Applied Mathematical Modelling 37 (2013) 3575–3588

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Applied Mathematical Modelling

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Size dependent forced vibration of nanoplates with considerationof surface effects

Abbas Assadi ⇑Department of Mechanical Engineering, Amirkabir University Technology (AUT), Hafez Avenue, Tehran 34311-15916, Iran

a r t i c l e i n f o

Article history:Received 18 July 2011Received in revised form 2 June 2012Accepted 5 July 2012Available online 1 September 2012

Keywords:Rectangular nanoplatesSurface effectsSize dependent natural frequenciesSize dependent forced vibrationGeneral external loading

0307-904X/$ - see front matter � 2012 Elsevier Inchttp://dx.doi.org/10.1016/j.apm.2012.07.049

⇑ Tel.: +98 912 823 9049/919 557 3209; fax: +98E-mail addresses: assadi@aut.ac.ir, assadimech@

a b s t r a c t

In this article, an analytical method is presented to study the size dependent forced vibra-tion of rectangular nanoplates under general external loading using a generalized form ofKirchhoff plate model. The effects of surface properties including surface elasticity, surfaceresidual stresses and surface mass density are considered which are bases for size depen-dent behaviors due to increase in surface to volume ratios at smaller scales. At first, a com-plete discussion is given for size dependent natural frequencies which are then used inforced vibration analyses. It is shown that the surface properties compact the frequencyspectrums of the nanoplates. Saving generality and using the superposition principle,closed form solution is derived for time response of nanoplates under general harmonicloads. As a result, some elliptic curves are obtained for which the surface properties willnot change the time response of nanoplates when a point load is applied on any point ofthese curves. It is observed that, for actuations inside these ellipses, the surface effectsreduce the vibration amplitude while increase it for actuations outside the curves. Sensitiv-ity of the problem to the excitation frequencies is also studied and various examples aregiven to illustrate the trend of size dependencies.

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1. Introduction

Nanostructures have widely been used for development of Nanoelectromechanical systems (NEMS) such as nanosensors,nanoactuators and nanoresonators in the last decades [1–3]. On the other hand, due to shortcomings of classical theories indescription of the proper mechanical behavior of nanostructures, the researchers have attempted to provide appropriatemodels and theories for this purpose. For more explanation, departure of the results of the well-known classical theoriesfrom those of experimental studies at nanoscales must return to the effects of additional material properties or constitutiveequations of nanomaterials [4]. It is shown that there are some additional material properties on external boundary layers ofelastic media that are different from those of bulk materials. This concept is demonstrated by satisfaction of equilibriumequations at both surface and bulk materials by Gurtin and Murdoch through a continuum approach [5]. The surface prop-erties cannot be overlooked in the study of nanostructures and nanomaterials due to the large value of surface to volumeratios at that scale. As a fundamental approach, atomistic calculations and molecular dynamics (MDs) methods can be usedto confirm the existence of additional properties at the surface of materials [6]. It must be noted that, the nature of the sur-face properties is dependent on materials types and their crystallographic directions that even change the effective elasticmodulus of nanomaterials in different manners [7]. Basically, since the surface to volume ratio of an elastic boy decreases

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3576 A. Assadi / Applied Mathematical Modelling 37 (2013) 3575–3588

at larger scales, the surface effects decreases and the concept of size dependent mechanical behavior of nanostructures isintroduced [8].

Gurtin and his coworkers showed that the strain independent part of surface stresses which is called the ‘‘surface residualstresses’’ do not change the natural frequencies of microscaled crystals [9] due to self-equilibrating condition. Vibration andbuckling of unidirectional nanostructures such as nanowires have been studied with consideration of the surface effects andgeneralization of the classical beam theories [10,11]. In this area, Farshi et al. modified the Timoshenko beam model to in-clude the additional surface properties at both inner and outer surfaces of a nanotube for their transverse frequency analysisusing Laplace-Young equation to simulate the effect of residual stresses as external transverse loadings [12]. Wang studiedthe flexural vibration of fluid conveying nanotubes with consideration of surface properties [13]. Fu et al. studied the sizedependent response of nanobeams under electrostatic actuation incorporating surface effects [14]. Yang studied the sizedependent tensile deformation of nanotubes incorporating surface energies [15]. Some new interesting mechanical behav-iors resulted from surface properties are introduced in the literature that cannot be described by classical theory of elasticity.For example, Wang et al. used a gradually vanishing special Airy stress function to show the surface buckling of a nanobeamwith stiffer surface materials under pure bending [16]. In addition, non-uniform distribution of surface residual stress overthe nanostructure surfaces can cause self-deformations in them without applying any external loadings [17]. In this area, Onet al. studied the effect of locally distributed surface properties on the natural frequencies of nanobeams [18].

Two dimensional ultrathin films and nanoplates with different shapes have been fabricated in different environmentalconditions [19,20]. Regarding the nanoplates’ applications, the researchers have attempted to obtain their effective materialproperties in the presence of surface effects [21,22]. Using surface free energy concept, Fedorchenko et al. proved that theeffective elastic modulus of a nanoplate is not constant at theirs different thicknesses while it is a size dependent parameter[23]. Lu et al. generalized the Kirchhoff and Mindlin plate theories to include the effect of surface properties in describing thesize dependent static bending, vibration and buckling of nanoplates [24]. Nonlinear mechanical behavior of nanoplates hasalso been formulated to cover the surface properties using Hamilton’s principal [25,26]. In this framework of study, Assadiet al. used size and temperature dependent elastic modulus beside the effect of surface properties to obtain the size depen-dent dynamic behavior of nanoplates in different thermal environments [27]. In another work, Assadi and Farshi studied thefree vibration of circular nanoplates with consideration of surface effects [28]. Sheng et al. used the state space method forthree dimensional elasticity analysis of nanoplates so that they considered them as a laminated structure with additionalsurface layers [29].

