Couple stress formulation. The solution for the static bending is obtained numerically. The solution for

icro-micrEMS irimen

iors which occur in micron- and sub-micron-scale structures. However, some non-classical continuum theories such as high-er-order gradient theories and the couple stress theory have been developed such that they are acceptably able to interpretthe size-dependencies.

In 1960s some researchers such as Mindlin, Touplin and Koiter introduced the couple stress elasticity theory whichproposes two higher-order material length scale parameters in addition to the two classical Lame constants [1214] in

0020-7225/$ - see front matter 2010 Elsevier Ltd. All rights reserved.

Corresponding author. Tel.: +98 21 66165523; fax: +98 21 66000021.E-mail address: asghari@sharif.edu (M. Asghari).

International Journal of Engineering Science 48 (2010) 17491761

Contents lists available at ScienceDirect

International Journal of Engineering Sciencedoi:10.1016/j.ijengsci.2010.09.025Fleck et al. [7] indicated that decrease of wires diameter results in a noteworthy enhancement of the torsional hardening.Stolken et al. [8] reported a notable increase of plastic work hardening caused by the decrease of beam thickness in the mi-cro-bending test of thin nickel beams. Also, size-dependent behaviors have been detected in some kinds of polymers. Forinstance, during micro-bending tests of beams made of epoxy polymers, Lam et al. [9] observed a signicant enhancementof bending rigidity caused by the beam thickness reduction. McFarland et al. [10] detected a considerable difference betweenthe stiffness values predicted by the classical beam theory and the stiffness values obtained during a bending test of poly-propylene micro-cantilever. These experiments reveal that the size-dependent behavior is an inherent property of materialswhich appears for a beam when the characteristic size such as thickness or diameter is close to the internal material lengthscale parameter [11].

The classical continuum mechanics theories are not capable of prediction and explanation of the size-dependent behav-Timoshenko beamNonlinear analysis

1. Introduction

Microbeams are widely used in msensors [1], electro-statically excitedthickness of beams used in MEMS, Nbehaviors in micro-scales have expethe free-vibration is presented analytically utilizing the method of multiple scales, oneof the perturbation techniques.

2010 Elsevier Ltd. All rights reserved.

and nano-electromechanical systems (MEMS and NEMS) such as vibration shocko-actuators [24], micro-switches [5] and atomic force microscopes (AFM) [6]. Thes in the order of microns and sub-microns. The size-dependent static and vibrationtally been validated. For example in the micro-torsion test of thin copper wires,A nonlinear Timoshenko beam formulation based on the modiedcouple stress theory

M. Asghari , M.H. Kahrobaiyan, M.T. AhmadianDepartment of Mechanical Engineering, Sharif University of Technology, Tehran, Iran

a r t i c l e i n f o

Article history:Received 10 July 2010Received in revised form 23 August 2010Accepted 24 September 2010

Keywords:Non-classic continuum theories

a b s t r a c t

This paper presents a nonlinear size-dependent Timoshenko beam model based on themodied couple stress theory, a non-classical continuum theory capable of capturing thesize effects. The nonlinear behavior of the new model is due to the present of inducedmid-plane stretching, a prevalent phenomenon in beams with two immovable supports.The Hamilton principle is employed to determine the governing partial differential equa-tions as well as the boundary conditions. A hingedhinged beam is chosen as an exampleto delineate the nonlinear size-dependent static and free-vibration behaviors of the derived

journal homepage: www.elsevier .com/locate / i jengsci

1750 M. Asghari et al. / International Journal of Engineering Science 48 (2010) 17491761the constitutive equations. As some applications, Zhou and Li [15] employed this theory to investigate the static and dynamictorsion of a micro-bar. Also, the size-effects in Timoshenko beamsmodeled on the basis of the couple stress theory have beeninvestigated by Asghari et al. [16].

Yang et al. [17] argued that in addition to the classical equilibrium equations of forces and moments of forces, anotherequilibrium equation should be considered for the material elements. This additional equation is the equilibrium of mo-ments of couples. Then, they concluded that this additional equilibrium equation implies the symmetry of the couple stresstensor. Accordingly, they modied the constitutive equations of the couple stress theory and present the new constitutiveequations with only one material length scale parameter.

