Static and dynamic analysis of micro beams based on strain gradient elasticity theory

International Journal of Engineering Science 47 (2009) 487–498

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International Journal of Engineering Science

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Static and dynamic analysis of micro beams based on strain gradientelasticity theory

Shengli Kong, Shenjie Zhou *, Zhifeng Nie, Kai WangSchool of Mechanical Engineering, Shandong University, Jinan 250061, PR China

a r t i c l e i n f o

Article history:Received 4 April 2008Received in revised form 4 July 2008Accepted 22 August 2008Available online 7 October 2008

Communicated by Prof. M. Kachanov

Keywords:Strain gradientBernoulli–Euler beamSize effectNatural frequencyNon-classical boundary conditions

0020-7225/$ - see front matter � 2008 Elsevier Ltddoi:10.1016/j.ijengsci.2008.08.008

* Corresponding author. Tel.: +86 531 88396708;E-mail address: [email protected] (S. Zhou).

a b s t r a c t

The static and dynamic problems of Bernoulli–Euler beams are solved analytically on thebasis of strain gradient elasticity theory due to Lam et al. The governing equations of equi-librium and all boundary conditions for static and dynamic analysis are obtained by a com-bination of the basic equations and a variational statement. Two boundary value problemsfor cantilever beams are solved and the size effects on the beam bending response and itsnatural frequencies are assessed for both cases. Two numerical examples of cantileverbeams are presented respectively for static and dynamic analysis. It is found that beamdeflections decrease and natural frequencies increase remarkably when the thickness ofthe beam becomes comparable to the material length scale parameter. The size effectsare almost diminishing as the thickness of the beam is far greater than the material lengthscale parameter.

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1. Introduction

Micro beams have become prevalent in the fields of MEMS such as those in sensors [1,2] and actuators [3,4] in whichthicknesses of beams are typically on the order of microns and sub-microns. A large number of those applications utilizethe mechanical properties of thin films materials for targeted performance and the MEMS devices can be made of metalsand polymers, in addition to traditional silicon-based materials [5–7].

In the recent decades, the size dependence of material deformation behavior in micron-scale had been experimentally dem-onstrated in some metals and polymers. First, the phenomenon were observed in some metals and polymers deformed plasti-cally [8–10]. For example, in the micro-torsion experiment of thin copper wires in 1994, Fleck et al. observed that the torsionalhardening increases by a factor of 3 as the wire diameter decreases from 170 to 12 lm [11]. In the micro-indentation experi-ments in 1995, Ma et al. observed that the measured indentation hardness of silver single crystal increases by a factor of morethan two as the penetration depth of the indenter decreases from 2.0 to 0.1 lm [12]. In 1999, Chong and Lam observed that hard-ness for both thermosetting epoxy resin and thermoplastic polycarbonate exhibits strong size dependence [13]. In the microbending test of thin nickel beams in1998, Stolken and Evans observed that the plastic work hardening shows a great increaseas the beam thickness decreases form 50 to 12.5 lm [14]. Currently, the size dependences were also observed in some polymersdeformed elastically. For example, in the micro bending testing of epoxy polymeric beams in 2003, Lam et al. observed that thenormolized bending rigidity increases about 2.4 times as the thickness of the beam reduces from 115 to 20 lm [15]. In the microbending testing of polypropylene micro-cantilevers in 2005, McFarland et al. observed that the measured stiffness values areseen to be at least four times larger than the classical beam theory stiffness predictions and the deformation is also in the range

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488 S. Kong et al. / International Journal of Engineering Science 47 (2009) 487–498

of linear elastic [16,17]. The above experimental results show that the size dependence is intrinsic to certain materials. Hence,accurate understanding and characterization of materials response at small scale is very important in the optimal design ofMEMS as well as in evaluating their reliability and performance [18–20].

Due to lacking internal material length scale parameters, conventional strain-based mechanics theories fail to character-ize those size effects phenomenon when the structural size is in micron- and sub-micron-scale. However, these size depen-dences can be successfully modeled by employing higher-order continuum theory, in which constitutive equations introduceadditional material length scale parameters in addition to classical material parameters.

