Multifield vibration analysis of a porosity-tailored bidirectional functionally graded piezoelectric nano-plate on variable elastic foundations under hygro-thermoelectric effects

Abstract

This study investigates the multifield vibration behavior of a porosity-dependent bidirectional functionally graded piezoelectric nano-plate (FGPN) subjected to hygrothermal and thermoelectric loading. The material composition is defined by sigmoid and power-law distributions along both transverse and axial directions, accommodating even, uneven, and symmetrically centered porosity patterns. The model incorporates nonclassical elasticity theory and von Kármán nonlinear strains, with the governing equations formulated using a modified first-order shear deformation theory and derived through the energy principle. A higher-order finite element formulation, coupled with a modified Newton–Raphson procedure, ensures robust computational accuracy, validated through convergence tests. The analysis delves into the influence of porosity distribution, bidirectional material variations, non-uniform thickness, thickness ratios, variable elastic foundations, and boundary conditions on vibrational behavior. Additionally, the study explores the interplay of hygrothermal and electrical loading conditions in diverse configurations. The findings highlight the pivotal role of bidirectional material gradation in shaping the vibrational response of porous FGPN structures, offering valuable insights for the design of nano-plates in hygrothermal and thermoelectric applications.

Graphical abstract

Keywords

Porosity / Bidirectional exponent / nano-plate / functionally graded piezoelectric plate / FEM / Elastic foundations

1 Introduction

Piezoelectric materials find diverse applications in smart structures, micro-/nano-electromechanical systems (MEMS/NEMS), and other fields due to their electromechanical coupling effects [1]. Traditional piezoelectric structures typically comprised varying piezoelectric and elastic layers. Despite their numerous advantages, working with these structures poses challenges, especially related to interface characteristics. An example of such challenges is the sudden material property changes at interfaces, leading to stress jumps between layers, matrix, and interfacial cracking under electromechanical loading. To address these limitations, a new kind of piezoelectric materials, termed functionally graded (FG) piezoelectric (FGP) materials, has been presented. FGPs are characterized by smooth spatial variations in material properties [2]. Theoretical and experimental studies have demonstrated that the consistency and stability of piezoelectric-based structures and devices can be significantly improved by employing FGP materials.

Currently, FGP plates and beams are extensively utilized in MEMS and NEMS applications, including ultrasonic transducers, sensors, and actuators [3,4]. Numerous studies have explored the static and dynamic behaviors of functionally graded materials (FGMs) employing power-law material distributions, particularly with FGM plates subjected to thermomechanical loading [5,6]. Governing equations obtained were solved using a combination of numerical and analytical methods [7,8]. In recent years, advancements in nanostructure fabrication technologies have increasingly integrated FGMs and smart materials into micro/nano-scale structures, such as piezo-actuators, acoustic applications, sensors, thin films, and energy harvesters, to augment their performance [9,10]. Notably, in these applications, the size effect holds significant importance, warranting careful examination of the thermomechanical responses of such nanoscale structures [11,12].

Eltaher et al. [13] conducted a study on the vibration behavior of FG size-dependent nanobeam using a finite element (FE) approach. Their investigations on size-dependent beams were based on a nonlocal continuum model. Natarajan et al. [14] explored the size-dependent, flexural-free vibration of FG nanoplate using an Isogeometric-based FE approach. Additionally, Liu et al. [15] examined the free vibration response of piezo-nanoplate based on the nonlocal and Kirchhoff theory. In their study, the plates were simply supported and subjected to thermomechanical and electromechanical loadings, with final equations derived using Hamilton’s principle and solved analytically. Jandaghian and Rahmani [16] presented the free vibration characteristics of simply supported nanoscale FGP plates using nonlocal Kirchhoff plate theory.

Advancements in FGMs have led to their widespread production and utilization. However, the inherent production process unavoidably introduces porosity within the material. This porosity alters the properties of the original FGMs, making it imperative to thoroughly investigate its impact on the thermomechanical behaviors of FG-based structures.

Li et al. [17] and Kumar and Harsha [18,19] investigated the porosity effect on bending and dynamic responses of FG plates using the first-order shear deformation theory (FSDT) and the energy principle. Additionally, Zenkour and Aljadani [20] explored the influence of porosity on the buckling analysis of functionally graded piezoelectric nano-plate (FGPN) subjected to thermomechanical loading, employing nonlocal theory and higher-order shear deformation theory. In an allied study, Kumar and Harsha [21] discussed the vibration characteristics of porosity-dependent FGP plates under thermo-electromechanical loading with various support constraints.

Numerous researchers have presented their findings evaluating the mechanical response of FGMs. However, solely employing transverse or axial variation of properties in FG structures might not be sufficient for smart structures, acoustic sensors, and certain aerospace applications, especially when temperature, moisture, and electromechanical loading occur along multiple directions. Hence, the concept of bidirectional variations in properties in FGP structures, involving material distributions along both the transverse and axial directions, has been proposed to address these requirements. To explore the bidirectional behavior of FG-based smart structures and FGMs, Kumar and Harsha [22] presented the free vibration behavior of bidirectional porous FGP plates subjected to thermomechanical loading. Lieu et al. [23] examined the bending and vibration behaviors of bidirectional FGMs plates with variable thicknesses using the isogeometric approach. Recently, the use of piezoelectric smart FGMs in diverse environments, encompassing thermomechanical, electrical, and moisture conditions, has become a significant point of interest for researchers. Tang and Ding [24] examined the nonlinear vibration behavior of bidirectional FG beams under hygrothermal loading, while Harsha and Kumar [25] studied the vibration response of a three-layer bidirectional FGP plate subjected to thermal, electrical, and mechanical loading. Brischetto and Cesare [26] investigate the three-dimensional vibration responses of multilayered FGP plates and shells. Alzahrani et al. [27] examined the thermomechanical bending behavior of nanoplate under high-moisture environmental conditions. In a recent study, Zenkour and Alghanmi [28] examined the bending responses of an FG plate with a piezoelectric layer under hygrothermal and thermoelectric loading with boundary conditions (BCs).

Scientists have extensively explored the interactions between smart structures and elastic foundations. Recent studies [29–31] have investigated various elastic foundations and their interactions with plate structures. A substantial body of literature exists focusing on different aspects of FG-based structures, including their static, buckling, and dynamic characteristics under diverse loading conditions. Sobhy [32] presented the thermal analysis of FG plates resting on variable elastic foundations under thermomechanical loading. Additionally, Mudhaffar et al. [33] examined the bending behavior of FG plates resting on viscoelastic foundations and subjected to hygrothermal loading. Ansari et al. [34] delved into the free vibration behavior of FG nanoplate resting on elastic foundations. Other researchers explored the vibrational response of nonlocal piezoelectric nanoplate [35], FG plates and beams [36], and FGP nanoshells [37] when exposed to thermal, electrical [38], and mechanical loading. The final equations were solved using a range of methods, either analytical [39–41] or numerical [42–44]. The study of size-dependent behavior and vibrational analysis of nanostructures and FGP materials has been a subject of increasing interest. Various researchers have explored the influence of thermal, electrical, and mechanical loading on these structures. For instance, Ebrahimi and Salari [45,46] investigated the effect of non-uniform temperature distributions on the nonlocal vibration and buckling of inhomogeneous, size-dependent beams, highlighting the impact of thermal gradients on structural stability and dynamic behavior. The study of FG nanobeams by Ehyaei et al. [47] further contributes by examining different BCs and nonlocal vibration behavior, expanding the understanding of size effects on FGMs. Research by Kumar and Harsha [48] and Salari et al. [49] highlights the role of material gradation and porosity on the vibration and buckling responses of piezoelectric plates and nanobeams. Further, the nonlinear dynamic behavior of thermally post-buckled nanobeams was studied by Salari et al. [50], considering temperature-dependent porous nanobeams under nonlocal theory. This analysis contributes to understanding the complexities of nanostructures under thermally induced stresses. Moreover, the electroelastic behavior of smart FGP plates has been studied in the context of both static and vibration responses under electromechanical loading [51,52]. The effect of porosity and elastic foundation on frequency and buckling responses of bidirectional FGP plates was also addressed by Kumar and Harsha [53], which emphasizes the role of BCs and material gradation in real-world applications.

Ebrahimi and Salari [54,55] conducted a semi-analytical vibration analysis of FG size-dependent nanobeams, taking into account various BCs, and their study contributed valuable insights into the influence of geometrical and material variations on vibrational modes. The work by Ashoori et al. [56] focused on the axisymmetric vibration of FG circular plates, addressing the effects of bifurcation and limit point instability in such plates. This work emphasized the impact of size-dependency on the stability of nanostructures and provided a more comprehensive framework for nonlinear vibration analysis. Similarly, Salari et al. [57] studied the nonlinear thermal stability and snap-through buckling of geometrically imperfect graded nanobeams on nonlinear elastic foundations, highlighting the importance of nonlinear material properties in the stability of nanostructures. Salari and Sadough Vanini [58,59] expanded upon this by analyzing the small/large amplitude vibration, static and snap-through instability of temperature-dependent FG porous circular nanoplates. Furthermore, Ghadiri et al. [60] examined the electro-thermo-mechanical vibration behavior of embedded single-walled boron nitride nanotubes using the nonlocal third-order beam theory, contributing to the growing understanding of multi-field interactions in nano-materials. Sharma et al. [61] focused on uncertainty quantification in the buckling strength of variable stiffness laminated composite plates under thermal loading, shedding light on how thermal effects impact the performance of composite materials [62]. In the realm of nanostructures, Ebrahimi et al. [63] presented a vibration analysis of size-dependent nano beams using the nonlocal Timoshenko beam theory, which has become a fundamental approach in understanding the behavior of nanostructures under size-dependent effects. Nasri et al. [64] explored the nonlinear bending and buckling analysis of three dimensional (3D) printed meta-sandwich curved beams with auxetic honeycomb cores, contributing to the design of lightweight and stable metastructures. For FGP plates, Kumar and Harsha [65] performed a response analysis under thermo-electro environments, considering the effects of thermal and electrical loading on structural behavior. Additionally, Prakash et al. [66] analyzed the thermoelastic behavior of thin FG sigmoidal porous plates on variable Winkler’s foundation, further advancing the study of smart materials [67,68].

Ghasemi et al. [69] explored the nonlinear vibration and post-buckling behavior of 3D-printed tubular metastructures, providing important insights into the effects of structural geometry on vibration characteristics under thermal and mechanical loading [70]. In the area of composite materials, Sharma et al. [71,72] performed a stochastic frequency analysis of laminated composite plates with curvilinear fibers, highlighting the impact of uncertainty on buckling strength. Ashoori et al. [73] focused on thermo-electro-mechanical vibrations of FGP nanobeams, a study relevant for understanding the dynamic response of smart materials under thermal and electrical loads. Kumar et al. [74] extended these studies to analyze the thermoelectric buckling response of nonuniform sigmoid FGP plates, with a focus on the influence of elastic foundations and porosity. Moreover, Sharma et al. [75] investigated dynamic responses of variable stiffness composite structures including damage effects, further advancing the design of robust composite materials [76]. Nishad et al. [77] utilized a non-uniform rational B-splines (NURBS) based isogeometric approach and higher-order shear deformation theory to analyze the stochastic critical buckling speed of rim-driven rotating composite plates, shedding light on rotational dynamics and buckling behavior in composite structures. Sharma et al. [78] explored the static and free vibration analyses of smart variable stiffness laminated composite plates with delamination, emphasizing dynamic control and the role of piezoelectric layers in controlling the vibrations and stability of these materials. In the realm of auxetic structures, Ghasemi et al. [79] integrated analytical and machine learning methods to investigate the nonlinear bending and post-buckling behavior of 3D-printed auxetic tubes, pushing the boundaries of traditional analysis methods in structural mechanics [79].

