbolic shear distribution function F\left(\mathit{\text{z}}\right) is used:

(9)F\left(\mathit{\text{z}}\right)=48.3\mathit{\text{z}}+\left(\frac{5}{4}\text{π}-5{\text{π}}^3\right)h\cdot {\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}}^{-1}\left(\frac{\mathit{\text{z}}}{\text{π}h}\right).

The linear strain parts of the FPF piezoelectric nanoplate are defined via displacements in Eq. (8) as follows:

\begin{array}{}\text{(10a)}& {\epsilon }_{xx}=u_{0,x}-\mathit{\text{z}}{\beta }_{,xx}^{\mathrm{b}}-F\left(\mathit{\text{z}}\right){\beta }_{,xx}^{\mathrm{s}},\end{array}

\begin{array}{}\text{(10b)}& {\epsilon }_{yy}=v_{0,y}-\mathit{\text{z}}{\beta }_{,yy}^{\mathrm{b}}-F\left(\mathit{\text{z}}\right){\beta }_{,yy}^{\mathrm{s}},\end{array}

\begin{array}{}\text{(10c)}& {\epsilon }_{xy}=u_{0,y}+v_{0,x}-2\mathit{\text{z}}{\beta }_{,xy}^{\mathrm{b}}-2F\left(\mathit{\text{z}}\right){\beta }_{,xy}^{\mathrm{s}},\end{array}

\begin{array}{}\text{(10d)}& {\gamma }_{x\mathit{\text{z}}}=\mathcal{G}\left(\mathit{\text{z}}\right){\beta }_{,x}^{\mathrm{s}};{\gamma }_{y\mathit{\text{z}}}=\mathcal{G}\left(\mathit{\text{z}}\right){\beta }_{,y}^{\mathrm{s}},\end{array}

where \mathcal{G}\left(\mathit{\text{z}}\right)=1+F_{,\mathit{\text{z}}}.

The strain gradient components in the x- and y-directions are explored in the present work, whereas the strain gradient component in the z-direction is zero. This demonstrates that strain gradient components along the x- and y-axes are higher than those along the depth direction. The strain gradient components are then described by:

(11)\begin{array}{rl}& {\chi }_{xx\mathit{\text{z}}}={\epsilon }_{xx,\mathit{\text{z}}}=-{\beta }_{,xx}^{\mathrm{b}}-F_{,\mathit{\text{z}}}{\beta }_{,xx}^{\mathrm{s}},\\ & {\chi }_{yy\mathit{\text{z}}}={\epsilon }_{yy,\mathit{\text{z}}}=-{\beta }_{,yy}^{\mathrm{b}}-F_{,\mathit{\text{z}}}{\beta }_{,yy}^{\mathrm{s}}.\end{array}

Taking into account the flexoelectricity and size-dependent effects, the stress parts and electrical displacement vector of a nanoscale dielectric material are given by sources [57,58].

(12)\{\begin{array}{l}{\left(1-{\mu }^2{\mathrm{\nabla }}^2\right)\sigma }_{ij}=c_{ijkl}{\epsilon }_{kl}-e_{kij}{\overline{E}}_k,\\ \left(1-{\mu }^2{\mathrm{\nabla }}^2\right){\tau }_{ijm}=-f_{kijm}{\overline{E}}_k,\\ {\Phi }_i=e_{ijk}{\epsilon }_{jk}+{\kappa }_{ij}{\overline{E}}_k+f_{ijkl}{\chi }_{jkl},\end{array}

where the electrical displacement component is represented by {\Phi }_i and the moment stress component or higher-level stress tensor is represented by {\tau }_{ijm}. Precise equations for stress and electrical displacement components, in terms of the strain components, are given by:

\begin{array}{}\text{(13a)}& \left(1-{\mu }^2{\mathrm{\nabla }}^2\right){\sigma }_{xx}=C_{11}{\epsilon }_{xx}+C_{12}{\epsilon }_{yy}-e_{31}{\overline{E}}_{\mathit{\text{z}}},\end{array}

\begin{array}{}\text{(13b)}& \left(1-{\mu }^2{\mathrm{\nabla }}^2\right){\sigma }_{yy}=C_{12}{\epsilon }_{xx}+C_{22}{\epsilon }_{yy}-e_{31}{\overline{E}}_{\mathit{\text{z}}},\end{array}

\begin{array}{}\text{(13c)}& \{\begin{array}{rl}& \left(1-{\mu }^2{\mathrm{\nabla }}^2\right){\sigma }_{xy}=C_{66}{\epsilon }_{xy},\\ & \left(1-{\mu }^2{\mathrm{\nabla }}^2\right){\sigma }_{x\mathit{\text{z}}}=C_{55}{\epsilon }_{x\mathit{\text{z}}},\\ & \left(1-{\mu }^2{\mathrm{\nabla }}^2\right){\sigma }_{y\mathit{\text{z}}}=C_{44}{\epsilon }_{y\mathit{\text{z}}},\end{array}\end{array}

\begin{array}{}\text{(13d)}& \left(1-{\mu }^2{\mathrm{\nabla }}^2\right){\tau }_{xx\mathit{\text{z}}}=-f_{14}{\overline{E}}_{\mathit{\text{z}}},\left(1-{\mu }^2{\mathrm{\nabla }}^2\right){\tau }_{yy\mathit{\text{z}}}=-f_{14}{\overline{E}}_{\mathit{\text{z}}},\end{array}

(14){\Phi }_{\mathit{\text{z}}}=e_{31}\left({\epsilon }_{xx}+{\epsilon }_{yy}\right)+{\kappa }_{33}{\overline{E}}_{\mathit{\text{z}}}+f_{14}\left({\chi }_{xx\mathit{\text{z}}}+{\chi }_{yy\mathit{\text{z}}}\right),

where f_{14} is defined as in Ref. [59]: P_{\mathit{\text{z}}}=f_{14}({\chi }_{xx\mathit{\text{z}}}+{\chi }_{yy\mathit{\text{z}}}), is the polarization derived by the strain gradient components in the present structure, and it is reliant on the coordinate z owing to the derivative of F\left(\mathit{\text{z}}\right) in the component of the strain gradient.

The initial constants C_{ij} is defined:

\begin{array}{rl}{C}_{11}=& C_{22}=\frac{E\left(\mathit{\text{z}}\right)}{2\left(1+v\right)}\left(\frac{2}{1-v_{\mathrm{u}}^2}\right)\\ & \cdot \left(1+v_{\mathrm{u}}+\frac{\left(v_{\mathrm{u}}-v\right)\left(1+v_{\mathrm{u}}\right)}{1-2v}\left(1-\frac{S_2}{S_1}\right)\right),\end{array}

\begin{array}{rl}{C}_{12}=& \frac{E\left(\mathit{\text{z}}\right)}{2\left(1+v\right)}\left(\frac{2}{1-v_{\mathrm{u}}^2}\right)\\ & \cdot \left(\left(1+v_{\mathrm{u}}\right)v_{\mathrm{u}}+\frac{\left(v_{\mathrm{u}}-v\right)\left(1+v_{\mathrm{u}}\right)}{1-2v}\left(1-\frac{S_2}{S_1}\right)\right),\end{array}

\begin{array}{}\text{(15c)}& C_{66}=C_{55}=C_{44}=\frac{E\left(\mathit{\text{z}}\right)}{2\left(1+v\right)}.\end{array}

\begin{array}{}\text{(16a)}& S_1=\frac{E\left(\mathit{\text{z}}\right)}{1+v}\left(1+\frac{v_{\mathrm{u}}}{1-2v_{\mathrm{u}}}+\frac{\left(v_{\mathrm{u}}-v\right)\left(1+v_{\mathrm{u}}\right)}{1-2v}\right),\end{array}

\begin{array}{}\text{(16b)}& S_2=\frac{E\left(\mathit{\text{z}}\right)}{\left(1+v\right)}\left(\frac{v_{\mathrm{u}}}{1-2v_{\mathrm{u}}}+\frac{\left(v_{\mathrm{u}}-v\right)\left(1+v_{\mathrm{u}}\right)}{1-2v}\right).\end{array}

In the absence of free electric charges, the electric displacement should satisfy the Gaussian rule in electrostatics [19].

(17){\Phi }_{\mathit{\text{z}},\mathit{\text{z}}}\left(\pm \frac{h}{2}\right)=0.

Under the open-circuit condition, the electric displacement on the surface is equal to zero. Hence, the internal electric field can be obtained from Eq. (17) as follows:

(18)\begin{array}{rl}{\overline{E}}_{\mathit{\text{z}}}=& -\frac{e_{31}}{{\kappa }_{33}}\left({\epsilon }_{xx}+{\epsilon }_{yy}\right)\\ & +\frac{f_{14}}{{\kappa }_{33}}\left({\beta }_{,xx}^{\mathrm{b}}+{\beta }_{,yy}^{\mathrm{b}}+F_{,\mathit{\text{z}}}{\beta }_{,xx}^{\mathrm{s}}+F_{,\mathit{\text{z}}}{\beta }_{,yy}^{\mathrm{s}}\right),\end{array}

where {\overline{E}}_{\mathit{\text{z}}} depends on the value of f_{14}, and this demonstrates that the flexoelectric phenomenon will have a considerable effect on the function of {\overline{E}}_{\mathit{\text{z}}} along the thickness. In this research, a combination of the nonlocal theory and application of the flexoelectric effect was applied for nanostructures of any shape for the first time. This is the first work in which the refined higher-order shear plate theory incorporates the IGA method for calculating flexoelectricity effects.

For the FPF piezoelectric nanoplate with flexoelectric effects, the equation of motion of the FPF piezoelectric nanoplate is given in the source [55].

