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Flexoelectricity and Surface Effects on Coupled Electromechanical Responses of Graphene Reinforced

Functionally Graded Nanocomposites: A unified size-dependent semi-analytical framework

S. Naskar a,*, 1, K. B. Shingareb,*, S. Mondalc,*, T. Mukhopadhyayd,*

aFaculty of Engineering and Physical Sciences, University of Southampton, Southampton, UK

bDepartment of Aerospace Engineering, Indian Institute of Technology Bombay, Bombay, India

cDepartment of Mechanical Engineering, National Institute of Technology Durgapur, Durgapur, India

dDepartment of Aerospace Engineering, Indian Institute of Technology Kanpur, Kanpur, India

1Email Address: S.Naskar@soton.ac.uk

Abstract

Owing to inhomogeneous strain and high surface-to-volume ratio in nanostructures, it is imperative to account

for the flexoelectricity as well as surface effect while analyzing the size-dependent electromechanical

responses of nano-scale piezoelectric materials. In this article, a semi-analytical ‘single-term extended

Kantorovich method (EKM)’ and ‘Ritz method’ based powerful framework is developed for investigating the

static and dynamic electromechanical responses of graphene reinforced piezoelectric functionally graded (FG)

nanocomposite plates, respectively. The residual surface stresses, elastic and piezoelectric surface modulus,

and direct flexoelectric effects are taken into account while developing the unified governing equations and

boundary conditions. The modified Halpin Tsai model and rules of mixture are implemented to predict the

effective bulk properties. Our results reveal that the static deflection and resonance frequency of the proposed

FG nanoplates are significantly influenced due to the consideration of flexoelectricity and surface effects.

While such outcomes emphasize the fact that such effects cannot be ignored, these also opens up the notion of

on-demand property modulation and active control. The effects are more apparent for nanoplates of lesser

thickness, but they diminish as plate thickness increases, leading to the realization and quantification of a size-

dependent behavior. Based on the developed unified formulation, a comprehensive numerical investigation is

further carried out to characterize the electromechanical responses of nanoplates considering different critical

parameters such as plate thicknesses, aspect ratios, flexoelectric coefficients, piezoelectric multiples,

distribution, and weight fraction of graphene platelets along with different boundary conditions. With the recent

advances in nano-scale manufacturing, the current work will provide the necessary physical insights in

modeling size-dependent multifunctional systems for active control of mechanical properties and harvesting

electromechanical energy.

Keywords: Flexoelectricity and surface effect; Size-dependence in composite materials; Graphene reinforced

functionally graded materials; Extended Kantorovich method; Ritz method; Electromechanical responses.

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1. Introduction

In recent decades, structures made of smart materials (piezoelectric) such as beams, wires, plates,

membranes and shells have intrigued the researchers’ interest in developing the micro-/nano-electromechanical

systems (M-/NEMS) for structural control and health monitoring applications such as smart actuators, sensors,

capacitor, generator and distributors with capabilities of self-monitoring and -controlling (Beeby et al., 2006;

Reddy, 1999; Song et al., 2006; Trindade and Benjeddou, 2009; Wang et al., 2006; Deng et al., 2014; Ghasemi

et al., 2018). In 1880, French scientists Jacques and Pierre Curie observed piezoelectricity effects for the first

time (Curie and Curie, 1880). Later, Gabriel Lippmann deduced mathematical relations for the converse

piezoelectric effect from the fundamental thermodynamic principles (Lippmann, 1881), which was not

estimated by Curie brothers. The piezoelectric phenomena are well-known for generating electrical response

when subjected to uniform mechanical deformation, known as the direct piezoelectric effect. The reverse is

also true when the electrical field is applied, known as the converse piezoelectric effect (electromechanical

coupling in non-centrosymmetric crystals). The inversion centre is not present in non-centrosymmetric

crystalline materials, which results in the generation of polarization when it is exposed to mechanical load. In

elementary structures such as beams, wires, plates, membranes and shells, piezoelectric materials can be

employed as a viable option for the application of distributed sensors and actuators due to the presence of

unique electro-mechanical couplings as discussed above. For instance, both the static as well as dynamic

response of functionally graded (FG) piezoelectric bimorph and sandwich composite beam structures are

extensively studied using the commonly available piezoelectric materials such as polyvinylidene fluoride

(PVDF) and Lead zirconate titanate (PZT-5H) (Beheshti-Aval and Lezgy-Nazargah, 2010; Komijani et al.,

2014; Lezgy-Nazargah et al., 2013; Reddy and Cheng, 2001; Vidal and Polit, 2008). PVDF is a ferroelectric

polymer that exhibits tailorable piezoelectric, dielectric, ferroelectric properties as well (Dang et al., 2003).

Recently, a novel class of active metamaterials has been developed by exploiting the electromechanical

coupling of piezoelectric materials (Singh et al., 2021). The FG hybrid composite shell with carbon nanotubes

(CNTs) as reinforcement element was investigated by Thomas and Roy (2017) using the Rayleigh damping

model. They reported that after the incorporation of CNTs, the frequency response of composite shells showed

enhanced magnitude because of enriched stiffness and damping performance which results in a decrease in its

amplitude. Abolhasani et al., (2017) first prepared graphene reinforced PVDF nanocomposite and

experimentally investigated its crystallinity, polymorphism, morphology, and electrical outputs. Since 2017,

the pioneering works on the emerging area of FG graphene-based composites and their structures such as

beams, plates, arches, and shells are being carried out by several researchers (Naskar, 2018a; Naskar et al.,

2019, 2018b). For instance, Feng et al. (2017) studied the nonlinear bending behavior of a novel class of multi-

layered FG graphene platelets (GPLs)-based composite nanobeams with non-uniform distribution of GPLs

along thickness direction. They found the most effective technique to decrease the deflections of beams by

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incorporating more GPLs in square shape with smaller amount of single graphene to its upper and bottom

surfaces. Yang et al. (2018) investigated the free vibration and buckling response of FG GPLs-reinforced

porous composite nanoplates based on Chebyshev-Ritz method and first-order shear deformation theory

(FSDT). To attain enhanced vibration and buckling response of nanocomposite plates, a comprehensive

parametric analysis was performed by considering different weight fraction, geometric parameters of GPLs

nanofillers and the porosity coefficient. Zhao et al. (2020) systematically presented a brief review to study the

graphene-based composites and newly FG graphene-reinforced nanocomposite using different

micromechanical models. They also reviewed different theories required for investigating the mechanical

analyses of FG composites structures with advantages, limitations and future technical challenges. In the case

of FG graphene-based polymer composite nanoplate, Kitipornchai and his co-authors studied the free and

forced vibration (Song et al., 2017), bending (Yang et al., 2017), and buckling responses (Song et al., 2018).

From these studies, they have concluded that one can tailor the desired mechanical response including bending

deflection, buckling, and post-buckling, as well as the natural frequency of the composite plates by altering the

nonuniformity in the distribution pattern of GPLs. Based on HSDT, Shen and his co-authors investigated the

nonlinear bending (Shen et al., 2017a), vibration (Chen et al., 2017), and buckling and post-buckling (Shen et

al., 2017b) behavior of graphene-based layered composite plates including thermal loading. (Kiani, 2018)

examined the free vibration of composite plates incorporated with GPLs to study large amplitudes with the

help of iso-geometric finite element (FE) modeling. Researchers (Karsh et al., 2019; Shingare and Kundalwal,

2019, 2020; Shingare and Naskar, 2021a; Trinh et al., 2020; Vaishali et al., 2020; Naskar et al., 2017) studied

the electromechanical response of hybrid graphene-based nanocomposites (GNC) including beam, plate, wire,

and shell by incorporating piezoelectric graphene nanofiber in a polyimide matrix. In such studies, they

assumed graphene as nanofiber and found the effect of size-dependent phenomena (piezoelectricity,

flexoelectricity, and surface effect) on these non-FGM GNC structures. Using analytical and numerical models,

they were able to examine the piezoelastic and dielectric properties of GNC. They showed a substantial

enrichment in the structural response of GNC structures by accounting for these size-dependent properties and

also revealed that one should not ignore these effects at the nanoscale. Kundalwal et al. (2020) investigated the

stress transfer characteristics and mechanical properties of composites including nano- and micro-scale

reinforcements via micromechanical pull-out model and molecular dynamic (MD) simulations.

In addition to piezoelectricity, flexoelectricity is also a noteworthy phenomenon, specifically in nano-

and microscales (Hamdia et al., 2018; Li et al., 2021). This is the formation of electric polarization () due to

a strain gradient inside all-dielectric material whether it is non-centrosymmetric (piezoelectric material)

or centrosymmetric structure. Schematically, this can be expressed by the following relation:

; where

and are direct flexoelectric (non-zero for all-dielectric materials) and piezoelectric

constant (zero for non-piezoelectric materials), respectively (Chandratre and Sharma, 2012; Shu et al., 2019).