Additionally, Ezzati et al. [80] reviewed graphene origami-enabled metamaterials, highlighting the potential of graphene-based structures in advanced metamaterial design for nano-mechanical applications. Sharma et al. [81] further contributed to dynamic control by analyzing smart damaged variable stiffness laminated composite plates with piezoelectric layers, providing new insights into smart material [82] control systems. Singh et al. [83] analyzed the dynamic behavior of FG gears, demonstrating improved stress distribution and vibration resistance. Kumar et al. [84] extended this to porous FGM plates, examining vibration responses with variable thickness and elastic foundations. At the nanoscale, van Minh et al. [85] investigated flexo-magnetic nanoplates, while Pham et al. [86] explored fluid-infiltrated porous piezoelectric nanoplates using an isogeometric approach, emphasizing flexoelectric effects. Thuy [87] further studied porous skew and elliptical nanoplates, reinforcing FGMs’ adaptability in nanostructures. Additionally, Kumar et al. [88] examined the buckling of piezoelectric graded plates under thermal loading, revealing the impact of porosity and foundation stiffness. Collectively, these studies underscore the significance of FGMs and smart materials in modern engineering [89], necessitating advanced modeling techniques for optimal design. These studies collectively emphasize the importance of integrating advanced materials, machine learning techniques, and dynamic control mechanisms to enhance the performance of composite structures in engineering applications. After a comprehensive assessment of the literature on the vibration response of FG-based structures, several significant conclusions emerged.

1) A review of existing studies reveals that the predominant focus has been on investigating the free vibration and static response of FG-based structures under various thermal, electrical, and mechanical loading.

2) Considering the author’s knowledge, there is a noticeable absence of research addressing the vibration behaviors of sigmoid bidirectional FGPN with various configurations, particularly when considering nanoplate resting on variable elastic foundations subjected to hygrothermal and thermoelectric loading.

3) This study endeavors to bridge this research gap by introducing an effective solution technique to address this concern. The objective of this work is to comprehend the hygrothermal nonlinear vibration behaviors of FGPN variable thickness bidirectional plates featuring diverse porosity distributions.

This study aims to explore the free vibration response of a porosity-dependent bidirectional sigmoid FGPN plate placed on variable elastic foundations under hygrothermal, thermoelectric, and electromechanical loading with diverse BCs. The material properties vary bidirectionally according to the power law (PL) and sigmoid law (SL) both transversely and axially. Additionally, various porosity distributions such as even, uneven, and symmetrically centered types are considered. Governing equations for the porous bidirectional FGPN were derived using the energy principle and solved using the higher-order finite element (HOFE) approach. The investigation thoroughly explores the influence of porous exponent variation, bidirectional material exponent, electrical loading, hygrothermal loading, and variable elastic foundations on the fundamental frequency of the porous bidirectional FGPN with different configurations. The outcomes of this study hold significant promise for enhancing smart structures, NEMS, nano-sensors, actuators, and energy harvesters.

2 Functionally graded piezoelectric nano-plate formulation

Consider an FGPN square plate of width a and variable thickness h and the plate resting on a Winkler−Pasternak foundation, as shown schematically in Fig. 1. A reference Cartesian coordinate system \{{\xi }_1,{\xi }_2,{\xi }_3\} is preferred such that the mid-plane of the plate occupies the region 0\le {\xi }_1, {\xi }_2\le a, {\xi }_3=0. The variable thickness of the plate is described by h({\xi }_1,{\xi }_2)=h_0+(h_1-h_0)({\xi }_1/a)+(h_2-h_0)({\xi }_2/a) where h_0, h_1 and h_2 are the thickness of the plate at ({\xi }_1,{\xi }_2)=\left(0,0\right), (a,0), and (0,a), respectively. By taking h_1=h_2=h_0 (Case-I), h_1\ne h_0, h_2=h_0 (Case-II), h_1=h_0, h_2\ne h_0 (Case-III), and h_1, h_2\ne h_0 (Case-IV), we obtain the plates with uniform thickness, linear variation of the thickness in the {\xi }_1 direction, linear variation of the thickness in the {\xi }_2 direction, and linear variation of the thickness in both {\xi }_1 and {\xi }_2 directions, respectively (Fig. 2 and Table 1).

2.1 Material properties distribution

Two piezoelectric base materials, PZT-4 and PZT-5H, are adopted to form the FGPN plate with the gradation along the axial ({\xi }_1) and transverse ({\xi }_3) directions following either the PL or SL. Even, uneven, and symmetric center distributions are considered porosity distribution, as illustrated in Fig. 1. The material properties of the bidirectional FGPN plate, computed based on the rule of mixture, are given, for the perfect plate (PP), even plate (EP), uneven plate (UPP), and symmetric center porous plate (SCPP), by

(1)\begin{array}{rl}& \mathrm{P}\mathrm{P}:\mathfrak{R}\left({\xi }_1,{\xi }_3\right)={\mathfrak{R}}_{\mathrm{P}\mathrm{T}5}+\left({\mathfrak{R}}_{\mathrm{P}\mathrm{T}4}-{\mathfrak{R}}_{\mathrm{P}\mathrm{T}5}\right)V_{\mathrm{P}\mathrm{T}4},\\ & \mathrm{E}\mathrm{P}:\mathfrak{R}\left({\xi }_1,{\xi }_3\right)={\mathfrak{R}}_{\mathrm{P}\mathrm{T}5}+\left({\mathfrak{R}}_{\mathrm{P}\mathrm{T}4}-{\mathfrak{R}}_{\mathrm{P}\mathrm{T}5}\right)V_{\mathrm{P}\mathrm{T}4}\\ & -0.5\gamma \left({\mathfrak{R}}_{\mathrm{P}\mathrm{T}4}+{\mathfrak{R}}_{\mathrm{P}\mathrm{T}5}\right),\\ & \mathrm{U}\mathrm{P}\mathrm{P}:\mathfrak{R}\left({\xi }_1,{\xi }_3\right)={\mathfrak{R}}_{\mathrm{P}\mathrm{T}5}+\left({\mathfrak{R}}_{\mathrm{P}\mathrm{T}4}-{\mathfrak{R}}_{\mathrm{P}\mathrm{T}5}\right)V_{\mathrm{P}\mathrm{T}4}\\ & -0.5\gamma \left({\mathfrak{R}}_{\mathrm{P}\mathrm{T}4}+{\mathfrak{R}}_{\mathrm{P}\mathrm{T}5}\right)\left(1-\frac{2\left|{\xi }_3\right|}{h}\right),\\ & \mathrm{S}\mathrm{C}\mathrm{P}\mathrm{P}:\mathfrak{R}\left({\xi }_1,{\xi }_3\right)={\mathfrak{R}}_{\mathrm{P}\mathrm{T}5}+\left({\mathfrak{R}}_{\mathrm{P}\mathrm{T}4}-{\mathfrak{R}}_{\mathrm{P}\mathrm{T}5}\right)V_{\mathrm{P}\mathrm{T}4}\\ & -\gamma \cos \left(\frac{\pi {\xi }_3}{h}\right)\left\{{\mathfrak{R}}_{\mathrm{P}\mathrm{T}5}+\left({\mathfrak{R}}_{\mathrm{P}\mathrm{T}4}-{\mathfrak{R}}_{\mathrm{P}\mathrm{T}5}\right)V_{\mathrm{P}\mathrm{T}4}\right\},\end{array}\}

where \mathfrak{R}({\xi }_1,{\xi }_3) denotes a generic material property (e.g., elastic, piezoelectric and dielectric constants, mass density, thermal expansion coefficient, and thermal conductivity) of the FGPN plate along the axial and transverse directions; {\mathfrak{R}}_{\mathrm{P}\mathrm{T}4} and {\mathfrak{R}}_{\mathrm{P}\mathrm{T}5} are generic material properties of PZT-4 and PZT-5H, respectively; \gamma is the porous exponent; and V_{\mathrm{P}\mathrm{T}4} is the volume fraction of PZT-4 given by

(2)\begin{array}{l}\begin{array}{ll}\mathrm{P}\mathrm{L}:V_{\mathrm{P}\mathrm{T}4}={\left(0.5+{\displaystyle \frac{{\xi }_3}{h}}\right)}^n{\left({\displaystyle \frac{{\xi }_1}{a}}\right)}^m,& -h/2\le {\xi }_3\le h/2,\end{array}\\ \mathrm{S}\mathrm{L}:V_{\mathrm{P}\mathrm{T}4}=\{\begin{array}{ll}1-{\displaystyle \frac{1}{2}}{\left(1-{\displaystyle \frac{2{\xi }_3}{h}}\right)}^n{\left({\displaystyle \frac{{\xi }_1}{a}}\right)}^m,& 0\le {\xi }_3\le h/2,\\ {\displaystyle \frac{1}{2}}{\left(1+{\displaystyle \frac{2{\xi }_3}{h}}\right)}^n{\left({\displaystyle \frac{{\xi }_1}{a}}\right)}^m,& -h/2\le {\xi }_3\le 0,\end{array}\end{array}\}

where n and m denoting the PL exponents along the transverse and axial directions. Examples of the variation of material properties of the FGPN plate with and without consideration of the porosity (i.e., PP, EP, UPP, and SCPP) are reported in Figs. 3–6. Figures 3 and 4 show the variation of thermomechanical properties of the FGPN along the transverse direction by neglecting the variation in the axial direction for PL and SL, respectively, whereas Figs. 5 and 6 show the variation of bidirectional thermomechanical properties for PL and SL, respectively.

2.2 Theoretical formulation

According to the nonlocal theory for linear piezoelectricity, stresses and electric displacement at any point \mathit{x} are affected by strains and electric fields at that point and all other points {\mathit{x}}^{\prime } within the body. The nonlocal constitutive relations can be stated as [13,90,91]:

(3)\begin{array}{rl}{\sigma }_{ij}(\mathit{x})=& {{\displaystyle \int }}_Vƛ\left(\left|{\mathit{x}}^{\prime }-\mathit{x}\right|,\tau \right)[C_{ijkl}\left({\epsilon }_{kl}\left({\mathit{x}}^{\prime }\right)-{\alpha }_{ij}\mathrm{\Delta }T-{\beta }_{ij}\mathrm{\Delta }H\right)\\ & -e_{kij}{E}_k\left({\mathit{x}}^{\prime }\right)]\mathrm{d}{\mathit{x}}^{\prime },\\ D_i(\mathit{x})=& {{\displaystyle \int }}_Vƛ\left(\left|{\mathit{x}}^{\prime }-\mathit{x}\right|,\tau \right)[e_{ikl}{\epsilon }_{kl}\left({\mathit{x}}^{\prime }\right)+{\kappa }_{ki}{E}_k\left({\mathit{x}}^{\prime }\right)+p_{ij}\mathrm{\Delta }T\\ & +{\chi }_{ij}\mathrm{\Delta }H]\mathrm{d}{\mathit{x}}^{\prime },\end{array}

where {\sigma }_{ij}, D_i, {\epsilon }_{ij}, and E_i are components of the stress, electric displacement, strain, and electric field, respectively; \mathrm{\Delta }T=T-T_{\mathrm{r}\mathrm{e}\mathrm{f}} is the temperature change with T and T_{\mathrm{r}\mathrm{e}\mathrm{f}} denoting the temperature field and reference temperature, respectively; \mathrm{\Delta }H=H-H_{\mathrm{r}\mathrm{e}\mathrm{f}} is the moisture change with H and H_{\mathrm{r}\mathrm{e}\mathrm{f}} denoting the applied moisture field and reference moisture concentration, respectively; C_{ijkl}, e_{kij}, and {\kappa }_{ki} are elastic, piezoelectric, and dielectric constants, respectively; and p_{ij}, {\chi }_{ij}, {\alpha }_{ij}, and {\beta }_{ij} are the pyroelectric properties, hygroelectric properties, coefficients of thermal expansion and moisture concentrations, respectively. Hereafter, the standard indicial notation and summation convention apply, and components of any vectors and tensors are referred to as the Cartesian coordinate system \{{\xi }_1,{\xi }_2,{\xi }_3\}. Equation (3) ƛ\left(\right|{\mathit{x}}^{\prime }-\mathit{x}|,\tau ) signifies the nonlocal attenuation function, which |{\mathit{x}}^{\prime }-\mathit{x}| denotes the Euclidean distance and \tau =\overline{a}{e}_0/l is the scaling constant representing the small-scale influence on the response of nanosize structures with e_0 denoting a physical parameter that has been found experimentally and \overline{a} and l representing the internal and external length scales, respectively. It is worth noting that the nonlocal constitutive laws in an integral form given by Eq. (3) present a nontrivial challenge in the solution procedure. To overcome such difficulty, Eq. (3) is transformed to the following differential form [91]:

(4)\begin{array}{rl}& (1-{\mu }_0{\mathrm{\nabla }}^2){\sigma }_{ij}=\left[C_{ijkl}{\epsilon }_{kl}-e_{kij}{E}_k-C_{ijkl}{\alpha }_{ij}\mathrm{\Delta }T-C_{ijkl}{\beta }_{ij}\mathrm{\Delta }H\right],\\ & (1-{\mu }_0{\mathrm{\nabla }}^2)D_i=\left[e_{ikl}{\epsilon }_{kl}+{\kappa }_{ki}{E}_k+p_{ij}\mathrm{\Delta }T+{\chi }_{ij}\mathrm{\Delta }H\right],\end{array}

where {\mu }_0=(\overline{a}{e}_0{)}^2 denotes the small-scale parameter that includes the small-scale effects into the nanostructure constitutive equations and {\mathrm{\nabla }}^2 is the Laplacian operator. Also, the thickness of the FGPN plate is much smaller than its length and width. Therefore, the nonlocal behavior of the plate in the thickness direction is ignored. Hence, the constitutive relations Eq. (4), when applied to the FGPN plate, become

(5)\begin{array}{rl}& \left(1-{\mu }_0{\mathrm{\nabla }}^2\right)\left\{\begin{array}{c}{\sigma }_{11}\\ {\sigma }_{22}\\ {\sigma }_{23}\\ {\sigma }_{13}\\ {\sigma }_{12}\end{array}\right\}=\left[\begin{array}{ccccc}{\overline{C}}_{11}& {\overline{C}}_{12}& 0& 0& 0\\ {\overline{C}}_{12}& {\overline{C}}_{22}& 0& 0& 0\\ 0& 0& {\overline{C}}_{44}& 0& 0\\ 0& 0& 0& {\overline{C}}_{55}& 0\\ 0& 0& 0& 0& {\overline{C}}_{66}\end{array}\right]\\ & \cdot \left(\left\{\begin{array}{c}{\epsilon }_{11}\\ {\epsilon }_{22}\\ {\epsilon }_{23}\\ {\epsilon }_{13}\\ {\epsilon }_{12}\end{array}\right\}-\left\{\begin{array}{c}{\overline{\alpha }}_{11}\\ {\overline{\alpha }}_{22}\\ 0\\ 0\\ 0\end{array}\right\}\mathrm{\Delta }T-\left\{\begin{array}{c}{\overline{\beta }}_{11}\\ {\overline{\beta }}_{22}\\ 0\\ 0\\ 0\end{array}\right\}\mathrm{\Delta }H\right)-\left[\begin{array}{ccc}0& 0& {\overline{e}}_{31}\\ 0& 0& {\overline{e}}_{32}\\ 0& {\overline{e}}_{24}& 0\\ {\overline{e}}_{15}& 0& 0\\ 0& 0& 0\end{array}\right]\left\{\begin{array}{c}{E}_1\\ E_2\\ E_3\end{array}\right\},\end{array}

(6)\begin{array}{rl}& \left(1-{\mu }_0{\mathrm{\nabla }}^2\right)\left\{\begin{array}{c}{D}_1\\ D_2\\ D_3\end{array}\right\}=\left[\begin{array}{ccccc}0& 0& 0& {\overline{e}}_{15}& 0\\ 0& 0& {\overline{e}}_{24}& 0& 0\\ {\overline{e}}_{31}& {\overline{e}}_{32}& 0& 0& 0\end{array}\right]\left\{\begin{array}{c}{\epsilon }_{11}\\ {\epsilon }_{22}\\ {\epsilon }_{23}\\ {\epsilon }_{13}\\ {\epsilon }_{12}\end{array}\right\}\\ +& \left[\begin{array}{ccc}{\overline{\kappa }}_{11}& 0& 0\\ 0& {\overline{\kappa }}_{22}& 0\\ 0& 0& {\overline{\kappa }}_{33}\end{array}\right]\left\{\begin{array}{c}{E}_1\\ E_2\\ E_3\end{array}\right\}+\left\{\begin{array}{c}{\overline{p}}_{11}\\ {\overline{p}}_{22}\\ {\overline{p}}_{33}\end{array}\right\}\mathrm{\Delta }T+\left\{\begin{array}{c}{\overline{\chi }}_{11}\\ {\overline{\chi }}_{22}\\ {\overline{\chi }}_{33}\end{array}\right\}\mathrm{\Delta }H,\end{array}

where {\overline{C}}_{ij}, {\overline{e}}_{ij} and {\overline{\kappa }}_{ij} are the reduced elastic, piezoelectric, dielectric constants, respectively, pyroelectric and hygroelectric parameters defined by

(7)\begin{array}{rl}& {\overline{C}}_{11}=C_{11}-\frac{C_{13}^2}{C_{33}},{\overline{C}}_{12}=C_{12}-\frac{C_{13}{C}_{23}}{C_{33}},{\overline{C}}_{22}=C_{22}-\frac{C_{23}^2}{C_{33}},\\ & {\overline{C}}_{44}=C_{44},{\overline{C}}_{55}=C_{55},{\overline{C}}_{66}=C_{66},{\overline{e}}_{24}=e_{24},{\overline{e}}_{15}=e_{15},\\ & {\overline{e}}_{31}=e_{31}-\frac{C_{13}{e}_{33}}{C_{33}},{\overline{e}}_{32}=e_{32}-\frac{C_{23}{e}_{33}}{C_{33}},{\overline{\kappa }}_{11}={\kappa }_{11},{\overline{\kappa }}_{22}=\\ & {\kappa }_{22},{\overline{\kappa }}_{33}={\kappa }_{33}+\frac{e_{23}^2}{C_{33}},{\overline{\alpha }}_{11}={\alpha }_{11}-\frac{C_{13}{\alpha }_{33}}{C_{33}},{\overline{\alpha }}_{22}={\alpha }_{22}-\\ & \frac{C_{13}{\alpha }_{33}}{C_{33}},{\overline{\beta }}_{11}={\beta }_{11}-\frac{C_{13}{\beta }_{33}}{C_{33}},{\overline{\beta }}_{22}={\beta }_{22}-\frac{C_{13}{\beta }_{33}}{C_{33}},\\ & {\overline{p}}_{11}=p_{11},{\overline{p}}_{22}=p_{22},{\overline{p}}_{33}=p_{33}+\frac{e_{33}{\alpha }_{33}}{C_{33}},\\ & {\overline{\chi }}_{11}={\chi }_{11},{\overline{\chi }}_{22}={\chi }_{22},{\overline{\chi }}_{33}={\chi }_{33}+\frac{e_{33}{\beta }_{33}}{C_{33}}.\end{array}\}

By following the framework of FSDT, the displacement field of the FGPN plate, whose components are denoted by u_i, can be expressed as [90,91]:

(8)\begin{array}{l}{u}_1({\xi }_1,{\xi }_2,{\xi }_3,t)=u_0\left({\xi }_1,{\xi }_2,t\right)+{\xi }_3{\phi }_1\left({\xi }_1,{\xi }_2,t\right),\\ u_2({\xi }_1,{\xi }_2,{\xi }_3,t)=v_0\left({\xi }_1,{\xi }_2,t\right)+{\xi }_3{\phi }_2\left({\xi }_1,{\xi }_2,t\right),\\ u_3({\xi }_1,{\xi }_2,{\xi }_3,t)=w_0\left({\xi }_1,{\xi }_2,t\right),\end{array}\}

where u_0, v_0, and w_0 are components of the mid-plane displacement in the {\xi }_1, {\xi }_2, and {\xi }_3 directions, respectively. {\phi }_1 and {\phi }_2 are the bending rotations about the {\xi }_1 and {\xi }_2 axes, respectively. The von Kármán nonlinear strains related to the above displacement field are assumed by

(9)\epsilon =\left\{\begin{array}{c}{\epsilon }_{11}\\ {\epsilon }_{22}\\ {\epsilon }_{12}\\ {\epsilon }_{13}\\ {\epsilon }_{23}\end{array}\right\}=\left\{\begin{array}{c}{\displaystyle \frac{\mathrm{\partial }{u}_0}{\mathrm{\partial }{\xi }_1}}+{\displaystyle \frac{1}{2}}{\left({\displaystyle \frac{\mathrm{\partial }{w}_0}{\mathrm{\partial }{\xi }_1}}\right)}^2\\ {\displaystyle \frac{\mathrm{\partial }{v}_0}{\mathrm{\partial }{\xi }_2}}+{\displaystyle \frac{1}{2}}{\left({\displaystyle \frac{\mathrm{\partial }{w}_0}{\mathrm{\partial }{\xi }_2}}\right)}^2\\ {\displaystyle \frac{\mathrm{\partial }{u}_0}{\mathrm{\partial }{\xi }_2}}+{\displaystyle \frac{\mathrm{\partial }{v}_0}{\mathrm{\partial }{\xi }_1}}+\left({\displaystyle \frac{\mathrm{\partial }{w}_0}{\mathrm{\partial }{\xi }_1}}{\displaystyle \frac{\mathrm{\partial }{w}_0}{\mathrm{\partial }{\xi }_2}}\right)\\ {\displaystyle \frac{\mathrm{\partial }{w}_0}{\mathrm{\partial }{\xi }_1}}+{\phi }_1\\ {\displaystyle \frac{\mathrm{\partial }{w}_0}{\mathrm{\partial }{\xi }_2}}+{\phi }_2\end{array}\right\}+{\xi }_3\left\{\begin{array}{c}{\displaystyle \frac{\mathrm{\partial }{\phi }_1}{\mathrm{\partial }{\xi }_1}}\\ {\displaystyle \frac{\mathrm{\partial }{\phi }_2}{\mathrm{\partial }{\xi }_2}}\\ \left({\displaystyle \frac{\mathrm{\partial }{\phi }_1}{\mathrm{\partial }{\xi }_2}}+{\displaystyle \frac{\mathrm{\partial }{\phi }_2}{\mathrm{\partial }{\xi }_1}}\right)\\ 0\\ \\ 0\end{array}\right\}.

Equation (9) is also written as:

(10)\epsilon =\left\{\begin{array}{c}{\epsilon }_1\\ {\epsilon }_2\end{array}\right\}=\left\{\begin{array}{c}{\epsilon }_1^0+{\xi }_3{\epsilon }_1^1\\ {\epsilon }_2^0\end{array}\right\},{\epsilon }_1=\left\{\begin{array}{c}{\epsilon }_{11}\\ {\epsilon }_{22}\\ {\epsilon }_{12}\end{array}\right\},{\epsilon }_2=\left\{\begin{array}{c}{\epsilon }_{13}\\ {\epsilon }_{23}\end{array}\right\}.

The in-plane strain components {\epsilon }_{11},{\epsilon }_{22},{\epsilon }_{12} vary linearly in the transverse direction whereas the transverse shear strains {\epsilon }_{13},{\epsilon }_{23} are constant.