Another theory that is frequently used for scale dependent mechanics of nanostructures is nonlocal theory of elasticitywhich is introduced by Eringen [30]. According to this theory, the stress components of a solid body are related to the straintensor of the entire body which can be stated by integral equations. Choosing special kernel functions for these integralequations would yield a simplified form of stress-strain relation which is able to reflect the size effects where the secondderivative of stress or strain is very large. Using this theory of elasticity, Lu et al. developed a beam models to study the scaleeffects on vibration of single and double walled carbon nanotubes (CNTs) [31]. Wang et al. evaluated the classical and non-local beam theories beside the molecular dynamics method for transverse vibration of nanotubes and illustrated the short-comings of the classical beam models [32]. In this field of study, Wang et al. studied the scale dependent inplane wavepropagation in isolated nanoplates and flexural wave propagation in embedded nanoplates [33,34]. They also studied theflexural vibration in double layered nanoplates by nonlocal theory of elasticity [35]. Murmu and Pradhan used the nonlocalelasticity for size dependent mechanical behavior of graphene sheets [36]. Using a similar framework, Assadi and Farshistudied the scale dependent stability of graphene based laminated composite sheets subjected to non-uniform inplane load-ing by nonlocal elasticity [37].

In this study, the effect of surface properties on forced vibration of rectangular nanoplates is investigated through an ana-lytical method. The plate is assumed to be under general external loading. Complete numerical results are given for the ob-tained size dependent natural frequencies and time response amplitude. Preliminary, it is seen that the surface effects makethe natural frequencies corresponding to different modes to vibration nearer to each other. Furthermore, it is shown that theeffect of surface properties on vibration amplitude can be controlled with the excitation location.

2. Problem statement

Nanoplates are common nanostructures which can be used in development of NEMS devices such as sensors, actuators,and resonators. On the other hand, design of nanoplates for these devices requires proper theoretical frameworks for theirmechanical behaviors. In addition to this concept, since the nanostructures are too small in size, they are always subjected toexternal actuations from different noise sources. In this manner, understanding the dynamic behavior of nanoplates helpsthe researchers to design proper NEMS devices for different applications with lowest noises. In this article it is tried to givea general framework for dynamic analysis of nanoplates with consideration of additional surface properties. Fig. 1 shows ananoplate which is subjected to general external loading q(x,y, t) where a, b and h are its length, width and thickness respec-tively. The elastic modulus, Poisson’s ratio and mass density of the bulk part of nanoplate are respectively indicated by E, tand q. In the present work ks and ls are the surface Lame constants while s is the surface residual stress which is uniformlydistributed on the upper and lower surfaces of the nanoplate. The surface mass density of the nanoplate is also indicated byqs.

Fig. 1. Geometry of a rectangular nanoplate with schematic upper and lower surface thin layers represented for additional surface properties.

A. Assadi / Applied Mathematical Modelling 37 (2013) 3575–3588 3577

To study the forced vibration of nanoplates, Kirchhoff plate model is used for which the displacement components in xand y directions are given with respect to the transverse deflection w as follows [24,28]

u ¼ �z@w@x

v ¼ �z@w@y

: ð1Þ

The resulting strain components in Cartesian coordinate which are always considered the same for bulk and surface partsof nanoplates can be derived from the relations of Eq. (1) as follows [38]

exx ¼ �z@2w@x2 ; eyy ¼ �z

@2w@y2 ; exy ¼ �2z

@2w@x@y

: ð2Þ

Next, according to the generalized Hook’s law for linear elastic materials, the stress components for the bulk part of thenanoplate can simply be obtained as

rbxx ¼

E1� t2 ðexx þ teyyÞ ¼ �

Ez1� t2

@2w@x2 þ t

@2w@y2

!; ð3:1Þ

rbyy ¼

E1� t2 ðeyy þ texxÞ ¼ �

Ez1� t2

@2w@y2 þ t

@2w@x2

!; ð3:2Þ

rbxy ¼

E2ð1þ tÞ exy ¼ �

Ez1þ t

@2w@x@y

: ð3:3Þ

As discussed in the previous section, the external layers of all the elastic solids contain additional material properties.Based on Gurtin-Murdoch theory of elastic solid surfaces [5], the linearized form of stress-strain relation for the surfaceof nanoplates are given as follows in which a, b, c = 1, 2.

rsab ¼ sdab þ ðls � ssÞ @ua

@xbþ @ub

@xa

� �þ ðks þ sÞ @uc

@xcdab þ s @ua

@xb; ð4:1Þ

rsaz ¼ s @w

@xa: ð4:2Þ

In Eq. (4) ks and ls are the surface Lame constants. Substituting the strain components from Eq. (2) into Eq. (4) gives theadditional surface stress components in terms of transverse displacement w as follows:

rs�xx ¼ sþ ð2ls þ ksÞ @u

@xþ ðks þ ssÞ @v

@y¼ s� h

2ð2ls þ ksÞ @

2w@x2 þ ðk

s þ sÞ @2w@y2

!; ð5:1Þ

rs�yy ¼ sþ ð2ls þ ksÞ @v

@yþ ðks þ sÞ @u

@x¼ s� h

2ð2ls þ ksÞ @

2w@y2 þ ðk

s þ ssÞ @2w@x2

!; ð5:2Þ

rs�xy ¼ ls @u

@yþ @v@x

� �� s @v

@x¼ �ð2ls � sÞ h

2@2w@x@y

; ð5:3Þ

rsxz ¼ s @w

@x; rs

yz ¼ s @w@y

: ð5:4Þ

3578 A. Assadi / Applied Mathematical Modelling 37 (2013) 3575–3588

In which superscripts + and � represent the upper and lower surfaces of the nanoplate respectively. Next, the bendingmoment resultants acting on the cross sections of a nanoplate will be obtained by the following integral equations:

Mxx ¼ ðrsþxx � rs�

xx Þh2þZ h=2

�h=2rb

xxzdz; ð6:1Þ

Myy ¼ ðrsþyy � rs�

yy Þh2þZ h=2

�h=2rb

yyzdz; ð6:2Þ

Mxy ¼ ðrsþxy � rs�

xy Þh2þZ h=2

�h=2rb

xyzdz: ð6:3Þ

In this stage, as can be observed from Eqs. (6), the bending moment resultants acting on the nanoplate’s cross sectionshave two distinct parts; the first part is resulted from bulk stress components and the second part from the surface stresses.Consequently, substituting from Eqs. (5) and (3) into Eqs. (6) and then simplifying the relations give the bending momentresultants with respect to the transverse displacement w as follows.

Mxx ¼ � Dþ h2ð2ls þ ksÞ4

!@2w@x2 � tDþ h2ðks þ sÞ

4

!@2w@y2 ; ð7:1Þ

Myy ¼ � Dþ h2ð2ls þ ksÞ4

!@2w@y2 � tDþ h2ðks þ sÞ

4

!@2w@x2 ; ð7:2Þ

Mxy ¼ � ð1� tÞDþ h2ð2ls � sÞ4

!@2w@x@y

; D ¼ Eh3

12ð1� t2Þ : ð7:3Þ

In these equations D is the classical flexural rigidity of the nanoplate without consideration of the surface effects. It is tobe noted that the Poisson’s ratio of the bulk and surface parts remains the same since the displacement field over the nano-plate’s thickness is considered to be continuous. Next, in order to derive the equilibrium equations of the nanoplate it is re-quired to find the shear forces acting on the cross sections. The additional shear stress on the cross sections from Eqs. (5) and(4) must be added to the classical terms of shear stress resultants. This concept must be considered for both lower and uppersurfaces of the nanoplates as follows:

Qx ¼@Mxx

@xþ @Mxy

@yþ rsþ

xz þ rs�xz ¼

@Mxx

@xþ @Mxy

@yþ 2s @w

@x; ð8:1Þ

Qy ¼@Myy

@yþ @Mxy

@xþ rsþ

yz þ rs�yz ¼

@Myy

@yþ @Mxy

@xþ 2s @w

@y: ð8:2Þ

The other equilibrium equation for a rectangular nanoplate in transverse direction that contains the external loading q isgiven as [38]:

@Qx

@xþ @Q y

@yþ qðx; y; tÞ ¼

Z h=2

�h=2qðzÞ @

2w@t2 dz ¼ ð2qs þ qhÞ @

2w@t2 : ð9Þ

Substituting from Eqs. (8) into Eq. (9) gives another form of this equilibrium equation in terms of bending moment resul-tants as follows:

2s @2w@x2 þ

@2w@y2

!þ @

2Mxx

@x2 þ 2@2Mxy

@x@yþ @

2Myy

@y2 þ qðx; y; tÞ ¼ ð2qs þ qhÞ @2w@t2 : ð10Þ

Finally, substituting from Eqs. (7) into Eq. (10) and then reformulating the relations, the generalized differential equationfor forced vibration of rectangular nanoplates will be derived as a pure function of transverse displacement w as follows:

Dþ 2ls þ ks

4h2

� �r2r2wþ qh 1þ 2qs

qh

� �@2w@t2 � 2sr2w ¼ qðx; y; tÞ: ð11Þ

In whichr2 = @2/@x2 + @2/@y2 is the Laplacian operator in Cartesian coordinate. In the next sections, this equation is solvedto obtain the size dependent natural frequencies and time response of nanoplates under external loading q.

A. Assadi / Applied Mathematical Modelling 37 (2013) 3575–3588 3579

3. General solution of the problem

In this section, saving the generality of the problem, Eq. (11) is solved for a rectangular nanoplate with simply supportedboundary condition at all the edges (SSSS boundary condition). Here the Navier solution is used to obtain the time responseof the nanoplates and the general series solution of the problem is given in Eq. (12) [38]. As a result, from this equation, it isseen that the vibration mode shapes of a nanoplate are preserved and they are not dependent to the effect of surface prop-erties. [24]. This may not be true for other boundary conditions or other geometries of nanoplates for which the vibrationmode shapes can be affected by the surface properties and therefore be dependent on the size of nanoplates [28].

wðx; y; tÞ ¼X1n¼1

X1m¼1

Amn sinðknxÞ sinðkmyÞgmnðtÞ; kn ¼npa; km ¼

mpb: ð12Þ

Substituting Eq. (12) into Eq. (11) yields to the following relation in which Fourier series coefficients Amn are unknownparameters that must be determined to obtain the real time response of the nanoplate corresponding to external load q.

