Static and dynamic response of graphene nanocomposite plates withflexoelectric effectK.B. Shingare, S.I. Kundalwal⁎
Applied and Theoretical Mechanics (ATOM) Laboratory, Discipline of Mechanical Engineering, Indian Institute of Technology Indore, Simrol, Indore 453 552, India
A R T I C L E I N F O
Keywords:GrapheneFlexoelectricityPiezoelectricityNanocompositeMicromechanicsMechanics of materialsFinite elementElastic properties
A B S T R A C T
In this work, the electromechanical behaviors of graphene reinforced nanocomposite (GRNC) plates withflexoelectric effect were studied by using Kirchhoff's plate theory, Navier's solution and extended linear piezo-electricity theory in conjunction with the mechanics of materials (MOM) and finite element models. The staticand dynamic responses of simply supported flexoelectric GRNC nanoplates under different loadings such asuniformly distributed, varying distributed, inline and point loads were investigated. The developed MOM and FEmodels envisage that the effective piezoelectric constants of a GRNC account for the actuating capability in itstransverse direction due to the applied electric field in the plane. The elastic properties of pristine and defectivegraphene sheets were also estimated via molecular dynamics (MD) simulations and the obtained results arefound in good agreement with the existing experimental and numerical results. Our results reveal that theflexoelectric effect on the static and dynamic responses of GRNC nanoplate is substantial and cannot be ne-glected. The electromechanical response of GRNC plates can be engineered to attain the desired deflectioncharacteristics and resonant frequencies for a range of nanoelectromechanical systems using different boundaryconditions as well as geometrical parameters such as aspect ratio/thickness of nanoplate and volume fraction ofgraphene.
1. Introduction
Graphene, one-atom-thick planer layer of carbon atoms discoveredand characterized by Novoselov et al. (2004), fascinated rigorous re-search interests and applications in nanoelectromechanical systems(NEMS) owing to its remarkable electro-thermo-mechanical propertiessuch as high Young's modulus ( ∼ 1.1 TPa) and high flexibility, elec-trical conductivity ( ∼ 6000 S/cm), thermal conductivity ( ∼ 5000 W/m/K) and scale-dependent electronic properties (Zhang et al., 2005;Balandin et al., 2008; Lee et al., 2008; Gupta and Batra 2010; Vermaet al., 2014; Cui et al., 2016; Alian et al., 2017; Kundalwal et al., 2017).In the twenty-first century, graphene is considered as one of the moststriking 2D material to form next generation NEMS due to its uniquemultifunctional properties with size-dependent physical structure. Mostrecently, the piezoelectric effect in non-piezoelectric graphene sheets isfound by Kundalwal et al. (2017) using the flexoelectric concept viaquantum mechanics calculations. This study revealed that the presenceof strain gradient in non-piezoelectric graphene sheet does not onlyaffects the ionic positions, but also the asymmetric redistribution of theelectron density, which induces strong polarization in the graphenesheet. This recent finding on the strain gradient polarization in non-
piezoelectric graphene added a new functionality in it and this couldlead to develop graphene-based energy harvesters.
Extensive research has been dedicated to the introduction of gra-phene as the modifiers to the conventional bulk composites in order toimprove their multifunctional properties. On the other hand, the use ofsmall quantity of graphene in the conventional matrices is found toalter and improve the thermo-electro-mechanical properties of resultingnanocomposite significantly. For instance, Ji et al. (2016) summarizedand reviewed the numerous methods of synthesis of graphene-basednanocomposites and showed the significant enhancement of mechan-ical properties of nanocomposite with mere 0.3 wt.% of graphene overthat of the pure aluminum matrix (Ji et al., 2016). Kandpal et al. (2017)experimentally investigated the enhancement of piezo-potential re-sponse such as output voltage of nanogenerator with the accumulationof graphene nanoplates (GNPs) into the polymer nanocomposite, whichcan be used as an energy harvester. García-Macías et al. (2018) showedthat the GNP-based composite is superior than carbon nanotube (CNT)-based composite in terms of the load carrying capacity and stiffeningeffect. Graphene as well as its derivatives such as graphite oxide andgraphene oxide (GO) also highlight promising way towards the bulk useof graphene for commercial applications and NEMS (Cui et al., 2016).
https://doi.org/10.1016/j.mechmat.2019.04.006Received 15 January 2019; Received in revised form 6 April 2019
For instance, Bhavanasi et al. (2016) reported the energy harvestingperformance with the proficient transfer of electromechanical energyfor the film made of bilayer GO and PVDF-TrFE. They observed that GOfilm improved the voltage output and power density of about 2 and 2.5times, respectively, when compared to the PVDF-TrFE film without GO.Few studies elucidated the methods of production, applications, in-ventions and limitations of composites made of graphene and its deri-vatives (Mohan et al., 2018; Sreenivasulu et al., 2018). They focused onseveral components of fortifying, scattering strategies and blendedcomposites utilizing graphene.
Thin composite plates are important types of structural elementsthat have potential applications in NEMS due to their linear behaviorand high sensitivity. For instance, Gao and Shen (2003) demonstratedthat the piezoelectric actuators can significantly reduce the vibrationsof composite plates using the FE analysis. A nonlinear bending analysisfor simply supported antisymmetric cross-ply laminated compositeplates incorporated with the piezoelectric actuators under the com-bined mechanical, electrical and thermal loadings presented byShen (2004). Ray and Pradhan (2006) reported the 1–3 piezoelectricvertically reinforced laminated composite to improve the performanceof hybrid damping using the procedure of active constrained layerdamping (ACLD). Experimental study by Parashar and Mertiny (2012)reported that the buckling capacity of the plate increases when thegraphene reinforced composite is enriched with only low percentage ofgraphene. They reported 26% enhancement in the buckling capacity ofgraphene-based composite plate under the unidirectional compressionwith only 6% volume fraction of graphene. Using the FE method,Chandra et al. (2012) highlighted the enhancement of the natural fre-quencies and mode shapes of graphene/epoxy composite plates. Theyfound that as the plate aspect ratio increases, the natural frequency ofthe graphene/epoxy composite plates decreases. Li and Narita (2014)proposed an active control method to reduce the wind-induced vibra-tion of the laminated composite plates using a velocity feedback controlstrategy. Zhang and Jiang (2014) studied the effect of static bulk flex-oelectricity on the piezoelectric nanoplate to predict its static and dy-namic behavior. Kundalwal and Ray (2016) investigated the influenceof CNT waviness on the smart damping of fuzzy fiber reinforced com-posite plates incorporated with the ACLD patches using 3D FE analysis.Sadeghzadeh (2016) studied the multilayer graphene reinforced plateusing multiscale approach and reported that the spaced multilayergraphene sheets are more efficient than the stacked multilayer gra-phene sheets having interlayer distance 0.34 nm, that is, no metallicnanoparticles or fullerenes exist between two adjacent graphene layers.Song et al. (2006) studied the free and forced vibrations of multilayerGNP-reinforced composite plates subjected to the axial compressionand transverse loadings. They revealed that the small amount of GNPssignificantly increases the critical buckling load of composite plate andreduces its vibration. Feng et al. (2017) examined the nonlinearbending behavior of polymer nanocomposite beams reinforced withmulti-layered GNPs that are non-uniformly dispersed in the thicknessdirection of beam. They found that the bending performance of polymermatrix nanocomposite significantly improved by adding small amountof GNPs. The micromechanical model and multiscale approach weredeveloped by Shen et al. (2017) for analyzing the post-buckling beha-vior of functionally graded (FG) graphene-based laminated compositeplates under the uniaxial compression in thermal environments. Theinfluence of surface effects on the nanoscale plate was investigated byLiu and Rajapakse (2013) and Sapsathiarn and Rajapakse (2017) tostudy its static and dynamic behavior. They also derived a solution forthe nanoscale rectangular plates using the FE approach.Karimi et al. (2017) investigated the effect of different parameters suchas nonlocal and surface layers on the in-phase and out-of-phase naturalfrequencies of double-layer piezoelectric nanoplate subjected to thethermo-electro-mechanical loadings. Zhao et al. (2017) investigated thebending and vibration behavior of FG GNP-reinforced trapezoidalplates using the FE method. They also predicted the effective material
properties such as Young's modulus, mass density and Poisson's ratio ofthe GNP-nanocomposite using modified Halpin–Tsai model and therules of mixture. Most recently, Rouzegar and Abbasi (2018) developedFE model to study the bending analysis of laminated plates integratedwith the piezoelectric fiber-reinforced composite actuators.