In order to determine the length scale parameter l for a specic material, some typical experiments such as micro-bendtest, micro-torsion test and specially micro/nano-indentation test can be carried out (see [7,8,10,1820]).

The modied couple stress theory has been utilized to develop the size-dependent formulations for beams by someresearchers. For example, Park and Gao [21] analyzed the static mechanical properties of an EulerBernoulli beam modeledon the basis of the modied couple stress theory and interpreted the outcomes of epoxy polymeric beam bending test. Also,Kong et al. [11] studied the natural frequencies of the beam based on the modied couple stress theory. In addition, the size-dependent natural frequencies of uid-conveying microtubes [22], the size-dependent buckling behavior of micro-tubules[23] and the size-dependent resonant frequencies and sensitivities of AFM microcantilevers [24] have been investigatedbased on the modied couple stress theory. A new Timoshenko beam model based on the modied couple stress theorywas formulated by Ma et al. [25]. They assessed the size-dependent static and free-vibration behavior of a simply-supportedTimoshenko beam as a case study.

In beams used in MEMS and NEMS with two immovable supports, we face with the nonlinear phenomena in the largeamplitude deections. The source of the nonlinearity is the induced mid-plane stretching during the transverse deections.This nonlinearity causes the static and vibration results to be changed signicantly [2,2628]. Hence, the abovementionedlinear investigations on the couple stress beams are not appropriate in these conditions. Recently, a size-dependent nonlin-ear EulerBernoulli beam model has been presented by Xia et al. [29] on the basis of the modied couple stress theory. Theystudied the nonlinear size-dependent static bending, buckling and the free vibration of beams.

Considering the presentation of the linear modied couple stress EulerBernoulli beam [11], the linear modied couplestress Timoshenko beam [25], and the nonlinear modied couple stress EulerBernoulli beam [29] formulations, our goal inthis paper is the establishing of the nonlinear modied couple stress Timoshenko beam formulation as the next step in thesequential mentioned works. This paper rigorously derive the governing equations and boundary conditions for the modiedcouple stress based nonlinear Timoshenko beam. The formulation of the manuscript is theoretically more complicated andconsequently possesses a more theoretical value. On the other hand, this complexity makes it capable to produce moreappropriate results and simulations, as will be discussed in the following.

The derived formulation of the manuscript enjoys the following three properties together, for the rst time in theliterature:

(A) The size of elements in micro- and nano-electromechanical systems (MEMS and NEMS) is very small and as aresult, using non-classical continuum theories such as the modied couple stress theory for modeling thematerial behavior is crucial for getting appropriate and accurate results in analyzing or designing these elements.Accordingly, in recent years many works have been presented in the literature based on the modied couplestress theory (e.g. [15,16,2124]). In other words, the experimentally validated small scale effects [1114]are very important in MEMS and NEMS elements and classical theories are not capable to treat themappropriately.

(B) It has experimentally been observed that the effects of nonlinearities are very signicant on the behavior of micro-and nanomechanical resonators, even in not so much large amplitudes [3032]. Hence, using the nonlinear formu-lations for simulating and designing of such MEMS devices seems to be essential such that many researchers havebeen attracted to study the nonlinear effects. As an example, Zhang et al. [33] has shown, both analytically andexperimentally, that the nonlinearities have a large impact on the dynamic response of the micro-resonant oscilla-tors. As another example, Adams et al. [34] have investigated nonlinear effects on the harmonic resonance of theseelements.

(C) Utilizing the Timoshenko beam, the effects of shear deformation and the rotary inertia are included in the formulation,while these effects are ignored in EulerBernoulli beams. These effects are not negligible in even thin beams that arevibrating at high frequencies. Due to their dimensions, resonance frequencies of micro- and nano-scale resonators areextremely high, and consequently modeling them by Timoshenko model have great merits over the EulerBernoullimodel to produce accurate and reliable results. As an example, the existence of the resonance frequency due to theshear deformation have been experimentally validated by Barr (see [35]), while the EulerBernoulli model is not capa-ble to predict it, but the Timoshenko theory is.

