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On characterizing the viscoelastic electromechanical responses of functionally graded
graphene-reinforced piezoelectric laminated composites: Temporal programming based
on a semi-analytical higher-order framework
S. Mondal1, K. B. Shingare1, T. Mukhopadhyay1, S. Naskar1,*
1Faculty of Engineering and Physical Sciences, University of Southampton, Southampton, UK
*Email Address: S.Naskar@soton.ac.uk (S. Naskar)
Abstract
The electromechanical responses of single and multi-layered piezoelectric functionally graded graphene-
reinforced composite (FG-GRC) plates are studied based on an accurate higher-order shear deformation theory
(HSDT) involving quasi-3D sinusoidal plate theory and linear piezoelectricity. These FG-GRC plates are
composed of randomly oriented graphene nanoplatelets (GPLs) reinforcing fillers and the piezoelectric PVDF
matrix considering two different distribution patterns such as linear- and uniform- distribution (LD and UD)
of GPLs across the thickness. The modified Halpin-Tsai (HT) and Rule of mixture (ROM) models are utilized
to determine the effective material properties of FG-GRCs. The analytical model of FG-GRCs is extended
further to analyze the time-dependent linear viscoelastic electromechanical behaviour of the system based on
Biot model of viscoelasticity in the framework of inverse Fourier algorithm. The viscoelastic electromechanical
responses include the static deformation and electric responses of simply supported FG-GRC plates which are
investigated by considering transverse mechanical and external electrical loading, as well as other critical
parameters like aspect ratio and weight fraction of GPLs. The numerical results reveal that the
electromechanical response of FG-GRC plates can be enriched due to the addition of a small weight fraction
of GPLs. The coupled multiphysics-based computational framework proposed here for predicting the
viscoelastic electromechanical behaviour of laminated composites can be exploited for stimulating and
developing a wide range of micro-electro-mechanical systems (MEMS) and devices incorporating time-
dependent programming features.
Keywords
Active composite laminates; Graphene-reinforced piezoelectric laminates; Functionally graded materials;
Piezoelectric effect; Viscoelastic electromechanical responses
1. Introduction
Owing to stimulating multi-functional properties, two-dimensional (2D) materials and their derivatives
have emerged to be significantly vital nanomaterials (Saumya et al., 2023). The quest for exploiting
extraordinary mechanical properties of 2D graphene (modulus of elasticity, ~1 TPa and strength, 130 GPa) and
2
high specific surface area (2630 m2/g) led to the opening of an evolving area of research for developing
graphene-based nanocomposites. It can be utilized as an ideal reinforcement candidate for tailoring
multifunctional hybrid composites, leading to multifarious applications. The mechanical and interfacial
properties of graphene-based composites are significantly enhanced due to the addition of controlled loading
contents of graphene. Extensive research is carried out for the prediction of electromechanical behaviour
including static and dynamic analysis of graphene-reinforced composite (GRCs) structures such as beam, plate,
rod and shell by introducing piezoelectric nanoscale graphene fiber in a non-piezoelectric polyimide matrix
(Mukhopadhyay et al., 2021) (Naskar et al., 2022) (Shingare and Naskar, 2023). Graphene is considered as a
nanoscale fiber, wherein the effect of size-dependent properties is investigated such as strain and electric field
gradient as well as piezoelectric, flexoelectric and surface effects on these GRC structures. The effective piezo
elastic and relative permittivity properties of GRC are also probed by using analytical and finite element (FE)
micromechanical models. They show a significant enhancement in mechanical and electrical behaviour of GRC
structures as it incorporates flexoelectric and surface effects in comparison to the piezoelectric effect. Despite
its importance in material design and engineering, research on the behaviors of graphene-reinforced composite
is still inadequate.
In the last few years, a new class of materials has emerged as an excellent choice of researchers due to
its unique tailorable variation in material properties mainly across the thickness, known as functionally graded
materials (FGMs) (Singh et al. 2023a). FGMs show great potential to be used in several areas like automotive,
aerospace, aviation and many other engineering domains as shown in Fig. 1. The quest for high performance
and exceptional properties including high stiffness, light in weight, durability, and high load-bearing/resistance
capacity are the main causes for adopting FGMs (Karsh et al., 2019, 2018; Trinh et al., 2020). The influences
of the graded components on the deformation and strength of thick-walled FGM tubes when subjected to
internal pressure were examined by Fukui and Yamanaka (1992). Further, Fukui et al. (1993) extended their
earlier research by considering these FGM tubes when subjected to uniform thermal loading and examined the
effects of gradation of components on residual stresses. By curtailing the circumferential compressive stress at
the inner surface of FGM tube, they also proposed an optimal composition. To examine stress intensity factors
and transient thermal stresses of FGMs considering cracks, Fuchiyama et al., (1993) utilized an eight-noded
quad-axisymmetric element. In order to get more accurate results, they emphasized that the temperature-
dependent properties must be accounted for in the investigation. Utilizing optimization and sensitivity
techniques, Tanaka et al. (1993) considered FGM property profiles based on the lessening of thermal stresses.
Mondal et al. (2022) formulated the closed-form solutions of three-dimensional crack-tip stress fields for a
FGM medium under thermo-mechanical loading and investigated the scope of delaying its failure mechanisms.
Lu et al. (2006) used stroh-like formalism to analyze a simply supported laminate of functionally graded
piezoelectric material (FGPM). Behjat et al. (2011) carried out FE formulation in their paper to investigate
3
static bending, free vibration and dynamic behaviour of FGPM plates where material properties are graded
along thickness direction based on power law. Das and Sarangi (2016) modeled and performed the analysis of
an FG beam within ANSYS environment using Solid 186 element. Based on the Navier solution and first-order
shear deformation theory (FSDT), Song et al. (2017a) reported the free and forced vibration analyses of FG
graphene platelet-reinforced composite (FG GPLRC) laminates. Afterward, by utilizing the same theories,
Song et al. (2017b) investigated the buckling and post-buckling behavior of bi-axially compressed FG GPLRC
plates. Shen et al. (2017) derived an analytical solution for buckling and post-buckling analyses of FG GPLRC
plates rested on elastic foundation. In addition to this, the results of thermal buckling and the post-buckling of
GPLRC plates were analyzed by Wu et al. (2017) using the differential quadrature-based iteration method.
They showed the increment and decrement of thermal buckling and post-buckling resistance considering
different parameters such as weight fraction of GPLs along with their distribution, width-to-thickness, and
aspect ratios. Gholami and Ansari (2017) investigated the large deflection and geometrically nonlinear analysis
of FG GPLRC plates using analytical solutions. Based on the Navier solution and FSDT, Song et al. (2018)
introduced the static and compressive buckling analyses of the FG GPLRC plate. They also reported that the
shear correction factor is essential to confirm the accuracy of the mathematical framework that they presented.
