INTRODUCTION

Functionally graded piezoelectric materials (FGPMs) represent a modern class of piezoelectric composites in which the electro‑elastic characteristics vary continuously along the thickness direction. Numerous studies have focused on the mechanical and vibrational responses of both piezoelectric and functionally graded structural elements. For instance, Liew et al. [1] developed a finite‑element framework for piezo‑thermoelastic analysis of functionally graded plates equipped with embedded piezoelectric sensor and actuator layers. Their work examined bending and torsional vibration control of FG plates exposed to thermal gradients by employing self‑sensing actuators. Zhao et al. [2] explored the free vibration behavior of functionally graded plates using the element‑free kp‑Ritz technique, demonstrating that the volume‑fraction exponent plays a crucial role in determining natural frequencies. Behjat and Khoshravan [3] investigated nonlinear static bending and free vibration responses of FGPM plates, concluding that for all values of the volume‑fraction index "n", the deflection of the plate increases noticeably under mechanical loading. Farsangi and Saidi [4] provided an exact analytical formulation for the vibration of rectangular FG plates with surface‑bonded piezoelectric layers, showing that increasing the thickness‑to‑length ratio enhances the natural frequencies across mechanical and electrical boundary conditions. Komijani et al. [5] reported that the mechanical behavior of unsymmetrical piezoelectric beams differs fundamentally from that of symmetrical configurations. Lezgy et al. [6] introduced a conforming three‑noded beam element capable of predicting the static, modal, and transient responses of FGM beams. Sharma and Kumar [7] examined the flexural vibration characteristics of short‑circuited FGPM annular plates subjected to shear‑induced excitation, and their analytical formulation showed strong agreement with numerical results obtained from COMSOL simulations. Their study indicated that the natural frequencies rise as the thickness‑to‑diameter ratio increases, and also grow with the volume‑fraction index up to a value of 5. Selim et al. [8] proposed an efficient methodology combining Reddy’s higher‑order shear deformation theory (HSDT) with the element‑free IMLS‑Ritz scheme to analyze free vibration and active vibration suppression of FGPM plates incorporating piezoelectric layers. Their findings revealed that plates with open‑circuit piezoelectric layers consistently exhibit higher natural frequencies than those with closed‑circuit electrical conditions due to electromechanical coupling effects. Nourmohammadi and Behjat [9] explored the bending response of FGPM plates under mechanical, electrical, and thermal loading using the first‑order shear deformation theory (FSDT), showing that the relationship between deflection and the volume‑fraction index varies depending on the type of applied load. Mikaeeli and Behjat [10] introduced a three‑dimensional Element‑Free Galerkin (EFG) formulation to study the static behavior of thick FGPM rectangular plates with arbitrary gradation profiles, applying it to model multi‑field piezoelectric and non‑homogeneous problems. Lezgy et al. [11] investigated the use of FGPMs in passive vibration damping of laminated composite beams, demonstrating that the R–L shunt circuit provides the most effective damping performance regardless of the material gradient. Nguyen‑Quang et al. [12] extended the CS‑DSG3 technique employing three‑node triangular elements to perform static analysis, free vibration assessment, and dynamic control of FGM plates integrated with piezoelectric sensors and actuators. Their results confirmed that the CS‑DSG3 approach relies solely on simple three‑node linear triangular elements, making it highly suitable for automatic mesh generation in complex geometrical domains. Ruocco and Mallardo [13] formulated a new finite strip model combined with the Kantorovich method to study the buckling and vibrational behavior of imperfect nanoplates. Thang et al. [14] explored the bending and buckling responses of bidirectional functionally graded composite nanobeams, whereas Tang and Dai [15] examined the stability behavior of laminated piezoelectric cylindrical shells subjected to various environmental influences. Fan et al. [16]–[19] employed the isogeometric analysis framework to model microscale plates of different geometries incorporating FG porosity and size‑dependent characteristics. Rahimi et al. [20] provided analytical predictions for the free vibration and bending performance of FGP‑GPLRC cylindrical shells, while Ton‑That et al. [21] assessed the bending, vibrational, and buckling behavior of FGP‑GPLRC plates. Wang and Liu [22] introduced piezoelectric composite materials (PCMs) consisting of a substrate combined with piezoelectric layers to illustrate the active elastic behavior of composite systems, showing that their elastic properties are influenced by geometry, constituent materials, and applied voltage. Tran et al. [23] studied the free vibration characteristics of piezoelectric FGP‑GPLRC plates under different boundary conditions, demonstrating that incorporating a lightweight metal‑foam core together with graphene platelets enhances stiffness and provides an efficient configuration for sandwich plates. Alshenawy et al. [24] formulated a three‑dimensional strain‑gradient shell model to analyze the micro‑scale‑dependent nonlinear stability of FG piezoelectric micro‑shells under thermo‑electromechanical loading, revealing that the inclusion of microstructural strain‑gradient tensors increases the critical mechanical and electrical loads associated with buckling. Bastami and Behjat [25] investigated the buckling and free vibration behavior of piezoelectric nanoplates resting on an elastic foundation using Kirchhoff plate theory, concluding that both the critical buckling load and the natural frequency decrease as the nonlocal parameter increases. Kumar et al. [26] examine the stress response, deformation characteristics, and vibrational performance of porous functionally graded piezoelectric tapered plates subjected to thermoelectric and mechanical loads by employing advanced finite element. Their findings are offering guidance for the development of smart structural components and biomedical sensing devices. In another study, the authors conduct a free vibration investigation of porous functionally graded plates using three‑dimensional degenerated shell elements formulated on Reissner–Mindlin theory, achieving lower computational cost while still accurately representing the variation of material properties through the thickness [27]. Kumar and Harsha [28] propose a higher‑order finite element formulation based on first‑order shear deformation theory to evaluate the hygrothermal static behavior of nanoscale, multidirectional nanofunctionally graded piezoelectric (NFGP) plates supported by spatially varying elastic foundations. In a related work, the bending response, deflection pattern, and vibrational characteristics of porous multidirectional functionally graded circular piezoelectric plates resting on variable elastic foundations under thermo‑electro‑mechanical loading are analyzed. The study reveals that the porosity level has a pronounced effect on static deflection, radial stress distribution, and the natural frequencies [29]. Kumar et al. [30] explore the buckling performance of porous PZT‑4/PZT‑5H smart graded plates supported by Winkler–Pasternak foundations when exposed to combined thermomechanical and thermoelectric loadings. Another investigation addresses the static and dynamic responses of exponential functionally graded piezoelectric materials under thermo‑electro‑mechanical effects using first‑order shear deformation theory (FSDT) and Hamilton’s principle [31]. The nonlinear vibration characteristics of porous multidirectional piezoelectric functionally graded nonuniform plates placed on orthotropic variable elastic foundations and subjected to hygrothermal environments are studied by Kumar et al. [32]. Free vibration analyses of porous functionally graded material (FGM) sandwich plates in thermal fields are also performed [33]. Two configurations—sandwich plates with FGM face sheets and homogeneous cores, and those with homogeneous face sheets and FGM cores—are considered, incorporating nonlinear temperature variation across the thickness for the vibration assessment. Harsha et al. [34] examine the thermoelectric vibration and buckling characteristics of a bidirectional functionally graded piezoelectric porous plate supported by an elastic foundation and subjected to general boundary conditions. Their work analyzes the dynamic behavior of functionally graded piezoelectric plates through a finite element framework and proposes an innovative multilayer configuration. The study evaluates the structural response under combined mechanical and electrical loading scenarios while considering variations in boundary constraints. Particular attention is given to how geometric configuration, gradation profiles, and power‑law indices influence system performance. The accuracy of the formulation is demonstrated through comparison with previously published results. The findings highlight the significant impact of material distribution, plate thickness, and aspect ratio on the dynamic response, offering valuable guidance for the design of structures used in aerospace systems, vibration mitigation, and energy‑harvesting technologies.