Dþ h2ð2ls þ ksÞ4

!X1n¼1

X1m¼1

Amnðk2n þ k2

mÞ2 sinðknxÞ sinðkmyÞgmnðtÞ þ 2s

X1n¼1

X1m¼1

Amnðk2n þ k2

mÞ sinðknxÞ sinðkmyÞgmnðtÞ

þ ð2qs þ qhÞX1n¼1

X1m¼1

Amn sinðknxÞ sinðkmyÞ€gmnðtÞ ¼ qðx; y; tÞ: ð13Þ

For further simplification of this series equation, orthogonality of the vibration mode shapes of nanoplates must be used.For this purpose, multiplying Eq. (13) by sin(n0px/a) sin(m0py/b) and then integrating the obtained relation over the nano-plate’s area gives a time dependent ordinary differential equation for the nanoplate as follows:

ð2qs þ qhÞ€gmnðtÞ þ Dþ h2ð2ls þ ksÞ4

!ðk2

n þ k2mÞ

2 þ 2sðk2n þ k2

mÞ( )

gmnðtÞ

¼ 4abAmn

Z 1

n¼1

Z 1

m¼1sinðknxÞ sinðkmyÞqðx; y; tÞdxdy: ð14Þ

Due to superposition principle for linear systems when a problem is solved for a point load excitation, the obtained solu-tions can be integrated over the surface areas to derive the time response of nanoplates under any arbitrary distributed load-ings. Therefore, saving the generality of the problem, assume that the concentrated point load q0 sin (xt) is applied at x0 andy0 on the nanoplate in transverse direction. Then using the definition of Dirac delta function, Eq. (14) can be rewritten asfollows:

ð2qs þ qhÞ€gmnðtÞ þ Dþ h2ð2ls þ ksÞ4

!ðk2

n þ k2mÞ

2 þ 2sðk2n þ k2

mÞ( )

gmnðtÞ ¼4q0

abAmnsinðknx0Þ sinðkmy0Þ sinðxtÞ: ð15Þ

In order to obtain the steady state time response of this ordinary differential equation, a function in the form of the righthand side of Eq. (15) must be selected for gmn(t). Substituting this function into Eq. (15) gives all its involved constants. Doingthis procedure, gmn(t) will be given by the following relation.

gmnðtÞ ¼4q0

abAmnð2qsþqhÞ sinðknx0Þ sinðkmy0Þx2

mn �x2 sinðxtÞ: ð16Þ

In which xmn is represented for natural frequencies of the nanoplate corresponding to mode shape sin(npx/a) sin(mpy/b)which can be calculated from Eq. (17). This is later used for free vibration or frequency analysis of nanoplates [27].

x2mn ¼

14ð4Dþ h2ð2ls þ ksÞÞðk2

n þ k2mÞ

2 þ 8sðk2n þ k2

mÞ2qs þ qh

: ð17Þ

Finally, substitution of Eq. (16) into Eq. (12) gives the time response of nanoplates corresponding to a harmonic point loadas follows:

wðx; y; tÞ ¼X1n¼1

X1m¼1

4q0

abð2qs þ qhÞsinðknx0Þ sinðkmy0Þ

x2mn �x2 sinðknxÞ sinðkmyÞ sinðxtÞ: ð18Þ

As discussed earlier, the time response of a nanoplate related to any type of harmonic actuation can simply be obtainedusing superposition principle. Mathematically this can be done by integration of Eq. (18) over the area S⁄ which is underexternal actuations as follows:

Wðx; y; tÞ ¼X1n¼1

X1m¼1

4 sinðknxÞ sinðkmyÞabð2qs þ qhÞ

ZZs�

�qðx0y0Þ sinðknx0Þ sinðkmy0Þx2

mn �x2 dx0dy0� �

sinðxtÞ: ð19Þ

3580 A. Assadi / Applied Mathematical Modelling 37 (2013) 3575–3588

Next, another reformulation of Eq. (18) with non-dimensional parameters is given by the following relation. It mustbe noted that from Eq. (17) xmn contains the effective flexural rigidity of the nanoplate which is not seen in Eqs. (20) directly.

Wðx; y; tÞ ¼X1n¼1

X1m¼1

2qs

qhþ 1

� ��1 x2

x2mn �x2 sinðknx0Þ sinðkmy0Þ sinðknxÞ sinðkmyÞ sinðxtÞ ð20:1Þ

Wðx; y; tÞ ¼ qhabx2

4q0wðx; y; tÞ ð20:2Þ

It must be noted that Eq. (18) may not entirely be normalized so that some geometric parameters remain if xmn is substi-tuted from Eq. (17) into Eq. (20.1). This matter reflects the size dependency of the natural frequencies and time responses. Inthis work, for the numerical calculations, time response amplitude of the nanoplate at point load at (x0, y0) is consideredwhich is obtained from Eqs. (20) as follows:

Wðx0; y0; tÞ ¼2qs

qhþ 1

� ��1X1n¼1

X1m¼1

x2

x2mn �x2 sin2ðknx0Þ sin2ðkmy0Þ sinðxtÞ: ð21Þ

It is seen that this real time response contains the natural frequencies xmn of the nanoplate that are size dependent. Inorder to have a complete discussion on the problem, a separate section is dedicated for size dependent free vibration analysisof nanoplates. A wide range of numerical examples are given to illustrate the size dependency of natural frequencies andtheir ratios.

