Nonlocal strain gradient free vibration analysis of sandwich functionally graded porous nanoshell integrated with piezoelectric surface layers taking into account flexoelectric effect

Abstract

In addition to accounting for non-gradient nonlocal elastic stress, a nonlocal strain gradient theory (NSGT) also considers the nonlocality of higher-order strain gradients; thus, it is applicable to small-scale structures and can account for both hardening and softening effects. An analytical model is constructed in this research endeavor to depict the free vibration characteristics of sandwich functionally graded porous (FGP) doubly-curved nanoshell integrated with piezoelectric surface layers consists of three distinct layers, taking into account flexoelectrici effect based on NSGT and novel refined high-order shear deformation hypothesis. The novelty of this study is that the two nonlocal coefficients and material length scale of the core layer are variable along thickness, like other material characteristics. The equilibrium equation of motion of the doubly-curved nanoshell is derived from Hamilton’s principle, then the Galerkin method is applied to derive the natural vibration frequency values of the doubly-curved nanoshell with different boundary conditions (BCs). The influence of parameters such as flexoelectric effect, nonlocal and length scale factors, elastic medium stiffness factor, porosity factor, and BCs on the free vibration esponse of the nanoshell is detected and comprehensively studied. This paper is claimed to provide a theoretical predicition on the impact of the size-small dependent and flexoelectric effect upon the oscillation of FGP nanoshell integrated with piezoelectric surface layers, thus sheding light on understanding the underlying physics of electromechanical coupling at the nanoshell to some extent.

Graphical abstract

Keywords

analytical solution / flexoelectric effect / piezoelectric / free vibration / nonlocal strain gradient theory

1 Introduction

Due to the rapid growth of the electronics industry in the early 20th century, scientists have developed a keen interest in dielectric materials that exhibit unique electrical properties, such as ferroelectric, piezoelectric, and flexoelectric effect. The reason for this is the ongoing trend of downsizing in current electronic equipment, which allows for the consideration of material structures at micro and nano dimensions. Flexoelectricity is a characteristic of a dielectric substance in which it displays an inherent electrical polarization caused by a gradient in strain [1,2]. Flexoelectricity is a phenomenon that is closely associated with piezoelectricity. However, whereas piezoelectricity pertains to the polarization caused by consistent strain, flexoelectricity particularly refers to the polarization resulting from strain that varies across different points inside the material. The presence of non-uniform strain disrupts centrosymmetry, allowing for the occurrence of flexoelectric effect in crystal formations that are centrosymmetric, unlike in piezoelectricity [3]. Flexoelectricity and Ferroelasticity are distinct phenomena. Inverse flexoelectricity may be characterized as the creation of a strain gradient resulting from polarization. Conversely, converse flexoelectricity is the phenomenon in which a polarization gradient causes deformation in a material [4,5]. The application potential of materials with unique and captivating electrical properties in current electronics sectors is vast [6]. This places the onus of mechanics on the duty of examining the mechanical performance of these structures as a foundation for computation and design in practical engineering.

A sandwich material refers to a composite made up of many layers of materials that possess distinct qualities, allowing for the use of the benefits offered by each individual layer. Currently, scientists are focused on the downsizing of electronic equipment, namely micro- and nano-sized microchips and semiconductors. They are also studying unique electrical phenomena like piezoelectric and flexoelectric phenomena, as well as the ferroelectric effect. In 2014, Zhang et al. [7] explored the flexoelectric effect’s impact on the electroelastic reactions and vibrational characteristics of a piezoelectric nanoplate. Zhou et al. [8] studied the electro-mechanical reactions of a flexoelectric bilayer circular nano-plate, taking into account the surface effect. Duc and his team [9] explored oscillation and static buckling response of flexoelectric non-uniform thickness nanoplates. van Ke et al. [10] performed a study on the flexoelectric phenomenon and its impact on the free oscillation and static bending characteristics of piezoelectric sandwich functionally graded porous (FGP) nanoplates using the nonlocal strain gradient theory (NSGT). Ebrahimi and coworkers [11–13] analyzed the mechanical response of embedded flexoelectric nanoplates, including the effect of surface phenomenon. Sahu and Biswas [14] studied how the mass loading impact affects the surface wave in a piezoelectric-flexoelectric dielectric plate that is fixed onto a stiff base reinforced with fibers. Yue [15] studied the nonlinear oscillation of a flexoelectric nanoplate with a surface elasticity electrode during dynamic electric loading. Ghasemi et al. [16–18] demonstrated an isogeometric analytic formulation for flexoelectric material topology optimization that is based on level sets. Thai and his team [19] employed finite element modeling to examine the mechanical properties of piezoelectric nanoplates, in which the influence of flexoelectric effect was taken into calculations. Liu and Liang [20] gave a presentation on the electromechanical coupling characteristics of flexoelectric nanostructures: a global sensitivity analysis. Giannakopoulos and Zisis [21] analyzed a stable fracture in a two-dimensional system, taking into account the influence of the flexoelectric effect on surface waves and flexoelectric metamaterials. Wang and Li [22] conducted a study on the impact of flexoelectric effects on the free oscillation of piezoelectric nanoplates during free vibration. Yang et al. [23] conducted an investigation on the impact of flexoelectric effects on the natural frequencies of piezoelectric nanoplates during free vibration. Wang and coelleages [24] discussed the flexoelectric phenomenon’s effect of the electroelastic fields of a piezoelectric nanosheet in their presentation. Ghobadi et al. [25] discussed the study of vibrations in a flexoelectric nano-plate that is affected by a magnetic field. The analysis takes into account the nonlinearity and size-dependency of the plate, as well as its functionally graded (FG) properties. van Minh and van Ke [26] gave a detailed analysis of the mechanical behavior of piezoelectric nanoplates with varying thicknesses, including the influence of the flexoelectric phenomenon. Zhang and Jiang [27] investigated the impact of size on the relationship between electromechanical coupling fields in a bending of the piezoelectric nanoplate, specifically considering the influence of surface effects and flexoelectricity. Phung [28] conducted a static bending study on nanoplates placed on an imperfect elastic foundation, in which the flexoelectric effect was taken into account. A study on the static and dynamical behavior of graphene nanocomposite plates exhibiting the flexoelectric effect was carried out by Shingare and Kundalwal [29]. van Lieu and Luu [30] conducted a thorough examination of the static bending, free and forced oscillation behavior of organic nanobeams in a thermal-controlled setting. Zhang and coworkers [31] introduced a dynamic model of a sandwich micro-shell that takes into account modified coupling stress and thickness-stretching. Most recently, Zhang’s research group [32–36] has worked on nanostructures integrated into small energy storage devices. Several different approaches to studying the mechanical response of mechanical systems were also implemented by the group [37–41]. For nonlocal elastic theory, Ren et al. [42,43] introduced an approach using higher order nonlocal operators to solve partial differential equations. Rabczuk and coworkers [44] introduced a technique using nonlocal operators to solve partial differential equations, specifically applied to the issue of electromagnetic waveguides. An energy approach [45–47] was also introduced by these author to deal with mechanical evaluation of micro and nano structures. Ghasemi [48] proposed a multiscale material framework for the analysis of liquid droplets in solid soft composites with varying characteristics.

For analytical methods, the advantage is that it can provide accurate and explicit solutions to problems analyzing the mechanical behavior of structures. This study has been used by numerous mechanics to examine micro- or nano-scale shell formations, while considering unique electrical phenomena. Zeng et al. [49] conducted a static stability study of nanoscale piezoelectric shells with flexoelectric reaction using the pair stress hypothesis. Xie and coworkers [50] proposed a mathematical model to analyze the behavior of nanoscale doubly curved shells with flexoelectric properties. Babadi and his team [51–53] introduced a model that describes the behavior of truncated conical nano/microshells with varying properties depending on size, using a continuum approach that incorporates flexoelectric and flexomagnetic effects. Babadi and coelleages [54] examined the impact of the flexoelectric phenomenon on the static and free oscillation behavior of FG piezo-flexoelectric cylindrical shells, taking into account their size dependency. Liu et al. [55] conducted a study on the stability of piezoelectric nanoshells made of FG materials under thermo-electromechanical load. The study focused on analyzing the transition of buckling modes using a nonlocal couple stress-based approach.

The literature indicates that a great deal of work has been done on the static and dynamic analysis of nanostructure with a one layer of piezoelectric materials, taking into account the flexoelectric effect. Studies on the interaction between layers of piezoelectric material and other smart material layers, taking into account the flexoelectric effect, are scarce, and the majority of research on nanostructures focuses on beam and plate structures. To the authors’ understanding, the investigation of free vibration analysis of sandwich FGP doubly-curved nanoshell integrated with piezoelectric surface layers resting on a Pasternak elastic foundation has not been undertaken. Therefore, this work represents one of the first endeavors to conduct a free vibration analysis of a sandwich FGP doubly-curved nanoshell integrated with piezoelectric surface layers resting on a Pasternak elastic foundation based on flexoelectric effect and NSGT using analytical method with various boundary conditions (BCs). The two-curvature nanoshell structure has three distinct layers of materials exhibiting varying mechanical characteristics. Specifically, the outermost layers are composed of piezoelectric materials, while the core layer employs materials whose mechanical properties change under Voigt’s law, based on the thickness. Two coefficients that reduce or increase the stiffness of the nanoshell, including nonlocal and length-scale coefficients, are considered to change along the nanoshell thickness direction, and three different porosity rules are novel points in this study. Hamilton’s principle is the foundation for the nanoshell motion balance equations, which are formulated using the novel refined high-order shear theory. To determine the natural frequency values of doubly-curved nanoshells subjected to various BCs and four fundamental two-curvature shell types (spherical, cylindrical, saddle, elliptical, and flat), the Galerkin method is utilized. On the Matlab software, a calculation program is implemented. A comparison is made between the precision of the program and published, dependable outcomes in specific instances of the model described in the article. This paper is claimed to provide a theoretical predicition on the impact of the size-small dependent and flexoelectric effects upon the vibration of FGP nanoshell integrated with piezoelectric surface layers, thus sheding light on understanding the underlying physics of electromechanical coupling at the nanoshell to some extent. While the article does address novel aspects of the research, it is important to note that its limitations restrict its discussion to linear deformation and a few other effects, excluding geometric nonlinear factors. This also constitutes one of the proposed avenues for future research discussed in the article.

The following paragraph has been added to enhance readability. Section 1 of the paper presents an analytical framework by describing the current situation regarding worldwide research and its limitations. Section 2 provides a full explanation of the geometric model, materials, NSGT that includes flexoelectric phenomena, and the fundamental components that make up the shell motion equilibrium equation. In Section 3, the Galerkin–Vlasov model is presented as a solution to the issue of specific oscillations. The survey’s precision and quantitative results are reported in Section 4. The article’s key findings and uniqueness are concisely summarized in Section 5.

