nonlocal vibration of a piezoelectric polymeric nanoplate carrying nanoparticle via mindlin plate...

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907 originally published online 5 June 2013 2014 228:Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science

A Haghshenas and A Ghorbanpour AraniNonlocal vibration of a piezoelectric polymeric nanoplate carrying nanoparticle via Mindlin plate theory

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Original Article

Nonlocal vibration of a piezoelectricpolymeric nanoplate carryingnanoparticle via Mindlin plate theory

A Haghshenas1 and A Ghorbanpour Arani1,2

Abstract

This paper is concerned with the vibration characteristics of an embedded nanoplate-based nanoelectromechanical

sensor made of polyvinylidene fluoride (PVDF) carrying a nanoparticle with different masses at any position. The

nanoplate is surrounded by elastic medium which is simulated as Pasternak foundation. The PVDF nanoplate is subjected

to an applied voltage in the thickness direction. In order to satisfy the Maxwell equation, electric potential distribution is

assumed as a combination of a half-cosine and linear variation. Adopting the nonlocal Mindlin plate theory, the governing

equations are derived based on the energy method and Hamilton’s principle which are then solved by Galerkin method

to obtain the natural frequency of the nanoplate. A detailed parametric study is conducted to elucidate the influences of

the nonlocal parameter, external electric voltage, position and mass of nanoparticle, temperature changes and dimension

of nanoplate and elastic medium. Results indicate that the frequency is increased as the nanoparticle comes closer to the

center of the nanoplate; also increasing mass of the nanoparticle decreases the frequency of the system. This study might

be useful for the design of PVDF nanoplate-based resonator as nanoelectromechanical sensor.

Keywords

Nonlocal vibration, nanoparticle, piezoelectric plate, Mindlin theory

Date received: 29 January 2013; accepted: 8 May 2013

Introduction

It is commonly believed in the scientific communitythat nanotechnology will spark a series of industrialrevolutions in the following decades. Nanotechnologyis a field of applied science concerned with the controlof matter at dimensions of roughly 1–100 nm. At theparticle size of 1–100 nm, nano-scale materials mayhave different molecular organizations and propertiesthan the same chemical substances in a larger size. Inrecent years, nanostructural carbon materials havereceived considerable interest of scientific commu-nities due to their superior properties. Amongcarbon based nanomaterials, single-layered graphenesheets (SLGSs) have attracted many researchers dueto their strong mechanical strength (Young’s modu-lus¼ 1.0 TPa), large thermal conductivity (thermalconductivity¼ 3000W/km), excellent electric con-ductivity (electric conductivity up to 6000 S/cm),high surface area and unusual optical properties.1–3

These superior properties have made SLGSspromising candidates in many applications such asnanosensors, nanoelectronics, nanocomposites, bat-teries, nanooscillators, nanoactuators, nanoresona-tors, nano-optomechanical systems, supercapicitores,fuel cells, solar cells and hydrogen storage.4,5

The application of the SLGS like mass sensor wasinvestigated by Sakhaee-Pour et al.6 Recently, thevibration characteristics of the graphene sheets(GSs) have attracted attention of many researchersdue to their superior vibrational behaviors. Forinstance, Ansari et al.7 studied vibrational behaviorsof SLGS based on the first order shear deformationtheory (FSDT) and the differential equations weresolved by generalized differential quadrature methodfor various boundary conditions. Vibration analysisof embedded orthotropic GSs was investigated byPradhan and Kumar.8

On the other hand, in modern technology, the useof piezoelectric materials as smart structure is of con-siderable interest. The piezoelectric materials producean electric field when subjected to deformationand experience deformation when subjected to anelectric field. The coupling nature of piezoelectric

1Faculty of Mechanical Engineering, University of Kashan, Kashan, Iran2Institute of Nanoscience & Nanotechnology, University of Kashan,

Kashan, Iran

Corresponding author:

A Haghshenas, Faculty of Mechanical Engineering, University of Kashan,

Kashan, Iran.

