* All authors have contributed equally
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
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
Keywords: Flexoelectricity and surface effect; Size-dependence in composite materials; Graphene reinforced
functionally graded materials; Extended Kantorovich method; Ritz method; Electromechanical responses.
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
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:
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).
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
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
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
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.
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-
Fig. 2. (a) Geometry and coordinate system of thin FGPM nanoplates under open circuit condition, (b) upper
and lower surface of the nanoplate.
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):
Parabolic distribution (PD):
Linear distribution (LD):
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:
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):
whereas and indicate moduli in the longitudinal and transverse directions respectively and the values
can be estimated from Eq. (4). Here,
are the reinforcing efficiency of GPLs considered in longitudinal
and transverse directions, respectively.
where the parameters
can be expressed by:
The parameters and indicate the respective Young moduli of GPLs and PVDF matrix. The filler
of GPLs are given by the following equation:
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
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
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 ():
Lower surface ():
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).
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:
. 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:
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):
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:
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
surfaces are insulated (Wang and Zhou, 2013). It also satisfies Gauss law of dielectrics
free electric charge density is zero. From Eq. (11f), for the zero electric displacement , the
internal electric field can be expressed as follows:
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:
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):
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:
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.
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.
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.
are obtained as follows:
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:
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:
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:
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:
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
From Eq. (29), the following ODE of is obtained.
For solving Eq. (30a), the requisite boundary conditions in terms of from the mechanical boundary
conditions shown in figure 4 can be given as:
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:
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:
The Ritz solution of the displacement W(x,y) for the square plate is assumed in the following form (Reddy,
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
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 .
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
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
Fig. 6. Convergence plot for transverse deflection of UD-FGPM nanoplates with respect to aspect ratio
considering three iterations.
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.
Location of Max.
Magnitude of Max. deflection
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
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 aspect ratio,
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
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,
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
() 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).
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
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,
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
flexoelectric effects. In later sections, we have also pointed out few limitations of our present model in
predicting its electromechanical characteristics.
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
Table 4. Geometric and material properties of constituents of FGPM (Arefi et al., 2018; Z. Zhao et al., 2020).
Elastic and geometrical properties
Charac. length ()
1 or 0
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
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
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
observed, due to consideration of the flexoelectric effect there is a significant reduction in deflection of FGPM
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.
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
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
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.
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
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
depends on size parameters of the structure but also on the value of its flexoelectric coefficient, GPLs weight
fraction and piezoelectric multiple.
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
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
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
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:
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
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
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.
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
Fig. 25. 3D representation of deflection of flexo-surface FGPM nanoplates with different boundary conditions:
(a) and (b) SSSS. .
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 %.
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
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
Thickness (h) (nm)
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
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, , ).
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,
Fig. 30. Mode shapes of flexo-surface FGPM nanoplates with CCCC and SSSS boundary conditions
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
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
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.
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).
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
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
The stiffness coefficients and their algebraic expressions in Eq. (22) are expressed as follows:
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:
Abolhasani, M.M., Shirvanimoghaddam, K., Naebe, M., 2017. PVDF/graphene composite nanofibers with
enhanced piezoelectric performance for development of robust nanogenerators. Compos. Sci. Technol.
138, 49–56. https://doi.org/10.1016/j.compscitech.2016.11.017
Aitken, Z.H., Fan, H., El-Awady, J.A., Greer, J.R., 2015. The effect of size, orientation and alloying on the
deformation of AZ31 nanopillars. J. Mech. Phys. Solids 76, 208–223.
Akbarzadeh Khorshidi, M., 2020. Validation of weakening effect in modified couple stress theory:
Dispersion analysis of carbon nanotubes. Int. J. Mech. Sci. 170.