2 Theoretical formulation

2.1 Geometrical and material of sandwich functionally graded porous double-curved shallow nanoshell

A sandwich FGP doubly-curved nanoshell consists of three distinct layers, each with its own set of mechanical properties. The upper and lower layers, denoted as h_{\mathrm{f}2} ={\mathit{\text{z}}}_4- {\mathit{\text{z}}}_3 and h_{\mathrm{f}1} ={\mathit{\text{z}}}_2- {\mathit{\text{z}}}_1, are constructed entirely of a homogeneous piezoelectric material. The hollow material comprising the core layer exhibits variable mechanical properties. Fig.1 illustrates FGP with thickness h_{\mathrm{c}}\left(h_{\mathrm{c}}=h-h_{\mathrm{f}1}-h_{\mathrm{f}2}\right).

The sandwich FGP double-curved shallow nanoshell has a primary radius of curvature R_x along the x-axis and a principal radius of curvature R_y along the y-axis. The geometric characteristics of the sandwich FGP double-curved shallow nanoshell with double curvature are presented in Fig.1. The lengths of the curves are denoted by a and b, while h represents the thickness. The whole nanoshell is positioned on Pasternak foundation with two coefficients: Winkler layer stiffness k_{\mathrm{w}} and shear layer stiffness k_{\mathrm{s}}.

The sandwich FGP double-curved shallow nanoshell consists of three components of material, each with a certain thickness. This includes two skin layers constructed of a uniform and symmetric piezoelectric material. Concurrently, the central layer of the shell is composed of a substance that exhibits different mechanical characteristics based on the thickness of the core layer. Microscopic porosities emerge throughout the production process of this core layer. Consequently, the shell’s mechanical characteristics vary in accordance with its thickness, as shown in the following manner [56]

(1)

where P denotes the effective material characteristics such as Young’s modulus E\left(\mathit{\text{z}}\right), mass density \rho, and the ratio of Poisson \upsilon, respectively. The symbols for the materials of the skin layer, the core layer’s metal, and the ceramic layer are, respectively, f, m, and c. The symbols n_{\mathrm{z}} is called the coefficient controlling the volume-fraction of ceramic and metal, \vartheta is called the pore volume and \xi is called the porosity parameter. According to Ref. [56], the relation between \varthetaand \xi through the pore laws are determined as follows.

FGP 1 (Even porosity distribution):

(2)\vartheta =\frac{\xi }{2}.

FGP 2 (Uneven porosity distribution):

(3)\vartheta =\frac{\xi }{2}\left(1-\frac{2\left|\mathit{\text{z}}-{\mathit{\text{z}}}_3-{\mathit{\text{z}}}_2\right|}{{\mathit{\text{z}}}_3-{\mathit{\text{z}}}_2}\right).

FGP 3 (Linear porosity distribution):

(4)\vartheta =\frac{\xi }{2}\left(\frac{\mathit{\text{z}}-{\mathit{\text{z}}}_2}{{\mathit{\text{z}}}_3-{\mathit{\text{z}}}_2}\right).

When the power-law index n_{\mathit{\text{z}}} changes, the elastic modulus E\left(\mathit{\text{z}}\right) of the sandwich FGP double-curved shallow nanoshell changes along the thickness z with four types of pores (\xi =0.3) as shown in Fig.2. Here the core layer of the sandwich FGP double-curved shallow nanoshell is made of \mathrm{A} \mathrm{l}/{\mathrm{A}\mathrm{l}}_2{\mathrm{O}}_3 material, the piezoelectric layer is made of PZT-5H material.

2.2 Relationship between displacement and strain

Based on suggestions as in Ref. [57], the displacement components (u_1,u_2,u_3) is expressed as follows according to the refined higher-order shear deformation theory

(5)\{\begin{array}{rl}& u_1\left(x,y,\mathit{\text{z}},t\right)=\left(1+\frac{\mathit{\text{z}}}{R_x}\right)u_0-\mathit{\text{z}}{w}_{,x}^{\mathrm{b}}-F\left(\mathit{\text{z}}\right)w_{,x}^{\mathrm{s}},\\ & u_2\left(x,y,\mathit{\text{z}},t\right)=\left(1+\frac{\mathit{\text{z}}}{R_y}\right)v_0-\mathit{\text{z}}{w}_{,y}^{\mathrm{b}}-F\left(\mathit{\text{z}}\right)w_{,y}^{\mathrm{s}},\\ & u_3\left(x,y,\mathit{\text{z}},t\right)=w^{\mathrm{b}}+w^{\mathrm{s}},\end{array}

where u_1, u_2, and u_3 present the displacement components in x-, y-, and z-directions, respectively; u_0 and v_0 are in-plane displacement components, respectively; w^{\mathrm{b}} and w^{\mathrm{s}} are the transverse displacements due to bending and shear phenomena, respectively. In this article, we use a new trigonometric thickness distribution function F \left(\mathit{\text{z}}\right) to improve the accuracy of the calculation results as follows:

(6)\begin{array}{rl}F\left(\mathit{\text{z}}\right)=& \mathit{\text{z}}-\frac{1}{4\text{π}}\mathrm{s}\mathrm{e}\mathrm{c}{\left(\frac{1}{12}\right)}^2(2\text{π}\mathit{\text{z}}\times \mathrm{c}\mathrm{o}\mathrm{s}\left(\frac{1}{6}\right)\\ & +h\times \mathrm{c}\mathrm{o}\mathrm{s}\left(\frac{\mathit{\text{z}}}{3h}\right)\times \mathrm{s}\mathrm{i}\mathrm{n}\left(\frac{2\text{π}\mathit{\text{z}}}{h}\right)).\end{array}

It is noted that:

R_x\to \mathrm{\infty } ,R_y\to \mathrm{\infty }: a sandwich FGP nanoshell becomes a sandwich FGP nanoplate;

R_x\to \mathrm{\infty } ,R_y=R: a sandwich FGP cylindrical shallow nanoshell;

R_x= R_y= R: a sandwich FGP spherical shallow nanoshell;

R_x= -R_y=R: a sandwich FGP saddle shallow nanoshell.

According to the displacements components in Eq. (5), the nonzero linear strains of the sandwich FGP doubly-curved nanoshell integrated with piezoelectric surface layers is written by

(7){\epsilon }_{xx}=\left(1+\frac{\mathit{\text{z}}}{R_x}\right)u_{0,x}+\frac{w^{\mathrm{b}}+w^{\mathrm{s}}}{R_x}-\mathit{\text{z}} w_{,xx}^{\mathrm{b}}-F\left(\mathit{\text{z}}\right) w_{,xx}^{\mathrm{s}},

(8){\epsilon }_{yy}=\left(1+\frac{\mathit{\text{z}}}{R_y}\right)v_{0,y}+\frac{w^{\mathrm{b}}+w^{\mathrm{s}}}{R_y}-\mathit{\text{z}} w_{,yy}^{\mathrm{b}}-F\left(\mathit{\text{z}}\right) w_{,yy}^{\mathrm{s}},

(9){\epsilon }_{xy}=u_{0,y}+v_{0,x}-2\mathit{\text{z}}\left(w_{,xy}^{\mathrm{b}}-\frac{u_{0,y}}{2R_x}-\frac{v_{0,x}}{2R_y}\right)-2F\left(\mathit{\text{z}}\right) w_{,xy}^{\mathrm{s}},

(10){\gamma }_{x\mathit{\text{z}}}= g\left(\mathit{\text{z}}\right)w_{,x}^{\mathrm{s}}, {\gamma }_{y\mathit{\text{z}}}= g\left(\mathit{\text{z}}\right)w_{,y}^{\mathrm{s}},

where {\epsilon }_{xx} ,{\epsilon }_{yy},{\epsilon }_{xy},{\gamma }_{y\mathit{\text{z}}},{\gamma }_{x\mathit{\text{z}}} are the linear strain components along the x-axis, y-axis, angular strain in the xy-plane, yz-plane and xz-plane, respectively, and g \left(\mathit{\text{z}}\right)=1- F_{,\mathit{\text{z}}}\left(\mathit{\text{z}}\right).

2.3 Flexoelectric effects combine nonlocal strain gradient theory

According to NSGT [58] and the hybrid flexoelectric phenomenon [26], stress should consider both the nonlocal elastic stress and the strain gradient stress. Given this information, the subsequent equation can be employed to elucidate the correlation between stress and displacement:

(11){\left(1-\mu {\mathrm{\nabla }}^2\right)\sigma }_{ij}=\left(1-l{\mathrm{\nabla }}^2\right)\left(C_{ijkl}{\epsilon }_{ij}-e_{kij}{\mathrm{{\rm Y}}}_k\right),

(12)\left(1-\mu {\mathrm{\nabla }}^2\right){\tau }_{ijm}=- \left(1-l{\mathrm{\nabla }}^2\right)f_{ijkl}{\mathrm{{\rm Y}}}_k,

(13)\left(1-\mu {\mathrm{\nabla }}^2\right){\mathrm{\Phi }}_i=\left(1-l{\mathrm{\nabla }}^2\right)\left(e_{ijk}{\epsilon }_{jk}+{\kappa }_{ij}{\mathrm{{\rm Y}}}_k+f_{ijkl}{\epsilon }_{ij,k}\right),

where C_{ijkl} are the material properties of the elastic; {\mathrm{{\rm Y}}}_k is an electric field component; {\sigma }_{ij}, {\epsilon }_{ij}, and e_{kij} represent the stress, strain components and piezoelectric constant tensor, respectively; {\mathrm{\Phi }}_i is the electric displacement tensor; f_{ijkl} is the electric field–strain gradient coupling tensor representing the higher-order electromechanical coupling; {\epsilon }_{ij,k} is the derivative of {\epsilon }_{ij} with respect to k; {\tau }_{ijm} is the higher-level stress vector, and {\kappa }_{ij} is the permittivity constant component, \mu and l are nonlocal and length-scale parameters, respectively. As l=0, one obtains the theory of the Eringen’s nonlocal elasticity [59,60]. When \mu =0 we obtain second-order strain gradient theory (SGT) [61] and symbols {\mathrm{\nabla }}^2= {\displaystyle \frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{x}^2}}+{\displaystyle \frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{y}^2}} is the Laplacian operator.

The derivative iteration of the traditional NSGT is restricted to the analysis of homogeneous and isotropic nanostructures. Along the direction of thickness, sandwich FGP nanostructures display material features, including nonlocal and length-scale variables, that vary. Therefore, it is critical to consider the variability of the nonlocal and length-scale quantities. Using the following formulas, one can express the constitutive relations of the nonlocal strain gradient in the sandwich FGP double-curved shallow nanoshell. It is assumed that the nonlocal and length-scale parameters exhibit variation along the thickness orientation of the nanoshell.

