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European Journal of Mechanics A/Solids 46 (2014) 22e29

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European Journal of Mechanics A/Solids

journal homepage: www.elsevier .com/locate/ejmsol

Effects of surface piezoelectricity and nonlocal scale on wavepropagation in piezoelectric nanoplates

L.L. Zhang a, J.X. Liu b,*, X.Q. Fang b, G.Q. Nie b

a Institute of Engineering Mechanics, Beijing Jiaotong University, Beijing 100044, PR ChinabDepartment of Engineering Mechanics, Shijiazhuang Tiedao University, Shijiazhuang 050043, PR China

a r t i c l e i n f o

Article history:Received 11 September 2013Accepted 27 January 2014Available online 7 February 2014

Keywords:Piezoelectric nanoplatesSurface piezoelectricityNonlocal dispersion

* Corresponding author. Tel.: þ86 311 87936066.E-mail address: liujx02@hotmail.com (J.X. Liu).

http://dx.doi.org/10.1016/j.euromechsol.2014.01.0050997-7538/� 2014 Elsevier Masson SAS. All rights re

a b s t r a c t

In this paper, the dispersion characteristics of elastic waves propagating in a monolayer piezoelectricnanoplate is investigated with consideration of the surface piezoelectricity as well as the nonlocal small-scale effect. Nonlocal electroelasticity theory is used to derive the general governing equations byintroducing an intrinsic length, and the surface effects exerting on the boundary conditions of thepiezoelectric nanoplate are taken into account through incorporation of the surface piezoelectricitymodel and the generalized YoungeLaplace equations. The dispersion relations of elastic waves based onthe current formulation are obtained in an explicit closed form. Numerical results show that both thenonlocal scale parameter and surface piezoelectricity have significant influence on the size-dependentproperties of dispersion behaviors. It is also found that there exists an escape frequency above whichthe waves may not propagate in the piezoelectric plate with nanoscale thickness.

� 2014 Elsevier Masson SAS. All rights reserved.

1. Introduction

A nanostructure is defined as a material system or object whereat least one of the dimensions lies below 100 nm. At this scale, thephysical and mechanical properties of nanostructures exhibitobvious size effect, which are quite different from their bulkcounterparts. The existing experimental observations (Poncharalet al., 1999; Stan et al., 2007; Cuenot et al., 2004; C.Q. Chen et al.,2006) and simulation results (Shenoy, 2005; Dai and Park, 2013;Diao et al., 2006; Agrawal and Espinosa, 2011) show that thesmall size effect and high surface effect are the predominant size-dependent features when the characteristic size of structuresscales down to the nanoscale, and these two effects become moreand more pronounced as the size decreases. Note that each indi-vidual effect mentioned above does not suffice to fully describe theoverall size-dependence, hence, it is necessary to consider thecooperative effects of surface and small size for further under-standing the actual properties of nanostructures. As well all know,the controlled experiments on nanomaterials are extremely diffi-cult and the atomistic simulations are time consuming andrestricted by computation capacities. For this reason, severalmodified continuum models, consisting mainly of high-order

served.

continuum theories and surface elasticity, have been naturallyproposed to be alternative and cost-effective methods for calcula-tions. From literature survey (Arash et al., 2012; Liu and Yang, 2012;Haftbaradaran and Shodja, 2009; Kaviani and Mirdamadi, 2013;Duan et al., 2005; G.F. Wang et al., 2006), it is found that thehigh-order continuum theories, such as nonlocal theory (Eringen,2002), couple stress theory (Yang et al., 2002), strain gradienttheory (Aifantis, 1999) etc., and the surface elasticity (Gurtin andMurdoch, 1975) are commonly applied to characterize the smallsize effect and surface effect in the nanostructures, respectively.However, very few efforts on the mechanical responses of nano-structures are conducted by the combination of these two kinds ofapproaches. Narendar and Gopalakrishnan (2012) and Narendaret al. (2012) superposed the surface effect and the nonlocalelastic theory to study the dispersive behaviors of flexural wave innanosized tubes and plates. In the context of strain gradient theorywith interface effect, Zhang and Sharma (2005) analyzed the strainand stress distribution of quantum dots structure.