Email: [email protected]

Proc IMechE Part C:

J Mechanical Engineering Science

2014, Vol. 228(5) 907–920

! IMechE 2013

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materials has wide applications in electro-mechanicaland electric devices, such as electro-mechanical actu-ators, sensors and transducers.9 Based on FSDT,Sheng and Wang10 approximated solution for func-tionally graded (FG) laminated piezoelectric cylin-drical shell under thermal shock and movingmechanical loads. In another attempt, they11 pre-sented the coupling equations to govern the electricpotential and the displacement of the FG cylindricalshell with surface-bonded PZT piezoelectric layerbased on FSDT, Hamilton’s principle and Maxwellequation. Polyvinylidene fluoride (PVDF) is a piezo-electric polymer. It has superior properties, such asexcellent dimensional stability, abrasion and corro-sion resistance, high strength and capability of main-taining its mechanical properties at elevatedtemperature.12 In contrast to GS, PVDF is a smartmaterial, since it has piezoelectric property.

In order to investigate the mechanical behaviors ofnanostructures, the higher order continuum theorieswhich are scale dependent theories such as nonlocalelasticity, nonlocal piezoelasticity, modified couplestress theory and strain gradient theory have beenrecently employed.13–22

It is because the classical theory is a scale independ-ent theory and cannot handle the small scale effects.Among these, nonlocal elasticity theory which wasfirst introduced by Eringen23 has been widely utilizedto study the mechanical manner in the micron andnano-scale structures. In fact, this theory expressesthe stress at a defined point as a function of thestrain at all the points of the continuum. There aremany works in the literature that have used thistheory.19–22 Pradhan and Murmu24 studied the smallscale effect on the buckling of embedded SLGS basedon nonlocal plate theory. They found that the buck-ling loads of SLGS are strongly dependent on thesmall scale coefficients. Explicit analytical expressionsfor critical buckling stresses in a SLGS based on thenonlocal elasticity were investigated by Ansari andRouhi.25 They concluded that with the appropriateselection of a nonlocal parameter, the nonlocal rela-tions are capable of yielding excellent results from thestatic deflection of SLGS under a uniformly distribu-ted load. Also, their results showed that the import-ance of the small length scale is dependent on theboundary conditions of SLGS. Nonlocal plate

model for nonlinear vibration of SLGSs in thermalenvironment was presented by Shen et al.26 Theirresults indicated that with properly selected smallscale parameters and material properties, the nonlocalplate model can provide a remarkably accurate pre-diction of the GS behavior under nonlinear vibrationin thermal environments. Murmu et al.27 presentedthe effect of in-plane magnetic field on the transversevibration of a magnetical SLGS using equivalent con-tinuum nonlocal elastic plate. Zhang et al.28 studiedthe effect of environmental temperature on mechan-ical properties of multi-walled carbon nanotubes(CNTs) by means of a molecular structural mechanicsmodel.

Furthermore, dynamic response of CNT andSLGS based sensor is an important problem due totheir prominent use as sensors. Axial vibration behav-ior of single-walled carbon nanotube-based masssensor was presented by Aydogdu and Filiz29 usingnonlocal elasticity theory. In another attempt, Shenet al.30 studied the vibration of CNT-based biosensorusing Timoshenko beam model. They concluded thatthe rotary inertia decreases the fundamental fre-quency, while an increase in the diameter of theattached bio-object reduces the natural frequency,but causes frequency shift to rise. It is worth mention-ing that very few studies studied the GS-based masssensor base on nonlocal elasticity theory. Shen et al.31

investigated the vibration of SLGS-based nanoelec-tromechanical sensor via nonlocal Kirchoff platetheory. They showed that when the mass of theattached nanoparticle increases or its location iscloser to the plate center, the natural frequencydecreases, but frequency shift increases.

However, to the best of author’s knowledge noreport has been found in the literature on the vibra-tion analysis of PVDF nanoplate-based nanoelectro-mechanical sensor. Motivated by this idea, in order toimprove optimum design of nanostructures, thispaper aims to study the vibration of an embeddedPVDF nanoplate-based nanoelectromechanicalsensor based on nonlocal Mindlin plate theory.Herein the nanoplate is polarized in thickness direc-tion and subjected to external electric voltage. Ananoparticle with different masses can be located atany position on the PVDF nanoplate. Moreover, theGalerkin method is applied to obtain the natural

Figure 1. Schematic of an embedded PVDF nanoplate with an attached nanoparticle. PVDF: polyvinylidene fluoride.

908 Proc IMechE Part C: J Mechanical Engineering Science 228(5)



frequency. The influences of external applied electricvoltage, mass and location of the nanoparticle, smallscale parameter and elastic medium are also discussedin detail.

