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Nonlinear free vibration of size-dependent functionally graded microbeams

Liao-Liang Ke a,⇑, 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

Article history:Received 2 December 2010Received in revised form 20 December 2010Accepted 21 December 2010Available online 15 January 2011

Keywords:Functionally graded materialsNonlinear vibrationCouple stress theorySize effectMicrobeam

a b s t r a c t

Nonlinear free vibration of microbeams made of functionally graded materials (FGMs) isinvestigated in this paper based on the modified couple stress theory and von Kármán geo-metric nonlinearity. The non-classical beam model is developed within the framework ofTimoshenko beam theory which contains a material length scale parameter related tothe material microstructures. The material properties of FGMs are assumed to be gradedin the thickness direction according to the power law function and are determined byMori-Tanaka homogenization technique. The higher-order nonlinear governing equationsand boundary conditions are derived by using the Hamilton principle. A numerical methodthat makes use of the differential quadrature method together with an iterative algorithmis employed to determine the nonlinear vibration frequencies of the FGM microbeams withdifferent boundary conditions. The influences of the length scale parameter, material prop-erty gradient index, slenderness ratio, and end supports on the nonlinear free vibrationcharacteristics of the FGM microbeams are discussed in detail. It is found that both the lin-ear and nonlinear frequencies increase significantly when the thickness of the FGM micro-beam is comparable to the material length scale parameter.

� 2010 Elsevier Ltd. All rights reserved.

1. Introduction

Functionally graded materials (FGMs) have received considerable attention because of their novel thermo-mechanicalproperties, which enable them to be widely used in many scientific and engineering fields, such as aerospace, automobile,electronics, optics, chemistry, biomedical engineering, nuclear engineering and mechanical engineering (Miyamoto, Kaysser,Rabin, Kawasaki, & Ford, 1999; Suresh & Mortensen, 1998). With the development of the material technology, FGMs havebeen employed in micro/nano-electro-mechanical system (MEMS/NEMS) and atomic force microscopes (AFMs) to achievehigh sensitivity and desired performance (Fu, Du, Huang, Zhang, & Hu, 2004; Hasanyan1, Batra1, & Harutyunyan, 2008;Lü, Chen, & Lim, 2009a,b; Mahmud, Liu, & Nam, 2008; Witvrouw & Mehta, 2005). Beams are the core structures widely usedin MEMS, NEMS and AFMS with the order of microns or sub-microns, and their properties are closely related to their micro-structures. The size-dependent deformation behavior has been experimentally observed in the micro-torsion test and micro-bending test of the microbeams (Chong & Lam, 1999; Chong, Yang, Lam, & Tong, 2001; Fleck, Muller, Ashby, & Hutchinson,1994; Lam, Yang, Chong, Wang, & Tong, 2003; Stolken & Evans, 1998). Therefore, it is necessary to take into account the sizeeffect resulting from the underlying microstructures in both theoretical and experimental studies of the FGM microbeams.

Since the classical elasticity theory which does not include a material length scale is unable to account for the size effect,various size-dependent continuum theories that can capture size effect, such as couple stress elasticity (Mindlin & Tiersten,

0020-7225/$ - see front matter � 2010 Elsevier Ltd. All rights reserved.doi:10.1016/j.ijengsci.2010.12.008

⇑ Corresponding author. Tel.: +86 10 515684070; fax: +86 10 51682094.E-mail address: llke@bjtu.edu.cn (L.-L. Ke).

International Journal of Engineering Science 50 (2012) 256–267

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1962; Toupin, 1962), nonlocal elasticity (Eringen, 1972), strain gradient elasticity (Aifantis, 1999) and surface elasticity (Gur-tin, Weissmuller, & Larche, 1998), have been proposed and employed to study the mechanical characteristics of structures ateither micro or nano scale (Abbasion, Rafsanjani, Avazmohammadi, & Farshidianfar, 2009; Ke, Xiang, Yang, & Kitipornchai,2009a; Kong, Zhou, Nie, & Wang, 2009; Lazopoulos, 2009; Lü et al., 2009a,b). The present research endeavors to use the cou-ple stress theory to analyze the size effect on the FGM microbeams.