The electrical Gibbs free energy density is externalized by U. Under the open-circuit condition, U reduces to zero:

(19)\begin{array}{rl}U=& \frac{1}{2}{\int }_{S_{\mathrm{e}}}({\mathcal{N}}_{xx}{u}_{0,x}-{\mathcal{M}}_{xx}{\beta }_{,xx}^{\mathrm{b}}-{\mathcal{L}}_{xx}{\beta }_{,xx}^{\mathrm{s}}+{\mathcal{N}}_{yy}{v}_{0,y}\\ & -{\mathcal{M}}_{yy}{\beta }_{,yy}^{\mathrm{b}}-{\mathcal{L}}_{yy}{\beta }_{,yy}^{\mathrm{s}}+{\mathcal{N}}_{xy}\left(u_{0,y}+v_{0,x}\right)\\ & -{2\mathcal{M}}_{xy}{\beta }_{,xy}^{\mathrm{b}}-{2\mathcal{L}}_{xy}{\beta }_{,xy}^{\mathrm{s}}+{\mathcal{Q}}_{xz}{\beta }_{,x}^{\mathrm{s}}+{\mathcal{Q}}_{yz}{\beta }_{,y}^{\mathrm{s}}\\ & -{\mathcal{X}}_{xx}{\mathcal{W}}_{,xx}^{\mathrm{b}}-{\mathcal{S}}_{xx}{\beta }_{,xx}^{\mathrm{s}}-{\mathcal{X}}_{yy}{\beta }_{,yy}^{\mathrm{b}}-{\mathcal{S}}_{yy}{\beta }_{,yy}^{\mathrm{s}})\mathrm{d}x\mathrm{d}y.\end{array}

The resultant forces and moments: {\mathcal{N}}_i,{\mathcal{M}}_i,{\mathcal{L}}_i,{\mathcal{Q}}_i,{\mathcal{S}}_i, and {\mathcal{X}}_i are determined as follows:

\begin{array}{rl}& \left\{\begin{array}{c}{\mathcal{N}}_{xx}\\ {\mathcal{N}}_{yy}\\ {\mathcal{N}}_{xy}\end{array}\right\}={\int }_{-\frac{h}{2}}^{+\frac{h}{2}}\left\{\begin{array}{c}{\sigma }_{xx}\\ {\sigma }_{yy}\\ {\sigma }_{xy}\end{array}\right\}\mathrm{d}\mathit{\text{z}},\\ & \left\{\begin{array}{c}{\mathcal{M}}_{xx}\\ {\mathcal{M}}_{yy}\\ {\mathcal{M}}_{xy}\end{array}\right\}={\int }_{-\frac{h}{2}}^{+\frac{h}{2}}\left\{\begin{array}{c}{\sigma }_{xx}\\ {\sigma }_{yy}\\ {\sigma }_{xy}\end{array}\right\}\widetilde{\mathit{\text{z}}}\mathrm{d}\mathit{\text{z}},\\ & \left\{\begin{array}{c}{\mathcal{L}}_{xx}\\ {\mathcal{L}}_{yy}\\ {\mathcal{L}}_{xy}\end{array}\right\}={\int }_{-\frac{h}{2}}^{+\frac{h}{2}}\left\{\begin{array}{c}{\sigma }_{xx}\\ {\sigma }_{yy}\\ {\sigma }_{xy}\end{array}\right\}\tilde{F}\mathrm{d}\mathit{\text{z}},\end{array}

\begin{array}{rl}& \left\{\begin{array}{c}{\mathcal{Q}}_{x\mathit{\text{z}}}\\ {\mathcal{Q}}_{y\mathit{\text{z}}}\end{array}\right\}={\int }_{-\frac{h}{2}}^{+\frac{h}{2}}\left\{\begin{array}{c}{\sigma }_{xx}\\ {\sigma }_{y\mathit{\text{z}}}\end{array}\right\}{\tilde{\mathcal{G}}}^2\mathrm{d}\mathit{\text{z}},\\ & \left\{\begin{array}{c}{\mathcal{X}}_{xx}\\ {\mathcal{X}}_{yy}\end{array}\right\}={\int }_{-\frac{h}{2}}^{+\frac{h}{2}}\left\{\begin{array}{c}{\tau }_{xx\mathit{\text{z}}}\\ {\tau }_{yy\mathit{\text{z}}}\end{array}\right\}\mathrm{d}\mathit{\text{z}},\\ & \left\{\begin{array}{c}{\mathcal{S}}_{xx}\\ {\mathcal{S}}_{yy}\end{array}\right\}={\int }_{-\frac{h}{2}}^{+\frac{h}{2}}\left\{\begin{array}{c}{\tau }_{xx\mathit{\text{z}}}\\ {\tau }_{yyz}\end{array}\right\}{\tilde{F}}_{,\mathit{\text{z}}}\mathrm{d}\mathit{\text{z}},\end{array}

where \widetilde{\mathit{\text{z}}}=\mathit{\text{z}}-t_0,\tilde{F}=F\left(\widetilde{\mathit{\text{z}}}\right),\tilde{\mathcal{G}}=\mathcal{G}\left(\widetilde{\mathit{\text{z}}}\right), and t_0 is the distance from the mean plane to the neutral plane of the FPF piezoelectric nanoplate. This distance is expressed below:

(21)t_0={\int }_{-\frac{h}{2}}^{\frac{h}{2}}E\left(\mathit{\text{z}}\right)\mathit{\text{z}}\mathrm{d}\mathit{\text{z}}/{\int }_{-\frac{h}{2}}^{\frac{h}{2}}E\left(\mathit{\text{z}}\right)\mathrm{d}\mathit{\text{z}}.

Here we present the outcomes of the stress−strain relationship when Eq. (10) is returned into Eqs. (13) and the subsequent results are substituted into Eqs. (20).

\begin{array}{}\text{(22a)}& \left\{\begin{array}{c}\begin{array}{c}{\mathcal{N}}_{xx}\\ {\mathcal{N}}_{yy}\\ {\mathcal{N}}_{xy}\end{array}\\ \begin{array}{c}{\mathcal{M}}_{xx}\\ {\mathcal{M}}_{yy}\\ {\mathcal{M}}_{xy}\end{array}\\ \begin{array}{c}{\mathcal{L}}_{xx}\\ {\mathcal{L}}_{yy}\\ {\mathcal{L}}_{xy}\end{array}\end{array}\right\}=\left[\begin{array}{ccccccccc}{\mathcal{A}}_{11}& {\mathcal{A}}_{12}& 0& {\mathcal{B}}_{11}& {\mathcal{B}}_{12}& 0& {\mathcal{B}}_{11}^s& {\mathcal{B}}_{12}^s& 0\\ {\mathcal{A}}_{12}& {\mathcal{A}}_{22}& 0& {\mathcal{B}}_{12}& {\mathcal{B}}_{22}& 0& {\mathcal{B}}_{12}^s& {\mathcal{B}}_{22}^s& 0\\ 0& 0& {\mathcal{A}}_{66}& 0& 0& {\mathcal{B}}_{66}& 0& 0& {\mathcal{B}}_{66}^s\\ {\overline{\mathcal{A}}}_{11}& {\overline{\mathcal{A}}}_{12}& 0& {\mathcal{F}}_{11}& {\mathcal{F}}_{12}& 0& {\mathcal{F}}_{11}^s& {\mathcal{F}}_{12}^s& 0\\ {\overline{\mathcal{A}}}_{12}& {\overline{\mathcal{A}}}_{22}& 0& {\mathcal{F}}_{12}& {\mathcal{F}}_{22}& 0& {\mathcal{F}}_{12}^s& {\mathcal{F}}_{22}^s& 0\\ 0& 0& {\overline{\mathcal{A}}}_{66}& 0& 0& {\mathcal{F}}_{66}& 0& 0& {\mathcal{F}}_{66}^s\\ {\overline{\mathcal{B}}}_{11}^s& {\overline{\mathcal{B}}}_{12}^s& 0& {\overline{\mathcal{F}}}_{11}^s& {\overline{\mathcal{F}}}_{12}^s& 0& {\mathcal{H}}_{11}& {\mathcal{H}}_{12}& 0\\ {\overline{\mathcal{B}}}_{12}^s& {\overline{\mathcal{B}}}_{22}^s& 0& {\overline{\mathcal{F}}}_{12}^s& {\overline{\mathcal{F}}}_{22}^s& 0& {\mathcal{H}}_{12}& {\mathcal{H}}_{22}& 0\\ 0& 0& {\overline{\mathcal{B}}}_{66}^s& 0& 0& {\overline{\mathcal{F}}}_{66}^s& 0& 0& {\mathcal{H}}_{66}\end{array}\right]\left\{\begin{array}{c}\begin{array}{c}{u}_{0,x}\\ v_{0,y}\\ u_{0,y}+v_{0,x}\end{array}\\ \begin{array}{c}-{\beta }_{,xx}^{\mathrm{b}}\\ -{\beta }_{,yy}^{\mathrm{b}}\\ -2{\beta }_{,xy}^{\mathrm{b}}\end{array}\\ \begin{array}{c}-{\beta }_{,xx}^{\mathrm{s}}\\ -{\beta }_{,yy}^{\mathrm{s}}\\ -2{\beta }_{,xy}^{\mathrm{s}}\end{array}\end{array}\right\},\end{array}

\begin{array}{rl}& \left\{\begin{array}{c}{\mathcal{Q}}_{x\mathit{\text{z}}}\\ {\mathcal{Q}}_{y\mathit{\text{z}}}\end{array}\right\}=\left[\begin{array}{cc}{\mathcal{A}}_{44}^{\mathrm{s}}& 0\\ 0& {\mathcal{A}}_{55}^{\mathrm{s}}\end{array}\right]\left\{\begin{array}{c}{\beta }_{,x}^{\mathrm{s}}\\ {\beta }_{,y}^{\mathrm{s}}\end{array}\right\},\\ & \left\{\begin{array}{c}{\mathcal{X}}_{xx}\\ {\mathcal{X}}_{yy}\end{array}\right\}=\left[\begin{array}{ccc}{\mathcal{C}}_{11}& {\mathcal{C}}_{12}& {\mathcal{C}}_{13}\\ {\mathcal{D}}_{11}& {\mathcal{D}}_{12}& {\mathcal{D}}_{13}\end{array}\right]\left\{\begin{array}{c}{-u}_{0,x}-v_{0,y}\\ {\beta }_{,xx}^{\mathrm{b}}+{\beta }_{,yy}^{\mathrm{b}}\\ {\beta }_{,xx}^{\mathrm{s}}+{\beta }_{,yy}^{\mathrm{s}}\end{array}\right\},\end{array}