As a higher-order continuum theory, the classical couple stress elasticity theory was originated by Mindlin and othersincluding Toupin and Koiter in 1960s and contains four material constants (two classical and two additional) for isotropicelastic materials [21–24]. Some related research works had been performed to model the static and dynamic problems basedon the classical couple stress theory [25,26]. In 1994, Fleck and Hutchinson extended and reformulated the classical couplestress theory and renamed it the strain gradient theory, in which for homogeneous isotropic and incompressible materials,two additional higher-order material length scale parameters are introduced for couple stress theory and three additionalhigher-order material length scale parameters are introduced for stretch and rotation gradient theory [27–30]. Recently, amodified couple stress theory for elasticity had been elaborated by Yang et al. in 2002, in which constitutive equations in-volve only one additional internal material length scale parameter besides two classical material constants [31]. This theoryhad been applied to analysis many boundary value problems successfully [32,33]. In 2003, Lam et al. proposed a modifiedstrain gradient elasticity theory in which a new additional equilibrium equations to govern the behavior of higher-orderstresses, the equilibrium of moments of couples is introduced, in addition to the classical equilibrium equations of forcesand moments of forces [15,34]. Moreover, there are only three independent higher-order materials length scale parametersfor isotropic linear elastic materials in the present theory. Then an elastic bending theory for thin plane-strain beams wasdeveloped. Static bending solutions for cantilever beams were derived based on the new higher-order bending theory andthe constitutive equations were expressed as the moment and higher-order moment in terms of the curvature and the cur-vature gradients [15,34].

To facilitate the application of the simplified strain gradient elasticity theory to problems that favor a displacement for-mulation such as finite element method and dealing with experimental results, the displacement forms of the solutions for astatic bending beam are preferred and derived below in this paper. Furthermore, the solutions of a beam for free flexuralvibrations are also presented.

The object of this work is to establish static and dynamic models for Bernoulli–Euler beams by using both the basic equa-tions of the strain gradient elasticity theory and a variational principle. The material is assumed to obey the strain gradientelasticity theory. The outline of this paper is as follows. The basic equations of strain gradient elasticity theory are reviewedin Section 2. Then governing equations and all possible boundary conditions for static analysis and dynamic analysis of abeam are obtained simultaneously with the aid of a variational principle in Sections 3 and 4, respectively. Compared to solu-tions of classical theory and modified coupled stress theory, the newly obtained solutions include three material length scaleparameters and can capture the size-dependent phenomenon. Additionally, two representative boundary values problems(one for static and another for dynamic) are solved and two corresponding numerical examples are presented. The papergives a conclusion in Section 5.

2. A review of strain gradient theory for elasticity

A modified strain gradient elasticity theory was proposed by Lam et al. in 2003, in which a new additional equilibriumequations to govern the behavior of higher-order stresses, the equilibrium of moments of couples is introduced, in additionalto the classical equilibrium equations of forces and moments of forces [15,34]. In the constitutive equations of this theory,there are only three independent higher-order materials length scale parameters in addition to two classical material param-eters for isotropic linear elastic materials. Then the strain energy U in a deformed isotropic linear elastic material occupyingregion X is given by

U ¼ 12

ZXðrijeij þ pici þ sð1Þijk gð1Þijk þms

ijvsijÞdv; ð1Þ

where the deformation measures are defined as follows,

eij ¼12ðoiuj þ ojuiÞ; ð2Þ

emm;i ¼ ci ¼ oiemm; ð3Þ

gð1Þijk ¼13ðoiejk þ ojeki þ okeijÞ �

115

dijðokemm þ 2omemkÞ ð4Þ

� 115½djkðoiemm þ 2omemiÞ þ dkiðojemm þ 2omemjÞ�

vsij ¼

12ðeipqopeqj þ ejpqopeqiÞ; ð5Þ

S. Kong et al. / International Journal of Engineering Science 47 (2009) 487–498 489

where oi is differential operator, ui is the displacement vector, eij is the strain tensor, emm,i is the dilatation gradient vector, gð1Þijk

is the deviatoric stretch gradient tensor, vsij is the symmetric rotation gradient tensor, dij and eijk are the Knocker delta and the

alternate tensor (In the above and in subsequent equations, the index notation will be used with repeated indices denotingsummation from 1 to 3).