The review of literature presented on graphene-based compositeplates clearly indicates that graphene is the most attracting 2D material,vastly studied in the last decade. Nevertheless, to date, to the best of thecurrent authors’ knowledge, there is no single study for investigatingthe static and dynamic response of a flexible piezoelectric graphenereinforced nanocomposite (GRNC) nanoplate considering the flexo-electric effect, which can offer many opportunities for developing nextgeneration NEMS. This provided the motivation for the present re-search. The aim of our research is to further broaden the knowledgebase of existence of strain and electric field gradients in thin nanoplates.Before proceeding further, we discuss the concept of flexoelectricity.
1.1. Flexoelectricity
Over the last two decades, the flexoelectricity phenomenon hasreceived much attention from both fundamental and application pointof view with aim of developing NEMS. Recently, the piezoelectric na-noplate based NEMS have attracted the enormous interest in the re-search community due to their potential use as a numerous nano-technology-based devices such as sensors, actuators, transistors, energyharvesters, nano-generators, electric switches and distributors (Denget al. 2014; Zhang and Jiang, 2014; Sapsathiarn and Rajapakse, 2017).The interfaces in nonpolar materials and symmetry breaking at thesurfaces permit novel forms of electromechanical coupling such asflexoelectricity and surface piezoelectricity, which can be induced onlyin the nanoscale materials. For a better understanding of flexoelec-tricity, first the concept of piezoelectricity and its mathematical relationis described; as follows:
P di ijk jk (1)
In the above relation, Pi denotes the polarization vector, ɛjk is thestrain tensor, and dijk is the piezoelectric tensor. This phenomenonexists in the material where the inversion symmetry plays a significantrole. As it was mentioned earlier, by breaking the inversion symmetryeven in the centrosymmetric crystals one can introduce the polarizationdue to the nonuniform strain gradient, and this phenomenon is wellknown as the flexoelectric effect. In 1964, Kogan Kogan, 1964) reporteda theoretical understanding of electron-phonon coupling which is re-sponsible for the flexoelectricity in centrosymmetric crystals. He hasprovided the estimation for the lower bounds of felxoelectric coeffi-cients with the help of relation between the electronic charge (e) inelectron volt and lattice parameter (a). Hence, the approximated valueof the flexoelectric coefficient (e/a) is interpreted 10 C/m9
(Nguyen et al., 2013). This phenomenon presents in all dielectric andinsulating materials (Tagantsev, 1986; Ma and Cross, 2006; Marangantiet al., 2006). The constitutive relation ((1) for the polarization vectoraccounting the flexoelectric effect may be re-written as
+P d fddxi ijk jk ijkl
jk
l (2)
where fijkl is the flexoelectric tensor and ddx
jk
lis strain gradient.
The organization of the article is as follows: Section 2 presents thedevelopment of modified mechanics of materials (MOM) model to de-termine the effective properties of GRNC. In this section, FE modelswere also developed, and they provided interesting numerical valuesfor validating the predictions of MOM model. In Section 3, we pre-sented an analytical model based on Kirchhoff's plate theory and Na-vier's solution for the simply supported GRNC nanoplates with flexo-electric effect. We derived closed form solutions for nanoplates withdifferent boundary and loading conditions which were not considered
K.B. Shingare and S.I. Kundalwal Mechanics of Materials 134 (2019) 69–84
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in the literature. In Section 4, the elastic properties of pristine anddefective graphene layers were evaluated using the molecular dynamics(MD) simulations. Section 5 presents the validations of our findings aswell as results for analyzing the static and dynamic behavior of GRNCnanoplates with flexoelectric effect considering different boundaryconditions and geometrical parameters. In Section 6, main inferencesare drawn from the current study.
2. Effective properties of GRNC
In this section, assuming a graphene sheet as a piezoelectric con-tinuum, the piezoelectric and elastic (piezoelastic) as well as dielectricproperties of GRNC were estimated. To simplify computational efforts,bulk properties of 2D graphene layer were obtained by considering it asa continuum plate (Roberts et al., 2010; Politano and Chiarello, 2013,and references therein; Hosseini-Hashemia et al., 2018). Using theconcept of continuum elasticity, analytical and numerical solutionswere obtained for graphene layers under strains (Gupta and Batra,2010; Gradinar et al., 2013; Verma et al., 2014; Bahamon et al., 2015).This suggests that the displacement of each carbon atom in homo-geneously deformed graphene layer is given by the deformation of thecontinuum medium, on which the atom is embedded.
A novel GRNC is reinforced with the multilayers of piezoelectricgraphene sheets and polyimide matrix. Such GRNC can be considered ascomposed of rectangular representative volume elements (RVEs) com-prising both graphene and polyimide matrix, as shown in Fig. 1, and welimited the development of our micromechanical model to a single RVE.We assumed that no slippage occurs between a graphene sheet and thesurrounding matrix, and the resulting nanocomposite is linearly elasticand homogeneous (Gao and Li, 2005; Song and Youn, 2006; Jiang et al.,2009; Kundalwal and Ray, 2011, 2013). The top and bottom surfaces of
the GRNC lamina are electroded and the electric field is applied alongits thickness. The conducting electrodes maintain constant electrostaticpotentials on both the upper and lower surfaces of GRNC lamina. SuchGRNC lamina demonstrates the inverse piezoelectric effect and may beconsidered as a capacitor having two parallel plates in which the gra-phene layers and polyimide matrix act as the dielectric mediums. Theeffective piezoelastic and dielectric properties of GRNC can be de-termined by using the analytical and numerical micromechanicalmodels.
Fig. 1. (a) Schematic representation of GRNC lamina and (b) cross-sections of an RVE of GRNC.
Fig. 2. Finite element meshing of RVE of GRNC.