\begin{array}{}\text{(22c)}& \left\{\begin{array}{c}{\mathcal{S}}_{xx}\\ {\mathcal{S}}_{yy}\end{array}\right\}=\left[\begin{array}{ccc}{\mathcal{C}}_{11}& {\mathcal{C}}_{12}& {\mathcal{C}}_{13}\\ {\mathcal{D}}_{11}& {\mathcal{D}}_{12}& {\mathcal{D}}_{13}\end{array}\right]\left\{\begin{array}{c}{-u}_{0,x}-v_{0,y}\\ {\beta }_{,xx}^{\mathrm{b}}+{\beta }_{,yy}^{\mathrm{b}}\\ {\beta }_{,xx}^{\mathrm{s}}+{\beta }_{,yy}^{\mathrm{s}}\end{array}\right\},\end{array}

in which {\mathcal{A}}_{kl},{\mathcal{B}}_{kl},{\mathcal{B}}_{kl}^{\mathrm{s}},{\overline{\mathcal{A}}}_{kl},{\mathcal{F}}_{kl},{\mathcal{F}}_{kl}^{\mathrm{s}},{\overline{\mathcal{B}}}_{kl}^{\mathrm{s}},{\overline{\mathcal{F}}}_{kl}^{\mathrm{s}},{\mathcal{H}}_{kl},{\mathcal{A}}_{kl}^{\mathrm{s}},{\mathcal{C}}_{kl}, and {\mathcal{D}}_{kl} are the stiffness parameters of the nanoplate material and are given in Appendix A:

The elastic energy of the medium is computed by the below expression, from sources [60–63]:

(23)U_{\mathrm{f}}=\frac{1}{2}{\int }_{S_{\mathrm{e}}}\left(k_{\mathrm{w}}{\left({\beta }^{\mathrm{b}}+{\beta }^{\mathrm{s}}\right)}^2+k_{\mathrm{s}}{\left({\beta }_{,x}^{\mathrm{b}}+{\beta }_{,x}^{\mathrm{s}}\right)}^2+k_{\mathrm{s}}{\left({\beta }_{,y}^{\mathrm{b}}+{\beta }_{,y}^{\mathrm{s}}\right)}^2\right)\mathrm{d}x\mathrm{d}y.

The potential energy can be calculated using the following expression:

(24)V=\frac{1}{2}{\int }_{S_{\mathrm{e}}}\left(-q\left(x,y\right)\left({\beta }^{\mathrm{b}}+{\beta }^{\mathrm{s}}\right)\right)\mathrm{d}x\mathrm{d}y.

The kinetic energy is expressed as follows:

(25)\begin{array}{rl}T=& \frac{1}{2}{\int }_{S_{\mathrm{e}}}{\int }_{-h/2}^{h/2}\rho \left(\mathit{\text{z}}\right)({({\dot{u}}_0-\widetilde{\mathit{\text{z}}}{\dot{\beta }}_{,x}^{\mathrm{b}}-\tilde{F}{\dot{\beta }}_{,x}^{\mathrm{s}})}^2\\ & +{({\dot{v}}_0-\widetilde{\mathit{\text{z}}}{\dot{\beta }}_{,y}^{\mathrm{b}}-\tilde{F}{\dot{\beta }}_{,y}^{\mathrm{s}})}^2+{(({\dot{\beta }}^{\mathrm{b}}+{\dot{\beta }}^{\mathrm{s}}))}^2)\mathrm{d}\mathit{\text{z}}\mathrm{d}x\mathrm{d}y.\end{array}

By substituting Eqs. (13), (20), (24)–(26) into (19) and integrating by parts, the equations of motion of the FPF piezoelectric nanoplate can be derived:

(26){\mathcal{N}}_{xx,x}+{\mathcal{N}}_{xy,y}=I_0{\ddot{u}}_0-I_1{\ddot{\beta }}_{,x}^{\mathrm{b}}-J_1{\ddot{\beta }}_{,x}^{\mathrm{s}}-I_{0\mu }{\mathrm{\nabla }}^2{\ddot{u}}_0+I_{1\mu }{\mathrm{\nabla }}^2{\ddot{\beta }}_{,x}^{\mathrm{b}}+J_{1\mu }{\mathrm{\nabla }}^2{\ddot{\beta }}_{,x}^{\mathrm{s}},

(27){\mathcal{N}}_{yy,y}+{\mathcal{N}}_{xy,x}=I_0{\ddot{v}}_0-I_1{\ddot{\beta }}_{,y}^{\mathrm{b}}-J_1{\ddot{\beta }}_{,y}^{\mathrm{s}}-I_{0\mu }{\mathrm{\nabla }}^2{\ddot{v}}_0+I_{1\mu }{\mathrm{\nabla }}^2{\ddot{\beta }}_{,y}^{\mathrm{b}}+J_{1\mu }{\mathrm{\nabla }}^2{\ddot{\beta }}_{,y}^{\mathrm{s}},

(28)\begin{array}{rl}& {\mathcal{M}}_{xx,xx}+{2\mathcal{M}}_{xy,xy}+{\mathcal{M}}_{yy,yy}+{\mathcal{X}}_{xx,xx}+{\mathcal{S}}_{xx,yy}\\ =& -\left(1-{\mu }^2\left(h/2\right){\mathrm{\nabla }}^2\right)q\left(x,y\right)\\ & +\left(1-{\mu }^2\left(-h/2\right){\mathrm{\nabla }}^2\right)\left(k_{\mathrm{w}}\left({\beta }^{\mathrm{b}}+{\beta }^{\mathrm{s}}\right)-k_{\mathrm{s}}{\mathrm{\nabla }}^2\left({\beta }^{\mathrm{b}}+{\beta }^{\mathrm{s}}\right)\right)\\ & +I_0\left({\ddot{\beta }}^{\mathrm{b}}+{\ddot{\beta }}^{\mathrm{s}}\right)+I_1\left({\ddot{u}}_{0,x}{+\ddot{v}}_{0,y}\right)-I_2{\mathrm{\nabla }}^2{\ddot{\beta }}^{\mathrm{b}}-J_2{\mathrm{\nabla }}^2{\ddot{\beta }}^{\mathrm{s}}\\ & -I_{0\text{μ}}{\mathrm{\nabla }}^2\left({\ddot{\beta }}^{\mathrm{b}}+{\ddot{\beta }}^{\mathrm{s}}\right)-I_{1\text{μ}}{\mathrm{\nabla }}^2\left({\ddot{u}}_{0,x}{+\ddot{v}}_{0,y}\right)\\ & +I_{2\text{μ}}{{\mathrm{\nabla }}^2\mathrm{\nabla }}^2{\ddot{\beta }}^{\mathrm{b}}+J_{2\text{μ}}{{\mathrm{\nabla }}^2\mathrm{\nabla }}^2{\ddot{\beta }}^{\mathrm{s}}{\ddot{\beta }}_{,y}^{\mathrm{s}},\end{array}

(29)\begin{array}{rl}& {\mathcal{L}}_{xx,xx}+{2\mathcal{L}}_{xy,xy}+{\mathcal{L}}_{yy,yy}+{\mathcal{Q}}_{xz,x}+{\mathcal{Q}}_{yz,y}+{\mathcal{X}}_{yy,xx}+{\mathcal{S}}_{yy,yy}\\ =& -\left(1-{\mu }^2\left(h/2\right){\mathrm{\nabla }}^2\right)q\left(x,y\right)\\ & +\left(1-{\mu }^2\left(-h/2\right){\mathrm{\nabla }}^2\right)\left(k_{\mathrm{w}}\left({\beta }^{\mathrm{b}}+{\beta }^{\mathrm{s}}\right)-k_{\mathrm{s}}{\mathrm{\nabla }}^2\left({\beta }^{\mathrm{b}}+{\beta }^{\mathrm{s}}\right)\right)\\ & +I_0\left({\ddot{\beta }}^{\mathrm{b}}+{\ddot{\beta }}^{\mathrm{s}}\right)+J_1\left({\ddot{u}}_{0,x}{+\ddot{v}}_{0,y}\right)-J_2{\mathrm{\nabla }}^2{\ddot{\beta }}^{\mathrm{b}}-K_2{\mathrm{\nabla }}^2{\ddot{\beta }}^{\mathrm{s}}\\ & -I_{0\mu }{\mathrm{\nabla }}^2\left({\ddot{\beta }}^{\mathrm{b}}+{\ddot{\beta }}^{\mathrm{s}}\right)+J_{1\mu }{\mathrm{\nabla }}^2\left({\ddot{u}}_{0,x}{+\ddot{v}}_{0,y}\right)\\ & +J_{2\mu }{{\mathrm{\nabla }}^2\mathrm{\nabla }}^2{\ddot{\beta }}^{\mathrm{b}}+K_{2\mu }{\mathrm{\nabla }}^2{\mathrm{\nabla }}^2{\ddot{\beta }}^{\mathrm{s}},\end{array}

where I_i,J_j,K_k,I_{i\text{μ}},J_{j\text{μ}},\mathrm{a}\mathrm{n}\mathrm{d}{K}_{k\text{μ}} are the mass-related parameters and the nonlocal parameters are defined below:

(30){(I}_0,I_1,I_2,J_1,J_2,K_2)={\int }_{-\frac{h}{2}}^{\frac{h}{2}}\left(1,\widetilde{\mathit{\text{z}}},{\widetilde{\mathit{\text{z}}}}^2,\tilde{F},\tilde{z}\tilde{F},{\tilde{F}}^2\right)\rho \left(\mathit{\text{z}}\right)\mathrm{d}\mathit{\text{z}},

