Article history:

Received 28 January 2011

Received in revised form

Accepted 20 August 2011

Handling Editor: L.G. ThamAvailable online 22 September 2011

This paper develops a Mindlin microplate model based on the modied couple stress

theory for the free vibration analysis of microplates. This non-classical plate model

eorydressticityhave

single-and double walled carbon nanotubes. Lu et al. [20] proposed the nonlocal Kirchhoff and Mindlin plate models basedon Eringens theory of nonlocal continuum mechanics. Jomehzadeh and Saidi [21] analyzed the effect of the nonlocal

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Journal of Sound and Vibration

Journal of Sound and Vibration 331 (2012) 941060022-460X/$ - see front matter & 2011 Elsevier Ltd. All rights reserved.

doi:10.1016/j.jsv.2011.08.020n Corresponding author. Tel.: 86 10 51684070; fax: 86 10 51682094.E-mail address: llke@bjtu.edu.cn (L.-L. Ke).been proposed and employed to study the mechanical characteristics of microscale structures. Using nonlocal elasticitytheory, Ke et al. [18] and Yang et al. [19] investigated the size effect on the nonlinear free vibration characteristics of thewhen the structural size is in micron or sub-micron scale. This is largely due to the fact that the classical continuum thdoes not admit the size-dependence in the elastic elds of defects such as dislocations, lattice defects and voids. To adthis issue, various size-dependent continuum theories which can capture the size effect, such as couple stress elas[12,13], nonlocal elasticity [14], strain gradient elasticity [15], surface elasticity [16] and micropolar elasticity [17],Microplates, microbeams and microbars have found extensive applications in micro/nano-electro-mechanical systems(MEMS/NEMS) and atomic force microscopes (AFMs) [16]. These structures are of the order of microns or sub-microns.Their material properties have been experimentally observed to exhibit the microstructure-dependent size effect in themicro-torsion and micro-bending tests of microscale structures [711]. It was found that when the thickness of thesestructures is close to the internal material length scale parameter, the size effect is signicant and has to be taken intoaccount when studying their static and dynamic behaviors.

It is well known that the classical continuum theory fails to characterize the size effect on the mechanical behaviors1. Introductionparameter.structures and is capable of interpreting the size effect that the classical Mindlin plate

model is unable to describe. The higher-order governing equations of motion and

boundary conditions are derived using the Hamilton principle. The p-version Ritz

method is employed to determine the natural frequencies of the microplate with

different boundary conditions. A detailed parametric study is conducted to study the

inuences of the length scale parameter, side-to-thickness ratio and aspect ratio on

the free vibration characteristics of the microplate. It is found that the size effect is

signicant when the thickness of microplate is close to the material length scale

& 2011 Elsevier Ltd. All rights reserved.13 May 2011contains an internal material length scale parameter related to the material micro-Free vibration of size-dependent Mindlin microplates based on themodied couple stress theory

Liao-Liang Ke a,n, Yue-Sheng Wang a, Jie Yang b, Sritawat Kitipornchai c

a Institute of Engineering Mechanics, Beijing Jiaotong University, Beijing 100044, PR Chinab School of Aerospace, Mechanical and Manufacturing Engineering, RMIT University, PO Box 71, Bundoora, Victoria 3083, Australiac Department of Building and Construction, City University of Hong Kong, Kowloon, Hong Kong

a r t i c l e i n f o a b s t r a c t

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

L.-L. Ke et al. / Journal of Sound and Vibration 331 (2012) 94106 95parameter on the large amplitude vibration of multilayered graphene sheets. Lazopoulos [22], Papargyri-Beskou andBeskos [23], and Papargyri-Beskou et al. [24,25] developed the microbeam and microplate models based on the straingradient elasticity theory. Lu et al. [26] developed a general thin plate theory using the surface elasticity. Huang [27] andLu et al. [28] analyzed the mechanical behavior of nano-scaled lms including surface effects. Assadi et al. [29] discussedthe size-dependent dynamic analysis of nanoplates including the effect of the surface elasticity and residual stresses.Ramezani et al. [30] presented a brief review of the micropolar continuum theory and introduced the concept of energypairs in the micropolar continuum. They later analyzed the micropolar elastic beams [31] and constitutive equations formicropolar hyper-elastic materials [32].