Basic equations

Consider a PVDF nanoplate as depicted in Figure 1 inwhich geometrical parameters of length a, width b andthickness h are indicated. The PVDF nanoplate isrested on an elastic medium which is simulated byPasternak foundation. As is well known, this founda-tion model is both capable of transverse shear loads(Kg) and normal loads (Kw). The PVDF nanoplate issubjected to external electric voltage ’ in thicknessdirection.

Nonlocal piezoelasticity theory

Based on the theory of nonlocal piezoelasticity, thestress tensor and the electric displacement at a refer-ence point depend not only on the strain componentsand electric-field components at same position butalso on all other points of the body. The nonlocalconstitutive behavior for the piezoelectric materialcan be given as follows32,33:

�nlij ðxÞ ¼

Zv

�ð x� x0�� , �Þ�lij dVðx0Þ, 8x 2 V ð1Þ

Dnlk ¼

Zv

�ð x� x0�� , �ÞDl

k dVðx0Þ, 8x 2 V ð2Þ

where �nlij and �lij are, respectively, the nonlocal stresstensor and local stress tensor, Dnl

k and Dlk are, respect-

ively, the components of the nonlocal and local elec-tric displacement. �ð x� x0j j, �Þ is the nonlocalmodulus. x� x0j j is the Euclidean distance, and� ¼ e0a=l is defined that l is the external characteristiclength, e0 denotes constant appropriate to each mater-ial and a is an internal characteristic length of thematerial. Consequently, e0a is a constant parameterwhich is obtained with molecular dynamics, experi-mental results, experimental studies and molecularstructure mechanics. The constitutive equation ofthe nonlocal elasticity can be written as32,33:

ð1� �r2Þ�nlij ¼ �lij ð3Þ

where the parameter � ¼ ðe0aÞ2 denotes the small

scale effect on the response of structures in nanosizeand r2 is the Laplacian operator in the above equa-tion. Similarly, equation (2) can be written as32,33:

ð1� �r2ÞDnlk ¼ Dl

k: ð4Þ

Piezoelectric nanoplate carrying a nanoparticle

In a piezoelectric material, application of an electricfield will cause a strain proportional to the mechanicalfield strength and vice versa. The constitutive equa-tion for stresses � and strains " matrix on the mech-anical side, as well as flux density D and field strengthE matrix on the electrostatic side may be arbitrarilycombined as follows33:

�xx

�yy

�yz

�zx

�xy

8>>>>>>>><>>>>>>>>:

9>>>>>>>>=>>>>>>>>;¼

C11 C12 0 0 0

C21 C22 0 0 0

0 0 C44 0 0

0 0 0 C55 0

0 0 0 0 C66

2666666664

3777777775

"xx

"yy

�yz

�xz

�xy

8>>>>>>>><>>>>>>>>:

9>>>>>>>>=>>>>>>>>;

�

0 0 e31

0 0 e32

0 e24 0

e15 0 0

0 0 0

2666666664

3777777775

Exx

Eyy

Ezz

8>><>>:

9>>=>>; ð5Þ

Dxx

Dyy

Dzz

8>><>>:

9>>=>>; ¼

0 0 0 e15 0

0 0 e24 0 0

e31 e32 0 0 0

2664

3775

"xx

"yy

�yz

�xz

�xy

8>>>>>>>><>>>>>>>>:

9>>>>>>>>=>>>>>>>>;

þ

211 0 0

0 222 0

0 0 233

2664

3775

Exx

Eyy

Ezz

8>><>>:

9>>=>>; ð6Þ

where Cij, eij and 2ij denote elastic, piezoelectric anddielectric coefficients, respectively.

Also, electric field Ei ði ¼ x, y, zÞ in term of electricpotential (’) is given as follow33:

E ¼ �r�: ð7Þ

The electric potential distribution in the thick-ness direction of the PVDF nanoplate in theform proposed by Refs. 32–34 as the combinationof a half-cosine and linear variation whichsatisfies the Maxwell equation is adopted asfollows:

�ðx, y, z, tÞ ¼ �cos�z

h

� �’ðx, y, tÞ þ

2zV0

hei�t ð8Þ

where ’ðx, y, tÞ is the time and spatial distribution ofthe electric potential caused by bending which mustsatisfy the electric boundary conditions, V0 is external

Haghshenas and Arani 909



electric voltage and � is the natural frequency of thePVDF nanoplate which is zero for buckling analysis.