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2.1. Mechanic of materials (MOM) model for GRNC
The effective piezoelastic and dielectric properties of GRNC can beobtained by modifying the existing mechanics of materials (MOM)model (Kundalwal and Ray, 2011). Fig. 1(b) demonstrates an RVE ofthe GRNC lamina in which the graphene layers are incorporated alongits thickness direction. Smith and Auld (1991) used the strength ofmaterials approach to predict the effective piezoelastic properties of the1–3 piezoelectric composite in which PZT fibers of square cross-sectionwere surrounded by the epoxy matrix. Note that the effective piezo-elastic properties predicted by Smith and Auld (1991) are most feasiblefor controlling the thickness mode oscillations of thin composite plates.Our micromechanical analysis was confined to the RVE of GRNC (seeFig. 1(b)) for determining the effective properties of bulk GRNC.
We assumed plane strain deformations in the GRNC lamina andnormal stresses can be induced in it due to the applied electric field (E3)along the 3-axis of the GRNC. The constitutive equations for the con-stituents of GRNC can be written as follows:
= ={ } [C ]{ } {e }E and { } [C ]{ }g g g g3
m m m (3)
= ={ } , { }r
1r
2r
3r
23r
13r
12r
r
1r
2r
3r
23r
13r
12r (4)
= = =[C ]
C C C 0 0 0C C C 0 0 0C C C 0 0 00 0 0 C 0 00 0 0 0 C 00 0 0 0 0 C
, {e }
eee000
, r g and mr
11r
12r
13r
12r
22r
23r
13r
23r
33r
44r
55r
66r
g
31g
32g
33g
In Eqs. (3) and (4), the respective g and m superscripts represent thegraphene and polyimide matrix. The superscript r is used to indicate thecorresponding constituent phase; 1
r, 2r, and 3
r are the normal stressesin the directions 1, 2, and 3, respectively; 1
r, 2r, and 3
r are the re-spective normal strains; 12
r , 13r , and 23
r are the shear stresses; 12r , 13
r ,and 23
r are the shear strains; Cijr (i,j= 1,2 and 6) are the elastic coeffi-
cients of rth phase; and e31g , e32
g , and e33g are the piezoelectric coefficients
of a graphene.Using isostrain and isostress conditions (Smith and Auld, 1991;
Benveniste and Dvorak, 1992; Ray, 2006), the perfect bonding amongsta graphene layer and the matrix can be modelled by satisfying thefollowing:
= = and
1g
2g
3g
23g
13g
12g
1m
2m
3m
23m
13m
12m
1NC
2NC
3NC
23NC
13NC
12NC
(5)
Using the rules of mixture, we can write:
+ =v vg
1g
2g
3g
23g
13g
12g
m
1m
2m
3m
23m
13m
12m
1NC
2NC
3NC
23NC
13NC
12NC
(6)
in which the superscript NC denotes the quantities of RVE of GRNC, andvg and vm are the volume fractions of a graphene layer and the poly-imide matrix, respectively. Using Eqs. (3–6), the stress and strain vec-tors of homogenized GRNC can be written in terms of the respectivestress and strain vectors of constituent phases as follows:
From Eq. (9), the effective piezoelectric coefficients of the GRNCcan be identified as =e e (1)31
NC , =e e (2)32NC , and =e e (3)33
NC .Consequently, the effective dielectric constant ( 33
NC) of GRNC is derivedby using the following relation (Ray and Pradhan 2006):
= + + +v v e v v /(v C v C )33NC
g 33g
m 33m
31g
g m m 11g
g 11m (10)
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2.2. Finite element (FE) modeling of GRNC
The MOM model developed in the preceding section is based on theisofield conditions in which the RVE of GRNC was imposed to theuniform stresses and strains. To validate such assumptions, the FEmodels can be developed which do not require any such assumption. Inthis section, FE models were developed using the commercial softwareANSYS 15.0. The FE approach allows the fully coupled electro-mechanical analysis and we obtained the homogenized effective ma-terial properties by modeling 3-D RVE of GRNC with 20 noded brickelements “solid 226″ having both displacement and electric potential(voltage) degrees of freedom (DOF). Using the material properties of agraphene layer and the polyimide matrix, a two-phase FE model of theRVE with its axis of transverse isotropy being the 3-axis is to be de-veloped for estimating the independent piezoelastic and dielectricconstants: C11
NC, C12NC, C23
NC, C33NC, e31
NC, e33NC, and 33
NC. These effective con-stants of the GRNC can be evaluated by applying the appropriateboundary constraints to its RVE.
The constitutive relations for the effective elastic constants (CijNC),
piezoelectric constants (eijNC), and dielectric constants ( ij
NC) of GRNCcan be written as:
=
¯¯¯¯¯¯DDD
C C C 0 0 0 0 0 eC C C 0 0 0 0 0 eC C C 0 0 0 0 0 e
0 0 0 C 0 0 0 e 00 0 0 0 C 0 e 0 00 0 0 0 0 C 0 0 00 0 0 0 e 0 0 00 0 0 e 0 0 0 0
e e e 0 0 0 0 0
¯¯¯¯¯¯EEE
11
22
33
23
13
12
1
2
3
11NC
12NC
13NC
31NC
12NC
22NC
13NC
31NC
13NC
13NC
33NC
33NC
44NC
15NC
55NC
15NC
66NC
15NC
11NC
15NC
22NC
31NC
31NC
33NC
33NC
11
22
33
23
13
12
1
2
3
(11)
The averaged stresses {¯ij}, strains { ij}, electrical displacements{Di}, and electric field {Ei} can be evaluated using the following rela-tions when the RVE is subjected to the electromechanical loads:
= =
= =
{ ¯ } { }V , {¯ } { }V
{D } {D }V , {E } {E }Vij
1V 1
nij e ij
1V 1
nij e
i1V 1
ni e i
1V 1
ni e (12)
where V denotes the volume of the RVE, Ve is the volume of finiteelement, and n is the total number of finite elements. The quantityhaving overbar symbolizes the volume averaged quantity. It is ob-served from Eq. (11) that if at any point in the GRNC only one normalstrain exists while the other strain components are zero, then threenormal stresses are nonzero. The ratio of one of these three normalstresses and the strain yields respective effective elastic constant.Hence, with one FE numerical simulation, we can obtain three effec-tive elastic constants at a point in the RVE. Detailed procedure ofdevelopment FE models under appropriate boundary conditions isdescribed elsewhere Kundalwal and Ray, 2012; Kundalwal et al.,2018.
3. Electromechanical response of GRNC nanoplates
3.1. Governing equations for GRNC nanoplates
In this section, the governing equations for simply supported (SS)piezoelectric GRNC nanoplate subjected to the uniformly distributedmechanical load (q0) are derived to investigate its static bending anddynamic behavior. Fig. 3 shows the schematic of SS GRNC nanoplatehaving thickness h, length a, and width b. A Cartesian coordinatesystem was used to describe the nanoplate with thickness along the z-axis and the mid plane of the undeformed nanoplate coincides with thex–y plane.