Using the element-free IMLS-Ritz method and FSDT, Guo et al. (2018) studied the vibration of GPLs
reinforced layered composite quadrilateral plates. Gholami and Ansari (2018) used higher-order SDT to find
out nonlinear harmonically excited vibration of rectangular FG GPLRC plates based on the variational
differential quadrature (DQ). Using the transformed DQ method, Malekzadeh et al. (2018) investigated the
vibration of FG GPLRC eccentric annular plates integrated with layers of piezoelectric material. Some of the
research investigations in this field combine FGMs and composite structures for achieving a wide range of
performances (Barati and Zenkour, 2019; Natarajan et al., 2014) (Assadi and Farshi, 2011). A major interest
of current research activities involves the presence of homogeneous strain and electric field in FGM plates
concerning the piezoelectric effect, which we discuss in the following paragraph.
In the search for emerging lightweight multi-functional structures, it was revealed that if piezoelectric
materials are utilized as distributed sensors/actuators which can be attached to or incorporated into the structure
then it accomplishes self-monitoring and self-controlling competencies (Smith and Auld, 1991). These
structures are usually named as “smart structures”. Piezoelectric materials generate the electric response to an
applied mechanical load by virtue of the direct piezoelectric effect while it deforms due to the electric load by
virtue of the inverse effect (Kuai et al., 2013; Maranganti et al., 2006; Tita et al., 2015). For developing high-
performing structures, the use of piezoelectric materials as distributed actuators and sensors is related to these
direct and inverse effects, respectively. In recent advances, the FG structures integrated with piezoelectric
actuators and sensors have received much interest from the application as well as the fundamental research
point of view to develop MEMS and technology-based energy harvesters
4
Fig 1. A bird's eye view concerning detailed flowchart of viscoelastic electromechanical analysis of
piezoelectric FG-GRC plates for application in various technologically demanding industries.
(Beeby et al., 2006; Yan and Jiang, 2017, 2011). For a better understanding of piezoelectricity phenomena, the
concept of piezoelectric effect is described using mathematical relation: . In this, , and
represent the electric displacement vector, the strain tensor and the piezoelectric tensor, respectively. Such
piezoelectricity phenomena are found to be present in materials where the inversion symmetry plays a vital
role, meaning the material should be non-centrosymmetric.
Consideration of viscoelasticity in smart piezoelectric composite materials makes it more realistic in
terms of accurate electromechanical response prediction due to the fact that many of the polymers used in
composite structures are inherently viscoelastic in nature. Time and frequency domain analyses of the
viscoelastic effect have been reported in composite structures. Aboudi and Cederbaum, 1989 presented a
micromechanical analysis of unidirectional fibre composites considering the phases to be viscoelastic in nature.
5
Salehi and Aghaei, 2005 analyzed axisymmetric viscoelastic circular plates using a non-linear and non-
axisymmetric formulation. Wenzel et al., 2009 developed a model to analyze the deflection of viscoelastic
(polymeric) cantilevered beams under uniform (adsorption-induced) surface stress. García-Barruetabeña et al.,
2013 discussed the interconversion scheme of viscoelastic relaxation modulus from the time-domain and
frequency domain and vice-versa. Amoushahi and Azhari, 2014 studied a moderately thick viscoelastic plate
using linear finite strip formulations. Mukhopadhyay et al., 2019 incorporated the effect of viscoelasticity into
an irregular hexagonal honeycomb lattice following a bottom-up analytical framework in the frequency
domain. Jafari and Azhari, 2021 discussed the bending of thick viscoelastic Mindlin plates with different
geometries in the time-domain. Singh et al., 2023 showed the usage of extended Kantorovich method (EKM)
to analyze IPFG viscoelastic plates embedded with piezo sensory layer.
The review of literature presented on composite/FGM with consideration of graphene platelets clearly
specifies that graphene/its derivatives are one of the most promising nanofiller for multiphysical applications.
However, until now, to the best knowledge of the authors, there are no (or very few) studies investigating the
electric and mechanical response of functionally graded graphene-reinforced piezoelectric composite (FG-
GRC) plates with and without consideration of the viscoelastic effect. Such an investigation could offer many
exploitable prospects for developing next-generation MEMS and smart structures (note: hereinafter the “FGM”
is used for functionally graded material without piezoelectric effect, while “FGPM” is used for functionally
graded piezoelectric material). In this article, the electromechanical responses of single and multi-layered
piezoelectric functionally graded graphene-reinforced composite (FG-GRC) plates would be studied based on
an accurate higher-order shear deformation theory (HSDT) involving quasi-3D sinusoidal plate theory, linear
piezoelectricity and the effect of viscoelasticity. The results would be further validated with separate finite
element (FE) modeling extensively. The reason for taking HSDT as a benchmark over classical plate theories
(CPT) is due to its ability to incorporate thickness deformation () and transverse shear deformation.
The contribution of this work is aimed at predicting the viscoelastic electromechanical performance of
simply supported FG-GRC plates with and without consideration of piezoelectric effect using analytical and
FE approach under generic loading conditions (quasi 3D sinusoidal distributed load). In the process, we would
investigate different critical parameters such as weight fraction of GPLs and aspect ratios concerning direct
and inverse piezoelectric effects. An overview of the comprehensive analysis concerning the current research
work is systematically presented in Fig. 1. This article is structured as: Section 2: the basic mathematical
formulations based on quasi-3D sinusoidal plate theory and linear viscoelasticity are introduced; Section 3:
the details of FE models are presented; Section 4: the numerical results are discussed for single and multilayer
FG-GRC plates to investigate their electromechanical responses, including the effect of viscoelasticity; Finally,
the article is summarized with concluding remarks and critical perspectives in Section 5.
6
2. Theoretical formulation
2.1 Geometric consideration
Figure 2 shows a schematic of single-layered rectangular plates made of functionally graded
piezoelectric material (FGPM) with width , length and height and it is associated with Cartesian
coordinate system
. The FGPM plate is assumed to consist of
polyvinylidene fluoride (PVDF) matrix and graphene platelets (GPLs) reinforcement. The top surface of
FGPM plate is subjected to a transversely distributed load . This mechanical load is dependent on only two
in-plane spatial coordinates and while an external electric voltage is applied in between its top and bottom
plate surface. Variation of material properties is considered to be continuous in nature (Nomura and Sheahen,
1997) and it is varied only along its thickness direction i.e., z-axis. The present formulation can also be
applicable to multilayer laminates (with number of layers ) which is explained later in Section 4.2.
2.2 Kinematic relations
Considering the quasi-3D sinusoidal plate theory (Zenkour, 2007) and the shape function proposed by
Levy (1877), Stein (1986) and Touratier (1991), the displacement field of any point within the volume
of interest along three orthonormal directions can be expressed in the following form (Zenkour and Hafed,
2020):
(1a)
(1b)
(1c)
where and indicate the in-plane and out-of-plane displacements of any point respectively on the
mid-plane and ( indicate the respective rotations of the transverse normal about y and x-
axis. is for accounting the stretching effect of the plate. Contrary to typical first-order shear
deformation theory, the present trigonometric plate theory does not require any shear correction coefficient.