Based on Mindlin plate theory, the displacementfield can be expressed as35:

uxðx, y, z, tÞ ¼ z xðx, y, tÞ

uyðx, y, z, tÞ ¼ z yðx, y, tÞ

uzðx, y, z, tÞ ¼ wðx, y, tÞ

ð9Þ

where xðx, yÞ and yðx, yÞ are the rotations of thenormal to the mid-plane about x- and y-directions,respectively.

The von Karman strains associated with the abovedisplacement field can be expressed in the followingform:

"xx ¼ z@ x

@x, "yy ¼ z

@ y

@y, �yz ¼

@w

@yþ y, �xz ¼

@w

@x

þ x, �xy ¼ z@ x

@yþ@ y

@x

� �ð10Þ

where ð"xx, "yyÞ are the normal strain components andð�yz, �xz, �xyÞ are the shear strain components.

The variation statements

The governing differential equations of motion arederived using the Hamilton’s principle which isgiven as:

ZT0

ð�Uþ �V� �KÞdt ¼ 0 ð11Þ

where �U is the virtual strain energy, �V is the virtualwork done by external applied forces and �K is thevirtual kinetic energy. The strain energy of the PVDFnanoplate can be expressed as

U ¼1

2

Z�0

Z h=2

�h=2

�xx"xx þ �yy"yy þ �xy�xy þ �xz�xz�

þ �yz�yz �DxxExx �DyyEyy �DzzEzz

�dv

ð12Þ

Combination of equations (5) to (10) yields

U¼1

2

Z�0

Mxx@ x

@xþMyy

@ y

@x

�

þMxy@ x

@yþ@ y

@x

� �þQxz

@w0

@xþ x

� �

þQyz@w0

@yþ y

� ��dxdy

�1

2

Z�0

Z h=2

�h=2

Dxx cos�z

h

� ��@’

@x

� �

þDyy cos�z

h

� ��@’

@x

� ��Dzz

�

hsin

�z

h

� �’þ

2V0

h

� �dzdxdy

ð13Þ

where the stress resultant–displacement relations canbe written as

ðNxx,Nyy,NxyÞ, ðMxx,Myy,MxyÞ ¼

Z h=2

�h=2

�xx, �yy, �xy

ð1, zÞdz ð14Þ

Qxx,Qyy

¼ Ks

Z h=2

�h=2

�xz, �yz

dz ð15Þ

in which Ks is shear correction coefficient.The kinetic energy of nanoplate is given by:

K ¼

2

Zv

@w

@t

� �2" #

dv: ð16Þ

The external work due to surrounding elasticmedium can be written as36,37:

W ¼

Zf wdA ð17Þ

where f is related to Pasternak foundation. Finally,using Hamilton’s principles (equation (11)), integrat-ing by parts and setting the coefficient of mechanicaland electrical to zero lead to the following motionequations:

@Mxx

@xþ@Mxy

@y�Qxx ¼ �I2

@2 x

@t2ð18Þ

@Mxy

@xþ@Myy

@y�Qyy ¼ �I2

@2 y

@t2ð19Þ

@Qxx

@xþ@Qyy

@yþ 1� �r2� �

kww� kgr2w

�þ ðNxm þNxe þNxTÞ

@2w

@x2þNyT

@2w

@y2� q x, tð Þ

�

¼ �I0@2w

@t2

ð20Þ

Z h=2

�h=2

cos�z

h

� �@Dxx

@xþcos

�z

h

� �@Dyy

@xþ�

hsin

�z

h

� �Dzz

�dz¼0

ð21Þ

where I2 ¼112 h

3 and I0 ¼ h are mass moments ofinertia; kw and kg are Winkler and shear coefficientsof Pasternak medium, respectively. Also, mechanicalforce is zero (i.e.Nxm ¼ 0) and electrical force isNxe ¼ 2e31V0. Also,

q x, tð Þ ¼ �M�ðx� x0Þ�ð y� y0Þ@2w

@t2ð22Þ




where x0 and y0 are the location of the nanoparticle inx and y directions, respectively; M is the mass of thenanoparticle and �ðxÞ is the impulse function, which isdenoted as

�ðx� x0Þ ¼1 x ¼ x00 x 6¼ x0

�ð23Þ

Finally, the motion equations can be written as

1

12�24he31

@’

