Multilevel Fully Integrated Electromechanical Property Modulation of Functionally Graded Graphene‐Reinforced Piezoelectric Actuators: Coupled Effect of Poling Orientation

This article explores the coupled static and dynamic electromechanical responses of single and multilayered functionally graded (FG) graphene platelet (GPL)‐reinforced piezoelectric composite (GRPC) plates by developing a 3D finite‐element model. The bending and eigenfrequency of piezoelectric FG composite plates are investigated, wherein an active behavior is proposed to be exploited in terms of the functional design of poling angle for a more elementary level property modulation. The numerical results reveal that the mechanical behavior concerning deflection and resonance frequency of FG‐GRPC plates can be significantly enhanced and modulated due to the influence of piezoelectricity and a small fraction of GPLs along with the consideration of poling angle in a multiscale fully integrated computational framework. The notions of on‐demand property modulation, actuation, and active control are established here by undertaking a comprehensive numerical analysis considering the coupled influences of poling orientations, different distributions, patterns, and weight fractions of GPLs along with different electromechanical loadings. Against the backdrop of the recent advances in microscale manufacturing, the current computational work will provide necessary physical insights in modeling piezoelectric multifunctional FG composites for active control of mechanical properties and harvesting electromechanical energy in a range of devices and systems.


Introduction
Functionally graded materials (FGMs) are a unique type of materials in which their composition, as well as material properties, are varied continuously and smoothly in one or DOI: 10.1002/adts.202200756 more directions. [1] FGMs and composite provide efficient ways to strengthen the existing structure and enhance its performance which leads to their wide application in the mechanical, aerospace, and defense industries. [2][3][4][5][6][7] FGMs have superior application-specific material properties such as the ability to withstand high temperatures, and high mechanical stresses while maintaining the weight as low as possible (depending on the material and scheme of the graded structure). The functionally graded (FG) laminate structure can be utilized as a smart structure such that when it senses the input stimuli or deformation it will actuate itself and subsequently return to its original shape. One approach to making a laminate structure to be smart is by integrating a piezoelectric layer with an FG layer. The electromechanical properties of a piezoelectric layer which acts as an actuator or sensor are used to govern the multiphysical structural response of a smart structure. [8] Different methods were developed in the past to evaluate the electromechanical performance of smart FG plates. For instance, Reddy and Cheng [9] developed a 3D solution for a coupled plate made of FG and piezoelectric material by using an asymptotic technique. Yang et al. [10] studied a semianalytical method based on the Galerkin approach to determine the large amplitude vibration of a prestressed rectangular FG piezoelectric plate under thermoelectromechanical loading conditions. Ray and Sachade [11] developed an exact solution for FG plate incorporated with piezoelectric fiber-reinforced composite (PFRC) considering simply supported boundary conditions under electromechanical loading for studying static analysis. Based on the state-space formulation, Bian et al. [12] developed an exact solution to investigate the effect of surface piezoelectric sensors and actuators on FG beams. Xiang and Shi [13] investigated the static analysis of FG piezoelectric sensors or actuators when subjected to electrothermal loading using the theory of piezoelasticity. By using the 3D element-free Galerkin method, Mikaeeli and Behjat [14] performed a static analysis of thick FG piezoelectric plates. Das and Nath [15] developed a coupled electromechanical zigzag theory for FG plates incorporated with piezoelectric layers for free and static vibrations. However, it is widely known that the analytical or exact solution for smart FG plates mentioned in the above literature is only valid for simple geometry as well as specific types of boundaries and loading conditions. Therefore, it is important to develop another method to explore the electromechanical response of these FG structures. The finite-element (FE) method is a widely used tool to solve complex geometry as well as general types of boundaries and loading conditions. For instance, Ray and Sachade [16] developed a FE model using first-order shear deformation theory (FSDT) for static analysis of FG plate attached with PFRC under electromechanical loading conditions. Su et al. [17] presented a FE approach for transient and free vibration analysis of FG piezoelectric plates using FSDT considering distinct boundary conditions.
The growing demand for high-performance actuators in engineering applications for smart structures has motivated researchers to enrich the electromechanical properties of piezoelectric materials. Dissimilar methods have been used by researchers [18,19] to enhance the performance of piezoelectric materials. For example, the concept of porosity in a piezoelectric material was introduced by Nomura et al. [20] Their experimental outcomes demonstrated that the porous PZT material increases the voltage about ≈5 times. Li et al. [21] examined the influence of volume fraction of porosity on elastic and piezoelectric properties of beam-shaped porous PZT actuator. Goldschmidtboeing and Woias [22] investigated that the triangular-shaped piezoelectric beams outperform rectangular-shaped piezoelectric beams in terms of maximum output power and tolerable excitation amplitude. Paquin and St-Amant [23] revealed that variable thickness beams enhance the electromechanical properties and performance of piezoelectric energy harvesting structures by ≈3.6 times. The experimental results of Zhang et al. [24] showed near about ≈35.3% improvement in longitudinal piezoelectric strain coefficient under alternating current (AC) polarization conditions instead of direct current (DC) polarization conditions. Naskar and Bhalla [25] reported experimental and computational assessment of optimized metal-wire based piezoelectric materials. Luo et al. [26] proposed a multiobjective topology optimization technique to improve the performance of piezoelectric actuators in the design domain. The topological layouts and placement of piezoelectric as well as non-piezoelectric materials were optimized so that they can have enough rigidity and sufficient flexibility. Uetsuji et al. [27] maximized the macrostructural piezoelectric response of polycrystalline piezoelectric ceramics using the multiscale FE method. In this, they showed that the 3D alternating structure and ordered layer structure are two distinct microstructures that improve piezoelectric properties. The layers were rotated by 120°in the ordered layer structure, increasing the longitudinal strain constant by 35% whereas adjacent grains were rotated by 180°in a 3D alternating structure to enhance the transverse strain constant by 284%. In the next paragraph, we have discussed the significance of the incorporation of graphene and its different derivatives into composite and FG material to enhance their overall performance.