Yang et al. [33] rst modied the classical couple stress theory [12,13] and proposed the modied couple stress theoryinvolving only one additional material length scale parameter. This feature makes the modied couple stress theory easierto use than the classical couple stress theory. Based on the modied couple stress theory [33], many investigatorsdeveloped the non-classical EulerBernoulli microbeam model, Timoshenko microbeam model and Kirchhoff microplatemodel to capture the size effect in microscale structures. Using the microbeam models, the elastic bending [34], linear freevibration [3538], nonlinear vibration [39] and postbuckling [40] problems of microbeams were examined. Based on theKirchhoff microplate model, Tsiatas [41], Yin et al. [42] and Jomehzadeh et al. [43] presented analytical solutions forthe static and dynamic problems of microplates. Recently, the modied couple stress theory is further used to establish thesize-dependent functionally graded microbeam models. Asghari et al. [44,45], Ke and Wang [46] and Ke et al. [47]investigated the bending, free vibration, static buckling, dynamic stability and nonlinear vibration of the functionallygraded EulerBernoulli and Timoshenko microbeams.

Although the size-dependent microbeam model and Kirchhoff microplate model have been developed and welldiscussed in the above mentioned investigations based on the modied couple stress theory, no literature has beenreported for the microplate model incorporating the effects of the transverse shear deformation and rotary inertia whichbecome more signicant and must be considered for thick and moderately thick microplates. In this paper, a size-dependent microplate model for the free vibration analysis of the microplate is developed based on the Mindlin platetheory and modied couple stress theory to account for the effects of transverse shear deformation, rotary inertia and sizeeffect. The higher-order governing equations and boundary conditions are derived using the Hamilton principle. Thep-version Ritz method is employed to determine the natural frequencies of the microplate with different boundaryconditions. The effects of the length scale parameter, side-to-thickness ratio and aspect ratio on the free vibrationcharacteristics are discussed in detail.

2. Theoretical formulations

The modied couple stress theory was rst presented by Yang et al. [33]. In this theory, the strain energy density isconsidered as the function of both strain tensor (conjugated to stress tensor) and curvature tensor (conjugated to couplestress tensor). Then, the strain energy PS in an isotropic linear elastic body occupying a region L can be written as

PS 1

2

ZLr:em:vdL, (1)

where e is the strain tensor; r is the Cauchy stress tensor; v is the symmetric curvature tensor andm is the deviatoric partof the couple stress tensor. These tensors are dened by [33]

e 12ruruT , (2)

v 12rhrhT , (3)

r ltreI2me, (4)

m 2l2mv, (5)where u is the displacement vector; l and m are Lames constants (m is also known as shear modulus); l is a material lengthscale parameter which is regarded as material property characterizing the effect of the couple stress and h is the rotationvector expressed as

h 12curl u: (6)Note that the material length scale parameter l is mathematically the square of the ratio of the modulus of curvature to

the modulus of shear and is physically a property measuring the effect of couple stress [36]. This parameter can bedetermined from torsion tests of slim cylinders [10] or bending tests [11] of thin beams in micron scale.

Consider a rectangular microplate of length a, width b, and constant thickness of h under plane stress state, dened inthe rectangular coordinate system (0rxra, 0ryrb, h/2rzrh/2) shown in Fig. 1. According to the Mindlin platetheory, the displacements ~U , ~V and ~W of an arbitrary point (x, y, z) are expressed as

~Ux,y,z,t zCxx,y,t, ~V x,y,z,t zCyx,y,t, ~W x,y,z,t Wx,y,t, (7)

L.-L. Ke et al. / Journal of Sound and Vibration 331 (2012) 9410696where W is the mid-plane displacement of the microplate in the z direction; Cx and Cy denote the transverse normalrotations about the y- and x-axes, respectively, and t is the time. Note that by setting Cxdw/dx and Cydw/dy, theKirchhoff plate theory can be recovered.

In view of Eqs. (2) and (7), the straindisplacement relations can be expressed as

exx z@Cx@x

, eyy z@Cy@y

, gxy z@Cx@y

@Cy@x

, (8)

gyz @W

@yCy, gxz

@W

@xCx, ezz 0 (9)

Substituting Eqs. (7) into Eq. (6) gives

yx 12

@ ~W

@y @

~V

@z

! 1

2

@W

@yCy

, (10)

yy 12

@ ~U

@z @

~W

@x

! 1

2Cx @W

@x

, (11)

yz 1

2

@ ~V

@x @

~U

@y

! z

2

@Cy@x

@Cx@y

(12)

Substituting Eqs. (10)(12) into Eq. (3) yields

wxx 1

2

@2W

@x@y @Cy

@x

!, wyy

1

2

@Cx@y

@2W

@x@y

!, wzz

1

2

@Cy@x

@Cx@y

, (13)

wxy 1

4

@2W

@y2 @

2W

@x2 @Cx

@x @Cy

@y

!, wxz

z

4

@2Cy@x2

@2Cx@x@y

!, wyz

z

4

@2Cy@x@y

@2Cx@y2

!(14)