As per Kirchhoff's plate theory, the plate displacement in terms oftransverse displacement w(x, y, t) can be expressed as (Zhao et al.,2012):
=u(x, y, z, t) z w(x, y, t)x (13)
=v(x, y, z, t) z w(x, y, t)y (14)
=w(x, y, z, t) w(x, y, t) (15)
in which u and v are the displacement components along the x and ydirections, respectively; t is the time; and w is the transverse dis-placement. Consequently, the nonzero strains can be written as follows:
= = =z wx
, z wy
, z wx yxx
2
2 yy2
2 xy2
(15a)
Assuming the electric field Ez exists only in the z-direction of thenanoplate, the in-plane dimensions and electric field components in thex y plane can be eliminated when they are compared with that in thethickness direction (Ying and Zhifei, 2005; Ray and Pradhan, 2006).Assuming infinitesimal deformation, considering the electric field,strain gradient coupling and purely nonlocal elastic effects the gen-eralized equation for the electric Gibbs free energy density function Ucan be written as follows (Hu and Shen, 2009):
= +
+ +
U 12
E E 12
C e E f E
r 12
g
kl k l ijkl ij kl ijk ij k ijkl i jkl
ijklm ij klm ijklmn ijk lmn (16)
rijklm represents the coupling between the strain and strain gradientwhile gijklmn represent the elements for purely nonlocal elastic effects,related to the strain gradient elastic theories. For the sake of simplicity,in centrosymmetric materials, the terms gijklmn and rijklm are assumed tobe zero. Under infinitesimal deformation, the constitutive equations forthe dielectric material considering the flexoelectric effect at nanoscalelevel can be written as (Hu and Shen, 2009):
= =U C e Eijij
ijkl kl ijk k(17)
= =U f Eijmijm
kijm k(18)
= = + +D UE
e E fii
jki jk ij j ijkl jkl (19)
where, ∈ ij, Cijkl and eijk are the dielectric, elastic and piezoelectrictensors, respectively; fijkl is the fourth order tensor representing thehigher order electric field-strain gradient coupling; σij is the traditionalstress tensor which is analogous to that in the classical theory of elas-ticity; Di and τijm are the electric displacement vector and higher-order
Fig. 3. Schematic of GRNC nanoplate subjected to the uniformly distributedload.
K.B. Shingare and S.I. Kundalwal Mechanics of Materials 134 (2019) 69–84
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stress tensor, respectively; Ei and ηjkl are the electric field vector andstrain gradient tensor, respectively, which can be obtained as follows:
=jkl jk,l (20)
Using the Eqs. (17)–(19), the constitutive relations can be re-formulated as:
= +C C e Exx 11 xx 12 yy 31 z (21)
= +C C e Eyy 12 xx 11 yy 31 z (22)
= 2Cxy 66 xy (23)
= f Exxz 14 z (24)
= f Eyyz 14 z (25)
= + + + +D e ( ) E f ( )z 31 xx yy 33 z 14 xxz yyz (26)
where f14 = f3113 = f3223 (Shu et al., 2011). For the sake of simplicity,the strain gradients other than =xxz
wx
22 and =yyz
wy
22 are as-
sumed to be zero, since the associated flexoelectric coefficients or straingradients are much smaller as compared to those along the thicknessdirection of GRNC nanoplate. Due to the absence of external electricfield, it can be clearly seen that the 3rd term in Eq. (26) indicates thepolarization induced in the nanoplate due to the strain gradients.
Using the equations of Gauss' Law of Electrostatics, in the absence offree electric charge, the electric displacement for the thin nanoplate canbe written as
=Dz
0z(27)
In case of open-circuit condition, on the surface of nanoplate theelectric displacement is zero. Therefore, from Eq. (27), the internalelectric field can be derived as follows:
= + + +E e wx
wy
z f wx
wyz
31
33
2
2
2
214
33
2
2
2
2 (28)
From Eq. (28) it can be observed that the first term e3133
related tothe piezoelectricity signifies the electric field induced due to the ap-plication of elastic strains, and the second term f14
33related to the
flexoelectricity signifies the electric field induced due to the applica-tion of elastic strain gradients. Considering the flexoelectricity, thepiezoelectricity associated internal electric field no longer remainsanti-symmetric respective to the midplane of the nanoplate in thedirection of its thickness. Then, taking the summation of the curva-tures at an arbitrary point in the nanoplate as a whole, Eq. (28) can berewritten as = +Ez
e z f31 1433
G with G= +wx
wy
22
22 . Note that the response
of the electric field Ez to G is intensely dependent on the z-coordinate.In addition, term +ze f31 14 is dependent on the thickness of nanoplateand its flexoelectric coefficient. If the piezoelectric effect is not takeninto account then the solution is attributed to the flis dependen effect(i.e., f14).
The governing equations for the SS nanoplate problems can be de-rived using Hamilton's variational principle; such as (Mindlin, 1968):
+ + =UdV K W dt 0t1
t2
V (29)
where V is the entire volume occupied by the GRNC nanoplate and U isthe electric Gibbs free energy density. In case of open-circuit condition,the relation of U can be written as
= +U 12
12ij ij ijk ijk (30)
If the vibration along the x y plane is ignored then the kineticenergy (K) is given by
=K 12
wt
dVV
2
(31)
where ρ is the mass density.The work done (W) due to the application of external load can be
determined as follows:
=W q wdydx0
a
0
b0 (32)
An energy formulation for continuum electro-elasticity is based onthe principle of minimum free energy, which is mainly suitable forcomplex materials with significant gradient effects and analysis ofstability (Liu et al., 2013). Therefore, the governing equation can bewritten as
+ + + + + + =qMx
Mx y
Mx y
My
Nx
Ny
h wt
02
xx2
2xy
2yx
2yy
2
2xxz2
2yyz2
2
2 0
(33)
Boundary conditions for SS rectangular GRNC nanoplates on all fouredges are prescribed and can be deduced as at =x 0 and =x a:
+
+ +
M N or
M or
or w
xx xxzwx
xywy
Mx
My
Nx
xx yx xxz(34)
at =y 0 and =y b:
+
+ +
M or
M N or
or w
yxwx
yy yyzwy
Mx
My
Ny
xy yy yyz
(35)
where Mxx, Mxy, Myx and Myy are the bending moments, andNxxz and Nyyz are the axial forces along the thickness; these can be ob-tained as follows:
= = = =M zdz, M zdz, M M zdzxx h/2
h/2xx yy h/2
h/2yy xy yx h/2
h/2xy
(36)
= =N dz, N dzxxz h/2
h/2xxz yyz h/2
h/2yyz (37)
Substituting Eqs. (15a) and (28) into the constitutive relations(21)–(25), the explicit expressions for the stresses and higher-orderstresses related to the transverse deflection (w) can be written as,
= + +
+
C e wx
z C e wy
z
e f wx
wy
xx 11312
33
2
2 12312
33
2
2
31 14
33
2
2
2
2 (38)
= + +
+
C e wx
z C e wy
z
e f wx
wy
yy 12312
33
2
2 11312
33
2
2
31 14
33
2
2
2
2 (39)
= 2C wx y
zxy 662
(40)
= = + +e f wx
wy
z f wx
wyxxz yyz
31 14
33
2
2
2
2142
33
2
2
2
2 (41)
Making the use of Eqs. (38)–(41) into Eqs. (36)–(37), the bendingmoments can be obtained in terms of w as follows:
= + +M C e h12
wx
C e h12
wyxx 11
312
33
3 2
2 12312
33
3 2
2 (42)
K.B. Shingare and S.I. Kundalwal Mechanics of Materials 134 (2019) 69–84
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= + +M C e h12
wx
C e h12
wyyy 12
312
33
3 2
2 11312
33
3 2
2 (43)
= =M M 2C h12
wx yxy yx 66
3 2
(44)
= = +N N f wx
wy
hxxz yyz142
33
2
2
2
2 (45)
After careful observations of Eqs. (38)–(45), it can be observed thatthe fl, it can be ob significantly influences the distribution of stress andhigher-order stresses. Accordingly, the higher-order stress vanisheswhen the flexoelectric effect is not considered and the conventionalbending moments are not influenced by the strain gradient polarization.Moreover, the introduction of flexoelectric effect is n effect yields thesummations of higher-order stresses.