Because of possessing the three mentioned important properties, the established formulation of this paper is more appro-priate than the available formulations. Using the derived formulations, the nonlinear size-dependent static and free-vibra-tion behaviors of a hingedhinged micro-beam are assessed. The obtained results are compared with those corresponding tothe linear modied couple stress theory as well as linear and nonlinear classical theories.

2. Preliminaries

In the modied couple stress theory, the strain energy density u for a linear elastic isotropic material in innitesimaldeformation is written as [17]

u 12rijeij mijvij i; j 1;2;3; 1

where

rij kemmdij 2leij; 2

eij 12 ruij ruTij; 3

mij 2l2lvij; 4

vij 12rhij rhTij; 5

in which rij, eij,mij and vij denote the components of the symmetric part of stress tensor r, the strain tensor e, the deviatoricpart of the couple stress tensor m and the symmetric part of the curvature tensor v, respectively. Also, u and h are the dis-placement vector and the rotation vector. The two Lame constants and the material length scale parameter are representedby k, l and l, respectively. The components of the innitesimal rotation vector hi are related to the components of the dis-placement vector eld ui as [11]

hi 12 curlui: 6

Consider a uniform homogeneous initially straight at rest beam along the x-direction with length L, initially under aknown axial load N0. The top and bottom surfaces are perpendicular to the z-direction. The centroid of each section is as-sumed to lie on the plane z = 0. The coordinate system, kinematic parameters and loading of a Timoshenko beam modeledon the basis of the modied couple stress theory are illustrated in Fig. 1. For a Timoshenko beam, the displacement eld afterthe loading (noting that the initial axial loading is not included in the loading) assumed as follows [36]

wheretions r

M. Asghari et al. / International Journal of Engineering Science 48 (2010) 17491761 1751Fig. 1. A Timoshenko beam, (a) kinematic parameters (b) loading, geometry and coordinate system.ux, uy and uz represent the displacements along x, y and z axes, respectively. Indeed, it is assumed that all cross sec-emain plane after deformation; however, they can undergo a rigid body displacement in xz plane and also a rotationux ux; t zwx; t; uy 0; uz wx; t; 7

aboutnotes

Inparambodytions

3. Gov

Byi.e. thexpre

Also, c

Substiture t

whereatoric

To obt

gura

1752 M. Asghari et al. / International Journal of Engineering Science 48 (2010) 174917612 0 A @t @t @t

Ubs ZA

udV 12

Z L0

ZArijeij mijvijdAdx

12

Z L0

ZA

E@u@x

z @w@x

12

@w@x

2 !28

It is noaxial lrxx + N ( )

1 @w @u 1 @w @u @w @w lAl @ w @w

where

The w

whereforce ostress

Th

Substi

@ w @w lAl @ w @ w 1 @ w

@x @x 4 @x @x @x @x 2 @x @x 2 @x @t

Eqs. (2the m

HeqA@2u

M. Asghari et al. / International Journal of Engineering Science 48 (2010) 17491761 1753EA@u@x

12

@w@x

2 ! k1t)

REA u

Z x0

12

@w@x

2dx

! xk1t k2t; 292lAl

@x2

@xMm

x0;L 0 or dhyjx0;L 2 d @x w

x0;L

0: 28

2)(24) represent the governing motion equations of a nonlinear size-dependent Timoshenko beammodeled based onodied couple stress theory and Eqs. (25)(28) express the corresponding boundary conditions.re, the axial body force f is neglected and as a consequence N0 is independent of x. Moreover, if the longitudinal inertia/@t2 is neglected, Eq. (22) can be double-integrated with respect to x as1 2 @2w @w" # ! 1 @w EI@w@x

Mrx0;L

0 or dwjx0;L 0; 27x0;L

lA @w

@x w

N0 EA @u

@x 12

@w@x

2" #( )@w@x

lAl2

4@3w@x3

@2w

@x2

! 12c V

! 0 or dwjx0;L 0; 26

N0 EA @u

@x 12

@w@x

2" # N

!x0;L

0 or dujx0;L 0; 25lA @2w2

@w !