• Assumptions:
1. Any straight lines perpendicular to mid-surface before deformation stay straight after its deformation.
2. There will be the contribution of bending and shear both in total transverse displacement.
(2)
3. Rotation function ( are approximated as respective slopes of shear transverse deflection:
7
Fig 2. Schematic of FGPM plate subjected to electromechanical loadings and associated cross-sections.
.
(3)
Considering these assumptions in Eq. (1), we get the following simpler forms:
(4a)
(4b)
(4c)
where
and
are derived from the assumed shape function. Neglecting
Von-Karmen non-linear terms in strain formulae, we can derive the following linear strain-displacement
relations from Eq. (4):
(5a)
(5b)
(5c)
(5d)
(5e)
8
(5f)
where
,
,
,
,
,
,
,
,
,
and
. We can observe here the
existence of non-zero transverse strains () which is also a characteristic of any typical shear deformation
plate theories.
Accounting Maxwell’s equation, the variation of electric potential (
through thickness can be
approximated by the following equation proposed by Quek and Wang (2000):
(6)
where denotes the distribution of electric potential induced in mid-plane. The electric field components E
can be given by:
(7)
2.3 Constitutive equations and function resultants
General constitutive relations for any piezoelectric material can be given by following two equations
of actuation- and sensing- law (Li et al., 2020):
where is electric displacement field and are the elastic constants under constant
electric field (Zenkour and Alghanmi, 2018) which can be given as follows:
9
(9a)
(9b)
(9c)
As this is an FGM system, all elastic and piezo coefficients such as elastic modulus (E), Poisson ratio (,
piezoelectric () and dielectric (coefficients are varying along the direction of the plate thickness (z).
The governing equations for the present static FGPM system are achieved from the principle of virtual
displacements that can be given as follows:
(10a)
The virtual strain energy is expressed as follows:
The virtual work done by the externally applied uniform transverse load and externally applied Electric
potential, V can be written as follows:
After substituting the strain-displacement relations and rearranging Eq. (10a), the following six governing
equations of motion can be obtained:
(11a)
(11b)
(11c)
(11d)
10
(11e)
(11f)
Here are the function (stress and moment) resultants whose definitions are given as follows:
(12a)
(12b)
(12c)
(12d)
Now if we substitute Eq. (8) in the aforementioned resultants and perform tabulation in the terms of
,
,
, we get the following matrix.
In Eq. (13), the stiffness coefficients can be defined by
11
(14)
2.4 Governing equations
Putting Eq. (13) in the governing Eq. (11), we get the following six partial differential equations.
(15a)
(15b)
(15c)
(15d)
12
(15e)
(15f)
In Eq. (15),
which corroborate
the piezoelectric coupling coefficients are defined by:
(16a)
(16b)
(16c)
(16d)
(16e)
(16f)
(16g)
13
Here, Navier’s method is implemented to get the analytical results, wherein the following boundary constraints
of the four ends simply supported (SSSS) plate are assigned.
Edge 1:
Edge 2:
Edge 3: x
Edge 4: x
To satisfy the aforementioned boundary conditions, () are expressed using an infinite series
which are given by:
(17a)
(17b)
(17c)
(17d)
(17e)
As Eq. (15) contains total six primary unknown variables, the mechanical ( and electrical load () are also
expressed based on double sine series as follows:
(17f)
(17g)
We consider that a uniform transverse load, is acting throughout the top surface of the plate. Thus
we have and value of can be determined from Fourier series expansion.
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(18)
Now, we have to assume a function (potential per unit surface) to tackle the external applied electric
voltage in the Eqn. (15e) and (15f). Similar to , this can also be expressed as double sine
series as follows.
(19)
Similarly, considering
i.e., independent of and , can be determined similar to .
(20)
Note here that while we have considered uniformly distributed mechanical load and electrical voltage, other
loading conditions can also be analyzed based on the analytical framework presented here. Substituting Eqs.
(17), (18), (19) and (20) in the partial differential Eqs. (15), we get the following six linear simultaneous
equations of .
(21a)
(21b)
(21c)
(21d)
15
(21e)
(21f)
Here
. Solving six simultaneous Eqs. (21), we can determine and
accordingly, the total transverse deflection of the plate can be calculated by adding these three effects
of bending, shear and stretching. By solving for , we can calculate the voltage component corresponding
to the applied load. It can be noted in this context that the above formulation is valid for single and multi-layer
(with the number of layers NL) functionally graded plates. Equivalent material properties are adopted for
utilizing the analytical framework presented here as described below.
2.5 Equivalent material properties
The PVDF has isotropic and piezoelectric properties, and it is presumed that GPLs are uniformly and
linearly distributed with a randomly distributed placement within the PVDF matrix. Such randomly oriented
fiber composite can be approximated to a quasi-isotropic laminate (Halpin and Karoos, 1978). Here the GPLs
are assumed as rectangular-shaped solid reinforcement of average width , length and thickness .
The electromechanical behaviour of the FGPM plate is discussed by considering two distributions of weight
fraction of GPLs along the thickness direction (z) which are as follows (Zhao et al., 2020):
• Linear distribution:
(22a)
• Uniform distribution:
(22b)
where
and
are the total weight fraction and characteristic value of GPLs weight fraction,
respectively. The total volume fraction of GPLs is calculated using the following relation:
(23)
where and denote the respective mass densities of GPLs and PVDF matrix. The modified Halpin-
Tsai model is utilised to estimate the effective material constants (properties). The Young modulus of a nearly
isotropic laminate system is as follows (Wang et al., 2020):
16
Fig. 3. Distribution of across the thickness of single-layered FGPM (x–z plane) (a) linear and (b) uniform
distribution.
(24)
whereas and indicate the longitudinal and transverse moduli and their values can be estimated from the
Eq. (25). Here
and
are the reinforcing efficiency of GPLs considered in longitudinal and transverse
directions, respectively.
(25)
where the parameters
and
can be expressed by
(26)
Here and indicate the respective Young moduli of GPLs and PVDF matrix, and filler geometric
factors
and
of GPLs are given by the following equations:
(27)
where ,, and are the respective thickness, length, and width of GPLs fillers. Other material
properties are calculated by the rule of mixture which is as follows:
(28a)
17
(28b)
(28c)
(28d)
where ,, and are the Poison ratio, coupling coefficient, electric
permittivity and piezoelectric multiple, respectively. Table 1 summarizes the values of all these coefficients.
Table 1: Geometric and material properties of constituents of FGPM (Li et al., 2020)
Elastic and geometrical properties
Piezoelectric constants
Dielectric constants
0.186
0.29
2.6 Time-dependent viscoelastic analysis
Here the analytical model presented in the preceding sections has been extended for analyzing the
functionally graded piezoelectric plates with time-dependent viscoelastic properties. For the sake of simplicity,
the complex elastic modulus of the plate in the frequency domain () is expressed as the Biot’s viscoelactic
model with only one term (Mukhopadhyay et al., 2019).