It should be pointed out that the whole analysis of the present paper is based on the assumed displacement and rotationelds, i.e. Eq. (7) and Eqs. (10)(12). The reason is that it can make the free vibration analysis for the size-dependent microplateeasily. Actually, nearly all the articles on the microplate or microbeam cited in this paper (see Refs. [3447]) started theiranalysis with the assumed displacement and rotation elds when using the modied couple stress theory. Of course, themodied couple stress theory can also be used without starting the assumed displacement and rotation elds. For example,Park and Gao [48] investigated the variational formulation of the modied couple stress theory and its application to a simpleshear problem. They did not start with the assumed displacement and rotation elds but solved the displacement and rotationelds from the governing equation.

Based on the modied couple stress theory, the strain energy PS of the Mindlin microplate can be written as

PS 1

2

ZLsijeijmijwijdL

1

2

ZLsxxexxsyyeyysxzgxzsxygxysyzgyzdL

12

ZLmxxwxxmyywyymzzwzz2mxzwxz2mxywxy2myzwyzdL (15)

h

a

b x, U

y, Vz, W

Fig. 1. Schematic conguration of a microplate.If PSC and PSNC are used to denote the strain energies associated with classical elastic theory and couple stress theory,respectively, PS can be expressed as

PS PSCPSNC , (16)where

PSC 1

2

ZLsxxexxsyyeyysxzgxzsxygxysyzgyzdL

12

Z a0

Z b0

Mxx@Cx@x

Myy@Cy@y

Mxy@Cx@y

@Cy@x

Qx

@W

@xCx

Qy

@W

@yCy

dxdy, (17)

L.-L. Ke et al. / Journal of Sound and Vibration 331 (2012) 94106 97PSNC 1

2

ZLmxxwxxmyywyymzzwzz2mxywxy2mxzwxz2myzwyzdL

12

Z a0

Z b0

Yxx2

@2W

@x@y @Cy

@x

! Yyy

2

@Cx@y

@2W

@x@y

! Yzz

2

@Cy@x

@Cx@y

(

Yxy2

@2W

@y2 @

2W

@x2 @Cx

@x @Cy

@y

! Txz

2

@2Cy@x2

@2Cx@x@y

! Tyz

2

@2Cy@x@y

@2Cx@y2

!)dxdy: (18)

In Eqs. (17) and (18), the shear forces (Qx, Qy), bending moments (Mxx, Myy, Mxy), couple moments (Yxx, Yyy, Yzz, Yxy), andhigh-order couple moments (Txz, Tyz) are expressed as

Qx,Qy Z h=2h=2

sxz,syzdz, Mxx,Myy,Mxy Z h=2h=2

sxx,syy,sxyzdz, (19)

Yxx,Yyy,Yzz,Yxy Z h=2h=2

mxx,myy,mzz,mxydz, Txz,Tyz Z h=2h=2

mxz,myzzdz: (20)

Assume

A11 Eh

1v2 , A44 A55 ksEh

21u , A66 Eh

21u , (21)

D11 D22 Eh3

121v2 , D12 vEh3

121v2 , D66 Eh3

241v , (22)

where ks5/6 is the shear correction factor of the rectangular microplate; E is Youngs modulus and u is Poissons ratio.Then, Eqs. (19) and (20) can be written as

Qx

Qy

( )

A44 0

0 A55

" #@W=@xCx@W=@yCy

( )(23)

Mxx

Myy

Mxy

8>:

9>=>;

D11 D12 0

D12 D22 0

0 0 D66

264

375

@Cx=@x@Cy=@y

@Cx=@y@Cy=@x

8>:

9>=>; (24)

Yxx

Yyy

Yzz

Yxy

8>>>>>>>:

9>>>>=>>>>;

2A66l21 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

26664

37775

@2W=@x@y@Cy=@x=2@Cx=@y@2W=@x@y=2@Cy=@x@Cx=@y=2

@2W=@y2@2W=@x2@Cx=@x@Cy=@y=4

8>>>>>>>:

9>>>>=>>>>;

, (25)

Txz

Tyz

( ) 2D66l2

1 0

0 1

@2Cy=@x2@2Cx=@x@y=4@2Cy=@x@y@2Cx=@y2=4

( )(26)