Novelty statement: This research develops a new multilayer model for functionally graded piezoelectric plates and explores their dynamic behavior under simultaneous mechanical and electrical excitations. By assessing the influence of structural configuration, boundary conditions, multiple piezoelectric layers, and power‑law gradation parameters, the study provides fresh insights that can enhance the design and optimization of FGPM‑based components for advanced engineering applications.

METHODS

Functionally graded piezoelectric plates

A variety of models have been presented in order to simulate the material distribution throughout the thickness of the plate. However, power law distribution is commonly used to simulate the material properties distribution. For FGM structures made of two different materials, material "a" in the top surface and material "b" in the bottom surface of the plate, the effective properties of the FGPM profile in the thickness direction of the plate can be written as [9]

where "Peff" is the effective material property of the FGPM plate, "Pa" is the upper surface property of the FGPM plate, "Pb" is the lower surface property of the FGPM plate, and "Va" is the volume fraction of the functionally graded piezoelectric upper material. Finally, the material properties can be expressed as:

where "z" shows the distance from the neutral axis of the plate and “h” represents the plate thickness. In this article, all of the material properties differ throughout the thickness by following the simple power law distribution (2.2) excluding Possion's ratio. The simple power-law model is widely adopted in functionally graded materials due to its mathematical simplicity and ease of implementation in finite element formulations. It offers flexibility by adjusting the exponent to represent different gradation profiles and ensures computational efficiency, making it a standard choice for analysis and comparison in FGM research.