And the corresponding stress measure, respectively, are

rij ¼ kdijemm þ 2le0ij; ð6Þ

pi ¼ 2ll20ci; ð7Þ

sð1Þijk ¼ 2ll21gð1Þijk ; ð8Þ

msij ¼ 2ll2

2vsij; ð9Þ

where e0ij is deviatoric strain defined as

e0ij ¼ eij �13emmdij; ð10Þ

where k and l are bulk and shear module, respectively, and l0, l1, l2 are additional independent material parameters associ-ated with dilatation gradients, deviatoric stretch gradients and rotation gradients, respectively.

3. Static analysis of micro beams based on strain gradient theory for elasticity

3.1. Governing equation and boundary conditions for bending

The governing equation of a beam in bending as well as all boundary conditions can be determined with the aid of a var-iational principle [35]

dðU �WÞ ¼ 0; ð11Þ

where U is the strain energy and W is the total work done by external forces and d indicates variation.Consider a straight beam, which is subjected to a static lateral load q(x) distributed along the longitudinal axis x of the

beam, as shown in Fig. 1. Thus, the loading plane coincides with the xz plane, and the cross-section of the beam parallelto the yz plane.

According to the Bernoulli–Euler hypothesis, the displacement field of a beam in bending can be written as [35]

u ¼ �zowðxÞ

oxv ¼ 0 w ¼ wðxÞ; ð12Þ

where u, v, w, are the x-, y-, and z-components of the displacement vector, respectively.By substituting Eq. (12) into Eq. (2), then the non-zero strain eij is

exx ¼ �zo2wox2 ð13Þ

And from Eqs. (2) and (3) it follows that

cx ¼ �zo3wox3 cy ¼ 0 cz ¼ �

o2wox2 ð14Þ

Fig. 1. Geometry and loading of a flexural beam.


By inserting Eq. (2) into Eq. (5), the non-zero strain gradients vsij are

vsxy ¼ vs

yx ¼ �12

o2wox2 ð15Þ

By using Eq. (2) in Eq. (4), the non-zero strain gradients gð1Þijk are

gð1Þ111 ¼ �25

zo3wox3 gð1Þ113 ¼ �

415

o2wox2 gð1Þ122 ¼

15

zo3wox3

gð1Þ133 ¼15

zo3wox3 gð1Þ212 ¼

15


15

zo3wox3

gð1Þ223 ¼1

15o2wox2 gð1Þ232 ¼

115

o2wox2 gð1Þ311 ¼ �

415

o2wox2

gð1Þ313 ¼15


115

o2wox2 gð1Þ331 ¼

15


15

o2wox2

ð16Þ

From Eqs. (10) and (13) it follows that

e0xx ¼23exx e0yy ¼ e0zz ¼ �

13exx ð17Þ

For a slender beam with a large aspect ratio, the Poisson effect is secondary and may be neglected to facilitate the formu-lation of a simple beam theory. By setting v = 0, as was done in classical beam theories, substituting Eqs. (13) and (17) intoEq. (6) then gives the non-zero stresses rij as

rxx ¼ Eexx ¼ �Ezo2wox2 ð18Þ

The use of Eq. (14) in Eq. (7) gives

px ¼ �2ll20z

o3wox3 py ¼ 0 pz ¼ �2ll2

0o2wox2 ð19Þ

and by inserting Eq. (15) into Eq. (9), the non-zero higher-order stresses msij are !

msxy ¼ ms

yx ¼ 2ll22 �

12

o2wox2 ¼ �ll2

2o2wox2 ð20Þ

Similarly, from Eqs. (16) and (8), the non-zero higher-order stresses sð1Þijk are

sð1Þ111 ¼ �45ll21z

o3wox3 sð1Þ113 ¼ �

815

ll21o2wox2 sð1Þ122 ¼

25ll2

1zo3wox3

sð1Þ133 ¼25ll2

1zo3wox3 sð1Þ212 ¼

25ll21z

o3wox3 sð1Þ221 ¼

25ll21z

o3wox3

sð1Þ223 ¼2

15ll2

1o2wox2 sð1Þ232 ¼

215

ll21o2wox2 sð1Þ311 ¼ �

815

ll21o2wox2

sð1Þ313 ¼25ll2


215

ll21o2wox2 sð1Þ331 ¼

25ll2


25ll2

1o2wox2

ð21Þ

Substituting Eqs. (13)–(15), (16), (19), (21) into Eq. (1), one obtains the following expression for the strain energy U