Akbarzadeh Khorshidi, M., 2018. The material length scale parameter used in couple stress theories is not a
material constant. Int. J. Eng. Sci. 133, 15–25. https://doi.org/10.1016/j.ijengsci.2018.08.005
Arefi, M., Mohammad-Rezaei Bidgoli, E., Dimitri, R., Tornabene, F., 2018. Free vibrations of functionally
graded polymer composite nanoplates reinforced with graphene nanoplatelets. Aerosp. Sci. Technol. 81,
Baskaran, S., He, X., Chen, Q., Fu, J.Y., 2011. Experimental studies on the direct flexoelectric effect in α -
phase polyvinylidene fluoride films. Appl. Phys. Lett. 98. https://doi.org/10.1063/1.3599520
Beeby, S.P., Tudor, M.J., White, N.M., 2006. Energy harvesting vibration sources for microsystems
applications. Meas. Sci. Technol. 17. https://doi.org/10.1088/0957-0233/17/12/R01
Beheshti-Aval, S.B., Lezgy-Nazargah, M., 2010. A finite element model for composite beams with
piezoelectric layers using a sinus model. J. Mech. 26, 249–258.
Chan, K.T., Pu, Z.Y., 2011. The dispersion characteristics of the waves propagating in a spinning single-
walled carbon nanotube. Sci. China Physics, Mech. Astron. 54, 1854–1865.
Chandratre, S., Sharma, P., 2012. Coaxing graphene to be piezoelectric. Appl. Phys. Lett. 100.
Chen, D., Yang, J., Kitipornchai, S., 2017. Nonlinear vibration and postbuckling of functionally graded
graphene reinforced porous nanocomposite beams. Compos. Sci. Technol. 142, 235–245.
Curie, J., Curie, P., 1880. Développement par compression de l’électricité polaire dans les cristaux hémièdres
à faces inclinées. Bull. la Société minéralogique Fr. 3, 90–93. https://doi.org/10.3406/bulmi.1880.1564
Dang, Z.M., Lin, Y.H., Nan, C.W., 2003. Novel Ferroelectric Polymer Composites with High Dielectric
Constants. Adv. Mater. 15, 1625–1629. https://doi.org/10.1002/adma.200304911
Deng, Q., Kammoun, M., Erturk, A., Sharma, P., 2014. Nanoscale flexoelectric energy harvesting. Int. J.
Solids Struct. 51, 3218–3225. https://doi.org/10.1016/j.ijsolstr.2014.05.018
Ebrahimi, F., Barati, M.R., 2019. Dynamic modeling of embedded nanoplate systems incorporating
flexoelectricity and surface effects. Microsyst. Technol. 25, 175–187. https://doi.org/10.1007/s00542-
Ebrahimi, F., Barati, M.R., 2017. Static stability analysis of embedded flexoelectric nanoplates considering
surface effects. Appl. Phys. A Mater. Sci. Process. 123. https://doi.org/10.1007/s00339-017-1265-y
Ebrahimi, F., Hosseini, S.H.S., 2020. Investigation of flexoelectric effect on nonlinear forced vibration of
piezoelectric/functionally graded porous nanocomposite resting on viscoelastic foundation. J. Strain
Anal. Eng. Des. 55, 53–68. https://doi.org/10.1177/0309324719890868
Feng, C., Kitipornchai, S., Yang, J., 2017. Nonlinear bending of polymer nanocomposite beams reinforced
with non-uniformly distributed graphene platelets (GPLs). Compos. Part B Eng. 110, 132–140.
Ghobadi, A., Beni, Y.T., Golestanian, H., 2020. Nonlinear thermo-electromechanical vibration analysis of
size-dependent functionally graded flexoelectric nano-plate exposed magnetic field. Arch. Appl. Mech.
90, 2025–2070. https://doi.org/10.1007/s00419-020-01708-0
Ghobadi, A., Golestanian, H., Beni, Y.T., Żur, K.K., 2021a. On the size-dependent nonlinear thermo-electro-
mechanical free vibration analysis of functionally graded flexoelectric nano-plate. Commun. Nonlinear
Sci. Numer. Simul. 95. https://doi.org/10.1016/j.cnsns.2020.105585
Ghobadi, A., Tadi Beni, Y., Kamil Żur, K., 2021b. Porosity distribution effect on stress, electric field and
nonlinear vibration of functionally graded nanostructures with direct and inverse flexoelectric
phenomenon. Compos. Struct. 259. https://doi.org/10.1016/j.compstruct.2020.113220
Ghorbanpour Arani, A., Soltan Arani, A.H., Haghparast, E., 2021. Flexoelectric and surface effects on
vibration frequencies of annular nanoplate. Indian J. Phys. 95, 2063–2083.