Couple stress theory is one of the higher-order continuum theories which contain material length scale parameters andcan explain microstructure-related size effects. The classical couple stress theory contains two classical and two additionalmaterial constants for isotropic elastic materials (Fleck & Hutchinson, 1997). The two additional constants are related to theunderlying microstructure of the material and are inherently difficult to determine. Yang, Chong, Lam, and Tong (2002) firstproposed the modified couple stress theory involving only one additional material length scale parameter. This featuremakes the modified couple stress theory easier to use. After their pioneering work, this theory has been developed andextensively used in many studies in the past eight years. Park and Gao (2006) and Kong, Zhou, Nie, and Wang (2008) inves-tigated the static and dynamic problems of size-dependent Euler–Bernoulli microbeams. Ma, Gao, and Reddy (2008) devel-oped a microstructure-dependent Timoshenko beam model based on the modified couple stress theory and discussed thestatic bending and free vibration of a simply supported microbeam. Tsiatas (2009) presented a new Kirchhoff plate modelfor the static analysis of the isotropic microplates. Recently, Kahrobaiyan, Asghari, Rahaeifard, and Ahmadian (2010) inves-tigated the size-dependent dynamic characteristics of atomic force microscope microcantilevers. Simsek (2010) analyzed thedynamic problem of an embedded microbeam carrying a moving microparticle. Xia, Wang, and Yin (2010) discussed the sizeeffect on the nonlinear bending, nonlinear vibration and postbuckling of the Euler–Bernoulli microbeams. Asghari, Kahrobai-yan, and Ahmadian (2010a) developed the size-dependent nonlinear Timoshenko beam formulation based on the modifiedcouple stress theory. These previous studies imply that the modified couple stress theory is an effective measure to gain aninsight into the size effect in microscale structures.

It should be noted that the above mentioned investigations dealt with the microbeams and microplates made of homo-geneous materials only. Very limited literature is available for the FGM structures at microscale. Asghari, Ahmadian, Kahro-baiyan, and Rahaeifard (2010b) investigated the free vibration of the FGM Euler–Bernoulli microbeams. They later developedthe formulation of a functionally graded Timoshenko beam based on the modified couple stress (Asghari, Rahaeifard, Kah-robaiyan, & Ahmadian, 2011). In their paper, the static bending and free vibration of the FGM cantilever and simply sup-ported microbeams were investigated. Ke and Wang (2011) examined the size effect on dynamic stability of functionallygraded Timoshenko microbeams. It should be pointed out that the problems for the macroscopical FGM structures have beenextensively studied by many investigators (Fares, Elmarghany, & Atta, 2009; Ke, Yang, & Kitipornchai, 2009c; Ke, Yang, Kitip-ornchai, & Xiang, 2009b; Kitipornchai, Ke, Yang, & Xiang, 2009; Navazi & Haddadpour, 2008) in which the classical contin-uum theory was employed.

This paper investigates the nonlinear free vibration of the FGM microbeams based on the modified couple stress theoryand von Kármán geometric nonlinearity. The non-classical beam model is developed within the framework of Timoshenkobeam theory which contains a material length scale parameter related to the material microstructures. A simple power lawfunction and Mori-Tanaka homogenization technique are employed to model the through-thickness variation in the materialproperties of the FGM microbeam. The higher-order nonlinear governing equations and boundary conditions are derived byusing the Hamilton principle. The differential quadrature method together with an iterative algorithm is conducted to deter-mine the nonlinear vibration frequencies of the FGM microbeams with different boundary conditions. A detailed parametricstudy is carried out to highlight the influences of the length scale parameter, material property gradient index, slendernessratio, and end supports on the nonlinear free vibration characteristics of the FGM microbeams.

2. The modified couple stress theory

In the modified couple stress theory, the strain energy density in an isotropic linear elastic material occupying region Kcan be written as (Yang et al., 2002)

PS ¼12

ZKðr : eþm : vÞdK; ð1Þ

where e is the strain tensor, r is the Cauchy stress tensor, v is the symmetric curvature tensor, and m is the deviatoric part ofthe couple stress tensor. These tensors are defined by Yang et al. (2002)

e ¼ 12½ruþ ðruÞT�; ð2Þ

v ¼ 12½rhþ ðrhÞT�; ð3Þ

r ¼ ktrðeÞIþ 2le; ð4Þ

m ¼ 2l2lv; ð5Þ

L.-L. Ke et al. / International Journal of Engineering Science 50 (2012) 256–267 257

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wherer ¼ ex@@xþ ey

@@yþ ez

@@z; u is the displacement vector; k and l re Lame’s constants (l is also known as shear modulus); l

is a material length scale parameter; h is the rotation vector expressed as

h ¼ 12

curlu: ð6Þ

Obviously, only one length scale parameter is involved in addition to the conventional Lame’s constants in the modifiedcouple stress theory. Note that mathematically, the additional parameter l is the square of the ratio of the curvature modulusto the shear modulus and physically, is a property measuring the effect of the couple stress (Ma et al., 2008). This parametercan be determined from torsion tests of slim cylinders (Chong et al., 2001) or bending tests (Stolken & Evans, 1998) of thinbeams in micron scale.