(31)\begin{array}{rl}& {(I}_{0\mu },I_{1\mu },I_{2\mu },J_{1\mu },J_{2\mu },K_{2\mu })\\ =& {\int }_{-\frac{h}{2}}^{\frac{h}{2}}\left(1,\widetilde{\mathit{\text{z}}},{\widetilde{\mathit{\text{z}}}}^2,\tilde{F},\widetilde{\mathit{\text{z}}}\tilde{F},{\tilde{F}}^2\right)\rho \left(\mathit{\text{z}}\right){\mu }^2\left(\mathit{\text{z}}\right)\mathrm{d}\mathit{\text{z}}.\end{array}

By applying the Galerkin approach in Eqs. (27)–(30) with weight expressions \delta u_0,{\delta v}_0,{\delta \beta }^{\mathrm{b}}, and {\delta \beta }^{\mathrm{s}}, respectively, using the divergence theorem, and then merging Eqs. (27)–(30), one obtains the below expression:

(32)\begin{array}{rl}& {\int }_{\psi }^{}\left({\sigma }_{ij,j}\delta {\epsilon }_{ij}+{\tau }_{iij,j}\delta {\chi }_{iij}\right)\mathrm{d}\psi \\ & +{\int }_{S_{\mathrm{e}}}\left(1-{\mu }^2\left(-h/2\right){\mathrm{\nabla }}^2\right)\left(k_{\mathrm{w}}{u}_i-k_{\mathrm{s}}{\mathrm{\nabla }}^2{u}_i\right)\delta u_i\mathrm{d}{S}_{\mathrm{e}}\\ & +{\int }_{\psi }^{}\left(1-{\mu }^2{\mathrm{\nabla }}^2\right)\rho {\ddot{u}}_i\delta u_i\mathrm{d}\psi \\ =& {\int }_{S_{\mathrm{e}}}^{}\left(1-{\mu }^2\left(h/2\right){\mathrm{\nabla }}^2\right)q_i\delta u_i\mathrm{d}{S}_{\mathrm{e}}\\ & +{\int }_{{\mathrm{\Lambda }}_{\mathrm{g}}}^{}{\sigma }_{ij}{n}_j\delta u_i\mathrm{d}{\mathrm{\Lambda }}_{\mathrm{g}}+{\int }_{{\mathrm{\Lambda }}_{\mathrm{g}}}^{}{\tau }_{iij}{n}_j\delta {\mathrm{\Phi }}_i\mathrm{d}{\mathrm{\Lambda }}_{\mathrm{g}},\end{array}

where {\mathrm{\Lambda }}_{\mathrm{g}} is the Neumann bound, \psi =S_{\mathrm{e}}x\left(-{\displaystyle \frac{h}{2}},{\displaystyle \frac{h}{2}}\right) is a region to integrate. In the present paper, the traction on the Neumann bound is disregarded [64–65]. Equation (33) is rewritten as follows:

(33)\begin{array}{rl}& {\int }_{\psi }^{}\left({\sigma }_{ij,j}\delta {\epsilon }_{ij}+{\tau }_{iij,j}\delta {\chi }_{iij}\right)\mathrm{d}\psi +{\int }_{S_{\mathrm{e}}}\left(1-{\mu }^2\left(-h/2\right){\mathrm{\nabla }}^2\right)\left(k_{\mathrm{w}}{u}_i-k_{\mathrm{s}}{\mathrm{\nabla }}^2{u}_i\right)\delta u_i\mathrm{d}{S}_{\mathrm{e}}\\ =& {\int }_{\psi }^{}\left(1-{\mu }^2{\mathrm{\nabla }}^2\right)\rho {\ddot{u}}_i\delta u_i\mathrm{d}\psi +{\int }_{S_{\mathrm{e}}}^{}\left(1-{\mu }^2\left(h/2\right){\mathrm{\nabla }}^2\right)q_i\delta u_i\mathrm{d}{S}_{\mathrm{e}}.\end{array}

Equation (34) can be rewritten as:

(34)\begin{array}{rl}& {\int }_{S_{\mathrm{e}}}\left({\mathit{\epsilon }}^{\mathrm{T}}{\mathit{D}}_{\mathrm{b}}\delta \mathit{\epsilon }+{\mathit{\gamma }}^{\mathrm{T}}{\mathcal{A}}^{\mathrm{s}}\delta \mathit{\gamma }+{\mathit{\Gamma }}_x^{\mathrm{T}}{\mathit{D}}_{\mathrm{f}}\delta \mathit{\phi }+{\mathbf{\Gamma }}_y^{\mathrm{T}}{\mathit{D}}_{\mathrm{f}}\delta \mathit{\phi }\right)\mathrm{d}{S}_{\mathrm{e}}+\left(k_{\mathrm{w}}{\int }_{S_{\mathrm{e}}}{u}_{\mathit{\text{z}}}^{\mathrm{T}}\delta u_{\mathit{\text{z}}}\mathrm{d}{S}_{\mathrm{e}}+\left(k_{\mathrm{w}}{\mu }^2\left(-h/2\right)+k_{\mathrm{s}}\right){\int }_{S_{\mathrm{e}}}\left(u_{\mathit{\text{z}},x}^{\mathrm{T}}\delta u_{\mathit{\text{z}},x}+u_{\mathit{\text{z}},y}^{\mathrm{T}}\delta u_{\mathit{\text{z}},y}\right)\mathrm{d}{S}_{\mathrm{e}}\right)\\ & +k_{\mathrm{s}}{\mu }^2\left(-h/2\right){\int }_{S_{\mathrm{e}}}\left({\mathrm{\nabla }u}_{\mathit{\text{z}},xx}^{\mathrm{T}}\mathrm{\nabla }\delta u_{\mathit{\text{z}},xx}+{\mathrm{\nabla }u}_{\mathit{\text{z}},yy}^{\mathrm{T}}\mathrm{\nabla }\delta u_{\mathit{\text{z}},yy}\right)\mathrm{d}{S}_{\mathrm{e}}={\int }_{S_{\mathrm{e}}}\left(\left(q(x,y)-{\mu }^2\left(h/2\right)\left(q_{,xx}+q_{,yy}\right)\right)\delta u_{\mathit{\text{z}}}-{q(x,y)\mu }^2\left(h/2\right)\left(\delta u_{\mathit{\text{z}},xx}+\delta u_{\mathit{\text{z}},yy}\right)\right)\mathrm{d}{S}_{\mathrm{e}}\\ & +{\int }_{S_{\mathrm{e}}}^{}{\mathit{u}}^{\mathrm{T}}{\mathit{H}}_{\mathrm{m}}\delta \ddot{\mathit{u}}\mathrm{d}{S}_{\mathrm{e}}+{\int }_{S_{\mathrm{e}}}^{}\left({\mathit{u}}_{,x}^{\mathrm{T}}{\mathit{H}}_{\mathrm{m}\text{μ}}\delta {\ddot{\mathit{u}}}_{,x}+{\mathit{u}}_{,y}^{\mathrm{T}}{\mathit{H}}_{\mathrm{m}\text{μ}}\delta {\ddot{\mathit{u}}}_{,y}\right)\mathrm{d}{S}_{\mathrm{e}},\end{array}

where

\begin{array}{rl}& \mathit{\epsilon }=\left\{\begin{array}{c}{\mathit{\epsilon }}^0\\ {\mathit{\epsilon }}^1\\ {\mathit{\epsilon }}^2\end{array}\right\},{\mathit{\epsilon }}^0=\left\{\begin{array}{c}{u}_{0,x}\\ v_{0,y}\\ u_{0,y}+v_{0,x}\end{array}\right\},{\mathit{\epsilon }}^1=-\left\{\begin{array}{c}{\beta }_{,xx}^{\mathrm{b}}\\ {\beta }_{,yy}^{\mathrm{b}}\\ {2\beta }_{,xy}^{\mathrm{b}}\end{array}\right\},\\ & {\mathit{\epsilon }}^2=-\left\{\begin{array}{c}{\beta }_{,xx}^{\mathrm{s}}\\ {\beta }_{,yy}^{\mathrm{s}}\\ {2\beta }_{,xy}^{\mathrm{s}}\end{array}\right\},{\mathit{\gamma }}^0=\left\{\begin{array}{c}{\beta }_{,x}^{\mathrm{s}}\\ {\beta }_{,y}^{\mathrm{s}}\end{array}\right\},\end{array}

\begin{array}{}\text{(35b)}& {\mathit{\Gamma }}_x=-\left\{\begin{array}{c}{\beta }_{,xx}^{\mathrm{b}}\\ {\beta }_{,xx}^{\mathrm{s}}\end{array}\right\},{\mathit{\Gamma }}_y=-\left\{\begin{array}{c}{\beta }_{,yy}^{\mathrm{b}}\\ {\beta }_{,yy}^{\mathrm{s}}\end{array}\right\},\mathit{\phi }=\left\{\begin{array}{c}{-u}_{0,x}-v_{0,y}\\ {\beta }_{,xx}^{\mathrm{b}}+{\beta }_{,yy}^{\mathrm{b}}\\ {\beta }_{,xx}^{\mathrm{s}}+{\beta }_{,yy}^{\mathrm{s}}\end{array}\right\}.\end{array}

(36){\mathit{D}}_{\mathrm{b}}=\left[\begin{array}{ccc}\mathcal{A}& \mathcal{B}& {\mathcal{B}}^{\mathrm{s}}\\ \overline{\mathcal{A}}& \mathcal{F}& {\mathcal{F}}^{\mathrm{s}}\\ {\overline{\mathcal{B}}}^{\mathrm{s}}& {\overline{\mathcal{F}}}^{\mathrm{s}}& \mathcal{H}\end{array}\right],{\mathit{D}}_{\mathrm{f}}=\left[\begin{array}{ccc}{\mathcal{C}}_{11}& {\mathcal{C}}_{12}& {\mathcal{C}}_{13}\\ {\mathcal{D}}_{11}& {\mathcal{D}}_{12}& {\mathcal{D}}_{13}\end{array}\right].