U ¼ 12

Z L

0½S � ðw00Þ2 þ K � ðwð3ÞÞ2�; ð22Þ

where

K ¼ Ið2ll20 þ45ll2

1Þ S ¼ EI þ 2lAl20 þ

43225

lAl21 þ lAl2

2

w0 ¼ o2wox2 wð3Þ ¼ o3w

ox3

ð23Þ

and I is the inertia moment of the beam and A is the cross section area of the beam.Taking the first variation of Eq. (22) results in, together with Eq. (23)

dU ¼ dZ L

0Fðw00;wð3ÞÞdx

¼Z L

0

d2

dx2

oFow00

� �� d3

dx3

oFowð3Þ

� �" #dwdxþ � d

dxoFow00

� �þ d2

dx2

oFowð3Þ

� �" #dw

" #L

0

oFow00� d

dxoF

owð3Þ

� �� dw0

� �L

0þ oF

owð3Þdw00

� �L

0

ð24Þ


where, for the present case, the Lagrangian function F is

F ¼ 12½S � ðw00Þ2 þ K � ðwð3ÞÞ2� ð25Þ

Eqs. (24) and (25) help to express the variation of the strain energy of the beam as

dU ¼Z L

0½S �wð4Þ � K �wð6Þ�dwdxþ ½�S �wð3Þ þ K �wð5Þ�dwjL0 þ ½S �w00 � K �wð4Þ�dw0jL0 þ K �wð3Þdw00jL0 ð26Þ

where

w0 ¼ owox

wð4Þ ¼ o4wox4 wð5Þ ¼ o5w

ox5 wð6Þ ¼ o6wox6 ð27Þ

On the other hand, the variation of the work done by the external force q(x), the boundary shear force V, and the boundaryclassical and non-classical bending moments M and Mh, respectively, reads

dV ¼Z L

0qðxÞdwðxÞdxþ ½Vdw�L0 þ ½Mdw0�L0 þ ½M

hdw00�L0 ð28Þ

In view of Eqs. (26) and (28), the variational Eq. (11) takes the form

dðU � VÞ ¼Z L

0½S �wð4Þ � K �wð6Þ � q�dwdxþ ½�S �wð3Þ þ K �wð5Þ � V �dwjL0 þ ½S �w00 � K �wð4Þ �M�dw0jL0

þ ½K �wð3Þ �Mh�dw00jL0 ¼ 0 ð29Þ

The above variational equation implies that each term must be equal to zero. Thus, the governing equation of the beam inbending is given by

S �wð4Þ � K �wð6Þ � q ¼ 0 ð30Þ

while the boundary conditions satisfy the following equations

½VðLÞ � ½K �wð5ÞðLÞ � S �wð3ÞðLÞ��dwðLÞ � ½Vð0Þ � ½K �wð5Þð0Þ � S �wð3Þð0Þ��dwð0Þ ¼ 0

½MðLÞ � ½S �w00ðLÞ � K �wð4ÞðLÞ��dw0ðLÞ � ½Mð0Þ � ½S �w00ð0Þ � K �wð4Þð0Þ��dw0ð0Þ ¼ 0

½MhðLÞ � K �wð3ÞðLÞ�dw00ðLÞ � ½Mhð0Þ � K �wð3Þð0Þ�dw00ð0Þ ¼ 0

ð31Þ

For boundary conditions: if one assumes the four classical boundary conditions to be w(0), w(L), w0(0) and w0(L) prescribedand the corresponding non-classical ones to be w00(0) and w00(L) prescribed, then dw(0) = dw(L) = 0, dw0(0) = dw0(L) = 0,dw00(0) = dw00(L) = 0 and Eq. (31) are all satisfied. In virtue of Eq. (31), one can get that, when dealing with the classical bound-ary conditions, either the deflection w or the boundary shear forces V = Kw(5) � Sw(3) and the strain w0 or the boundary clas-sical bending moments M = Sw00 � Kw(4) at the boundaries of the beam have to be specified. For the case of the non-classicalboundary conditions, one has to specify either the boundary strain gradient w00 or the boundary non-classical momentsMh = Kw(3).