Gurtin, M.E., Ian Murdoch, A., 1975. A continuum theory of elastic material surfaces. Arch. Ration. Mech.
Anal. 57, 291–323. https://doi.org/10.1007/BF00261375
Hamdia, K.M., Ghasemi, H., Zhuang, X., Alajlan, N., Rabczuk, T., 2018. Sensitivity and uncertainty analysis
for flexoelectric nanostructures. Comput. Methods Appl. Mech. Eng. 337, 95–109.
He, J., Lilley, C.M., 2008. Surface effect on the elastic behavior of static bending nanowires. Nano Lett. 8,
Hosseini, M., Jamalpoor, A., Fath, A., 2017. Surface effect on the biaxial buckling and free vibration of FGM
nanoplate embedded in visco-Pasternak standard linear solid-type of foundation. Meccanica 52, 1381–
Hu, X., Zhou, Y., Liu, J., Chu, B., 2018. Improved flexoelectricity in PVDF/barium strontium titanate (BST)
nanocomposites. J. Appl. Phys. 123. https://doi.org/10.1063/1.5022650
Huang, G.Y., Yu, S.W., 2006. Effect of surface piezoelectricity on the electromechanical behaviour of a
piezoelectric ring. Phys. Status Solidi Basic Res. 243. https://doi.org/10.1002/pssb.200541521
Jang, D., Meza, L.R., Greer, F., Greer, J.R., 2013. Fabrication and deformation of three-dimensional hollow
ceramic nanostructures. Nat. Mater. https://doi.org/10.1038/nmat3738
Jones, R., Milne, B.J., 1976. Application of the extended Kantorovich method to the vibration of clamped
rectangular plates. J. Sound Vib. 45, 309–316. https://doi.org/10.1016/0022-460X(76)90390-4
Kapuria, S., Kumari, P., 2011. Extended Kantorovich method for three-dimensional elasticity solution of
laminated composite structures in cylindrical bending. J. Appl. Mech. Trans. ASME 78, 1–8.
Kapuria, S., Kumari, P., 2012. Multiterm extended kantorovich method for three-dimensional elasticity
solution of laminated plates. J. Appl. Mech. Trans. ASME 79, 1–9. https://doi.org/10.1115/1.4006495
Kapuria, S., Kumari, P., 2013. Extended Kantorovich method for coupled piezoelasticity solution of
piezolaminated plates showing edge effects. Proc. R. Soc. A Math. Phys. Eng. Sci. 469, 20120565.
Karsh, P.K., Mukhopadhyay, T., Chakraborty, S., Naskar, S., Dey, S., 2019. A hybrid stochastic sensitivity
analysis for low-frequency vibration and low-velocity impact of functionally graded plates. Compos.
Part B Eng. 176, 107221. https://doi.org/10.1016/j.compositesb.2019.107221
Kerr, A.D., Alexander, H., 1968. An application of the extended Kantorovich method to the stress analysis of
a clamped rectangular plate. Acta Mech. 6, 180–196. https://doi.org/10.1007/BF01170382
Kiani, Y., 2018. Isogeometric large amplitude free vibration of graphene reinforced laminated plates in
thermal environment using NURBS formulation. Comput. Methods Appl. Mech. Eng. 332, 86–101.
Komijani, M., Reddy, J.N., Eslami, M.R., 2014. Nonlinear analysis of microstructure-dependent functionally
graded piezoelectric material actuators. J. Mech. Phys. Solids 63, 214–227.