(29)
where is the effective young modulus in the absence of viscoelasticity, given in Eq. 24 and .
and are the constants indicating “the strength” of viscosity and relaxation parameter respectively. (real
18
part) and (imaginary part) can be regarded as storage and loss modulus respectively. The amplitude
(dynamic modulus) and phase of the complex modulus can be derived as follows:
(30a)
(30b)
As the phase increases, the contribution of loss modulus will increase which in turn makes the material more
viscous (Mun et al., 2007). Considering all the limiting properties of , present in the existing literature,
it can be deduced that for all positive and , achieves minimum amplitude () when and
whereas the same will achieve maximum amplitude () when and . However,
for all limiting cases, the viscoelastic material properties tend to be pure elastic as the phase becomes
zero.
To employ this complex modulus in the present analytic model and for capturing the realistic time-
dependent viscoelastic behaviour of the plate, it is necessary to invert back the frequency domain representation
discussed above into the time-domain (). This inversion of young modulus from its frequency domain (refer
Eq. 29) to the time domain is carried out by the efficient inverse Fourier algorithm written in MATLAB
symbolic environment. All values of , from highest to lowest, are taken into account when inverting
frequency domain data into the time domain (). This inversion can be expressed as follows:
(31)
Here the function (also called relaxation modulus) encompasses the thickness direction () along with
the time parameter (). Handling these two parameters at the same time in the present analytical model can be
cumbersome. To mitigate this issue, we have performed the temporal analysis throughout the plate thickness
at each time step separately. The evolution of material properties at each time step is determined by a suitable
viscoelasticity model. Note that the FE validations (comparative results obtained from the analytical approach
and FEM) presented in this paper can be regarded as the validation for a particular time step and the
corresponding material properties in the context of time-dependent viscoelastic analysis. By ensuring the
accuracy of results at each time step corresponding to time-dependent material properties obtained based on
suitable viscoelastic models, the correctness of the overall temporal analysis is ascertained. In this context, it
can be noted that two viscoelastic parameters ( and ) need to be evaluated specific to the material under
consideration. The time-dependent variation of material properties depends on these parameters.
19
We approximate the integral in Eq. (14) by taking a summation through the plate’s thickness for each
time instant. As here some approximations are involved, the accuracy of the result is ensured by comparing the
value of each constant in Eq. (14) obtained from the present summation method and the normal integral method
at a particular time instant. Considering time-domain, Eq. (17a-f) can be rewritten as follows:
(32a)
(32b)
(32c)
(32d)
(32e)
(32f)
Afterwards, solving Eq. (21) in conjunction with Eq. (32), the time-dependent parameters
can be calculated for each time instant. Thus, in the proposed framework of
viscoelastic analysis, we first characterize the frequency-domain depth-wise material properties, which are then
inverted to time-domain variation of the depth-wise varying material properties and subsequently used to
analyze the electromechanical response of the plates at each time-step.
3. Finite element (FE) analysis
We have carried out a separate finite element analysis to validate the analytical framework as described
in the preceding section. The primary objective of including the finite element model in present paper is to give
an initial validation to our parent general analytic model (which is equally applicable for both viscoelastic and
non-viscoelastic structures). The elastic model, for which we have presented finite element validations, is
further extended to analyze the viscoelastic behavior where only the elastic modulus terms in the analytical
expressions are replaced by the viscoelastic parameters based on the correspondence principle. The essence of
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this principle is that the analytical expressions of the structure in the elastic regime can be readily extended to
the viscoelastic regime without any further change in the parent elastic framework. It is well-established in the
literature that a correct elastic framework for analyzing structural behaviour can be converted to time-
dependent viscoelastic analysis through this principle. The FEM analysis presented in the current paper is
intended to establish the accuracy of the elastic analysis, which in turn assures the correctness of the
viscoelastic analysis. That is why no additional FE calculations are added for viscoelastic structures separately.
In this context, it can be noted that the FE validations presented here can be regarded as the validation for a
particular time step and the corresponding material properties in the context of time-dependent viscoelastic
analysis. Once the electromechanical analysis is validated for a particular set of material properties, it can be
extended to the other time steps readily by considering the appropriate time-dependent material properties at
different other time steps. Thus finite element validation of the electromechanical analysis for a particular set
of material properties, as presented here, is sufficient for the time-dependent viscoelastic analysis performed
afterwards in this study.
The finite element analysis is carried out here considering single and multi-layered systems, wherein
the equivalent material properties are evaluated based on the approach presented in the preceding section. The
same geometry and the coordinate system are adopted in FE analysis as shown in Fig. 2. The CAD model is
prepared in the COMSOL multi-physics version 5.5 software package and FEM simulation has been performed
in COMSOL’s 3D “piezoelectricity multiphysics interface” which combines Solid Mechanics and
Electrostatics together with the constitutive relationships required to model piezoelectrics. In the geometry,
thickness direction is taken along the z-axis and material coordinate system are same as the spatial coordinate
system in COMSOL. For simulating FGPM (functionally graded piezoelectric material) in the present
structure, we have created a “blank material” within COMSOL material library whose elastic and piezoelectric
properties are given in accordance with Eqns. (24) and (28). As all the properties vary along the thickness
direction, we have adopted the COMSOL’s global coordinate variable z with lower limit of and upper
limit of to build the analytic functions for its material properties (density, elasticity matrix, coupling
matrix and relative permittivity matrix). The discretization of the rectangular plate is carried out using free
tetrahedral (tet) mesh (fine) elements where the maximum element size varies between 0.24 m to 0.03 m.
As the analysis is performed on an electromechanical structure, we have to incorporate electrical
boundary conditions in addition to mechanical loadings. Present FGPM rectangular plate is under the closed-
circuit condition where initially, the electric potential applied on the upper and lower surfaces are expected
to be zero (grounded) throughout the analysis (as demonstrated in Fig. 4). In COMSOL, the piezoelectric
polarization axis is not changed as its default direction is always along its spatial coordinate axis. In case of
mechanical loading, a uniformly distributed (UDL) unit force is applied throughout the top surface and the
bottom surface is kept free. The plate is modeled as simply supported along four edges (SSSS). The overall FE
21
Table 2: Initial dimensions of the plate.
Parameters
Values
Unit
Length,
3
m
Width,
1
m
Thickness to span ratio ()
0.01
-
Total No. of layer,
• Single (= 1)
• multilayer
-
Fig. 4. Schematic representation of FGPM plate su bjected to: (a) electrical (b) mechanical loads.
modeling is demonstrated using a flow diagram in Fig. 5. In this context, it can be noted that the finite element
validations presented here can be regarded as the validation for a typical time step considering the
corresponding material properties. In time-dependent viscoelastic analysis, the material properties vary at
different time steps that can be ascertained by the adopted viscoelastic model. Once the electromechanical
analysis is validated for a particular set of material properties, it can be extended to the other time steps readily
by considering the appropriate time-dependent material properties at different other time steps. The time-
dependent evolution of material properties in a viscoelastic analysis is discussed in the preceding section.
4. Results and discussions
This section presents numerical results concerning the electromechanical behavior of FGPM plates
based on the proposed analytical approach and comparative validation results using finite element simulations.
We would investigate three different configurations with single and multiple layers (UD, LD and UD/LD).