4. Free vibration analysis

In this section, it is tried to illustrate the size dependent behavior of natural frequencies of a rectangular nanoplates. Forthis purpose, the normalized natural frequencies (NNFs) are depicted in different examples versus the nanoplates’ geometricparameters and mode numbers. Here, NNF is the ratio of natural frequencies obtained from Eq. (17) to those of classical the-ories without consideration of surface effects. Consequently, deviation of NNF from unity shows the effect of surface prop-erties on natural frequencies. Conversely, as this parameter goes to unity the effect of surface properties diminishes.According to the definitions, NNF can be given by the following relation:

NNF ¼ 2qs

qhþ 1

� ��1

1þ h2ð2ls þ ksÞ4D

þ 8s4Dðk2

n þ k2mÞ

( ): ð22Þ

The numerical results of this section are given for aluminum nanoplates with the following material properties [14] inwhich Es is the surface elasticity:

E ¼ 68:5 GPa m ¼ 0:35 q ¼ 2700 kg=m3 s ¼ 0:910 N=m Es ¼ 6:090 N=m qs ¼ 5:46e� 7 kg=m2:

The surface Lame constants ks and ls in Eq. (22) are related to the surface elasticity Es and the surface Poisson’s rationts = t by the following relations:

ks ¼ Estð1þ tÞð1� 2tÞ ¼ 5:26 N=m ls ¼ Es

2ð1þ tÞ ¼ 2:26 N=m: ð23Þ

For the first numerical example, the normalized natural frequency NNF is evaluated for different combinations of thenanoplates’ length a and thickness h and vibration mode shape m.

From numerical results in Fig. 2a it is concluded that as the thickness of a nanoplate gets higher, the effect of surface prop-erties tends to decrease. On the other hand, the effect of surface properties is more for larger nanoplates. However, the curvesof Fig. 2a get closer to each other at higher values of a so that for a > 120 nm the NNF is saturated versus variation of a. Thismeans that the NNF is more sensitive to the nanoplate’s length at smaller scales. In addition, Fig. 2b shows the NNF withrespect to the length of square nanoplates with thickness h = 3 nm for different mode numbers of vibration. From this figure,it is observed that the effect of surface properties can be ignored at higher modes of vibration. It must be also mentioned thatthe NNF decreases considerably from mode I to mode II of vibration while this is not seen at higher modes where the cor-responding curves converge to each other at m > 4. For more clarification of the concept of size dependent behavior of naturalfrequencies, other complementary numerical examples for the NNF of aluminum nanoplates are given in Fig. 3.

As illustrated in Fig. 3 for mode numbers larger than 5 the effect of surface properties may rationally be ignored with aslight error while this is true for a wide range of nanoplates’ geometric parameters. For such cases, when NNF goes to unity,the forced vibration responses converge to those of classical theories without surface effects. In harmonic actuation exam-ples, the relations are reformulated to include the ratio of natural frequencies of nanoplates which is given in Eq. (24).Parameters with superscript ⁄ refer to the case of nanoplates when the effect of surface properties in not included.

Fig. 2. Plots illustrating the size dependency of NNF for a wide range of nanoplates’ geometric parameters and vibration mode numbers.

Fig. 3. Complementary numerical examples for size dependent behavior of natural frequencies.

A. Assadi / Applied Mathematical Modelling 37 (2013) 3575–3588 3581

xmn

xm0n0

� �2

¼ x�mn

x�m0n0

2sþ Deff x�mnx�

m0n0ðk2

m0 þ k2n0 Þ

2sþ Deff ðk2m0 þ k2

n0 Þg ¼ a

b

x�mn

x�m0n0¼ m2 þ g2n2

m02 þ g2n02Deff ¼ Dþ ð2l

s þ ksÞh2

4:

ð24Þ

Due to the formulations of this work, the only required parameter for size dependent forced vibration study of nanoplatesis the ratio of natural frequencies to the fundamental natural frequency x11. Using Eq. (22), the following relation is obtainedfor this ratio.

xmn

x11

� �2

¼ m2 þ g2n2

1þ g2

2sa2 þ Deff p2ðm2 þ g2n2Þ2sa2 þ Deff p2ð1þ g2Þ

: ð25Þ

From Eq. (25) it is understood that for a given value of g the ratio xmn/x11 is not the same for different values of a and h.This concept addresses the size dependent behavior of the natural frequencies. As a comparison, when xmn/x11 is smallerthan its corresponding value without surface effects obtained from classical plate theory, then the surface properties madethe natural frequencies of different vibration modes to get closer to each other. Table 1 shows the numerical results for xmn/x11 for aluminum nanoplates with thickness h = 2 nm for different modes of vibration, length a and aspect ratio g.

Fig. 4. The ratio of xmn/x11 for aluminum nanoplates with different geometries.

3582 A. Assadi / Applied Mathematical Modelling 37 (2013) 3575–3588

Before discussion on the results, another table is given for aluminum nanoplates with thickness h = 4 nm. Comparing theresults of Tables 1 and 2, it is seen that the surface effects are significantly lower for thicker nanoplates.

From these tables, it is concluded that the ratio xmn/x11 decreases when the surface effects are considered. In this man-ner, it can be seen that for greater surface effects (for example in thinner and larger nanoplates) the natural frequencies cor-responding to different modes of vibration will be compacted into a tighter spectrum. For better understanding of the givenresults, Fig. 4 shows the ratio of xmn/x11 with respect to the vibration mode numbers that are listed in Table 3. For example,a sample set of natural frequencies corresponding to these mode numbers is bolded in Table 1.

From Fig. 4 it is concluded that the frequency ratio xmn/x11 in the presence of surface effects even becomes one-third ofits corresponding classical value. Here, it must be noted that xmn/x11 from classical plate theory without surface effects hasa unique value for a given g. This is shown in Fig. 4 by single dotted curves for each value of g.