In recent years, graphene and its different forms of derivatives have received considerable attention from researchers due to their superior mechanical, electrical, and thermal properties as well as their low mass density. [28][29][30] Several studies have been conducted to explore the mechanical properties of graphene-based composites. [31][32][33] Kitipornchai et al. [34] used a continuum model to analyze vibration with different mode shapes of multilayered graphene sheets. They showed that by varying the number of layers in a multilayered graphene sheet, different resonance modes can be determined. Alibeigloo [35] used nonlocal continuum mechanics to conduct a 3D analysis of the frequency of multilayered graphene sheets incorporated in a polymer matrix. Li et al. [36] investigated the wave propagation properties in FG graphene-reinforced piezoelectric plates via a semianalytical method. Wang et al. [37] studied the effect of graphene nanoplatelets (GNPs) size on the thermo-mechanical properties of epoxy composites. Their results indicated that a larger GNPs size improves the tensile modulus but decreases the strength of epoxy nanocomposites. In addition to this, the use of graphene reinforcement in a polyvinylidene fluoride (PVDF) matrix can enhance the mechanical, piezoelectric, and stiffness properties of composite structures. Chandra et al. [38] developed a multiscale FE method for analyzing the vibrational mode shapes of graphene-based composites. They demonstrated that the mode shapes depend on length (size of structures) as well as boundary conditions. Rafiee et al. [39] conducted an experimental analysis to determine the mechanical properties of epoxy composite with a 0.1% weight fraction of graphene platelets (GPLs). The Young's modulus, toughness, and tensile strength of the epoxy matrix with GPLs were increased by 31%, 53%, and 40%, respectively. Layek et al. [40] conducted experiments on graphene-reinforced PVDF matrix with 0.75% volume fraction of GPLs. They reported that GPLs in PVDF composite shows an enhancement of 321% in Young's modulus and 124% in storage modulus. Yang et al. [41] carried out a comprehensive parametric analysis for studying the buckling and postbuckling response of multilayered composite beams subjected to axial loading. The effects of different distribution patterns, boundary conditions, weight fraction of GPLs, and size of GPLs nanofillers were examined for buckling analysis. Mao and Zhang [42] studied the buckling behavior of a multilayered composite plate when subjected to axial forces and external electric potential. In this, they demonstrated that even small addition of GPLs significantly improved the buckling strength of the FG graphene reinforced composites (FG-GRCs) plate. Zhao et al. [43] investigated the thermal, electrical, and mechanical buckling performance of FG-GRCs cylindrical nanoshells. The torsional buckling of FG porous nanocomposite cylindrical shells incorporated with GPLs was examined by Ghahfarokhi et al. [44] Song et al. [45] examined the free and forced vibrations of simply supported FG multilayered graphene polymer plates under mechanical loading conditions. Yang et al. [46] developed analytical solutions using 3D elasticity theory for bending of FG graphene reinforced polymer composite annular and circular plates, wherein they considered continuously varying GPLs weight fraction along the thickness direction under thermo-mechanical loading. Their results reported that the GPLs nanofillers have a noteworthy effect on the bending behavior of these plates. Shen et al. [47,48] explored the nonlinear buckling and bending analysis of FG-GRC plates when subjected to thermomechanical loading conditions with substantial enhancement in their respective properties. Since it is not possible to provide an exhaustive literature review on this subject here, a detailed cluster plot on the literature of GPLs is shown in Figure 1 for the last twenty years. The cluster plot shows how researchers across the globe are devoting their scientific efforts to various aspects of GPLs, indicating the importance and impact of this topic.
From a careful review of the literature, it can be concluded that the studies related to graphene-based composites are mainly focused on mechanical and thermal loading for buckling and bending analysis. To the authors' best knowledge, the static bending and modal analysis of functionally graded graphene reinforced piezoelectric composite (FG-GRPC) actuators under electromechanical loading and effect of poling orientation of piezoelectric actuator for the smart composite plate is an unexplored area of research. A multiscale approach of including the effect of poling angle in the design space along with the coupled electromechanical behavior would essentially expand the flexibility and scope of designing advanced mechanical systems significantly. In the current study, the bending and eigenfrequency of the piezoelectric actuator integrated with an FG plate would be investigated in a seamless multilevel computational framework. Three different design spaces would be investigated in the present work to enhance the electromechanical properties of the piezoelectric actu-ator: i) multilayered graphene reinforced piezoelectric actuator, ii) single-layered piezoelectric actuator with different graphene distribution patterns, and iii) effect of poling orientation of piezoelectric actuators. A FE method based on FSDT would be developed in this work for achieving the aforementioned objectives, and the outcomes would be validated with existing results using Zigzag theory and the exact solution method. The micromechanics model such as the rule of mixture (ROM) and Halpin-Tsai (HT) method would be utilized to compute the effective electromechanical properties of FG-GRPCs. The effects of the GPLs volume fraction in piezoelectric plates would be demonstrated for different GPL distribution patterns (refer to Figures 2 and 3). In both multilayered-and single-layered graphene distribution, the variation of material properties would be considered along the plate thickness. Taking inspiration from the literature (Uetsuji et al., [27] Kiran et al. [49] and Sharma et al. [50] ) about poling orientation, we would study this effect in case of graphene-based functionally graded materials. For instance, Kiran et al. [49] showed the poling tuning phenomenon, which increases the magnitude of the piezoelectric coefficient for polycrystalline ceramics. In this phenomenon, dipoles of piezoelectric materials are inclined at a specific angle for sensing and actuation applications. In order to achieve the dipole inclination at a specific angle during poling for polycrystalline ceramics, Sharma et al. [50] manipulate the placement of electrodes and performed FE simulation for three different configurations. In addition to functional gradation, the concept of poling orientation would be exploited in the current study to enhance the electromechanical properties of piezoelectric actuators further. The poling of the piezoelectric actuator would be implemented at a critical angle along the thickness direction, due to which, there could be substantial enhancement and modulation in electromechanical properties. In summary, this paper aims to investigate the multiscale expanded design space of different volume fractions and functional distributions, layers and thickness, and poling angles to enable the optimum design of multifunctional micro-and nano-electromechanical systems (M-/NEMS) including sensors, actuators, nanogenera-tors, and other structural control devices. A systematic representation of the complete analysis of the present work is demonstrated in Figure 2 for giving an introductory overview to the readers.