(37)\mathit{u}={\left\{\begin{array}{cccccccc}{u}_0& v_0& {\beta }^{\mathrm{b}}& {\beta }^{\mathrm{s}}& {\beta }_{,x}^{\mathrm{b}}& {\beta }_{,x}^{\mathrm{s}}& {\beta }_{,y}^{\mathrm{b}}& {\beta }_{,y}^{\mathrm{s}}\end{array}\right\}}^{\mathrm{T}}.

\begin{array}{}\text{(38a)}& {\mathit{H}}_{\mathrm{m}}=\left[\begin{array}{cccccccc}{I}_0& 0& 0& 0& I_1& J_1& 0& 0\\ 0& I_0& 0& 0& 0& 0& I_1& J_1\\ 0& 0& I_0& 0& 0& 0& 0& 0\\ 0& 0& 0& I_0& 0& 0& 0& 0\\ I_1& 0& 0& 0& I_2& J_2& 0& 0\\ J_1& 0& 0& 0& J_2& K_2& 0& 0\\ 0& I_1& 0& 0& 0& 0& I_2& J_2\\ 0& J_1& 0& 0& 0& 0& J_2& K_2\end{array}\right],\end{array}

\begin{array}{}\text{(38b)}& {\mathit{H}}_{\mathrm{m}\text{μ}}=\left[\begin{array}{cccccccc}{I}_{0\text{μ}}& 0& 0& 0& I_{1\text{μ}}& J_{1\text{μ}}& 0& 0\\ 0& I_{0\text{μ}}& 0& 0& 0& 0& I_{1\text{μ}}& J_{1\text{μ}}\\ 0& 0& I_{0\text{μ}}& 0& 0& 0& 0& 0\\ 0& 0& 0& I_{0\text{μ}}& 0& 0& 0& 0\\ I_{1\text{μ}}& 0& 0& 0& I_{2\text{μ}}& J_{2\text{μ}}& 0& 0\\ J_{1\text{μ}}& 0& 0& 0& J_{2\text{μ}}& K_{2\text{μ}}& 0& 0\\ 0& I_{1\text{μ}}& 0& 0& 0& 0& I_{2\text{μ}}& J_{2\text{μ}}\\ 0& J_{1\text{μ}}& 0& 0& 0& 0& J_{2\text{μ}}& K_{2\text{μ}}\end{array}\right].\end{array}

3.2 Exact solution

The displacement fields are drawn out to obey fully simply supported bound constraints of the rectangular FPF piezoelectric nanoplate based on Navier’s form [66–70]:

(39)u_0=\sum _{r=1}^{\mathrm{\infty }}\sum _{s=1}^{\mathrm{\infty }}{U}_{0rs}\mathrm{c}\mathrm{o}\mathrm{s}\left(\theta x\right)\mathrm{s}\mathrm{i}\mathrm{n}\left(\phi y\right),

(40)v_0=\sum _{r=1}^{\mathrm{\infty }}\sum _{s=1}^{\mathrm{\infty }}{V}_{0rs}\mathrm{s}\mathrm{i}\mathrm{n}\left(\theta x\right)\mathrm{c}\mathrm{o}\mathrm{s}\left(\phi y\right),

(41){\beta }^{\mathrm{b}}=\sum _{r=1}^{\mathrm{\infty }}\sum _{s=1}^{\mathrm{\infty }}{W}_{rs}^{\mathrm{b}}\mathrm{s}\mathrm{i}\mathrm{n}\left(\theta x\right)\mathrm{s}\mathrm{i}\mathrm{n}\left(\phi y\right),

(42){\beta }^{\mathrm{s}}=\sum _{r=1}^{\mathrm{\infty }}\sum _{s=1}^{\mathrm{\infty }}{W}_{rs}^{\mathrm{s}}\mathrm{s}\mathrm{i}\mathrm{n}\left(\theta x\right)\mathrm{s}\mathrm{i}\mathrm{n}\left(\phi y\right),

where \theta =r\text{π}/a,\phi =s\text{π}/b; \left\{U_{0rs},V_{0rs},W_{rs}^{\mathrm{b}},W_{rs}^{\mathrm{s}}\right\} are the unknown coefficients; r and s denote numbers of the modes of the FPF piezoelectric nanoplate. The external transverse force q\left(x,y\right) is developed in the double-Fourier sine series as [66]:

(43)q\left(x,y\right)=\sum _{r=1}^{\mathrm{\infty }}\sum _{s=1}^{\mathrm{\infty }}{Q}_{rs}\mathrm{s}\mathrm{i}\mathrm{n}\left(\theta x\right)\mathrm{s}\mathrm{i}\mathrm{n}\left(\phi y\right),

where

(44)Q_{rs}=\frac{4}{ab}{\int }_0^a{\int }_0^b{q}_0\mathrm{s}\mathrm{i}\mathrm{n}\left(\theta x\right)\mathrm{s}\mathrm{i}\mathrm{n}\left(\phi y\right)\mathrm{d}x\mathrm{d}y.

For the sinusoidally distributed load Q_{rs}=q_0 while for the uniformly distributed load Q_{\mathrm{r}\mathrm{s}}={\displaystyle \frac{{16q}_0}{rs{\text{π}}^2}}. Substituting Eqs. (40)–(43) into Eqs. (27)–(30), respectively, leads to:

(45)\begin{array}{rl}& \left[\begin{array}{cccc}{m}_{11}& m_{12}& m_{13}& m_{14}\\ m_{12}& m_{22}& m_{23}& m_{24}\\ m_{13}& m_{23}& m_{33}& m_{34}\\ m_{14}& m_{24}& m_{34}& m_{44}\end{array}\right]\left\{\begin{array}{c}{\ddot{U}}_{0rs}\\ {\ddot{V}}_{0rs}\\ {\ddot{W}}_{rs}^{\mathrm{b}}\\ {\ddot{W}}_{rs}^{\mathrm{s}}\end{array}\right\}\\ & +\left[\begin{array}{cccc}{k}_{11}& k_{12}& k_{13}& k_{14}\\ k_{12}& k_{22}& k_{23}& k_{24}\\ k_{13}& k_{23}& k_{33}& k_{34}\\ k_{14}& k_{24}& k_{34}& k_{44}\end{array}\right]\left\{\begin{array}{c}{U}_{0rs}\\ V_{0rs}\\ W_{rs}^{\mathrm{b}}\\ W_{rs}^{\mathrm{s}}\end{array}\right\}=\left\{\begin{array}{c}0\\ 0\\ \tilde{Q}\\ \tilde{Q}\end{array}\right\}.\end{array}

The coefficients of the matrices and vectors are given in Appendix B.

3.3 The isogeometric approach

Using NURBS basis functions [31,32,52,53] the displacement fields in the middle plane of the FPF piezoelectric nanoplate are expressed by:

(46)\begin{array}{rl}{\mathit{q}}^{\mathrm{h}}=& {\sum }_{I=1}^{ne}\left[\begin{array}{cccc}{R}_I(x,y)& 0& 0& 0\\ 0& R_I(x,y)& 0& 0\\ 0& 0& R_I(x,y)& 0\\ 0& 0& 0& R_I(x,y)\end{array}\right]\left\{\begin{array}{c}{u}_{0I}\\ v_{0I}\\ {\beta }_I^{\mathrm{b}}\\ {\beta }_I^{\mathrm{s}}\end{array}\right\}\\ =& {\sum }_{I=1}^{ne}{\mathit{R}}_I(x,y){\mathit{q}}_I,\end{array}

where ne=(p+1)(q+1) and is the number of control points per physical part; R_I is the NURBS basis function defined in Ref. [31]; {\mathit{q}}_I={\left\{u_{0I},v_{0I},{\beta }_I^{\mathrm{b}},{\beta }_I^{\mathrm{s}}\right\}}^{\mathrm{T}}denotes the unknown displacement part at the controlling point I.

By substituting Eq. (47) into Eq. (36), the matrices can be rewritten as:

\begin{array}{rl}& {\mathit{\epsilon }}^0={\sum }_{I=1}^{ne}{\mathit{B}}_I^1{\mathit{q}}_I,{\mathit{\epsilon }}^1={\sum }_{I=1}^{ne}{\mathit{B}}_I^2{\mathit{q}}_I,\\ & {\mathit{\epsilon }}^2={\sum }_{I=1}^{ne}{\mathit{B}}_I^3{\mathit{q}}_I,{\mathit{\gamma }}^0={\sum }_{I=1}^{ne}{\mathit{B}}_I^4{\mathit{q}}_I{\sum }_{I=1}^{ne}{\mathit{R}}_I(x,y){\mathit{q}}_I,\end{array}

\begin{array}{}\text{(47b)}& {\mathit{\Gamma }}_x={\sum }_{I=1}^{ne}{\mathit{B}}_I^5{\mathit{q}}_I,{\mathit{\Gamma }}_y={\sum }_{I=1}^{ne}{\mathit{B}}_I^6{\mathit{q}}_I,\mathit{\phi }={\sum }_{I=1}^{ne}{\mathit{B}}_I^7{\mathit{q}}_I.\end{array}

By substituting Eq. (47) into Eq. (38), displacement component \mathit{u} is drawn by:

(48)\mathit{u}={\sum }_{I=1}^{ne}{\mathit{N}}_I{\mathit{q}}_I;{\mathit{N}}_I=\left[\begin{array}{ccc}{\mathit{N}}_I^1;& {\mathit{N}}_I^2;& {\mathit{N}}_I^3\end{array}\right].

The displacement vector u_{\mathit{\text{z}}} is:

(49)u_{\mathit{\text{z}}}={\sum }_{I=1}^{ne}\left[\begin{array}{cccc}0& 0& R_I& R_I\end{array}\right]{\mathit{q}}_I={\sum }_{I=1}^{ne}{\mathit{B}}_I^8{\mathit{q}}_I,

where the strain matrix components are shown in Appendix C.