Subsequently, we present time-dependent viscoelastic results for the deformation and electric potential of
FGPM plates.
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Fig. 5. Flowchart describing the detailed FE analysis.
4.1 Electromechanical analysis of single FGPM plate (
A thin FGM plate is subjected to uniformly distributed load while it’s all
four edges are kept simply supported. We first concentrate on validating the analytical model using separate
FE analysis, wherein a convergence study is important to obtain credible results before proceeding further.
Therefore, the convergence study concerning FE analysis is carried out to investigate the influence of mesh
size or the number of elements on the transverse deflection of the plate. Different types of meshing such as:
extreme coarse, coarse, normal, fine and extreme fine are considered with the overall range of average element
size between 0.855 m to 0.0303 m. In Table 3, the results of convergence study with respect to maximum
center deflection of the plate and its mesh statistics used in FE analysis have been presented. Figure 6 illustrates
the variation of transverse deflection of the plate over its length. It can be observed that the results for transverse
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Table 3: Convergence study of FEM for Center deflection of Plate
Mesh Type
Number of
Domain
Element
Number of
Boundary
Element
Number of
Edge Element
Plate Center
Deflection
()
Extreme coarse
597
476
92
-21.966
Coarse
3196
2268
212
-32.929
Normal
7486
5252
312
-33.123
Fine
24560
16340
548
-33.199
Extreme Fine
301244
142524
1604
-33.200
Fig. 6. Mesh convergence analysis.
Fig. 7. Variation of transverse deflection of UD FGM plates along its length based on analytical and FE models.
24
(a) (b)
Fig. 8. (a) Variation of transverse deflection of SSSS FGPM plate along its length
. (b) Variation of transverse deflection of LD plates along its length.
deformations are converged for fine and extreme fine elements. Therefore, we followed the ‘fine’ type of
element for further analysis in FE modeling.
We further validate the FE model with literature to ensure its prediction accuracy. Figure 7
demonstrates the variation of transverse deflection of SSSS FGPM and FGM plates (i.e., with and without
accounting for the piezoelectric effect) over their lengths and a comparison of the available results based on
classical plate theory (CPT) (Reddy, 2006). For this purpose, we have considered a thin FGM plate with
uniform distribution (UD) of
and carried out a deflection analysis of UD FGM (refer to Figure 7). The
earlier existing analytical result for center deflection of the plate calculated from CPT is m
(Reddy, 2006) which shows a very close agreement with the current FE results. Due to consideration of the
piezoelectric effect, there is a decrement in maximum transverse deflection of the plate compared with non-
piezoelectric FGM plate. Having adequate confidence in the finite element model, we present further numerical
results based on analytical predictions and finite element analyses.
To check the accuracy of the current quasi-3D sinusoidal shear deformation plate theory with respect
to the FE model in case of FGPM LD plates subjected to uniform loading (), the comparative results are
plotted in Fig. 8(a). The results are observed to be in excellent agreement, corroborating the validity of the
proposed analytical framework further. In the following numerical results, we investigate different critical
effects on the electromechanical behaviour of FGPM plates, primarily based on the analytical approach. Figure
8(b) shows the variation of transverse deflection of SSSS FGM and FGPM plates over their lengths considering
25
linear distribution (LD) of GPLs weight fraction
. A similar trend is observed in UD cases (refer to
Figures 7) for transverse deflection, while the LD case shows higher deflection of the plate. A detailed
comparison of the results considering LD and UD cases is presented in Figure 9 based on analytical and FE
approaches. The results show that the incorporation of piezoelectricity stiffens the FGM plates for both
distributions.
Fig. 9. Effect of distribution of GPLs on the transverse deflection of plates.
(a) (b)
Fig. 10. Variation of transverse deflection of SSSS FGPM plate along its length . (a)
for different aspect ratio (
) (b) for different GPL volume fractions ().
26
Fig. 11. Variation of electric potential across the thickness of SSSS FGPM plate at its center
.
Fig. 12. Variation of electric potential across the thickness of an SSSS FGPM plate at its center
.
Figure 10(a) shows the variation of transverse deflection of the SSSS plate over its length by
considering different aspect ratios (a/h). For this, we considered
with LD case while the different
aspect ratios are considered as 10, 20, 50 and 100. It is noticed that the transverse deflection increases with
increasing aspect ratio. Figure 10(b) shows the variation of transverse deflection of SSSS FGPM plates along
their length by considering different
with the LD case. From this, it is noticed that the deflection of SSSS
27
(a) (b)
(c) (d)
Fig. 13. Variation of transverse deflection of SSSS FGPM plate along the length direction of the plate. (a) LD
distribution with a constant voltage of zero throughout the upper surface (along with different values of loads)
(b) UD distribution with a constant voltage of zero throughout the upper surface (along with different values
of loads) (c) LD distribution when a constant load of 100 Pa is applied on the structure (along with different
values of voltage) (d) UD distribution when a constant load of 100 Pa is applied on the structures (along with
different values of voltage).
plate is significantly influenced due to the incorporation of nanoparticles such as graphene. The transverse
deflection of the plate is reduced due to the addition of a large value of
. This effect of different
on
deflection is also intuitively true as the overall elastic modulus of the model increases if the percentage of GPLs
28
increases and consequently, it gets stiffer. Further, from Figures 10(a) and 10(b), it is clear that the results
obtained from both analytical and FE modelling are observed in excellent agreement.
Figure 11 shows the variation of electric potential generated due to piezoelectricity at the center of
SSSS FGPM plate with respect to its thickness by considering different
. From this, it can be noted that
the electric potential shows the maximum value at the middle of the plate thickness. Similar to transverse
deflection, the electric potential decreases due to the addition of different
. For
, the electric
potential shows larger values compared to the remaining three values of
. The electric potential generated
in SSSS FGPM plates also depends on the aspect ratios of the plate which are investigated in Figure 12 for
aspect ratios of 10, 20, 50 and 100 (considering
). It can be noted that the electric potential
increases with respect to the aspect ratio, while the peak voltage appears in the middle layer of the FGPM plate
with LD distribution
.
The numerical results are presented here (unless otherwise mentioned) considering unit load and zero
voltage, which lead to deflections in the micrometer range. The accuracy of the results is ensured through
separate finite element simulations. It can be noted that the developed semi-analytical framework is generic
enough to analyze larger values of load which would lead to higher deformations. However, to establish the
generic nature of the proposed computational framework, we have added two separate studies for single and
multi-layered FGPM structures (refer to Figures 13 and 18) where we vary the mechanical load and applied
voltage in a reasonable range (complying the small linear strain-displacement assumption). It is observed that
the deflections increase significantly with external loads up to the millimeter range. The impact of mechanical
load and non-zero voltages on the overall deflection of the plate for both LD and UD FG distributions is shown
in Figure 13. Both distributions in Figure 13(a-b) show a direct relationship between transverse deflection ()
and external load (). For instance, in FGPM structure with LD distribution, a maximum deflection of 3.313
mm is observed along the centroidal axis of the plate when a 100 Pa load is applied. In Figure 13(c-d), the
piezoelectric voltages are varied for both distributions while maintaining a constant external mechanical stress
of 100 Pa. Although the effect of voltage is minimal in UD distribution, the transverse deflection is directly
correlated with applied voltages in LD distribution. Overall, the UD distribution in FGPM structures exhibits
stiffer behavior than the LD distribution.