Since Wang’s group (Pan et al., 2001) successfully synthesizedthe nanobelts of semiconducting oxides, various piezoelectricnanomaterials (e.g. ZnO, ZnS, PZT, GaN, BaTiO3, etc.) have receivedtremendous attention from the research community (Fang et al.,2013). The enhanced piezoelectric effect, excellent resilience andunique semiconducting properties make them strong candidatesfor applications as sensors (Lao et al., 2007), resonators (Tanneret al., 2007), generators (Wang and Song, 2006) and transistor

L.L. Zhang et al. / European Journal of Mechanics A/Solids 46 (2014) 22e29 23

(G.F. Wang et al., 2006, X.D. Wang et al., 2006) in the nano-electromechanical systems (NEMS). To better realize the appliedmechanisms of these advanced devices and improve their per-formances, the prediction of their electromechanical coupling ef-fect and electroelastic responses to external loads becomes acritical issue. In the framework of classical continuum mechanics,several modified piezoelectric models with incorporation of sizeeffect have been attempted to interpret the size-dependent phe-nomena of piezoelectric nanostructures. Based on the nonlocalpiezoelasticity and Timoshenko beam theory, the linear andnonlinear thermo-electric-mechanical vibrations of the piezo-electric nanobeams subjected to an applied voltage and a uniformtemperature change were investigated (Ke and Wang, 2012; Keet al., 2012). Shen and Hu (2010) employed the electric enthalpyvariational principle to establish a comprehensive theory con-cerning with the strain-gradient-induced flexoelectricity, the sur-face effects and the electrostatic force. Ghorbanpour Arani et al.(2013) examined the electro-thermo-torsional buckling responseof a double-walled boron nitride nanotube (DWBNNT) by usingthe nonlocal shear deformable shell theory. As an extension of thesurface elasticity model, a conceptual idea of surface piezoelec-tricity was first proposed by Huang and Yu (2006). Subsequently,this model has been further developed and extensively used instatic and dynamic analysis of piezoelectric nanostructures. Forexample, the dynamic strength around a piezoelectric nano-particle (Fang et al., 2012a), the dynamic effective properties ofpiezoelectric medium with randomly distributed piezoelectricnano-fibers (Fang et al., 2012b), the bending of piezoelectricnanowires (Yan and Jiang, 2011a), as well as the vibration andbuckling of piezoelectric nanobeams (Yan and Jiang, 2011b). Inaddition, Wang and Wang (2012) studied the surface effects on theelectromechanical coupling behaviors of a piezoelectric nanowirevia nonlocal electroelasticity theory. Recently, C.L. Zhang et al.(2014) derived the two-dimensional basic equations of piezo-electric shells with nano-thickness by considering the surfacepiezoelectricity.

Beside the beam- and shell-like piezoelectric structuresmentioned above, nanoscale piezoelectric plates are also the keybuilding blocks among most novel nanodevices. Therefore, un-derstanding the electroelastic responses of piezoelectric nano-plates is essential for their design and applications. Byconsideration of the surface piezoelectricity, the dispersioncurves of anti-plane and plane waves propagating in a single-layered piezoelectric nanoplate were provided (Zhang et al.,2012; L.L. Zhang et al., 2014). Under the Kirchhoff hypothesis,Yan and Jiang (2012) and Liu et al. (2013) investigated the surfaceeffects and nonlocal piezoelasticity on the vibration behavior of asimply supported piezoelectric nanoplate, respectively. Thesetheoretical researches have demonstrated the significance ofconsidering both the surface effects and nonlocal small-scale instudying the mechanical and physical properties of piezoelectricnanoplates. For their applications in NEMS, the dispersive prop-erties of waves propagating in piezoelectric nanoplates need tobe accurately predicted. However, to date, the relevant literaturesare very limited except the works of Zhang et al. (2012) and L.L.Zhang et al. (2014), where only the influence of surface effectswas discussed. Motivated by this consideration, the objective ofthe present work is to investigate the cooperative effects ofsurface piezoelectricity and nonlocal small-scale on the propa-gation characteristics of elastic waves in an infinite piezoelectricnanoplate. An intrinsic nonlocal length is introduced to theconstitutive relations of piezoelectric materials, and the surfaceeffects are accounted into the boundary conditions of thepiezoelectric nanoplate via the surface piezoelectricity modeland the generalized YoungeLaplace equations. Based on the