The kinetic energy PK is given by

PK 1

2

Z a0

Z b0

I1@W

@t

2 I3

@Cx@t

2 I3

@Cy@t

2" #dxdy, (27)

where

I1 rh, I3 rh3

12: (28)

Using the Hamilton principle Z t00

dPSCdPSNCdPK dt 0, (29)

substituting Eqs. (17), (18), (27) into Eq. (29) leads toZ t00

ZG

@Qx@x

@Qy@x

12

@2Yxx@x@y

@2Yyy@x@y

@2Yxy@y2

@2Yxy@x2

!I1

@2W

@t2

" #dW

@Mxx@x

@Mxy@y

Qx1

2

@Yzz@y

@Yyy@y

@Yxy@x

@2Txz@x@y

@2Tyz@y2

!I3

@2Cx@t2

" #dCx

L.-L. Ke et al. / Journal of Sound and Vibration 331 (2012) 9410698 @Myy@y

@Mxy@x

Qy1

2

@Yxx@x

@Yzz@x

@Yxy@y

@2Txz@x2

@2Tyz@x@y

!I3

@2Cy@t2

" #dCydxdydt

Z t00

IC

Qx1

4Yxx,y

1

4Yyy,y

1

2Yxy,x

nx Qy

1

4Yxx,x

1

4Yyy,x

1

2Yxy,y

ny

dW

12Yxynx

1

4Yxx

1

4Yyy

ny

dW,x

1

4Yxx

1

4Yyy

nx

1

2Yxyny

dW,y

Mxx1

2Yxy

1

4Txz,y

nx Mxy

1

2Yyy

1

2Yzz

1

2Tyz,y

1

4Txz,x

ny

dCx

14TxznydCx,x

1

4Txznx

1

2Tyzny

dCx,y

1

2Txznx

1

4Tyzny

dCy,x

1

4TyznxdCy,x

Mxy1

2Yxx

1

2Yzz

1

4Tyz,y

1

2Txz,x

nx Myy

1

2Yxy

1

4Tyz,x

ny

dCydSdt 0: (30)

where G and C are the region of the microplate in xy plane and the boundary of the microplate in xy plane, respectively.From Eq. (30), one can obtain the following higher-order equations of motion:

@Qx@x

@Qy@x

12

@2Yxx@x@y

@2Yyy@x@y

@2Yxy@y2

@2Yxy@x2

! I1

@2W

@t2, (31)

@Mxx@x

@Mxy@y

Qx12

@Yzz@y

@Yyy@y

@Yxy@x

@2Txz@x@y

@2Tyz@y2

! I3

@2Cx@t2

, (32)

@Myy@y

@Mxy@x

Qy1

2

@Yxx@x

@Yzz@x

@Yxy@y

@2Txz@x2

@2Tyz@x@y

! I3

@2Cy@t2

, (33)

and the corresponding boundary conditions

W 0 or Qx1

4Yxx,y

1

4Yyy,y

1

2Yxy,x

nx Qy

1

4Yxx,x

1

4Yyy,x

1

2Yxy,y

ny 0, (34)

W,x 0 or1

2Yxynx

1

4Yxx

1

4Yyy

ny 0, (35)

W,y 0 or 14Yxx1

4Yyy

nx 1

2Yxyny 0, (36)

Cx 0 or Mxx 12Yxy 1

4Txz,y

nx Mxy 1

2Yyy1

2Yzz 1

2Tyz,y 1

4Txz,x

ny 0, (37)

Cx,x 0 or 14Txzny 0, (38)

Cx,y 0 or 1

4Txznx

1

2Tyzny 0, (39)

Cy 0 or Mxy1

2Yxx

1

2Yzz

1

4Tyz,y

1

2Txz,x

nx Myy

1

2Yxy

1

4Tyz,x

ny 0, (40)

Cy,x 0 or1

2Txznx

1

4Tyzny 0, (41)

Cy,y 0 or1

4Tyznx 0, (42)

where (nx, ny) denote the direction cosines of the outward unit normal to the boundary of the mid-plane. Note the orderof the governing equations is six for the classical Mindlin plate model, while it is increased to twelve for the presentnon-classical microplate model due to the inclusion of the size effect. By neglecting the couple stress terms, Eqs. (31)(33)can reduce to the governing equations for the classical Mindlin plate [49].