(14){\left(1-\mu \left(\mathit{\text{z}}\right){\mathrm{\nabla }}^2\right)\sigma }_{ij}=\left(1-l\left(\mathit{\text{z}}\right){\mathrm{\nabla }}^2\right)\left(C_{ijkl}{\epsilon }_{ij}-e_{kij}{\mathrm{{\rm Y}}}_k\right),

(15)\left(1-\mu \left(\mathit{\text{z}}\right){\mathrm{\nabla }}^2\right){\tau }_{ijm}=- \left(1-l\left(\mathit{\text{z}}\right){\mathrm{\nabla }}^2\right)f_{ijkl}{\mathrm{{\rm Y}}}_k,

(16)\left(1-\mu \left(\mathit{\text{z}}\right){\mathrm{\nabla }}^2\right){\mathrm{\Phi }}_i=\left(1-l\left(\mathit{\text{z}}\right){\mathrm{\nabla }}^2\right)\left(e_{ijk}{\epsilon }_{jk}+{\kappa }_{ij}{\mathrm{{\rm Y}}}_k+f_{ijkl}{\epsilon }_{ij,k}\right),

where the variation of the nonlocal and length-scale parameters along the thickness of the shell is expected to be similar to other material characteristics. Consequently, the effective nonlocal and length-scale factors of the sandwich FGP double-curved shallow nanoshell are determined using the following method.

(17)

(18)

This distinguishes the present study from other existing research on the examination of sandwich FGP doubly-curved nanoshell integrated with piezoelectric surface layers. The updated NSGT was reverted to the classical NSGT by configuring {\mu }_{\mathrm{c}}={\mu }_{\mathrm{m}}=\mu and l_{\mathrm{c}}=l_{\mathrm{m}}=l. Furthermore, the volume fraction index and porosity of the sandwich FGP nanostructures influence the effective nonlocal and length-scale parameters. Using theoretical methods to ascertain these coefficients presents a significant challenge. To ascertain this coefficient, experimental investigations or atomistic dynamics modeling are required. To facilitate the process of formula implementation and minimize complexity, the operators {\mathrm{\Lambda }}_1= 1-\mu \left(\mathit{\text{z}}\right){\mathrm{\nabla }}^2 and {\mathrm{\Lambda }}_2=1-l\left(\mathit{\text{z}}\right){\mathrm{\nabla }}^2 were incorporated.

2.4 Stress–strain relations

2.4.1 Functionally graded porous core layer

For the FGP core layer, the relationship between stress and strain is determined as follows

(19){\mathrm{\Lambda }}_1\left\{\begin{array}{c}{\sigma }_{xx}^{\mathrm{c}}\\ {\sigma }_{yy}^{\mathrm{c}}\\ \begin{array}{c}\begin{array}{c}{\sigma }_{xy}^{\mathrm{c}}\\ {\sigma }_{x\mathit{\text{z}}}^{\mathrm{c}}\end{array}\\ {\sigma }_{y\mathit{\text{z}}}^{\mathrm{c}}\end{array}\end{array}\right\}={\mathrm{\Lambda }}_2\left[\begin{array}{ccccc}{C}_{11}^{\mathrm{c}}& C_{12}^{\mathrm{c}}& 0& 0& 0\\ C_{12}^{\mathrm{c}}& C_{22}^{\mathrm{c}}& 0& 0& 0\\ 0& 0& C_{66}^{\mathrm{c}}& 0& 0\\ 0& 0& 0& C_{55}^{\mathrm{c}}& 0\\ 0& 0& 0& 0& C_{44}^{\mathrm{c}}\end{array}\right]\left\{\begin{array}{c}\begin{array}{c}{\epsilon }_{xx}\\ {\epsilon }_{yy}\\ {\epsilon }_{xy}\end{array}\\ {\epsilon }_{x\mathit{\text{z}}}\\ {\epsilon }_{y\mathit{\text{z}}}\end{array}\right\},

where

(20)\begin{array}{rl}& C_{11}^{\mathrm{c}}=C_{22}^{\mathrm{c}}=\frac{E\left(\mathit{\text{z}}\right)}{1-{v\left(\mathit{\text{z}}\right)}^2},C_{12}^{\mathrm{c}}=\frac{v\left(\mathit{\text{z}}\right)E\left(z\right)}{1-{v\left(\mathit{\text{z}}\right)}^2},\\ & C_{66}^{\mathrm{c}}=C_{55}^{\mathrm{c}}=C_{44}^{\mathrm{c}}=\frac{E\left(\mathit{\text{z}}\right)}{2\left(1+v\left(\mathit{\text{z}}\right)\right)}.\end{array}

2.4.2 Piezoelectric face layer

The flexoelectric effect is accounted for by using the expanded version of linear piezoelectricity theory, which takes into consideration the connection between strain gradient and polarization. We assume that the electric field only occurs in the z-direction for a thin piezoelectric layer with a large ratio of curved dimension to thickness because the electric field components in the x–y plane are small in comparison to those in the thickness direction. The subsequent equation represents the gradient of strain.

(21){\chi }_{xx\mathit{\text{z}}}={\epsilon }_{xx,\mathit{\text{z}}}=\frac{u_{0,x}}{R_x}-w_{,xx}^{\mathrm{b}}-F_{,\mathit{\text{z}}}\left(\mathit{\text{z}}\right)w_{,xx}^{\mathrm{s}},

(22){\chi }_{yy\mathit{\text{z}}}={\epsilon }_{yy,\mathit{\text{z}}}=\frac{v_{0,y}}{R_y}-w_{,yy}^{\mathrm{b}}-F_{,\mathit{\text{z}}}\left(\mathit{\text{z}}\right)w_{,yy}^{\mathrm{s}}.

Equations (14)–(16) allow us to rewrite the constitutive equation of the sandwich FGP doubly-curved nanoshell integrated with piezoelectric surface layers with flexoelectricity effect as follows:

(23){\mathrm{\Lambda }}_1\left\{\begin{array}{c}{\sigma }_{xx}^{\mathrm{f}}\\ {\sigma }_{yy}^{\mathrm{f}}\\ \begin{array}{c}\begin{array}{c}{\sigma }_{xy}^{\mathrm{f}}\\ {\sigma }_{x\mathit{\text{z}}}^{\mathrm{f}}\end{array}\\ {\sigma }_{y\mathit{\text{z}}}^{\mathrm{f}}\end{array}\end{array}\right\}={\mathrm{\Lambda }}_2\left(\left[\begin{array}{ccccc}{C}_{11}^{\mathrm{f}}& C_{12}^{\mathrm{f}}& 0& 0& 0\\ C_{12}^{\mathrm{f}}& C_{22}^{\mathrm{f}}& 0& 0& 0\\ 0& 0& C_{66}^{\mathrm{f}}& 0& 0\\ 0& 0& 0& C_{55}^{\mathrm{f}}& 0\\ 0& 0& 0& 0& C_{44}^{\mathrm{f}}\end{array}\right]\left\{\begin{array}{c}\begin{array}{c}{\epsilon }_{xx}\\ {\epsilon }_{yy}\\ {\epsilon }_{xy}\end{array}\\ {\epsilon }_{x\mathit{\text{z}}}\\ {\epsilon }_{y\mathit{\text{z}}}\end{array}\right\}-\left\{\begin{array}{c}{e}_{31}{\mathrm{{\rm Y}}}_{\mathit{\text{z}}}\\ e_{31}{\mathrm{{\rm Y}}}_{\mathit{\text{z}}}\\ \begin{array}{c}\begin{array}{c}0\\ 0\end{array}\\ 0\end{array}\end{array}\right\}\right),

(24){\mathrm{\Lambda }}_1\left\{\begin{array}{c}{\tau }_{xx\mathit{\text{z}}}\\ {\tau }_{yy\mathit{\text{z}}}\end{array}\right\}=-{\mathrm{\Lambda }}_2\left\{\begin{array}{c}{f}_{14}{\mathrm{{\rm Y}}}_{\mathit{\text{z}}}\\ f_{14}{\mathrm{{\rm Y}}}_{\mathit{\text{z}}}\end{array}\right\},

(25){\mathrm{\Lambda }}_1{\mathrm{\Phi }}_{\mathit{\text{z}}}={\mathrm{\Lambda }}_2\left(e_{31}\left({\epsilon }_{xx}+{\epsilon }_{yy}\right)+{\kappa }_{33}{\mathrm{{\rm Y}}}_{\mathit{\text{z}}}+f_{14}\left({\chi }_{xx\mathit{\text{z}}}+{\chi }_{yy\mathit{\text{z}}}\right)\right),

where f_{14}= f_{3113} and f_{14}= f_{3223} are introduced for convenience [9,19,26,28,62]. Here, we make the following assumptions: there is no external electric field exerted on the shell, and the electric displacement is equal to the electric polarization.

In electrostatics, the electric displacement must adhere to the Gaussian rule when there are no free electric charges present.

(26){\mathrm{\Phi }}_{\mathit{\text{z}}}=0.

When there is no current flowing, the electric displacement on the surface is zero. Therefore, the internal electric field may be derived from Eq. (26) as

(27)\begin{array}{rl}{\mathrm{{\rm Y}}}_{\mathit{\text{z}}}=& -\frac{e_{31}}{{\kappa }_{33}}\left({\epsilon }_{xx}+{\epsilon }_{yy}\right)+\frac{f_{14}}{{\kappa }_{33}}\left(w_{,xx}^{\mathrm{b}}+w_{,yy}^{\mathrm{b}}-\frac{u_{0,x}}{R_x}-\frac{v_{0,y}}{R_y}\right)\\ & +\frac{f_{14}{F}_{,\mathit{\text{z}}}\left(\mathit{\text{z}}\right)}{{\kappa }_{33}}\left(w_{,xx}^{\mathrm{s}}+w_{,yy}^{\mathrm{s}}\right),\end{array}

It can be seen that the electric field {\mathrm{{\rm Y}}}_{\mathit{\text{z}}} depends on the function F\left(\mathit{\text{z}}\right), and the derivative of the function F\left(\mathit{\text{z}}\right) with respect to z and the radius of curvature R_x, R_y. and the flexoelectric coefficient f_{14}. This issue is quite intriguing and very few published studies have addressed this issue.

2.5 Equations of motion general

The mathematical equations regarding motion for the sandwich FGP doubly-curved nanoshell integrated with piezoelectric surface layers lying on a Pasternak meidum are as follows, when Hamilton’s principle [63] is applied

(28){\int }_{t_1}^{t_2}\delta \left(U_{\mathrm{p}}+{\mathcal{U}}_{\mathrm{f}}+V-\mathcal{K}\right) \mathrm{d}t=0,

The variables U_{\mathrm{p}},{\mathcal{U}}_{\mathrm{f}}, V, and \mathcal{K} represent the strain energy, strain energy of nanoshell, external potential energy, and kinetic energy of the sandwich FGP doubly-curved nanoshell integrated with piezoelectric surface layers, respectively.