existence of nonzero solution in homogeneous linearity equa-tions, explicit expressions of the dispersion relations will beobtained to show how the influence of surface piezoelectricityand nonlocal parameter upon the dispersion curves of elasticwaves changes with the plate thickness.

2. Nonlocal piezoelectric model

Different from the classical (local) theory of piezoelectricity(Fang and Liu, 2013), the nonlocal piezoelectric model implies thatthe stress and electric displacement at a reference point x dependnot only on the strain and electric-field components at that positionbut also on those at all other points x0 of the body. Mathematically,the nonlocal constitutive equations for the piezoelectric materialcan be given as follows (Zhou et al., 2006; Liang, 2008):

tijðxÞ ¼ZV

aðjx0 � xj; sÞsijðx0ÞdVðx0Þ; cx˛V; (1)

SiðxÞ ¼ZV

aðjx0 � xj; sÞDiðx0ÞdVðx0Þ; cx˛V; (2)

sijðx0Þ ¼ cijkl 3klðx0Þ � ekijEkðx0Þ; Diðx0Þ ¼ eikl 3klðx0Þ þ kikEkðx0Þ;(3)

3klðx0Þ ¼ 12

vukðx0Þvx0l

þ vulðx0Þvx0k

!; Ekðx0Þ ¼ �v4ðx0Þ

vx0k; (4)

where tij and Si are, respectively, the nonlocal stress and nonlocalelectric displacement. The classical stress sij and classical electricdisplacement Di are related to the strain 3kl and electric field Ek,with cijkl, ekij and kik being the elastic, piezoelectric, and dielectricconstants. The terms uk and 4 are the mechanical displacementand electric potential, and the volume integral is over the region Voccupied by the body. The nonlocal modulus a is a function of theEuclidean distance jx0 � xj and the material constant s ¼ e0a/l.Here, e0 is a non-dimensional material property for calibrating themodel, which can be determined by experimental results orcomparison with calculations based on lattice dynamics; a is aninternal characteristics length (e.g., lattice parameter, granulardistance), and l is an external characteristics length (e.g., cracklength, wave length).

The interesting properties of the nonlocal modulus a have beendiscussed in detail by Eringen (2002), and themost notable of theseis that a is Green’s function of a linear differential operator L, i.e.,

Laðjx0 � xj; sÞ ¼ dðjx0 � xjÞ; (5)

where dð$Þ is the Dirac delta function.Applying L into Eqs. (1) and (2), and utilizing Eq. (5), the

equivalent differential form of the nonlocal constitutive relationscan be obtained as

Ltij ¼ sij; LSi ¼ Di: (6)

In the nonlocal piezoelectric model, it is well known that all filedequations involving stress and electric displacement must be basedon the nonlocal components tij and Si. However, if L is the differ-ential operator with constant coefficient, the equilibrium equationswithout body forces and free charges can be expressed in terms ofthe local stress and electric displacement as

L.L. Zhang et al. / European Journal of Mechanics A/Solids 46 (2014) 22e2924

sij;j ¼ rL€ui; Di;i ¼ 0; (7)

where r is the mass density of the bulk layer, the comma followedby the subscript i indicates differentiation with respect to the cor-responding space coordinate xi, and the dot denotes differentiationwith respect to time t.

Fig. 1. Schematic of an infinite piezoelectric plate with nano-thickness.