3. Nonlinear vibration analysis of the FGM microbeams

Fig. 1 shows an FGM microbeam with length L and thickness h that is made from a mixture of ceramics and metals. It isassumed that the top surface (z = �h/2) of the microbeam is metal-rich whereas the bottom surface (z = h/2) is ceramic-rich.The local effective material properties of the FGM microbeam vary in the thickness direction and are estimated through theMori–Tanaka homogenization technique. Let K and V denote the bulk modulus and the volume fraction of the phase mate-rials with the subscripts ‘‘m’’ and ‘‘c’’ standing for metal and ceramic phases, respectively. The effective bulk modulus Ke andshear modulus le can be calculated by (Fares et al., 2009; Navazi & Haddadpour, 2008)

Ke � Km

Kc � Km¼ Vc

1þ VmðKc � KmÞ=ðKm þ 4lm=3Þ ; ð7Þ

le � lm

lc � lm¼ Vc

1þ Vmðlc � lmÞ=½lm þ lmð9Km þ 8lmÞ=6ðKm þ 2lmÞ�: ð8Þ

The volume fractions of ceramic and metal phases are related by Vm + Vc = 1 and Vc is defined by a simple power law functionas

VcðzÞ ¼ ð0:5þ z=hÞn; ð9Þ

where n is the gradient index of the material property.The effective values of Young’s modulus E and Poisson’s ratio t are expressed in terms of Ke and le as

EðzÞ ¼ 9Kele

3Ke þ le; ð10Þ

tðzÞ ¼ 3Ke � 2le

6Ke þ 2le: ð11Þ

The effective mass density q is given by the rule of mixture as

qðzÞ ¼ qcVc þ qmVm: ð12Þ

According to the Timoshenko beam theory, the displacements of an arbitrary point in the beam along the x-, y- and z-axes,denoted by eUðx; y; z; tÞ, eV ðx; y; z; tÞ and fW ðx; y; z; tÞ respectively, are (Ma et al., 2008)eUðx; y; z; tÞ ¼ Uðx; z; tÞ þ zWðx; z; tÞ; eV ðx; y; z; tÞ ¼ 0; fW ðx; y; z; tÞ ¼Wðx; z; tÞ; ð13Þ

where U(x, z, t) and W(x, z, t) are displacement components in the mid-plane; W(x, z, t) is the rotation of the beam cross-sec-tion; and t is time.

The von Kármán type nonlinear strain–displacement relations are given by (Asghari et al., 2010a; Xia et al., 2010)

exx ¼@U@xþ 1

2@W@x

� �2

þ z@W@x

; exz ¼12

@W@xþW

� �; eyy ¼ ezz ¼ exy ¼ eyz ¼ 0: ð14Þ

L

z

h x

Fig. 1. Schematic configuration of an FGM microbeam.

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Putting Eq. (13) into Eq. (6) gives the rotation vector

hy ¼12

W� @W@x

� �; hx ¼ hz ¼ 0: ð15Þ

Substituting Eq. (15) into Eq. (3) yields

vxy ¼14

@W@x� @

2W@x2

!; vxx ¼ vyy ¼ vzz ¼ vxy ¼ vyz ¼ 0: ð16Þ

From Eqs. (1), (14)–(16), the strain energy PS of the FGM Timoshenko microbeam can be written as

PS ¼12

ZKðrijeij þmijvijÞdK ¼ 1

2

ZKðrxxexx þ 2rxzexz þ 2mxyvxyÞdK

¼ 12

Z L

0

Z h=2

�h=2rxx

@U@xþ 1

2@W@x

� �2" #

þ rxxz@W@xþ rxz

@W@xþW

� �þ 1

2mxy

@W@x� @

2W@x2

!( )dzdx

¼ 12

Z L

0Nx

@U@xþ 1

2@W@x

� �2" #

þMx@W@xþ Q x

@W@xþW

� �þ 1

2Yxy

@W@x� @

2W@x2

!( )dx; ð17Þ

where the normal resultant force Nx, shear force Qx, bending moment Mx, and couple moment Yxy are expressed as

Nx ¼Z

ArxxdA ¼ A11

@U@xþ 1

2@W@x

� �2" #

þ B11@W@x

; Q x ¼Z

ArxzdA ¼ ksA55

@W@xþW

� �; ð18aÞ

Mx ¼Z

ArxxzdA ¼ B11

@U@xþ 1

2@W@x

� �2" #

þ D11@W@x

; Yxy ¼Z

AmxydA ¼ 1

212A55

@W@x� @

2W@x2

!; ð18bÞ

where A is the cross-sectional areas of the microbeam; ks = 5/6 is the shear correction factor. The kinetic energy PT is givenby

PT ¼12

Z L

0I1

@U@t

� �2

þ 2I2@U@t

� �@W@t

� �þ I3

@W@t

� �2

þ I1@W@t

� �2" #

dx: ð19Þ

The stiffness components and inertia related terms in Eqs. (18) and (19) are defined as

fA11;B11;D11g ¼Z h=2

�h=2

EðzÞ½1� tðzÞ�½1þ tðzÞ�½1� 2tðzÞ� f1; z; z

2gdz; ð20aÞ

A55 ¼Z h=2

�h=2

EðzÞ2½1þ tðzÞ�dz; fI1; I2; I3g ¼

Z h=2

�h=2qðzÞf1; z; z2gdz: ð20bÞ

The equations for the motion of the FGM Timoshenko microbeam can be derived from the Hamilton’s principleZ t