When the higher-order material parameters l0 and l1 equal to zero, then the constitutive relation reduces to that of mod-ified couple stress theory and the corresponding governing equation reads

ðEI þ lAl22Þwð4Þ � q ¼ 0 ð32Þ

and the corresponding boundary conditions can be written as

½VðLÞ þ ðEI þ lAl22Þwð3ÞðLÞ�dwðLÞ � ½Vð0Þ þ ðEI þ lAl2

2Þwð3Þð0Þ�dwð0Þ ¼ 0

½MðLÞ � ðEI þ lAl22Þw00ðLÞ�dw0ðLÞ � ½Mð0Þ � ðEI þ lAl2

2Þw00ð0Þ�dw0ð0Þ ¼ 0ð33Þ

These results conform to solutions based on the modified couple stress theory [32]. Moreover, when all three higher-ordermaterial parameters, e.g., l0, l1, l2, equal to zero, then the constitutive relation reduces to that of classical theory.

3.2. Solution of boundary value problems in bending

This sub-section deals with the solution of a boundary value problem for bending. Considering a cantilever beam of lengthL with its built-in end at x = 0, subjected to a static concentrated force P, as shown in Fig. 2, the deflection w of the beam inbending satisfies

S �wð4Þ � K �wð6Þ ¼ 0 ð34Þ

Fig. 2. Schematic diagram of the rectangular cantilever beam.


Then the solution of Eq. (34) is

wðxÞ ¼ C1þ C2xþ C3x2 þ C4x3 þ C5effiffiSpffiffi

Kp x þ C6e�

ffiffiSpffiffi

Kp x ð35Þ

The classical boundary conditions at the both ends of the beam are

wð0Þ ¼ 0; w0ð0Þ ¼ 0

K �wð5ÞðLÞ � S �wð3ÞðLÞ ¼ P

S �w00ðLÞ � K �wð4ÞðLÞ ¼ 0

ð36Þ

For non-classical boundary conditions, there are two possible boundary conditions at the fixed end. One of the higher-orderboundary conditions can be written as

w00ð0Þ ¼ 0

K �wð3ÞðLÞ ¼ 0 namely wð3ÞðLÞ ¼ 0ð37Þ

Use of the above boundary conditions (36) and (37) into Eq. (35), enables one to determine the constants as

C1 ¼ LKP

S2 C2 ¼PffiffiffiffiKp ffiffiffi

Sp

effiffiSpffiffi

Kp LL�

ffiffiffiSp

e�ffiffiSpffiffi

Kp LLþ 2

ffiffiffiffiKp� �

S2 effiffiSpffiffi

Kp L þ e�

ffiffiSpffiffi

Kp L

� �

C3 ¼ 12

LPS

C4 ¼ �16

PS

C5 ¼PK

ffiffiffiffiKp�

ffiffiffiSp

e�ffiffiSpffiffi

Kp LL

� �

S52 e

ffiffiSpffiffi

Kp L þ e�

ffiffiSpffiffi

Kp L

� � C6 ¼ �PK

ffiffiffiffiKpþ

ffiffiffiSp

effiffiSpffiffi

Kp LL

� �

S52 e

ffiffiSpffiffi

Kp L þ e�

ffiffiSpffiffi

Kp L

� �

ð38Þ

Another of the higher-order boundary conditions can be written as

K �wð3Þð0Þ ¼ 0 namely wð3Þð0Þ ¼ 0


Using of the above boundary conditions (36) and (39) into Eq. (35), then enable one to determine the constants as

C1 ¼PK

32 e

ffiffiSpffiffi

Kp L þ e�

ffiffiSpffiffi

Kp L � 2

� �

S52 e

ffiffiSpffiffi

Kp L � e�

ffiffiSpffiffi

Kp L

� � C2 ¼ �KP

S2

C3 ¼ 12

LPS

C4 ¼ �16

PS

C5 ¼ �PK

32 �1þ e�

ffiffiSpffiffi

Kp L

� �

S52 e

ffiffiSpffiffi

Kp L � e�

ffiffiSpffiffi

Kp L

� � C6 ¼ �PK

32 �1þ e

ffiffiSpffiffi

Kp L

� �

S52 e

ffiffiSpffiffi

Kp L � e�

ffiffiSpffiffi

Kp L

� �

ð40Þ

From Eqs. (38) and (40), one can observe that all constants, Ci (i = 1, 2, . . ., 6) relate to coefficients S and K which are functionsof three higher-order material length scale parameters l0, l1 and l2.