Using the open-circuit condition, the change in the strain potential energy \delta U_{\mathrm{p}}of the sandwich FGP doubly-curved nanoshell integrated with piezoelectric surface layers is calculated as follows

(29)\begin{array}{rl}\delta U_{\mathrm{p}}=& {\int }_S(N_{xx}\left(\delta u_{0,x}+\frac{\delta w^{\mathrm{b}}+\delta w^{\mathrm{s}}}{R_x}\right)-M_{xx}\left(\delta w_{,xx}^{\mathrm{b}}-\frac{\delta u_{0,x}}{R_x}\right)\\ & -L_{xx}\delta w_{,xx}^{\mathrm{s}}+N_{yy}\left(\delta v_{0,y}+\frac{\delta w^{\mathrm{b}}+\delta w^{\mathrm{s}}}{R_y}\right)\\ & -M_{yy}\left(\delta w_{,yy}^{\mathrm{b}}-\frac{{\delta v}_{0,y}}{R_y}\right)-L_{yy}\delta w_{,yy}^{\mathrm{s}}+N_{xy}\left(\delta u_{0,y}+\delta v_{0,x}\right)\\ & -M_{xy}\left(2\delta w_{,xy}^{\mathrm{b}}-\frac{{\delta u}_{0,y}}{R_x}-\frac{\delta v_{0,x}}{R_y}\right)-{2L}_{xy}\delta w_{,xy}^{\mathrm{s}}\\ & +Q_{x\mathit{\text{z}}}\delta w_{,x}^{\mathrm{s}}+Q_{y\mathit{\text{z}}}\delta w_{,y}^{\mathrm{s}}-X_{xx}\left(\delta w_{,xx}^{\mathrm{b}}-\frac{\delta u_{0,x}}{R_x}\right)\\ & -S_{xx}\delta w_{,xx}^{\mathrm{s}}-X_{yy}\left(\delta w_{,yy}^{\mathrm{b}}-\frac{\delta v_{0,y}}{R_y}\right)-S_{yy}\delta w_{,yy}^{\mathrm{s}})\mathrm{d}x\mathrm{d}y.\end{array}

The elastic foundation’s strain energy is represented by

(30)\delta U_{\mathrm{f}}={\int }_S\left(k_{\mathrm{w}}\left(w^{\mathrm{b}}+w^{\mathrm{s}}\right)-k_{\mathrm{s}}{\mathrm{\nabla }}^2\left(w^{\mathrm{b}}+w^{\mathrm{s}}\right)\right)\delta \left(w^{\mathrm{b}}+w^{\mathrm{s}}\right)\mathrm{d} S.

The external potential energy is defined by

(31)\begin{array}{rl}\delta V=& {\int }_S\left(-q_{\mathit{\text{z}}}-N_x^0\left(w_{,xx}^{\mathrm{b}}+w_{,xx}^{\mathrm{s}}\right)-N_y^0\left(w_{,yy}^{\mathrm{b}}+w_{,yy}^{\mathrm{s}}\right)\right)\\ & \cdot \delta \left(w^{\mathrm{b}}+w^{\mathrm{s}}\right)\mathrm{d}S.\end{array}

The vertical force is represented by q_{\mathit{\text{z}}}, whereas the horizontal forces are designated by N_x^0, N_y^0, and N_{xy}^0, respectively. Below are the precise definitions of these forces

(32)N_x^0= N_y^0= -N^0, N_{xy}^0 =0.

Kinetic energy is calculated by

(33)\begin{array}{rl}\delta \mathcal{K}=& {\int }_S\sum _{i=1}^3{\int }_{{\mathit{\text{z}}}_i}^{{\mathit{\text{z}}}_{i+1}}{\rho }_i(\left({\dot{u}}_0-\widetilde{\mathit{\text{z}}}{\dot{w}}_{,x}^{\mathrm{b}}-\tilde{F}{\dot{w}}_{,x}^{\mathrm{s}}\right)\delta \left({\dot{u}}_0-\widetilde{\mathit{\text{z}}}{\dot{w}}_{,x}^{\mathrm{b}}-\tilde{F}{\dot{w}}_{,x}^{\mathrm{s}}\right)\\ & +\left({\dot{v}}_0-\widetilde{\mathit{\text{z}}}{\dot{w}}_{,y}^b-\tilde{F}{\dot{w}}_{,y}^{\mathrm{s}}\right)\delta \left({\dot{v}}_0-\widetilde{\mathit{\text{z}}}{\dot{w}}_{,y}^{\mathrm{b}}-\tilde{F}{\dot{w}}_{,y}^{\mathrm{s}}\right)\\ & +\left({\dot{w}}^{\mathrm{b}}+{\dot{w}}^{\mathrm{s}}\right)\delta \left({\dot{w}}^{\mathrm{b}}+{\dot{w}}^{\mathrm{s}}\right))\mathrm{d}\mathit{\text{z}}\mathrm{d}x\mathrm{d}y.\end{array}

The equations of motion for the sandwich FGP doubly-curved nanoshell integrated with piezoelectric surface layers, using the suggested assumptions, are calculated in the following manner

(34)\begin{array}{rl}{\delta u}_0:& N_{xx,x}+N_{xy,y}+\frac{M_{xx,x}+X_{xx,x}}{R_x}+\frac{M_{xy,y}}{R_x}\\ & =I_{01}{\ddot{u}}_0-I_{11}{\ddot{w}}_{,x}^{\mathrm{b}}-J_{11}{\ddot{w}}_{,x}^{\mathrm{s}},\end{array}

(35)\begin{array}{rl}{\delta v}_0:& N_{yy,y}+N_{xy,x}+\frac{M_{xy,x}}{R_y}+\frac{M_{yy,y}+X_{yy,y}}{R_y}\\ & =I_{02}{\ddot{v}}_0-I_{12}{\ddot{w}}_{,y}^{\mathrm{b}}-J_{12}{\ddot{w}}_{,y}^{\mathrm{s}},\end{array}

(36)\begin{array}{rl}{\delta w}^{\mathrm{b}}:& M_{xx,xx}+{2M}_{xy,xy}+M_{yy,yy}+X_{xx,xx}+X_{yy,yy}-\frac{N_{xx}}{R_x}-\frac{N_{yy}}{R_y}\\ & -k_{\mathrm{w}}\left(w^{\mathrm{b}}+w^{\mathrm{s}}\right)-c_{\mathrm{w}}\left({\dot{w}}^{\mathrm{b}}+{\dot{w}}^{\mathrm{s}}\right)+\left(k_{\mathrm{s}}-N_0\right){\mathrm{\nabla }}^2\left(w^{\mathrm{b}}+w^{\mathrm{s}}\right)\\ =& q\left(x,y\right)+I_0\left({\ddot{w}}^{\mathrm{b}}+{\ddot{w}}^{\mathrm{s}}\right)+I_{11}{\ddot{u}}_{0,x}+I_{12}{\ddot{v}}_{0,y}\\ & -I_2{\mathrm{\nabla }}^2{\ddot{w}}^{\mathrm{b}}-J_2{\mathrm{\nabla }}^2{\ddot{w}}^{\mathrm{s}},\end{array}

(37)\begin{array}{rl}{\delta w}^{\mathrm{s}}:& L_{xx,xx}+{2L}_{xy,xy}+L_{yy,yy}+S_{xx,xx}+S_{yy,yy}+Q_{x\mathit{\text{z}},x}+Q_{y\mathit{\text{z}},y}\\ & -\frac{N_{xx}}{R_x}-\frac{N_{yy}}{R_y}-k_{\mathrm{w}}\left(w^{\mathrm{b}}+w^{\mathrm{s}}\right)+\left(k_{\mathrm{s}}-N_0\right){\mathrm{\nabla }}^2\left(w^{\mathrm{b}}+w^{\mathrm{s}}\right)\\ =& I_0\left({\ddot{w}}^{\mathrm{b}}+{\ddot{w}}^{\mathrm{s}}\right)+J_{11}{\ddot{u}}_{0,x}+J_{12}{\ddot{v}}_{0,y}-J_2{\mathrm{\nabla }}^2{\ddot{w}}^{\mathrm{b}}-K_2{\mathrm{\nabla }}^2{\ddot{w}}^{\mathrm{s}},\end{array}

where the dot (˙) superscript convention is utilized to denote differentiation with respect to the time variable t. The symbols (I_i, J_j, K_k) symbolize the mass moment of inertia and they are defined as follows

(38)\begin{array}{rl}& \left\{I_0,I_{01},I_{02},I_{11},I_{12}\right\}\\ =& {\int }_{\frac{-h}{2}}^{-\frac{h}{2}+h_{\mathrm{f}1}}\left\{1,{\left(1+\frac{\widetilde{\mathit{\text{z}}}}{R_x}\right)}^2,{\left(1+\frac{\widetilde{\mathit{\text{z}}}}{R_y}\right)}^2,\widetilde{\mathit{\text{z}}}\left(1+\frac{\widetilde{\mathit{\text{z}}}}{R_x}\right),\widetilde{\mathit{\text{z}}}\left(1+\frac{\widetilde{\mathit{\text{z}}}}{R_y}\right)\right\}{\rho }_{\mathrm{f}1}\mathrm{d}\mathit{\text{z}}\\ & +{\int }_{\frac{-h}{2}+h_{\mathrm{f}1}}^{\frac{h}{2}-h_{\mathrm{f}2}}\left\{1,{\left(1+\frac{\widetilde{\mathit{\text{z}}}}{R_x}\right)}^2,{\left(1+\frac{\widetilde{\mathit{\text{z}}}}{R_y}\right)}^2,\widetilde{\mathit{\text{z}}}\left(1+\frac{\widetilde{\mathit{\text{z}}}}{R_x}\right),\widetilde{\mathit{\text{z}}}\left(1+\frac{\widetilde{\mathit{\text{z}}}}{R_y}\right)\right\}\rho \left(\mathit{\text{z}}\right)\mathrm{d}\mathit{\text{z}}\\ & +{\int }_{\frac{h}{2}-h_{\mathrm{f}2}}^{\frac{h}{2}}\left\{1,{\left(1+\frac{\widetilde{\mathit{\text{z}}}}{R_x}\right)}^2,{\left(1+\frac{\widetilde{\mathit{\text{z}}}}{R_y}\right)}^2,\widetilde{\mathit{\text{z}}}\left(1+\frac{\widetilde{\mathit{\text{z}}}}{R_x}\right),\widetilde{\mathit{\text{z}}}\left(1+\frac{\widetilde{\mathit{\text{z}}}}{R_y}\right)\right\}{\rho }_{\mathrm{f}2}\mathrm{d}\mathit{\text{z}},\end{array}