lAl2

@4w4

@3w3

! @ N0 EA @u 1 @w

2 !" #@w

( ) 1 @c q m0 @

2w2 ; 24EI@x2

lA@x

w 4 @x3

@x2

2c m2

@t2; 23U 2 0

EI@x

EA@x

2 @x: N0 2 @x @x lA @x w 4 @x2 @x ;dx; 18

m0 ZAqdA qA; I

ZAz2 dA; m2

ZAqz2 dA qI: 19

ork done by external loads including body forces, body couples and boundary surface tractions is expressed as

dW Z L0f du qdw cdhydx N du V dwMr dwMm dhy

xLx0

; 20

N; V ; Mr and Mm are, respectively, the axial resultant force of normal stresses rxx + N0/A, the transverse resultantf shear stresses, the resultant moment of normal stresses rxx, and the resultant moment around y-axis due to couplees mxy at sections.e Hamilton principle can be written asZ t2

t1

dT dU dWdt 0: 21

tution of Eqs. (17), (18) and (20) into (21) results in

@

@xN0 EA @u

@x 12

@w@x

2 !" # f m0 @u

2

@t2; 22

2 2 3 2 ! 2T 12

Z L0

m0@u@t

2m2 @w

@t

2m0 @w

@t

2dx; 17

Z L 2 2 !28< 2" # 2 2 2 !29=ted that the reference conguration is the one coincident with the status of the beam having length L and carrying theoad N0. As a result, the component rxx is not the total stress acting on beam elements. The total normal axial stress is0/A. Since

RA zdA 0, the kinetic energy Tand the total potential energy U = Ubs + Uis can be rewritten as

where k1 and k2 are some functions. In immovable supports cases, we have u(0, t) = u(L, t) = 0, and in these conditions Eq. (29)result

So, on

Indcauseand is

Sub

N@w lA @w w lAl @ w @ w 1 c V 0 or dwj 0; 33

Eqs. (beam

m L

wherethe corresponding EulerBernoulli beam with immovable ends. For example, for a hingedhinged beam j is taken as j = p

ing the governing equations and boundary conditions for a Timoshenko beam with a uniform rectangular cross-section withheigh

x0;1

1754 M. Asghari et al. / International Journal of Engineering Science 48 (2010) 17491761j4E h @~x2

@~xMm

~x0;1 0; or dhyj~x0;1 2 d @~x w ~x0;1 0; 406l l

2@2 ~w @w" #

~

! 1 @ ~w 1j4

@w@~x

eMr ~

0 or dwj~x0;1 0; 39~N@~x

j4E h @~x

w j4E h @~x3

@~x2

2~c ~V

~x0;1 0 or d ~wj~x0;1 0; 38

1j4

Lr

2@2w@~x2

12lj4E

Lr

2 Lh

2@ ~w@~x

w

3lj4E

Lr

2 lh

2@3 ~w@~x3

@2w

@~x2 12

Lr

2~c @

2w@s2

; 37

@ ~w 12l L 2

@ ~w

3l l 2

@3 ~w @2w !

1! t h and width b (see Fig. 1b) in terms of the newly dened variables yields

12lj4E

Lh

2@2 ~w@~x2

@w@~x

! 3lj4E

lh

2@4 ~w@~x4

@3w

@~x3

! eN @2 ~w

@~x2 12

@~c@~x

~q @2 ~w

@s2; 36

!while for clampedcamped and clampedhinged beams, this value would be j = 4.73 and j = 3.92, respectively [38]. Rewrit-0

j is a constant that depends on the boundary conditions. The term j2EI=m0L

4q

represents the rst linear frequency of(28) and (33). In these equations, letting l = 0, the governing equations and boundary conditions of a nonlinear Timoshenkobeam modeled by the classical beam theory are achieved (see [37]). Similarly, letting N = 0, the governing equations andboundary conditions of a linear Timoshenko beam modeled by the modied couple stress theory are obtained (see [25]).It is helpful to normalize the governing equations and boundary conditions. For this purpose, the following dimensionlessvariables are dened