Theoretical Formulations
We emphasize on the word "multiscale" in the current paper since the study includes the properties at multiple length scales involving microscale parameters to lamina-level design space and the laminate-level effective electromechanical behavior. The notions of on-demand property modulation, actuation and active control are established here by undertaking a comprehensive numerical analysis considering the coupled influences of poling orientations, different distributions, patterns, and weight fractions of GPLs along with different electromechanical loadings. We essentially try to integrate such parameters and features across different length scales through efficient computational schemes such as homogenization and finite-element analysis. In the present study, a simply supported smart composite plate is considered for the finite-element-based numerical analyses. This smart plate is further categorized into two types: i) multilayered functionally graded graphene-reinforced piezoelectric composite (FG-GRPC) plate and ii) single-layered FG-GRPC plate with different distribution patterns as illustrated in Figure 3.

Constitutive Equations
In this section, the static and dynamic analysis of smart FGM plates including deflection, stress, and modal analysis are ex-plained using finite-element (FE) modeling. The generalized constitutive relations for piezoelectric material can be expressed as [51] where , S, E f and D are stress, strain, electric field, and electrical displacement vector, respectively. C, e and denote Young's modulus, piezoelectric stress coefficient, and electrical permittivity matrix. The first-order shear deformation theory (FSDT) is used to define the displacements u, v and w at any point in a plate along x, y and z axes, respectively [16] u(x, y, z) = u 0 (x, y) www.advancedsciencenews.com where u 0 , v 0 and w 0 are the generalized displacements at the midplane; x and y are bending rotations about the y-and x-axes, respectively. The equation of motion for an element can be obtained by applying Hamilton's principle wherein V P ,Tand W denote the potential energy, kinetic energy, and external work done; is a variational operator. After simplification, the coupled electromechanical FE equations for the piezoelectric laminated structure are given by [52] [ In the following subsections, we provide a brief overview of the determination of effective properties which are used in the FE analysis discussed above.

Determination of Effective Material Properties
In this section, the required effective properties of smart FGM and piezoelectric composite are determined using micromechanical models such as rules of the mixture (ROM) and Halpin-Tsai (HT) models.