4

Sharma et al. (2021) reported the substantial enrichment in resultant coupling in the presence of flexoelectricity

and piezoelectricity both in an electrically poled material sample. Besides the piezoelectric and flexoelectric

effects, the surface effect is a size-dependent property that has a significant impact on the elastic response of

structural building block elements. Due to the high surface-to-volume ratio, it plays a crucial part in forecasting

static and dynamic characteristics of nanostructures (Gurtin and Ian Murdoch, 1975; He and Lilley, 2008; Liu

and Rajapakse, 2010; Miller and Shenoy, 2000; Shenoy, 2005). For instance, Gurtin and Murdoch initially

proposed a fundamental theory of surface elasticity, namely, GM surface elasticity theory in order to account

for the surface effects (Gurtin and Ian Murdoch, 1975). Rajapakse and co-authors (Liu et al., 2011; Liu and

Rajapakse, 2010, 2013; Sapsathiarn and Rajapakse, 2017) explored the effect of different surface parameters

such as surface elasticity, surface energy, levy parameters, etc., on different structural elements subjected to

different loadings (point and UDL) and boundary conditions (cantilever, simply-supported and clamped-

clamped). Yan and Jiang (2012a) and Yan and Jiang (2012b) investigated the influences of surface parameters

on the static bending, vibration, and buckling behavior of a non-FGM nanoplate where they mentioned two

cases, traction free boundary condition and without in-plane movement of plate’s mid-plane, and reported that

the residual surface stress becomes more noticeable in the latter case. By using Mindlin and Kirchhoff plate

theories, Ebrahimi and Hosseini (2020) studied the effect of flexoelectricity on nonlinear forced vibration of

piezoelectric FG porous composite nanoplate subjected to electric voltage and external parametric excitation

without considering surface effect. They also reported that electric voltage has no influence on the performance

of piezoelectric and flexoelectric properties of the material on vibrational response. In this, they didn’t consider

the surface effects and static response of nanocomposite plates. Shingare and Naskar (2021b) studied the effect

of piezoelectricity and surface on a hybrid graphene-based composite plate to study its static and dynamic

responses, but didn’t consider the effect of flexoelectricity. From the study of extensive literature in related

fields, it is noticed that the classical continuum mechanics is not able to consider the small-scale effect of nano-

scaled structures due to the absence of any material length scale parameters. Considering the inadequacies of

classical continuum theories to incorporate size effects, the higher-order non-classical continuum theories,

which give more precise outcomes by taking size effects into account, have been strongly suggested. Moreover,

due to the time-consuming nature of MD simulation presented by Chan and Pu (2011) and Mehralian et al.

(2017) and limitations of experimentation (Li et al., 2018) for determining length scale parameters, in recent

years, several non-classical elasticity theories such as non-local elasticity theory, shear deformation theory,

modified strain gradient elasticity theory and modified coupled stress theory (MCST) have been suggested.

For instance, based on nonlocal elasticity theory of Eringen in conjunction with surface elasticity theories,

Ebrahimi and Barati (2017) studied the electromechanical buckling response of non-FG flexoelectric

nanoplates. They compared their results for higher buckling loads with and without considering flexoelectric

effects and reported that the flexoelectric nanoplate shows enhanced results at smaller thicknesses. More

5

recently, Ghobadi et al. (2020) and Ghobadi et al. (2021a, 2021b) developed a continuous-based thermo-

electromechanic model based on assumptions of Kirchhoff plate’s theory and the modified flexoelectricity

theory in conjunction with the strain gradient theory in order to study the size-dependent nonlinear free

vibration of FG flexoelectric nanoplate subjected to a thermo-electro-mechanical loading. They also

investigated the effect of the diverse distribution of porosity on the static and nonlinear dynamic responses of

a sandwich FG nanostructure. The nonlinear governing differential equations of the nanoplate and their

respective boundary conditions were solved by using Hamilton’s principle and variation method, and the

governing equations were solved by using Galerkin’s and perturbation methods. Furthermore, the advantage

of modified coupled stress theory developed by Yang et al. (2002) over the earlier version of couple stress

theory is that the former one needs only one material length scale parameter as compared to later which needs

two parameters. Earlier, the MCST was frequently utilised for micro-scale structures, not for nanoscales.

Furthermore, contrary to this, Akbarzadeh Khorshidi (2018) correctly showed that if experimental data within

the relevant range is available, the material length scale parameter can be determined for micro- or nano-scaled

thickness, and it is also concluded that MCST covers both micro and nano ranges if the material length scale

parameter is determined in these ranges. Again, Akbarzadeh Khorshidi (2020) confirmed the conclusion of his

aforementioned paper by comparing the results of MCST for two single-walled CNTs with MD simulation

results by Wang and Hu (2005). Therefore, in present study, authors used the MCST as it is capable for

considering the higher-order electro-mechanical coupling effects besides size effects. By taking inspiration

from above mentioned work and approaches, authors proposed the MCST for advanced graphene-reinforced

FG nanocomposites using the powerful frameworks of semi-analytical ‘single-term extended Kantorovich

method (EKM)’ and ‘Ritz method’ for investigating the static and dynamic electromechanical responses

considering flexoelectric as well as surface effects. It should be noted that the size effect is considered in this

formulation based on the MCST for analyzing the anisotropic nanostructures and can also be used for isotropic

structures. Besides, this formulation can also be converted into the classical plate formulation.

From a careful review of literature, it can be noticed that researchers have worked on different types of

theories such as Euler Bernoulli beam, Kirchhoff’s plate theory, weighted residual method, and approximated

Ritz method for studying the mechanical behaviour of different structures. These methods consume significant

computational time for convergence of results and hence, it is important to consider more efficient methods

such as EKM for evaluating mechanical behaviour with a higher convergence rate. In 1968, reported the very

effective EKM for obtaining semi-analytical solutions to 2-D elasticity problems including bivariate PDEs.

Another advantage of EKM is that one can choose the priori function arbitrarily irrespective of whether it

fulfills the boundary conditions of the concerned geometry or not. The iterations over the two axes are carried

out in repetition till the convergence is attained, which turns out to be faster compared to Galerkin’s and Ritz’s

methods. For instance, Kapuria and Kumari (2011, 2012, 2013) employed the powerful EKM in the 3-D

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elasticity problem of transversely loaded laminated structures. They also envisaged the coupled

electromechanical behaviour, comprising the edge effects of single-layer piezoelectric sensors and hybrid

laminates, when subjected to electromechanical loadings conditions.

With the tremendous recent advances in nano-scale manufacturing capabilities (Jang et al., 2013), while

the literature categorically reveals the crucial influences of size-dependent properties such as piezoelectricity,

flexoelectricity, and surface effect on the static and dynamic electromechanical behaviour of different structural

elements, the aspect of effective and efficient modeling of the coupled behavior becomes a priority for better

understanding of the physical behavior and subsequent engineering applications. However, the coupled

electromechanical problems of functionally graded piezoelectric materials (FGPM) considering surface and

flexoelectric effects in open-circuit have not been explored in a unified efficient framework so far. Therefore,

the objective of the present work is to provide a unified mathematical formulation for the open-circuit electric

boundary condition of the proposed composite nanostructure, as well as to analyze its size-dependent behaviors

for various FGPM distributions: (i) linear distribution (LD), (ii) uniform distribution (UD) and (iii) parabolic

distribution (PD). This paper hereafter is organized as: Section 2 presents the theoretical formulation to analyze

the static and dynamic behavior of FGPM nanoplates subjected to electromechanical loading considering both

flexoelectric as well as surface effects. Here two different semi-analytical models such as EKM for

flexoelectric and surface effects as well as Ritz method would be incorporated for developing an efficient

computational framework; Section 3 deals with the results and discussion on the effect of flexoelectricity and

surface parameters on the static and dynamic behavior of FGPM nanoplates (referred to as “flexo-surface

FGPM nanoplates”). Section 4 presents the summary of the results and concluding remarks. A comprehensive

overview of the current research work is systematically presented in Fig. 1. These results would offer new

insights to engineer the domain configurations for tailoring the desired static and dynamic electromechanical

responses of the novel graphene reinforced FG materials considering surface and flexoelectric effects. This

would be demonstrated by comparison of different sets of results such as (i) conventional nanoplate (without

flexo and surface effects), (ii) flexo FGPM nanoplate (considering only flexoelectric effect), and (iii) flexo-

surface FGPM nanoplate (considering flexo and surface effects). Thus, the present study aims to complete a

gap in our knowledge about the consideration of flexoelectric and surface effects for FGPM nanostructures.