@xþ �c11h

2 @2 x

@x2þ �c12h

2 @2 y

@x@y

� �

þ1

12�c66h

3 @2 x

@y2þ@2 y

@x@y

� �

�Ksh

��2e‘15

@’

@xþ c55� x þ c55�

@w

@x

� �

� 1� e0að Þ2r2

� �I2@2 x

@t2,

ð24Þ

h3

12c66

@2 x

@x@yþ@2 y

@x2

� �þ

h

12�24e‘32

@’

@x

�

þ �c12h2 @

2 x

@x@yþ �c11h

2 @2 y

@y2

�

�Ksh

��2e‘24

@’

@yþ c44� y þ c44�

@w

@y

� �

� 1� e0að Þ2r2

� �I2@2 y

@t2,

ð25Þ

Ksh

��2e15

@2’

@x2þ c55�

@ x

@xþ c55�

@2w

@x2� 2e24

@2’

@y2

�

þ c44�@ y

@yþ c44�

@2w

@y2

�þ 1� e0að Þ

2r2

� �

� �c11h�xT� c12h�yTþ 2e31V0

� � @2w@x2

�M�ðx� x0Þ�ð y� y0Þ@2w

@t2

þ �c12h�xT� c1h�yT0

� � @2w@y2þ Kww

� Kg@2w

@x2þ@2w

@y2

� �� I0

@2w

@t2

�,

ð26Þ

1

2�h4e‘15h

2 @ x

@xþ 4e‘31h

2 @ x

@xþ 4e‘15h

2 @2w

@x2

�

þ 4e‘24h2 @ y

@yþ 4e‘32h

2 @ y

@yþ 2‘11 �h

2 @2’

@x2

� 2‘33 �3’þ 4e‘24h

2 @2w

@y2þ 2‘22 �h

2 @2’

@y2:

ð27Þ

Galerkin method

In order to study the free vibration of the nonlocalPVDF nanoplate, Galerkin method is used to solvethe motion equations. For simply supported bound-ary condition and zero electric potential along theedges of the surface electrodes, the assumed shapefunction can be written as31:

d ðx, yÞ ¼XNx

m¼1

XNy

n¼1

d sinm�x

a

� �sin

n�y

b

� �ei!t ð28Þ

where m and n are the wave numbers in x and y dir-ections, respectively; Nx and Ny are the number of inthe x and y directions, respectively; also d is the con-stant amplitude of the function which is

d ¼ ½w, x, y, ’�: ð29Þ

Table 1. Material properties

of PVDF.

PVDF

C11 ¼ 238:24 GPað Þ

C22 ¼ 23:60 GPað Þ

C12 ¼ 3:98 GPað Þ

C44 ¼ 2:15 GPað Þ

C55 ¼ 4:40 GPað Þ

C66 ¼ 6:43 GPað Þ

e31 ¼ �0:13ðC=m2Þ

e32 ¼ �0:145ðC=m2Þ

e24 ¼ �0:276ðC=m2Þ

e15 ¼ �0:009ðC=m2Þ

211 ¼ 1:1068� 10�8 F=mð Þ

PVDF: polyvinylidene fluoride.

Table 2. A comparison between the present work and

vibration analysis of SLGS without nanoparticle.

ðe0aÞ2 ðnmÞ2

�nl=�l

Present

work

�nl=�l

Ref. 18

�nl=�l

Ref. 30

�nl=�l

Ref. 4

0 1 1 1 1

1 0.9089 0.9139 0.9139 0.9139

2 0.8367 0.8467 0.8467 0.8468

3 0.7889 0.7925 0.7925 0.7926

SLGS: single-layered graphene sheet.




For a non-trivial solution of the system, the coeffi-cients of d must equal zero. Substituting equation (28)into equations (24) to (27) multiplying both sides ofthe resulting equation with integrating it over thedomain (0,a) and (0,b) yields the fourth order matrixequation as follows:

L11 L12 L13 L14

L21 L22 L23 L24

L31 L32 L33 L34

L41 L42 L43 L44

2664

3775

w x

y

’

2664

3775 ¼

0000

2664

3775 ð30Þ

where Lijði, j ¼ 1, 2, 3, 4Þ is defined in Appendix A.The roots of the resulting equation are the desirednatural frequencies corresponding to the assumedshape function.