(39)\begin{array}{rl}& \left\{I_2,J_{11},J_{12},J_2,K_2\right\}\\ =& {\int }_{\frac{-h}{2}}^{-\frac{h}{2}+h_{\mathrm{f}1}}\left\{{\widetilde{\mathit{\text{z}}}}^2,\tilde{F}\left(1+\frac{\widetilde{\mathit{\text{z}}}}{R_x}\right),\tilde{F}\left(1+\frac{\widetilde{\mathit{\text{z}}}}{R_y}\right),\widetilde{\mathit{\text{z}}}\tilde{F},{\tilde{F}}^2\right\}{\rho }_{\mathrm{f}1}\mathrm{d}\mathit{\text{z}}\\ & +{\int }_{\frac{-h}{2}+h_{\mathrm{f}1}}^{\frac{h}{2}-h_{\mathrm{f}2}}\left\{{\widetilde{\mathit{\text{z}}}}^2,\tilde{F}\left(1+\frac{\widetilde{\mathit{\text{z}}}}{R_x}\right),\tilde{F}\left(1+\frac{\widetilde{\mathit{\text{z}}}}{R_y}\right),\widetilde{\mathit{\text{z}}}\tilde{F},{\tilde{F}}^2\right\}\rho \left(\mathit{\text{z}}\right)\mathrm{d}\mathit{\text{z}}\\ & +{\int }_{\frac{h}{2}-h_{\mathrm{f}2}}^{\frac{h}{2}}\left\{{\widetilde{\mathit{\text{z}}}}^2,\tilde{F}\left(1+\frac{\widetilde{\mathit{\text{z}}}}{R_x}\right),\tilde{F}\left(1+\frac{\widetilde{\mathit{\text{z}}}}{R_y}\right),\widetilde{\mathit{\text{z}}}\tilde{F},{\tilde{F}}^2\right\}{\rho }_{\mathrm{f}2}\mathrm{d}\mathit{\text{z}}.\end{array}

And the stress resultants N_{ij},M_{ij},L_{ij} ,Q_{ij},X_{ij}, and S_{ij} are defined as follows

\begin{array}{}\text{(40a)}& \left(N_{xx},N_{yy},N_{xy}\right)=\sum _{i=1}^3{\int }_{{\mathit{\text{z}}}_i}^{{\mathit{\text{z}}}_{i+1}}\left({\sigma }_{xx},{\sigma }_{yy},{\sigma }_{xy}\right)\mathrm{d}\mathit{\text{z}},\end{array}

\begin{array}{}\text{(40b)}& \left(M_{xx},M_{yy},M_{xy}\right)=\sum _{i=1}^3{\int }_{{\mathit{\text{z}}}_i}^{{\mathit{\text{z}}}_{i+1}}\left({\sigma }_{xx},{\sigma }_{yy},{\sigma }_{xy}\right)\widetilde{\mathit{\text{z}}}\mathrm{d}\mathit{\text{z}},\end{array}

\begin{array}{}\text{(40c)}& \left(L_{xx},L_{yy},L_{xy}\right)=\sum _{i=1}^3{\int }_{{\mathit{\text{z}}}_i}^{{\mathit{\text{z}}}_{i+1}}\left({\sigma }_{xx},{\sigma }_{yy},{\sigma }_{xy}\right)\tilde{F}\mathrm{d}\mathit{\text{z}},\end{array}

\begin{array}{}\text{(40d)}& \left(Q_{x\mathit{\text{z}}},Q_{y\mathit{\text{z}}}\right)=\sum _{i=1}^3{\int }_{{\mathit{\text{z}}}_i}^{{\mathit{\text{z}}}_{i+1}}\left({\sigma }_{x\mathit{\text{z}}},{\sigma }_{y\mathit{\text{z}}}\right)\tilde{g}\mathrm{d}\mathit{\text{z}}\end{array}

\begin{array}{}\text{(40e)}& \left(X_{xx},X_{yy}\right)=\sum _{i=1}^3{\int }_{{\mathit{\text{z}}}_i}^{{\mathit{\text{z}}}_{i+1}}\left({\tau }_{xx\mathit{\text{z}}},{\tau }_{yy\mathit{\text{z}}}\right)\mathrm{d}\mathit{\text{z}},\end{array}

\begin{array}{}\text{(40f)}& \left(S_{xx},S_{yy}\right)=\sum _{i=1}^3{\int }_{{\mathit{\text{z}}}_i}^{{\mathit{\text{z}}}_{i+1}}\left({\tau }_{xx\mathit{\text{z}}},{\tau }_{yy\mathit{\text{z}}}\right){\tilde{F}}_{,\mathit{\text{z}}}\left(\mathit{\text{z}}\right)\mathrm{d}\mathit{\text{z}}.\end{array}

By substituting Eqs. (19), (23), and (24) into Eq. (40), the internal force components of the sandwich FGP doubly-curved nanoshell integrated with piezoelectric surface layers are precisely confirmed as follows

\begin{array}{}\text{(41a)}& {\mathrm{\Lambda }}_1\left\{\begin{array}{c}\begin{array}{c}{N}_{xx}\\ N_{yy}\\ N_{xy}\end{array}\\ \begin{array}{c}{M}_{xx}\\ M_{yy}\\ M_{xy}\end{array}\\ \begin{array}{c}{L}_{xx}\\ L_{yy}\\ L_{xy}\end{array}\end{array}\right\}={\mathrm{\Lambda }}_2\left[\begin{array}{ccc}\begin{array}{ccc}{\mathcal{A}}_{11}& {\mathcal{A}}_{12}& 0\\ {\mathcal{A}}_{12}& {\mathcal{A}}_{22}& 0\\ 0& 0& {\mathcal{A}}_{66}\end{array}& \begin{array}{ccc}{\mathcal{B}}_{11}& {\mathcal{B}}_{12}& 0\\ {\mathcal{B}}_{12}& {\mathcal{B}}_{22}& 0\\ 0& 0& {\mathcal{B}}_{66}\end{array}& \begin{array}{ccc}{\mathcal{B}}_{11}^{\mathrm{s}}& {\mathcal{B}}_{12}^{\mathrm{s}}& 0\\ {\mathcal{B}}_{12}^{\mathrm{s}}& {\mathcal{B}}_{22}^{\mathrm{s}}& 0\\ 0& 0& {\mathcal{B}}_{66}^{\mathrm{s}}\end{array}\\ \begin{array}{ccc}{\overline{\mathcal{B}}}_{11}& {\overline{\mathcal{B}}}_{12}& 0\\ {\overline{\mathcal{B}}}_{12}& {\overline{\mathcal{B}}}_{22}& 0\\ 0& 0& {\overline{\mathcal{B}}}_{66}\end{array}& \begin{array}{ccc}{\mathcal{F}}_{11}& {\mathcal{F}}_{12}& 0\\ {\mathcal{F}}_{12}& {\mathcal{F}}_{22}& 0\\ 0& 0& {\mathcal{F}}_{66}\end{array}& \begin{array}{ccc}{\mathcal{F}}_{11}^{\mathrm{s}}& {\mathcal{F}}_{12}^{\mathrm{s}}& 0\\ {\mathcal{F}}_{12}^{\mathrm{s}}& {\mathcal{F}}_{22}^{\mathrm{s}}& 0\\ 0& 0& {\mathcal{F}}_{66}^{\mathrm{s}}\end{array}\\ \begin{array}{ccc}{\overline{\mathcal{B}}}_{11}^{\mathrm{s}}& {\overline{\mathcal{B}}}_{12}^{\mathrm{s}}& 0\\ {\overline{\mathcal{B}}}_{12}^{\mathrm{s}}& {\overline{\mathcal{B}}}_{22}^{\mathrm{s}}& 0\\ 0& 0& {\overline{\mathcal{B}}}_{66}^{\mathrm{s}}\end{array}& \begin{array}{ccc}{\overline{\mathcal{F}}}_{11}^{\mathrm{s}}& {\overline{\mathcal{F}}}_{12}^{\mathrm{s}}& 0\\ {\overline{\mathcal{F}}}_{12}^{\mathrm{s}}& {\overline{\mathcal{F}}}_{22}^{\mathrm{s}}& 0\\ 0& 0& {\overline{\mathcal{F}}}_{66}^{\mathrm{s}}\end{array}& \begin{array}{ccc}{\mathcal{H}}_{11}& {\mathcal{H}}_{12}& 0\\ {\mathcal{H}}_{12}& {\mathcal{H}}_{22}& 0\\ 0& 0& {\mathcal{H}}_{66}\end{array}\end{array}\right]\left\{\begin{array}{c}\begin{array}{c}{u}_{0,x}+{\displaystyle \frac{w^{\mathrm{b}}+w^{\mathrm{s}}}{R_x}}\\ v_{0,y}+{\displaystyle \frac{w^{\mathrm{b}}+w^{\mathrm{s}}}{R_y}}\\ u_{0,y}+v_{0,x}\end{array}\\ \begin{array}{c}{\displaystyle \frac{u_{0,x}}{R_x}}-w_{,xx}^{\mathrm{b}}\\ {\displaystyle \frac{v_{0,y}}{R_y}}-w_{,yy}^{\mathrm{b}}\\ {\displaystyle \frac{u_{0,y}}{R_x}}+{\displaystyle \frac{v_{0,x}}{R_y}}-2w_{,xy}^{\mathrm{b}}\end{array}\\ \begin{array}{c}-w_{,xx}^{\mathrm{s}}\\ -w_{,yy}^{\mathrm{s}}\\ -2w_{,xy}^{\mathrm{s}}\end{array}\end{array}\right\},\end{array}

\begin{array}{}\text{(41b)}& {\mathrm{\Lambda }}_1\left\{\begin{array}{c}{Q}_{x\mathit{\text{z}}}\\ Q_{y\mathit{\text{z}}}\end{array}\right\}={\mathrm{\Lambda }}_2\left[\begin{array}{cc}{\mathcal{A}}_{44}^{\mathrm{s}}& 0\\ 0& {\mathcal{A}}_{55}^{\mathrm{s}}\end{array}\right]\left\{\begin{array}{c}{w}_{,x}^{\mathrm{s}}\\ w_{,y}^{\mathrm{s}}\end{array}\right\},\end{array}

\begin{array}{rl}{\mathrm{\Lambda }}_1\left\{\begin{array}{c}{X}_{xx}\\ X_{yy}\end{array}\right\}=& {\mathrm{\Lambda }}_2\left[\begin{array}{ccc}{\mathcal{C}}_{11}& {\mathcal{C}}_{12}& {\mathcal{C}}_{13}\\ {\mathcal{C}}_{11}& {\mathcal{C}}_{12}& {\mathcal{C}}_{13}\end{array}\right]\\ & \cdot \left\{\begin{array}{c}{-u}_{0,x}-v_{0,y}-\left({\displaystyle \frac{1}{R_x}}+{\displaystyle \frac{1}{R_y}}\right)\left(w^{\mathrm{b}}+w^{\mathrm{s}}\right)\\ w_{,xx}^{\mathrm{b}}+w_{,yy}^{\mathrm{b}}-{\displaystyle \frac{u_{0,x}}{R_x}}-{\displaystyle \frac{v_{0,y}}{R_y}}\\ w_{,xx}^{\mathrm{s}}+w_{,yy}^{\mathrm{s}}\end{array}\right\},\end{array}