Substituting Eqs. (48)–(50) into Eq. (35), the forced oscillation equation of the FPF piezoelectric nanoplate can be written in the form below:

(50)\mathit{M}\ddot{\mathit{q}}+\mathit{C}\dot{\mathit{q}}+\mathit{K}\mathit{q}=\mathit{F}.

The stiffness matrix K is:

(51)\mathit{K}={\mathit{K}}^{\mathrm{p}}+{\mathit{K}}^{\mathrm{f}\mathrm{l}\mathrm{e}\mathrm{x}\mathrm{o}}+{\mathit{K}}^{\mathrm{f}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{d}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}},

where {\mathit{K}}^{\mathrm{p}},{\mathit{K}}^{\mathrm{f}\mathrm{l}\mathrm{e}\mathrm{x}\mathrm{o}},{\mathit{K}}^{\mathrm{f}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{d}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}} are the stiffness matrix components and are calculated as follows:

(52){\mathit{K}}^{\mathrm{p}}={\int }_{S_{\mathrm{e}}}^{}{\left\{\begin{array}{c}{\mathit{B}}^1\\ {\mathit{B}}^2\\ {\mathit{B}}^3\\ {\mathit{B}}^4\end{array}\right\}}^{\mathrm{T}}\left[\begin{array}{cccc}\mathcal{A}& \mathcal{B}& {\mathcal{B}}^{\mathrm{s}}& \mathbf{0}\\ \overline{\mathcal{A}}& \mathcal{F}& {\mathcal{F}}^{\mathrm{s}}& \mathbf{0}\\ {\overline{\mathcal{B}}}^{\mathrm{s}}& {\overline{\mathcal{F}}}^{\mathrm{s}}& \mathcal{H}& \mathbf{0}\\ \mathbf{0}& \mathbf{0}& \mathbf{0}& {\mathcal{A}}^{\mathrm{s}}\end{array}\right]\left\{\begin{array}{c}{\mathit{B}}^1\\ {\mathit{B}}^2\\ {\mathit{B}}^3\\ {\mathit{B}}^4\end{array}\right\}\mathrm{d}{S}_{\mathrm{e}},

(53){\mathit{K}}^{\mathrm{f}\mathrm{l}\mathrm{e}\mathrm{x}\mathrm{o}}={\int }_{S_{\mathrm{e}}}^{}\left({\left({\mathit{B}}^5\right)}^{\mathrm{T}}{\mathit{D}}_{\mathrm{f}}{\mathit{B}}^7+{\left({\mathit{B}}^6\right)}^{\mathrm{T}}{\mathit{D}}_{\mathrm{f}}{\mathit{B}}^7\right)\mathrm{d}{S}_{\mathrm{e}},

(54)\begin{array}{rl}{\mathit{K}}^{\mathrm{f}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{d}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}=& {\int }_{S_{\mathrm{e}}}^{}(k_{\mathrm{w}}{\left({\mathit{B}}^8\right)}^{\mathrm{T}}{\mathit{B}}^8+\left(k_{\mathrm{w}}{\mu }^2\left(-\frac{h}{2}\right)+k_{\mathrm{s}}\right)\\ & \cdot \left({\left({\mathit{B}}_{,x}^8\right)}^{\mathrm{T}}{\mathit{B}}_{,x}^8+{\left({\mathit{B}}_{,y}^8\right)}^{\mathrm{T}}{\mathit{B}}_{,y}^8\right)+k_{\mathrm{s}}{\mu }^2\left(-\frac{h}{2}\right)\\ & \cdot ({\left({\mathit{B}}_{,xx}^8\right)}^{\mathrm{T}}{\mathit{B}}_{,xx}^8+{\left({\mathit{B}}_{,yy}^8\right)}^{\mathrm{T}}{\mathit{B}}_{,yy}^8\\ & +\left({\left({\mathit{B}}_{,xx}^8\right)}^{\mathrm{T}}{\mathit{B}}_{,yy}^8+{\left({\mathit{B}}_{,yy}^8\right)}^{\mathrm{T}}{\mathit{B}}_{,xx}^8\right)))\mathrm{d}{S}_{\mathrm{e}}.\end{array}

The mass matrix M is:

(55)\mathit{M}={\int }_{S_{\mathrm{e}}}^{}\left({\mathit{N}}^{\mathrm{T}}{\mathit{H}}_{\mathrm{m}}\mathit{N}+({\mathit{N}}_{,x}^{\mathrm{T}}{\mathit{H}}_{\mathrm{m}\text{μ}}{\mathit{N}}_{,x}+{\mathit{N}}_{,y}^{\mathrm{T}}{\mathit{H}}_{\mathrm{m}\mu }{\mathit{N}}_{,y})\right)\mathrm{d}{S}_{\mathrm{e}}.

The load vector F is:

(56)\begin{array}{rl}\mathit{F}=& -{\int }_{S_{\mathrm{e}}}(q(x,y)\left({\left({\mathit{B}}^8\right)}^{\mathrm{T}}-{\mu }^2\left(\frac{h}{2}\right)\left({\left({\mathit{B}}_{,xx}^8\right)}^{\mathrm{T}}+{\left({\mathit{B}}_{,yy}^8\right)}^{\mathrm{T}}\right)\right)\\ & -{\mu }^2\left(\frac{h}{2}\right){\left({\mathit{B}}^8\right)}^{\mathrm{T}}\left(q_{,xx}+q_{,yy}\right))\mathrm{d}{S}_{\mathrm{e}}.\end{array}

The damping matrix \mathit{C} [71] is

(57)\mathit{C}={ϵ}_1\mathit{M}+{ϵ}_2\mathit{K},

where {ϵ}_1 and {ϵ}_2 are the damping constant coefficients computed based on the structural damping scale and natural frequency of vibration. To ease the calculation [71], one commonly considers the first two components {\omega }_1 and {\omega }_2, and assuming that the structural drag coefficients are constant with {\xi }_1={\xi }_2=\xi, the Rayleigh constants {ϵ}_1 and {ϵ}_2are found as follows:

(58){ϵ}_1=\frac{2\xi }{{\omega }_1+{\omega }_2}{\omega }_1{\omega }_2,{ϵ}_2=\frac{2\xi }{{\omega }_1+{\omega }_2}.

Moreover, the direct integration method in source [72], called Newmark-beta, was used to solve the dynamic behavior of the FPF piezoelectric nanoplate [19]. This present algorithm is presented as:

1) The initial proviso is allocated:

(59){\mathit{q}}_{\mathbf{0}}=\mathbf{0},{\dot{\mathit{q}}}_{\mathbf{0}}=\mathbf{0},

(60)\stackrel{⌣}{\mathit{A}}=\mathit{M}+{\alpha }_1\mathrm{d}t\mathit{C}+{\alpha }_2{\mathrm{d}t}^2\mathit{K},

where \mathrm{d}t is the integral step and the difference coefficients are:

(61){\alpha }_1=0.5,{\alpha }_2=0,25.

2) Acceleration at the step (n+1) is computed:

(62)\begin{array}{rl}{\ddot{\mathit{q}}}_{n+1}=& \frac{1}{\stackrel{⌣}{\mathit{A}}}(\mathit{F}-\mathit{C}\left({\dot{\mathit{q}}}_n+\left(1-{\alpha }_1\right)\mathrm{d}t{\ddot{\mathit{q}}}_n\right)\\ & -\mathit{K}({\mathit{q}}_n+\mathrm{d}t{\dot{\mathit{q}}}_n+(\frac{1}{2}-{\alpha }_2){\mathrm{d}t}^2{\ddot{\mathit{q}}}_n)).\end{array}

3) The below equation is employed to obtain the velocity at step (n+1):

(63){\dot{\mathit{q}}}_{n+1}={\dot{\mathit{q}}}_n+\left(1-{\alpha }_1\right)\mathrm{d}t{\ddot{\mathit{q}}}_n+{\alpha }_2\mathrm{d}t{\ddot{\mathit{q}}}_n.

4) The displacement component at step (n+1) is computed as:

(64){\mathit{q}}_{n+1}={\mathit{q}}_n+\mathrm{d}t{\dot{\mathit{q}}}_n+{\mathrm{d}t}^2\left(\frac{1}{2}-{\alpha }_2\right){\ddot{\mathit{q}}}_n+{\alpha }_2{\mathrm{d}t}^2{\ddot{\mathit{q}}}_{n+1}.

The above algorithm is shown in Fig.3.

The boundary constraints are examined as:

1) Simply supported (S):

(65)u_0={\beta }^{\mathrm{b}}={\beta }^{\mathrm{s}}=0,(x=0,a),v_0={\beta }^{\mathrm{b}}={\beta }^{\mathrm{s}}=0,(y=0,b).

2) Clamped (C)

(66)u_0=v_0={\beta }^{\mathrm{b}}={\beta }^{\mathrm{s}}={\beta }_{,x}^{\mathrm{b}}={\beta }_{,x}^{\mathrm{s}}={\beta }_{,y}^{\mathrm{b}}={\beta }_{,y}^{\mathrm{s}}=0\mathrm{a}\mathrm{t}\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{e}\mathrm{d}\mathrm{g}\mathrm{e}\mathrm{s}.

3) Free (F): all degrees of freedom at the boundary are nonzero.

4 Numerical evaluation and discussion

A series of algorithms are established employing Matlab 2018a to explore the static bending, free and forced vibration of rectangular and elliptical FPF piezoelectric nanoplates, taking into account flexoelectric effects and variable nonlocal parameters. For ease of comparison with other available precise results, we provide the following non-dimensional descriptions.