3. Formulation of the problem

As shown in Fig. 1, we divide a piezoelectric nanoplate withthickness 2h into three laminated layers, including two surfacelayers and a bulk layer. Refer to a rectangular Cartesian coordinatesystem (x1,x2,x3), where x3 along the transverse direction and thex1ex2 plane sitting on the mid-plane of the undeformed plate. Thesix-fold axes of the transversely isotropic piezoelectric body par-allel to the x3-direction, and Gþ and G� representing the upper andlower surfaces are defined by x3 ¼ �h, respectively.

Consider a plane strain problem (x2ex3 plane) in which all fieldquantities are assumed to be constant in the x1-direction. For thetwo-dimensional case, the differential operator L ¼ 1� ðe0aÞ2V2

can be employed to represent the state of the body faithfully.Consequently, the equilibrium Eq. (7) are reduced to

s22;2 þ s23;3 ¼ rh1� ðe0aÞ2V2

i€u2;

s23;2 þ s33;3 ¼ rh1� ðe0aÞ2V2

i€u3;

D2;2 þ D3;3 ¼ 0;

(8)

where V2 ¼ v2=vx22 þ v2=vx23 is the Laplacian operator. To be noted,the nonlocal parameter e0a denotes the small-scale effect on theresponse of nanostructures, and the nonlocal components in Eq. (6)approaches to the corresponding local components in the limitwhen the internal characteristic length is considerably less than theexternal characteristics length, that is, e0a / 0.

The interior bulk material obeys the same constitutive relationsas those for the conventional piezoelectric materials. Therefore, theexisting components of stress and electric displacement arewrittenaccording to Eqs. (3) and (4) as

s22 ¼ c11u2;2 þ c13u3;3 þ e314;3;s33 ¼ c13u2;2 þ c33u3;3 þ e334;3;

s23 ¼ c44�u2;3 þ u3;2

�þ e154;2;

D2 ¼ e15�u2;3 þ u3;2

�� k114;2;D3 ¼ e31u2;2 þ e33u3;3 � k334;3:

(9)

Substitution of Eq. (9) into Eq. (8), yields

c11u2;22 þ c44u2;33 þ ðc13 þ c44Þu3;23 þ ðe31 þ e15Þ4;23 ¼ rh1� ðe0aÞ2V2

i€u2;

ðc44 þ c13Þu2;23 þ c44u3;22 þ c33u3;33 þ e154;22 þ e334;33 ¼ rh1� ðe0aÞ2V2

i€u3;

ðe15 þ e31Þu2;23 þ e15u3;22 þ e33u3;33 � k114;22 � k334;33 ¼ 0:

(10)

Assume the solutions of Eq. (10) correspond to the free wavepropagating in the x2 direction,which canbe characterized in the form

u2 ¼ A1 expðikbx3Þexp½ikðx2 � ctÞ�;u3 ¼ A2 expðikbx3Þexp½ikðx2 � ctÞ�;4 ¼ A3 expðikbx3Þexp½ikðx2 � ctÞ�;

(11)

where A1, A2, and A3 are constants, b is an unknownparameter to bedetermined, k is the angular wave number, c is the phase velocity,u ¼ kc stands for the circular frequency of the propagating waves,and i ¼

ffiffiffiffiffiffiffi�1

pis the imaginary unit.

Substituting Eq. (11) into Eq. (10), one can obtain a system oflinear and homogeneous equations for A1, A2, and A3 as

½M�fAg ¼ 0; (12)

where [M] is a symmetric matrix whose elements are given by

M11 ¼ c11 þ c44b2 � rc2h1þ ðe0aÞ2k2

�1þ b2

�i;

M12 ¼ M21 ¼ ðc13 þ c44Þb; M13 ¼ M31 ¼ ðe31 þ e15Þb;M22 ¼ c44 þ c33b2 � rc2

h1þ ðe0aÞ2k2

�1þ b2

�i;

M23 ¼ M32 ¼ e15 þ e33b2; M33 ¼ k11 þ k33b2:

(13)

According to Eq. (12), the amplitude ratios can be determined byusing Cramer’s rule, as