Results and discussion

In the following section, the influences of nonlocalparameter, dimensions of nanoplate, elastic medium,mass and position of the attached nanoparticle, wavenumber in y direction, temperature change and

0 0.5 1 1.5 2 2.5

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Nonlocal Parameter (nm)

Fre

quen

cy (

GH

z)

a/b=1.0

a/b=1.5

a/b=2.0

a/b=2.5

a/b=3.0

Figure 2. Effect of length-to-width ratio (a=b) on the frequency with respect to nonlocal parameter.

0 0.5 1 1.5 2 2.50.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4


Fre

quen

cy (

GH

z)

h/a=0.010h/a=0.015h/a=0.020h/a=0.025h/a=0.030

Figure 3. Effect of thickness-to-length ratio (h=a) on the frequency with respect to nonlocal parameter.




external electric voltage on the frequency of thePVDF nanoplate are studied and discussed in detail.The mechanical and electrical characteristics of aPVDF nanoplate are shown in Table 1.12

To the best of the authors’ knowledge, no pub-lished literature is available for comparison of thetransverse vibration response of the PVDF nanoplatecarrying a nanoparticle. However, the present results

0 0.5 1 1.5 2 2.50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4


Fre

quen

cy (

GH

z)

Without elastic mediumWith Winkler mediumWith Pasternak medium

Figure 4. Effect of surrounding elastic medium on the frequency with respect to nonlocal parameter.

0 0.5 1 1.5 2 2.50.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45


Fre

quen

cy (

GH

z)

M=10-18

M=10-19

M=10-20

M=10-21

M=10-22

Figure 5. Frequency vs. nonlocal parameter for different mass of attached nanoparticle values.




can be validated by the other published literatures inthe transverse vibration of the nanoplates. In thisregard, the simplified result of this paper is comparedwith the work of Shen et al.,38 Pradhan andPhadikar39 and Pradhan and Kumar40 neglecting elec-trical terms (i.e. piezoelectric coefficient in equation (5)

and electric displacement (i.e. equation (6))) and thenanoparticle. The results are shown in Table 2 whichindicates nonlocal to local frequency for differentnonlocal parameters. As the observed results are thesame as those reported in Refs. 4,18,30 indicating val-idation of the present work.

0 0.5 1 1.5 2 2.50.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9


Fre

quen

cy (

GH

z)

n=1n=2n=3n=4n=5

Figure 6. Frequency vs. nonlocal parameter for different values of wave numbers in y direction.

0 0.5 1 1.5 2 2.5

0.2

0.25

0.3

0.35

0.4

0.45

0.5


Fre

quen

cy (

GH

z)

V0= -1 volt

V0= -0.5 volt

V0= 0 volt

V0= 0.5 volt

V0= 1 volt

Figure 7. Effect of external voltage on the frequency with respect to nonlocal parameter.




Figure 2 depicts the frequency versus nonlocal par-ameter for different values of length to width a=b ofthe PVDF nanoplate. It is seen that the frequencydecreases with increasing the nonlocal parameter.This is because increasing the nonlocal parameterimplies decreasing interaction force between nanorods

atoms which leads to a softer structure. Also with theincrease of a=b, the frequency is decreased.Furthermore, the effect of the a=b ratio becomesimportant at lower nonlocal parameter values.Indeed, the effect of a=b decreases with increasingthe nonlocal parameter. It is worth mentioning that

0 0.5 1 1.5 2 2.50.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4


Fre

quen

cy (

GH

z)

T=-100T=-50T= 00T= 50T= 100

Figure 9. Effect of temperature change on the frequency with respect to nonlocal parameter.

0 0.5 1 1.5 2 2.50

0.05

0.1

0.15

0.2

0.25

0.3

0.35


Fre

quen

cy (

GH

z)

ζ=0.25,η=0.25

ζ=0.50,η=0.25

ζ=0.50,η=0.50

Figure 8. Effect of the position of the attached nanoparticle on the frequency vs. nonlocal parameter.




small scale effects on the frequency become more dis-tinguished at lower a=b values.

Figure 3 illustrates the effects of h=a which isdefined as the thickness to length on the frequencyof the PVDF nanoplate with respect to nonlocal

parameter. The results indicate that the frequencyincreases at a decreasing rate as a=b is increased.