(67)\begin{array}{rl}& \left(w_1,W_{\mathrm{c}}\left(t\right)\right)=\left(w_{\mathrm{m}\mathrm{a}\mathrm{x}},w_{\mathrm{m}\mathrm{a}\mathrm{x}}\left(t\right)\right)\cdot \frac{{10}^{2}E{h}^3}{q_0{a}^4},\\ & {\sigma }_{xx}^{\ast \ast }\left(\mathit{\text{z}}\right)=\frac{10h}{q_{0}a}{\sigma }_{xx}\left(\frac{a}{2},\frac{b}{2},\mathit{\text{z}}\right),\\ & {\sigma }_{xy}^{\ast \ast }\left(\mathit{\text{z}}\right)=\frac{10h}{q_{0}a}{\sigma }_{xy}\left(0,0,\mathit{\text{z}}\right),\\ & {\sigma }_{x\mathit{\text{z}}}^{\ast \ast }\left(\mathit{\text{z}}\right)=\frac{10h}{q_{0}a}{\sigma }_{x\mathit{\text{z}}}\left(0,\frac{b}{2},\mathit{\text{z}}\right),\\ & {\mathrm{\Omega }}_1^{\ast \ast }={\omega }_{11}\frac{a^2}{{\text{π}}^2}\sqrt{\rho h/D_{\mathrm{c}}},\\ & {\mathrm{\Omega }}_1={\omega }_{11}\frac{a^2}{h}\sqrt{{\rho }_1/E_1}.\end{array}

(68)K_{\mathrm{w}}=\frac{k_{\mathrm{w}}{a}^4}{D_{\mathrm{c}}},K_{\mathrm{s}}=\frac{k_{\mathrm{s}}{a}^2}{D_{\mathrm{c}}},D_{\mathrm{c}}=\frac{E_1{h}^3}{12(1-v^2)},f_{14}^{\ast \ast }={10}^7{f}_{14}.

The present work only considers cases for which external voltage can be neglected, for which the rectangular FPF piezoelectric nanoplate is subjected to a sinusoidally dispensed load q\left(x,y\right)=q_0\mathrm{s}\mathrm{i}\mathrm{n}\left({\displaystyle \frac{\text{π}x}{a}}\right)\mathrm{s}\mathrm{i}\mathrm{n}\left({\displaystyle \frac{\text{π}y}{b}}\right), and for which the elliptical FPF piezoelectric nanoplate is subjected to a uniformly distributed load q\left(x,y\right)=q_0 with the greatest magnitude q_0. Hence, to understand the electrical field and the polarization (caused by strain gradient P_{\mathit{\text{z}}}=f_{14}({\chi }_{xx\mathit{\text{z}}}+{\chi }_{yy\mathit{\text{z}}})) corresponding to the presented mechanical force, the function parameters of the electrical field and polarization are expressed as:

(69){\overline{E}}_{\mathit{\text{z}}}^{\ast }\left(\mathit{\text{z}}\right)=\frac{{\overline{E}}_{\mathit{\text{z}}}}{q_0},{\overline{P}}_{\mathit{\text{z}}}^{\ast }\left(\mathit{\text{z}}\right)=\frac{{1\mathrm{e}7P}_{\mathit{\text{z}}}}{q_0}.

4.1 Verification evaluations

In this subsection, the convergence and accuracy of formulas and calculation programs are verified through numerical comparison with reliable analytical publications. Tab.1 and Fig.4 present the convergence investigation of the non-dimensional natural frequency {\mathrm{\Omega }}_1 of SSSS homogenous square nanoplate. Classical plate (\mu =0\mathrm{n}\mathrm{m}) and nanoplate (\mu =2\mathrm{n}\mathrm{m}) are exposed to sinusoidally distributed load with length-to-thickness ratio a/h=10. To evaluate the convergence rate of the proposed IGA approach, six cases of meshes are studied for each example of the plate. As seen in Tab.1 and Fig.4, although the quick convergence rate of the analysis is conducted for polynomial orders p=3 and p=4, solutions using p=2 exhibit a comparatively moderate convergence rate toward analytical solutions, as described by Sobhy [66]. Based on the aforementioned findings, the 11 × 11 cubic (p=4) NURBS element mesh is adequate for all examples. For the ellipse plate, the convergence study also uses the 11 × 11 cubic (p=4) NURBS element mesh as the rectangular plate.

The dimensionless natural frequency {\widehat{\text{Ù}}}_1 for a rectangular plate made of porous material with an asymmetric distribution are shown in Tab.2. This table contains the findings of the investigation in the case of liquid saturation. A square plate made of porous material that has an unequal and asymmetrical distribution (E = 69\mathrm{G}\mathrm{P}\mathrm{a}; \rho =2260\mathrm{k}\mathrm{g}/{\mathrm{m}}^3;v=0,25) with SSSS and CCCC boundary conditions. We consider the length-to-thickness ratio a/h=5,20. The outcomes are compared with those of Ebrahimi and Habibi [73] using the FEM based on the HSDT. It can be seen that the findings of this work are congruent with the results of Ebrahim and Habibi (the largest error between the two methods is only 1.1872 percent).

Next, the non-dimensional deflection and natural frequency of SSSS rectangular nanoplates, taking into account flexoelectric effects, are considered. The geometric parameters of nanoplate including h=20\mathrm{n}\mathrm{m}, a=b=50h, and physical properties C₁₁ = 102 GPa, C₁₂ = 31 GPa, C₆₆ = 35.5 GPa; e₃₁ = −17.05 C/m², k_{33}=-\mathrm{1.76.}{10}^{-8}\mathrm{C}/(\mathrm{V}\cdot \mathrm{m}),f_{14}={10}^{-7}\mathrm{C}/\mathrm{m}. The plate is subjected to a uniformly distributed load q_0=0.05\mathrm{M}\mathrm{P}\mathrm{a}. Fig.5 and Fig.6 show the deflection and natural frequency results of the rectangular nanoplate taking into account flexoelectric effects, compared with those of Yang et al. [74] using Navier’s solution based on the classical shear deformation plate theory. It can be observed that the present work’s results are completely congruent with those of Yang et al. [74]. As a result, the proposed approach offers reliability for subsequent investigations.

The deflection of the mid-point by time is studied to specify the correctness of the dynamical problem. Let us consider the isotropic square plate with the following characteristics: E = 151 GPa, v = 0.3, \rho =3000\mathrm{k}\mathrm{g}/{\mathrm{m}}^3, a = 0.2 m, h = 0.01 m. The plate is subjected to a uniformly distributed load q₀ = −1 M N/m² for a specified time, then the applied load is removed. A dimensionless period tf=t\sqrt{E_{\mathrm{m}}/\left({\rho }_{\mathrm{m}}{a}^2\right)}(E_{\mathrm{m}}=70\mathrm{G}\mathrm{P}\mathrm{a},{\rho }_{\mathrm{m}}=2700\mathrm{k}\mathrm{g}/{\mathrm{m}}^3) produces a dimensionless vertical displacement W^{\ast \ast }=w{\displaystyle \frac{E_{\mathrm{m}}h}{q_0{a}^2}} as shown in Fig.7. From this figure, the parameter f_{14}=0, it can be seen that the present approach is identical to those in the work of Reddy [75] using Navier’s solution and the third-order shear plate theory.

4.2 Static bending and free vibration

In this subsection, the effect of the following parameters are given in tables and figures: length-to-thickness ratio a/h; nonlocal parameter {\mu }_1; porosity parameter {\lambda }_0; the Biot parameter of effective stress \alpha; Skempton parameter \beta; two elastic foundation parameters K_{\mathrm{w}} and K_{\mathrm{s}}; width-to-length ratio b/a; and the boundary conditions on the dimensionless values of the FPF piezoelectric nanoplate including w_1, {\mathrm{\Omega }}_1, {\sigma }_{xx}^{\ast \ast }\left(\mathit{\text{z}}\right), {\sigma }_{xy}^{\ast \ast }\left(\mathit{\text{z}}\right), {\sigma }_{x\mathit{\text{z}}}^{\ast \ast }\left(\mathit{\text{z}}\right), {\overline{E}}_{\mathit{\text{z}}}^{\ast }\left(\mathit{\text{z}}\right), and {\overline{P}}_{\mathit{\text{z}}}^{\ast }\left(\mathit{\text{z}}\right). PZT-5H is employed to generate the piezoelectric nanoplate, which has physical characteristics: C₁₁ = 102 GPa, C₁₂ = 31 GPa, C₆₆ = 35.5 GPa; e₃₁ = −17.05 C/m², k_{33}=-\mathrm{1.76.}{10}^{-8}\mathrm{C}/(\mathrm{V}\cdot \mathrm{m}), and f_{14}={10}^{-7}\mathrm{C}/\mathrm{m}. For rectangular plates, the data are taken as follows: h = 10 nm, {\displaystyle \frac{a}{h}}=10,{\displaystyle \frac{b}{a}}=1,{\lambda }_0=0.2,{\mu }_1=0.2a\mathrm{n}\mathrm{m}, K_{\mathrm{w}}=20, K_{\mathrm{s}}=5, \alpha =0.3, \beta =0. For the elliptical FPF piezoelectric nanoplate (as Fig.8), the radius R_x=0.5a, other parameters taken as the rectangular FPF piezoelectric nanoplate. The boundary condition of the rectangular nanoplate used is arbitrary, while for the elliptical FPF piezoelectric nanoplate, the two main boundary conditions used are SSSS and CCCC.

First, the effects of the nonlocal parameter {\mu }_1 and {\overline{E}}_{\mathit{\text{z}}}^{\ast }\left(h/2\right) on w_1, {\mathrm{\Omega }}_1, {\sigma }_{xx}^{\ast \ast }(h/2), {\sigma }_{xy}^{\ast \ast }(-h/2), {\sigma }_{x\mathit{\text{z}}}^{\ast \ast }\left(0\right) of the elliptical FPF piezoelectric nanoplate with flexoelectric effects are described in Tab.3. For both porous and perfect plates, the findings reveal that when the parameter {\mu }_1 increases, the plates become softer. In addition, it can be seen that with the two types of porosity investigated in the present work, Type 1 produces better hardness than Type 2.