Z ¼ A1A3

¼ M12M23�M13M22M11M22�M12M21

;

ƛ ¼ A2A3

¼ M13M21�M11M23M11M22�M12M21

:(14)

To obtain a nontrivial solution it is required that the determi-nant of the coefficient matrix [M] must be equal to zero. Thiscondition results in a sixth-order polynomial eigenvalue problemin b. For every value of k and c there exists six b, each of which yieldsan independent solution of the partial differential Eq. (10). Sinceone solution is insufficient to represent all wave componentspropagating in the piezoelectric bulk, six allowed solutions bytaking a linear superposition are desired, i.e.,

u2 ¼ P6n¼1

ZnA3n expðikbnx3Þexp½ikðx2 � ctÞ�;

u3 ¼ P6n¼1

ƛnA3n expðikbnx3Þexp½ikðx2 � ctÞ�;

4 ¼ P6n¼1

A3n expðikbnx3Þexp½ikðx2 � ctÞ�:

(15)

L.L. Zhang et al. / European Journal of Mechanics A/Solids 46 (2014) 22e29 25

According to the theory of surface piezoelectricity, the surfacelayer possesses its own material properties and constitutive re-lations, which are different from those of the bulk counterpart. Thesurface stress ss

bgand the surface electric displacement Ds

bcan be

provided as (Yan and Jiang, 2012; Zhang et al., 2012)

ss11 ¼ s011 þ cs11u1;1 þ cs12u2;2 þ es314;3;

ss22 ¼ s022 þ cs12u1;1 þ cs11 3s22 þ es314;3;

ss12 ¼ s012 þ cs66�u1;2 þ u2;1

�;

Ds1 ¼ D0

1 � ks114;1; Ds2 ¼ D0

2 � ks114;2;

(16)

where s0bg

is the residual surface stress, D0bis the residual surface

electric displacement, cs11, cs12 and cs66 are the surface elastic con-

stants, es31 and ks11 are the surface piezoelectric and dielectricconstants, and the subscripts b and g range from 1 to 2. Moreover,the presence of surface effects also gives rise to the nonclassicalboundary conditions on the bulk part, which can be described

K1n ¼ ��ik�cs11Zn þ es31bn

�� c44ðZnbn þ ƛnÞ � e15hn � ikrsc2Zn

�expðikbnhÞ;

K2n ¼ �� ðc13Zn þ c33ƛnbn þ e33bnÞ=hn � ikrsc2ƛnexpðikbnhÞ;

K3n ¼ ��ikks11 þ e31Zn þ e33ƛnbn � k33bn

�hnexpðikbnhÞ;

K4n ¼ ��ik�cs11Zn þ es31bn

�þ c44ðZnbn þ ƛnÞ þ e15hn � ikrsc2Zn

�expð�ikbnhÞ;

K5n ¼ �ðc13Zn þ c33ƛnbn þ e33bnÞ=hn � ikrsc2ƛnexpð�ikbnhÞ;

K6n ¼ ��ikks11 � e31Zn � e33ƛnbn þ k33bn

�hnexpð�ikbnhÞ:

(25)

accurately by the generalized YoungeLaplace equations (T.Y. Chenet al., 2006; L.L. Zhang et al., 2014), i.e.,

ts11;1 þ ts12;2Ht13 ¼ rs€u1; at x3 ¼ �h; (17)

ts21;1 þ ts22;2Ht23 ¼ rs€u2; at x3 ¼ �h; (18)

s011u3;11 þ s022u3;22Ht33 ¼ rs€u3; at x3 ¼ �h; (19)

Ss1;1 þ Ss2;2HS3 ¼ 0; at x3 ¼ �h; (20)

where rs is the mass density of the surface layer, and tsbg

and Ssb

denote the nonlocal surface stress and surface electric displace-ment, respectively. Note that Eqs. 17e20 can reduce to the con-ventional traction-free and electrically open-circuited cases byletting all the surface-related parameters vanish.