Multilayered Functionally Graded Graphene Reinforced Piezoelectric Composite Plate
The perfectly bonded multilayered functionally graded graphene reinforced piezoelectric composite (FG-GRPC) plate with equal thickness is considered for the present study. Uniformly distributed graphene platelets (GPLs) nanofillers are used in each layer of the FG-GRPC plate. To make FG-GRPC plate structure symmetric, even number of graphene reinforced piezoelectric layers with different distribution of GPLs in each layer, namely, U, O, and X patterns are considered as illustrated in Figure 3. In the U pattern, the volume fraction of GPLs remains the same in every layer whereas, in the O and X pattern, the GPLs volume fraction changes linearly along the thickness direction. In the X pattern, the top and bottom layers exhibit the highest percentage of GPLs and the middle layer exhibits a lower percentage of GPLs. In the O pattern, the topmost and bottommost layers exhibit the lowest percentage of GPLs while the middle layer exhibits a higher percentage of GPLs (exactly opposite to the X pattern) as illustrated in Figure 3. The coordinate system of the multilayered FG-GRPC plate is considered at the middle and the polarization direction of the piezoelectric material is along the z-axis (positive direction). The volume fraction of GPL distribution for different patterns in each layer (j = 1, 2, …, N − 1, N) are given by U pattern: where Here, v gpl is the volume fraction of GPLs in the whole plate, and v j is the volume fraction of GPLs in each layer of the FG-GRPC plate.

Single-Layered FG-GRPC Plate
We have considered uniform or nonuniform distribution of GPLs within the piezoelectric matrix to form an FG-GRPC plate in such a way that the material properties are the function of its thickness.
In the present study, three different distribution GPLs are considered such as uniform, linear, and parabolic distribution (referred as UD, LD, and PD, respectively) as illustrated in Figure 3, and their respective expressions for determining volume fraction are given in equations (15)(16)(17). In the case of UD, the value of v j remains constant throughout the plate thickness. In case of LD, the value of v j is zero at the bottommost surface and maximum at the topmost surface, whereas, in case of PD, the value of v j is zero at mid-plane and maximum at the top and bottom surface.
Linear distribution: Parabolic distribution: The parameter v gpl is the characteristic volume fraction of GPLs in the plate, and v * gpl is the total volume fraction of GPLs in the plate.

Effective Material Properties
To compute the effective properties of GRPC such as Young's modulus and other multiphysical properties, different micromechanical models are proposed in the literature namely: rulesof-mixture (ROM), modified rules-of-mixture (mROM), Halpin-Tsai (HT), and Mori-Tanaka (MT) models. Shokrieh et al. [53] compared the values of effective tensile modulus computed using MT and HT models with experimental results. The experimental results demonstrated that the HT model was more precise and consistent than the MT model. Moreover, using numerical and experimental study, Layek et al. [40] reported that the HT model was more precise to calculate the value of effective Young's modulus for graphene reinforced PVDF matrix when v j < 1%. Therefore, in the current study, we have adopted the HT model to evaluate the performance of the GRPC layer which can act as an actuator for an FG plate. Due to the risk of forming agglomeration in a GRPC plate, the value of v j is kept below 1%. Hence, the value of effective Young's modulus (E j ) is given by Here, E p and E G denote respective Young's modulus of piezoelectric material and GPLs. l gpl and t gpl represent the length and thickness of the rectangular GPLs. Note that P and G subscript denote the piezoelectric and graphene material properties, respectively. The value of effective Poisson's ratio ( j ), density ( j ), piezoelectric coefficient (e lm,j ), and electrical permittivity ( lm,j ) are calculated using ROM and expressed as follows The ROM micromechanical model is adopted here because it is one of the simplest models with the lowest computational cost as well as widely used in literature to predict the effective property such as piezoelectric coefficient and electric permittivity. [36,43,45,54,55] In this, we considered different assumptions, i.e., there is no slippage or perfect bond between the graphene platelets and PVDF matrix and it is considered that matrix is free from the voids. Therefore, one can use ROM for nanoscale fibers or fillers having higher material properties at low volume fraction which is to be introduced in low material properties matrix by considering the above assumptions. It can be noted that several experimental studies used ROM for calculating the effective properties of graphene-reinforced composites. [56,57]

Poling Orientation
The working principle of most piezoelectric materials primarily depends upon three electromechanical coupling modes known as transverse(d 31 ), longitudinal (d 33 ), and shear (d 15 ) modes. In transverse mode, a mechanical force is applied to the piezoelectric material in 1 st direction due to which an electric field is generated along 3 rd direction as demonstrated in Figure 4a. The transverse mode (d 31 )is most commonly used in the bending analysis of piezoelectric plates. In longitudinal mode (d 33 ), a mechanical force is applied in the thickness (3 rd ) direction and an electric field is also generated in the thickness direction as illustrated in Figure 4b. In shear mode (d 15 ), the shear strain is applied on 1-3 plane of a piezoelectric material which causes an electric field in the 1 st direction as shown in Figure 4c. The magnitude of the shear piezoelectric constant is much higher as compared to longitudinal and transverse piezoelectric coefficients. However, the shear mode (d 15 ) is not recommended for piezoelectric applications due to its complex manufacturing and poling process.
In the present study, the transverse mode (d 31 ) with poling orientation as an input is proposed to enhance the performance of the piezoelectric actuator. In conventional poling of piezoelectric materials, the electric field is induced along the 3 rd direction due to the activation of d 31 mode as illustrated in Figure 4d. In poling orientation phenomena, the poling of the piezoelectric material is performed at a particular angle with respect to the 3 rd direction, as a result of which an electric field is induced along 3 rd direction due to the activation of all the three modes, i.e., d 31 , d 33 and d 15 as illustrated in Figure 4e. To determine the overall effect of poling angle on an FG smart plate, there is a need to investigate how the piezoelectric strain coefficient (d) changes concerning the poling angle. The value of polarization vector in an old coordinate system (conventional poling) is the product of stress tensor { } and piezoelectric strain coefficient matrix [d] where {P} is transformed polarization vector in a new coordinate system and [r] is the direction cosine matrix in poling orientation phenomena. The direction cosine matrix and piezoelectric coefficient matrix for tetragonal symmetry are expressed as [r] = Adv. Theory Simul. 2023, 6, 2200756 where The direction cosine matrix [ ] is used to transform the stress tensor from the old coordinate system to the new coordinate system. Therefore, the transformed stress matrix { ′ } is given by By substituting Equation (33) into Equation (26) and comparing the value of the polarization vector and transformed polarization vector from Equations (35) and (24) [ The effective piezoelectric coefficient after poling orientation phenomena is given by