0ðdPS � dPTÞdt ¼ 0: ð21Þ

Substituting Eqs. (17) and (19) into Eq. (21) and integrating by parts yield the following higher-order equations of motion

@Nx

@x¼ I1

@2U@t2 þ I2

@2W@t2 ; ð22Þ

@Q x

@xþ @

@xNx@W@x

� �þ 1

2@2Yxy

@x2 ¼ I1@2W@t2 ; ð23Þ

@Mx

@x� Q x þ

12@Yxy

@x¼ I2

@2Ui

@t2 þ I3@2Wi

@t2 ; ð24Þ

and the corresponding boundary conditions at beam ends

U ¼ 0 or Nx ¼ 0; x ¼ 0; L; ð25aÞ

W ¼ 0 or Q x þ Nx@W@xþ 1

2@Yxy

@x¼ 0; x ¼ 0; L; ð25bÞ

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W ¼ 0 or Mx þ12

Yxy ¼ 0; x ¼ 0; L; ð25cÞ

@W@x¼ 0 or Yxy ¼ 0; x ¼ 0; L: ð25dÞ

It is observed in Eq. (25) that there are eight boundary conditions in the current non-classical FGM Timoshenko beam modelwhich requires more boundary conditions than the classical FGM Timoshenko beam model (Ke et al., 2009b).

By substituting Eq. (18) into Eqs. (22)–(24) and introducing the following dimensionless quantities

f ¼ xL; ðu;wÞ ¼ ðU;WÞ

h; ðI1; I2; I3Þ ¼

I1

I10;

I2

I10h;

I3

I10h2

!; w ¼ W; g ¼ L

h; ð26aÞ

l0 ¼lh; ða11; a55; b11; d11Þ ¼

A11

A110;

A55

A110;

B11

A110h;

D11

A110h2

!; s ¼ t

L

ffiffiffiffiffiffiffiffiffiA110

I10

s; ð26bÞ

where A110 and I10 are the values of A11 and I1 of a homogeneous metal microbeam (i.e. Vm = 1), the equations of motion canbe expressed in dimensionless form as

a11@2u

@f2 þ1g@w@f

@2w

@f2

!þ b11

@2w

@f2 ¼ �I1@2u@s2 þ�I2

@2w@s2 ; ð27Þ

ksa55@2w

@f2 þ g@w@f

!þ Z0 þ

14g

l20a55

@3w

@f3 �1g@4w

@f4

!¼ �I1

@2w@s2 ; ð28Þ

b11@2u

@f2 þ1g@w@f

@2w

@f2

!þ d11

@2w

@f2 � ksga55@w@fþ gw

� �þ 1

4l20a55

@2w

@f2 �1g@3w

@f3

!¼ �I2

@2u@s2 þ�I3

@2w@s2 ; ð29Þ

where

Z0 ¼a11

g@u@f

@2w

@f2 þ3

2g@w@f

� �2@2w

@f2 þ@2u

@f2

@w@f

" #þ b11

g@2w

@f2

@w@fþ @w@f

@2w

@f2

" #: ð30Þ

The associated boundary conditions can also be written in dimensionless form as

u ¼ w ¼ w ¼ dwdf¼ 0; at f ¼ 0;1; ð31Þ

for a clamped–clamped (C–C) microbeam and

u ¼ w ¼ 0; b11@u@fþ 1

2g@w@f

� �2" #

þ d11@w@f¼ 0;

@w@f� 1

g@2w

@f2 0; at f ¼ 0;1; ð32Þ

for a hinged–hinged (H–H) microbeam.It is found from Eqs. (27)–(29) that the orders of Eqs. (27)–(29) in the current FGM Timoshenko beam model are higher

than those in the classical Timoshenko beam model (Ke et al., 2009b). By setting the material length parameter to be zeroand neglecting the nonlinear terms, Eqs. (27)–(29) reduce to the governing equations for the classical FGM Timoshenkobeam (Ke et al., 2009b). If both effects of material property gradient and nonlinear terms are neglected in Eqs. (27)–(29),the governing equations presented by Ma et al. (2008) can be recovered.

4. Solution technique

The nonlinear vibration of the FGM microbeams is solved with the help of the differential quadrature method (Shu, 2000).Using the discretization rules of the differential quadrature method, displacement components {u, w and w} and their kthderivatives with respect to x can be approximated by

fu;w;wg ¼XN

m¼1

lmðxÞfum;wm;wmg; ð33Þ

@k

@xkfu;w;wg

�����x¼xi

¼XN

m¼1

CðkÞim fum;wm;wmg; ð34Þ

260 L.-L. Ke et al. / International Journal of Engineering Science 50 (2012) 256–267

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where um = u(xm, t); wm = w(xm, t); wm = w(xm, t); N is the total number of nodes distributed along the x-axis; lm(x) is theLagrange interpolation polynomials; and CðkÞim is the weighting coefficients whose recursive formula can be found in (Shu,2000). The sampling points are determined as follows:

x0 ¼ 0; x1 ¼ 0:0001; xN�1 ¼ 0:9999; xN ¼ 1:0; xj ¼12

1� cospðj� 3Þ

N � 4

� �� ; ð35Þ

where j = 3, 4, . . . ,N � 2.Substituting Eqs. (33) and (34) into Eqs. (27)–(29) leads to a set of ordinary differential equations

a11

XN

m¼1

Cð2Þim um þ1gXN

m¼1

Cð1Þim wm

XN

m¼1

Cð2Þim wm

!þ b11

XN

m¼1

Cð2Þim wm ¼ �I1€ui þ�I2 €wi; ð36Þ

ksa55

XN

m¼1

Cð2Þim wm þ gXN

m¼1

Cð1Þim wm

!þ �Z0 þ

14g

l20a55

XN

m¼1

Cð3Þim wm �1gXN

m¼1

Cð3Þim wm

!¼ �I1 €wi; ð37Þ

b11

XN

m¼1

Cð2Þim um þ1gXN

m¼1

Cð1Þim wm

XN

m¼1

Cð2Þim wm

!þ d11

XN

m¼1

Cð2Þim wm � ksa55gXN

m¼1

Cð1Þim wm þ gwi

!

þ 14

l20a55

XN

m¼1

Cð2Þim wm �1gXN

m¼1

Cð3Þim wm

!¼ �I2€ui þ�I3

€wi; ð38Þ

where i = 1, 2, . . . , N; the over dot denotes partial derivative with respect to dimensionless time s;

Z0 ¼a11

gXN

m¼1

Cð1Þim um

XN

m¼1

Cð2Þim wm þ3

2gXN

m¼1

Cð1Þim wm

!2XN

m¼1

Cð2Þim wm þXN

m¼1

Cð2Þim um

XN

m¼1

Cð1Þim wm

24 35þ b11

gXN

m¼1

Cð2Þim wm

XN

m¼1

Cð1Þim wm þXN

m¼1

Cð1Þim wm

XN

m¼1

Cð2Þim wm

!ð39Þ

The associated boundary conditions can be approximated in the similar way. For a C–C FGM microbeam,

u1 ¼ w1 ¼ w1 ¼ 0;XN

m¼1

Cð1Þ1mwm ¼ 0; at f ¼ 0; ð40aÞ

uN ¼ wN ¼ wN ¼ 0;XN

m¼1

Cð1ÞNmwm ¼ 0; at f ¼ 1; ð40bÞ

and for an H–H FGM microbeam,

u1 ¼ w1 ¼ 0;XN

m¼1

Cð1Þ1mwm �1gXN

m¼1

Cð2Þ1mwm ¼ 0;

b11

XN

m¼1

Cð1Þ1mum þ1

2gXN

m¼1

Cð1Þ1mwm

!224 35þ d11

XN

m¼1

Cð1Þ1mwm ¼ 0; at f ¼ 0; ð41aÞ

uN ¼ wN ¼ 0;XN

m¼1

Cð1ÞNmwm �1gXN

m¼1

Cð2ÞNmwm ¼ 0;

b11

XN

m¼1

Cð1ÞNmum þ1

2gXN

m¼1

Cð1ÞNmwm

!224 35þ d11

XN

m¼1

Cð1ÞNmwm ¼ 0; at f ¼ 1; ð41bÞ

Denoting the unknown dynamic displacement vector as

d ¼ fuigT ; fwigT ; fwigT

n oT; i ¼ 1;2; � � �N; ð42Þ

Eqs. (36)–(38) and the associated boundary conditions (40) and (41) can be expressed in the matrix form as

KL þ12

KNL1 þ13

KNL2

� �dþM€d ¼ 0; ð43Þ

where M is the mass matrix; KL is the linear stiffness matrix; KNL1 and KNL2 are nonlinear stiffness matrices that are linearand quadratic functions in d, respectively. KL, KNL1, KNL2 and M are 3N � 3N symmetric matrices.

L.-L. Ke et al. / International Journal of Engineering Science 50 (2012) 256–267 261

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Expand the dynamic displacement vector d in the form of

d ¼ d�eixl; ð44Þ

where x ¼ XLffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiI10=A110

prepresents the dimensionless frequency; X is the nonlinear vibration frequency of the FGM micro-

beams; d� ¼ ffu�i gT; fw�i g

T; fw�i g

TggT is the nonlinear vibration mode shape vector. Then, substituting Eq. (44) into Eq. (43)yields the nonlinear eigenvalue equations as follows