Using Eqs. (42)–(45) into Eq. (33), the governing equation can bewritten in terms of w as follows:
+ + + + = qD wx
wy
2(D 2D ) wx y
h wt11
4
4
4
4 12 664
2 2
2
2 0 (46)
with
= + +
= + +
=
h
h
D C
D C
D C
11 11e h
12f
12 12e h
12f
66 66h12
312
33
3 142
33
312
33
3 142
33
3
(47)
3.2. Exact solution for static bending response of GRNC nanoplates
For static bending response of the piezoelectric GRNC nanoplates,the governing Eq. (46) can be re-written as follows Reddy (2003):
+ + + = qD wx
wy
2(D 2D ) wx y11
4
4
4
4 12 664
2 2 0 (48)
It may be noted that in the absence of flexoelectricity, the governingEq. (48) follows the conventional classical Kirchhoff plate theory con-sidering linear piezoelectricity. According to the conventional platetheory, for solving the governing Eq. (48) of the SS GRNC nanoplate,the following Fourier series can be used to determine w (x, y)
== =
x yw(x, y) A sin sinm 1 n 1
mn(49)
where = =,ma
nb and Amn are the coefficients to be calculated for
each m and n half wave numbers that should be satisfied everywhere inthe domain of a nanoplate. It has previously corroborated that Eq. (49)satisfies the boundary conditions given in Eqs. (34) and (35). Using theexpression of Fourier series, the uniformly distributed load q0 (x, y) canbe obtained as follows,
== =
xq(x, y) Q sin sin ym 1 n 1
mn(50)
with
=w(x, y) Qd
andmn
mn
where
= =Q16q
mnm, n 1, 3, 5, ......mn
02 (51a)
= + + +db
D ma
D nb
2(D 2D ) mnabmn
411
412
412 66
2
(51b)
Substituting Eqs. (49)–(50) into Eq. (48), we can derive the solution
for the SS nanoplate to obtain its transverse deflection.
=+ + += … = … { }( ) ( ) ( )
w(x, y)16q sin x sin y
mn D D 2(D 2D )0
6m 1,3,5 n 1,3,5 11
ma
412
nb
412 66
mnab
2
(52)
3.3. Exact solution for dynamic response of GRNC nanoplates
Using Eq. (46), the governing equation for the GRNC nanoplate canbe written as
+ + + + =D wx
wy
2(D 2D ) wx y
h wt
0114
4
4
4 12 664
2 2
2
2 (53)
Similar to the conventional plate model, the harmonic solution forw (x,y,t) is derived as
== =
w(x, y, t)sin n y
be
m 1 n 1
B sin m xa i t
mnmn
(54)
where Bmn is a constant indicating the mode shape amplitude; m and nare the half wave numbers; ωmn is the resonant frequency; and =i 1 .
Making use of Eq. (54) into Eq. (53) yields the nanoplate resonantfrequency; as follows:
+ + + =D ma
nb
2(D 2D ) ma
nb
h 0114 4
12 662 2
mn2
(55)
Henceforth, the resonant frequency for nanoplate can be obtainedfor different order numbers m and n as:
= + + +h
D ma
nb
2(D 2D ) ma
nbmn
2
114 4
12 662 2
(56)
From above it is clear that the resonant frequency can be de-termined using the traditional theory of piezoelectricity if the flexo-electric effect is neglected.
4. Elastic properties of graphene
In the literature, piezoelectric properties of graphene sheets con-taining non-centrosymmetric pores are available (Kundalwal et al.,2017) but their elastic properties need to be estimated. Therefore, wefirst determined the elastic properties of (i) pristine graphene sheet and(ii) defective graphene sheets containing 4.5% and 20% vacancies inform of non-centrosymmetric pores using MD simulations. Schematicsof such graphene layers are shown in Fig. 4. All MD simulations runswere conducted with large-scale atomic/molecular massively parallelsimulator (LAMMPS) (Plimpton 1995), and the molecular interactionsin CNTs are described in terms of Adaptive Intermolecular ReactiveEmpirical Bond Order (AIREBO) force fields (Stuart et al., 2000). Theatomic volume was determined from the relaxed graphene sheet withthe thickness (t) of 3.4 Å (Huang et al., 2006; Pei et al., 2010). Thestress in the graphene sheet was computed by averaging the obtainedstress of each carbon atom in it. Then, the stress-strain curves duringthe tensile loading were obtained, and Young's modulus (E) and Pois-son's ratio of pristine and defective graphene sheets were determined.Detailed procedure of MD simulations is described elsewhere(Kundalwal and Choyal, 2018). The predicted elastic properties ofgraphene are summarized in Table 1.
The predictions of pristine graphene agree well with the existingresults obtained by using different modeling techniques and potentialsas well as experimental estimates (Lee et al., 2008; Jing et al., 2012;Dewapriya et al., 2015). In case of defective graphene, we validated ourresults with those reported by Jing et al. (2012). They used COMPAASforce field to model the defective graphene sheets containing vacancies
K.B. Shingare and S.I. Kundalwal Mechanics of Materials 134 (2019) 69–84
75
which were functionalized by hydrogen atoms on the dangling bonds.The percentage of reduction in Young's modulus ∼1.6% in case ofgraphene containing 6 carbon atom vacancies (that is, 4.5% vacancies)is found to be close with that reported by Jing et al. (2012). They re-ported the percentage of reduction ∼1.53% for functionalized gra-phene with 6 missing carbon atoms. In case of 20% vacancies, theelastic properties of graphene are not much significantly affected. Thisis attributed to the hydrogenation and saturation of the dangling bondsat the edges and porosity in graphene sheet (Jing et al., 2012).
5. Result and discussion
In this section, we discussed the numerical outcomes of effectivepiezoelastic and dielectric properties of GRNC estimated by means ofMOM and FE models developed in the previous sections. The piezo-electric properties of graphene sheets containing non-centrosymmetricpores were taken from Ref. Kundalwal et al. (2017). The graphene sheetunder consideration has 224 carbon atoms and accordingly, the normalpiezoelectric coefficient (e33) was determined as 0.221 C/m2 when thevalue of = 15.2Å. The material properties of graphene and polyimidematrix are summarized in Table 1.