Eqs. (34)(45) provides all possible boundary conditions (classical and non-classical) for the size-dependent rectangularmicroplate. These boundary conditions refer to either the prescribed boundary deformations or tractions or a combinationof both. For example, the boundary conditions for a rectangular microplate with all edges clamped (CCCC) are

W Cx Cy W,x Cx,x Cy,x 0, at x 0,a, (43)

L.-L. Ke et al. / Journal of Sound and Vibration 331 (2012) 94106 99W Cx Cy W,y Cx,y Cy,y 0, at y 0,b, (44)and those for a rectangular microplate with all edges simply supported (SSSS) are

W Cy Mxx Yxy Tyz Txz 0, at x 0,a, (45)

W Cx Myy Yxy Tyz Txz 0, at y 0,b, (46)Note that we can also write the other types of boundary conditions for the rectangular microplate according to

Eqs. (34)(42). In the present paper, our concern is to analyze the free vibration characteristics of the CCCC and SSSSmicroplates. Therefore, we do not list all types of boundary conditions for the rectangular microplate for brevity.

It is observed that the governing equations and boundary conditions of the non-classical microplate are more complexthan those of the classical microplate. Analytical solution of the present problem is very difcult. A numerical method thatmakes use of the p-version Ritz method [5052] is employed in this paper to determine the natural frequencies of themicroplate.

3. Solution procedure

By introducing the following dimensionless quantities:

z1 x

a, z2

y

b, w W

h, cx Cx, cy Cy, I1,I3

I1I1

,I3I1h2

, (47)

l0 l

h, d11,d22,d12,d66

D11A11h2

,D22A11h2

,D12A11h2

,D66A11h2

, (48)

a44,a55,a66 A44A11

,A55A11

,A66A11

, Z1

a

h, Z2

b

h, l a

b, (49)

and substituting Eqs. (23)(26), (28) into Eqs. (17), (18) and (27), the strain energy and kinetic energy can be expressed indimensionless form as

PnSC 1

2

Z 10

Z 10

d11@cx@z1

2d22l2

@cy@z2

22ld12

@cx@z1

@cy@z2

d66 l@cx@z2

@cy@z1

2(

a55@W

@z1Z1cx

2a44 l

@W

@z2Z1cy

2)dz1dz2, (50)

PnSNC l202

Z 10

Z 10

a662

1

Z2@2w

@z1@z2 @cy

@z1

!2 a66

2l@cx@z2

1Z2

@2w

@z1@z2

!28:

9>=>; 0, (65)

where Kclassical and Knoclassical are the stiffness matrices corresponding to PnSC and Pn

SNC , respectively, and M is the massmatrix. Kclassical, Knoclassical and M are 3N3N symmetric matrices whose elements are given in Appendix A. Unknownvectors A, B and C are given by

A fA1,A2,. . .AmgT, B fB1,B2,. . .BmgT, C fC1,C2,. . .CmgT, m 1,2,. . .,N, (66)where the superscript T denotes the transposition of matrix. By solving Eq. (65), the dimensionless natural frequencyand the associate mode shapes of the microplate can be obtained.

4. Numerical results

The numerical results for the free vibration of size-dependent microplates with all edges either simply supported (SSSS)or clamped (CCCC) are presented in Table 4 and Figs. 25. The inuences of the dimensionless length scale parameter h/l,side-to-thickness ratio a/h and aspect ratio a/b on the free vibration characteristics are discussed in detail. Unlessotherwise stated, the microplate is made of epoxy with the following material properties: v0.38, r1220 kg/m3,E1.44 GPa and l17.6 mm [11,36]. The material length scale parameter l17.6 mm for the epoxy microplate is based onthe reported experimental work by Lam et al. [11]. They fabricated the micron-sized epoxy beams and conducted thebending tests using a nanoindenter to determine the material length scale parameter which is depended on the materialmicrostructure (e.g. dislocations, lattice defects and voids). It should be pointed out that, so far, no experimental datais available for the microplate to determine the material length scale parameter in open literature. Therefore, theexperimental data for the epoxy microbeam is approximately used for the epoxy microplate to quantitatively analyze thesize effect on the free vibration characteristics of the microplate in the present analysis.

02

4

6

8

10

Present model:12

Classical model:12

Freq

uenc

y (M

Hz)

h/l0

0

2

4

6

8

10

Present model:12

Classical model:12

Freq

uenc

y (M

Hz)

h/l2 4 6 8 10 12 140 2 4 6 8 10 12 14

Fig. 2. The rst two natural frequencies of microplates (b/a1, a/h10) as a function of h/l: (a) SSSS and (b) CCCC.