In order to derive the relations between the nonlocal and clas-sical components from Eqs. (1) and (2), the two-dimensionalnonlocal modulus is chosen as

aðjx0 � xjÞ ¼ 1

2pðe0aÞ2K0

0@

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�x02 � x2

�2 þ �x03 � x3�2q

e0a

1A (21)

with K0ð$Þ being the modified Bessel function of the second kind.Substituting Eq. (21) into Eqs. (1) and (2) and integrating over

the domain V, one can obtain

�tijSi

�¼ 1

hn

�sijDi

�; (22)

"tsbgSsb

#¼"s0bg

D0b

#þ 1hn

"ssbg � s0

bg

Dsb � D0

b

#; (23)

where hn ¼ 1þ ðe0aÞ2k2ð1þ b2nÞ.In what follows, we set the residual surface stress and surface

electric displacement to be zero since they have aminor effect uponwave propagation in contrast to the surface piezoelectricity (L.L.Zhang et al., 2014). As the current interest is of plane-strain na-ture, the mechanical boundary condition along the x1-direction,namely Eq. (17) should be excluded. Upon substitution of Eqs. (9),(15)e(16) and (22)e(23), Eqs. 18e20 become a linear system ofalgebraic equations for unknown constants A3n(n ¼ 1 � 6), i.e.,

½K�fA3ng ¼ 0; (24)

where [K] is a 6 � 6 coefficient matrix with

By imposing the existence of nontrivial solutions of Eq. (24), therelation between the phase velocity and frequency can be deter-mined by

jKj ¼ 0: (26)

It should be mentioned that Eq. (26) is so-called dispersionequation, and the numerical methods are generally used to find theroots of phase velocity for a given frequency.

4. Results and discussion

In the previous studies (Huang and Yu, 2006; Ke and Wang,2012; Ke et al., 2012; Fang et al., 2012a, 2012b; Yan and Jiang,2011a, 2011b, 2012; Wang and Wang, 2012; Zhang et al., 2012;Liu et al., 2013), PZT with both semiconducting and piezoelectricproperties has proven to be a versatile choice for nanoscalepiezoelectric structures. Here, PZT-5H is selected as an examplematerial with c11 ¼ 126 GPa, c13 ¼ 83.9 GPa, c33 ¼ 117 GPa,c44 ¼ 23 GPa, e15 ¼ 17 C/m2, e31 ¼ �6.5 C/m2, e33 ¼ 23.3 C/m2,k11 ¼1.505�10�8 C/Vm, k33 ¼ 1.302� 10�8 C/Vm, and r¼ 7500 kg/m3 for the bulk part. To date, the surface properties that can bedetermined from related experiments and atomistic simulationsare not completely available for PZT-5H owing to the lack of suchwork. In the current work, the estimated values cs11 ¼ 7:56N=m,es31 ¼ �3� 10�8C=m, ks11 ¼ 1:505� 10�17C=V, andrs ¼ 7.5 � 10�6 kg/m2 are taken as reasonable approximations ofsurface material parameters. For the sake of brevity, the non-dimensional circular frequency U ¼ uh/csh, phase velocity c/cshand wave number kh are introduced in the following numericalanalysis.

It is evident that the dispersion Eq. (26) in the local theory canbe recovered by setting the nonlocal parameter e0a ¼ 0. Tographically show the effects of surface and nonlocal small-scale onthe propagation characteristics of elastic wave in a monolayerpiezoelectric nanoplate, the phase velocity dispersion curves of the

L.L. Zhang et al. / European Journal of Mechanics A/Solids 46 (2014) 22e2926

first three modes with different nonlocal parameters are plotted inFig. 2, in which csh ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðc44 þ e215=k11Þ=r

qis the velocity of shear

bulk wave propagating in the piezoelectric medium. Whether thesurface piezoelectricity or nonlocal scale is considered, one cannote the dispersion curves deviate obviously from the case of themacroscopic piezoelectric plate. The reason causing this