4.2 Electromechanical analysis of multilayer FGPM plate
In this Section, we extend the same theoretical approach discussed in Section 2 for presenting numerical
results concerning layered composite structures. We have considered a perfectly bonded double-layered FGPM
plate in which GPLs nanofibers are assumed to be randomly oriented in respective matrixes in both
the layers with UD/LD configuration. The total thickness is taken as and it is equally divided in both layers.
Distribution of GPLs weight fraction is taken differently in two layers i.e., a top layer having UD distribution
29
Fig. 14. Distribution of weight fraction of GPLs
across layered FGPM plate thickness.
Fig. 15. Variation of transverse deflection of layered FGPM plate along its length
.
and a bottom layer having LD distribution of GPLs. This layer-wise variation of the FGPM plate is depicted
in Figure 14. Other boundary conditions remain the same as considered in the single-layer plate. Unit
mechanical load and zero external voltage are applied on the outer surface of the plate as before. Both the
analytical and FE analyses are performed to check the accuracy of the present theory in predicting the behavior
of the layered structure.
30
(a) (b)
Fig. 16. (a) Effect of weight fraction of GPLs
on the transverse deflection of SSSS layered plate
. (b) Effect of aspect ratio on the transverse deflection of SSSS layered plate
Fig. 17. Variation of electric potential
across the thickness of layered FGPM plate at the centre
.
Figure 15 shows the variation of transverse deflection of layered FGPM plate along its length. The
analytical and FE results for total transverse deflection () of the plate under mechanical load
and electric potential are compared. From this figure, it is observed that theoretical and FE results are
31
(a) (b)
Fig. 18. Effect of external mechanical load and electrocortical loading on the overall deformation of layered
FGPM structures. (a) Variation of the plate’s deflection along length direction under a constant voltage of zero
and different values of mechanical load (b) Variation of plate’s deflection along length direction under a
constant mechanical load of 100 Pa and different values of voltage.
in very good agreement for the multi-layered structures. The maximum deflection is found at the half-length
(i.e., 1.5 m) of the plate which is the same in the case of a single-layered plate. From this, we can conclude that
number of layers has negligible effects on the overall deflection pattern of the plate if other parameters are kept
the same. In figure 15, it is also observed that in the case of multi-layered plates, there is an increment in the
magnitude of maximum deflection by 7.94 compared to the single-layered plate. A similar trend of the
result is observed in existing literature (Lu et al., 2006), which validates our present formulation further.
Subsequently, insightful parametric analysis is performed by varying weight fraction of GPLs
and plate aspect ratio , as shown in figures 16(a) and 16(b). The deflection of layered plate decreases as
the value of
and increases. It is due to increased stiffness which is discussed in the earlier section.
In figure 17, the variation of electric potential along the thickness direction is generated due to the direct
piezoelectric effect. It can be observed that the variation is continuous, there is no discontinuity at the interface
of the two layers.
Similar to single-layered FGPM distributions (refer to Figure 13), a parametric analysis has been
conducted here to check the dependency of external loading parameters on the structure’s overall deformation
in a multi-layered FGPM system. A significant increase in the maximum transverse deflection in comparison
32
to its unit-loading state is observed in Figure 18(a) where the deformation increases to the millimeter range
with the increase of mechanical loading. Figure 18(b) shows the length-wise change in the plate’s deflection
with applied voltage. A non-uniform symmetric deformation pattern with respect to plate’s centroidal axis can
be observed where an increasing trend of deflection with voltage is noticed near the central zone and a reverse
decreasing trend of deflection with voltage is obtained near the supporting edge of the plate. Contrary to the
uniform trend seen in single layered plates (Figure 13), the double layered (UD/LD) plate here exhibits such
non-uniform deformation trend with voltages along its length.
4.3 Time-dependent electromechanical analysis of single and multi-layer viscoelastic FGPM plates
In this section, the time-dependent dynamic behavior of the structure’s responses has been investigated
by the incorporation of viscoelastic effect. Before obtaining the final results, the effect of viscoelasticity on the
effective elastic modulus, along the thickness has been checked by plotting it in the time domain. As
mentioned earlier, this modulus not only depends on time () but also the thickness direction () since the
present structure is depth-wise functionally graded. For each vertical point at a particular section of the plate,
we obtain a time-variation curve for the effective elastic modulus. For the sake of brevity, in Table 4, the effect
of viscoelasticity on Young’s modulus at three locations of the thickness ( at
, and
) in
LD distribution is shown in both the frequency domain and the corresponding transformed time domain. Note
that the parameters and in Eq. (29) are crucial for conceptualizing viscoelasticity of the present structure.
It is important to keep in mind that the exact values of both parameters ( and ) in general, depend on different
physical experimental outcomes of the relevant viscoelastic system. In particular, they can be obtained from
the curve fitting of experimental data concerning creep test of the material. Such experimental implementations
were performed in the existing literature (Endo and de Carvalho Pereira, 2017) (Rouleau et al., 2013)(Enelund
and Olsson, 1999). As the present study doesn’t include any experimental work, we have adopted reasonable
parametric values for obtaining the numerical results. Analytically these two parameters, present in complex
elasticity modulus () in the frequency domain, come from the viscoelastic kernel function in the time-
domain. This function can be obtained by constructing various equivalent lumped spring-dashpot damping
models for viscoelastic material such as Maxwell model, Voigt model, Standard linear model, Generalised
Maxwell model and Prony series model. Whereas, in the frequency or Laplace domain, various existing
viscoelastic models complying Kramers-Kronig relations can be used to derive such as Biot model,
Gaussian model, Fractional derivative, Half cosine model etc. Among them, the Biot’s standard classical model
of viscoelasticity (Biot, 1955)(Biot, 1954) has been chosen here by which complex elasticity modulus ()
can be obtained without any significant accuracy loss. To examine the effect of the two parameters on overall
strength of viscoelastic model, a parametric study (refer to Figure 19(a-d)) is presented. First, we have
investigated the influence of these two parameters on the present viscoelastic system and afterwards, a suitable
33
(a) (b)
(c) (d)
Fig. 19. (a) Effect of the parameter of the amplitude of mid-plane-Young-modulus in the frequency domain
at constant . (b) Effect of the parameter of the amplitude of mid-plane-Young-modulus in the
frequency domain at constant
. (c) Effect of the parameter of the magnitude of viscoelastic phase
angle at mid-plane in the frequency domain at constant . (d) Effect of the parameter of the magnitude
of viscoelastic phase angle at mid-plane in the frequency domain at constant
.
combination of their values are chosen to produce numerical results in succeeding sections. It can be noted that
the temporal framework presented here is generic and any suitable value of and , obtained based on
experimental investigations, can be used for exploration of the viscoelastic behaviour. In Figure 19(a-d), the
34
amplitude of Young modulus at mid-plane and its associated phase angle are varied in a reasonable frequency
range for different values of and . It can be seen in Figure 19(a-b) that the amplitude increases as the values
of and reduce and increase respectively. They all show converging patterns in the values after certain
frequencies. This can be explained with the help of Figure 19(c-d). Figure 19(c) depicts the variation of phase
angle in frequency domain for different at a constant , whereas the same is plotted in Figure 19(d)
for different taking as constant i.e.