0

1

2

3

4

5

SSSS: h/l = 1.0 h/l = 1.5 h/l = 2.0 h/l = 3.0

Fund

amen

tal f

requ

ency

1 (

MH

z)

a/h5

0

1

2

3

4

5

6

7

CCCC: h/l = 1.0 h/l = 1.5 h/l = 2.0 h/l = 3.0

Fund

amen

tal f

requ

ency

1 (

MH

z)

a/h10 15 20 25 305 10 15 20 25 30

Fig. 3. The fundamental frequency of microplates (b/a1) as a function of a/h: (a) SSSS and (b) CCCC.

0

1

2

3

SSSS: h/l = 1.0 h/l = 1.5 h/l = 2.0 h/l = 3.0 h/l = 4.0

Fund

amen

tal f

requ

ency

1 (

MH

z)

a/b0

0

1

2

3

4

5

CCCC: h/l = 1.0 h/l = 1.5 h/l = 2.0 h/l = 3.0 h/l = 4.0

Fund

amen

tal f

requ

ency

1 (

MH

z)

a/b1 2 3 4 5 60 1 2 3 4 5 6

Fig. 4. The fundamental frequency of microplates (a/h10) as a function of b/a: (a) SSSS and (b) CCCC.

L.-L. Ke et al. / Journal of Sound and Vibration 331 (2012) 94106 101

L.-L. Ke et al. / Journal of Sound and Vibration 331 (2012) 941061020.0

0.2

0.4

0.6

0.8

1.0

SSSS: h/l = 1.0 h/l = 2.0 h/l = 3.0 classical plate

w

0.00.0

0.2

0.4

0.6

0.8

1.0

CCCC: h/l = 1.0 h/l = 2.0 h/l = 3.0 classical plate

w

0.2 0.4 0.6 0.8 1.00.0 0.2 0.4 0.6 0.8 1.04.1. Verication and convergence

In order to verify the convergence of the present numerical method, Table 1 presents the rst three natural frequenciesof SSSS or CCCC microplates with varying total numbers of the degree of polynomial (ab10h, h3l). Excellentconvergence is observed for both SSSS and CCCC microplates. Good accuracy is achieved when the degree of polynomialp8. Thus, p8 (i.e. N45) is used in all subsequent calculations.

As mentioned before, if the effect of material length scale parameter is neglected, free vibration results of the classicalMindlin plate given by Reddy [49] can be recovered. Table 2 gives the rst three natural frequencies for the SSSS Mindlinplate with ab10h and v0.3. The natural frequency is normalized as oOa2=h

r=E

p. The analytical solutions [49]

are also provided for a direct comparison. Excellent agreement is achieved between the present results and thoseanalytical ones [49].

Table 3 lists the rst three natural frequencies for the SSSS microplate with the size effect. The present results agreewell with the analytical solutions [42] which are based on Kirchhoff microplate model. The difference is observed to bequite small when the side-to-thickness ratio a/hZ20 but relatively large when a/ho20. This is because the transverseshear deformation which is very important in thick (a/h5) or moderately thick (a/h10) microplates is included in theMindlin plate theory but neglected in the Kirchhoff plate theory.

x/a x/a

Fig. 5. Effect of the dimensionless length scale parameter h/l on the fundamental mode shapes of the microplates (b/a1, a/h10) at y/b0.5: (a) SSSSand (b) CCCC.

Table 1The rst three natural frequencies (MHz) with varying total numbers of the degree of polynomial (ab10h and h3l).

p SSSS CCCC

O1 O2 O3 O1 O2 O3

3 9.12554 24.88314 37.87749 16.38935 31.80591 44.44525

4 9.12554 21.72854 37.18004 16.36839 30.60669 44.07486

5 9.12274 21.72155 33.31123 15.95886 30.58992 42.76942

6 9.12274 21.70758 33.29026 15.95607 30.1133 42.72888

7 9.12274 21.70618 33.28607 15.95607 30.10771 42.72190

8 9.12274 21.70618 33.28607 15.95607 30.10771 42.72190

Table 2Comparisons of the rst three dimensionless natural

frequencies for SSSS Mindlin plate (ab10h and v0.3).

Mode no. Present Reddy [49]

1 5.7784 5.7690

2 13.8157 13.7640

3 21.2304 21.1210

L.-L. Ke et al. / Journal of Sound and Vibration 331 (2012) 94106 1034.2. Free vibration analysis

Table 4 shows the effect of dimensionless length scale parameter h/l on the rst three natural frequencies for themicroplate with a/h10. As h/l decreases from 2 to 1, the frequencies increase signicantly for both SSSS and CCCCmicroplates. It is indicated that size dependence of the material properties increases the stiffness of the microplate andhence increases the values of frequencies. These results are quite similar to what have been observed in free vibrationanalysis of Kirchhoff microplates by Yin et al. [42]. Obviously, the frequencies of the CCCC microplate are larger than thoseof the SSSS microplate. In addition, for a given value of a/h, the decrease of the plate thickness can also lead to the increaseof the frequency of the microplate even if the size effect is excluded (i.e. the case for the classical plate theory). This can beveried by Fig. 2. Therefore, the increase for the frequency is caused by the combined effects of the material length scaleparameter and the decrease of the plate thickness with the xed a/h.