Fig. 2. Phase velocity dispersion curves of the first three modes with different nonlocalparameters. (a) the first mode (b) the second mode and (c) the third mode.

phenomenon is the magnitude of phase velocity decreasesconsiderably for a given frequency. By inspection of Fig. 2, it is clearthat the stronger nonlocal effect indicated by the larger value of e0ainduces the smaller phase velocity. It is also found that the couplinginfluence of surface piezoelectricity and nonlocal small-scale ismore pronounced in comparisonwith each individual effect, whichmeans the necessity of combining the surface piezoelectricitymodel and nonlocal theory in the dispersion analysis of thepiezoelectric nanoplates.

As is well known, the first and second modes in the trans-versely isotropic piezoelectric plate resemble that of the lowest-order antisymmetric (A0) and symmetric (S0) modes, and theyexist under arbitrary low frequency. With increasing the fre-quency, the phase velocity of the A0 mode rises from zero whilethe S0 mode reduces from a constant. As U / N, the phase ve-locities of the classical A0 and S0 modes converge toward to theRayleigh surface wave velocity of the conventional piezoelectricplate. Different from the classical situation, the phase velocity ofthe A0 mode predicted by the surface piezoelectricity or nonlocalmodel reaches its maximum at a certain frequency U0, and thendescends continually in analogy to the S0 mode with increase ofthe frequency. A close inspection of the Fig. 2(a) reveals that thelarger the nonlocal parameter, the smaller the value of U0. Anotherimportant finding is that the dispersion curves show slight dif-ference between the local and nonlocal models for small values ofU, and the difference becomes more pronounced at higher fre-quencies. Similar observation for the effect of surface piezoelec-tricity on the A0 mode has also been obtained from Fig. 2(a).However, owing to the existence of the surface effects, the phasevelocity of the S0 mode drops sharply in the whole frequencyregion.

The third mode propagates only above individual cut-off fre-quency Uc, near where the magnitude of phase velocity descendsrapidly from infinity. From Fig. 2(c), it is seen that the separateinfluence of surface piezoelectricity and nonlocal small-scale canmarkedly diminish the value of Uc. In contrast to the classical case,the phase velocity of the third mode based on the surface ornonlocal piezoelectricity model is no longer asymptotic to the bulkshear wave velocity for U / N. Instead, it declines constantly asincreasing the frequency.

Different from the macroscpoic situation, the electroelasticproperties and responses of piezoelectric nanoplate are closelyrelated to its size due to the presence of surface effects or nonlocalsmall-scale, which has been demonstrated in many studies (Yanand Jiang, 2012; Zhang et al., 2012; L.L. Zhang et al., 2014; Liuet al., 2013). Therefore, the impact of thickness change on thedispersion characteristics of piezoelectric nanoplates is presentedin the following numerical analysis, and the size-dependent phe-nomena are also been observed.

The cut-off frequency of the third mode versus nonlocalparameter with different plate thicknesses is illustrated in Fig. 3.Obviously, the value of Uc decreases with the increase of thenonlocal parameter. In the conventional piezoelectric plate, thedispersive behavior is independent of the size. However, as theFig. 3 suggests, the curves predicted by surface piezoelectricitytheory or nonlocal model exhibit distinct size-dependent charac-teristics once the plate thickness reduces to nanometers. That is tosay, the thinner nanoplate corresponds to the smaller cut-off fre-quency. With the value of e0a increasing, surface piezoelectricityhas a weaker effect on the dispersive response. This phenomenonimplies that the nonlocal small-scale play a more prominent rolethan surface piezoelectricity when the nonlocal parameter is larger.In addition, one can also observe that the smaller the plate thick-ness is, the more susceptible the cut-off frequency is to changes innonlocal parameter.

Fig. 4. Wave number dispersion curves of the first two modes with different nonlocalparameters. (a) the first mode and (b) the second mode.

Fig. 5. Escape frequency of the first two modes versus nonlocal parameter withdifferent plate thicknesses.

Fig. 3. Cut-off frequency of the third mode versus nonlocal parameter with differentplate thicknesses.