The influences of the elastic medium on the fre-quency of system are shown in Figure 4. As can beseen considering elastic medium caused to increase of

15 20 25 30 35 400.335

0.34

0.345

0.35

0.355

0.36

0.365

Length (nm)

Fre

quen

cy r

atio

T=-100T=-50T= 00T= 50T= 100

Figure 10. Frequency ratio vs. PVDF nanoplate length for different values of temperature change.


15 20 25 30 35 400.335

0.34

0.345

0.35

0.355

0.36

0.365

0.37

0.375

0.38

Length (nm)

Fre

quen

cy r

atio

(G

Hz)

V0= -1 volt

V0= -0.5 volt

V0= 0 volt

V0= 0.5 volt

V0= 1 volt

Figure 11. Frequency ratio vs. PVDF nanoplate length for different values of external applied voltages.





nanoplate frequency. This is due to the fact that con-sidering elastic medium leads to a stiffer structure.The effect of the Pasternak type is higher than theWinkler type on the phase velocity. It is perhaps

due to the fact that the Winkler-type is capable todescribe just normal load of the elastic mediumwhile the Pasternak-type describes both transverseshear and normal loads of the elastic medium.

15 20 25 30 35 400.335

0.34

0.345

0.35

0.355

0.36

0.365

Length (nm)

Fre

quen

cy r

atio

ζ=0.50,η=0.50

ζ=0.50,η=0.25

ζ=0.25,η=0.25

Figure 13. Effect of the attached nanoparticle position on the frequency ratio with respect to the PVDF nanoplate length.


15 20 25 30 35 400.33

0.335

0.34

0.345

0.35

0.355

0.36

0.365

Length (nm)

Fre

quen

cy r

atio

M=10-22

M=10-21

M=10-20

M=10-19

M=10-18

Figure 12. Frequency ratio vs. PVDF nanoplate length for different values of the attached nanoparticle mass values.





Figure 5 demonstrates the graph of frequencyversus the nonlocal parameter for different nanopar-ticle mass values. It can be found that the frequency ofthe system increases as the nanoparticle mass isdecreased. Also, the effects of the nanoparticle massvalue become prominent at lower nonlocal parametervalues. Meanwhile, at the higher values of nonlocalparameter, the frequency does not vary much bychanging the mass nanoparticle values. In the otherword, one can say that the effect of the nanoparticlemass becomes considerable at lower values of nonlo-cal parameter.

The wave number in y direction (n) effects on thefrequency of the system with respect to nonlocal par-ameters is depicted in Figure 6. It is found that thefrequency increases significantly as the values of n areincreased. Additionally, the frequency does not varysignificantly at lower n values.

The variations of frequency with respect to thenonlocal parameter for various external electric volt-ages are illustrated in Figure 7. It is shown that apply-ing negative electric voltage can increase thefrequency of the PVDF nanoplate and vice versa.This is because the imposed negative electric voltageincreases the stiffness of the system, while positiveimposed electric voltages decrease the stability and,therefore, the frequency of the PVDF nanoplate.Moreover, the effect of the external voltages becomesmore prominent at higher values of nonlocal param-eter. Hence, the imposed external applied voltage is aneffective controlling parameter for vibration ofa PVDF nanoplate-based nanoelectromechanicalsensor.

Figure 8 shows the effects of the nanoparticle pos-ition , �ð Þ on the frequency of the system. It should benoted that ¼ x0=a, � ¼ y0=b. The results demon-strate that the frequency increases as the nanoparticlecomes close to the center of PVDF nanoplate.Therefore, position of the nanoparticle can be aneffective parameter for controlling the vibrationalbehavior of the system.

In realizing the influence of temperature change T,Figure 9 shows how frequency changes with respect tononlocal parameter. It is found that the frequency ofthe system decreases with increasing temperaturechange. It is also due to the fact that increasing tem-perature change decreases the stiffness of the system.

Figures 10 to 13 are plotted to show the effects ofdifferent parameters such as temperature change,external applied voltage, nanoparticle position, massof the nanoparticle and length of the nanoplate on thefrequency ratio. For this purpose, frequency ratio isdefined as follows:

Frequency ratio ¼Nonlocal frequency !nlð Þ

Local frequency !lð Þ

It is seen that increasing the nanoplate lengthincreases the frequency ratio of PVDF nanoplate.

Indeed, the differences between the local and nonlocalfrequencies become more visible at higher nanoplatelength values.