Convergence Analysis
In the present study, the FE method is used to obtain the electromechanical response of a simply supported FG plate. Before analyzing the numerical findings of the smart FGM system, convergence and validation analysis are needed to be performed to check the effect of mesh size and reliability of the results. The FE modeling presented in this paper is classified into three phases: preprocessing, solver, and postprocessing (COM-SOL Multiphysics). In the preprocessing phase, a suitable physics and study module is chosen for various modeling and simulation studies. The piezoelectric physics module which combines electrostatics and solid mechanics, as well as the stationary and eigenfrequency study are used here. Then, the geometric and material properties of the FGM and piezoelectric layer are provided as input. After assigning material properties, the simply supported boundary conditions (unless otherwise mentioned) and electromechanical loading conditions are applied on the smart FGM plate. The discretization or meshing of the composite plate is carried out using the "free tetrahedral" element. The optimum mesh is determined by changing mesh size ranges from extra coarse to extra fine for calculating the error percentage or convergence analysis of deflection under mechanical as well as electromechanical loading conditions as illustrated in Table 1 and Figure 5a. Because of the very high accuracy of the results, which in turn gives validation of the finite-element model, fine meshing with the number of elements (122 531) and the number of edge elements (676) is chosen. In the solver phase, a set of electromechanical equations is solved, which creates the nodal solutions of the simply supported plate. After obtaining nodal solutions, the postprocessor enables us to study FE results such as mode shape, stress, and displacement.

Verification of Finite-Element Model
In this subsection, we have considered a standard benchmark problem to evaluate the performance of a piezoelectric layer that acts as an actuator for FG plate by using the finite-element ap-  proach. For this, a composite plate made of an FG material having 3 mm thickness (h), attached with a thin plate of PFRC with 250 μm thickness is taken. The length (l) and breadth (b) of both FG and PFRC squared plates are considered as 30 mm. The PFRC is made of PZT-5H and epoxy fiber. The PFRC layer is po-larized in this study along the z-axis; reversing the polarization direction reverses the direction of actuation phenomena. This essentially means that depending on the polarization direction, the static and dynamic behavior of the system presented here can be modulated. Figure 5c illustrates that the value of Young's modu- where E 0 is Young's modulus of FG plate at bottom of the surface, is the inhomogeneity parameter, and z is the function of thickness for the FG plate. The Young's modulus of FG plate at the bottom surface is 300 GPa. The electromechanical properties of the orthotropic PFRC layer are summarized in Table 2. [11] The same electromechanical properties of the PFRC material have been considered to validate the results as in the literature, where they have taken e 31 mode and other stress coefficients as zero. From this validation study, it would be observed in the following results that our FE model is in good agreement with the results reported in the existing literature. Thereafter, we have implemented the same FE modeling approach for determining further results in the manuscript, while the material and other parameters may change in the results section for a comprehensive investigation. We have also checked our results using stress-and strain-charge forms, and the results are found to be in good agreement. In this context, it can be noted that the studies reported in the literature indicate that stress charge form can be converted to strain charge form for the same piezoelectric materials and the results will remain unchanged. [58,59] For mechanical validation, a sinusoidal varying mechanical force (F 0 = 40 N m −2 ) in the downward direction is applied on the top surface of a piezoelectric layer. For electromechanical validation, a sinusoidal varying mechanical force (F 0 = 40 N m −2 , downward), as well as an electrical voltage, (V 0 = 100 V) are applied on the top surface of a piezoelectric layer. For both cases, the ratio of Young's modulus from top to bottom surface is 0.1 and length to thickness ratio (s) of 10 is considered. The numerical results are evaluated by considering simply supported boundary conditions and function of mechanical and electric potential force (as in Equations (41) and (42)) as To present the stress and displacement results for the smart FG plate, nondimensional parameters are used as ( x , y , xy The results obtained for smart FG piezoelectric plate under mechanical and electrical loading are compared with existing results, as summarized in Tables 3 and 4. The estimated results are found in excellent agreement with Zigzag theory and exact method. Therefore, based on the results presented in Tables 1, 3, and 4, one can adopt this FE procedure to perform further electromechanical analyses by varying different parameters.