L.L. Zhang et al. / European Journal of Mechanics A/Solids 46 (2014) 22e29 27

The wave number dispersion with circular frequency fordifferent nonlocal parameters is depicted in Fig. 4. It can be seenthat the local (e0a ¼ 0) wave number follows a nonlinear variationat lower frequencies, and it varies linearly at higher frequencies.However, the calculation obtained from nonlocal theory showshigh nonlinearity in the entire frequency bandwidth. Here, thelinear and nonlinear variations indicate the waves are non-dispersive and dispersive in nature, respectively. Since the size ofthe nanostructure is very small, the elastic waves can not propagatein the region of higher frequency. This property can be accuratelycaptured by introducing the nonlocality. As shown in Fig. 4, thewave number goes to infinity at a certain frequency Ue for the caseof e0as 0, and the larger the nonlocal parameter is, the smaller thevalue of Ue becomes. This frequency is called as the escape fre-quency, above which the corresponding phase velocity shrinks tozero and the waves stop propagating. Therefore, the higher escapefrequency corresponds to the wider frequency bandwidth in thenanoscaled structures. For a given nonlocal parameter, it is alsofound that the first twomodes vanish at the same escape frequency.

The escape frequency of the first two modes as a function of thenonlocal parameter is presented with different plate thicknesses inFig. 5. As the nonlocal parameter increases, one can see the escapefrequency decreases from infinity to a very small value. Accord-ingly, it is deduced that a larger nonlocal parameter leads to anarrower frequency bandwidth. With the increase in plate thick-ness, the values of Ue also increase and this trend becomes moresignificant when the nonlocal parameter is smaller. The surfacepiezoelectricity has little impact on the variation of escape fre-quency. For a given nanomaterial, it can be reasonably concludedthat the frequency bandwidth only depends on its size andnonlocality.

Taking the second mode for example, the variation of phasevelocity with the natural logarithms of plate thickness is shown inFig. 6 for different frequencies. Note that the surface layer andnonlocal models are only suitable for accurately simulating thephysical properties of small sized structures. If the characteristicslength of structures is large enough, both effects of surface andnonlocal scale will vanish. As expected, with the increase in theplate thickness, the predicted velocity in the current formulationtends to approach a constant as obtained by the local plate modelwithout consideration of the surface effects. It is also mentionedthat there exists a critical thickness hc ¼ e4.7 (about 110 nm), below

Fig. 6. Phase velocity of the second mode versus plate thickness with differentfrequencies.

L.L. Zhang et al. / European Journal of Mechanics A/Solids 46 (2014) 22e2928

which the influences of surface piezoelectricity and nonlocal scalemust be taken into account.

5. Conclusions

Combining the surface piezoelectricity model and nonlocalelectroelasticity theory, the propagation characteristic of elasticwaves in an infinite piezoelectric plate with nanometer thicknesshas been studied theoretically and quantitatively in this work. Theimpacts of nonlocal scale parameter and surface piezoelectricity onthe dispersion curves are analyzed in detail as the plate thicknessvaries. Different from the results obtained from the classical con-tinuum theories, the dispersion behaviors predicted by the currentformulation exhibit significant size-dependent properties.Regardless of the consideration of surface piezoelectricity ornonlocal small-scale, numerical examples show that the phasevelocities of dispersion modes decline considerably. Nonlocal effectresults in an escape frequency above which the elastic waves maystop propagating. With the nanoplate thickness decreasing ornonlocal parameter increasing, the phase velocity, the cut-off fre-quency as well as the escape frequency decrease obviously. Thisstudy with more accurate modeling methodology is envisaged toprovide the useful guidance for the design and applications ofpiezoelectric nanoplate based devices in NEMS.

Acknowledgments

This work was supported by the National Natural ScienceFoundations of China (Grant Nos. 11272222, 11172185, 11272221).The first author also would like to acknowledge the support pro-vided by the Fundamental Research Funds for the Central Univer-sities (No. C12JB00170).

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