~x xL; ~w w

L; s j2

EI

4

st; 352L 0 @x

23) and (32) represent the governing equations of the dynamic behavior of a nonlinear size-dependent Timoshenkowith two immovable supports (see Fig. 1b). Also, the corresponding boundary conditions are presented in Eqs. (27),@x @x 4 @x3 @x2 2x0;L

x0;L

where

N N0 bN N0 EA Z L @w 2dx: 34s in

k1t EA2LZ L0

@w@x

2dx; k2 0: 30

e can get

EA@u@x

12

@w@x

2 ! k1t EA2L

Z L0

@w@x

2dx : bN : 31

eed, bN is the variation of the axial force in the beam with respect to the initial conguration due to the extensiond by the transverse deection, noting that it is a constant value all over the beam. It is called the mid-plane stretchingthe source of nonlinearity in the considered problem.stitution of Eq. (31) into (24) and (26) leads to

lA @2w

@x2 @w

@x

! lAl

2

4@4w@x4

@3w

@x3

! N @

2w@x2

12

@c@x

q m0 @2w@t2

; 32 2 3 2 ! !

where

~c L2c

j4EI; ~q L

3qj4EI

; eV L2Vj4EI

; eMr;m LMr;mj4EI ;eN eN0 6j4 Lh 2 Z 1

0

@ ~w@~x

2d~x; eN0 L2N0j4EI ;

41

and parameter r denotes the gyration radius that can dened as r I=Ap . For a beamwith rectangular cross-section, like theone depicted in Fig. 1b, the gyration radius will be: r I=Ap h=2 3p .4. Static bending

In this section, the static bending of a nonlinear size-dependent Timoshenko beam is delineated. Here, we have @/os = 0and @=@~x d=d~x. Consequently, the governing equations (36) and (37) reduce to:

differeories

5. Fre

Inon the

M. Asghari et al. / International Journal of Engineering Science 48 (2010) 17491761 1755Fig. 2. A hingedhinged beam considered for case studies.the ratios become 1.32 and 1.02, respectively. For the maximum rotation angle, a similar trend is observed.

e vibration

this section, as a case study, the free vibration of a hingedhinged nonlinear size-dependent Timoshenko beams basedmodied couple stress theory is assessed. Here, we assume ~c ~q 0, noting that parameters ~c and ~qwere appeared indiminishes as the ratio increases. For example, the maximum deections of the classical linear and nonlinear Timoshenkobeam are respectively 2.17 and 1.69 times greater than the maximum deection of the present model when h/l = 2. For h/l = 10,nce between the results evaluated by the modied couple stress theory and those predicted by the classical beam the-is signicant when the ratio of the beam thickness to the material length scale parameter h/l is small, however, it1j4

d2wd~x2

12lj4E

Lh

2 d ~wd~x

w

3lj4E

lh

2 d3 ~wd~x3

d2wd~x2

! 12~c 0; 42

12lj4E

Lh

2 d2 ~wd~x2

dwd~x

! 3lj4E

lh

2 d4 ~wd~x4

d3wd~x3

! eN d2 ~w

d~x2 12d~cd~x

~q 0: 43

A hingedhinged beamwith uniform rectangular cross-section subjected to a constant transverse distributed force-per-unit-length is illustrated in Fig. 2. The corresponding boundary conditions are expressed as

~w0 ~w1 dw0d~x

dw1d~x

d2 ~w0d~x2

d2 ~w1d~x2

0: 44It is assumed that j = p, ~c 0; eN0 0; ~q 4:17 and L/h = 10. Also, it is considered that the beam is made of epoxy with thefollowing mechanical properties: E = 1.44 GPa, l = 521.7 MPa and l = 17.6 lm [21,25]. The nonlinear boundary value prob-lem presented in Eqs. (42)(44) has numerically been solved using a nite-difference method that implements a collocationformula. The normalized static deection and rotation angle of cross-sections of the nonlinear size-dependent Timoshenkohingedhinged beam are depicted in Figs. 3 and 4, respectively. Additionally, in these gures, the results of current model arecompared with those of linear modied couple stress Timoshenko beam as well as linear and nonlinear classical Timoshenkobeams. The gures show that the nonlinear theories generally evaluate the static deection less than do the linear theories.In addition, it is observed that the deections evaluated by the modied couple stress theory are less than those predicted bythe classical beam theories. Hence, it can be inferred that the nonlinear and modied couple stress theories predict the beamstiffer than do the linear and classical beam theories.