KL þ12

KNL1 þ13

KNL2

� �d� �x2Md� ¼ 0: ð45Þ

Previous studies (Ke, Yang, & Kitipornchai, 2010; Kitipornchai et al., 2009) have shown that due to the presence of bend-ing–extension coupling effect in the FGM beams (i.e.B11 – 0), the macroscopical FGM beams vibrate with the different ampli-tudes at positive and negative cycles. It is believed that FGM microbeams have the similar nonlinear vibration behavior sincethey also have this coupling effect. The numerical method suggested by Kitipornchai et al. (2009) will then be used to solvethe nonlinear free vibration of the FGM microbeams. Based on the fact that the energy required in each deflection cycle is thesame, the nonlinear frequency of the FGM microbeams can be obtained by computing the period of both positive and neg-ative deflection cycles. Eq. (45) can be solved by employing a direct iterative algorithm described below (Kitipornchai et al.,2009):

Step 1. The linear eigenvalue and the associated eigenvector are obtained from Eq. (45) by neglecting the nonlinear matricesKNL1 and KNL2. We first assume that the given amplitude wmax is positive. Then, the eigenvector is appropriatelyscaled up such that the maximum transverse displacement is equal to the given vibration amplitude wmax.

Step 2. Using the eigenvector to calculate KNL1 and KNL2, a new eigenvalue and eigenvector are obtained from the updatedeigensystem (45).

Step 3. The eigenvector is scaled up again and step 2 is repeated until the relative error between the eigenvalues obtainedfrom two consecutive iterations is within 0.1%. Then, the nonlinear half-cycle frequency x1 is obtained for the posi-tive deflection cycle. The strain energy PþS max in the positive deflection cycle is computed from Eq. (17).

Step 4. Given a negative vibration amplitude wmin, repeat steps 1-3 to find the strain energy P�S max in the negative deflectioncycle. If P�S max ¼ PþS max, the nonlinear half-cycle frequency x2 is obtained for the negative deflection cycle; otherwisenew value of wmin must be chosen and the iteration procedure is continued till P�S max ¼ PþS max:

Table 2Effect of the dimensionless length scale parameter h/l on nonlinear frequency ratio xNL/xL for the FGM microbeams with n = 2.0 and L/h = 12.

Boundary conditions h/l xL wmax

0.2 0.4 0.6 0.8 1.0

H–H Classical beam 0.3638 1.0284 1.1751 1.4293 1.7136 2.012210 0.3701 1.0283 1.1693 1.4134 1.6934 1.98536 0.3812 1.0275 1.1595 1.3906 1.6598 1.94203 0.4285 1.0248 1.1277 1.3127 1.5370 1.78082 0.4968 1.0209 1.0972 1.2348 1.4119 1.60961.5 0.5784 1.0169 1.0740 1.1749 1.3101 1.46681.0 0.7625 1.0108 1.0445 1.1024 1.1827 1.2811

C–C Classical beam 0.7401 1.0155 1.0603 1.1303 1.2202 1.325510 0.7548 1.0150 1.0580 1.1257 1.2128 1.31516 0.7801 1.0141 1.0546 1.1183 1.2009 1.29803 0.8885 1.0108 1.0424 1.0927 1.1589 1.23782 1.0416 1.0079 1.0311 1.0685 1.1183 1.17881.5 1.2194 1.0058 1.0228 1.0504 1.0877 1.13341.0 1.6022 1.0034 1.0132 1.0293 1.0514 1.0789

Table 1Comparisons of first five natural frequencies (MHz) for isotropic homogeneous microbeams with consideration of size effect.

Mode h/l = 10 h/l = 3 h/l = 1

Present Ma et al. (2008) Present Ma et al. (2008) Present Ma et al. (2008)

1 0.03765 0.03765 0.1391 0.1391 0.6724 0.67232 0.1397 0.1396 0.5163 0.5163 2.4533 2.45303 0.2838 0.2838 1.0519 1.0519 4.9374 4.93684 0.4514 0.4514 1.6814 1.6814 7.8615 7.86065 0.6306 0.6306 2.3677 2.3676 11.1057 11.1045

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After the nonlinear half-cycle frequencies x1 and x2 have been obtained through the above described process, the cor-responding half-cycle periods can be given as T1 = p/x1 and T2 = p/x2. Finally, the nonlinear frequency of the FGM micro-beams is computed from

x ¼ 2pT1 þ T2

: ð46Þ

5. Numerical results

Numerical results for the nonlinear free vibration of H–H and C–C FGM microbeams are presented in Tables 2–4 and Figs.2–4. The effects of the dimensionless length scale parameter h/l, material gradient index n, slenderness ratio L/h, and endsupported conditions on the nonlinear free vibration of the FGM microbeams are discussed in detail. The FGM microbeamsconsidered here consist of aluminum (Al) and ceramic (SiC) with the material properties (Fares et al., 2009) Em = 70 GPa,vm = 0.3, qm = 2702 kg/m3 for Al and Ec = 427 GPa, vc = 0.17, qc = 3100 kg/m3 for SiC. The top surface of the beam is metal-richwhereas the bottom surface is ceramic-rich. It should be pointed out that the length scale parameter of a homogeneousepoxy beam has been experimentally obtained as l = 17.6 lm by Lam et al. (2003). However, so far, no experimental datais available for the FGM microbeams in open literature. In order to quantitatively analyze the size effect of the FGM micro-beams, the length scale parameters of the FGM microbeams are approximately assumed to be l = 15 lm in the followingexamples.