3.3. Numerical examples

To illustrate the newly derived solution of the cantilever beam problem, some numerical results have been obtained andpresented in Fig. 3a–d, where the deflection distributions along the beam given by the current strain gradient elasticity the-

Fig. 3. Deflections of cantilever beams along the beams based on strain gradient elastic theory, modified couple stress theory and classical theory,respectively, for various values of thickness h. (a) h = 20 lm; (b) h = 50 lm; (c) h = 100 lm; (d)=200 lm.


ory solution, modified couple stress theory solution and classical theory solution are shown. Boundary conditions 1 (BC1)consist of Eqs. (36) and (37) and Boundary conditions 2 (BC2) consist of Eqs. (36) and (39). For illustration purpose, the beamconsidered here is taken to be made of epoxy (see [15,34]) and material properties used in the calculations are taken to beE = 1.44 GPa, v = 0.38, and l = 17.6 lm. For comparison purpose, assume that all three material length scale parameters arethe same, i.e., l0 = l1 = l2 = l. the cross-sectional shape is kept to be the same by letting b/h = 2 for all cases and the values of Pand h have been so chosen that the beam remains elastic everywhere. Four sets of geometries of beams are shown in Table 1.

From the curves in Fig. 3a–d, it can be seen that the deflection predicted by the classical beam theory is about 12 timesthan that predicted by the present strain gradient elastic beam theory (with BC1 and BC2) and about 4.4 times than thatpredicted by the modified couple stress theory when the thickness of the beam is approximately equal to the material lengthscale parameter (h = 20 lm). However, the difference among the four sets of predicted values is diminishing when the thick-ness of the beam becomes far greater than the material length scale parameter, thereby indicating that the size effect is onlysignificant when the thickness of the beam is comparable to the material length scale parameter. It also can be seen that themaximum difference in deflections with two various boundary conditions (BC1 and BC2) at the free end, predicted by thepresent strain gradient elastic beam theory, is less than 3.8%. This conclusion conforms to that given by Lam et al.[15,34]. Moreover, compared to the deflections from modified couple stress theory, the deflections from present strain

Table 1Four sets of micro beams for various geometries

Sets 1 2 3 4

Thickness, h (lm) 20 50 100 200Width, b = 2h (lm) 40 100 200 400Length, L = 20h (lm) 400 1000 2000 4000


gradient elasticity theory are less and the stiffen effects are reasonable that strain gradient elasticity theory introduce addi-tional dilatation gradient tensor and the deviatoric stretch gradient tensor in addition to the rotation gradient tensor.

4. Dynamic analysis of micro beams based on strain gradient theory for elasticity

4.1. Governing equation, initial conditions and boundary conditions for free vibrations of beams

The governing equation of the beam in bending as well as initial conditions and all boundary conditions can be deter-mined with the aid of Hamilton’s principle [36]

dZ t2

t1ðT � V � UÞdt

� �¼ 0 ð41Þ

where U is the strain energy, V is the kinetic energy of beam and W is the total work done by external forces and d indicatevariation.

According to aforementioned expression for the static analysis of beam, the first variant of corresponding strain energy Uis

dU ¼Z L

0½S �wð4Þ � K �wð6Þ�dwdxþ ½�S �wð3Þ þ K �wð5Þ�dwjL0 þ ½S �w00 � K �wð4Þ�dw0jL0 þ K �wð3Þdw00jL0 ð42Þ

On the other hand, the first variant of kinetic energy of the beam has the form

dT ¼ d12

Z L

0qA

owot

� �2

dx

" #ð43Þ

where q is the material density.Finally, the first variation of the work done by the external force q, the boundary shear force V and the boundary classical

and double moments M and Mh, respectively, reads

dV ¼Z L

0qdwdxþ ½Vdw�jL0 þ ½Mdw0�jL0 þ ½M

hdw00�jL0 ð44Þ

Thus, in view of Eqs. (42), (43) and (44), (41) takes the form
Z t2
t1

Z L

0½S �wð4Þ � K �wð6Þ þ qA €wþ q�dwdxdt þ

Z t2

t1½V � S �wð3Þ þ K �wð5Þ�dwjL0dt

þZ t2

t1½M þ S �w00 � K �wð4Þ�dw0jL0dt þ

Z t2

t1½Mh þ K �wð3Þ�dw00jL0dt �

Z L

0½qA _wdw�jt2

t1dx ¼ 0 ð45Þ

where

_w ¼ owot

€w ¼ o2wot2 ð46Þ

The above variational equation implies that each term must be equal to zero. Thus, the governing equation of the beam inbending is given by