Table 1Ratios of xmn/x11 for aluminum nanoplates with h = 2 nm.

m = 1 m = 2 m = 3 m = 4 m = 5 m = 1 m = 2 m = 3 m = 4 m = 5

n = 1 1.000 1.658 2.516 3.571 4.838 1.000 1.281 1.666 2.124 2.644n = 2 2.489 2.980 4.007 5.250 2.124 2.420 2.806 3.275n = 3 3.715 4.700 5.928 3.476 3.818 4.247n = 4 Symmetric 5.685 6.868 Symmetric 5.125 5.530n = 5 g = 1; a = 100 nm 8.065 g = 0.5; a = 100 nm 7.118

n = 1 1.000 1.826 3.073 4.762 6.906 1.000 1.321 1.797 2.412 3.165n = 2 2.582 3.801 5.478 7.618 2.412 2.834 3.412 4.142n = 3 5.001 6.668 8.802 4.463 5.022 5.737n = 4 Symmetric 8.329 10.50 Symmetric 7.238 7.946n = 5 g = 1; a = 50 nm 12.58 g = 0.5; a = 50 nm 10.77

Table 2Ratios of xmn/x11 for aluminum nanoplates with h = 4 nm.

m = 1 m = 2 m = 3 m = 4 m = 5 m = 1 m = 2 m = 3 m = 4 m = 5

n = 1 1.000 1.941 3.430 5.484 8.112 1.000 1.353 1.899 2.628 3.542n = 2 2.839 4.313 6.361 8.987 2.628 3.138 3.843 4.744n = 3 5.777 7.820 10.44 5.143 5.838 6.731n = 4 Symmetric 9.861 12.48 Symmetric 8.611 9.501n = 5 g = 1; a = 100 nm 15.10 g = 0.5; a = 100 nm 13.05

n = 1 1.000 2.240 4.293 7.164 10.85 1.000 1.462 2.221 3.276 4.628n = 2 3.473 5.524 8.394 12.08 3.246 4.028 5.079 6.429n = 3 7.574 10.44 14.13 7.029 8.079 9.428n = 4 Symmetric 13.31 17.00 Symmetric 12.28 13.62n = 5 g = 1; a = 50 nm 20.69 g = 0.5; a = 50 nm 19.02

Table 3The vibration mode numbers corresponding to horizontal axes of Fig. 4.

Frequency number 1 2 3 4 5 6 7 8 9Natural frequency x11 x12 x22 x23 x33 x34 x44 x45 x55

A. Assadi / Applied Mathematical Modelling 37 (2013) 3575–3588 3583

As discussed earlier, the effect of surface properties is more at thinner and larger nanoplates. This trend of size dependentbehavior can also be seen from Fig. 4 while xmn of thinner nanoplates get nearer to their fundamental natural frequency.Using the presented frequency analysis, it is tried to use the conclusions for size dependent forced vibration analysis of nano-plates in the next section.

5. Forced vibration analysis

The general time response of nanoplates under concentrated force actuation is given in Eqs. (20). Reformulation of Eqs.(20) yields to the following normalized relation that contains two frequency ratios of xmn/x11 and k = x/x11.

Fig. 5.corresp

Wðx0; y0; tÞ ¼2qs

qhþ 1

� ��1X1n¼1

X1m¼1

k2

xmnx11

� �2� k2

sin2ðknx0Þ sin2ðkmy0Þ sinðxtÞ: ð26Þ

In this equation, the frequency ratio xmn/x11 can be taken from Tables 1 and 2. On the other hand, the corresponding timeresponse for a nanoplate without consideration of surface effects can be simply obtained from Eq. (26) in which the surfaceproperties are put equal to zero. In the following formulations, all the parameters with superscript ⁄ refer to those of classicaltheories without surface effects. The frequency ratio x�mn=x�11 in Eq. (27) are substituted from Eq. (24) and the correspondingtime response for the nanoplate is obtained in the absence of surface effects as follows:

W�ðx0; y0; tÞ ¼X1n¼1

X1m¼1

k2

x�mnx�

11

� �2� k2

sin2ðknx0Þ sin2ðkmy0Þ sinðxtÞ

¼X1n¼1

X1m¼1

k2

m2þg2n2

12þg2

� �2� k2

sin2ðknx0Þ sin2ðkmy0Þ sinðxtÞ: ð27Þ

Next, it is intended to obtain the amplitude ratio of time responses at point (x0, y0) for nanoplates with and without sur-face effects. From Eq. (26) and Eq. (27) it is concluded that sin (xt) = 1 corresponds to vibration amplitude of the nanoplatesat point (x0,y0). Therefore the objective function for size dependent forced vibration analysis of nanoplates can be stated asfollows:

Amplitude Ratio ¼ max½Wðx0; y0; tÞ�max½W�ðx0; y0; tÞ�

ð28Þ

In the following numerical examples the Amplitude Ratio (AR) is plotted for various geometries, excitation frequenciesand location of point loads for aluminum nanoplate to show the size dependent forced vibration responses. At first, AR isdepicted in a three dimensional graph in Fig. 5 for a square nanoplate with a = b = 100 nm and h = 2 nm. In this example,

(a) Three dimensional plot of Amplitude Ratio (AR) versus the location of point load (x0,y0). (b) Schematic plot of a rectangular nanoplate with itsonding Neutral Ellipse.

3584 A. Assadi / Applied Mathematical Modelling 37 (2013) 3575–3588

horizontal axes show the location of excitation force. For this case k is chosen equal to 0.1. As an important conclusion, it isobserved that for a nanoplate with the assumed parameters AR can be either more or less than unity for different excitations.This illustrates that the effect of additional surface properties can decrease or increase the vibration amplitude of the nano-plates depending on the location of excitation load.