Effective Properties of Graphene Reinforced Piezoelectric Composite (GRPC) Plates
This section investigates the actuator performance of graphene reinforced piezoelectric composite (GRPC) plates subjected to electromechanical loading. This GRPC plate is made of GPLs nanofillers and PVDF piezoelectric matrix and their electromechanical properties are enlisted in Table 5. [42] The HT and ROM are implemented to calculate the effective electromechanical properties of the composite. The volume fraction of GPLs is kept below or equal to 1% due to the risk of forming agglomeration in view of manufacturability. The length (l gpl ), width (b gpl ), and thickness (t gpl ) of rectangular GPLs used in the analysis are 2.5 m, 1.5 m, and 1.5 nm. The effective properties of elastic modulus, density, piezoelectric coefficient, and electrical permittivity of GRPC plate against the GPLs volume fraction (V gpl ) are illustrated in Figure 6a-d, wherein it can be observed that as the value of V gpl increases, the electromechanical properties of GRPC plates improve significantly. A linear variation is observed in effective properties and they follow iso-field conditions (i.e.,the combination of iso-stress and iso-strain conditions). The main reason behind the improvement of electromechanical properties is the high electromechanical properties of GPLs nanofillers. Note that the effective properties discussed in this subsection would be central to investigating further static and dynamic responses of the smart composite plates, as presented in the following subsections.

Static Behavior of Multilayered FG-GRPC Plates
In this subsection, a simply supported multilayered smart FG-GRPC plate is considered where GRPC plate is integrated with FG plate. Three different types of layered patterns, i.e., U, O, and Table 3. Deflection and stress analysis of the smart FG plate subjected to mechanical loading.
Zigzag theory [15] Exact method [11] Present study  X patterns are investigated considering the same volume fraction of GPLs. The results obtained from these patterns lead to critical insights that can be utilized in different micro-and nanoelectromechanical systems (M-/NEMS). In these figures, we have illustrated nondimensional deflections in a) Z-direction Z(w) and b) X-direction X(u) as well as stresses in c) X-direction ( x ), d) Ydirection ( y ), and e) shear stresses ( xy ) in the X-Y plane. The FG-GRPC plate with the X pattern exhibits the highest deflection and stress, whereas the O-pattern plate exhibits the lowest deflection and stress as shown in Figure 7. It is due to the fact that, in the case of X pattern, a layer with a high value of V gpl is lying at the top and bottom surface. As a result, the upper and bottom layer has high magnitude of the transverse piezoelectric stress coefficients and elastic modulus. The high magnitude of the piezoelectric stress coefficient is primarily responsible for the high value of deflection, while the high value of the elastic modulus is primarily responsible for the low value of deflection. In electromechanical analysis, piezoelectric stress coefficient is the predominant factor over elastic modulus. As a result, there is a high magnitude of nondimensional deflection and stress in the X pattern. Whereas in the O pattern, a layer with a small value of V gpl is lying at the upper and bottom surface. Therefore, it results in a lower value of the transverse piezoelectric stress coefficient and elastic modulus of the upper and bottom layer of the O pattern as compared to its middle layers. Thus, the predominant factor, i.e., piezoelectric stress coefficient will cause less magnitude of deflection and stresses. In the U pattern, every layer has the same GPLs volume fraction. Hence, the magnitudes of deflections and stresses are found in between X-and O-patterns. Note that the deflections and stresses are measured at the midsection of the plate (critical) and the material distributions away from the neutral axis play more crucial roles in the analysis.

Static Behavior of Single-Layered FG-GRPC Plates
In this subsection, we discuss the results related to a singlelayered piezoelectric plate with different distribution patterns of GPLs which can act as an actuator for the FG plate. We have considered uniform, linear, and parabolic distribution (referred to as UD, LD, and PD, respectively). The composite plate with PD distribution has the highest magnitude of nondimensional deflections and stresses followed by UD and LD as shown in Figure 8.
In the case of PD, more electro-active matter lies near the top and bottom surfaces. As a result, in case of PD, the magnitude of elastic modulus and piezoelectric stress coefficient is much higher, resulting in a larger magnitude of deflections due to a similar coupled effect dominated by piezoelectric stress coefficient, as discussed in the preceding section. However, in case of LD, the value of V gpl is zero at the bottom surface and maximum at the top surface. As a result, the magnitudes of elastic modulus and piezoelectric stress coefficient are much lower, leading to a Adv. Theory Simul. 2023, 6, 2200756 www.advancedsciencenews.com www.advtheorysimul.com lower magnitude of deflections and stresses. Following a similar logic, the UD case shows results in between LD and PD.