Practically, the fiber volume fraction in the composite can varytypically from 0.2 to 0.7 and hence, this range was considered to ana-lyze the effect of graphene volume fraction on the piezoelastic and di-electric properties of GRNC. Unless otherwise mentioned, the pristinegraphene was considered to determine the piezoelastic and dielectricproperties of GRNC. First, the effective piezoelastic and dielectricproperties of GRNC were determined using the MOM and FE ap-proaches as shown in Figs. 5–10. In FE simulations, the governingequations were solved by using a linear perturbation for piezoelectricanalysis and the sparse direct solver was used for structural analysis.First, the FE mesh convergence was carried out to study the effect ofelement size on the effective properties of GRNC for obtaining the re-liable results.
Fig. 5 demonstrates the variation of effective elastic constant C11NC of
the GRNC against the volume fraction of graphene (vg). It may be ob-served that the values of C11
NC are overestimated by the MOM modelcompared to the FE results, especially for higher values of vg. The sametrends of results were obtained for the values of C12
NC but are not shownhere for the sake of brevity; some predictions of C12
NC are summarized inTable 2. It is well known fact that the transverse elastic properties ofcomposite are mostly the function of matrix material properties,therefore, the predictions of both the models are in good agreement forthe lower values of vg or higher values of vm. The calculated values ofC22
NC are found to be identical to those of C11NCand are not shown here.
This is attributed to the fact that the constructional feature of GRNCdemonstrates the transversely isotropic behavior with the axis of sym-metry along the 3-direction. Fig. 6 shows the variation of effectiveelastic constant C23
NC of the GRNC against the volume fraction of gra-phene (vg). The predicted values of C23
NC by the MOM approach are
slightly lower than that of predictions by FE model and this indicatesthat the Poisson's effect in GRNC is accurately captured by the former.Due to the exerted load along the axis of symmetry (i.e., 3-direction),the extension-extension coupling occurs between the different normalstress (σ33) and normal strain (ɛ22), and FE simulations captured suchcoupling accurately. The predictions of values of C13
NC are found to besame as those of C23
NC and are not shown here for the sake of brevity.It may be observed from Figs. 5 and 6 that the predictions by both
the models differ as the graphene volume fraction increases. Such dis-crepancy exists because the transverse properties of composite arematrix dependent and hence the discrepancy between the predictionsby models increases as the matrix volume fraction decreases. Thisclearly indicates that the MOM model cannot accurately model isostressconditions (Eq. (5)) applied to the RVE of GRNC. Likewise, other re-searchers also observed the discrepancy in the predictions of transverseproperties of composite (Pettermann and Suresh, 2000; Odegard,2004). Fig. 7 depicts the variation of effective axial elastic constant C33
NC
of the GRNC with the volume fraction of graphene (vg). It can be ob-served that the values of C33
NC vary almost linearly with the values of vg
and both the models predict indistinguishable results. This comparisonalso ensures the validity of rules of mixture as well as the assumptionsadopted to develop MOM model, especially the isostrain condition. Theexisting experimental studies also reported the same for the axialproperties of graphene-based nanocomposite (Zhao et al., 2010; Khanet al., 2012; Ji et al., 2016; García-Macías et al., 2018). Note that theeffective longitudinal or axial elastic constant is usually estimated byusing the isostrain condition, which models actual experiments almostexactly, along the axis of symmetry and therefore, the predictions ofvalues of C33
NC are identical to that of Voigt-upper bound predictions. Itcan be observed from Figs. 5 and 7 that the magnitude of values of C33
NC
is significantly higher than that of the values of C11NC for a given value
of vg. This indicates that the axial stiffness of GRNC lamina is enhancedby aligning the graphene layer in the same direction. The effectiveelastic constant C66
NC is a function of elastic constants C11NC and C12
NC andhence the predictions C66
NC are not shown here.Figs. 8–10 demonstrate the variations of effective piezoelectric
coefficients e31NC and e33
NC as well as the axial dielectric constant 33NC of
GRNC against the volume fraction of graphene (vg). It can be observedfrom Fig. 8 that the value of e31
NC increases with the increase in thegraphene volume fraction. Since the GRNC is transversely isotropicmaterial with the axis of symmetry being aligned along the 3-direction,the values of e32
NC of the GRNC are found to be identical those of e31NC. It
can be seen from Figs. 9 and 10 that both the models predict almostidentical and linear estimates for the values of e33
NCand 33NC for a vast
range of volume fraction of graphene, respectively. Comparison of re-sults obtained by the MOM and FE models reveals that the formermodel yields conservative predictions for the most of the piezoelasticproperties of GRNC. Therefore, we considered the properties of GRNC(with 0.5 graphene volume fraction) predicted by the MOM model insubsequent sections to investigate the static and dynamic response of
Fig. 4. Armchair graphene sheet with trapezoidal pore subjected to axial stress: (a) 4.5% and (b) 20% vacancies.
K.B. Shingare and S.I. Kundalwal Mechanics of Materials 134 (2019) 69–84
76
GRNC nanoplates.To investigate the effect of flexoelectricity on the electromechanical
response of GRNC nanoplates, we considered both pristine and defectedTable1
Mat
eria
lpro
pert
ies
ofgr
aphe
nela
yer
and
poly
imid
em
atri
x.
Mat
eria
lE
(GPa
)μ
e 31
(C/m
2 )e 3
3(C
/m2 )
33(F
/m)
Pris
tine
Gra
phen
e98
5(p
rese
nt)
0.26
5(p
rese
nt)
−0.
221
(Kun
dalw
alet
al.,
2017
)0.
221
(Kun
dalw
alet
al.,
2017
)1.
106
x10
−10
(Muñ
oz-H
erná
ndez
etal
.,20
17)
Gra
phen
ew
ith4.
5%va
canc
y96
9(p
rese
nt)
0.26
5(p
rese
nt)
−0.
027
(Kun
dalw
alet
al.,
2017
)0.
027
(Kun
dalw
alet
al.,
2017
)1.
106
×10
−10
Gra
phen
ew
ith20
%va
canc
y89
0(p
rese
nt)
0.26
5(p
rese
nt)
−0.
12(K
unda
lwal
etal
.,20
17)
0.12
(Kun
dalw
alet
al.,
2017
)1.
106
×10
−10
Poly
imid
e4.
2(O
dega
rdet
al.,
2005
)0.
4(O
dega
rdet
al.,
2005
)–
–3.
009
×10
−11
(Lie
tal
.,20
15)
Fig. 5. Variation of effective elastic constant (C11NC) of the GRNC with the gra-
phene volume fraction (vg).
Fig. 6. Variation of effective elastic constant (C23NC) of the GRNC with the gra-
phene volume fraction (vg).
Fig. 7. Variation of effective elastic constant (C33NC) of the GRNC with the gra-
phene volume fraction (vg).
K.B. Shingare and S.I. Kundalwal Mechanics of Materials 134 (2019) 69–84
77
graphene (with 4.5% and 20% vacancies) sheets. Considering the vo-lume fraction of graphene as 0.5, the effective properties of GRNC weredetermined using the MOM approach, as summarized in Table 2.