From Fig. 5a it is concluded that the Amplitude Ratio is less than unity when the nanoplate is being excited in a point nearto the central areas. Conversely, the AR becomes larger when the nanoplate is excited in a point closer to its edges. Accord-ingly, as illustrated in Fig. 5a and due to the continuity of the problem, a continuous path must exist over the nanoplate forwhich AR = 1. Here this path is called the ‘‘Neutral Ellipse’’ and it is shown in Fig. 5b. This means that, if a harmonic concen-trated load is applied on any point of this ellipse, then the surface effects do not change the time response of the nanoplate. Itis to be noted that for square nanoplates this ellipse becomes a circle (Neutral Circle). Further examples are given for differentsizes of square nanoplates to show the size dependent behavior of AR generally for different excitation locations. Although,other numerical results follow the general trend of Fig. 5 but they differ in the values of AR and the size of Neutral Ellipses.The curves of AR in Fig. 6 are plotted with respect to the excitation locations corresponding to the point numbers on path A–Ain Fig. 5b.

As another important conclusion, from Fig. 6 it is observed that in general the AR is lower for smaller nanoplates. In addi-tion, for excitations inside the corresponding neutral ellipses, the effect of surface properties is larger for smaller nanoplates.Conversely, for excitations outside these ellipses, the surface effects are higher for larger nanoplates. Furthermore, it is con-cluded that the ratio of R/a (mean diameter of neutral ellipse/nanoplate’s length) in Fig. 5b is higher at smaller nanoplates. Ingeneral, by increasing the size of nanoplates at larger scales (larger than a > 300 nm) the relative size of neutral ellipse be-comes approximately independent of the nanoplate’s size and R converges to 0.30a.

It is also observed that for smaller nanoplates, R/a increases for thicker ones. On this stage, it must be noted that for somecases, (for example a = b = 50 nm and h = 4 nm) AR becomes less than unity over the entire path A–A and all the nanoplate isplaced into its corresponding neutral ellipse. Conversely for larger nanoplates, the size of their neutral ellipse gets smaller for

Fig. 6. Plots of Amplitude Ratio over path A–A for square nanoplates with (a) h = 2 nm and (b) h = 4 nm.

Fig. 7. Plots of Amplitude Ratio over path A–B for square nanoplates with (a) h = 2 nm and (b) h = 4 nm.

A. Assadi / Applied Mathematical Modelling 37 (2013) 3575–3588 3585

thicker nanoplates and for most of excitation location AR is greater than unity. Complementary numerical examples are gi-ven in Fig. 7 in which the same problem of Fig. 6 is solved for excitations over A–B path in Fig. 5b.

From Fig. 7 it is seen that, although the AR curves are similar over any arbitrary outward path from the nanoplate’s center,but variation of AR for excitations over path A–B is more than that of path A–A. in addition, the sensitivity of AR to the exci-tation location is much higher for excitations near the corner of nanoplates.

Next, it is tried to show the size dependent behavior of AR for rectangular nanoplates with different aspect ratios g = a/b.The numerical results in Fig. 8 are given for excitations at the nanoplate’s center point (x0 = a/2; y0 = b/2) and k = 0.1.

From Fig. 8 it is seen that each curve has a local maximum point so that for most of the cases, the surface properties hastheir maximum or minimum effects depending on the nanoplate’s size and thickness. In this case, two different cases of sizedependent behavior must be discussed as follows:

1. For curves of Fig. 8 which are completely under the thick horizontal lines, the surface properties have their lowest effectat the local maximum of corresponding curves (for example, curves of a = 100 nm in Fig. 8). This condition occurs at smal-ler nanoplates and the surface effects get higher as the nanoplate becomes a plate strip.

2. For curves that intersect the horizontal thick lines, the local maximum points of the curves may not refer to aspect ratiosin which the surface effects have their maximum values. For example, for nanoplate with a = 150 nm and h = 2 nm theeffect of surface properties is maximum at a/b = 1 or a/b =1 while for a = 300 nm and h = 2 nm the aspect ratio a/b = 5corresponding to the curve’s local maximum point is where the surface effect is maximum.

Fig. 8. Variation of AR with respect to the nanoplate’s aspect ratio g for different plate sizes.

Fig. 9. Variation of AR with respect to k = x/x11 for different excitation locations over path A–B.

3586 A. Assadi / Applied Mathematical Modelling 37 (2013) 3575–3588

In addition, it is seen that AR is less sensitive to g at longer nanoplates while the curves of Fig. 8 are saturating when thenanoplates are so elongated so that they become plate strips.

All the presented numerical examples were evaluated for k = x/x11 = 0.1. Further results are given in Fig. 9 for other exci-tation frequencies for the special case of nanoplates with a = b = 100 nm and h = 2 nm. Here it is to be noted that combinationof the results of Fig. 9 and other previous figures gives many ideas for size dependent time response of nanoplates underexternal actuations with different conditions which is left to the readers.

From this figure it is seen that, as the nanoplate is excited nearer to its resonant frequency x11, the effect of surface prop-erties reduces so that the AR converges to unity. On the other hand, the sensitivity of AR to k gets higher at excitations nearerto k = 1. This behavior seems to be independent of the excitation location. As an important conclusion, the general variationtrend of AR versus k is independent of the nanoplate’s size and thickness. According to the previous numerical results, even ifFig. 9 is given for the especial case of a nanoplate with a = b = 100 nm and h = 2 nm, it gives a general form of AR curves ver-sus g for any other nanoplates’ sizes.