Tab.4 and Tab.5 show the effect of the Skempton parameter \beta on the static bending response of the nanoplate. The tables show that the reduction in displacement and the rise in the natural frequency correlate with increasing Skempton parameter \beta. However, the influence of \beta on this values is not much, between \beta =0 and \beta =0.8 lead to the results changing from 3% to 5%, while the normal stress value changes about 10%.

Effects of the length-to-thickness ratio a/h (square nanoplate) or the diameter-to-thickness ratio 2R/h (elliptical nanoplate) on the static bending of FPF piezoelectric nanoplates are shown in Fig.9 and Fig.10, respectively. The findings show that when the flexoelectric effect is present, the nanoplate becomes stiffer, decreasing vertical displacement and increasing the natural frequency It can be seen that the flexoelectric effect increases the stiffness of the material matrix (Readers can find details in Appendix A and the electrical stiffness matrix as in Eq. (54)) Without the influence of flexoelectric phenomena, displacement as well as frequency results in two different porosity cases are significantly different.. However, when the flexoelectric effect is present, the results of the vertical deflection as well as the natural frequency of the plates with cases of varying porosities are very close to each other. The electric field {\overline{E}}_{\mathit{\text{z}}}^{\ast }\left(h/2\right) and the polarization {\overline{P}}_{\mathit{\text{z}}}^{\ast }(h/2) both rise continuously with the ratio of a/h (2R/h), whereas the displacement and natural frequency only significantly alter when a/h\le 10,\left(2R/h\le 10\right). Little change occurs beyond this range. In addition, it can be found that the flexoelectric effect also increases significantly the bending of nanoplates with a thickness less than 5 nm.

Fig.11 and Fig.12 depict the effect of the elastic foundation stiffness, the Skempton parameter \beta and the porosity parameter on the displacement and the natural frequency behaviors of the square and the elliptic FPF piezoelectric nanoplates. The stiffness of the overall structure is enhanced by the elastic medium’s increasing stiffness, leading to the vertical displacement decrease and the natural frequency rise. When the two parameters of foundation stiffness are increased, the overall stiffness value rises approximately linearly.

Next, Fig.13–Fig.16 illustrate the values of {\sigma }_{xx}^{\ast \ast }\left(\mathit{\text{z}}\right), {\sigma }_{xy}^{\ast \ast }\left(\mathit{\text{z}}\right),{\sigma }_{x\mathit{\text{z}}}^{\ast \ast }\left(\mathit{\text{z}}\right), {\overline{E}}_{\mathit{\text{z}}}^{\ast }\left(\mathit{\text{z}}\right), and {\overline{P}}_{\mathit{\text{z}}}^{\ast }\left(\mathit{\text{z}}\right) along the thickness of SSSS FPF piezoelectric nanoplates taking into account flexoelectric effects. It can be seen that the stresses, the electric field {\overline{E}}_{\mathit{\text{z}}}^{\ast }\left(\mathit{\text{z}}\right) and the polarization {\overline{P}}_{\mathit{\text{z}}}^{\ast }\left(\mathit{\text{z}}\right) throughout the length rise as the nonlocal parameter grows, as shown in Fig.13 ({\sigma }_{xx}^{\ast \ast }\left(\mathit{\text{z}}\right)=0 at \mathit{\text{z}}\approx 0.1h and {\sigma }_{xy}^{\ast \ast }\left(\mathit{\text{z}}\right)=0 at \mathit{\text{z}}\approx 0.01h). The electric field {\overline{E}}_{\mathit{\text{z}}}^{\ast }\left(\mathit{\text{z}}\right) fluctuates with plate thickness and reaches its highest value at the bottom face of the plate. {\overline{P}}_{\mathit{\text{z}}}^{\ast }\left(\mathit{\text{z}}\right) is more symmetric and achieves its minimal value at \mathit{\text{z}}=0. Changes in the parameter f_{14} alter the electric field of the plate structure, which stiffens the structure and reduces the values of all stresses (Fig.14). The stresses are less impacted by the change of the Skempton parameter \beta than the porosity parameter (Fig.15 and Fig.16). The vibration mode shapes and frequency values of the first ten modes of the elliptical nanoplate are depicted in Fig.17. It can be observed that even applying the symmetry boundary, the ellipse do not give a symmetric oscillation pair.

4.3 Transient analysis

In this section, the effects of triangular loads, step loads, explosive loads, and sinusoidal loads over time on the FPF piezoelectric nanoplates, with the flexoelectric effects are investigated (using the same input data as in Subsection 4.2). The mathematical model of the force functions acting on the plate over time, and the equation \mathit{F}\left(x,y,t\right)=q\left(x,y\right)P\left(t\right), where P\left(t\right) represents the typical time-varying loads of the step, triangular, sinusoidal, and explosion kinds, are shown in Fig.18.

(70)P\left(t\right)=[\begin{array}{l}\{\begin{array}{l}\begin{array}{l}1,0\le t\le t_1,\\ 0,t>t_1,\end{array}\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{p}\mathrm{l}\mathrm{o}\mathrm{a}\mathrm{d}\end{array}\left(\mathrm{I}\right)\\ \{\begin{array}{l}\begin{array}{l}1-{\displaystyle \frac{t}{t_1}},0\le t\le t_1,\\ 0,t>t_1,\end{array}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{a}\mathrm{n}\mathrm{g}\mathrm{u}\mathrm{l}\mathrm{a}\mathrm{r}\mathrm{l}\mathrm{o}\mathrm{a}\mathrm{d}\end{array}\left(\mathrm{I}\mathrm{I}\right)\\ \{\begin{array}{l}\begin{array}{l}\mathrm{s}\mathrm{i}\mathrm{n}\left({\displaystyle \frac{\tilde{\omega }t}{t_1}}\right),0\le t\le t_1,\\ 0,t>t_1,\end{array}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{u}\mathrm{s}\mathrm{o}\mathrm{i}\mathrm{d}\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{o}\mathrm{a}\mathrm{d}\end{array}\left(\mathrm{I}\mathrm{I}\mathrm{I}\right)\\ \{\begin{array}{l}{\mathrm{e}}^{-\gamma t},0\le t\le t_1,\\ 0,t>t_1,\end{array}\mathrm{e}\mathrm{x}\mathrm{p}\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{i}\mathrm{v}\mathrm{e}\mathrm{b}\mathrm{l}\mathrm{a}\mathrm{s}\mathrm{t}\mathrm{l}\mathrm{o}\mathrm{a}\mathrm{d}\left(\mathrm{I}\mathrm{V}\right)\end{array}

where t_1 denotes the duration during which the structure is subjected to load, and t is the overall survey time. t_1=0.6t; \tilde{\omega }=0.5{\omega }_1 and {\omega }_1 is the first natural frequency. The explosive load coefficient \gamma =0.5{\omega }_1. The structural drag coefficient \xi was set to 0.01 throughout the survey, and the entire survey duration was t = 2 ns.

Dynamic responses in time, of displacement W_{\mathrm{c}}\left(t\right), stress {\sigma }_{xx}^{\ast \ast }(h/2,t), electric field {\overline{E}}_{\mathit{\text{z}}}^{\ast }\left(h/2,t\right) and the polarization {\overline{P}}_{\mathit{\text{z}}}^{\ast }\left(h/2,t\right), for FPF piezoelectric elliptical (R_y/R_x=1.5) nanoplate, with and without flexoelectric effect, and under sinusoidal load, are described in Fig.19. From this figure, it can be seen that under the impact of dynamic load, the response of vertical displacement over time is the same as the rule of applied load. The flexoelectric effect is amplified by increasing the parameter f_{14}, then the displacement W_{\mathrm{c}}\left(t\right) and stress {\sigma }_{xx}^{\ast \ast }\left(h/2,t\right) during forced oscillation and free oscillation are greatly decreased when the load is removed. The findings demonstrate that this effect may be used to effectively regulate structural vibrations, expanding designers and engineers frameworks for creating piezoelectric nanostructures. In contrast to the reduction in oscillation caused by an increase in the parameter f_{14} andthe nonlocal parameter {\mu }_1\left(\mathrm{n}\mathrm{m}\right) makes the structure less stiff thus raising the nanoplate’s vertical displacement and stress levels when subjected to dynamic loads (Fig.20).

The increase of the porosity parameter leads to the displacement W_{\mathrm{c}}\left(t\right) and stress {\sigma }_{xx}^{\ast \ast }\left(h/2,t\right) of the FPF piezoelectric nanoplate increase (Fig.21). Fig.22 depicts the influence of the Skempton parameter \beta on the dynamic response of a FPF foam circular nanoplate subjected to an explosive blast load. It is evident that an increase in the Skempton parameter \beta decreases the displacement and stress values of the plate by a little amount.

5 Conclusions

In this paper, an IGA approach based on the novel refined higher-order shear deformation plate theory and nonlocal theory has been applied to analyze the static bending, free and forced vibrations of FPF piezoelectric rectangular and elliptical nanoplates resting on a Pasternak’s foundation. The nonlocal coefficient is assumed to change with the thickness, as do the two different porous patterns of behavior, which are new findings in this study. Numerical comparison with authoritative sources confirms the proposed method’s accuracy and reliability. Following a review and assessment of the impact of the input parameters, the following significant conclusions are drawn.

1) The stiffness of the plate is increased by the parameter f_{14}, therefore the maximum deflection decreases as f_{14} rises.

2) The maximum deflection value rises when the nonlocal parameter {\mu }_1 and porosity {\lambda }_0 increase. The natural frequency also rises as {\lambda }_0 becomes larger.

3) The stiffness of the plate increases with the rise of the Skempton parameter \beta and the stiffness of the elastic medium, although the effect of the factor \beta is negligible.

4) Maximum deflection rises with decreasing plate thickness, although this effect is attenuated more by flexoelectricity the thinner the plate. These are intriguing and practical study findings.

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