Fig. 2 plots the rst two natural frequencies for microplates as a function of h/l with b/a1 and a/h10. The results forthe classical Mindlin microplate [49] are also plotted in Fig. 2 for comparison. The frequencies predicted by the presentnon-classical Mindlin plate model are always higher than those by classical Mindlin plate model. It is observed that the

Table 3Comparisons of the rst three natural frequencies (in MHz) for SSSS microplates with size effect (a/b1 and h2l).

Mode no. a/h5 a/h10 a/h20 a/h30

Present Ref. [42] Present Ref. [42] Present Ref. [42] Present Ref. [42]

1 1.4701 1.6816 0.4042 0.4204 0.1040 0.1051 0.04650 0.04671

2 3.2093 4.2040 0.9603 1.0510 0.2568 0.2627 0.1157 0.1168

3 4.6857 6.7264 1.4720 1.6816 0.4058 0.4203 0.1844 0.1868

Table 4Size effect on the rst three natural frequencies (MHz) for the microplate with a/h10.

Boundary condition h/l b/a1 b/a0.5

O1 O2 O3 O1 O2 O3

SSSS 1.0 1.2431 2.8989 4.3947 2.9223 4.4222 8.2063

1.5 0.6295 1.4889 2.2773 1.4997 2.2919 4.3419

2.0 0.4042 0.9603 1.4720 0.9662 1.4799 2.8214

CCCC 1.0 2.0733 3.8810 5.4493 4.7581 6.0524 10.4102

1.5 1.0769 2.0433 2.8758 2.5244 3.2115 5.5668

2.0 0.6997 1.3356 1.8753 1.6462 2.0966 3.6196difference between the frequencies given by the non-classical Mindlin plate model and classical Mindlin plate model issignicantly large when the thickness of the microplate is small (ho6l), but it becomes insignicant and negligible whenthe thickness of the microplate becomes large (hZ6l), especially for the fundamental frequency at a large value of h/l(h/lZ10). This indicates that the size effect is pronounced and must be taken into account only when the thickness ofmicroplate has a similar value to the material length scale parameter.

Fig. 3 shows the fundamental frequency for the microplate as a function of a/hwith b/a1 and h/l 1.0, 1.5, 2.0 and 3.0.For a given value of a/h, the fundamental frequency increases as h/l decreases. The fundamental frequency of both SSSS andCCCC microplates decreases rapidly with increasing a/h because a higher side-to-thickness ratio indicates that themicroplate is thinner thus has a lower stiffness. The size effect on the fundamental frequency is considerably large whena/hr20 (i.e. thick and moderately thick microplates) but becomes relatively small when a/h420 (i.e., thin microplates).

Fig. 4 examines the fundamental frequency for microplates as a function of b/a with a/h10 and h/l 1.0, 1.5, 2.0, 3.0and 4.0. The fundamental frequency increases rapidly as the aspect ratio a/b increases from 0.5 to 1, and then it changesslightly as a/b increases from 1 to 6. Obviously, the size effect on the microplate with b/ar1 is larger than that on themicroplate with b/a41.

Fig. 5 displays the effect of the dimensionless length scale parameter h/l on the fundamental mode shapes (w) for themicroplate with b/a1 and a/h10 at y/b0.5. The dimensionless length scale parameter h/l nearly has no effect on thefundamental mode shape (w) for the SSSS microplate, but it is relative large for the CCCC microplate. The maximumdisplacement w occurs at the central point of both SSSS and CCCC microplates.

5. Conclusions

Based on the Mindlin plate theory and modied couple stress theory, a size-dependent microplate model whichcontains an internal material length scale parameter related to the material microstructures is developed for the free

bb

Z 1 Z 12 @f

xi@fxj a66 @f

xi@fxj d66 @2f

xi

@2fxj d66l2 @2fxi @

2fxj !