2. Theoretical formulations

In the present section, the governing differential equations for thin square FGPM nanoplates subjected

to electromechanical loading and boundary conditions are developed to study the static and dynamic responses

using two different semi-analytical solution methods: (i) Extended Kantorovich method (EKM) and (ii) Ritz

method, respectively. These solution approaches in the static and dynamic domains have been chosen here

7

Fig. 1. Detailed flowchart of electromechanical analysis of FGPM flexo-surface nanoplates.

based on the consideration of computational convenience as per published literature (Jones and Milne, 1976;

Singhatanadgid and Singhanart, 2019), which is further discussed later in this section.

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2.1 Geometrical consideration

Figure 2(a) represents a thin square undeformed FGPM nanoplate of length , width , uniform

thickness and its associated rectangular coordinate system , where axis defines its out-of-plane

direction and the in-plane axis is lying in the mid-plane . It is assumed that the piezoelectric

polarization direction is along the –axis. This FGPM nanoplate is subjected to a uniform transverse

(downwards) loading over its upper surface and placed in the open-circuit electric boundary conditions. In

the context of FGM system, the variation of material property is expected to be continuous (Vatanabe et al.,

2014) and limited to the thickness direction .

Here the FGM system with a regular shaped (square) geometry is considered for analysis because it

exhibits a more prominent flexoelectric effect due to the large strain gradient and can reduce the geometry

dependency requirement of flexoelectricity (Sharma et al., 2021). In this paper, both flexoelectric and surface

effects are considered. The upper

and lower surface

of the plate are denoted by and

which are schematically shown in figure 2(b). Here the whole FGPM system can be divided into two regions,

the surfaces and the bulk region.

2.2 Micromechanical models – effective material properties

2.2.1 Material properties of the bulk region

The present FGPM system consists of the graphene nanoplatelets (GPLs)-based nanocomposite where

a piezoelectric polymer is used as the matrix phase. Polyvinylidene fluoride (PVDF) is a good choice for this

composite as it shows excellent piezoelectric and dielectric properties. The GPLs are assumed as rectangular-

shaped solid reinforcement of average width , length and thickness , where these are non-

(a) (b)

Fig. 2. (a) Geometry and coordinate system of thin FGPM nanoplates under open circuit condition, (b) upper

and lower surface of the nanoplate.

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Fig. 3. Distribution of across the thickness of FGPM nanoplates ( plane): (a) uniform, (b) linear and

(c) parabolic pattern.

uniformly dispersed with varying weight fractions across the thickness of the composite plate. To determine

the effective elastic properties of the present nanocomposite incorporating geometrical parameters (Shingare

and Naskar, 2021a), Halpin-Tsai (HT) model is adopted whereas the effective piezoelectric and dielectric

properties are determined by the rule of mixture (ROM).

In this work, three distribution patterns of GPLs are considered where the weight fraction of GPLs

varies as per the following relations (Yang et al., 2017; Z. Zhao et al., 2020):

Uniform distribution (UD):

(1a)

Parabolic distribution (PD):

(1b)

Linear distribution (LD):

(1c)

where

and are the total weight fraction (%) and characteristic value of GPLs weight fraction (%),

respectively. These three distributions are depicted in figure 3 schematically. The total volume fraction of

GPLs is calculated with the help of the following relation:

(2)

where and denote the respective mass densities of GPLs and PVDF matrix.

The elastic modulus of the system from the HT model is determined as follows (Wang et al., 2020):

10

(3)

whereas and indicate moduli in the longitudinal and transverse directions respectively and the values

can be estimated from Eq. (4). Here,

and

are the reinforcing efficiency of GPLs considered in longitudinal

and transverse directions, respectively.

(4)

where the parameters

and

can be expressed by:

(5)

The parameters and indicate the respective Young moduli of GPLs and PVDF matrix. The filler

geometric factors

and

of GPLs are given by the following equation:

(6)

Due to the existence of the piezoconductive effect of graphene which is greatly dependent on its layer number

(Xu et al., 2015), in the current mathematical model, it is supposed that the piezoelectric properties of GPL are

times stronger than PVDF (Mao and Zhang, 2018). Other material properties are derived from the ROM as

follows:

(7a)

(7b)

(7c)

(7d)

where ,, and are the Poisson’s ratio, coupling coefficient, electric permittivity, and

piezoelectric multiple, respectively. Regarding the intrinsic flexoelectric coefficient of the present FGPM

composite system, due to the unavailability of sufficient literature and difficulties in the experimental

determination of (Shu et al., 2014) , we assume it constant throughout the bulk region under the assumption

of crystal with cubic symmetry (). Also due to the incorporation of reinforcement elements

(e.g. GPLs) in the PVDF matrix, there will be an increase in the value of (Hu et al., 2018). This is why the

11

flexoelectric coefficient () is set within a range in the current model. The range of its values is taken as the

same as that of the other two components of the flexoelectric tensor (i.e., and ) of PVDF-based polymers

(Baskaran et al., 2011; Zhou et al., 2017). This range is also mentioned in section 3.2. However, note that the

analytical model developed here is equally applicable for both varying and constant flexoelectric systems.

2.2.2 Material properties concerning the surface layers and

Due to lack of proper atomistic experiments and to deal with the problem of zero thickness outer layers,

one characteristic length () is assumed to estimate the surface material constants of the present FGPM system.

The surface constants of the upper and lower surface layers are related to their corresponding bulk constants

by the following relations (Pan et al., 2011; Shingare and Kundalwal, 2020):

Upper surface ():

(8a)

Lower surface ():

(8b)

From Eq. (8), it is evident that in the FGPM system, the surface properties of the upper and lower surface are

different (Hosseini et al., 2017). Here, the value of the residual surface stress

is considered as zero (Zhang

et al., 2012).

2.3 Constitutive relations of bulk and surface layer

2.3.1 Bulk region

To incorporate the flexoelectricity phenomena along with its inbuilt piezoelectricity within the bulk

region, an extended linear theory of piezoelectricity is adopted under the assumption of infinitesimal

deformation. Hence, the general expression of the electric Gibbs free energy density function for the bulk

region can be given as follows (Liang et al., 2013):

where , , , and are the element of material property tensors – permittivity (rank 2 tensor), elastic

stiffness (rank 4 tensor), piezoelectric coupling (rank 3 tensor), and flexocoupling (rank 4 tensor), respectively.

and are the electric field vector and strain component, respectively. In Eq. (9), tensor couples strain

gradient and electric field whereas and are higher-order coupling terms which couple strain and its strain

gradient, and strain gradient and strain gradient, respectively. Here the comma in the subscript of indicates

its differentiation with respect to one spatial variable. Under the aforementioned assumption, the higher-order

coupling terms and are neglected to simplify the current formulation. Following generalized constitutive

relations for the bulk region are derived using Eq. (9).

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(10a)

(10b)

(10c)

where and are the classical stress and strain tensor. is the hyper stress tensor generated due to

consideration of flexoelectricity; is the higher-order strain gradient tensor. denotes fourth-order

flexocoupling tensor; and are the electric displacement and electric field vector. and denote the

second-order permittivity and third-order piezoelectric tensor. In contrast to the conventional piezoelectric

relation in Eq. (10c), it is observed that there is also a contribution of flexoelectricity in the electric

displacement of the nanoplate.

While considering the flexoelectric effect, Shu et al. (2011) reported the direct flexoelectric coefficient

tensor for a cubic crystal, and these can be expressed as:

where,,

. In addition to this, as the thickness of the proposed FGPM plate is

considered as very small as compared to length and width, the electric displacement and electric field in the x-

and y- directions are considered as zero (; ). This indicates the electric field and

electric displacement to be present only along the z-direction and the strain variations are also

considered along z-direction only i.e., and . In other words, in Eq. (10c), ‘k’ and ‘n’ will be ‘3’ and

‘l’ and ‘m’ will be either 1 or 2 (i.e., ). So, it is evident that the flexoelectric coefficients

except will be zero . As the present thin FGPM plate is under the 2D assumption, all strains must

be in-plane only. Therefore, and will be non-zero . Consequently, the second-order

permittivity tensor is also considered in z-direction only . Later the values of non-zero

coefficients are given in Table 4 for the present study.

For the present FGPM system, Eq. (10) can be rewritten by considering bi-subscript notations (Wang

and Li, 2021) and cubic crystal symmetry as follows:

(11a)

13

(11b)

(11c)

(11d)

(11e)

(11f)

where , and are classical in-plane strains. In this, all material constants are considered as the

functions of due to the thickness-wise FGM system under consideration.