Figs. 5 and 6 illustrate the maximum values of the static deection and the rotation angle of the cross-sections of thebeam, which belong to the mid point and end cross-sections, respectively, versus the ratio of the beam thickness to the mate-rial length scale parameter, h/l. Furthermore, the results of the present model are compared with those of linear modiedcouple stress Timoshenko beam as well as linear and nonlinear classical Timoshenko beams. The gures indicate that the

1756 M. Asghari et al. / International Journal of Engineering Science 48 (2010) 17491761x / LEqs. (3with o

wherethat s

Sub0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 100.010.02

0.03

0.04

0.05

0.06

w /

L6) and (37). In order to obtain the governing ordinary differential equation from the partial one, the Galerkin methodne-term approximation is utilized:

~w~x; s W~xgs; w~x; s W~xns; 45W~x and W~x represent the rst and of course the most dominant vibrational mode shapes of a Timoshenko beam

atisfy the boundary conditions.stituting Eq. (45) into (36) and (37), then integrating along the beam length, one arrives at the following expressionsZ 10

W~x 12lj4E

Lh

2W 00~xgs W0~xns 3l

j4Elh

2W 4~xgs W000~xns eNW 00~xgs( )dx Z 1

0W2~xgsdx;

46Z 10W~x 1

j4Lr

2W00~xns 12l

j4ELr

2 Lh

2W 0~xgs W~xns 3l

j4ELr

2 lh

2W 000~xgs

(

W00~xns 12

Lr

2~c

)dx

Z 10W2~xnsdx; 47

Fig. 3. The static deection of the hingedhinged beam.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

x / L

(rad)

Fig. 4. The rotation angle of cross-sections for the hingedhinged beam.

M. Asghari et al. / International Journal of Engineering Science 48 (2010) 17491761 1757h / lwheremakin

in wh

with1 2 3 4 5 6 7 8 9 1000.010.02w

0.03

0.04

0.05

0.06

max

/Lthe prime and dot symbols refer to derivation with respect to ~x and s, respectively. Combining Eqs. (46) and (47), andg some mathematical simplication, we get

{g a1 a2g2g a3g a4g3 0; 48ich

a1 b0b3 b5b6

; a2 b1b3 ; a3

b0b5 b2b4b3b6

; a4 b1b5b3b6 ; 49

b0 6lj4E 2Lh

2 Z 10

W 00Wd~x 12

lh

2 Z 10

W 4Wd~x

" # eN0 Z 1

0W 00Wd~x; 50

b1 6j4Lh

2 Z 10W 02 d~x

Z 10

W 00Wd~x

; 51

b2 6lj4E 2Lh

2 Z 10W0Wd~x 1

2lh

2 Z 10W000Wd~x

!; 52

Fig. 5. The maximum static deection of the hingedhinged beam.

1 2 3 4 5 6 7 8 9 100

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

h / l

()

max

rad

Fig. 6. The maximum rotation angle of cross-sections for the hingedhinged beam.

It can(44)).

Eq. (4this sethe prfoundFollow

where

the fo

Substi

1758 M. Asghari et al. / International Journal of Engineering Science 48 (2010) 17491761gs eg1T0; T1; T2 e2g2T0; T1; T2 e3g3T0; T1; T2 : 69tuting Eqs. (66)(69) into (48) and then equating the coefcients of like powers of e to zero, we get

D40 a1D20 a3

g1 0; 70

D40 a1D20 a3

g2 4D30D1g1 2a1D0D1g1; 71

D40 a1D20 a3

g3 6D20D21 4D30D2

g1 a1 D21 2D0D2

g1 a2g21D20g1 4D30D1g2 2a1D0D1g2 a4g31:72The response of the system now is expanded asdds4