(where is the maximum magnitude of considered
frequency spectrum). Existence of critical frequencies can be observed, wherein the physical significance of
such critical frequency can be explained in the light of spring-dashpot lumped model of viscoelasticity. At very
low and high frequencies, the model behaves as pure elastic, whilst in the vicinity of the critical frequency, the
viscous effects become maximum. That is why in Figure 19(a-b), at very low frequency, amplitude of Young
modulus takes a constant non-zero value and after a certain frequency, it again converges to a constant non-
zero value. From Figure 19(c-d), it can be concluded that the parameter controls the critical frequency
whereas the parameter shows its influence on magnitude of phase angle of the present viscoelastic model.
After the afore-explained parametric study investigating the influence of viscoelastic parameters, we can now
proceed for obtaining numerical results in the current context by taking a combination of these two parameters
and In the frequency domain, the variation of the dynamic modulus and its phase is obtained by taking the
parameters as
and for the present paper. Except for Young’s moduli, Poisson's ratio and
other piezoelectric properties are considered to have a negligible effect of viscoelasticity based on published
literature (Salehi and Aghaei, 2005, Salehi and Safi-djahanshahi, 2010, Barrett and Gotts, 2004). It has also
been verified in MATLAB that the values of elastic modulus in all three cases (refer to Table 4) at low
frequency () are the same as the non-viscous elastic modulus mentioned in the prior sections. Such
observations are in coherence with the existing literature (Mukhopadhyay et al., 2019, Malekmohammadi et
al., 2014). In Table 4, the variation of phase angle in all three locations has also been plotted and it is observed
that all are the same and achieve their peak value at a certain critical frequency. So, the phase angle variation
is independent of thickness direction, though the amplitude of elastic moduli keeps changing with the thickness
direction. The phase angle decreases on both sides of the frequency spectrum i.e. at the lowest and highest
frequency of the plot. At low frequencies, the present viscoelastic plate will behave more like a normal elastic
plate. After applying the inverse Fourier algorithm, the time domain plots are obtained along the thickness. As
time increases, elastic moduli keep decreasing and after a certain time, it gets converged at their non-viscous
elastic moduli value (
) ). Such a trend has also been observed in the existing literature
(García-Barruetabeña et al., 2013).
35
Table 4: Variation of the effective Young Modulus in frequency and corresponding time domain at three
locations of the thickness (LD distribution).
Position
Frequency Domain
Time Domain
36
(a) (b)
Fig. 20. Time-dependent variation of plate’s deflection (a) in UD, LD and LD/UD cases in terms of normalized
form (b) in LD case in terms of absolute form. The normalization is carried out here with respect to the
deflection value at initial time step (t = 0), as shown in the inset. In the axis titles of the inset figures, Nor.
representes Normalized.
The obtained depth-wise varying elastic moduli in the time domain are now embedded within the
analytic model to obtain the time-dependent responses of the viscoelastic FGPM plate, subjected to the
aforementioned boundary conditions (, SSSS). Figure 20(a) depicts the effect of viscoelasticity
on the plate’s deflection under the unit uniform static transverse load. Here the normalized maximum
deflections ( along the centroidal axis of the plate are plotted for the single-layered (LD and UD
distribution) and multi-layered (LD/UD) FGPM plate. In terms of final (saturated) normalized deflection, it is
observed that the multi-layered (LD/UD) plate is between the rest two distributions where LD is having the
highest value of the same. It indicates the fact that the increment of the plate’s deflection with time is highest
in LD distribution, while LD/UD plate is in between the two. Although, in terms of absolute deflection, the
deflection in LD/UD plate at its saturation stage is higher than the rest two. In each case, this maximum
deflection converges at a certain deflection which is almost the same as that of pure elastic case. For instance,
in LD distribution, the FGPM plate starts bending with a maximum deflection of 30.51 under the constant
load of 1 and after 0.01 sec, its deflection increases to a deflection of around 30.8 . The bar charts in
Figures 20-22 are showing the initial deflections () in . Due to the incorporation of viscoelasticity,
we can see a time lag in achieving its prior elastic deflection. A similar trend is observed in other distributions
as well. But the rate of change in the deflection in three distributions is found to be different depending on the
37
(a) (b)
(c)
Fig. 21. Effect of viscoelasticity on the maximum transverse deflection of the FGPM plate with four different
volume fractions of GPL (a) LD (b) UD (c) LD/UD. The normalization is carried out here with respect to the
deflection value at initial time step (t = 0), as shown in the inset. In the axis titles of the inset figures, Nor.
representes Normalized.
distribution (Multi-layered FGPM > Single-layered LD FGPM > Single-layered UD FGPM). For the sake of
clarity, the time variation (discussed in Figure 20(a)) in terms of absolute deflection for only LD distribution
is shown in Figure 20(b) (similar plots can be readily obtained for the other distributions). The trends are in
good coherence with the existing literature on non-FGPM viscoelastic plate structures (Jafari and Azhari, 2021,
Salehi and Aghaei, 2005, Wenzel et al., 2009).
38
(a) (b)
(c)
Fig. 22. Effect of viscoelasticity on the maximum transverse deflection of the FGPM plate with four different
plate aspect ratios () (a) LD (b) UD (c) LD/UD. The normalization is carried out here
with respect to the deflection value at initial time step (t = 0), as shown in the inset. In the axis titles of the inset
figures, Nor. representes Normalized.
Figure 21 shows the viscoelastic effect on the plate’s central deflection at four different volume
fractions i.e.
of GPLs in the PVDF matrix. With the increase of volume fraction, the
plate’s elastic deflection at and the steady state () both decrease. Moreover, the rates of
deformation are found to be decreased with the increase of GPL’s volume fraction. For the increment of
deflection in each FGM distribution, all three show different trends with respect to volume fractions. For
instance, in LD plate, the increment goes up as the volume fraction increases, whereas in UD case, the opposite
39
(a) (b)
Fig. 23. Time-dependent variation of plate’s mid-line electric potential (
and ) (a) in
UD, LD and LD/UD cases in terms of normalized form (b) in LD case in terms of absolute form. The
normalization is carried out here with respect to the potential value at initial time step (t = 0), as shown in the
inset.
trend is observed. A little exception is identified in LD/UD plate as here the normalized deflection rises with
volume fractions except the one with 1% volume fraction whose increment is higher than rest three in Figure
21(c).