2.3.2 Surface layers ( and )

Similar to the bulk region, the general constitutive relations for the surfaces (and ) of FGPM

nanoplates can be derived from the surface energy density function which are given as follows (Huang

and Yu, 2006):

(12a)

(12b)

wherein

and

are the surface residual stress and surface electric field, respectively. Here the surface

quantities are indicated by superscript ‘s’. According to Zhang et al. (2013), the equations of strain () and

electric field () in case of surface effects are the same as that of the bulk one. Based on the present FGPM

system, one can rewrite the surface stresses as follows:

(13a)

(13b)

(13c)

(13d)

2.4 Governing equations for static and dynamic analysis

As the present model is under open-circuit (sensor type) condition, the electric displacement on

the surfaces of FGPM nanoplate is zero (Zhang et al., 2013) and it can be achieved if the top and bottom-most

14

surfaces are insulated (Wang and Zhou, 2013). It also satisfies Gauss law of dielectrics

where

free electric charge density is zero. From Eq. (11f), for the zero electric displacement , the

internal electric field can be expressed as follows:

(14)

The same electric field relation as above is also applicable for the surface constitutive equations.

In this mathematical formulation, the classical Kirchhoff plate theory is used for the thin FGPM

nanoplate. As mentioned in the existing literature (Yan and Jiang, 2012a), to investigate the surface effect

effectively, the mid-plane extensional deformations are constrained to zero. The displacement fields

can be expressed as:

(15a)

(15b)

(15c)

Neglecting Von-Karmen non-linear terms in strain formulae, the following linear strain-displacement relations

and internal electric field are derived from Eq. (14) and (15):

(16a)

(16b)

(16c)

(16d)

(16e)

(16f)

(16g)

(16h)

15

where

,

,

,

and

.

Equations (16c) and (16e) are found in coherence with the Kirchhoff hypothesis (Reddy, 2003).

The governing equations for the present FGPM system are achieved from the principle of virtual

displacements that can be given as follows:

(17)

The virtual strain energy for the bulk region is expressed as follows:

The virtual strain energy considering both the surface layers can be written as follows:

The virtual work done by the externally applied uniform transverse load and loads induced by the traction

jump and in-plane forces (Yan and Jiang, 2012; Zhang et al., 2014) can be written as follows:

Finally, the virtual kinetic energy considering the motions in all three directions is expressed in the following

equation. Though we have shown all three directions for generality, only vibrational motion in the transverse

plane of the nanoplate is considered in the final results.

16

where and are the mass inertia terms and

are the function resultants (stress and

moment) whose definitions are given as follows:

By substituting Eq. (16) in the aforementioned resultants, the following matrix is derived.

(22)

In Eq. (22), the stiffness coefficients and their algebraic expressions are given in Appendix A. Now, if we

substitute Eqs. (18), (19), (20) and (21) into the Hamilton Eq. (17) and apply the principles of variational

calculus, the following governing equation of the present FGPM system incorporating both flexoelectricity and

surface effect can be derived.

Here,

,

are obtained as follows:

17

.

2.5 Solution methodology based on EKM and Ritz approach

In the present study, two semi-analytical solution methods are adopted for getting the solutions of the

governing equation (23) which are the single-term extended Kantorovich method (EKM) and the Ritz method.

These two methods are applied separately for the static and dynamic cases of FGPM nanoplates considering

flexoelectricity as well as surface effects, and the results are validated in later sections. For the static analysis,

the reason for selecting the EKM approach is its accuracy and rapid convergence rate. Its solution is also

independent of the initially chosen functions. The traditional Navier approach can only be applied for all edges

simply-supported (SSSS) plate whereas, the Levy method needs at least two simply supported edges of the

concerned plate. In the Ritz and Galerkin method, the final solutions are dependent on initial guess (algebraic

polynomials or basis) functions. It would be found in later sections of this paper that for a square thin plate

under the 2D assumption, a single-term EKM solution is sufficient to provide accurate results. On

the other hand, the dynamic analysis is performed using the Ritz approach instead of EKM because the EKM

method is computationally more intensive than other methods due to the existence of several vibrational

frequencies of any continuous system and the presence of symmetric and antisymmetric vibration modes in

many cases (Singhatanadgid and Singhanart, 2019). In the dynamic scenario, in each iteration of EKM there

exists two unknown variables (second unknown single variable function and eigenfrequency) in the ODE

obtained after substituting the first known priori function. To resolve this, we also have to take into

consideration the symmetry and antisymmetry conditions about one direction of the structure and for that its

final closed-form solution is dependent on the mode of vibration whether it is symmetric or

antisymmetric(Jones and Milne, 1976). As in this paper, the focus is given to the analysis of the behavioural

aspects of FGM nanoplate of different distributions within a semi-analytical framework, Ritz method is

preferred in the dynamic scenarios. Before discussing the mathematical formulations of the analytical

approaches, the boundary conditions and their mathematical representation need to be mentioned as both the

techniques are based on the geometric and essential boundary conditions of the problem. We considered SSSS

and clamped-clamped (CCCC) conditions for square plates which are schematically shown in figure 4.

2.5.1 Static analysis based on EKM

Substituting all the stiffness coefficients of Eq. (22) and strain-displacement relations of Eq. (16) into

the governing equation (23), the following simplified form of Eq.(23) in terms of displacement can be written:

18

(a) (b)

Fig. 4. FGPM nanoplates subjected to mechanical boundary conditions: (a) CCCC and (b) SSSS.

In Eq. (24), all the coefficients of each term and their algebraic expressions are given in Appendix B. It

is observed that Eq. (24) is non-linear in nature. Under infinitesimal deformation assumptions and to linearize

the calculation process, we have neglected these four non-linear terms ( and ) in further

calculations. However, mathematical error due to neglecting the non-linearities becomes minimal if

symmetrical distributions of FGM and non-flexoelectric () surface nanoplate is considered. For

instance, it is observed from the present model that in the absence of , the magnitudes of , , and

are zero for UD and PD distribution whereas it is in the order of for LD distribution. The magnitudes

become more as plate thickness increases. To account for this, the results and discussions in section 3 are

mostly focused on thicknesses and distributions (UD and PD) within a reasonable range. The following

equation is the weak form of nanoplate under bending which is used in the EKM method:

In single-term EKM, first, it is essential to assume a solution into two bivariate functions for the Eq. (25) in a

separable form which is shown below:

(26)

19

Here either or is taken as a priori function. To start the first iteration, in the first step, a function

is chosen as priori for . Another advantage of the EKM is that one can choose this priori function arbitrarily

irrespective of whether it satisfies the boundary conditions of the problem or not. Now, if the updated

function is substituted into the weak form (Eq. 25), one ordinary differential equation (ODE) of

will be obtained which can be solved using any standard method of differential calculus. Following is the

ODE and its associated boundary conditions after the first step.

From Eq. (27), the following ODE of is obtained:

where

(28b)

(28c)

(28d)

(28e)

(28f)

(28g)

(28h)

For solving Eq. (28a), the mechanical boundary conditions (CCCC, SSSS, CSCS, CSSC) in terms of

mentioned in figure 4 can be used as listed below:

20

CCCC:

(28i)

SSSS:

(28j)

CSCS:

(28k)

CSSC:

(28l)

Here, CCCC represents fully clamped plate; SSSS represents fully simply supported plate; CSSC represents

the plate with adjacent two edges clamped and remaining two edges simply supported; and CSCS represents

plate with two opposite edges clamped and remaining two edges simply supported, whereas C denotes the

clamped and S denotes the simply supported edge. In the present paper, we showed the utilization of the EKM

only for CCCC and SSSS boundary, but this method can easily be extended to any arbitrary boundary

conditions (Kumari and Shakya, 2017), just by changing Eqs. (28) and (30).

After solving Eq. (28), the obtained (let the solution is ) is introduced as priori known

function for in the next step of EKM whereas is taken as an unknown function that needs to be solved.

In the same way as before, we can derive ODE of after substituting updated in the weak form

(Eq. 25).

From Eq. (29), the following ODE of is obtained.

where

(30b)

(30c)

(30d)

21

(30e)

(30f)

(30g)

(30h)

For solving Eq. (30a), the requisite boundary conditions in terms of from the mechanical boundary

conditions shown in figure 4 can be given as:

CCCC:

(30i)

SSSS:

(30j)

CSCS:

(30k)

CSSC:

(30l)

After solving Eq. (30), the obtained (let the solution is ) from the aforementioned first

iteration is then used as the priori function for g(y) in the next iteration step where is taken as an unknown

function. Likewise, one can perform multiple iterations using the same solution technique based on Eq. (28)

and (30) until the converged results, i.e., converged is obtained. Generally, after two or three

iterations, the solution gets converged.