Effect of Poling Orientation on the Static Response of Smart FG Plates
In poling orientation analysis, the poling of piezoelectric material is investigated at a particular angle along the thickness direction, due to which all the three coupling modes, i.e., d 31 ,d 33 and d 15 activate simultaneously and start contributing in the effective d 31 (d eff 31 ). A piezoelectric layer of PZT-7A material on the uppermost surface of the smart FG plate is considered for analyzing the effect of poling orientation on the smart FG plate. The important governing parameter for the actuation phenomena of a smart FG plate is the effective transverse piezoelectric constant. The value of the effective coefficient (d eff 31 ) is the function of poling angle as well as piezoelectric coefficients including longitudinal (d 33 ), transverse (d 31 ), and shear (d 15 ) effects. As the poling angle changes from 0°to 90°, the individual contributions of these effects are shown in Figure 9. Due to a compound effect, the magnitude of d eff 31 increases with poling angle up to a critical value and changes the trend thereafter, as shown in Figure 9. The poling angle that corresponds to the maximum magnitude of d eff 31 is known as the optimum poling angle ( optimum ). The magnitude Table 6. Piezoelectric properties of PZT-7A (pm V −1 ) and variation of d eff 31 with change in poling angle ( optimum = degree°). of d eff 31 continues to increase until it reaches 47°and at 47°, the value of d eff 31 reaches its maximum magnitude. From Figure 9, it is observed that the PZT-7A material shows a substantial enhancement of 60.78% for d eff 31 at an optimum poling angle 47°. The material properties and the calculation of d eff 31 coefficient with change in poling angle are described in Table 6.
In this subsection, a smart FG plate is analyzed parametrically with respect to poling angle for displacement and stress. The electromechanical loading is applied on the topmost surface of the piezoelectric layer and their effect is calculated on the smart FG plate. A strain is developed in the piezoelectric layer when an electrical voltage is applied on the topmost surface of the piezoelectric material. The strain produced in the piezoelectric layer is directly proportional to the piezoelectric constant and electric voltage. In poling orientation phenomena, the poling angle changes the piezoelectric coefficients, and thus the strain generated in the piezoelectric layer also changes. The magnitude of d eff 31 coefficient continues to increase up to 47°, resulting in an increased magnitude of nondimensional deflection and stress up to 47°poling angle. For better actuation in different applications, the maximum magnitude of nondimensional deflection is found at 47°poling angle in a thickness direction compared to the longitudinal direction as shown in Figures 10a,b. At an optimum poling angle of 47°, an improvement of 60.78% in the actuation performance is obtained. A higher magnitude of normal stress and shear stress is generated at an optimum poling angle as illustrated in Figure 10c-e. It can therefore be noted that poling angle can play a vital role in achieving desired electromechanical response of piezoelectric structures such as actuators, sensors, or nanogenerators.

Eigenfrequency Study of Multilayer and Single-Layer FG-GRPC Plates
The mode shapes and corresponding natural frequencies of different distribution patterns of FG-GRPC plates (multilayer and single-layer) are investigated in this subsection. Since we are considering free vibration analysis, the input voltage and force are considered zero. As the plate is a continuous system, in principle, it has an infinite number of eigenfrequency. Here, only the first few modes are discussed (refer to Figure 11) as the value eigenfrequency of these modes is most dominant in most of the practical applications. [60] Considering the same volume fraction of the GPLs, the first four mode shapes and corresponding natu- ral frequencies of different types of single-layered and multilayered FG-GRC with different distribution and patterns of GPLs are compared for conventional poling orientation, as illustrated in Figures 12 and 13. The composite plates with X pattern and PD distribution exhibit the highest natural frequency compared to the other distributions in respective plate configurations. As the magnitude of V gpl increases, the natural frequencies also increase. It can be noted that a coupled effect of stiffness and mass distributions along with the piezoelectric contributions results in such outcomes in the natural frequencies.

Effect of Poling Orientation on the Free Vibration Characteristics of Smart FG Plates
Similar to the case of static responses, the free vibration behavior of these active plates can further be modulated and enhanced for various applications based on consideration of the poling angle.
In poling orientation, the elastic modulus of a piezoelectric layer remains constant, and the magnitude of the piezoelectric strain coefficient changes with a change in poling angle. The piezoelectric strain coefficient continues to decrease until it reaches 47°A dv. Theory Simul. 2023, 6, 2200756 www.advancedsciencenews.com www.advtheorysimul.com and at 47°, the value reaches its minimum point. The natural frequencies of a smart composite plate increase as the poling angle is increased up to 47°, and reaches the maximum value at 47°, after which these start decreasing as illustrated (refer to Figure 14). Note that the capability of maximizing and modulating natural frequencies in an expanded design space beyond the consideration of conventional parameters would critically improve the multifunctional performances of a range of devices and systems.