It has been mentioned above that if both effects of material gradient and nonlinearity are neglected, the linear free vibra-tion of the homogeneous microbeams presented in Ma et al. (2008) can be recovered. Table 1 lists the first five natural fre-quencies for H–H isotropic homogeneous Timoshenko microbeam with j = 5/6, L/h = 10, v = 0.38, q = 1220 kg/m3,E = 1.44 GPa and l = 17.6 lm. The analytical solutions given by Ma et al. (2008) are also provided for direct comparison.The present results agree very well with the analytical solutions (Ma et al., 2008).

Table 2 shows the effect of the dimensionless length scale parameter h/l on the nonlinear frequency ratio xNL/xL for theFGM microbeams with n = 2.0 and L/h = 12. xNL and xL are the dimensionless nonlinear and linear frequencies, respectively.All these beams exhibit a typical ‘hard-spring’ behavior, i.e., the nonlinear frequency ratio increases as the vibration ampli-tude is increased. It is observed that both the linear and nonlinear frequencies of the size-dependent Timoshenko beam mod-el are larger than those of the classical Timoshenko beam model. This is because the size dependence of the materialproperties increases the bending rigidity of the microbeams and hence increases the values of frequencies (Kong et al.,2008). A decrease in h/l from 6.0 to 1.0 leads to a higher linear frequency and lower nonlinear frequency ratio for C–C

Table 3Effect of material property gradient index n on nonlinear frequency ratio xNL/xL for the FGM microbeams with h/l = 2 and L/h = 12.

Boundary conditions n xL wmax

0.2 0.4 0.6 0.8 1.0

H–H SiC 0.8006 1.0280 1.1075 1.2287 1.3805 1.55380.6 0.5869 1.0238 1.1056 1.2446 1.4236 1.62241.2 0.5328 1.0211 1.0984 1.2378 1. 4170 1.61682.0 0.4968 1.0209 1.0972 1.2348 1.4119 1.6096Al 0.3186 1.0340 1.1265 1.2671 1.4407 1.6372

C–C SiC 1.7518 1.0071 1.0281 1.0619 1.1072 1.16250.6 1.2482 1.0080 1.0313 1.0688 1.1188 1.17951.2 1.1201 1.0081 1.0315 1.0692 1.1195 1.18052.0 1.0416 1.0079 1.0311 1.0685 1.1183 1.1788Al 0.6901 1.0086 1.0336 1.0739 1.1274 1.1922

Table 4Effect of the slenderness ratio L/h on nonlinear frequency ratio xNL/xL for the FGM microbeams with n = 2.0 and h/l = 2.

Boundary conditions L/h xL wmax

0.2 0.4 0.6 0.8 1.0

H–H 8 0.7318 1.0217 1.1006 1.2408 1.4199 1.618310 0.5924 1.0212 1.0982 1.2365 1.4147 1.612012 0.4968 1.0209 1.0972 1.2348 1.4119 1.609616 0.3748 1.0207 1.0964 1.2334 1.4097 1.6067

C–C 8 1.4754 1.0087 1.0340 1.0746 1.1284 1.193410 1.2234 1.0083 1.0322 1.0706 1.1218 1.183812 1.0416 1.0079 1.0311 1.0685 1.1183 1.178816 0.7993 1.0076 1.0302 1.0666 1.1151 1.1740

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and H–H FGM microbeams. However, both linear frequency and nonlinear frequency ratios for h/l = 10 are almost equal tothose computed from the classical Timoshenko beam model. These results indicate that the size effect is large only when thethickness of the FGM microbeam is comparable to the material length scale parameter (h 6 6l), but is diminishing when thethickness of the microbeam becomes larger (h P 10l).

Table 3 investigates the effect of material property gradient index n on nonlinear frequency ratio xNL/xL for the FGMmicrobeams with h/l = 2 and L/h = 12. The results for homogeneous SiC microbeams and Al microbeams are also listed.The linear frequency of H–H and C–C FGM microbeams decreases with increasing gradient index. At the given amplitude,an increase in n leads to decreased xNL/xL of H–H FGM microbeam while for H–H FGM microbeam, xNL/xL increases at firstand then decreases slightly as n increases from 0.6 to 2. Obviously, both linear frequency and nonlinear frequencies of theFGM microbeams are larger than those of the Al microbeams and smaller than those of the SiC microbeams.

Table 4 examines the effect of the slenderness ratio L/h on the nonlinear frequency ratio xNL/xL for the FGM microbeamswith n = 2.0 and h/l = 2. With an increase in the slenderness ratio L/h from 8 to 16, both linear frequency and nonlinear fre-quency ratios decrease. For the long FGM microbeams (L/h P 12), in particular, the effect of L/h on the nonlinear frequencyratio is very small and negligible.