2.5.2 Dynamic analysis based on Ritz method

In dynamic analysis, free vibration of the present FGPM nanoplate incorporating both

flexoelectricity and surface effect is performed using the Ritz method. In this section, in-plane vibration (

) is neglected for the purpose of simplicity. The weak form of governing equation is derived from Eq.

(17) and can be written as follows:

22

After substituting the stiffness coefficients of Eq. (22) and strain-displacement relations (Eq. 16) into Eq. (31)

and considering the time-dependent harmonic function of , the final equation of the weak form can

be written as follows:

where is the frequency of the FGPM nanoplate. The coefficients and their algebraic expressions are

given in Appendix B. is taken from harmonic (Euler) relation of the mid-plane displacement which is

given as follows:

(33)

The Ritz solution of the displacement W(x,y) for the square plate is assumed in the following form (Reddy,

2006):

(34)

Selection of and depend on the boundary conditions of the problem geometry. Further, and

may be infinity (i.e., Eq. (34) signifying an infinite series). Hence, the standard approximate functions for

and for SSSS, CCCC, CSCS, SCSS etc. boundary conditions are chosen from the existing literature (Reddy,

2006) of plate theory (note that we have focused on CCCC and SSSS boundary conditions only for presenting

numerical results).

CCCC:

(35a)

SSSS:

(35b)

CSCS:

(35c)

23

SCSS

(35d)

Substituting Eqs. (35a) and (35b) in the weak form and by arranging the expression in the matrix form, the

following equation is derived, from which the frequencies () of the system can be calculated using different

combinations of and .

(36)

where

For the sake of simplicity in formulation, we neglected the non-linear terms in the present study. In this

context (regarding the non-linearity case), it is also possible to include nonlinear parameters within the

computational framework. In other words, we can easily include geometric nonlinearity in the system and also

nonlinearity in the Eqs. 25 and 32 for making the system more accurate. In that case, the solution will involve

simultaneous nonlinear partial differential equations.

3. Numerical results and discussions

3.1 Validation and convergence studies

Before analyzing the numerical findings of the FGPM system, two different types of convergence and

validation studies are needed to be performed to check the reliability of the results. In this investigation, two

different approximate methods which are discussed in earlier section 2.5 are followed for studying the effects

of flexoelectric and surface properties on an FGPM nanoplate. Among these, as the EKM approach is newly

introduced here in the field of electromechanical analysis, to check the efficiency of this method for FGPM

2D-plate problems, the convergence study is carried out by comparing the deflection curve, i.e., transverse

24

deflection along one edge (axis) of rectangular UD-FGPM nanoplates. Figure 5 shows that the

convergence of the present iteration-based method is very rapid with a maximum of two or three iteration steps

that are enough to get the converged results. Therefore, we have carried out a maximum of three iterations in

the further analyses. As per the discussion in earlier sections, the initial guess function for is needed to

start the EKM method.

Fig. 5. Convergence plot for transverse deflection of UD-FGPM nanoplates with respect to aspect ratio

considering function,.

Fig. 6. Convergence plot for transverse deflection of UD-FGPM nanoplates with respect to aspect ratio

considering three iterations.

25

In Fig. 6, the results are calculated using three different types of function – polynomial, exponential

and trigonometric, wherein it can be concluded that the final result is unaffected by the initial guesses. We get

almost the same center deflection from each initial choice. The magnitudes of the center deflections are shown

in Table 1. From this, it is found that the maximum deflection is occurring at the center of the plate for each

function and the difference between obtained values are in the order of to which strongly

confirm the convergence efficiency of the present EKM approach. The above two conclusions are precisely

matching with the discussions presented in the preceding sections as per Kerr and Alexander (1968). In the

following subsections, the validation study is performed with respect to existing literature (Shingare and

Naskar, 2021c; Yang et al., 2015) considering two illustrative cases for the static and dynamic analysis of

flexo-surface FGPM nanoplates.

Table 1. Convergence study for transverse deflection of UD-FGPM nanoplates under three iterations.

Priori Function

Location of Max.

deflection

Magnitude of Max. deflection

(mm)

0.5

17.198756520806104

0.5

17.198756538984320

0.5

17.198757361498956

0.5

17.198756518719623

3.1.1 Case 1: Static and dynamic response of non-FGM flexoelectric nanoplates

We have first examined the static and dynamic responses of a non-FGM flexoelectric nanoplate from

(Shingare and Naskar, 2021c) using the present model where we just neglect the surface effects and

) on the upper and lower surface. This nanoplate is made of hybrid graphene reinforced piezoelectric

composite (GRPC) material where all the edges are simply supported and a transverse load of 0.1MPa is acting

upon it. Table 2 lists the material parameters and dimensions of the GRPC nanoplate used in this case study.

For getting static and dynamic responses, Shingare and Naskar (2021c) used the Navier approach which we

validate here for our present single-term extended Kantorovich method (EKM) and Ritz method, respectively.

Figure 7 shows the static response of center deflection of hybrid GRPC nanoplate along the length whereas the

dynamic response is shown by varying resonant frequency with plate aspect ratio in Fig. 8. From figures 7 and

8, it can be observed that the results obtained from EKM and Ritz solutions are found to be in excellent

agreement with the results estimated by Shingare and Naskar (2021c). Concerning the issue of non-linearity

26

mentioned in previous Section 2.5.1, these figures also clearly indicate that the effect of neglecting non-linear

terms is acceptable even in flexoelectric cases ().

Table 2. Material properties and dimensions of hybrid GRPC nanoplates.

Plate thickness,

20 nm

Plate aspect ratio,

45

112.43 GPa

3.34 GPa

2.03 GPa

-6.9337

3.264

Fig. 7. Comparison of two different models for the static transverse deflection of nanoplates along the length.

3.1.2 Case 2: Static and electric potential response of non-FGM nanoplates considering surface effects

Here, we have examined and validated the surface effect of a 2D non-FGM nanoplate. As our problem

statement is on open-circuit condition and due to lack of literature on open-circuit, we validate our results

indirectly with Yan and Jiang (2012a) which is based on closed-circuit conditions. This validation is based on

one simple observation of the internal electric field in both cases. If we compare Eq. (14) with the electric field

27

Fig. 8. Comparison of two different models for the resonant frequency of nanoplates in terms of aspect ratio.

Table 3. Material and surface parameters of 2D non-FGM plates.

Plate aspect ratio,

30

Material

PZT-5H

126 GPa

55 GPa

2.03 GPa

-6.5

23.3

1.30

7.56

-3.0

1.0

the internal electric field () is the same. We have also validated the aforementioned statement by plotting the

electric potential distribution on the upper and lower layer of nanoplate in figures 9 and 10 from our analytical

model considering two instances (UD and LD distributions) and observed that potential is zero on the upper

28

() equation mentioned by Yan and Jiang (2012a), we can conclude that non-flexo () open-circuit case

is equivalent to zero voltage () close-circuit case for the present nanoplate problem because in both cases

surface () and the lower surface is equivalent to a ground node. The material properties and surface

parameters used in this case are enlisted in Table 3.

In figure 11, it can be observed that there is a dependency of plate thickness on its deflection in the

presence of positive residual surface stress () and with the increase of thickness, the effect of

surface stress diminishes. From figure 11, it can also be concluded that the results obtained using the EKM

solution are found in good agreement with the results estimated by Yan and Jiang (2012a). Now if we include

flexoelectric property () in our present analytical model along with surface effects, there will be the

inclusion of some non-zero potential in upper surfaces and it is observed that the value of this potential (V) is

very less (nearly zero), but it is more than the previous non-flexo cases. Figure 12 shows the potential

distribution of flexo-surface FGPM nanoplates considering UD and LD cases. The same trend is also observed

in the case of a flexoelectric GNC nanowire (Kundalwal et al., 2020).

(a) (b)

Fig. 9. Electric potential distribution on upper surface of nanoplates considering: (a) UD and (b) LD.

3.2 Static response of FGPM nanoplates

The classical thin plate theory is adopted in this investigation by neglecting nonlinear terms to

determine the electromechanical response (static, dynamic, and electrical behavior) of the FGPM nanoplate.

The FGPM nanoplate is initially subjected to a uniformly distributed load . Even though

substantial advances are reported over the past couple of decades, still various complexities exist related to the

flexoelectric coefficient () for PVDF based structures. Several authors have stated that the range of its

29

magnitude typically varies in between to (Baskaran et al., 2011; Zhou et al., 2017). The

material properties of GPLs and PVDF and related dimensional parameters are summarized in Table 4. Here,

(a) (b)

Fig. 10. Electric potential distribution on lower surface of nanoplates considering: (a) UD and (b) LD.