L.-L. Ke et al. / Journal of Sound and Vibration 331 (2012) 94106104Cij 0 0

a66l @z2 @z2

4 @z1 @z14Z1Z2 @z1@z2 @z1@z2

4Z22 @z22 @z

22

dz1dz2,

Cbcij Z 10

Z 10

a66l2

@fyi@z1

@fxj@z2

a66l4

@fyi@z2

@fxj@z1

d664Z21

@2fyi@z21

@2fxj@z1@z2

d66l2

4Z1Z2@2fyi@z1@z2

@2fxj@z22

!dz1dz2,

Cccij Z 10

Z 10

a66@fyi@z1

@fyj@z1

a66l2

4

@fyi@z2

@fyj@z2

d664Z21

@2fyi@z21

@2fyj@z21

d664Z22

@2fyi@z1@z2

@2fyj@z1@z2

!dz1dz2vibration analysis of microplates. The higher-order governing equations and boundary conditions are derived using theHamilton principle. The p-version Ritz method is employed to determine the natural frequencies of the microplate withdifferent boundary conditions. The effects of the length scale parameter, side-to-thickness ratio and aspect ratio on the freevibration characteristics are discussed. It is concluded that the size effect on the free vibration characteristics of themicroplate is signicant when the thickness of microplate has a similar value to the material length scale parameter. Thefrequencies increase signicantly for both SSSS and CCCC microplates as the length scale parameter h/l decreases. The sizeeffect on the fundamental frequency is signicantly large for thick and moderately thick microplates, but becomesrelatively small for thin microplates.

Acknowledgements

The work described in this paper was supported by National Natural Science Foundation of China (No. 11002019), Ph.D.Programs Foundation of Ministry of Education of China (No. 20100009120018) and Fundamental Research Funds for theCentral Universities (No. 2009JBM073).

Appendix A

The symmetric matrices Kclassical, Knoclassical and M have the structures of

Kclassical Kaaij K

abij K

acij

Kbbij Kbcij

Kccij

2664

3775, Knoclassical

Caaij Cabij C

acij

Cbbij Cbcij

Cccij

2664

3775, M

Maaij Mabij M

acij

Mbbij Mbcij

Mccij

2664

3775, (A1)

where i,j 1,2,. . .,N. The elements of symmetric stiffness matrix Kclassical are

Kaaij Z 10

Z 10

a55@fwi@z1

@fwj@z1

a44l2@fwi@z2

@fwj@z2

!dz1dz2, Kabij

Z 10

Z 10a55Z1f

xi

@fwj@z1

dz1dz2,

Kacij Z 10

Z 10a44lZ1f

yi

@fwj@z2

dz1dz2, Kbbij Z 10

Z 10

d11@fxi@z1

@fxj@z1

d66l2@fxi@z2

@fxj@z2

a55Z21fxifxj !

dz1dz2,

Kbcij Z 10

Z 10

d12l@fyi@z2

@fxj@z1

d66l@fyi@z1

@fxj@z2

!dz1dz2,

Kccij Z 10

Z 10

d22l2 @f

yi

@z2

@fyj@z2

d66@fyi@z1

@fyj@z1

a44Z21fyi fyj

!dz1dz2

The elements of symmetric stiffness matrix Knoclassical are

Caaij Z 10

Z 10

a66Z22

@2fwi@z1@z2

@2fwj@z1@z2

a664

l2@2fwi@z22

@2w

@z21

!1

Z22

@2fwi@z22

1Z22

@2fwj@z21

!" #dz1dz2

Cabij Z 10

Z 10

a66l2Z2

@fxi@z2

@2fwj@z1@z2

a66Z14

@fxi@z1

1

Z22

@2fwj@z22

1Z22

@2fwj@z21

!" #dz1dz2;

Cacij Z 10

Z 10

a662Z2

@fyi@z1

@2fwj@z1@z2

a66lZ14

@fyi@z2

1

Z22

@2w

@z22 1Z22

@2fwj@z21

!" #dz1dz2,

L.-L. Ke et al. / Journal of Sound and Vibration 331 (2012) 94106 105The elements of the symmetric mass matrix M are

Maaii Z 10

Z 10I1f

wi f

wj dz1dz2, M

bbii

Z 10

Z 10I3f

xif

xj dz1dz2, M

ccii

Z 10

Z 10I3f

yi f

yj dz1dz2, M

abij Macij Mbcij 0

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L.-L. Ke et al. / Journal of Sound and Vibration 331 (2012) 94106106

Free vibration of size-dependent Mindlin microplates based on the modified couple stress theoryIntroductionTheoretical formulationsSolution procedureNumerical resultsVerification and convergenceFree vibration analysis

ConclusionsAcknowledgementsAppendix AReferences