Theoretical formulations

Displacements and Strain Relations

According to the FSDT, the displacement field in the plate can be expressed as [35]:

where the variables u, v, and w denote displacements along the x, y, and z axes, respectively. Also, u0, v0, and w0 denote the displacements of a point in the mid-plane, and ϕx and ϕy are the rotations of a transverse normal about the x-axis and y-axis, respectively. And strain field can be written as:

Note that the strains εx,εy, and γxy are varied linearly along the thickness while transverse shear strains γyz and γxz are constant along the thickness. Equation (3.3) can be written as:

where εi0and εsj0are in-plane and shear strains, respectively, and ki0is the mid-surface curvature. The electric field is defined as the following equation:

where Rφ is the electric field operator. In this article, it is assumed that the electric field has a constant value along the thickness [36].

Constitutive equations of piezoelectric material

Based on the linear theory of piezoelectricity, the constitutive equation for piezoelectric plate can be written as [22]:

where, σijand εkl are the stress and strain tensors respectively, CijklEis the elasticity matrix, eijk is the piezoelectric coefficient tensor, Ek is the electrical field vector Dl is the electrical displacement vector, ξlkis the electrical permittivity.

Final equations of motion

The equations of motion for the plate are obtained by applying the extended Hamilton’s principle [37]:

Where δ is the variational symbol, and T is the kinetic energy and H is the electromechanical energy of each layer in the sandwich, and W is the work done by externally applied loads, both mechanical and electrical. It should be noted that dissipative terms are included in the electromechanical energy of the core (Hv) via the complex modulus approach. The kinetic and electromechanical energies are defined as:

where Ω represents the volume domain of the layer, ρ is mass per unit volume of the material, σ,ε, E and D are the stress, strain, electric field and electric displacement vectors, respectively, and u̇ is the time derivative of the displacement field vector u={u,ν,w}T.

By integrating in thickness direction, and replacing the strain field and assuming that (Ex,Ey=0), (3.7) can be written as:

By combining (3.5) and (3.8) the final Equation of electromechanical energy can be written as follow:

Here O is the coupled tensile stiffness matrix, F demonstrate the coupled tensile-bending stiffness matrix and R is the dielectric tensile stiffness matrix that can be defined as follow:

The tensors in (3.10) are defined as:

Kinetic energy in (3.7) using integral by part finally can be written as:

Finite element model

Using (3.2) the strain relation can be written as:

where, [Lm], [Lb], [Ls] are strain operators based on the displacements-strain relations and can be defined as follows:

In (3.6) the work done by external mechanical force and applied electric charge are defined as [38]:

where FΩ, FS,FP are the body force, surface traction and concentrated force respectively.

The last integration refers to applied electrical charge. By combining (3.9), (3.11) and (3.6) and assume that only concentrated force be applied on the plate. (3.15) can be written in matrix form as follow:

Where submatrices M, u and φ are the mass element, mechanical degree of freedom and electrical potential degree of freedom respectively. Kuu, Kuφand Kφφindicate the elastic, piezoelectric, and permittivity stiffness matrices, Fu is the applied external force tensor.

The above matrices defined as follow:

Where the m matrix defined as [39]:

And I can be defined as:

Here ρ is the density.

Newmark method

In this article, the Newmark method is used to solve the final time dependent equation of motion. In this method, the acceleration is assumed to vary in a specific manner over each time step. The finite difference relationships for the Newmark-β method are [40]:

where ui+1,ui,u̇i,u̇i+1,üi and üi+1 are demonstrate the displacement at time step “i”, displacement at time step “i+1”, velocity at time step “i”, velocity at time step “i+1”, Acceleration at time step “i” and Acceleration at time step “i+1”, respectively. The parameters β and γ define the variation of acceleration over a time step and determine the stability and accuracy characteristics of the method. Typical selection for γ and β are γ= 0.5, and 1/6 ≤ β ≤ 1/4 and here Δt defined as:

For linear systems, it is possible to modify Newmark’s original formulations in matrix form as:

RESULTS AND DISCUSSION

Comparison studies

To support the findings of this study, two examples were used to compare the data with results from previous studies by Selim et al. [8] and Nguyen-Quang et al. [12].