The mass density of the GRNC (ρnc) is calculated using the simplerules of mixture: = +( v ) ( v )nc g g m m . We have taken the values of ρgand ρm as 2200 kg/m3 and 1330 kg/m3, respectively (Yolshina et al.,2016; Odegard et al., 2005), and the calculated ρnc is 1765 kg/m3. Themagnitude of the applied uniformly distributed load on the GRNC na-noplate is taken as =q 0.050 MPa. Recent experimental studies reportedthat the flexoelectric coefficients of certain ceramics and polymers aremuch larger than the previous estimates. However, in experimentalmeasurements of certain crystals, elastomers, polymers and ceramics,the predicted flexoelectric coefficient was found to be in the range of10 10 C/m6 10 ; where e is the electron charge and a is the latticeconstant (Kogan, 1963; Ma and Cross, 2003; Nguyen et al., 2013). Forinstance, the experimentally predicted values of flexoelectric coeffi-cients of polymers are found to be in the range from 10−8 to 10−9 C/m(Chu and Salem, 2012; Jiang et al., 2013; Zhang et al., 2015). Hence,unless otherwise mentioned, we considered the flexoelectric coefficient10–9 C/m for our calculations.
5.1. Effect of flexoelectricity on static response of GRNC nanoplate
The variation of normalized bending stiffness (D /D11 110 ) of GRNC
nanoplate with respect to its thickness (h) is plotted in Fig. 11, whereD11 and D11
0 are the bending stiffnesses of GRNC nanoplate with andwithout flexoelectric effect, respectively. Note that the value of D11
depends only on the thickness of nanoplate and is independent of its in-plane dimensions, as seen from Eq. (47). Fig. 11 reveals that thebending stiffness of GRNC nanoplate with flexoelectric effect is ∼10times higher than that of the conventional nanoplate (i.e., withoutflexoelectric effect) when the value of h is 1 nm. This difference is no-ticeable and cannot be ignored for predicting the electromechanicalresponse of thin nanostructures. As the thickness of nanoplate increases,the effect of flexoelectricity starts diminishing and this finding agreewell with the results obtained by Zhou et al. (2016). From this, it can beconcluded that the stiffness of nanoplate depends on the size or shape ofnanostructure (constant or varying cross section). As expected, thenormalized bending stiffness approaches unity when the fldepends oneffect vanishes. It may also be observed from Fig. 11 that the effect offlexoelectricity on the normalized bending stiffness of GRNC nanoplateis size dependent as it can be clearly seen from the Eq. (47).
The effect of flexoelectricity on the static bending of GRNC nano-plate is examined here. Figs. 12 and 13 demonstrate the variation oftransverse deflections of SS GRNC nanoplates with and without flexo-electric effect. The dimensions of the square GRNC nanoplate are takenas: =h 4 nm, and = =a b 50 h. Our selection of in-plane dimensions ofnanoplate is based on the fact that the theory of Kirchhoff plate pro-vides better results when the aspect ratio of a plate is in the range of5–80 (Yang et al., 2015). It can be observed from Figs. 12 and 13 thatthe maximum deflection of the GRNC nanoplate occurs at its center i.e.,at =x a/2 and =y b/2 for both the cases (with and without flexo-electric effect). The maximum deflection of GRNC nanoplate increasesif its in-plane dimensions are increased (a= b= 60 h). It may also beobserved that the deflection of GRNC nanoplate with flexoelectric effectis lower than that of the conventional plate for both the cases of in-plane dimensions. In addition, Figs. 12(a) and (b) represent the 3-Drepresentation of deflection of GRNC nanoplate with and withoutflexoelectric effect. The stiffness of GRNC nanoplate significantly im-proves due to the incorporation of flexoelectric effect over that ofconventional nanoplate. These results clearly demonstrate the im-portance of flexoelectricity in the static bending of GRNC nanoplateswhich cannot be neglected at nanoscale level.
In the previous sets of results, the static response of GRNC nano-plates subjected to the UDL is studied. However, the different types ofloadings may influence the deflection behavior of nanoplates.Therefore, three different cases were considered for GRNC nanoplateunder: (i) hydrostatic load (UVL), (ii) point load, and (iii) inline load.
Fig. 8. Variation of effective piezoelectric constant (e31NC) of the GRNC with the
graphene volume fraction (vg).
Fig. 9. Variation of effective piezoelectric constant (e33NC) of the GRNC with the
graphene volume fraction (vg).
Fig. 10. Variation of effective dielectric constant ( 33NC) of the GRNC with the
graphene volume fraction (vg).
K.B. Shingare and S.I. Kundalwal Mechanics of Materials 134 (2019) 69–84
78
These cases represent the practical situation of different types of load-ings applied to the thin plates. We considered the equivalent magnitudeof loading in all situations. Table 3 illustrates the effect of three types ofloading conditions on the deflection of GRNC nanoplates. As expected,the maximum deflection of the nanoplate occurs at its center irrespec-tive of the type of loading in both the cases (with and without flex-oelectricity). It is evident from this table that the consideration offlexoelectric effect results in the lowering the deflection of GRNC na-noplates compared to that of conventional plates. For example, thereduction in the static deflection of GRNC nanoplate having 4 nmthickness is found to be ∼38.0% in all the loading cases. As expected,the maximum deflection of nanoplate occurs in case of the application
of point load on it; the magnitude of maximum deflection of GRNCnanoplate observed in the following order: Point load > In-lineload > UDL > VDL.
So far, the deflection characteristics of SS GRNC nanoplate werestudied by considering its thickness as 4 nm. To explore the effect ofthickness of GRNC nanoplate on its static behavior, once again the fourdiscrete types of loading conditions are considered: UDL, VDL, inlineload, and point load. Table 4 summarizes the values of maximum de-flections of GRNC nanoplates. The reductions in static deflections ofGRNC nanoplates, irrespective of the type of loading, are found to be∼71.0%, ∼37.0%, ∼21.0%, ∼13%, and ∼9.0% corresponding to2 nm, 4 nm, 6 nm, 8 nm, and 10 nm thicknesses of nanoplate. It can be
Table 2Effective properties of GRNC (vg = 0.5).
Material C11(GPa) C12(GPa) C66(GPa) e31(C/m2) e33(C/m2) 33(F/m)
Fig. 11. Effect of variation of plate thickness on the normalized bending stiff-ness (D /D )11 11
0 of GRNC nanoplate under UDL.
Fig. 12. Effect of variation of plate aspect ratios (x/a and y/b) on the deflection of GRNC nanoplate: (a) without flexoelectricity and (b) with flexoelectricity underUDL = = =h a b( 4 nm; 50 h).
Fig. 13. Effect of variation of plate aspect ratio (x/a) on the deflection of GRNCnanoplate under UDL = = =h a b( 4 nm; 50 h and 60 h).
K.B. Shingare and S.I. Kundalwal Mechanics of Materials 134 (2019) 69–84
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clearly seen that the influence of flexoelectricity on the maximum de-flection of a nanoplate diminishes as its thickness increases and tends toapproach the results of maximum deflection of the conventional GRNCnanoplate indicating that the flexoelectric effect is size dependent. It
can be concluded from the above discussion that the effect of flex-oelectricity is more prominent for thin plates and this finding agree wellwith the results obtained by other researchers (Yan and Jiang, 2012;Zhang and Jiang, 2014). In the literature, nanofabrication techniques
Table 3Effect of flexoelectricity on the central deflection of GRNC nanoplate under different loadings = = =h a b( 4 nm; 50 h).