With consideration of the direct and converse piezoelectric effects, the electric potential field \mathrm{\Theta } is assumed to satisfy the adopted Maxwell equation in the following form:

(11)\mathrm{\Theta }\left({\xi }_1,{\xi }_2,{\xi }_3,t\right)=\frac{2{\xi }_3}{h}V-\cos \left(\frac{\pi {\xi }_3}{h}\right){\mathrm{\Theta }}_0\left({\xi }_1,{\xi }_2,t\right),

where {\mathrm{\Theta }}_0 is an unknown function of the in-plane coordinates {\xi }_1 and {\xi }_2, and V is the applied voltage. The electric field is associated with the electric potential \mathrm{\Theta } by

(12)\left\{E_1,E_2,E_3\right\}=-\left\{\frac{\mathrm{\partial }}{\mathrm{\partial }{\xi }_1},\frac{\mathrm{\partial }}{\mathrm{\partial }{\xi }_2},\frac{\mathrm{\partial }}{\mathrm{\partial }{\xi }_3}\right\}\mathrm{\Theta }.

2.3 Hygrothermal loading

For the hygrothermal vibration problem, four different distributions of hygrothermal loading in the transverse direction of the FGPN plate are considered, as detailed below [67,92].

2.3.1 Uniform hygrothermal rise (UHTR)

In this case, the hygrothermal rise undergoes uniform variation across the plate thickness. In particular, the temperature and moisture concentration are taken as:

(13)\begin{array}{l}T\left({\xi }_3\right)=T_{\mathrm{f}\mathrm{n}},\\ H\left({\xi }_3\right)=H_{\mathrm{f}\mathrm{n}},\end{array}\}

where T_{\mathrm{f}\mathrm{n}} and H_{\mathrm{f}\mathrm{n}} are the constant final temperature and moisture concentration, respectively.

2.3.2 Linear hygrothermal rise (LHTR)

In this case, the hygrothermal rise assumes the linear variation across the plate thickness, i.e., the temperature and moisture concentration are taken as:

(14)\begin{array}{l}T\left({\xi }_1,{\xi }_2,{\xi }_3\right)=T_{\mathrm{P}\mathrm{T}5}+\left(T_{\mathrm{P}\mathrm{T}4}-T_{\mathrm{P}\mathrm{T}5}\right)\left({\displaystyle \frac{{\xi }_3}{h}}+0.5\right),\\ H\left({\xi }_1,{\xi }_2,{\xi }_3\right)=H_{\mathrm{P}\mathrm{T}5}+\left(H_{\mathrm{P}\mathrm{T}4}-H_{\mathrm{P}\mathrm{T}5}\right)\left({\displaystyle \frac{{\xi }_3}{h}}+0.5\right),\end{array}\}

where T_{\mathrm{P}\mathrm{T}4},T_{\mathrm{P}\mathrm{T}5} and H_{\mathrm{P}\mathrm{T}4},H_{\mathrm{P}\mathrm{T}5} are the temperature and moisture concentration at the top and bottom of the plate, respectively.

2.3.3 Nonlinear hygrothermal rise (NHTR)

The temperature and moisture concentration for this case are taken as:

(15)\begin{array}{l}T\left({\xi }_1,{\xi }_2,{\xi }_3\right)=T_{\mathrm{P}\mathrm{T}5}+\left(T_{\mathrm{P}\mathrm{T}4}-T_{\mathrm{P}\mathrm{T}5}\right){\left({\displaystyle \frac{{\xi }_3}{h}}+0.5\right)}^p,\\ H\left({\xi }_1,{\xi }_2,{\xi }_3\right)=H_{\mathrm{P}\mathrm{T}5}+\left(H_{\mathrm{P}\mathrm{T}4}-H_{\mathrm{P}\mathrm{T}5}\right){\left({\displaystyle \frac{{\xi }_3}{h}}+0.5\right)}^p,\end{array}\}

where , p\ne 1 denotes the hygrothermal exponent and is considered in the present study.

2.3.4 Sinusoidal hygrothermal rise (SHTR)

In this case, the hygrothermal rise assumes the sinusoidal variation in the transverse direction and is given by

(16)\begin{array}{l}T\left({\xi }_1,{\xi }_2,{\xi }_3\right)=T_{\mathrm{P}\mathrm{T}5}+\left(T_{\mathrm{P}\mathrm{T}4}-T_{\mathrm{P}\mathrm{T}5}\right)\left[1-\cos {\displaystyle \frac{\pi }{2}}\left({\displaystyle \frac{{\xi }_3}{h}}+0.5\right)\right],\\ H\left({\xi }_1,{\xi }_2,{\xi }_3\right)=H_{\mathrm{P}\mathrm{T}5}+\left(H_{\mathrm{P}\mathrm{T}4}-H_{\mathrm{P}\mathrm{T}5}\right)\left[1-\cos {\displaystyle \frac{\pi }{2}}\left({\displaystyle \frac{{\xi }_3}{h}}+0.5\right)\right].\end{array}\}

3 Variational formulation

Applying the Hamilton principle to FGPN plate yields

(17){\int }_0^t\left(\delta E_{\mathrm{s}}+\delta E_{\mathrm{e}\mathrm{f}}+\delta E_{\mathrm{w}}-\delta E_{\mathrm{k}}\right)\mathrm{d}t=0,

where E_{\mathrm{s}}, E_{\mathrm{e}\mathrm{f}}, E_{\mathrm{k}}, E_{\mathrm{w}} are the electromechanical strain energy, energy stored in the elastic foundations, kinetic energy, and applied external work done, respectively. The strain energy E_{\mathrm{s}} stored in the FGPN plate is expressed as:

(18)\begin{array}{rl}\delta E_{\mathrm{s}}=& {{\displaystyle \int }}_V({\sigma }_{11}\delta {\epsilon }_{11}+{\sigma }_{22}\delta {\epsilon }_{22}+k_{\mathrm{s}\mathrm{f}}{\sigma }_{23}\delta {\epsilon }_{23}+k_{\mathrm{s}\mathrm{f}}{\sigma }_{13}\delta {\epsilon }_{13}\\ & +{\sigma }_{12}\delta {\epsilon }_{12}-D_1\delta E_1-D_2\delta E_2-D_3\delta E_3)\mathrm{d}V.\end{array}

Substituting Eqs. (9) and (11) into Eq. (17) yields:

(19)\begin{array}{rl}\delta E_{\mathrm{s}}=& {\displaystyle \underset{0}{\stackrel{a}{\int }}{\displaystyle \underset{0}{\stackrel{a}{\int }}(\left(N_1\delta u_{0,1}+N_2\delta v_{0,2}+N_{12}\left(\delta u_{0,2}+\delta v_{0,1}\right)\right)}}\\ & +\left(M_1\delta {\phi }_{1,1}+M_2\delta {\phi }_{2,2}+M_{12}\left(\delta {\phi }_{1,2}+\delta {\phi }_{2,1}\right)\right)\\ & +\frac{N_1}{2}\delta {\left(w_{0,1}\right)}^2+\frac{N_2}{2}\delta {\left(w_{0,2}\right)}^2+N_{12}\left(\delta w_{0,1}+\delta w_{0,2}\right)\\ & +Q_{13}\left(\delta {\phi }_1+\delta w_{0,1}\right)+Q_{23}\left(\delta {\phi }_2+\delta w_{0,2}\right))\mathrm{d}{\xi }_1\mathrm{d}{\xi }_2\\ & +{\displaystyle \underset{0}{\stackrel{a}{\int }}{\displaystyle \underset{0}{\stackrel{a}{\int }}{\displaystyle \underset{-h/2}{\stackrel{h/2}{\int }}[-D_1\mathrm{c}\mathrm{o}\mathrm{s}\left(\pi {\xi }_3/h\right)\delta \left({\displaystyle \frac{\mathrm{\partial }{\Theta }_0}{\mathrm{\partial }{\xi }_1}}\right)}}}\\ & -D_2\mathrm{c}\mathrm{o}\mathrm{s}\left(\pi {\xi }_3/h\right)\delta \left({\displaystyle \frac{\mathrm{\partial }{\Theta }_0}{\mathrm{\partial }{\xi }_2}}\right)\\ & -D_3(\pi /h)\mathrm{s}\mathrm{i}\mathrm{n}\left(\pi {\xi }_3/h\right)\delta {\Theta }_0]\mathrm{d}{\xi }_3\mathrm{d}{\xi }_1\mathrm{d}{\xi }_2,\end{array}

where k_{\mathrm{s}\mathrm{f}} is the shear correction factor (k_{\mathrm{s}\mathrm{f}}=5/6 is utilized in the present study). The above equations consists of force and moment resultants are expressed as:

(20)\begin{array}{rl}& \left\{\begin{array}{c}{N}_1\\ N_2\\ N_{12}\end{array}\right\}={\displaystyle \underset{-h/2}{\overset{h/2}{\int }}\left\{\begin{array}{c}{\sigma }_{11}\\ {\sigma }_{22}\\ {\sigma }_{12}\end{array}\right\}\mathrm{d}{\xi }_3,\left\{\begin{array}{c}{Q}_1\\ Q_2\end{array}\right\}=k_{\mathrm{s}\mathrm{f}}{\displaystyle \underset{-h/2}{\overset{h/2}{\int }}\left\{\begin{array}{c}{\sigma }_{13}\\ {\sigma }_{23}\end{array}\right\}\mathrm{d}{\xi }_3,}}\\ & \left\{\begin{array}{c}{M}_1\\ M_2\\ M_{12}\end{array}\right\}={\displaystyle \underset{-h/2}{\overset{h/2}{\int }}\left\{\begin{array}{c}{\sigma }_{11}\\ {\sigma }_{22}\\ {\sigma }_{12}\end{array}\right\}{\xi }_3\mathrm{d}{\xi }_3,}\end{array}\}

(21)\begin{array}{rl}& {\left\{N_1{N}_2{N}_{12}\right\}}^T=A{\epsilon }_1^0+B{\epsilon }_1^1,\\ & {\left\{M_1{M}_2{M}_{12}\right\}}^T=B{\epsilon }_1^0+F{\epsilon }_1^1,\\ & {\left\{Q_{13}{Q}_{23}\right\}}^T=A^s{\epsilon }_2^0+Q_i^{El},i=1,2.\end{array}\}

Substituting Eq. (9) into Eqs. (5) and (6). The force and moment resultants in the nonlocal model can be rewritten as

(22)\begin{array}{c}\left\{\begin{array}{c}{N}_1\\ N_2\\ N_{12}\end{array}\right\}-{\mu }_0{\mathrm{\nabla }}^2\left\{\begin{array}{c}{N}_1\\ N_2\\ N_{12}\end{array}\right\}=A{\epsilon }_1^0+B{\epsilon }_1^1,\\ \left\{\begin{array}{c}{M}_1\\ M_2\\ M_{12}\end{array}\right\}-{\mu }_0{\mathrm{\nabla }}^2\left\{\begin{array}{c}{M}_1\\ M_2\\ M_{12}\end{array}\right\}=B{\epsilon }_1^0+F{\epsilon }_1^1,\\ \left\{\begin{array}{c}{Q}_1\\ Q_2\end{array}\right\}-{\mu }_0{\mathrm{\nabla }}^2\left\{\begin{array}{c}{Q}_1\\ Q_2\end{array}\right\}=A^s{\epsilon }_2^0+Q_i^{El},\end{array}\}

(23)\begin{array}{rl}& (A,B,F)={\displaystyle \underset{-h/2}{\overset{h/2}{\int }}{\left({\overline{C}}_{ij}\right)}_b\left\{1,{\xi }_3,{\xi }_3^2\right\}\mathrm{d}{\xi }_3,}\\ & \left(A^s\right)=k_{\mathrm{s}\mathrm{f}}{\displaystyle \underset{-h/2}{\overset{h/2}{\int }}{\left({\overline{C}}_{ij}\right)}_s\mathrm{d}{\xi }_3.}\end{array}