Physical Realization of Poling Orientation Control
For achieving poling orientation phenomena for a piezoelectric layer experimentally, different electrostatic simulations using FE modeling are performed further. Conventional poling is used to align the randomly oriented dipoles in order to achieve a significant magnitude of net polarization. In conventional poling, two electrodes are placed parallel to each other on the top and bottom surfaces of a piezoelectric layer. The top surface of the piezoelectric layer is connected to the ground electrode, while the bottom surface is connected to the voltage electrode which is exposed to a voltage greater than its coercive field as illustrated in Figure 15a. The applied electric field forces the dipoles to be oriented in the same direction as the applied field, i.e., from higher potential to lower potential. The color contour represents the magnitude of the electric field, while the direction of the arrow represents the direction of the electric field as illustrated in Figure 15b.
It has been observed that the orientation of dipoles is influenced by the placement of electrodes. Two different types of electrode configuration, i.e., rectangular and L-shaped electrodes are used at diagonally opposite corners of the piezoelectric layer to achieve different poling orientations as illustrated in Figure 15cf. It can be observed that as the shape of the electrode varies from rectangular to L-shaped, the value of the poling angle changes. To control the value of a poling angle more accurately, the length and shape of the electrode can be varied through a robust optimization framework.

Concluding Remarks and Perspective
The work presented in this paper deals with the enhancement of electromechanical properties of the piezoelectric actuator integrated with a functionally graded (FG) plate for bending and eigenfrequency analysis. A seamless multilevel computational framework has been proposed for modulating static and dynamic features of the microplate-based actuator, wherein the elementary level effect of poling angle is coupled with the distribution of graphene platelets (GPLs) and other conventional design parameters for enhancing the electromechanical performance significantly. Subsequently, it is shown how different poling angles can be achieved physically by appropriate positioning of the electrodes. Thus, for significantly enhancing the electromechanical performances, we try to integrate and exploit the design possibilities at multiple levels in the following bottom-up order: 1) placement of electrodes, 2) poling angle, 3) distribution and weight fraction of GPLs, and 4) plate level geometries and electromechanical loading.
In the present study, two different plate structures are considered for the finite-element-based numerical analyses: i) single-layered GPL distribution patterns and ii) multilayered GPLs reinforced piezoelectric actuator. The finite-element (FE) model presented in this paper is based on first-order shear deformation theory (FSDT), which is extensively validated with existing results determined from Zigzag theory and exact solution under electromechanical loading conditions. To compute the effective properties of GPLs reinforced piezoelectric plate, the rule of mixture (ROM) and Halpin-Tsai (HT) models are used, which are subsequently coupled with the effective piezoelectric properties for enhancing the electromechanical behavior of the composite actuators. The major contributions and outcomes delineated from the present work are summarized below: • The effective properties including elastic modulus, piezoelectric coefficient, and electrical permittivity increase against the volume fraction of GPLs. It is due to the high magnitude of electromechanical properties of GPLs compared to the surrounding matrix. • In bending and eigenfrequency analysis, the single-layered piezoelectric actuators with different GPL distribution patterns, such as uniform, linear and parabolic distribution (i.e., UD, LD, and PD), are considered for the same volume fraction of GPLs. In all cases, the piezoelectric plate with PD exhibits the stiffest elastic behavior and higher natural frequency compared to the other distributions (UD and LD). • In multilayered FG-GRPC, three different types of layered patterns, i.e., U, O, and X patterns are considered for the analysis.
Since, the top and bottom layers contain a high-volume fraction of GPLs, therefore, X pattern shows higher value of deflection and natural frequency compared to U and O patterns. • We have adopted the mathematical equations for the poling orientation to enhance the piezoelectric actuator performance. In this method, poling is carried out at an angle along with the    thickness of the piezoelectric plate which improves the magnitude of the transverse piezoelectric coefficient. • At a poling angle of 47°, PZT-7A material shows the highest magnitude of the transverse piezoelectric coefficient. The numerical results reveal that the poling orientation phenomenon can enhance the deflection-based performance of a piezoelectric actuator by ≈60.78%. • In different poling orientations, the elastic modulus of the piezoelectric layer remains constant and piezoelectric strain coefficient changes with a change in poling angle. As a result, the natural frequency of a smart composite plate increases as the poling angle increases up to 47°where it reaches the maximum value, and then decreases significantly. However, the effect of poling angle on natural frequency is much lesser compared to deflection. • For achieving dipoles or poling orientation for a piezoelectric layer in practical applications, the length and shape of the electrode can be varied, which is demonstrated as an integral part of this computational study.
In summary, the bending and eigenfrequency of piezoelectric FG composite plates are investigated in this article, wherein an active behavior has been proposed to be exploited in terms of the functional design of poling angle for a more elementary level property modulation coupled with the gradation schemes and weight fractions of GPLs. The numerical results reveal that the mechanical behavior in terms of deflection and resonance frequency of FG-GRPC plates can be significantly enhanced and modulated due to the influence of piezoelectricity and a small quantity of GPLs along with the consideration of poling angle in a multiscale fully integrated computational framework. With the recent advances in experimental capabilities and microscale manufacturing techniques, this article will offer the necessary physical insights in modeling the electromechanical coupling in multifunctional piezoelectric materials, systems, and devices for prospective applications in sensors, actuators, nanogenerators, active controllers, robotics, and energy harvesters.

Data availability statement
The data that support the findings of this study are available from the corresponding author upon reasonable request.