S �wð4Þ � K �wð6Þ þ qA €wþ q ¼ 0 ð47Þ

and initial conditions can be written as

½qA _wdw�jt2t1 ¼ 0 ð48Þ

and all boundary conditions satisfy the equations

½VðLÞ � S �wð3ÞðLÞ þ K �wð5ÞðLÞ�dwðLÞ � ½Vð0Þ � S �wð3Þð0Þ þ K �wð5Þð0Þ�dwð0Þ ¼ 0

½MðLÞ þ S �w00ðLÞ � K �wð4ÞðLÞ�dw0ðLÞ � ½Mð0Þ þ S �w00ð0Þ � K �wð4Þð0Þ�dw0ð0Þ ¼ 0

½MhðLÞ þ K �wð3ÞðLÞ�dw00ðLÞ � ½Mhð0Þ þ K �wð3Þð0Þ�dw00ð0Þ ¼ 0 ð49Þ

When the higher-order material parameters l0 and l1 equal to zero, then the constitutive relation reduces to that of modifiedcouple stress theory and the corresponding governing equation reads

ðEI þ lAl22Þwð4Þ þ qA €wþ q ¼ 0 ð50Þ

and the corresponding boundary conditions can be written as


½VðLÞ � ðEI þ lAl22Þwð3ÞðLÞ�dwðLÞ � ½Vð0Þ � ðEI þ lAl2

2Þwð3Þð0Þ�dwð0Þ ¼ 0

½MðLÞ þ ðEI þ lAl22Þw00ðLÞ�dw0ðLÞ � ½Mð0Þ þ ðEI þ lAl2

2Þw00ð0Þ�dw0ð0Þ ¼ 0ð51Þ

These results conform to solutions based on the modified couple stress theory [33]. Moreover, when all three higher-ordermaterial parameters, e.g., l0, l1, l2, equal to zero, then the constitutive relation reduces to that of classical theory.

4.2. Solution of boundary value problems for free vibrations of beams

This sub-section deals with the solution of a boundary value problem for free flexural vibrations of a micro beam. Con-sidering a cantilever beam of length L with its built-in end at x = 0, as shown in Fig. 1, with the external force is zero, theequation of motion of the beam satisfies

Swð4Þ � Kwð6Þ þm €w ¼ 0 ð52Þ

where m = qA.In order to study the free flexural motion of the beam, assuming a solution of the form

wðx; tÞ ¼ w0ðxÞei/t ð53Þ

Substituting Eq. (53) into Eq. (52) yields

Swð4Þ0 ðxÞ � Kwð6Þ0 ðxÞ �m/2w0 ¼ 0 ð54Þ

which has a solution of the form

w0ðxÞ ¼X6

i¼1

Ciekix ð55Þ

where exponents ki(i = 1,2, . . ., 6) are in general complex and roots of the algebraic equation

Skð4Þ � Kkð6Þ �m/2 ¼ 0 ð56Þ

and Ci (i = 1,2, . . ., 6) are constants to be determined with the aid of the corresponding boundary conditions.For example, for the case of a cantilever beam of length L the classical boundary conditions are

wð0Þ ¼ 0; w0ð0Þ ¼ 0

K �wð5ÞðLÞ � S �wð3ÞðLÞ ¼ 0; S �w00ðLÞ � K �wð4ÞðLÞ ¼ 0ð57Þ

while for non-classical boundary conditions, a possible choice is

w00ð0Þ ¼ 0


The above boundary conditions satisfy Eq. (49). Thus the six boundary conditions, exclusively in terms of displacements, aregiven by Eqs. (57) and (58).

Inserting the above boundary conditions to Eq. (55), one obtains the following system of equations in matrix form to besolved for the constants Ci.