The variation of energy stored in the elastic foundations is stated as:

(24)\delta E_{\mathrm{e}\mathrm{f}}={\displaystyle \underset{0}{\overset{a}{\int }}{\displaystyle \underset{0}{\overset{a}{\int }}\left[q_{\mathrm{e}\mathrm{f}}\right]\delta w_0\mathrm{d}{\xi }_1\mathrm{d}{\xi }_2.}}

The kinetic energy E_{\mathrm{k}} is defined similarly to that of non-smart plates as:

(25)\begin{array}{rl}\delta E_{\mathrm{k}}=& {{\displaystyle \int }}_V\rho \left({\dot{u}}_1^2+{\dot{u}}_2^2+{\dot{u}}_3^2\right)\mathrm{d}V={\displaystyle \underset{0}{\stackrel{a}{\int }}{\displaystyle \underset{0}{\stackrel{a}{\int }}\{I_0\left({\dot{u}}_0^2+{\dot{v}}_0^2+{\dot{w}}_0^2\right)}}\\ & +2I_1\left({\dot{u}}_0{\dot{\phi }}_1+{\dot{v}}_0{\dot{\phi }}_2\right)+I_2\left({\dot{\phi }}_1^2+{\dot{\phi }}_2^2\right)\}\mathrm{d}{\xi }_1\mathrm{d}{\xi }_2,\end{array}

where the inertia terms I_0, I_1, and I_2 are defined as:

(26)\left(I_0,I_1,I_2\right)={\displaystyle \underset{-h/2}{\overset{h/2}{\int }}\left(1,{\xi }_3,{\xi }_3^2\right)\rho \mathrm{d}{\xi }_3.}

The applied external work done E_w, with the effect of mechanical loads, elastic foundation and hygrothermal load, is given by

(27)\begin{array}{rl}& \delta E_{\mathrm{w}}={\displaystyle \underset{0}{\overset{a}{\int }}{\displaystyle \underset{0}{\overset{a}{\int }}\left[\left(-q+\left(F_{{\xi }_1}\frac{{\mathrm{\partial }}^2{w}_0}{\mathrm{\partial }{\xi }_1^2}+F_{{\xi }_2}\frac{{\mathrm{\partial }}^2{w}_0}{\mathrm{\partial }{\xi }_2^2}\right)\right)\delta w_0\mathrm{d}A\right]}}\\ & \mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}\\ & \left\{F_{{\xi }_1},F_{{\xi }_2}\right\}=-\{\left(N_1^{\mathrm{M}\mathrm{e}}+N_1^{\mathrm{E}\mathrm{l}}-N_1^{\mathrm{T}\mathrm{h}}-N_1^{\mathrm{H}\mathrm{g}}\right),\\ & \left(N_2^{\mathrm{M}\mathrm{e}}+N_2^{\mathrm{E}\mathrm{l}}-N_2^{\mathrm{T}\mathrm{h}}-N_3^{\mathrm{H}\mathrm{g}}\right)\}\end{array}

where q are the normal transverse loading, N_1^{\mathrm{M}\mathrm{e}}, N_2^{\mathrm{M}\mathrm{e}}, and N_{12}^{\mathrm{M}\mathrm{e}} are the mechanical force, N_1^{\mathrm{E}\mathrm{l}} and N_2^{\mathrm{E}\mathrm{l}} are the electrical forces, N_1^{\mathrm{T}\mathrm{h}} and N_2^{\mathrm{T}\mathrm{h}} are the thermal forces, N_1^{\mathrm{H}\mathrm{g}} and N_2^{\mathrm{H}\mathrm{g}} are the hygro forces. The definitions of these forces are as follows:

(28)\begin{array}{l}{N}_1^{\mathrm{M}\mathrm{e}}=N_2^{\mathrm{M}\mathrm{e}}=N^{\mathrm{M}\mathrm{e}},N_{12}^{\mathrm{M}\mathrm{e}}=0,\\ N_1^{\mathrm{E}\mathrm{l}}=N_2^{\mathrm{E}\mathrm{l}}=N^{\mathrm{E}\mathrm{l}},N^{\mathrm{E}\mathrm{l}}={\displaystyle \underset{-h/2}{\overset{h/2}{\int }}{\overline{e}}_{3i}\left(\frac{2V}{h}\right)\mathrm{d}{\xi }_3,i=1,2,}\\ N_1^{\mathrm{T}\mathrm{h}}=N_2^{\mathrm{T}\mathrm{h}}=N^{\mathrm{T}\mathrm{h}},N^{\mathrm{T}\mathrm{h}}={\displaystyle \underset{-h/2}{\overset{h/2}{\int }}{\overline{C}}_{1i}\overline{\alpha }\left({\xi }_3\right)\left(\mathrm{\Delta }T\right)\mathrm{d}{\xi }_3,i=1,2,}\\ N_1^{\mathrm{H}\mathrm{g}}=N_2^{\mathrm{H}\mathrm{g}}=N^{\mathrm{H}\mathrm{g}},N^{\mathrm{H}\mathrm{g}}={\displaystyle \underset{-h/2}{\overset{h/2}{\int }}{\overline{C}}_{1i}\overline{\beta }\left({\xi }_3\right)\left(\mathrm{\Delta }H\right)\mathrm{d}{\xi }_3,i=1,2.}\end{array}\}

This study considered the FGPN porous plate resting on the variable elastic foundations, i.e., Winkler−Pasternak foundations. q_{\mathrm{e}\mathrm{f}} is the load induced by the Winkler−Pasternak foundation, and the generalized representation of the Winkler−Pasternak foundation is as follows (as shown in Fig. 1(c)):

(29)q_{\mathrm{e}\mathrm{f}}=k_{\mathrm{w}}{w}_0-k_{\mathrm{p}}{\mathrm{\nabla }}^2{w}_0,

where k_{\mathrm{w}} and k_{\mathrm{p}} are Winkler and Pasternak parameters, respectively, and {\mathrm{\nabla }}^2 denotes the two-dimensional Laplace operator. In the current study, k_{\mathrm{p}} is assumed to be constant, whereas k_{\mathrm{w}} possesses the linear, quadratic, or sinusoidal variation along the axial direction (i.e., {\xi }_1 direction), as shown in Figs. 1(d)–1(f); in particular, they are defined as:

(30)\begin{array}{rl}& k_{\mathrm{w}}=\frac{12K_{\mathrm{w}}{a}^4}{{\left(C_{11}\right)}_{\mathrm{P}\mathrm{T}4}{h}^3}(1+\varsigma ),\\ & \varsigma =\{\begin{array}{lr}\psi \left({\xi }_1/a\right),& \mathrm{L}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{a}\mathrm{r}\left(\mathrm{L}\mathrm{F}\right),\\ \psi {\left({\xi }_1/a\right)}^2,& \mathrm{P}\mathrm{a}\mathrm{r}\mathrm{a}\mathrm{b}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{c}\left(\mathrm{P}\mathrm{F}\right),\\ \psi \sin \pi \left({\xi }_1/a\right),& \mathrm{S}\mathrm{i}\mathrm{n}\mathrm{u}\mathrm{s}\mathrm{o}\mathrm{i}\mathrm{d}\mathrm{a}\mathrm{l}\left(\mathrm{S}\mathrm{F}\right),\end{array}\\ & k_{\mathrm{p}}=\frac{12K_{\mathrm{p}}{a}^4}{{\left(C_{11}\right)}_{\mathrm{P}\mathrm{T}4}{h}^3},\end{array}

where K_{\mathrm{w}}, K_{\mathrm{p}}, and \varsigma are given constants.

By substituting Eq. (22) into Eq. (19), then substituting Eqs. (19), (24), (25), and (27) into Eq. (17) and integrating them by parts, and then rearranging all involved terms, it finally yields the equilibrium equations of the FGPN plate are as follows:

(31)\begin{array}{rl}& \frac{\mathrm{\partial }{N}_1}{\mathrm{\partial }{\xi }_1}+\frac{\mathrm{\partial }{N}_{12}}{\mathrm{\partial }{\xi }_2}=\left(1-{\mu }_0{\mathrm{\nabla }}^2\right)\left(I_0{\ddot{u}}_0^2+I_1{\ddot{\phi }}_1^2\right),\\ & \frac{\mathrm{\partial }{N}_{12}}{\mathrm{\partial }{\xi }_1}+\frac{\mathrm{\partial }{N}_2}{\mathrm{\partial }{\xi }_2}=\left(1-{\mu }_0{\mathrm{\nabla }}^2\right)\left(I_0{\ddot{v}}_0^2+I_1{\ddot{\phi }}_2^2\right),\\ & \frac{\mathrm{\partial }{Q}_1}{\mathrm{\partial }{\xi }_1}+\frac{\mathrm{\partial }{Q}_2}{\mathrm{\partial }{\xi }_2}+\left(1-{\mu }_0{\mathrm{\nabla }}^2\right)(\left(N^{\mathrm{M}\mathrm{e}}+N^{\mathrm{E}\mathrm{l}}-N^{\mathrm{T}\mathrm{h}}-N^{\mathrm{H}\mathrm{g}}\right),\\ & {\mathrm{\nabla }}^2{w}_0+\left(q-q_{\mathrm{e}\mathrm{f}}\right))=\left(1-{\mu }_0{\mathrm{\nabla }}^2\right)I_0{\ddot{w}}_0^2,\\ & \frac{\mathrm{\partial }{M}_1}{\mathrm{\partial }{\xi }_1}+\frac{\mathrm{\partial }{M}_{12}}{\mathrm{\partial }{\xi }_2}-Q_1=\left(1-{\mu }_0{\mathrm{\nabla }}^2\right)\left(I_1{\ddot{u}}_0^2+I_2{\ddot{\phi }}_1^2\right),\\ & \frac{\mathrm{\partial }{M}_{12}}{\mathrm{\partial }{\xi }_1}+\frac{\mathrm{\partial }{M}_2}{\mathrm{\partial }{\xi }_2}-Q_2=\left(1-{\mu }_0{\mathrm{\nabla }}^2\right)\left(I_1{\ddot{v}}_0^2+I_2{\ddot{\phi }}_2^2\right),\\ & \frac{\mathrm{\partial }{\overline{D}}_1}{\mathrm{\partial }{\xi }_1}+\frac{\mathrm{\partial }{\overline{D}}_2}{\mathrm{\partial }{\xi }_2}+{\displaystyle \underset{-h/2}{\overset{h/2}{\int }}{D}_3\left(1-{\mu }_0{\mathrm{\nabla }}^2\right)\left(\frac{\pi }{h}\right)\sin \left(\frac{\pi {\xi }_3}{h}\right)\mathrm{d}{\xi }_3=0,}\end{array}\}

(32)\begin{array}{l}{\overline{D}}_1={\displaystyle \underset{-h/2}{\overset{h/2}{\int }}{D}_1\left(1-{\mu }_0{\mathrm{\nabla }}^2\right)\left[\cos \left(\frac{\pi {\xi }_3}{h}\right)\right]\mathrm{d}{\xi }_3,}\\ {\overline{D}}_2={\displaystyle \underset{-h/2}{\overset{h/2}{\int }}{D}_2\left(1-{\mu }_0{\mathrm{\nabla }}^2\right)\left[\cos \left(\frac{\pi {\xi }_3}{h}\right)\right]\mathrm{d}{\xi }_3.}\end{array}\}

Lastly, by multiplying Eq. (31) with the corresponding displacement variables, bending rotations, and electric potential, the resulting expression is integrated over the element domain. The same terms from the right-hand side of Eq. (31) are then added, yielding the variational (or weak) form of the equation and finally obtains the variational or weak forms as follows:

(33)\begin{array}{rl}& {\int }_S[\left(N_1\delta u_{0,1}+N_2\delta v_{0,2}+N_{12}\left(\delta u_{0,2}+\delta v_{0,1}\right)\right)\\ -& \left(M_1\delta {\phi }_{1,1}+M_2\delta {\phi }_{2,2}+M_{12}\left(\delta {\phi }_{1,2}+\delta {\phi }_{2,1}\right)\right)\\ +& Q_{13}\left(\delta {\phi }_1+\delta w_{0,1}\right)+Q_{23}\left(\delta {\phi }_2+\delta w_{0,2}\right)\\ -& \left(1-{\mu }_0{\mathrm{\nabla }}^2\right)\left(\left(N^{\mathrm{M}\mathrm{e}}+N^{\mathrm{E}\mathrm{l}}-N^{\mathrm{T}\mathrm{h}}-N^{\mathrm{H}\mathrm{g}}\right){\mathrm{\nabla }}^2{w}_0+\left(q-q_{\mathrm{e}\mathrm{f}}\right)\right)\delta w_0\\ -& \left(1-{\mu }_0{\mathrm{\nabla }}^2\right)I_2\left({\dot{\phi }}_1\delta {\dot{\phi }}_1+{\dot{\phi }}_2\delta {\dot{\phi }}_2\right)\\ -& \left(1-{\mu }_0{\mathrm{\nabla }}^2\right)(I_0\left({\dot{u}}_0\delta {\dot{u}}_0+{\dot{v}}_0\delta {\dot{v}}_0+{\dot{w}}_0\delta {\dot{w}}_0\right)\\ +& 2I_1\left({\dot{\phi }}_1\delta {\dot{u}}_0+{\dot{\phi }}_2\delta {\dot{v}}_0+{\dot{u}}_0\delta {\dot{\phi }}_1+{\dot{v}}_0\delta {\dot{\phi }}_2\right))\\ +& \delta {\mathrm{\Theta }}_0{\displaystyle \underset{-h/2}{\overset{h/2}{\int }}[\left(1-{\mu }_0{\mathrm{\nabla }}^2\right)D_{1,1}\cos \left(\pi {\xi }_3/h\right)}\\ +& \left(1-{\mu }_0{\mathrm{\nabla }}^2\right)D_{2,2}\cos \left(\pi {\xi }_3/h\right)\\ +& \left(1-{\mu }_0{\mathrm{\nabla }}^2\right)D_3(\pi /h)\sin \left(\pi {\xi }_3/h\right)]\mathrm{d}{\xi }_3]\mathrm{d}{\xi }_1\mathrm{d}{\xi }_2=0.\end{array}

Each side of the FGPN plate is subjected to free (F), clamped (C), or simply supported (S) conditions, as shown in Fig. 7(a). The BCs for each type are given below.

1) BCs parallel to {\xi }_1 axis:

F-conditions: u_0=v_0=w_0={\phi }_1={\phi }_2=\mathrm{\Theta }\ne 0;

S-conditions: u_0=w_0={\phi }_1=\mathrm{\Theta }=N_2=M_2=Q_2=0 and v_0,{\phi }_2\ne 0;

C-conditions: u_0=v_0=w_0={\phi }_1={\phi }_2=\mathrm{\Theta }=0.

2) BCs parallel to {\xi }_2 axis:

F-conditions: u_0=v_0=w_0={\phi }_1={\phi }_2=\mathrm{\Theta }\ne 0;

S-conditions: v_0=w_0={\phi }_2=\mathrm{\Theta }=N_1=M_1=Q_1=0 and u_0,{\phi }_2\ne 0;

C-conditions: u_0=v_0=w_0={\phi }_1={\phi }_2=\mathrm{\Theta }=0.

4 Solution technique

A HOFE method is utilized to determine the solution of the governing equation established above. In particular, the FGPN plate is discretized by Q9, as shown in Fig. 7(b). Each node of the mesh element contains six degrees of freedom (DOFs), viz., three displacements \left\{d_{\mathrm{d}}\right\}, i.e., \left(u_0,v_0,w_0\right)two bending rotations\left\{d_{\mathrm{b}}\right\}, i.e.,({\phi }_1,{\phi }_2) and one electric potential \left(\mathrm{\Theta }\right). These vectors connected with the i\mathrm{t}\mathrm{h} (i = 1,2,…,9) node of the element can be stated as:

(34)\begin{array}{rl}& \left\{d_{\mathrm{d}i}\right\}={\left[u_{0i}{v}_{0i}{w}_{0i}\right]}^{\mathrm{T}},\left\{d_{{\mathrm{b}}_i}\right\}={\left[{\phi }_1{\phi }_2\right]}^{\mathrm{T}},\\ & \left\{d_{\mathrm{d}}\right\}=\left[N_{\mathrm{d}i}\right]\left\{d_{\mathrm{d}i}^{\mathrm{e}}\right\},\left\{d_{\mathrm{b}}\right\}=\left[N_{\mathrm{b}i}\right]\left\{d_{\mathrm{b}i}^{\mathrm{e}}\right\},\left\{\mathrm{\Theta }\right\}=\left[N_{\mathrm{\Theta }i}\right]\left\{{\mathrm{\Theta }}_i^{\mathrm{e}}\right\},\\ & \left[u_{0i}\right]=\left[N_{\mathrm{u}i}\right]\left\{d_{\mathrm{d}i}^{\mathrm{e}}\right\},\left[v_{0i}\right]=\left[N_{\mathrm{v}i}\right]\left\{d_{\mathrm{d}i}^{\mathrm{e}}\right\},\\ & \left[w_{0i}\right]=\left[N_{\mathrm{w}i}\right]\left\{d_{\mathrm{d}i}^{\mathrm{e}}\right\},\left[{\phi }_{1i}\right]=\left[N_{{\phi }_{1i}}\right]\left\{d_{\mathrm{b}i}^{\mathrm{e}}\right\},\left[{\phi }_{2i}\right]=\left[N_{{\phi }_{2i}}\right]\left\{d_{\mathrm{b}i}^{\mathrm{e}}\right\},\end{array}\}

(35)\begin{array}{rl}& \left\{d_{\mathrm{d}i}^{\mathrm{e}}\right\}={\left[{\left\{d_{\mathrm{d}1}^{\mathrm{e}}\right\}}^{\mathrm{T}}{\left\{d_{\mathrm{d}2}^{\mathrm{e}}\right\}}^{\mathrm{T}}\cdots {\left\{d_{\mathrm{d}9}^{\mathrm{e}}\right\}}^{\mathrm{T}}\right]}^{\mathrm{T}},\\ & \left\{d_{\mathrm{b}i}^{\mathrm{e}}\right\}={\left[{\left\{d_{\mathrm{b}1}^{\mathrm{e}}\right\}}^{\mathrm{T}}{\left\{d_{\mathrm{b}2}^{\mathrm{e}}\right\}}^{\mathrm{T}}\cdots {\left\{d_{\mathrm{b}9}^{\mathrm{e}}\right\}}^{\mathrm{T}}\right]}^{\mathrm{T}},\\ & \left\{{\mathrm{\Theta }}_i^{\mathrm{e}}\right\}=\left[{\mathrm{\Theta }}_1^{\mathrm{e}}{\mathrm{\Theta }}_2^{\mathrm{e}}\cdots {\mathrm{\Theta }}_9^{\mathrm{e}}\right],\left[N_{\mathrm{\Theta }i}\right]=\left[N_1{N}_2\cdots N_9\right],\\ & \left[N_{\mathrm{d}i}\right]=\left[N_{\mathrm{d}1}{N}_{\mathrm{d}2}\cdots N_{\mathrm{d}9}\right],\left[N_{\mathrm{b}i}\right]=\left[N_{\mathrm{b}1}{N}_{\mathrm{b}2}\cdots N_{\mathrm{b}9}\right],\end{array}\}

where N_i denotes the shape function and \left\{d_{\mathrm{d}}\right\} is the generalized displacement and \left\{d_{\mathrm{b}}\right\} rotational vector of the i\mathrm{t}\mathrm{h} node.

The system’s strain and electric potential vectors can be associated with the nodal displacement vector \left\{d\right\} and the nodal electric potential vector {\mathrm{\Theta }}_0. The infinitesimal engineering strain for bending and shear that is associated with displacement, bending rotations and electric field vectors in terms of the shape function matrices is as follows:

(36)\begin{array}{l}\left\{{\epsilon }_1\right\}=\left\{{\epsilon }_l\right\}+\left\{{\epsilon }_{\mathrm{n}\mathrm{l}}\right\}=\left[B_{\mathrm{d}\mathrm{b}}\right]\left\{d_{\mathrm{d}}\right\}+\left[B_{\mathrm{b}\mathrm{b}}\right]\left\{d_{\mathrm{b}}\right\}+{\displaystyle \frac{1}{2}}\left[B_1\right]\left[B_2\right]\left\{d_{\mathrm{d}}\right\},\\ \left\{{\epsilon }_2\right\}=\left[B_{\mathrm{d}\mathrm{s}}\right]\left\{d_{\mathrm{d}}\right\}+\left[B_{\mathrm{b}\mathrm{s}}\right]\left\{d_{\mathrm{b}}\right\},\\ \left\{E\right\}=\left[B_{\mathrm{\Theta }}\right]\left\{{\mathrm{\Theta }}^e\right\}.\end{array}\}

The shape function derivate appearing in the above Eq. (36) can be written as:

(37)\begin{array}{rl}& \left[B_{\mathrm{d}\mathrm{b}}\right]=\left[\begin{array}{ccc}{\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }{\xi }_1}}& 0& 0\\ 0& {\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }{\xi }_2}}& 0\\ {\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }{\xi }_2}}& {\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }{\xi }_1}}& 0\end{array}\right],\left[B_{\mathrm{b}\mathrm{b}}\right]=\left[\begin{array}{cc}{\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }{\xi }_1}}& 0\\ 0& {\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }{\xi }_2}}\\ {\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }{\xi }_2}}& {\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }{\xi }_1}}\end{array}\right],\left[B_{\mathrm{d}\mathrm{s}}\right]=\left[\begin{array}{ccc}0& 0& {\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }{\xi }_1}}\\ 0& 0& {\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }{\xi }_2}}\end{array}\right],\left[B_{\mathrm{\Theta }i}\right]=[\begin{array}{cccc}{B}_{\mathrm{\Theta }1}& B_{\mathrm{\Theta }2}& \cdots & B_{\mathrm{\Theta }9}\end{array}],\\ & \left[B_{\mathrm{b}\mathrm{s}}\right]=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right],\left[B_2\right]=\left[\begin{array}{ccc}0& 0& {\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }{\xi }_1}}\\ 0& 0& {\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }{\xi }_2}}\\ 0& 0& 0\end{array}\right],\left[B_1\right]=\left[\begin{array}{ccc}{\displaystyle \frac{\mathrm{\partial }{w}_0}{\mathrm{\partial }{\xi }_1}}& 0& 0\\ 0& {\displaystyle \frac{\mathrm{\partial }{w}_0}{\mathrm{\partial }{\xi }_2}}& 0\\ {\displaystyle \frac{\mathrm{\partial }{w}_0}{\mathrm{\partial }{\xi }_2}}& {\displaystyle \frac{\mathrm{\partial }{w}_0}{\mathrm{\partial }{\xi }_1}}& 0\end{array}\right].\end{array}\}

Substituting Eq. (34) into Eq. (33), assembling and rearranging the element equations based on the coefficient of the nodal displacement, bending rotations, and electric potential vector. Hence, the generalized FE governing equations can be stated as:

(38)\left[M\right]\left\{\ddot{d}\right\}+\left[K_{\mathrm{e}\mathrm{q}\mathrm{v}}\right]\left\{d\right\}=\left[F_{\mathrm{e}\mathrm{q}\mathrm{v}}\right],

where \left[M\right], \left[K_{\mathrm{e}\mathrm{q}\mathrm{v}}\right]and \left[F_{\mathrm{e}\mathrm{q}\mathrm{v}}\right]are the global mass matrix, coupled equivalent stiffness matrices and coupled loading vector, respectively. For the free vibration of the FGPN porous plate, the generalized Eq. (38) can be expressed as:

(39)\left[M\right]\left\{\ddot{d}\right\}+\left[K_{\mathrm{e}\mathrm{q}\mathrm{v}}\right]\left\{d\right\}=0,

(40)\begin{array}{rl}& \left[K_{\mathrm{e}\mathrm{q}\mathrm{v}}\right]=\left[K_{\mathrm{d}\mathrm{d}\mathrm{t}}\right]+\left[K_{\mathrm{d}\mathrm{b}}\right]+\left[K_{\mathrm{b}\mathrm{b}}\right]+\left[K_{\mathrm{d}\mathrm{\Theta }}\right]+\left[K_{\mathrm{b}\mathrm{\Theta }}\right]+\left[K_{\mathrm{\Theta }\mathrm{\Theta }}\right]\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h},\\ & \begin{array}{l}\left[K_{\mathrm{d}\mathrm{d}\mathrm{t}}\right]=\left[K_{\mathrm{d}\mathrm{d}}\right]+\left[K_{\mathrm{d}\mathrm{d}\mathrm{e}\mathrm{f}}\right],\left[K_{\mathrm{d}\mathrm{d}}\right]=\left[K_{\mathrm{d}1}\right]+\left[K_{\mathrm{d}2}\right]+\left[K_{\mathrm{d}\mathrm{n}\mathrm{l}1}\right],\\ \left[K_{\mathrm{d}\mathrm{b}}\right]=\left[K_{\mathrm{d}\mathrm{d}1}\right]+\left[K_{\mathrm{d}\mathrm{d}2}\right]+\left[K_{\mathrm{d}\mathrm{b}\mathrm{n}\mathrm{l}1}\right],\\ {\left[K_{\mathrm{d}\mathrm{\Theta }}\right]}^{\mathrm{T}}={\left[K_{\mathrm{d}\mathrm{\Theta }l}\right]}^{\mathrm{T}}+{\displaystyle \frac{1}{2}}{\left[K_{\mathrm{d}\mathrm{\Theta }\mathrm{n}\mathrm{l}}\right]}^{\mathrm{T}},\\ \left[K_{\mathrm{b}\mathrm{b}}\right]=\left[K_{\mathrm{b}\mathrm{b}1}\right]+\left[K_{\mathrm{b}\mathrm{b}2}\right],\left[K_{\mathrm{d}\mathrm{\Theta }}\right]=\left[K_{\mathrm{d}\mathrm{\Theta }\mathrm{l}}\right]+\left[K_{\mathrm{d}\mathrm{\Theta }\mathrm{n}\mathrm{l}}\right].\end{array}\}\end{array}

Equation (39) is written for the corresponding eigenvalue problem:

(41)\left(\left[K_{\mathrm{e}\mathrm{q}\mathrm{v}}\right]-{\mathrm{\Omega }}^2\left[M_{\mathrm{d}\mathrm{d}}\right]\right)\left\{d\right\}=0,

where \mathrm{\Omega } is the natural vibration frequency. In Eq. (41), various matrices and vectors are involved, and these are as follows: \left[K_{\mathrm{d}\mathrm{d}\mathrm{t}}\right], \left[K_{\mathrm{d}\mathrm{b}}\right], and \left[K_{\mathrm{b}\mathrm{b}}\right] are the elastic stiffness matrix, \left[K_{\mathrm{d}\mathrm{\Theta }}\right] and \left[K_{\mathrm{b}\mathrm{\Theta }}\right] are piezoelectric-elastic coupling matrix. \left[K_{d1}\right], \left[K_{\mathrm{d}2}\right], \left[K_{\mathrm{d}\mathrm{b}1}\right], and \left[K_{\mathrm{d}\mathrm{b}2}\right] are the linear stiffness matrix. \left[K_{\mathrm{d}\mathrm{n}\mathrm{l}1}\right] and \left[K_{\mathrm{d}\mathrm{b}\mathrm{n}\mathrm{l}1}\right] are the nonlinear stiffness matrix. \left[K_{\mathrm{d}\mathrm{\Theta }\mathrm{l}}\right] and \left[K_{\mathrm{d}\mathrm{\Theta }\mathrm{n}\mathrm{l}}\right] are the linear and nonlinear piezoelectric matric. \left[K_{\mathrm{d}\mathrm{d}\mathrm{e}\mathrm{f}}\right] is the foundation stiffness matrix. \left[K_{\mathrm{\Theta }\mathrm{\Theta }}\right] is the elemental electric stiffness matrix. The various stiffness matrices and associated terms in Eq. (33) are represented in Electronic Supplementary Material. The eigen frequency values of the FGPN plate are determined by solving Eq. (41) using the modified iterative technique or the Newton–Raphson technique [93,94]. The solution techniques involve various steps, and a comprehensive flow chart of the solution methods is shown in Fig. 8. The equation below presents the error convergence norms of the modified Newton–Raphson technique.

(42)\left(\sqrt{{\left(\left|{\mathrm{\Omega }}_{nt+1}-{\mathrm{\Omega }}_{nt}\right|\right)}^2}/\sqrt{{\left(\left|{\mathrm{\Omega }}_{nt+1}\right|\right)}^2}\right)\le \mathrm{E}\mathrm{r}\mathrm{r}\mathrm{o}\mathrm{r}\left(10^{-3}\right).

5 Numerical results and discussion

In this study, the free vibration response of the FGPN porous plate under hygrothermal and thermoelectric conditions has been conducted. The electromechanical material properties of the FGPN plate are detailed in Table 2. Various BCs, denoted as CCCC (BC-I), CFFF (BC-II), CFCF (BC-III), SSSS (BC-IV), SFSF (BC-V), and SCSF (BC-VI), along with different geometric configurations, Case-I, Case-II, Case-III, and Case-IV, are explored. The geometric parameter a/h for the FGPN plate is assumed to be 20.

Reference values for temperature and moisture concentrations are set at 300 K and 0, respectively. Material properties, including variations in the trend of elastic constants, thermal expansion, moisture concentrations, and other effective material properties, follow a similar trend as depicted in Fig. 2. The numerical investigation specifically explores the impact of bidirectional material exponents (n and m), thickness ratio (a/h), porosity distribution with porous exponent, electrical voltage, hygrothermal loading, and BCs on the free vibration responses.

For easy understanding, this analysis assumes the following parameters to be nondimensionalized. Frequency parameter: \omega =\mathrm{\Omega }\left(a^2/h\right)\sqrt{{\left(\rho /C_{11}\right)}_{\mathrm{P}\mathrm{T}4}}, whereby \mathrm{\Omega } is the natural frequency, while (\rho {)}_{\mathrm{P}\mathrm{T}4} and {\left(C_{11}\right)}_{\mathrm{P}\mathrm{T}4} are the mass density and elastic constants of the PZT-4 material, respectively.

5.1 Convergence and validations

In this segment, the convergence and confirmation of the current HOFE model are assessed by analyzing the free vibration outcomes. To evaluate convergence, the free vibration responses of the BC-I bidirectional FGPN plate are computed and detailed in Table 3. The plate’s assumed parameters are n = m = 2, without considering porosity and foundations. Analysis indicates that optimal convergence is achieved with a 16 × 16 mesh refinement, which adequately captures the nonlinear frequency of the FGPN plate. Existing literature offers validations of the present formulations for the FGPN plate when compared to established methods. However, the numerical approach for the nonlinear vibration responses of the FGPN plate resting on the foundations is limited in existing studies. To authenticate the cogency of the current study, the model is adapted to that of a porous non-uniform graded piezoelectric nanoplate with elastic foundations. In this section, two different literature numerical problems are explored to validate and correct the present formulations.

Example 1: This illustration displays the vibration response of the square nano FG plate with BCs and nonlocal parameters. The assumed geometric condition is a/h = 10 and 20, while the bidirectional material exponent is n = 5, m = 0 with BC-IV and BC-I. The properties of the nano FG plate are considered to be the same as those in Ref. [14]. Table 4 confirms the natural frequency of the nanoplate and indicates decent agreement with Ref. [14] results.

Example 2: The example results shown in Table 5 present the first two nondimensional frequencies of the piezoelectric nanoplate for certain values of the nonlocal parameters. The assumed dimensions and plate parameters are considered the same as those in Refs. [15,16]. From the results, it is deduced that the current study outcomes verify the results of previous works of literature.

5.2 Parametric study

Based on the verification analysis detailed in the previous section, this study delves into a numerical investigation of nonlinear free vibrations influenced by various parameters. These parameters include the thickness ratio, porous exponent, bidirectional material exponents, BCs, as well as thermoelectric and hygrothermal loading. Diverse configurations of the FGPN plates are considered to assess the frequency values when employing different BCs and porosity distributions.

Fundamental frequencies of the various FGPN plate configurations are estimated for different BCs, nonlocal parameters, bidirectional material exponents, and material distribution laws, namely, PL and SL, as presented in Tables 6 and 7. It is detected that the plate frequencies decrease with an increase in the bidirectional material exponent, attributed to the higher mechanical properties of PZT-4 compared to PZT-5H. Additionally, SL gradation demonstrates higher frequency values, whereas PL material gradation exhibits lower frequencies. Notably, BC-I plate conditions manifest higher frequencies than other BCs due to increased constraints, which enhance flexural rigidity along the edges, ultimately leading to higher frequency and greater edge restrictions in the FGPN plate.

Tables 8 and 9 display the nondimensional frequencies for the Case-I configuration of the FGPN plate, considering diverse porosity distributions and bidirectional material exponents. The tabulated data highlights larger frequency values for the SCPP plate, while the PP type of the BC-I plate demonstrates the lowest frequency values. Moreover, an increase in the porous exponent results in a reduction of nondimensional frequency values due to decreased stiffness in the graded material caused by increased porosity. Tables 8 and 9 also disclose that the nondimensional frequencies of the FGPN BC-I plate decrease as both the transverse and axial direction exponents increase. This trend is attributed to higher material exponents, which diminish the impact of the stronger ceramic phase, subsequently reducing plate stiffness and increasing the plate’s mass. Notably, both tables indicate that the obtained frequencies are higher for BC-I configurations but lower for BC-IV configurations, despite having the same plate configurations (i.e., Case-I and a/h = 100).

5.2.1 Impact of the bidirectional material and porous exponent

Figures 9 and 10 display the first five mode shapes of the BC-I NGFP plate for Case-I and Case-IV configuration types. Considering the assumed plate dimensions as a = b and a/h = 10, these figures showcase distinct vibration behaviors in various mode shapes. The lack of symmetry in the mode shapes is due to the plates’ differing configuration types. Figures 9 and 10 depict the deformation behavior during natural vibration conditions for both Case-I and Case-IV plates.

In Case-I, where the material properties are uniform across the plate, the mode shapes follow typical bending behavior. The first mode shows a symmetric displacement with a peak at the center, representing the fundamental frequency. The second mode exhibits two regions of displacement along the length, indicating bending. The third mode alternates in displacement, with peaks near the edges. The fourth mode becomes more complex, with multiple regions of displacement along the plate, while the fifth mode shows a high-frequency, intricate deformation pattern. In Case-IV, where the material properties vary linearly along both the axial and thickness directions, the vibrational behavior becomes more complex. In this case, the first mode still shows a central displacement, but the material gradation introduces subtle shifts in the vibration frequency compared to the uniform case. The second mode exhibits similar behavior to Case-I, but with slight alterations in the displacement profile due to the material gradation. In the third mode, the alternating displacement becomes more pronounced, and the linear variation of material properties causes shifts in the nodal points, altering the overall vibration behavior. As we move to the fourth mode, we observe a more complex deformation pattern with multiple nodal points. The influence of material gradation becomes evident as it changes the frequency response and mode shape when compared to the uniform plate in Case-I. Lastly, the fifth mode shows even more complexity, where the material variation introduces coupling between the modes, resulting in changes to both the frequencies and displacement patterns.