\begin{array}{rl}{\mathrm{\Lambda }}_1\left\{\begin{array}{c}{S}_{xx}\\ S_{yy}\end{array}\right\}=& {\mathrm{\Lambda }}_2\left[\begin{array}{ccc}{\mathcal{D}}_{11}& {\mathcal{D}}_{12}& {\mathcal{D}}_{13}\\ {\mathcal{D}}_{11}& {\mathcal{D}}_{12}& {\mathcal{D}}_{13}\end{array}\right]\\ & \cdot \left\{\begin{array}{c}{-u}_{0,x}-v_{0,y}-\left({\displaystyle \frac{1}{R_x}}+{\displaystyle \frac{1}{R_y}}\right)\left(w^{\mathrm{b}}+w^{\mathrm{s}}\right)\\ w_{,xx}^{\mathrm{b}}+w_{,yy}^{\mathrm{b}}-{\displaystyle \frac{u_{0,x}}{R_x}}-{\displaystyle \frac{v_{0,y}}{R_y}}\\ w_{,xx}^{\mathrm{s}}+w_{,yy}^{\mathrm{s}}\end{array}\right\},\end{array}

where {\mathcal{A}}_{ij}, {\mathcal{B}}_{ij}, {\mathcal{B}}_{ij}^{\mathrm{s}}, {\overline{\mathcal{B}}}_{ij} ,{\mathcal{F}}_{ij}, {\mathcal{F}}_{ij}^{\mathrm{s}}, {\overline{\mathcal{B}}}_{ij}^{\mathrm{s}}, {\overline{\mathcal{F}}}_{ij}^{\mathrm{s}}, {\mathcal{H}}_{ij} ,{\mathcal{A}}_{ij}^{\mathrm{s}},{\mathcal{C}}_{ij}, and {\mathcal{D}}_{ij} are the stiffness parameters of the sandwich FGP doubly-curved nanoshell integrated with piezoelectric surface layers material and they are given in Appendix A in Electronic Supplementary Material. By replacing the force components in the aforementioned system of balanced equations, we obtain a system of equations expressed in terms of the displacement components (Appendix B in Electronic Supplementary Material).

3 Analytical method

This article employs the Galerkin analytical approach to ascertain the free vibration properties of the sandwich FGP doubly-curved nanoshell integrated with piezoelectric surface layers. This approach has the benefit of providing precise outcomes for a wide range of BCs. Nevertheless, this analytical technique often yields solutions in integral format and is only applicable for computing smooth shell configurations with a flat representation of the nanoshell with rectangular planform, or those that possess symmetry. Subsequently, we will introduce the solution format of the approach and the equations used to get the coefficients of the stiffness and mass matrix of the shell construction.

3.1 Galerkin approach

This section presents the developed solutions for the sandwich FGP doubly-curved nanoshell integrated with piezoelectric surface layers under different BCs, as described by equations in Appendix B in Electronic Supplementary Material. The BCs for every given edge are determined by whether it is simply supported or clamped [64].

For fully clamped (C)

(42)\begin{array}{rl}{u}_0=& v_0=w^{\mathrm{b}}=w^{\mathrm{s}}=w_{,x}^{\mathrm{b}}=w_{,x}^{\mathrm{s}}=w_{,y}^{\mathrm{b}}=w_{,y}^{\mathrm{s}}=0\\ & \mathrm{a}\mathrm{t}x=0,a\mathrm{a}\mathrm{n}\mathrm{d}y=0,b.\end{array}

For simply (S) support

(43)u_0= w^{\mathrm{b}}= w^{\mathrm{s}}= w_{,x}^{\mathrm{b}}= w_{,x}^{\mathrm{s}}= 0\mathrm{a}\mathrm{t}y=0, b,

(44)v_0= w^{\mathrm{b}}= w^{\mathrm{s}}= w_{,y}^{\mathrm{b}}= w_{,y}^{\mathrm{s}}= 0\mathrm{a}\mathrm{t}x=0, a.

The displacement components at any position of the sandwich FGP doubly-curved nanoshell integrated with piezoelectric surface layers are expressed as solutions in the following manner

(45)u_0= \sum _m^{\mathrm{\infty }} \sum _n^{\mathrm{\infty }}{U}_{mn} X_{m,x}\left(x\right)Y_n\left(y\right),

(46)v_0= \sum _m^{\mathrm{\infty }} \sum _n^{\mathrm{\infty }}{V}_{mn} X_m\left(x\right)Y_{n,y}\left(y\right),

(47)w^{\mathrm{b}}= \sum _m^{\mathrm{\infty }} \sum _n^{\mathrm{\infty }}{W}_{mn}^{\mathrm{b}}{X}_m\left(x\right)Y_n\left(y\right),

(48)w^{\mathrm{b}}= \sum _m^{\mathrm{\infty }} \sum _n^{\mathrm{\infty }}{W}_{mn}^{\mathrm{b}}{X}_m\left(x\right)Y_n\left(y\right),

where the parameters U_{mn}, V_{mn}, W_{mn}^{\mathrm{b}}, and W_{mn}^{\mathrm{s}} are unknown in this context. The functions X_m\left(x\right) and Y_n\left(y\right) are given

(49)\begin{array}{rl}{X}_m\left(x\right)=& \mathrm{s}\mathrm{i}\mathrm{n}\left({\kappa }_{m}x\right)+A_m\mathrm{c}\mathrm{o}\mathrm{s}\left({\kappa }_{m}x\right)+{\mathrm{B}}_m\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}\left({\kappa }_{m}x\right)\\ & +C_m\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h}\left({\kappa }_{m}x\right),\end{array}

(50)\begin{array}{rl}{Y}_n\left(y\right)=& \mathrm{s}\mathrm{i}\mathrm{n}\left({\gamma }_{n}y\right)+A_n\mathrm{c}\mathrm{o}\mathrm{s}\left({\gamma }_{n}y\right)+B_n\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}\left({\gamma }_{n}y\right)\\ & +C_n\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h}\left({\gamma }_{n}y\right).\end{array}

The determination of the values of {\kappa }_m,{\gamma }_n, A_m, A_n, B_m,B_n, C_m, and C_n is contingent upon the distinct BCs of the shell. This study takes into account the following boundary circumstances: SSSS, CCCS, CCSS, CSCS, CCCC, and CSSS. The specification of every scenario is as follows [64]

CCCC:

\begin{array}{rl}& {\kappa }_1=\frac{4.7300}{a},{\kappa }_2=\frac{7.8532}{a},{\kappa }_i=\text{π}\frac{2i+1}{2a}(i>2),\\ & A_m=-\frac{\mathrm{s}\mathrm{i}\mathrm{n}\left({\kappa }_{m}a\right)-\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}\left({\kappa }_{m}a\right)}{\mathrm{c}\mathrm{o}\mathrm{s}\left(r_{m}a\right)-\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h}\left(r_{m}a\right)},B_m=-1,C_m=-A_m,\end{array}

\begin{array}{rl}& {\gamma }_1=\frac{4.7300}{b},{\gamma }_2=\frac{7.8532}{b},{\gamma }_i=\text{π}\frac{2i+1}{2b}(i>2),\\ & A_n=-\frac{\mathrm{s}\mathrm{i}\mathrm{n}\left({\gamma }_{n}b\right)-\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}\left({\gamma }_{n}b\right)}{\mathrm{c}\mathrm{o}\mathrm{s}\left({\gamma }_{n}b\right)-\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h}\left({\gamma }_{n}b\right)},B_n=-1,C_n=-A_n.\end{array}

CCSS:

\begin{array}{rl}& {\kappa }_1=\frac{4.7300}{a},{\kappa }_2=\frac{7.8532}{a},{\kappa }_i=\text{π}\frac{2i+1}{2a}\left(i>2\right),\\ & A_m=-\frac{\mathrm{s}\mathrm{i}\mathrm{n}\left({\kappa }_{m}a\right)-\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}\left({\kappa }_{m}a\right)}{\mathrm{c}\mathrm{o}\mathrm{s}\left({\kappa }_{m}a\right)-\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h}\left({\kappa }_{m}a\right)},B_m=-1,C_m=-A_m,\end{array}

\begin{array}{}\text{(52b)}& {\gamma }_n=\text{π}\frac{n}{b},A_n=0,B_n=0,C_n=0.\end{array}

CCCS:

\begin{array}{rl}& {\kappa }_1=\frac{4.7300}{a},{\kappa }_2=\frac{7.8532}{a},{\kappa }_i=\text{π}\frac{2i+1}{2a}(i>2),\\ & A_m=-\frac{\mathrm{s}\mathrm{i}\mathrm{n}\left({\kappa }_{m}a\right)-\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}\left({\kappa }_{m}a\right)}{\mathrm{c}\mathrm{o}\mathrm{s}\left({\kappa }_{m}a\right)-\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h}\left({\kappa }_{m}a\right)},B_m=-1,C_m=-A_m,\end{array}

\begin{array}{rl}& {\gamma }_1=\frac{3.9266}{b},{\gamma }_2=\frac{7.0685}{b},{\gamma }_i=\text{π}\frac{4i+1}{4b}(i>2),\\ & A_n=0,B_n=-\frac{\mathrm{s}\mathrm{i}\mathrm{n}\left({\gamma }_{n}b\right)}{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}\left({\gamma }_{n}b\right)},C_n=0.\end{array}

CSCS:

\begin{array}{rl}& {\kappa }_1=\frac{3.9266}{a},{\kappa }_2=\frac{7.0685}{a},{\kappa }_i=\text{π}\frac{4i+1}{4a}(i>2),\\ & A_m=0,B_m=-\frac{\mathrm{s}\mathrm{i}\mathrm{n}\left({\kappa }_{m}a\right)}{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}\left({\kappa }_{m}a\right)},C_m=0,\end{array}

\begin{array}{rl}& {\gamma }_1=\frac{3.9266}{b},{\gamma }_2=\frac{7.0685}{b},{\gamma }_i=\text{π}\frac{4i+1}{4b}(i>2),\\ & A_n=0,B_n=-\frac{\mathrm{s}\mathrm{i}\mathrm{n}\left({\gamma }_{n}b\right)}{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}\left({\gamma }_{n}b\right)},C_n=0.\end{array}

CSSS:

\begin{array}{rl}& {\kappa }_1=\frac{3.9266}{a},{\kappa }_2=\frac{7.0685}{a},{\kappa }_i=\text{π}\frac{4i+1}{4a}(i>2),\\ & A_m=0,B_m=-\frac{\mathrm{s}\mathrm{i}\mathrm{n}\left({\kappa }_{m}a\right)}{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}\left({\kappa }_{m}a\right)},C_m=-A_m,\end{array}

\begin{array}{}\text{(55b)}& {\gamma }_n=\text{π}\frac{n}{b},A_n=0,B_n=0,C_n=0.\end{array}

SSSS:

\begin{array}{}\text{(56a)}& {\kappa }_m=\text{π}\frac{m}{a},A_n=0,B_n=0,C_n=0,\end{array}

\begin{array}{}\text{(56b)}& {\gamma }_n=\text{π}\frac{n}{b},A_n=0,B_n=0,C_n=0.\end{array}

The subsequent equations are obtained by substituting Eqs. (45)–(48) for equations in Appendix B in Electronic Supplementary Material, multiplying each equation by the eigenfunction, integrating over the entire solution domain, and executing mathematical operations.