It can be easily concluded that changing some par-ameter values can alter the frequency ratio. FromFigures 10 and 11, one can conclude that increasingthe temperature change and external electric voltagesenhances the frequency ratio. As can be seen fromFigure 10, imposing negative temperature changedecreases the frequency ratio and vice versa. It isseen from Figure 11 that the difference betweenlocal and nonlocal frequencies becomes more prom-inent at higher external applied voltages. Also,Figure 12 shows that the frequency ratio is decreasedat an increasing rate by increasing the nanoparticlemass values. Furthermore, as can be seen fromFigure 13, the frequency ratio decreases as the nano-particle comes closer to the center of nanoplate.Indeed, when the nanoparticle comes closer to thecenter of the nanoplate, the differences between localand nonlocal frequencies decrease.

Conclusion

Vibration of PVDF nanoplates-based nanoelectrome-chanical sensor has applications in designing manynanoelectromechanical system devices such as strainsensor, mass and pressure sensors and atomic dustdetectors. Electro-thermal vibration of a piezoelectricpolymeric nanoplate with a nanoparticle subjected toexternal voltage is the main contribution of the pre-sent paper. The elastic medium is simulated by aPasternak foundation. The governing equations areobtained based on the nonlocal Mindlin plate theoryso that the effects of small scale, elastic medium, wavenumber, temperature change, dimensions of the nano-plate, position and mass of the nanoparticle andexternal applied voltage are discussed. The followingconclusions may be made from the results:

1. The frequency increases as the nanoparticle iscloser to the center of nanoplate.

2. Increasing external voltage decreases the fre-quency of the system, while the frequency isdecreased with increasing nanoparticle massvalues.

3. The elastic medium increases the frequency.4. Frequency does not vary significantly for lower

values of the wave number in y direction.5. The stiffness and, therefore, the frequency of the

PVDF nanoplate decrease by increasing the tem-perature change.

6. Frequency decreases as the nonlocal parameter isincreased.

7. The differences between local and nonlocal fre-quencies increase by increasing the length of nano-plate, external applied voltage, temperaturechange and nanoparticle mass.




Funding

The authors thank the referees for their valuable comments.

They also thank the Iranian nanotechnology DevelopmentCommittee for their financial support and University ofKashan.

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Appendix A

L11 ¼ �KshC55m

2�2b

4a�KshC44m

2�2a

4b

þab!2

4M sin2ðm��Þsin2ðn��Þ

�2e31V0 � C11h�xT� C12h�yT

4am2�2b

��C12h�xT� C11h�yT

4bn2�2aþ

kwab

4

þ kgm2�2b

4aþn2�2a

4b

� �þI0ab!

2

4

� e0að Þ2 2e31V0 � C11h�xT� C12h�yT

4a3m4�4b

�

�m2�2b!2

4Ma sin2ðm��Þsin2ðn��Þ

þ�C12h�xT� C11h�yT

4abm4�4n2

�kwm

2�2b

4a� kg

m4�4b

4a3þn2�2m4

4ab

� �

�I0m

2�2b!2

4aþ2e31V0 � C11h�xT� C12h�yT

4abm2n2�4

�n2�2a!2

4Mb sin2ðm��Þ sin2ðn��Þ

þ�C12h�xT� C11h�yT

4a3n4�4a�

kwn2�2a

4b

�kgm2n2�4

4abþa�4n4

4b3

� ��I0n

2�2a!2

4b

�ðA1Þ

L14 ¼Kshe15m

2�b

2aþKshe24n

2�a

2b

L22 ¼ �h3C11m

2�2b

48a�h3C66n

2�2a

48b

�KshC55ab

4þI2ab!

2

4

þ e0að Þ2 I2m

2�2b!2

4aþI2n

2�2a!2

4b

��,

L32 ¼ �h3C66m

2�2b

48a�h3C11n

2�2a

48b�KshC44ab

4

þI2ab!

2

4þ e0að Þ

2 I2m2�2b!2

4aþI2n

2�2a!2

4b

��,

L41 ¼ �h 211 m

2�2b

8a�h 233 �

2ab

8h�h 222 n

2�2a

8b,

L44 ¼ �he15m

2�b

2a�he24n

2�b

2b:

It should be noted that the other components ofthe matrix in equation (30) are zero.




nonlocal vibration of a piezoelectric polymeric nanoplate carrying nanoparticle via mindlin plate...

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