Overall, Case-I demonstrates the typical vibrational modes of a uniform plate, while Case-IV reveals how the linear variation of material properties influences the dynamic behavior of the nanoplate, altering the vibrational patterns and frequencies. This comparison emphasizes the role of material gradation in shaping the vibrational characteristics of the FGPN plate, showing how the material distribution can significantly impact the plate’s response to vibrational loading.

Figure 11 illustrates the impact of nondimensional frequency on the FGPN plate with bidirectional exponents and porosity distributions, considering a sigmoidal material distribution and a/h = 10 under BC-I conditions. Notably, the frequency values decrease as the bidirectional material exponent (transverse and axial direction exponents) increases, leading to reduced plate stiffness. Across various plate configurations, the results consistently highlight the Case-IV plate’s maximum porosity distribution effect on free vibration behavior.

Figure 12 explains the effect of the porous exponent on the nondimensional frequency, maintaining a constant material exponent with various porosity distributions. The findings consistently show a decrease in the plate’s frequency characteristics with an increase in the porous exponent, regardless of the porosity distributions. This decrease signifies a reduction in stiffness due to varying porosity levels. Notably, the SCPP-type porosity exerts a more significant impact on the FGPN plate’s nondimensionalized frequency compared to EP and UPP-type porous distributions.

5.2.2 Impact of the thickness ratio

Figure 13 showcases the frequency variations of the sigmoidal material distribution in the FGPN plate concerning different BCs and thickness ratios. The geometrical parameters are set as a = b, n = m = 2 with a/h values of 10, 20, 50, and 100. The results reveal that both porosity distribution and porous exponents play a substantial role in influencing the plate frequency. The decrease in frequency within this specific range is likely due to a combination of factors related to the plate’s dynamics, particularly the interplay between the thickness ratio and porosity distribution. One possible explanation is that as the a/h ratio increases, the plate undergoes a transition from bending-dominated to shear-dominated behavior, which leads to a reduction in the resonant frequency. This shift can be quite pronounced at intermediate values of a/h, causing the observed dip in frequency. Additionally, the effect of porosity plays a significant role in this behavior. As the porosity exponent varies, it introduces softening in the material, which reduces the plate’s stiffness. The computation involves porous exponent values ranging from 0 to 0.2, which notably impact individual porosity distributions such as EP, UPP, and SCPP. At a/h ratios between 10 and 30, this effect is more pronounced, leading to a sudden decrease in frequency. This is due to higher a/h ratios indicating thinner nanoscale plates, resulting in decreased stiffness.

5.2.3 Impact of the nonlocal parameter

The nondimensional frequency parameters of the FGPN plate with nonlocal parameters, different BCs and porosity distributions are presented in Fig. 14. The assumed parameters are SL material variations, Case-III configurations, a/h = 10, γ = 0.2, and n = m = 2. It is evident that nonlocal parameters have a noteworthy impact on the nondimensional frequency of the FGPN plate across different porosity distributions while maintaining constant bidirectional material exponents. Figure 14 illustrates a consistent decreasing trend in the frequency of the FGPN plate as nonlocal parameters increase. This decline is attributed to nonlocal parameters softening the plate, thereby increasing its flexibility. Notably, nonporous plates exhibit higher frequencies compared to porous plates. Among various porous distributions, the SCPP configuration displays a higher frequency and exerts a more pronounced effect than other porosity configurations.

Table 10 lists the first five fundamental frequencies of the FGPN plate, illustrating variations in nonlocal parameters and bidirectional material exponents with SL material distributions. Under the specifications of Case-III, a/h = 10, V = 0 V, ΔT = 0 K, and ΔC = 0, it’s detected that fundamental frequency declines with a rise in bidirectional direction exponents at a constant material exponent. However, as the frequency increases with the number of modes, this is due to the nonlocal parameter reducing the stiffness of the nanoplates, subsequently leading to a decrease in frequency.

5.2.4 Impact of the applied electrical loading

The fundamental frequency of the FGPN plate with bidirectional material exponent, porosity distributions and applied electrical loading are presented in Figs. 15 and 16. The presumed geometrical parameter is BC-I BCs, SL material distributions, Case-I (Fig. 15) and Case-III (Fig. 16), a/h = 10, γ = 0.2, n = m = 2, ΔT = 0 K and ΔC = 0. An increase in the bidirectional material exponent results in a reduction in the plate’s frequency by decreasing the content of PZT-4 within the plate. Moreover, the frequency of the plate changes based on the nature of the applied electrical loading, i.e., negative and positive values. Negative electrical loading generates a higher frequency than positive values due to the axial tensile and compressive forces induced by both types. The stiffness of the FGPN plate is significantly affected by applied electrical loading; axial compressive forces decrease the stiffness while axial tensile forces increase it.

Figure 17 illustrates the effect of voltage and temperature changes on the FGPN plate with different porosities and nonlocal parameters while maintaining a constant material exponent and thickness ratio. Notably, the plate’s nondimensional frequency decreases as thermoelectric loading increases and vice versa. This effect arises from the electrical loading softening the plate structures, particularly the temperature and axial compressive forces, leading to a decrease in frequency.

5.2.5 Effects of hygrothermal loading

This section explores the impact of hygrothermal loading on the BC-IV FGPN plate with different porosities while maintaining a constant bidirectional material exponent and thickness ratio. Figure 18 illustrates the fundamental frequency of the plate concerning temperature changes and moisture variations, all at constant porous exponents. The findings indicate a reduction in the nondimensional frequency as temperature and moisture changes increase. Both temperature and moisture changes occur along the plate’s transverse direction, affecting the plate’s thermal vibration and buckling behavior significantly.

In Fig. 19, the nondimensional frequencies of the BC-IV FGPN plate are examined under various hygrothermal loadings while maintaining a constant porous exponent but with different porosity distributions. The results show that increased temperature and moisture change values lead to a reduction in the plate’s frequency across diverse hygrothermal loadings. Notably, the SHTR-type hygrothermal loading has a more pronounced effect compared to other types of hygrothermal loading.

5.2.6 Impact of the variable foundation

In this subsection, the investigation involves the FGPN plate resting on various elastic foundations, including the linear (LF), parabolic (PF), and sinusoidal (SF) types, with variations along the axial direction. Figure 20 displays the influence of the Winkler-Pasternak foundation on the BC-I FGPN plate with dimensions a/h = 10, considering variations in the bidirectional material exponent. The results show a consistent increase in the plate’s nondimensional frequency with rising Winkler and Pasternak parameters and ξ values for both perfect and porous plates. This trend is attributed to the added stiffness provided by the foundation, which enhances the plate’s resistance to deformation. Notably, the Pasternak parameter exhibits a more significant effect compared to the Winkler parameter due to its ability to model shear interactions between neighboring points, providing superior lateral stiffness. Among the different foundation types, parabolic foundations yield higher frequency values compared to linear and sinusoidal types, owing to their concentrated support in critical regions of the plate. Additionally, increasing the material gradation exponent further amplifies the frequency by enhancing the effective stiffness of the structure. These findings highlight the crucial role of both foundation modeling and material design in optimizing the dynamic performance of smart structural components.

Tables 11 and 12 show the nondimensional frequency results of BC-I and BC-IV constrained FGPN plates resting on variable elastic foundations while maintaining a constant Pasternak parameter and a/h = 10. The findings indicate that the nondimensional frequency decreases with increasing bidirectional material exponents (ranging from 2 to 5) and increases with the elevation of the variable Winkler parameters. This observed trend is due to the foundation stiffness effect on the plate’s frequency. Further investigation into the effects of porosity on the plate’s nondimensional frequency while resting on variable foundations is detailed in Table 11. The findings confirm that SCPP porosity distributions exhibit a more significant impact on the plate frequency. Additionally, Table 12 presents the nondimensional frequency of various FGPN plate configurations resting on elastic variable foundations, suggesting that the Case-III plate configuration exhibits a higher frequency compared to other plate configurations.

Figure 21 displays the impact of the hygrothermal loading on the BC-IV FGPN plate resting on the variable elastic foundations at constant kp = 20, γ = 0.2, n = m = 2. The findings confirm that an increase in hygrothermal loading results in a decrease in the nondimensional frequency across all distinct variable elastic foundations. This behavior can be understood physically: hygrothermal effects introduce softening in the material, reducing its overall stiffness. Elevated temperatures and moisture content weaken the material’s bonding at the microscopic level, which naturally leads to a drop in the structure’s resistance to vibration, thus lowering its frequency. Notably, there are considerable differences in the nondimensional frequencies observed in LF, PF, and SF elastic foundations. Specifically, the PF variations of the elastic foundation yielded higher nondimensional frequency values in contrast to the linear and sinusoidal types. The reason behind this is related to how the stiffness is distributed across the plate in different foundations. In the parabolic foundation model, the stiffness is not uniform but increases more significantly toward the center of the plate, effectively offering greater resistance to deformation where it matters most. This enhanced localized stiffness results in an overall stiffer structural behavior, making the plate less sensitive to softening effects due to hygrothermal exposure compared to the linear and sinusoidal foundation models.

6 Conclusions

The investigation delves into the vibration characteristics of a bidirectional sigmoid FGPN plate affected by hygrothermal and thermoelectric loading while resting on variable elastic foundations. The study assumes bidirectional variations in thermomechanical properties along transverse and axial directions conforming to sigmoid and PL distributions and also, the plate considering different porosity distributions, i.e., EP, UPP, and SCPP types. Final equations were derived using the energies principle and modified FSDT-based displacement fields. The HOFE formulations employing six DOFs per node were utilized to solve the governing equations. The study’s accuracy was confirmed through sigmoid FGPN plate assessments and convergence tests. The study investigated the impacts of hygrothermal loading, electrical loading, temperature changes, bidirectional material exponents, porous exponent with various distributions, thickness ratio, variable elastic foundations, and BCs across different nanoplate configurations. The findings from this study present critical insights into the vibrational behavior of the nanoplate under different loadings and BCs. Here are the key observations derived from the investigation.

1) Nonlocal parameters and bidirectional material exponents: these factors have a diminishing effect on the FGPN plate’s frequency. Moreover, higher frequency modes are notably affected by nonlocal parameters.

2) Material distribution and thickness: the variation in thickness law with sigmoid material distribution is more influential than PL distribution. Axial direction material exponents exert a more significant effect than transverse direction material.

3) Electric and thermal loading: electrical loading has the most substantial impact on the FGPN plate’s frequency, while thermal loading exerts the least influence. The frequencies are more responsive to axial forces induced by electrical loading rather than temperature changes.

4) Porosity distributions: different porosity types affect the plate’s fundamental frequencies distinctly. The SCPP porosity type displays higher sensitivity toward frequency changes compared to other distributions.

5) Variable thickness and foundations: variable thickness significantly influences the plate’s fundamental frequencies, with the transverse variable thickness parameter exhibiting more impact. Parabolic foundations show higher frequencies compared to linear and sinusoidal foundations.

6) Hygrothermal loading: the increase in hygrothermal loading consistently reduces the frequency of nanoplate, with SHTR loading having more substantial effects than other types.

The bidirectional analysis plays a pivotal role in the accurate design and performance evaluation of smart structures and multifunctional devices, including piezoelectric sensors and actuators, morphing aircraft wings, flexible electronics, biomedical devices, and energy harvesting systems. By accounting for environmental factors such as hygrothermal loads, electro-mechanical coupling, and material property variations, the analysis ensures reliability and enhanced functionality under complex operating conditions. This research employed advanced FE modeling and incorporation of material gradation theories offer a flexible and robust framework. Notably, the solution strategies developed are generalizable, enabling seamless extension to a wide range of BCs, making the approach highly adaptable for multidisciplinary engineering applications involving FGMs and smart composites.

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