Kumar, A., Kiran, R., Kumar, R., Chandra Jain, S., Vaish, R., 2018. Flexoelectric effect in functionally
graded materials: A numerical study. Eur. Phys. J. Plus 133. https://doi.org/10.1140/epjp/i2018-11976-1
Kumari, P., Shakya, A.K., 2017. Two-dimensional Solution of Piezoelectric Plate Subjected to Arbitrary
Boundary Conditions using Extended Kantorovich Method, in: Procedia Engineering. pp. 1523–1530.
Kundalwal, S.I., Shingare, K.B., 2020. Electromechanical response of thin shell laminated with flexoelectric
composite layer. Thin-Walled Struct. 157. https://doi.org/10.1016/j.tws.2020.107138
Kundalwal, S. I., Shingare, K.B., Gupta, M., 2020. Flexoelectric effect on electric potential in piezoelectric
graphene-based composite nanowire: Analytical and numerical modelling. Eur. J. Mech. A/Solids 84.
Lezgy-Nazargah, M., Vidal, P., Polit, O., 2013. An efficient finite element model for static and dynamic
analyses of functionally graded piezoelectric beams. Compos. Struct. 104, 71–84.
Li, A., Wang, B., Yang, S., 2021. On some basic aspects of flexoelectricity in the mechanics of materials. Int.
J. Eng. Sci. 166. https://doi.org/10.1016/j.ijengsci.2021.103499
Li, Z., He, Y., Lei, J., Guo, S., Liu, D., Wang, L., 2018. A standard experimental method for determining the
material length scale based on modified couple stress theory. Int. J. Mech. Sci. 141, 198–205.
Liang, X., Hu, S., Shen, S., 2013. Bernoulli-Euler dielectric beam model based on strain-gradient effect. J.
Appl. Mech. Trans. ASME 80. https://doi.org/10.1115/1.4023022
Lippmann, G., 1881. Principe de la conservation de l’électricité, ou second principe de la théorie des
phénomènes électriques. J. Phys. Théorique Appliquée 10, 381–394.
Liu, C., Rajapakse, R.K.N.D., 2013. A size-dependent continuum model for nanoscale circular plates. IEEE
Trans. Nanotechnol. 12, 13–20. https://doi.org/10.1109/TNANO.2012.2224880
Liu, C., Rajapakse, R.K.N.D., 2010. Continuum models incorporating surface energy for static and dynamic
response of nanoscale beams. IEEE Trans. Nanotechnol. 9, 422–431.
Liu, C., Rajapakse, R.K.N.D., Phani, A.S., 2011. Finite element modeling of beams with surface energy
effects. J. Appl. Mech. Trans. ASME 78. https://doi.org/10.1115/1.4003363
Lu, P., He, L.H., Lee, H.P., Lu, C., 2006. Thin plate theory including surface effects. Int. J. Solids Struct. 43,
Mao, J.J., Zhang, W., 2018. Linear and nonlinear free and forced vibrations of graphene reinforced
piezoelectric composite plate under external voltage excitation. Compos. Struct. 203, 551–565.
Mehralian, F., Tadi Beni, Y., Karimi Zeverdejani, M., 2017. Nonlocal strain gradient theory calibration using
molecular dynamics simulation based on small scale vibration of nanotubes. Phys. B Condens. Matter
514, 61–69. https://doi.org/10.1016/j.physb.2017.03.030
Miller, R.E., Shenoy, V.B., 2000. Size-dependent elastic properties of nanosized structural elements.
Nanotechnology 11, 139–147. https://doi.org/10.1088/0957-4484/11/3/301
Naskar, S., 2018a. Spatial Variability Characterisation of Laminated Composites. University of Aberdeen.
Naskar, S., Mukhopadhyay, T., Sriramula, S., 2018b. Probabilistic micromechanical spatial variability
quantification in laminated composites. Compos. Part B Eng. 151, 291–325.
Naskar, S., Mukhopadhyay, T., Sriramula, S., 2019. Spatially varying fuzzy multi-scale uncertainty
propagation in unidirectional fibre reinforced composites. Compos. Struct. 209, 940–967.