(57)\begin{array}{rl}& \underset{M}{\underbrace{\left[\begin{array}{cccc}{m}_{11}& m_{12}& m_{13}& m_{14}\\ m_{21}& m_{22}& m_{23}& m_{24}\\ m_{31}& m_{32}& m_{33}& m_{34}\\ m_{41}& m_{42}& m_{43}& m_{44}\end{array}\right]}}\left\{\begin{array}{c}{\ddot{U}}_{mn}\\ {\ddot{V}}_{mn}\\ {\ddot{W}}_{mn}^{\mathrm{b}}\\ \underset{\ddot{d}}{\underbrace{{\ddot{W}}_{mn}^{\mathrm{s}}}}\end{array}\right\}\\ +& \underset{K}{\underbrace{\left[\begin{array}{cccc}{k}_{11}& k_{12}& k_{13}& k_{14}\\ k_{21}& k_{22}& k_{23}& k_{24}\\ k_{31}& k_{32}& k_{33}& k_{34}\\ k_{41}& k_{42}& k_{43}& k_{44}\end{array}\right]}}\left\{\begin{array}{c}{U}_{mn}\\ V_{mn}\\ W_{mn}^{\mathrm{b}}\\ \underset{d}{\underbrace{W_{mn}^{\mathrm{s}}}}\end{array}\right\}=\left\{\begin{array}{c}0\\ 0\\ 0\\ \underset{F}{\underbrace{0}}\end{array}\right\},\end{array}

where m_{ij} ,k_{ij} are the coefficients of the mass matrix and stiffness matrix of sandwich FGP doubly-curved nanoshell integrated with piezoelectric surface layers, respectively. And \ddot{d} and d vectors components are the acceleration and displacement vectors of amplitude.

3.2 Procedure for obtaining the natural oscillation frequency

For the free vibration problem: Set d=\overline{d}\mathrm{s}\mathrm{i}\mathrm{n}\left(\omega t\right), the value of the natural oscillation frequency of the shell is determined as follows

(58)\mathrm{d}\mathrm{e}\mathrm{t}\left(K-{\omega }^{2}M\right)=0,

where \omega is the natural oscillation frequency of the piezoelectric nanoshell.

4 Numerical results and discussion

To enhance efficiency and speed up calculations, a program is developed using Matlab software. This program utilizes the analytical formulas discussed in the previous section to compute the natural frequency of sandwich FGP doubly-curved nanoshell integrated with piezoelectric surface layers resting on a Pasternak elastic foundation and incorporates the flexoelectric effect. To ascertain the dependability and precision of the program, certain numerical comparisons are conducted with analytical papers in certain scenarios of the model described in the article. To enable the comparison of its numerical results with other accurate answers published in the literature, the following non-dimensional representations are presented.

(59)\begin{array}{rl}& {\mathrm{\Omega }}_{ij}={\omega }_{ij}\frac{b^2}{{\text{π}}^2}\sqrt{{\rho }_{\mathrm{c}}h/D_{\mathrm{c}}},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_{\mathrm{c}}{h}^3}{12(1-v^2)},{\overline{K}}_{\mathrm{w}}=\frac{k_{\mathrm{w}}{a}^4}{D_m},\\ & {\overline{K}}_{\mathrm{s}}=\frac{k_{\mathrm{s}}{a}^2}{D_m},D_m=\frac{E_m{h}^3}{12(1-v^2)}.\end{array}

The geometric and shell material characteristics were acquired by numerical investigations: a=10\mathrm{n}\mathrm{m},b=a,h=1\mathrm{n}\mathrm{m},h_{\mathrm{f}1}=h_{\mathrm{f}2} ={\displaystyle \frac{h}{4}}, n_{\mathit{\text{z}}}=1, \xi =0.2, K_{\mathrm{w}}= 20, K_{\mathrm{s}}=2, R_x= 5a, R_y= 5b, {\mu }_1= 1{\mathrm{n}\mathrm{m}}^2, {\mu }_{\mathrm{c}}= 2{\mathrm{n}\mathrm{m}}^2, l_{\mathrm{m}}= 1{\mathrm{n}\mathrm{m}}^2, l_{\mathrm{c}}= 3{\mathrm{n}\mathrm{m}}^2. Symbols of typical shell types and their radii of curvature are given: Spherical shell (SPS, R_x=5a, R_y=5b), cylindrical shell (CYS, R_x=5a, R_y=\mathrm{\infty }), saddle shell (HPS, R_x= 5a, R_y= -5b), elliptical shell (ELS, R_x=5a, R_y= 10b). The material characteristics of each layer are expressed as follows: \mathrm{A}\mathrm{l}/{\mathrm{A}\mathrm{l}}_2{\mathrm{O}}_3, the piezoelectric layer made of PZT-5H material is taken as in Refs. [26,56].

4.1 Verification examples

This part provides numerical examples to assess the precision and dependability of the model presented in the article, specifically in unique scenarios of the doubly-curved shell model, in comparison to trustworthy analytical publications. The first non-dimensional frequency {\mathrm{\Omega }}_1of the doubly-curved shell composed of FG material is confirmed, as shown in Tab.1. The validated shell types in this study consist of spherical shell, cylindrical shell, elliptical shell, and flat plate supported by a Pasternak-elastic medium with a variable grading indexn_{\mathit{\text{z}}}. The study conducted by Kiani et al. [65] provides the fundamental natural frequency results presented in Tab.1. These results were obtained using the first-order shear deformation theory and the Navier method. Alternatively, the fundamental natural frequency values can be calculated directly from the model described in the article when the thickness of the piezoelectric two-layer is set to zero. It is important to note that the effects of nanostructures and porosities were not taken into consideration in this study. The findings indicate that the fundamental natural frequency {\mathrm{\Omega }}_1 of doubly-curved the shell, as suggested in the paper, yields comparable outcomes to those reported by Kiani et al. [65].

Furthermore, we conduct a comparative analysis of the natural frequency of present work with the findings of Shen et al. [66] in Tab.2. Six orthotropic nanoplates are studied in this work, and each has different geometric properties. Shen et al. [66] used molecular dynamics (MD) simulation and a nonlocal theory-based analytical technique to accurately calculate the fundamental natural frequency of orthotropic SSSS nanoplates. The results of our investigation closely correspond to the findings of Shen et al. [66], as seen in both the MD simulation and analytical technique.

Next, compare the first non-dimensional frequency {\mathrm{\Omega }}_1 of an isotropic nanosheet using the SGT. We consider several values of \sqrt{{\mu }_2}(0,0.2,0.5,1 \mathrm{n}\mathrm{m}) and three BCs: SSSS, CSCS, and CCCC. The input data are collected in the following manner: the values of a, b, and h are given as 10, 10, and 0.34 nm, respectively. The findings are then compared to those of Babu and Patel [67], and are shown in Tab.3. This work used both the Navier method and the finite element approach to calculate the values of natural frequency {\mathrm{\Omega }}_1. The result table reveals that the SSSS and CSCS BCs provide almost similar results to those obtained by Babu and Patel [67]. However, the CCCC BC exhibit the highest inaccuracy of 2.82% when compared to Babu and Patel’s [67] findings.

The fundamental natural frequency (GHz) of the square piezoelectric nanosheets subjected to SSSS BC is being analyzed. The nanoplate has a thickness of 20 \mathrm{n}\mathrm{m} and dimensions of a=b=50h. The material characteristics are as follows: C_{11}=102 \mathrm{G}\mathrm{P}\mathrm{a},C_{12}=31\mathrm{G}\mathrm{P}\mathrm{a} ,C_{66}= 35.5\mathrm{G} \mathrm{P}\mathrm{a}; e_{31}=-17.05\mathrm{C}/{\mathrm{m}}^2, k_{33}=-\mathrm{1.76.}{10}^{-8}\mathrm{C}/\mathrm{V}\mathrm{m}, f_{14}= {10}^{-7}\mathrm{C}/\mathrm{m}. The fundamental natural frequency of the piezoelectric nanosheets is compared with Navier’s solution and classical plate theory in Ref. [23]. The comparison findings are shown in Fig.3. The statistics clearly demonstrate that the findings presented in the paper align perfectly with the results mentioned in Ref. [23]. Thus, the paper suggests using the provided strategy to guarantee the dependability and precision necessary for doing more research.

4.2 Study parameters

This section analyzes the influence of different input factors on the non-dimensional natural frequencies of the sandwich FGP doubly-curved nanoshell integrated with piezoelectric surface layers. The parameters include nonlocal and length-scale parameters, porosity parameters, grading indices, elastic foundation stiffness, ratio of material layers, and other BCs. The estimation of the non-dimensional primary natural frequency {\mathrm{\Omega }}_1\left({\mathrm{\Omega }}_{11}\right) and secondary natural frequency {\mathrm{\Omega }}_2\left({\mathrm{\Omega }}_{12}\right) of structures is crucial in studying resonance phenomena and identifying structural resistance coefficients in dynamic problems. Moreover, the arrangement of the oscillation patterns will highlight the regions of highest and lowest stiffness inside the structure throughout its operation. It is possible to determine the placements of measuring devices and calculate and adjust certain properties of the structure to achieve different degrees of stiffness.