6. On verification of the results

In this section it is tried to have a discussion about verification of the presented results in this article. For this purpose, aninverse method is used for calculation of the effective elastic modulus Eeff of the nanoplates from their natural frequencies. Asa key point, since the natural frequencies are directly used in the forced vibration responses, by their verification, the resultsof the forced vibration section will also be verified automatically. As derived earlier, the natural frequencies of a rectangularnanoplate can be obtained from the following relation:

x2mn ¼

14ð4Dþ h2ð2ls þ ksÞÞðk2

n þ k2mÞ

2 þ 8sðk2n þ k2

mÞ2qs þ qh

: ð29Þ

From another viewpoint, consider the same nanoplate for which the natural frequencies are measured from experimentalstudies equal to xmn. In addition, assume that the nanoplate is an elastic body that can be treated by classical plate theoryduring the experimentation. In this manner, it is obvious that during the considered experimental study, the nanoplateshows an effective elastic modulus that determines the value of xmn from the following relation:

x2mn ¼

Eeff h3

12ð1� t2Þðk2

n þ k2mÞ

2

qeff h: ð30Þ

At the first step, since the effective mass density qeff of the nanoplates can be measured without negotiation with thevibration problem, it can be set equal to q + 2qs/h directly. Eq. (30) is obtained from Eq. (29) in which all the surface param-eters are covered by the effective elastic modulus Eeff. For more explanation, it must be noted that, this virtual experimentalsetup is assumed to not be able to sense the existence of additional surface properties or any other similar effects such asnonlocalities. Consequently, the natural frequencies from Eq. (29) and Eq. (30) must be equal and the effective elastic mod-ulus Eeff will be obtained from this equality as follows:

Eeff ¼ Eþ 3ð1� t2Þð2ls þ ksÞh

þ 24sð1� t2Þh3ðk2

n þ k2mÞ: ð31Þ

Finally, the effective elastic modulus Eeff from Eq. (31) for nanoplates can be verified by the same parameter of nanowiresobtained from experiment or molecular dynamics method. In this section, the numerical results for the verification purposeare given for silver nanoplates. For more explanations, excellent experimental results are found for silver nanowires’ elasticmodulus in the literature. The materials constants for silver are equal to:

E ¼ 76 GPa m ¼ 0:3 s ¼ 0:89 N=m Es ¼ 1:22 N=m:

Since the experiments of Ref. [38] are given for bending of silver nanowires subject to a point load, here the nearest studyto this experiment relates to the fundamental natural frequency of nanoplates with m = n = 1 for which the nanoplate carriesthe minimum value of curvature variations. Substituting these values together with the given material parameters into Eq.(31) gives:

Eeff ¼ 76 1þ 0:06Hþ 0:03

Hh2

1a2 þ

1

b2

� ��1" #

: ð32Þ

In this equation H is the thickness of the nanoplate in nanometers. On the other hand, even if the experimental results inRef. [39] are given for nanowires and Eq. (33) can simply be reformulated for nanowires, but however, here square nano-plates with a = b are selected for which the Eq. (32) can be further simplified to:

Eeff ¼ 76þ 1:14H

4þ a2

h2

� �� 76þ 1:14w2

H; ð33Þ

Fig. 10. Comparison of the effective elastic modulus of silver nanoplates from this work and that of nanowires from experiment in Ref. [39].

A. Assadi / Applied Mathematical Modelling 37 (2013) 3575–3588 3587

where w = a/h is the nanoplates aspect ratio. Fig. 10 shows the corresponding results for comparison between the Eeff ofnanoplates and nanowires.

From Fig. 10 it is seen that even if the presented verification method may have some shortcomings which are left as openquestions for the future studies, but good agreement is observed between the effective elastic modulus Eeff of nanoplateswith the experimental results of Ref. [39] for silver nanowires. In this stage, the author regrets that any experimental resultcould not be found for aluminum nanowires but the above verification can satisfactorily guarantee the validity of the devel-oped framework for mechanical behavior of aluminum nanoplates also. On the other hand, since the derivation procedurespresented in this work were so straightforward and the natural frequencies are directly used in the forced vibration re-sponses, verification of Eeff is sufficient for verification of other analyses such as free vibration, and forced vibration.

7. Summary and conclusions

In summary, size dependent forced vibration of nanoplates under the effect of additional surface properties is studied viaanalytical method. The problem is solved for general harmonic loadings with the closed form solutions given for time re-sponses. Due to the superposition principal and saving the generality of the problem the numerical examples were givenfor a concentrated point load applied on nanoplate. At the first step, the free vibration problem of nanoplates is studiedfor their size dependent frequency analyses which were used later in forced vibration sections. In this manner, a completeset of numerical results are given for normalized natural frequencies of Aluminum nanoplates. In general it is observed thatas nanoplates get thicker and smaller, the effect of surface properties reduces. On the other hand it is observed that the nat-ural frequencies get closer to each other as the surface effects get higher. This concept is more observable for thinner andlarger nanoplates. For example in some cases the ratio of xmn/x11 becomes one-third of its corresponding value withoutconsideration of surface effects.

From forced vibration analysis it is concluded that there exist a Neutral Ellipse on the surface of nanoplates for which if aconcentrated load is applied on any point of this elliptic curve, then the surface properties has no effect on direct time re-sponses at the excitation location. The length of major and minor axes of this ellipse is approximately equal to half of thecorresponding plate edges’ length in their parallel directions. Furthermore, for excitations inside Neutral Ellipse the surfaceproperties reduces the maximum deflection of the nanoplate at the location of excitation while this behavior reverses forloads applied on outside of this elliptic curve. Further numerical examples showed that as the excitation frequency gets high-er the effect of surface properties decreases and can be ignored at the near resonance excitations. Finally through an inversemethod, it is shown that the effective elastic modulus of nanoplates was in good agreement with that of nanowires that areobtained from experimental results.

Appendix A. Supplementary data

Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.apm.2012.07.049.

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