Naskar, S., Mukhopadhyay, T., Sriramula, S., Adhikari, S., 2017. Stochastic natural frequency analysis of
damaged thin-walled laminated composite beams with uncertainty in micromechanical properties.
Compos. Struct. 160, 312–334. https://doi.org/10.1016/j.compstruct.2016.10.035
Pan, X., Yu, S., Feng, X., 2011. A continuum theory of surface piezoelectricity for nanodielectrics. Sci.
China Physics, Mech. Astron. 54, 564–573. https://doi.org/10.1007/s11433-011-4275-3
Reddy, J.N., 2006. Theory and Analysis of Elastic Plates and Shells, Theory and Analysis of Elastic Plates
and Shells. https://doi.org/10.1201/9780849384165
Reddy, J.N., 2003. Mechanics of Laminated Composite Plates and Shells, Mechanics of Laminated
Composite Plates and Shells. https://doi.org/10.1201/b12409
Reddy, J.N., 1999. On laminated composite plates with integrated sensors and actuators. Eng. Struct. 21,
Reddy, J.N., Cheng, Z.Q., 2001. Three-dimensional solutions of smart functionally graded plates. J. Appl.
Mech. Trans. ASME 68, 234–241. https://doi.org/10.1115/1.1347994
Sapsathiarn, Y., Rajapakse, R.K.N.D., 2017. Static and dynamic analyses of nanoscale rectangular plates
incorporating surface energy. Acta Mech. 228, 2849–2863. https://doi.org/10.1007/s00707-015-1521-1
Sharma, S., Kumar, R., Talha, M., Vaish, R., 2021. Flexoelectric Poling of Functionally Graded Ferroelectric
Materials. Adv. Theory Simulations 4. https://doi.org/10.1002/adts.202000158
Shen, H.S., Lin, F., Xiang, Y., 2017a. Nonlinear bending and thermal postbuckling of functionally graded
graphene-reinforced composite laminated beams resting on elastic foundations. Eng. Struct. 140, 89–97.
Shen, H.S., Xiang, Y., Lin, F., Hui, D., 2017b. Buckling and postbuckling of functionally graded graphene-
reinforced composite laminated plates in thermal environments. Compos. Part B Eng. 119, 67–78.
Shenoy, V.B., 2005. Atomistic calculations of elastic properties of metallic fcc crystal surfaces. Phys. Rev. B
- Condens. Matter Mater. Phys. 71, 1–11. https://doi.org/10.1103/PhysRevB.71.094104
Shingare, K.B., Kundalwal, S.I., 2020. Flexoelectric and surface effects on the electromechanical behavior of
graphene-based nanobeams. Appl. Math. Model. 81, 70–91. https://doi.org/10.1016/j.apm.2019.12.021
Shingare, K.B., Kundalwal, S.I., 2019. Static and dynamic response of graphene nanocomposite plates with
flexoelectric effect. Mech. Mater. 134, 69–84. https://doi.org/10.1016/j.mechmat.2019.04.006
Shingare, K.B., Naskar, S., 2021a. Probing the prediction of effective properties for composite materials.
Eur. J. Mech. A/Solids 87. https://doi.org/10.1016/j.euromechsol.2021.104228
Shingare, K.B., Naskar, S., 2021b. Effect of size-dependent properties on electromechanical behavior of
composite structures, in: UK Association for Computational Mechanics (UK ACM 2021). pp. 0–4.
Shingare, Kishor Balasaheb, Naskar, S., 2021c. Analytical solution for static and dynamic analysis of
graphene-based hybrid flexoelectric nanostructures. J. Compos. Sci. 5.