First, the influence of different theoretical models (Fig.4 and Fig.5) on the non-dimensional natural frequency results of sandwich FGP doubly-curved nanoshell includes nonlocal theory without flexoelectric effect (NL), nonlocal strain gradient theory without flexoelectric effect (NSGT), strain gradient theory without flexoelectric effect (SGT), nonlocal theory with flexoelectric effect (NL-Flexo), nonlocal strain gradient theory with flexoelectric effect (NSGT-Flexo), strain gradient theory with flexoelectric effect (SGT-Flexo). These theoretical models are all drawn from the general theoretical model of the article, thereby showing the comprehensiveness of this article. Fig.4 displays the outcomes of the non-dimensional first-two natural frequencies, obtained from various theories, as a dependent variable of the grading index n_{\mathit{\text{z}}} of the FGP core layer in a sandwich FGP cylindrical shallow nanoshell. According to the findings, the sandwich FGP doubly-curved nanoshell would achieve the highest natural frequency by using SGT in conjunction with the flexoelectric effect. When the NL is used, the structure will exhibit the lowest natural frequency. Furthermore, when the aforementioned theoretical models are used, the natural frequency will consistently diminish as the grading index n_{\mathit{\text{z}}} rises. The rationale for this phenomenon is that the augmentation of n_{\mathit{\text{z}}} results in an elevation of the metal concentration inside the core layer, thus resulting in a reduction of the overall rigidity of the sandwich FGP doubly-curved nanoshell. The reduction in natural frequency is most pronounced when 0\le n_{\mathit{\text{z}}}\le 2. When the value of n_{\mathit{\text{z}}} is more than 2, there is little alteration in the rise of the natural frequency. Fig.5 illustrates the variation in the non-dimensional natural frequency of a sandwich FGP spherical shallow nanoshell when theoretical models are used. This variation is dependent on the change in the ratio {\displaystyle \frac{h_{\mathrm{f}}}{h}} (where h_{\mathrm{f}1}=h_{\mathrm{f}2}=h_{\mathrm{f}}). The results graph clearly demonstrates that the continuous reduction in the non-dimensional natural frequency of the structure occurs as the ratio {\displaystyle \frac{h_{\mathrm{f}}}{h}} increases, when the sandwich FGP doubly-curved nanoshell is subjected to nonlocal stress theoretical models, SGT, and NSGT. The reason for this is because the piezoelectric layer components possess mechanical qualities that are less rigid compared to the core layer. As a result, the thickness h_{\mathrm{f}} rises, causing the shell structure to become more pliable. However, the combination of the aforementioned three theoretical models with the flexoelectric effect results in an increase in the initial {\displaystyle \frac{h_{\mathrm{f}}}{h}} ratio, leading to a maximum value of the natural frequency. Subsequently, the natural frequency steadily drops.

Following this, as illustrated in Fig.6, the non-dimensional first two natural frequency values of a sandwich FGP elliptical shallow nanoshell supported by an elastic foundation and exhibiting flexoelectric effect as a result of the length a and thickness h are detailed. In this case, the flexoelectric coefficient f_{14}^{\ast \ast }=0,0.1,0.3, 0.5,0.7,1 is altered. The thickness of the nanoshell is maintained at h=1\mathrm{n}\mathrm{m} in Fig.6(a) and Fig.6(b), while the length of the nanoshell varies between a=5÷50 \mathrm{n}\mathrm{m}. The relationship between the flexoelectric coefficient and the non-dimensional natural frequency becomes readily apparent: The natural frequency of the oscillation rises. Consequently, the manifestation of this coefficient results in an elevation of the nanoshell’s hardness. Additionally, it is worth mentioning that as the a/h ratio increases, the nanoshell structure becomes more susceptible to variations in the flexoelectric coefficient, resulting in a rapid escalation of both the non-dimensional first and second natural frequencies. The variation in the ratio {\displaystyle \frac{a}{h}}= 5÷50, a=10\mathrm{n} \mathrm{m} is illustrated in Fig.6(c) and Fig.6(d). As the ratio a/h increases, the size sensitivity to the flexoelectric effect becomes more pronounced, resulting in a continuous increase in the natural frequency of the number of specific oscillations. Furthermore, when the flexoelectric effect is disregarded, {\displaystyle \frac{a}{h}}> 10 yields a dimensionless natural frequency that is nearly constant.

The subsequent analysis examines the impact of the porosity coefficient and various forms of porosity, such as those with and without flexoelectric effect, on the non-dimensional first two natural frequencies of a sandwich FGP shallow nanoshell. The FGP hyperbolic shallow nanoshell is characterized by two distinct BCs, as seen in Fig.7. It is evident that, without taking into account the flexoelectric impact of the two piezoelectric layers, a rise in the porosity coefficient \xi \left(\xi =0÷0.8\right) occurs. The application of porosity rules FGP 2 and FGP 3 results in an increase in the non-dimensional first-two natural frequencies, whereas the implementation of uniform porosity laws FGP 1 leads to a drop in the non-dimensional natural frequencies. When taking into account the flexoelectric effect, it is shown that all three porosity laws result in a rise in both the non-dimensional first-two natural frequencies as the porosity coefficient increases. However, the noteworthy result is that the porosity rule FGP 1 experiences the greatest rise in its non-dimensional natural frequency during this period. This phenomenon is seen in both SSSS and CCCC BCs, and research suggests that it is also prevalent in most other BCs. The existence of porosities will result in a reduction in the structure’s rigidity. Nevertheless, due to the fact that porosities influence both mass and rigidity, they will alter the first non-dimensional frequency of the sandwich FGP shallow nanoshell, as illustrated in Fig.7. The effect of the elastic foundation stiffness coefficients K_{\mathrm{w}} and K_{\mathrm{s}} on the first non-dimensional frequency of a piezoelectric sandwich FGP hyperbolic shallow nanoshell is depicted in Fig.8. Six distinct BCs are examined in the study, one with and one without the flexoelectric effect. It is evident that the first non-dimensional frequency increases significantly as the two elastic foundation stiffness coefficients K_{\mathrm{w}} and K_{\mathrm{s}} (K_{\mathrm{w}}= 0÷100, K_{\mathrm{s}}= 0÷10) increase, excluding the flexoelectric effect. The non-dimensional natural frequency increases less under harsher BCs, such as CCCC or CCCS, particularly under gentler BCs like SSSS or CSSS. However, in the context of the flexoelectric effect, it can be observed that as the value of the coefficient K_{\mathrm{w}} increases, the value of K_{\mathrm{s}} barely contributes to the increase in the non-dimensional natural frequency of the shell. This demonstrates that the flexoelectric effect has a substantial impact on the rigidity of the shell structure, resulting in a large increase in sandwich FGP shallow nanoshell stiffness. This rise is so pronounced that it surpasses the effects of other factors that typically enhance shell stiffness, such as the stiffness of the elastic foundation. Fig.9 illustrates the first nine distinct vibrational configurations of the spherical doubly-curved nanoshell supported by an elastic foundation. Here, the nanoshell sizes in the x and y directions are equal because of the implementation of SSSS BCs. Consequently, it will produce contrasting oscillation patterns, namely mode 2 and mode 3, mode 5 and mode 6, and mode 7 and mode 8. Symmetric vibration modes will not occur when asymmetric BCs and asymmetric geometric dimensions are used.

Tab.4 provides a description of the impact of the radius of curvature R_x, R_y, and six distinct BCs of the sandwich FGP shallow nanoshell on the values of the first non-dimensional frequency {\mathrm{\Omega }}_1. The radius of curvature of the shell is determined by the ratios {\displaystyle \frac{a}{R_x}} and {\displaystyle \frac{b}{R_y}}, which take on the values of −0.2, −0.1, 0, 0.1, and 0.2. Additionally, six distinct BCs are considered: CCCC, CCCS, CCSS, CSCS, CSSS, and SSSS. The findings of the non-dimensional natural frequency suggest that the lowest natural frequency occurs when the ratio of a to R_x and b to R_y is −0.2, while the highest natural frequency is attained when the ratio is 0.2, for all examined BCs. The findings also indicate that a higher ratio of {\displaystyle \frac{a}{R_x}}\left({\displaystyle \frac{b}{R_y}}\right) will result in an elevation of the eigen-oscillation values. Nevertheless, these individual fluctuation values exhibit little disparity. The Galerkin solution is used for analyzing sandwich FGP shallow nanoshell under various BCs, which adds to its overall appeal. The primary advantage of this analytical approach is in its ability to serve as a benchmark for numerical methods, such as FEM, isogeometric analysis, and differential quadrature method, when computing structures of similar kind. In addition, the CCCC BC will result in the highest level of hardness for the sandwich FGP shallow nanoshell, with decreasing levels of hardness seen in the following sequence: CCCC, CCCS, CCSS, CSCS, CSSSS, and the lowest level of hardness is observed in SSSS.

Finally, the effects of nonlocal coefficients {\mu }_{\mathrm{m}}, {\mu }_{\mathrm{c}}, and material length-scale l_{\mathrm{m}}, l_{\mathrm{c}} on the first non-dimensional frequency values of doubly-curved nanoshells with different BCs are described in Tab.5 and Tab.6. It can be seen that theirst non-dimensional frequency values {\mathrm{\Omega }}_1are contradictory when the nonlocal and length-scale coefficients increase. On the one hand, the nonlocal coefficient {\mu }_{\mathrm{m}},{\mu }_{\mathrm{c}} increases, causing the stiffness of the structure to decrease. Meanwhile, the increase in the length-scale coefficient l_{\mathrm{m}},l_{\mathrm{c}} causes the stiffness of the nanostructure to increase. This shows that for nanometer-sized structures, there exist coefficients that increase or decrease hardness, and thus can control the vibrations of nanostructures easily.

5 Conclusions

In this study, the exact free vibration characteristics of sandwich FGP doubly-curved nanoshell integrated with piezoelectric surface layers with various BCs based on flexoelectric effect and NSGT are explored for the first time. The sandwich nanoshell consists of two piezoelectric components and a porous and FG core. The distinguishing feature of this study is that the two nonlocal coefficients and material length scale of the core layer may change over its thickness, akin to other material qualities. The equations that describe the balance of shell motion are derived by using Hamilton’s principle, NSGT, and a highly refined high-order shear hypothesis. The natural frequency of the shell is calculated using the Galerkin–Vlasov solution, taking into account arbitrary BCs. These models include the nonlocal stress theory, SGT, and NSGT. Furthermore, these models integrate the flexoelectric effect, with the nonlocal stress theory, SGT, and NSGT combining it. Nanostructures that use the SGT in combination with the flexoelectric effect model will demonstrate the utmost degree of stiffness. On the other hand, when the nonlocal stress theory model is used, the nanostructures will have the least amount of stiffness. Therefore, it is clear that in sandwich constructions, the adjustment of stiffness may be accomplished by increasing the flexoelectric coefficients of the piezoelectric layer. In addition, the overall stiffness of the structure may be increased by modifying the length-scale coefficient, while raising the nonlocal coefficient would make the structure more flexible. Moreover, the proportion of the piezoelectric layer’s thickness to the FGP core layer is a critical determinant in examining the control of the stiffness of the whole structure, whether it is to enhance or reduce it. An increase in the grading index will cause the overall rigidity of the structure to decrease, but an increase in the elastic base stiffness will cause the overall rigidity to rise. Furthermore, an increase in the pore coefficient might either increase or decrease the natural frequency of the shell, depending on the interaction between the mass and stiffness of the structure.

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