D40 4eD30D1 e2 6D20D21 4D30D2 : 68dds2

D20 2eD0D1 e2 D21 2D0D2 ; 674 8) is a nonlinear ordinary differential equation that can be solved analytically by employing perturbation techniques. Inction, in order to solve Eq. (48) analytically, the well-known method of Multiple Scales [39] is utilized. It is noted thatocedure of nding the solution of Eq. (48) presented here does not possess any novelty and similar procedures may bein the literature, e.g. [39]. The aim of presenting the details in the following is that the paper to be self-contained.ing the method of multiple scale, the scaled times can be introduced as

Tn ens n 1;2;3; . . .; 64e is a small dimensionless parameter in the order of amplitude of vibration. Introducing a new differential operators as

Dn @@Tn

n 1;2;3; . . .; 65llowing expressions can be written

dds

D0 eD1 ; 662 b4 p3E r 2 h 2 h ; 61

b5 6lp2ELr

2 12

lh

2 2 L

h

2" # 1p2

Lr

2; 62

b3 b6 1: 636l L 2 L 2 p2 l 2 !b2 p3E 2 h 2 h ; 60b1 3 h ; 59

6l L 2 p2 l 2 !L 2W~x 2

psinp~x; W~x

2

pcosp~x: 57

be readily found out that expressions in Eq. (57) satisfy the boundary conditions of a hingedhinged beam (see Eq.Substitution of Eq. (57) into (50)(56) yields

b0 6lp2E 2Lh

2 p

2

2lh

2" # p2 eN0; 58To assess the nonlinear size-dependent vibration behavior of Timoshenko beams, the frequency of the hingedhinged beamdepicted in Fig. 2 is going to be delineated. For the beam, the vibration mode shapes are written as60b Z 1

W2 d~x: 56b3 Z 10

W2 d~x; 53

b4 6lj4ELr

22

Lh

2 Z 10WW 0 d~x 1

2lh

2 Z 10WW 000 d~x

!; 54

b5 6lj4ELr

2 12

lh

2 Z 10W00Wd~x 2 L

h

2 Z 10W2 d~x

" # 1j4

Lr

2 Z 10W00Wd~x; 55

The so

where

Conse

wherethe re

where

where

Now,

M. Asghari et al. / International Journal of Engineering Science 48 (2010) 17491761 1759the initial conditions of Eq. (48) are considered as followsX1 x1 e2k1 Oe3; X2 x2 e2k2 Oe3: 86gs ea1 cosX1s b^1 a2 cosX2s b^2 Oe3; 85X1 and X2 are the system frequencies written as2 2

In view of Eq. (69), by inserting Eq. (84) into (73) and considering q2 = 0, the analytical solution of the governing nonlinearordinary differential equation is obtained asA1 1 a1 expie2k1s ib^1; A2 1 a2 expie2k2s ib^2: 84Substitution of Eq. (81) into (80) givesk2 2 2 2 1 18x2 2x22 a1 : 838x1 2x1 a13 a2x2 a4

a2 2 a2 x2 2x2 3a4 a2A1T2 12 a1T2 expib1T2; A2T2 12a2T2 expib2T2; 80

a1, a2, b1 and b2 are real. Substituting Eq. (80) into (78) and (79), then separating real and imaginary parts and solvingsulting differential equations, it is obtained that a1 and a2 are some constants, and also b1 and b2 can be expressed as

b1 k1T2 b^1; b2 k2T2 b^2; 81

k1 3 a2x21 a4

a21 2 a2 x21 2x22 3a4 a222

; 82Parameters A1 and A2 can be written in polar forms as:2ix1 2x21 a1

D2A1 3 a2x21 a4

A21A1 2 a2 x21 2x22 3a4 A1A2A2 0; 78

2ix2 2x22 a1

D2A2 3 a2x22 a4

A22A2 2 a2 x22 2x21 3a4 A1A1A2 0: 79Eliminating the secular terms of Eq. (77) results in a2 x22 2x21 3a4 A21A2 expi2x1 x2T0 A21A2 expi2x1 x2T0 cc:77 a2 x21 2x22 3a4 A1A22 expix1 2x2T0 A1A22 expix1 2x2T0n o n o a2x21 a4