½AðxÞ�fCg ¼ f0g ð59Þ

among those, {C} = {C1, C2, . . ., C6}T and for i = 1, 2, . . ., 6,

A1i ¼ 1; A2i ¼ ki A3i ¼ k2i

A4i ¼ k3i ekiL A5i ¼ ðk5

i K � k3i SÞekiL A6i ¼ ðk2

i S� k4i KÞekiL

ð60Þ

where the ki = ki (x) are the six roots of Eq. (56) and no summation is applied over repeated indices i.In order to have non-zero solution for {C} for the system (59), the condition

det½AðxÞ� ¼ 0 ð61Þ

should be satisfied. Eq. (61) is the frequency equation and provides all the natural frequencies of the cantilever beam. Thesolution of Eq. (61) is accomplished numerically using complex arithmetic due to the fact that ki’s are, in general, complex.Thus, for a sequence of values of x, one computes the corresponding values of the real function D(x) = ln(jdet[A(x)]j) andthose values of x for which D(x) attains minimum values, are the natural frequencies. For more details on this approach, onecan refer to related book and paper [19,37].

Once the natural frequencies (k = 1, 2, . . .,1) have been obtained from Eq. (61), with the aid of Eq. (55), one can go againto Eq. (59) to obtain the k modal shapes corresponding to the k natural frequencies in the form

Fig. 4.respectfourth


v0kðxÞ ¼X6

i¼1

Cikekikx ð62Þ

where there is no summation over the repeated index k and i.

4.3. Numerical examples

To illustrate the newly derived solution of the cantilever beam problem, some numerical results have been obtained andpresented in Fig. 4a–d, where the first four natural frequencies of a cantilever beam given by the current strain gradient elas-ticity theory solution, modified couple stress theory solution and classical theory solution are shown. For illustration pur-pose, the beam considered here is taken to be made of epoxy (see [15,34]) and material properties used in thecalculations are taken to be E = 1.44 GPa, v = 0.38, and material density q = 1000 kg/m3 (assumed here). For comparison pur-pose, assume that all three material length scale parameters are the same, i.e., l0 = l1 = l2 = l. The geometries of four sets ofbeams list in Table 1.

From the curves in Fig. 4a–d, it can be seen that natural frequencies predicted by the present strain gradient elastic beamtheory are about 3.5 times than that predicted by the classical beam theory and natural frequencies predicted by the mod-ified couple stress theory are about 2.4 times than that predicted by the classical beam theory when the thickness of beam isapproximately equal to the material length scale parameter (h = 20 lm). It is also shown that the difference among the threesets of predicted values is diminishing when the thickness of the beam becomes larger, thereby indicating that the size effectis only significant when the thickness of the beam is comparable to the material length scale parameter. Compared to naturalfrequencies from modified couple stress theory, the natural frequencies from this strain gradient elasticity theory are largerand the size effects are reasonable that the strain gradient elasticity theory introduce additional dilatation gradient tensorand the deviatoric stretch gradient tensor in addition to the rotation gradient tensor.

The first four natural frequencies of cantilever beams based on strain gradient elastic theory, modified couple stress theory and classical theory,ively, for various values of thickness h. (a) The first natural frequencies; (b) the second natural frequencies; (c) the third natural frequencies; (d) thenatural frequencies.


5. Conclusions

The static and dynamic problems of Bernoulli–Euler beams are solved analytically on the basis of strain gradient elasticitytheory due to Lam et al. The governing equations of equilibrium and all boundary conditions for static and dynamic analysisare obtained by a combination of the basic equations and a variational statement. The new model contains three materiallength scale parameters in addition to two classical material constants. Two boundary value problems for cantilever beamsare solved and the gradient elasticity effect on the beam bending response and its natural frequencies are assessed for bothcases. Two numerical examples of cantilever beams are presented for static and dynamic analysis. For static analysis ofmicrobeams, it is found that the deflection of the beam predicted by the classical beam theory is about 12 times than thatpredicted by the present strain gradient elastic beam theory when the thickness of the beam is approximately equal to thematerial length scale parameter (h = 20 lm, l = 17.6 lm). For dynamic analysis of microbeams, it is found that the naturalfrequencies of the beam predicted by the present strain gradient elastic beam theory are about 3.5 times than that predictedby the classical beam theory when the thickness of the beam is approximately equal to the material length scale parameter.However, the size effects are almost diminishing as the thickness of the beam is far greater than the material length scaleparameter for both static and dynamic analysis of microbeams.

Acknowledgements

The work reported here is funded by the National Natural Science Foundation of China (Grant No. 10572077), the ChineseMinistry of Education University Doctoral Research Fund (Grant No. 20060422013) and the Shandong Province Natural Sci-ence Fund (Grant No. Y2007F20).

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Static and dynamic analysis of micro beams based on strain gradient elasticity theory