K.B. Shingare and S.I. Kundalwal Mechanics of Materials 134 (2019) 69–84
80
such as layer-by-layer assembly and dispersion method, which mainlydeal with interaction between cation and anions of adjacent graphenelayers, are commonly used to fabricate graphene-based thin films(Gamboa et al., 2010). Using these techniques, the fabrication of thingraphene-based nanocomposite film can be achieved and its thicknesson the order of nm can be tailored by varying the number of graphenelayers (Yang et al., 2013; Prolongo et al., 2014 and 2018; Tzeng et al.,2015; Prolongo et al., 2018). Some challenges are associated with thesetechniques such as in every step of LBL technique the layered graphenestructure is required to be rinsed with deionized water followed by thedrying for a specific time that may lead to error and non-uniform de-position. Mature fabrication techniques have been evolving as the ap-plications of 2D graphene sheets became more defined and thin struc-tures made of their layers are being fabricated at relatively low-cost.
Fig. 14 shows the effect of variation of plate aspect ratio (a/h) onthe maximum deflection of GRNC nanoplates for different flexoelectriccoefficients. We kept thickness of GRNC nanoplate constant to study theeffect of its aspect ratio. It can be observed from Fig. 14(c) that theflexoelectric effect is more prominent when the values of aspect ratio ofplate and flexoelectric are 40 and 10–9 C/m, respectively. When thevalue of flexoelectric coefficient is 10–10 C/m then both the cases pro-vide almost same results (Fig. 14d). On the other hand, the flexoelectriceffect on the deflection behavior of nanoplates is negligible when thevalues of flexoelectric coefficients are 10–7 C/m and 10–8 C/m (Fig. 14aand b). It can also be observed that the flexoelectricity plays an im-portant role when the in-plane dimensions of the plates are on the orderof nm. However, flexoelectricity does not much influence the staticbehavior of nanoplate when its aspect ratio is less than 30 demon-strating the strong size-dependent behavior. We considered the value offlexoelectric coefficient as 10–9 C/m to study the effect of flexoelec-tricity on the dynamic response of GRNC nanoplates.
5.2. Effect of flexoelectricity on dynamic response of GRNC nanoplate
In this section, the investigations are carried out to study the effectof flexoelectricity on the dynamic response of GRNC nanoplates.Figs. 15 illustrates the effect of flexoelectricity on the resonant fre-quency of mode (1,1) of GRNC nanoplates against the plate aspect ratio.We kept the in-plane dimensions of plates constant (a= b= 100 and150 nm) and varied their thickness. It can be observed that the resonantfrequency is higher for the flexoelectric nanoplate over that of theconventional plate when the plate thickness is less than 3 nm. Theflexoelectricity does not much influence the resonant frequencies ofnanoplates having larger thickness (>4 nm) and this is due to the factthat the effect of size dependent flexoelectricity diminishes as thethickness of nanoplate increases. This figure also reveals that the re-sonant frequency is largely depends on the in-plane dimensions of na-noplate; resonant frequency of the nanoplate diminishes as its in-planedimensions increase.
So far, the effect of flexoelectricity on the resonant frequencies of
GRNC nanoplate was studied by varying its thickness from 1 to 15(Fig. 15). Here, the parametric results of resonant frequencies of GRNCnanoplates are presented to investigate the effect of flexoelectricityconsidering the plate thicknesses as 1 nm and 2 nm. Fig. 16 demon-strates the effect of flexoelectricity on the resonant frequency of GRNCnanoplate with mode (1,1). It can be observed that the resonant fre-quency decreases as the aspect ratio of plate increases. The effect offlexoelectricity is noteworthy in case of thin plate. For instance, re-sonant frequencies of GRNC nanoplate with flexoelectricity are en-hanced by ∼225% for the plate aspect ratios of 10 to 30 when the platethickness is 1 nm. On the contrary, when the aspect ratio is sufficientlylarge, the difference between the resonant frequencies is very small,therefore the flexoelectric effect can be neglected. The results shown inFigs. 15 and 16 are significant which indicate that the flexoelectricityplays an important role in the dynamics of thin plates and needs to beaccounted properly. It is observed from Figs. 11–16 that as the thick-ness of nanostructure increases the flexoelectric effect starts dimin-ishing, and this finding agree well with the results obtained by otherresearchers (Su et al., 2018; Shi and Wang, 2019; Yang, 2019).
6. Conclusions
Static and dynamic behaviors of graphene reinforced nanocompo-site (GRNC) plates with flexoelectric effect were investigated for thefirst time in the literature. First, we determined the elastic properties ofpristine and defective graphene sheets using MD simulations and theobtained results are found in good agreement with the existing ex-perimental and numerical results. Second, we developed the mechanicsof materials (MOM) and finite element (FE) models to predict the ef-fective piezoelastic and dielectric properties of GRNC. The MOM modelwas derived using the isofield conditions and rules of mixture. Goodagreements between the MOM and FE results were obtained as long asthe graphene volume fraction is small. Finally, we derived the exactsolutions for flexoelectric GRNC nanoplate based on Kirchhoff's platetheory, Navier's solution and extended linear theory of piezoelectricity.Based on this, the static and dynamic behaviors of simply supportedGRNC nanoplates under different types of loadings were investigated tostudy the role of flexoelectricity. It is found that the bending stiffness ofnanoplates having thickness less than 5 nm increases significantly dueto the incorporation of flexoelectric effect and such effect cannot beneglected for predicting static response of thin structures. Similarly, thedynamics response of GRNC nanoplates is enhanced due to the flexo-electric effect as the plate thickness reduces. Resonant frequencies ofGRNC nanoplates are enhanced by ∼225% for the plate aspect ratios of10–30 when the plate thickness is 1 nm. Our results indicate that theflexoelectricity plays an important role in the static and dynamics be-haviors of thin plates and needs to be accounted properly while mod-eling 2D nanostructures.
Table 4Effect of flexoelectricity on the maximum deflection of GRNC nanoplate subjected to different types of loading conditions.
Thickness h (nm) Maximum deflection
UDL (nm) VDL (nm) Point load (nm) In-line load (nm)
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Acknowledgments
This work was generously supported by the CSIR Senior ResearchFellowship awarded to the first author from the Council of Scientific &Industrial Research (CSIR) of India (141200/2K18/1).
Supplementary materials
Supplementary material associated with this article can be found, inthe online version, at doi:10.1016/j.mechmat.2019.04.006.
Fig. 14. Effect of variation of plate aspect ratio (a/h) on the maximum deflection of GRNC nanoplate, with fixed in-plane dimensions, considering differentflexoelectric coefficients: (a) 10–7 C/m (b) 10–8 C/m (c) 10–9 C/m and (d) 10–10 C/m.
Fig. 15. Effect of variation of plate thickness =h a( /x) on the resonant fre-quency of GRNC nanoplate.
Fig. 16. Effect of variation of plate aspect ratio =a( hx) on the resonant fre-quency of GRNC nanoplate =h( 1 nm and 2 nm).
K.B. Shingare and S.I. Kundalwal Mechanics of Materials 134 (2019) 69–84
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