Fig. 11. Variation of non-dimensional deflection of nanoplates with respect to thickness.

for sake of simplicity, we have adopted the cubic crystal symmetry for graphene reinforced polymer matrix

composites. The different influencing parameters on the static deflection of nanoplates are investigated and

discussed here in two sections, i.e., considering only flexoelectric effect and considering both surface and

30

flexoelectric effects. In later sections, we have also pointed out few limitations of our present model in

predicting its electromechanical characteristics.

(a) (b)

Fig. 12. Electric potential distribution on the upper surface of flexo-surface nanoplates considering: (a) UD

and (b) LD.

3.2.1 Consideration of only flexoelectric effect

In this section, we investigated the effect of various parameters such as plate thickness , aspect ratio

, in-plane dimensions , etc., considering the center deflection and deflection ratio ( and )

as our reference. The deflection ratio is given by:

Figure 13 shows the effects of thicknesses on a square FGPM nanoplate's deflection ratio with three different

distributions for CCCC boundary condition. Here the width-to-thickness ratio of nanoplate is kept constant,

i.e., . It can be observed, with the increase of thickness, the deflection ratio tends to unity which

indicates that the flexoelectric effect is more pronounced in case of a lesser thickness of the plate. From this, it

can be concluded that the phenomenon of flexoelectricity is size-dependent. In addition, the flexoelectric

FGPM nanoplates with LD and UD distribution show almost the same behaviour while in the case of PD, we

observe less flexoelectric effect than UD and LD because it achieves the saturation stage () faster. In

all three distributions, pure flexoelectricity stiffens the FGPM plate in terms of maximum static deflection.

The effects of thickness and in-plane dimensions of the nanoplate on its static bending response are now

investigated in the following figures. In figure 14, we have kept in-plane dimensions constant

31

Table 4. Geometric and material properties of constituents of FGPM (Arefi et al., 2018; Z. Zhao et al., 2020).

Elastic and geometrical properties

Piezoelectric constants

Surface parameters

Charac. length ()

1

1 or 0

0

Dielectric constants

Flexoelectric parameters

0.186

0.29

whereas, in figure 15, the thickness is kept constant. Figure 14 illustrates that all distributions have a

stronger flexoelectric impact if the aspect ratio is large, i.e., when the thickness is small. Here, the deflection

ratios of UD and LD both drop to 0.56, whereas PD only drops to 0.69 for aspect ratio, .

Fig. 13. Effect of thickness on deflection ratio of flexo FGPM nanoplates at constant aspect ratio

.

32

Figure 15 shows another important aspect of size-dependency of flexoelectric property. When the thickness is

kept constant, in-plane dimensions have nearly no effect on the deflection ratio for that particular range of

aspect ratios (10 to 100). In this case, we observe the deflection ratio of PD is more than UD. For the sake of

brevity, we have not presented the results for LD nanoplates as it shows similar behavior as that of UD

nanoplate. The reason for such thickness dependency is that the strain gradient has an inverse relationship with

the material dimension i.e., size of structures (Kumar et al., 2018) while it has a direct effect on flexoelectricity

(Kundalwal, et al., 2020). This is also the reason why we get more flexoelectric properties in the case of UD.

We also observed from our analytical model that the strain gradient is more in the case of UD or LD as

compared to PD-FGPM nanoplate. For example, the values of strain gradient ( at centre in UD and PD

for a square flexoelectric nanoplate () with are

and , respectively.

Fig. 14. Effect of aspect ratios on the deflection ratio ( of flexo FGPM plates at constant in-plane

dimension .

Figure 16 represents the maximum transverse deflections of UD- and PD-FGPM nanoplates with and

without consideration of flexoelectricity. Here, we have used a plate thickness of to study the

flexoelectric effect on the static bending deflection because both distributions don’t reach saturation at

as shown in figure 13. The difference in transverse deflection due to the flexoelectric effect is maximum

near the center of UD-FGPM nanoplates and is less for PD case which also validates the observations of figures

13, 14 and 15. From figure 16 (a), it can be concluded that PD-FGPM flexoelectric plate shows stiffer behavior

than the UD-FGPM plate, which is also in agreement with existing literature (Z. Zhao et al., 2020). Like

conventional FGM plates (Talha and Singh, 2010), figure 16 (b) also shows the center deflection of the SSSS

flexoelectric plate is more than the CCCC plate as the bending stiffness of the CCCC plate is higher. It is also

33

observed, due to consideration of the flexoelectric effect there is a significant reduction in deflection of FGPM

nanoplates.

Fig. 15. Effect of aspect ratios on the deflection ratio ( of flexo FGPM plates at a constant thickness

. The results corresponding to LD nanoplates are similar to that of the UD case.

(a) (b)

Fig. 16. Variation of transverse deflection of flexo FGPM nanoplates along longitudinal axis with different

distribution of GPLs. (a) the effect of flexoelectric effect over non-flexoelectric effect (b) SSSS and CCCC

boundary conditions. The results corresponding to LD nanoplates are similar to that of the UD case in both

cases.

Figure 17 shows the variation of transverse deflection of the clamped-clamped FGPM nanoplate

considering different distribution and flexoelectric coefficients ranging from to . It is

evident that the effect of flexoelectricity significantly affects the overall static deflections of the FGPM

34

nanoplate. The deflection continues to decrease as the flexoelectric coefficient increases in magnitude. We

observe the same trend of results in published literature (Shingare and Naskar, 2021c) which affirms the

validity of our present model (EKM) further. In addition to this, in all three distributions, there exists a

significant reduction in the magnitude of deflection in between and whereas

after , this reduction is almost negligible. From the observations of figures 16 (a) and 17, one

can report another prominent conclusion that as flexoelectric coefficient increases there is a significant

reduction in the static deflection (figure 17), while there will be one case where there will be no effect of

flexoelectricity, i.e., deflection with and without flexoelectric effect will be same for all aspect ratios. In that

case, the deflection ratio, . The same trend of results was also observed in the existing literature

(Shingare and Kundalwal, 2019) on the non-FGM graphene/polyimide nanocomposite structures.

(a) (b)

Fig. 17. Variation of transverse deflection of flexo FGPM nanoplates for various flexoelectric coefficients and

distributions: (a) UD and (b) PD. The results corresponding to LD nanoplates are similar to that of the UD

case.

Figures 18 and 19 illustrate the effect of two influencing parameters, the total weight fraction of GPLs

(

) in the PVDF matrix and piezoelectric multiple (), on the static bending deflection of a square

flexoelectric nanoplate. Due to the incorporation of more GPLs into the FGPM system or due to an increase in

the theoretical value of piezoelectric multiple (), there is an increase in the overall stiffness of composite

which directly influences the bending rigidity of nanoplates. This is one of the reasons for the decrement of

static deflection with an increment of GPLs weight fraction and piezoelectric multiple. After analyzing figures

17, 18 and 19, it is evident that the deflection ratio or effect of flexoelectricity on static deflection not only

35

depends on size parameters of the structure but also on the value of its flexoelectric coefficient, GPLs weight

fraction and piezoelectric multiple.

(a) (b)

Fig. 18. Variation of transverse deflection of flexo FGPM nanoplates for different weight fractions of GPLs

and distributions: (a) UD and (b) PD. The results corresponding to LD nanoplates are similar to that of the UD

case.

(a) (b)

Fig. 19. Variation of transverse deflection of flexo FGPM nanoplates for various piezoelectric multiples and

distributions: (a) UD and (b) PD. The results corresponding to LD nanoplates are similar to that of the UD

case.

36

(a) (b)

(c)

Fig. 20. Electric field variation along thickness of FGPM nanoplates for various distributions: (a) UD, (b) LD

and (c) PD.

Figure 20 depicts the distribution of internally generated electric field () along the thickness of CCCC

flexoelectric nanoplate for three different distributions. As per the discussion in section 2.3, there will be

contributions of both flexoelectricity and piezoelectricity in the induced electric field of the open-circuit case.

The results associated with electric filed considering piezoelectricity, flexoelectricity, and both piezoelectricity

and flexoelectricity are illustrated in Fig. 20 (a, b and c). In the case of UD-FGPM, a linear variation of the

electric field is observed whereas in LD- and PD-FGPM, a linear variation is observed except end and center

position, respectively. Such a linear variation in case of UD-FGPM is in coherence with the observations of

37

Yan and Jiang (2012b). In case of LD- and PD-FGPM, there exists a jump in the field at the end and middle

position, respectively, and the magnitude of LD or PD field is more than that of the UD case.