A31 exp3ix1T0 a2x22 a4

A32 exp3ix2T0

2ix2 2x22 a1

D2A2 3 a2x22 a4

A22A2 2 a2 x22 2x21

3a4 A1A1A2n o expix2T0D1A1 0; D1A2 0 ) A1 A1T2; A2 A2T2: 76quently, since the right side of Eq. (75) is zero, one can get q2 = 0. Now, substitution of Eq. (73) into (72) yields

D40 a1D20 a3

g3 2ix1 2x21 a1

D2A1 3 a2x21 a4

A21A1 2 a2 x21 2x22 3a4 A1A2A2n o expix1T0Elimination of the secular term [39] of the previous equation leads toD40 a1D20 a3

g2 2ix1 2x21 a1

D1A1 expix1T0 2ix2 2x22 a1

D1A2 expix2T0 cc: 75The smaller frequency x1 is associated with bending deformation, and the larger one x2 is associated with the shear defor-mation. The existence of these frequencies, which is the most important difference between a Timoshenko and an EulerBer-noulli beam, has been demonstrated experimentally [40]. Substitution of Eq. (73) into (71) givesx1 2 4 a3; x2 2 4 a3: 74

2 a1

a21

r2 a1

a21

rcc denotes the complex conjugate and

g1T0; T1; T2 A1T1; T2 expix1T0 A2T1; T2 expix2T0 cc; 73

lution of Eq. (70) is written as

creasenonlinnally, it is inferred that when h/l > 10, the non-classical theories evaluate the same values for frequency as do the classical

6. Con

Inclassicin bearive thexampbehavbeam

Refer

[1] F.Y64

[2] MAp

[3] MNo

[4] Min

[5] R.AA:

1760 M. Asghari et al. / International Journal of Engineering Science 48 (2010) 17491761clusionbeam theories.s, the difference increases until it reaches a constant value. This gure conrms that modeling beams based on theear and non-classical couple stress formulations results in stiffer behavior than linear and classical formulations. Fi-g0 ~wmax wmaxL ; _g0 g0 gv0 0: 87

Applying these conditions to Eq. (85) leads to

b^1 b^2 0; ea1 x22wmax

x22 x21

L; ea2 x

21wmax

x21 x22

L: 88

In order to delineate the size-dependent frequency of the present model, it is assumed that eN0 0, L/h = 10,~wmax wmax=L 0:06 and the beam is made of epoxy with the mechanical properties stated in previous section. In Fig. 7,the ratio of the rst frequency of the current model to the rst frequency of a linear classical Timoshenko beam, X1=X

L:C1

has been depicted versus the ratio of the beam thickness to the material length scale parameter h/l. Moreover, in this gure,the frequency of the current model has been compared with the frequency of a linear modied couple stress Timoshenkobeam. The gure shows that the difference between the results of the current model and those of a linear classical Timo-shenko beam is notable when h/l is small but it decreases to a constant value as h/l increases. For example, when h/l = 2,the ratio of the rst frequency of the current model to the rst frequency of the corresponding linear classical Timoshenkobeam is about 1.53. This ratio is 1.26 for h/l = 10. From this gure, it can also be seen that the difference between the fre-quency predicted by the linear and nonlinear modied couple stress theories is negligible when h/l is small but as h/l in-

1 2 3 4 5 6 7 8 9 101

1.5

2

2.5

3

3.5

h / l

.

11

/L

C

Fig. 7. The rst normalized frequency of the hingedhinged beam.this paper, a nonlinear Timoshenko beammodel has been developed based on the modied couple stress theory, a non-continuum theory capable of capturing the size effects. The source of nonlinearity is the mid-plane stretching arisenms with two immovable supports during nite deections. The Hamilton principle has been employed in order to de-e governing equations of motion (Eqs. (22)(24)) and the corresponding boundary conditions (Eqs. (25)(28)). As anle, a hingedhinged beam has been considered to delineate its size-dependent static and also the free-vibrationior based on the derived equations. The obtained numerical results based on the nonlinear couple stress Timoshenkoare compared with those of linear modied couple stress as well as linear and nonlinear classical beams.

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A nonlinear Timoshenko beam formulation based on the modified couple stress theoryIntroductionPreliminariesGoverning equationStatic bendingFree vibrationConclusionReferences