Shu, L., Li, F., Huang, W., Wei, X., Yao, X., Jiang, X., 2014. Relationship between direct and converse
flexoelectric coefficients. J. Appl. Phys. 116. https://doi.org/10.1063/1.4897647
Shu, L., Liang, R., Rao, Z., Fei, L., Ke, S., Wang, Y., 2019. Flexoelectric materials and their related
applications: A focused review. J. Adv. Ceram. https://doi.org/10.1007/s40145-018-0311-3
Shu, L., Wei, X., Pang, T., Yao, X., Wang, C., 2011. Symmetry of flexoelectric coefficients in crystalline
medium, in: Journal of Applied Physics. https://doi.org/10.1063/1.3662196
Singh, A., Mukhopadhyay, T., Adhikari, S., Bhattacharya, B., 2021. Voltage-dependent modulation of elastic
moduli in lattice metamaterials: Emergence of a programmable state-transition capability. Int. J. Solids
Struct. 208–209, 31–48. https://doi.org/10.1016/j.ijsolstr.2020.10.009
Singhatanadgid, P., Singhanart, T., 2019. The Kantorovich method applied to bending, buckling, vibration,
and 3D stress analyses of plates: A literature review. Mech. Adv. Mater. Struct. 26, 170–188.
Song, J., Zhou, J., Wang, Z.L., 2006. Piezoelectric and semiconducting coupled power generating process of
a single ZnO belt/wire. A technology for harvesting electricity from the environment. Nano Lett. 6,
Song, M., Kitipornchai, S., Yang, J., 2017. Free and forced vibrations of functionally graded polymer
composite plates reinforced with graphene nanoplatelets. Compos. Struct. 159, 579–588.
Song, M., Yang, J., Kitipornchai, S., 2018. Bending and buckling analyses of functionally graded polymer
composite plates reinforced with graphene nanoplatelets. Compos. Part B Eng. 134, 106–113.
Sugino, C., Ruzzene, M., Erturk, A., 2018. Merging mechanical and electromechanical bandgaps in locally
resonant metamaterials and metastructures. J. Mech. Phys. Solids 116, 323–333.
Talha, M., Singh, B.N., 2010. Static response and free vibration analysis of FGM plates using higher order
shear deformation theory. Appl. Math. Model. 34, 3991–4011.
Thomas, B., Roy, T., 2017. Vibration and damping analysis of functionally graded carbon nanotubes
reinforced hybrid composite shell structures. JVC/Journal Vib. Control 23, 1711–1738.
Trindade, M.A., Benjeddou, A., 2009. Effective electromechanical coupling coefficients of piezoelectric
adaptive structures: Critical evaluation and optimization. Mech. Adv. Mater. Struct. 16, 210–223.
Trinh, M.C., Mukhopadhyay, T., Kim, S.E., 2020. A semi-analytical stochastic buckling quantification of
porous functionally graded plates. Aerosp. Sci. Technol. 105, 105928.
Vaishali, Mukhopadhyay, T., Karsh, P.K., Basu, B., Dey, S., 2020. Machine learning based stochastic
dynamic analysis of functionally graded shells. Compos. Struct. 237, 111870.
Vatanabe, S.L., Rubio, W.M., Silva, E.C.N., 2014. Modeling of Functionally Graded Materials, in:
Comprehensive Materials Processing. Elsevier, pp. 261–282. https://doi.org/10.1016/B978-0-08-
Vidal, P., Polit, O., 2008. A family of sinus finite elements for the analysis of rectangular laminated beams.
Compos. Struct. 84, 56–72. https://doi.org/10.1016/j.compstruct.2007.06.009
Wang, B., Li, X.F., 2021. Flexoelectric effects on the natural frequencies for free vibration of piezoelectric
nanoplates. J. Appl. Phys. 129. https://doi.org/10.1063/5.0032343
Wang, L., Hu, H., 2005. Flexural wave propagation in single-walled carbon nanotubes. Phys. Rev. B -
Condens. Matter Mater. Phys. 71. https://doi.org/10.1103/PhysRevB.71.195412
Wang, X., Zhou, J., Song, J., Liu, J., Xu, N., Wang, Z.L., 2006. Piezoelectric field effect transistor and
nanoforce sensor based on a single ZnO nanowire. Nano Lett. 6, 2768–2772.