3.2.2 Consideration of flexoelectric and surface effects

Along with flexoelectricity, the surface effect is one of the important influencing factors in predicting

the electromechanical behavior of nanoplates. In this analysis, we also considered the deflection ratio ()

and center deflection as our reference. The deflection ratio by considering surface effect can be given as:

(a) (b)

Fig. 21. (a) Effect of thickness on deflection ratio of flexo-surface FGPM nanoplates at constant

aspect ratio and (b) Effect of aspect ratios on deflection ratio of flexo-surface

FGPM nanoplates for thickness . The results corresponding to LD nanoplates are similar to that

of the UD case in both cases.

In the static deflection of a plate, the sign of the surface residual stress is crucial. In case of negative

residual stresses, the mechanical buckling instability occurs at a certain range of thickness for the applied

transverse load and thus, it results in large deformation. As in this paper, the discussion is being limited

to the static deflection and dynamic behavior of nanoplates, we consider only non-negative residual surface

stresses in further analyses to avoid buckling.

Figures 21 (a and b) demonstrate the resultant effect of flexoelectricity and surface residual stress on

the static bending deflection of square flexo-surface FGPM nanoplate against its thickness and aspect ratio,

respectively. In figure 21(a), we kept the aspect ratio whereas, in figure 21(b), the thickness is kept

constant . In all these combinations, one can observe that the deflection ratio is less than 1 which

38

indicates that this combined effect stiffens the FGPM nanoplate. From our analysis, it can be observed that the

percentage reduction of deflection is more when we incorporate the surface effect. For example, for a square

UD-FGPM nanoplate , pure flexoelectricity reduces the static deflection of the

conventional piezoelectric nanoplate by 4.71% whereas the combined effect of surface and flexoelectricity

reduces it by 26 % – 91.5 % depending upon the sign and magnitude of residual surface stresses. In figure

21(a), with the reduction of surface residual stresses , the combined effect also diminishes. Unlike

pure flexoelectricity (refer to figure 13), when residual surface stress is zero, the deflection ratios of PD- and

UD- FGPM are inverted, i.e.,.

Fig. 22. Variation of maximum deflection of flexo-surface FGPM nanoplates against the thickness

. The results corresponding to LD nanoplates are similar to that of the UD case.

In figure 21(b), it can be observed, for zero residual stress, the deflection ratio is almost independent of in-

plane dimensions if the thickness is fixed, which is similar to the pure flexoelectricity case. But when the

residual surface stress is non-zero, there is a reduction in and it becomes dependent on in-plane

dimensions. It can be also seen that there exists one critical aspect ratio where PD and UD give

the same value of and after that, we get Surface effects become increasingly

apparent in all distributions as in-plane dimensions increase. Figure 22 represents the maximum (center)

deflection of flexo-surface FGPM nanoplate with respect to its thickness for UD and PD distribution with and

without considering residual surface stress . It can be seen that the effect of surface and

flexoelectricity decreases as there is an increment in the thickness of flexo-surface nanoplate, and for both the

residual surface stresses, the deflection is less for PD which is similar to the previously discussed pure

flexoelectricity case.

39

(a) (b)

Fig. 23. Variation of transverse deflection of flexo-surface FGPM nanoplates along the longitudinal axis (x)

for LD and UD distribution with surface effects: (a) and (b) . .

The results corresponding to LD nanoplates are similar to that of the UD case.

(a) (b)

Fig. 24. Variation of transverse deflection of CCCC and SSSS flexo-surface FGPM nanoplates along

longitudinal axis (x) with different distributions: (a) UD and (b) PD. . The results

corresponding to LD nanoplates are similar to that of the UD case.

Figure 23 investigates the effect of surface and flexoelectricity on flexo-surface FGPM nanoplate with

thickness and in-plane dimension as , respectively. As per the discussion in figure

22, it can be seen that PD-FGPM shows less deflection than UD-FGPM. Here, the effect of flexoelectricity

reduces when we increase the residual stress from . In figure 23(a) (), the difference in static

40

(a) (b)

Fig. 25. 3D representation of deflection of flexo-surface FGPM nanoplates with different boundary conditions:

(a) and (b) SSSS. .

(a) (b)

Fig. 26. Variation of transverse deflection of flexo-surface FGPM nanoplates for different weight fractions of

GPLs and distributions: (a) UD and (b) PD ). The results

corresponding to LD nanoplates are similar to that of the UD case.

deflection with and without considering flexoelectricity is more prominent than in figure 23(b) ).

Furthermore, if we compare the deflection reduction by considering only surface effect (), then also

there is a reduction in maximum deflection of nanoplate due to pure surface effect, which is in coherence with

the theory proposed by (Lu et al., 2006). For example, in a square UD-FGPM nanoplate

, pure surface effect reduces the deflection by 91.47 %, whereas the combined surface and flexoelectricity

reduce it near about the same magnitude ~91.52 %.

41

(a) (b)

Fig. 27. Variation of transverse deflection of flexo-surface FGPM nanoplates for various piezoelectric

multiples and distributions: (a) UD and (b) PD. ). The results

corresponding to LD nanoplates are similar to that of the UD case.

Figure 24 shows the influences of the mechanical boundary conditions on the transverse deflection of

flexo-surface FGPM nanoplate considering UD and PD distribution. In both cases and , it is

apparent that CCCC nanoplate is stiffer than SSSS nanoplate as discussed in section 3.2.1. Another notable

observation from this figure is that when we change residual surface stress from , the percentage

reduction of deflection in the SSSS surface nanoplate is much more than the CCCC plate which is true in the

case of both distributions. Figure 25 represents a 3D representation of the deformed shape of the PD-FGPM

plate for a better understanding of the deflection.

Figures 26 and 27 show the influences of the total weight fraction of GPLs in PVDF and piezoelectric

multiple () in static deflection incorporating surface and flexoelectric effects. For the case of these

variations and trends are similar as presented in figures 18 and 19. For the sake of brevity, we have omitted

those results here. It can be also observed that there is a reduction in transverse deflection of flexo-surface

FGPM nanoplate with the increment in weight fraction of GPLs in PVDF matrix as well as increment in the

value of piezoelectric multiples.

3.3 Dynamic response of FGPM nanoplates

In this section, we have performed free vibration analysis of FGPM structures with three different

distributions of graphene nanoplatelets incorporating both flexoelectric and surface effects to carry out

dynamic analysis. The material and dimensional parameters are same as that of the static case which is

summarized in Table 5 and the Ritz method is implemented to extract all the dynamic results. As this is a free

vibration case, the transverse load () is zero, i.e., dynamic analysis is independent of the externally applied

42

load. Further analysis is performed in terms of mode (1, 1) natural eigenfrequency of the structure as this is

fundamentally important for a range of applications.

Table 5. Resonant frequency () () of SSSS flexoelectric FGPM nanoplates

Distributions of

FGM

Thickness (h) (nm)

Navier solution

Ritz solution

UD

20

29.7605

29.7662

40

14.5722

14.5751

60

9.6763

9.6782

80

7.2471

7.2485

100

5.7939

5.7949

LD

20

29.7939

29.7996

40

14.5905

14.5933

60

9.6887

9.6905

80

7.2563

7.2578

100

5.8014

5.8025

PD

20

33.6339

33.6406

40

16.6312

16.6346

60

11.0645

11.0665

80

8.2923

8.2939

100

6.6315

6.6328

In Table 5, the resonant frequencies of one SSSS flexoelectric FGPM nanoplate are compared for

different thicknesses and GPLs distributions obtained from two different solutions – exact (Navier) and present

(Ritz) solution. It demonstrates that the current Ritz solution is capable of producing findings with high level

of accuracy (average % of error is 0.0193%). Having our semi-analytical framework validated with respect to

exact solutions, we further investigate different critical aspects of free vibration and the effects of multiple

influencing parameters. It can be noted in this context that the combined effect of surface and flexoelectricity

becomes more noticeable with decreasing the dimensions of structures (thickness). It is observed that there is

a significant increment in the magnitudes of eigenfrequencies due to the incorporation of surface effect and

43

(a) (b)

Fig. 28. Variations of resonant frequency of FGPM nanoplates with thickness considering different

distributions of GPLs: (a) UD and (b) PD. . The results corresponding to LD nanoplates are

similar to that of the UD case.

this percentage increment reduces in the higher mode of vibration. Our further analysis also reveals that, in

case of PD-FGPM flexoelectric nanoplates , an increase of 193.9% can be observed

in the mode (1,1) whereas this percentage reduces to 76.4% in the mode (3,3) when the surface effect is

considered. Figure 28 depicts the fluctuation in resonant frequency as a function of plate thickness for all three

distributions that include surface, as well as with and without the flexoelectric effect. As discussed in earlier

sections (refer to figure 22), the stiffness of PD-FGPM nanoplate is more as compared to UD and LD, and

Fig. 29. Variation of resonant frequency of flexo-surface FGPM nanoplates with thickness for CCCC and

SSSS boundary conditions (UD, , ).