In Figure 22, the aspect ratio of FGPM plates is varied over a reasonable range to observe the effect of
viscoelasticity on the plate’s deformation. Contrary to the trend observed in the aforementioned volume
fractions’ cases, both the deflections (at and ) are in direct relation with the plate’s aspect ratio.
The rate of deformation here is found to have a rapid increment with the increase of aspect ratio. Furthermore,
the increment in deflection with respect to its initial value () is observed to have a direct relation with
plate’s aspect ratio in all three distributions. In general, the numerical study considering viscoelastic behaviour,
besides giving a more realistic analysis, demonstrates a potential programmable time-dependent structural
behaviour (including temporal programming in smart stuctures and metamaterials (Sinha and Mukhopadhyay,
2023)), which could be crucial for analysing and designing the mechanical behaviour of a wide range of
polymer composites accurately.
The time-dependent generation of electric potential in the present FGPM viscoelastic system is shown
in Figure 23(a) where all three FGM distributions show rapid growth in their values with time. These depicted
40
potentials are extracted along the centroidal axis of the plate and with the consideration of GPL percentage of
1% and plate’s length-to-thickness ratio of 100. Though the double-layered plate gives a relatively higher
voltage (~70% higher) than the rest of the distributions after getting saturated, the increment of it with respect
to initial electric potential over time is in between the other two distributions where the uniformly distributed
(UD) FGPM plate shows the highest increment. Figure 23(b) shows the same time-variation of electric
potential in LD distribution but in its absolute form, leading to the same conclusion (similar plots can be readily
obtained for the other distributions). Essentially, the numerical results concerning electric potential for time-
dependent viscoelastic analysis shows that it is possible to harness more accumulated power with increasing
time.
5. Prospective engineering applications
Functionally graded piezoelectric material (FGPM) viscoelastic composite structures combine the
properties of piezoelectric materials and viscoelastic composites to create a class of materials with unique
prospective engineering applications. Such an analysis following efficient semi-analytical framework is
presented for the first time in this manuscript. We have discussed here a few critical real-life engineering
applications associated with FGPM viscoelastic composite structures.
• Shape Morphing Structures: FGPM viscoelastic composite materials can be engineered to change their
shape in response to electrical stimuli. These materials are used to create adaptive structures and
morphing surfaces in aerospace applications, where shape changes can improve aerodynamic
performance. Similar applications can be found in various other mechanical systems. The interesting
notion here is the capability of temporal programming as discussed in the manuscript.
• Energy Absorption: FGPM viscoelastic composites can be employed in impact-absorbing structures,
such as helmet liners and automotive crash pads. The combination of viscoelastic damping and
piezoelectric energy conversion helps dissipate energy during impact events, reducing the risk of injury
and damage.
• Smart Materials and Structures: FGPM viscoelastic composites can be utilized in the development of
smart materials and structures. By exploiting their piezoelectric properties, these materials can sense
changes in the operating environment and respond accordingly, enabling applications like active
vibration control, actuators, shape adaptation, programmable mechanical properties etc. In
microelectromechanical systems (MEMS), FGPM viscoelastic composites can be utilized in the
fabrication of tiny sensors, actuators, and resonators for applications in various mechanical, aerospace,
consumer electronics, and medical sectors.
41
• Structural Health Monitoring: FGPM viscoelastic composite structures can be used for structural health
monitoring (SHM) purposes. They can be embedded with sensors that utilize the piezoelectric effect to
detect changes in the systems's properties and identify structural damage or defects in real time.
• Soft Robotics: FGPM viscoelastic composites can be integrated into soft robotic systems to create
deformable structures with piezoelectric functionality. These materials would enable more flexible and
adaptable robots for delicate tasks and human-robot interaction.
• Vibration Damping and Control: FGPM viscoelastic composite structures can be employed in the
aerospace, automotive, and civil engineering industries for vibration damping and control. These
materials can be designed to have specific damping properties, reducing vibrations and minimizing
resonance effects in structures like aircraft wings, car panels, and buildings.
6. Concluding remarks
In this article, we have investigated the time-dependent viscoelastic electromechanical behaviour of
single and multi-layered piezoelectric functionally graded graphene-reinforced composite (FG-GRC) plates
(called FGM and FGPM plates). Higher-order shear deformation theory (HSDT) with quasi-3D plate
formulation that incorporates sinusoidal shape function and linear piezoelectricity are implemented along with
the Biot model of viscoelasticity in the framework of the inverse Fourier algorithm. The principle of virtual
work is adopted to derive the governing equations and boundary constraints for analytical solutions based on
Navier’s method. Further, finite element models are developed to confirm the accuracy and validity of the
analytical results. The electromechanical behaviour includes the static and electric response of FG-GRC
viscoelastic plates which are predicted considering transverse mechanical and external electrical loading under
simply supported (SSSS) conditions. Following major inferences are drawn from the numerical results:
• The transverse deflection of FGM and FGPM plates are significantly affected due to consideration of
piezoelectricity, weight fraction of GPLs (
) and different distribution patterns (such as linear and
uniform distribution) of GPLs along the thickness. It increases with respect to aspect ratio while
reduces for a larger value of
.
• The electric potential shows significant enrichment with higher values of ratio, while it shows a
decrement with respect to the addition of
.
• A rapid decrement in the elastic properties of FG-GRC plates can be observed with time due to the
consideration of the viscoelastic effect.
• The numerical results concerning electric potential for time-dependent viscoelastic analysis establishes
that it is possible to harness more accumulated power with increasing time before eventually reaching
the steady state condition. With regard to the aspect ratio of the plate, weight fractions of the GPL and
42
the distribution patterns, the steady-state values follow a similar general trend as the non-viscoelastic
scenario.
• All three distributions exhibit a rapid increment in transverse deformation and electric potential with
time, although the rates at which this increment occurs vary depending on the distribution of the
material properties (Multi-layered FGPM > Single-layered LD FGPM > Single-layered UD FGPM).
In summary, the current semi-analytical study demonstrates a potential time-dependent
electromechanical behaviour based on practically relevant viscoelastic modelling coupled with through-
thickness gradation, which could be crucial for analysing the structural behaviour of a wide range of ‘smart’
plate-like structures accurately and prospective temporal programming for a range of engineering applications
across the length scales. The present analytical solution approach can also be extended to obtain the viscoelastic
electromechanical responses of functionally graded smart shells and other complicated structural assemblies.
The combination of piezoelectricity and viscoelasticity in FGPM viscoelastic composite structures opens up a
wide range of engineering possibilities, from energy harvesting, vibration control, shape-morphing to advanced
robotics and smart structures. Ongoing research in this field continues to explore new applications and optimize
the performance of these materials for real-world scenarios.
Acknowledgments
TM and SN acknowledge the initiation grant received from the University of Southampton.
Conflict of Interest
The authors declare no conflict of interest.
ORCID ID
Soumyadeep Mondal : https://orcid.org/0000-0002-3368-7828
Kishor Shingare https://orcid.org/0000-0002-8255-9849
Tanmoy Mukhopadhyay https://orcid.org/ 0000-0002-0778-6515
Susmita Naskar https://orcid.org/0000-0003-3294-8333
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