Example 1 focuses on the dynamic response of a plate without the damping effect. The plate analyzed is a CFFF functionally graded material (FGM) plate composed of Aluminum Oxide and Ti–6Al–4V materials at room temperature (300 K), as described by Selim et al. [8]. Two G1195N piezoelectric layers are bonded to the top and bottom surfaces of the FGM plate. In this configuration, the actuator is located on the top surface, while the sensor is placed on the opposite bottom side. This configuration is shown in Fig. 1-a. Also Fig. 1-b describes the functionally graded piezoelectric plate that has other piezoelectric layers in top and bottom as sensor and actuator layers. The second configuration will be addressed in the next example. The plate has dimensions of 0.3 m in both length and width, with a thickness of 5 mm. Each piezoelectric layer has a thickness of 0.1 mm. The material properties are provided in Tables 1 and 2.

Initially, the plate is subjected to a uniformly distributed load of q=100 N/m² in the downward vertical direction. Subsequently, the load is removed, resulting in motion generated from the initial displacements. The tip deflection response of the plate is calculated with a time step of 0.001 s, and the system is solved using a state-space model. Figures 2 and 3 demonstrate a satisfactory agreement between the results obtained in this study and those reported by Selim et al. [8].

Fig. 1. a) FGM plate with Piezoelectric Layers b) FGPM plate with Piezoelectric Layer

Tab. 1

Material properties of G-1195N and PZT-4 [41]

Material	G-1195N	PZT-4
ε33 (F/m)	15e−9	1.15e−8
e32 (m/V)	254e−12	−1.22e−10
e31 (m/V)	254e−12	−1.22e−10
ρ (kg/m3)	7600	7600
ν	0.3	0.33
E22 (GPa)	63	81.3
E11 (GPa)	63	81.3

Tab. 2

Material properties of Aluminum Oxide and Ti-6Al-4V [41]

Material	Young’s moduli (GPa)	Poisson’s ratios	Density (kg/m3)
Ti-6Al-4V	105.7	0.29	4429
Aluminum oxide	320.2	0.26	3750
Aluminum	70	0.3	2707

Fig. 2. The tip deflection of the plate presented by Selim et al. [8]

Fig. 3. The tip deflection of plate in present study

Example 2 In this case, the dynamic response is compared with the results presented by Nguyen-Quang et al [12]. They considered the damping effect and modified (3.15) as follows:

In equation (4.1) the symbols M, C, K and F represent the mass matrix, structural damping matrix, stiffness matrix, and external force, respectively. Since the structural damping value is not available, its magnitude is estimated using the Rayleigh-Ritz method, which defines the stiffness matrix as:

The values of M and K have been defined earlier, α and β′ are the Rayleigh ratios, which are calculated as [35]:

where ω1 and ωm are the minimum and maximum value of natural frequency and ζ1 and ζm are the damping ratio for ω1 and ωm,respectively.

The FGPM plate with simply supported boundary conditions which is studied here consists of two materials surrounded by a piezoelectric layer Fig.1.(b). The plate is square-shaped, with each side measuring 200 millimeters. The thickness of the intermediate composite plate is one millimeter, and the thickness of the piezoelectric layer is 0.1 millimeters. The value of the power law constant, n, is assumed to be 2. Additionally, a distributed load q=100Nm2 is applied to the plate with the application of the load being sudden in nature. The dynamic response of the plate is obtained using the Newmark method with values of γ and β equal to 0.5 and 0.25, respectively. Moreover, the value of Δt is considered to be 0.0005. Fig.4 and Fig.5 in this paper compare the dynamic responses of the targeted plate with piezoelectric layers to the references [12]. As shown in this figure, the results obtained from this research exhibit good agreement with the reference results Nguyen-Quang [12].

Fig. 4. Dynamic response of FGPM plate presented by Nguyen-Quang et al [12]. Figure in colour only on the web site of the journal

Fig. 5. Dynamic response of FGPM plate present study. Figure in colour only on the web site of the journal

CONCLUSIONS

In this study, the dynamic response of functionally graded piezoelectric plates was investigated using the finite element method based on the first-order shear deformation theory. Through comprehensive analyses of various parameters and boundary conditions, several important conclusions were drawn.

This study emphasizes the importance of considering material composition, thickness, aspect ratio, and boundary conditions in the design and analysis of functionally graded piezoelectric plates. The findings can guide engineers and researchers in developing more efficient and reliable structures in areas such as aerospace, energy harvesting, and vibration control. Future research may focus on exploring additional parameters and advanced numerical techniques to further enhance the understanding and optimization of these materials. The outcomes of this study have direct relevance to the design of advanced smart structures, where functionally graded piezoelectric plates can be applied in aerospace components for vibration suppression, in energy harvesting devices for improved efficiency, and in precision sensors or actuators for biomedical and industrial applications. By clarifying how material gradation and electrical loading affect dynamic behavior, the findings provide practical guidance for engineers to optimize stiffness-to-weight ratios, enhance durability, and achieve reliable performance in real-world environments.