Wang, X., Zhou, K., 2013. Twelve-dimensional Stroh-like formalism for Kirchhoff anisotropic piezoelectric
thin plates. Int. J. Eng. Sci. 71, 111–136. https://doi.org/10.1016/j.ijengsci.2013.06.004
Wang, Y., Xie, K., Fu, T., 2020. Vibration analysis of functionally graded graphene oxide-reinforced
composite beams using a new Ritz-solution shape function. J. Brazilian Soc. Mech. Sci. Eng. 42.
Xu, K., Wang, K., Zhao, W., Bao, W., Liu, E., Ren, Y., Wang, M., Fu, Y., Zeng, J., Li, Z., Zhou, W., Song,
F., Wang, X., Shi, Y., Wan, X., Fuhrer, M.S., Wang, B., Qiao, Z., Miao, F., Xing, D., 2015. The
positive piezoconductive effect in graphene. Nat. Commun. 6. https://doi.org/10.1038/ncomms9119
Yan, Zhi, Jiang, L., 2012. Surface effects on the electroelastic responses of a thin piezoelectric plate with
nanoscale thickness. J. Phys. D. Appl. Phys. 45. https://doi.org/10.1088/0022-3727/45/25/255401
Yan, Z., Jiang, L.Y., 2012. Vibration and buckling analysis of a piezoelectric nanoplate considering surface
effects and in-plane constraints. Proc. R. Soc. A Math. Phys. Eng. Sci. 468, 3458–3475.
Yang, B., Kitipornchai, S., Yang, Y.F., Yang, J., 2017. 3D thermo-mechanical bending solution of
functionally graded graphene reinforced circular and annular plates. Appl. Math. Model. 49, 69–86.
Yang, F., Chong, A.C.M., Lam, D.C.C., Tong, P., 2002. Couple stress based strain gradient theory for
elasticity. Int. J. Solids Struct. 39, 2731–2743. https://doi.org/10.1016/S0020-7683(02)00152-X
Yang, J., Chen, D., Kitipornchai, S., 2018. Buckling and free vibration analyses of functionally graded
graphene reinforced porous nanocomposite plates based on Chebyshev-Ritz method. Compos. Struct.
193, 281–294. https://doi.org/10.1016/j.compstruct.2018.03.090
Yang, W., Liang, X., Shen, S., 2015. Electromechanical responses of piezoelectric nanoplates with
flexoelectricity. Acta Mech. 226, 3097–3110. https://doi.org/10.1007/s00707-015-1373-8
Zhang, Chunli, Chen, W., Zhang, Ch, 2013. Two-dimensional theory of piezoelectric plates considering
surface effect. Eur. J. Mech. A/Solids 41, 50–57. https://doi.org/10.1016/j.euromechsol.2013.02.005
Zhang, J., Wang, C., Adhikari, S., 2012. Surface effect on the buckling of piezoelectric nanofilms. J. Phys. D.
Appl. Phys. 45. https://doi.org/10.1088/0022-3727/45/28/285301
Zhang, Z., Yan, Z., Jiang, L., 2014. Flexoelectric effect on the electroelastic responses and vibrational
behaviors of a piezoelectric nanoplate. J. Appl. Phys. 116. https://doi.org/10.1063/1.4886315
Zhao, S., Zhao, Z., Yang, Z., Ke, L.L., Kitipornchai, S., Yang, J., 2020. Functionally graded graphene
reinforced composite structures: A review. Eng. Struct. https://doi.org/10.1016/j.engstruct.2020.110339
Zhao, Z., Ni, Y., Zhu, S., Tong, Z., Zhang, J., Zhou, Z., Lim, C.W., Xu, X., 2020. Thermo-Electro-
Mechanical Size-Dependent Buckling Response for Functionally Graded Graphene Platelet Reinforced
Piezoelectric Cylindrical Nanoshells. Int. J. Struct. Stab. Dyn. 20.
Zhou, Y., Liu, J., Hu, X., Chu, B., Chen, S., Salem, D., 2017. Flexoelectric effect in PVDF-based polymers.
IEEE Trans. Dielectr. Electr. Insul. 24, 727–731. https://doi.org/10.1109/TDEI.2017.006273