44

hence, it results in the highest natural frequency for PD distribution considering all residual surface stresses.

For example, in case of , a square UD or LD-FGPM nanoplate shows

281.1 fundamental frequency for mode (1,1), whereas in case of PD-FGPM, it is 290.3 . From figure

28, it is observed, due to the incorporation of surface and flexoelectricity, the natural frequency of FGPM

nanoplate increases, and the results are in very good agreement with existing literature (Ebrahimi and Barati,

2019). Nanoplate with non-zero residual surface stress () gives a higher resonant frequency than

that of zero residual surface stress. For example, in the case of in a square PD-FGPM nanoplate

, the fundamental frequency for mode (1,1) is 290.3 , whereas it is 113.4 for

. Similar trend of results were observed in the existing literature on annular nanoplate (Ghorbanpour

Arani et al., 2021). In figure 28, the difference among these three curves reduces with the increase of thickness.

This is due to the size-dependent effect of nano-scaled structures.

Figure 29 demonstrates the effect of mechanical boundary conditions such as SSSS and CCCC

nanoplate on its resonant frequency. Here, the plate aspect ratio and residual surface stress are 50 and 1 N/m,

respectively. Like conventional plates, the eigenfrequency in the case of CCCC plate is always higher than that

of SSSS plate. Figure 30 represents the first four mode shapes of a square PD-FGPM nanoplate

with CCCC and SSSS boundary conditions incorporating both surface and flexoelectric

effects. Figure 31 shows the variation of resonant frequency in terms of plate thickness for various aspect ratios

and weight fractions of GPLs in the PVDF matrix. For instance, increasing aspect ratio leads to a

decrease in the frequencies of the UD-FGPM nanoplate. The same trend of results is also observed in the case

of other remaining distributions. Incorporating more graphene platelets within the PVDF matrix as

reinforcement leads to an increase in its overall stiffness. Therefore, we get higher values of natural frequency

in the case of

than that of

. For example, in the case of , a square UD-

FGPM nanoplate shows an increase of ~20.01 % in natural frequency if we increase

GPL’s percentage from 1% to 4%. This percentage increases with the aspect ratio, indication a coupled effect

between GPL’s percentage and aspect ratio.

Physical realization of the nano-scale structures, as discussed in this paper, is of crucial importance. From an

experimental viewpoint, it is important to find out an appropriate fabrication technique to manufacture the nanocomposite

structures. Nanofabrication techniques such as layer-by-layer (LbL) assembly, dispersion and solution blending route

methods are widely used to fabricate multifunctional thin films (Gamboa et al., 2010). For instance, the assemblies of

multi-layers of graphene oxide (GO) and polyethylenimine (PEI) were presented by tailoring the thickness of assemblies

by varying the number of GO layers. In the case of bi-layer of GO and PEI, the thickness of assembly near about ~4.5-5

nm was achieved. In some other studies (Yang et al., 2013; Prolongo et al., 2014; Tzeng et al., 2015; Prolongo et al.,

2018), the thickness of assembly was achieved in the range of 8-10 nm using 4 to 30 GPLs. Using all these techniques,

45

CCCC nanoplate

SSSS nanoplate

(a)

(e)

(b)

(f)

(c)

(g)

(d)

(h)

Fig. 30. Mode shapes of flexo-surface FGPM nanoplates with CCCC and SSSS boundary conditions

.

46

Fig. 31. Variation of resonant frequency of flexo-surface FGPM nanoplates with thickness (h) for different

aspect ratios and GPL weight fractions (UD, ). Here a clamped boundary condition is considered.

the fabrication of thin nanocomposite can be achieved on the order of nanometer. Therefore, one can use these

techniques to fabricate graphene-based nanocomposite and achieve significant electromechanical response

considering flexoelectric and surface effects.

In view of the analytical and numerical study, in the present study, the GPLs are assumed as rectangular-

shaped solid reinforcement of average width , length and thickness , and the values of these

parameters are taken 1.8 nm, 3 nm, and 0.7 nm, respectively. We used material properties and geometrical

parameters of GPLs and PVDF from existing literature which are given in Table 4. Moreover, we make use of

‘classical Kirchhoff plate theory’ (Eq. 15) which generally gives accurate results if the concerned plate is thin

(). Now for instance, if we take a square plate with aspect ratio and length (1000 nm),

then it is evident that the GPLs with thickness 0.7 nm can easily be accommodated within the plate of 10 nm

thickness.

4. Concluding remarks

The coupled electromechanical behaviour of graphene reinforced functionally graded piezoelectric

material (FGPM) nanoplates is explored by taking into account the surface and flexoelectricity effects

concerning the static and dynamic responses. Owing to inhomogeneous strain and high surface-to-volume ratio

in nanostructures, it is important to account for the flexoelectricity as well as surface effect while analyzing

the size-dependent electromechanical responses of nano-scale piezoelectric materials. An efficient semi-

analytical framework is developed here for the FGPM nanoplates under the open-circuit condition, wherein

the single term extended Kantorovich method (EKM) is implemented for static analysis and the Ritz method

is adopted for dynamic analysis. The numerical results are extensively validated with existing literature for

47

checking the accuracy of the present model. It is noted that the novel EKM based framework for graphene

reinforced FGPM nanoplates provides rapid convergence for investigating the role of surface and

flexoelectricity effects, while the Ritz method is capable of predicting accurate dynamic behavior of the system

incorporating both the effects.

Based on the unified semi-analytical framework, a comprehensive investigation is carried out on static

deflection and free vibration considering different distributions of graphene platelets in FGPM nanoplates such

as linear distribution (LD), uniform distribution (UD), and parabolic distribution (PD). In all cases, the

parabolic distribution exhibits the stiffest behavior and higher natural frequency compared to the other

distributions. In this study, the electromechanical response of nanoplates is investigated considering different

crucial parameters such as plate thicknesses, aspect ratios, flexoelectric coefficients, piezoelectric multiples

and weight fraction of graphene platelets along with different boundary conditions. Our findings reveal that

the static deflection and dynamic resonance of FGPM are enhanced significantly due to consideration of

flexoelectricity and surface effects, leading to the realization that such effects cannot be neglected. Further,

these electromechanical effects can be exploited in designing novel materials and devices for active response

control and energy harvesting. The electromechanical effects are found to be more pronounced for nanoplates

of lesser thickness, and these diminish as plate thickness increases, indicating a novel size-dependent behaviour

that could potentially be of significant importance for micro and nano architected materials. The current

investigation further reveals that the stiffening behavior is greatly dependent on the sign and magnitude of

residual surface stress. In the absence of surface effect, FG-PD shows a high correlation to the thickness change

as compared to FG-UD and FG-LD. However, in the presence of a surface effect, this correlation can be

modulated in different distributions as per application-specific demands. In the presence of non-zero residual

stresses, the static and dynamic responses for different FGPM distributions are dependent on the in-plane

dimensions of the plates in addition to thickness. Such numerical outcomes essentially open up the avenues of

prospective exploitation and augmentation of the electromechanical responses in an expanded design space

including the factors like open- and close-circuit condition, strain/electric field gradient, electrical and

mechanical loading as well as converse piezoelectricity and flexoelectricity. With the recent advances in nano-

scale manufacturing and experimental capabilities, this article will provide the necessary physical insights in

modeling the size-dependent electromechanical coupling in multifunctional materials, systems and devices for

applications in sensors, actuators, nanogenerators, active controllers, nano-robotics and energy harvesters.

Acknowledgments

SN acknowledges the financial support through initiation grant from University of Southampton. KBS would

like to thank IIT Bombay for the Institute Post-doctoral Fellowship (IPDF). TM would like to acknowledge

the financial support from Science and Engineering Research Board (Grant no. SRG/2020/001398).

48

Conﬂict of Interest

The authors declare no conﬂict of interest.

Data availability statement

All relevant data have been included either in the manuscript or it will be made available upon reasonable

requests.

ORCID ID

Susmita Naskar https://orcid.org/0000-0003-3294-8333

Kishor Shingare https://orcid.org/0000-0002-8255-9849

Soumyadeep Mondal https://orcid.org/0000-0002-3368-7828

Tanmoy Mukhopadhyay https://orcid.org/0000-0002-0778-6515

Appendix A

The stiffness coefficients and their algebraic expressions in Eq. (22) are expressed as follows:

49

50

Appendix B

All the coefficients of each term and their algebraic expressions in Eq. (24) are expressed as follows:

Here, and are non-linear terms which are expressed as follows:

The coefficients and their algebraic expressions in Eq. (32) are expressed as:

51

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