Fig. 2 reveals the effect of the material gradient index n on the nonlinear frequency ratio versus amplitude curves for theFGM microbeams with h/l = 2 and L/h = 12. At the vibration amplitudes of the same magnitude but opposite sign, nonlinearfrequency ratios of H–H FGM microbeam (n = 0.6, 1.2, 2.0) are seen to be different, i.e., the curves are unsymmetrical. Thisindicates that the nonlinear frequency ratio of H–H FGM microbeams is dependent on not only the magnitude but alsothe sign of the vibration amplitude. The reason is that due to the bending–stretching coupling effect (i.e.B11 – 0), the energybalance equation does not yield equal and opposite roots. For C–C FGM microbeam and homogeneous microbeams, however,the nonlinear frequency ratio is independent on the sign of the vibration amplitude and their curves are symmetrical. Theseresults are quite similar to what have been observed in nonlinear free vibration analysis of the macroscopical FGM beams (Keet al., 2010; Kitipornchai et al., 2009).

(a)

(b)

-1.2 -0.9 -0.6 -0.3 0.0 0.3 0.6 0.9 1.21.0

1.1

1.2

1.3

1.4

1.5

1.6

1.7H-H:

SiC n = 0.6 n = 1.2 n = 2.0 Al

wmin

ω NL/ ω

L

wmax

-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.01.00

1.05

1.10

1.15

1.20C-C:

SiC n = 0.6 n = 1.2 n = 2.0 Al

wmin

ω NL/ ω

L

wmax

Fig. 2. Effect of material property gradient index n on nonlinear frequency ratio versus amplitude curves for the FGM microbeams with h/l = 2 and L/h = 12:(a) H–H; and (c) C–C.

264 L.-L. Ke et al. / International Journal of Engineering Science 50 (2012) 256–267

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Fig. 3 plots the nonlinear frequency ratio versus the amplitude curves for the FGM microbeams (n = 2.0, L/h = 12) with thedifferent dimensionless length scale parameter h/l. The results in this figure are consistent with those in Fig. 2. At a givenamplitude (positive or negative), the nonlinear frequency ratio increases with an increase in h/l.

Fig. 4 displays the effect of the dimensionless length scale parameter h/l on the fundamental nonlinear mode shapes forthe FGM microbeams with n = 2, L/h = 12 and wmax = 0.8. It is found that the length scale parameter h/l has an insignificanteffect on the nonlinear mode shape for all microbeams. The maximum amplitude occurs at the midpoint of the H–H and C–Cmicrobeams.

6. Conclusions

The nonlinear free vibration of the FGM microbeams is investigated in this paper based on the modified couple stress the-ory and Timoshenko beam theory. Mori-Tanaka homogenization technique are employed to model the through-the-thick-ness variation of the material properties in a simple power law function. The higher-order governing equations andboundary conditions are derived by using the Hamilton principle. The differential quadrature method is employed to discret-ize the nonlinear governing equations which are then solved by a direct iterative algorithm to obtain the nonlinear vibrationfrequencies of the FGM microbeams. The influences of the length scale parameter, material gradient index and slendernessratio on the nonlinear free vibration characteristics of the FGM microbeams are discussed.

It is found that

(1) The size effect on the nonlinear free vibration characteristics is large only when the thickness of the FGM microbeam iscomparable to the material length scale parameter, but it becomes insignificant as the thickness increases;

(2) An increase in length scale parameter h/l leads to a higher linear frequency and a lower nonlinear frequency ratio forC–C and H–H FGM microbeams;

(a)

(b)

-1.2 -0.9 -0.6 -0.3 0.0 0.3 0.6 0.9 1.21.0

1.2

1.4

1.6

1.8

2.0H-H:

classical beam h/l = 6 h/l = 3 h/l = 2 h/l = 1

wmin

ωN

L/ωL

wmax

-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.01.0

1.1

1.2

1.3

1.4C-C:

classical beam h/l = 6 h/l = 3 h/l = 2 h/l = 1

wmin

ωN

L/ωL

wmax

Fig. 3. Effect of the dimensionless length scale parameter h/l on nonlinear frequency ratio versus amplitude curves for the FGM microbeams with n = 2.0and L/h = 12: (a) H–H; and (c) C–C.

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(3) As the gradient index n increases, both linear frequency and nonlinear frequency ratio decrease for H–H and C–C FGMmicrobeams;

(4) Nonlinear frequency ratio of H–H FGM microbeam is dependent on the sign of the vibration amplitudes, i.e., their non-linear frequency ratio versus amplitude curves are unsymmetrical;

(5) The nonlinear frequency ratios of C–C FGM microbeam and all homogeneous microbeams are independent on the signof the vibration amplitude and their curves are symmetrical.

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).

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(a)

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