inq

of TSimoad

Timoshenko bending beamModied couple stress theoryMaterial length parameterScale effect

e st. Th

dependent variable and the use of only one constant to describe the materials micro-structural charac-

fw = q0sin(px/L) is adopted and explicit expression of analysis solution is obtained. Numerical results

e show

the problems of the scale effects, theories for microstructures needto be developed.

rotation variables than the strain gradient theory does for thestrain variables. In the couple stress theory, the variables relatedto micro-scale impurities or defects are formulated into rotationequilibrium equations. In the strain gradient theory, these vari-ables are formulated into higher order strain terms in geometricequations. In both cases, new parameters which describe the mate-rial scale characteristics are introduced as higher order term (4thorder) into the partial differential governing equation. Yet, in con-ventional continuum mechanics, this partial differential governing

q Contract/Grant sponsor: National Natural Sciences Foundation of China (No.11072156). Corresponding author at: Key Laboratory of Liaoning Province for Composite

Structural Analysis of Aerocraft and Simulation, Shenyang University of Aerospace,Shenyang 110136, China.

Composite Structures 93 (2011) 27232732

Contents lists availab

Composite S

journal homepage: www.elsevE-mail address: chenwj@dlut.edu.cn (W. Chen).has scale effects due to impurities, crystal lattice mismatch and mi-cro cracks at micro scales. With the material size scaling down tothe order of micro scales, the stiffness and the strength of metalmaterials can increase with the size decreasing, which is called sizeeffects. The size effects have been proved by many experiments inthe recent two decades. For example, Fleck et al. [1] observed thatthe scaled shear strength increases by a factor of three as the wirediameter decreases from 170 lm to 12 lm in the twisting of thincopper wires; Stolken and Evan [2] reported a signicant increasein the normalized bending hardening with the beam thicknessdecreasing in bending of ultra thin beams. Sun et al. [3] put for-ward a alternative view of the size effects in the nano-scale struc-tures. As conventional continuum theory cannot explain or solve

1960s, Toupin [4], Koiter [5] and Mindlin proposed couple stresstheory [6]. Between the 1980s and 1990s, Aifantis [7], Fleck andHutchinson [8,9] developed the strain gradient theory in plasticity.Gao et al. [10] further improved the strain gradient theory inplasticity. A modied couple stress theory has recently beenproposed by Yang et al. in which the couple stress tensor is sym-metric and only one internal material length scale parameter isconsidered [11].

The couple stress theory can be viewed as a special format ofstrain gradient theory which uses rotation as a variable to describecurvature, while the strain gradient theory uses strain as variableto describe curvature. Though both theories can describe the scaledefects at micro-scale, the couple stress theory contains fewer1. Introduction

Since the 1960s, experiments hav0263-8223/$ - see front matter 2011 Elsevier Ltd. Adoi:10.1016/j.compstruct.2011.05.032show that the present beam model can capture the scale effects of microstructure, and the deectionsand stresses of the present model of couple stress beam are smaller than that by the classical beammode.Additionally, the present model can be reduced to the classical composite laminated Timoshenko beammodel, Isotropic Timoshenko beam model of couple stress theory, classical isotropic Timoshenko beam,composite laminated BernoulliEuler beam model of couple stress theory and isotropic BernoulliEulerbeam of couple stress theory.

2011 Elsevier Ltd. All rights reserved.

n that micro-structure

Theories for microstructures include couple stress theory andstrain gradient theory. A series of research in the couple stress/strain gradient theories have been made. For example, in theKeywords:Composite laminated

teristics. The present model of beam can be viewed as a simplied couple stress theory in engineeringmechanics. An example as a cross-ply simply supported beam subjected to cylindrical bending loads ofA modied couple stress model for bendbeams with rst order shear deformation

Wanji Chen a,b,, Li Li a,c, Ma Xu aa State Key Laboratory for Structural Analysis of Industrial Equipment, Dalian UniversitybKey Laboratory of Liaoning Province for Composite Structural Analysis of Aerocraft andcPhysics and Biophysics Department, China Medical University, No. 92, The 2nd North R

a r t i c l e i n f o

Article history:Available online 31 May 2011

a b s t r a c t

Based on a modied coupldeformation is developedll rights reserved.g analysis of composite laminated

echnology, DaLian 116023, Chinaulation, Shenyang Aerospace University, Shenyang, LN 110136, China, Heping District, Shenyang 110001, China

ress theory, a model for composite laminated beam with rst order sheare characteristics of the theory are the use of rotationdisplacement as

le at ScienceDirect

tructures

ier .com/locate /compstruct

2trucequation is a 2nd order equation. Generally speaking, the couplestress/strain gradient theory for microstructures can be classiedinto two respective theories, C1 theory and C0 theory. For C1 theorythe displacements and rotations/strains are dependent variables.For C0 theory, the displacements and rotations/strains are indepen-dent variables. In the application in the engineering, the micro-structures such as sensors and actuators in micro-electromechan-ical systems (MEMS) and nano-electromechanical systems (NEMS)are often consist in the components of beam, plate and membraneet al. According to the application in engineering, the beam, plateand shell theories based on couple stress/strain gradient theoryshould be developed.

The researchers have focused on the beam theory on micro-scale in recent years. A number of papers have been publishedfor attempting to develop microstructure-dependent non-localTimoshenko beam models and apply them to analyze nanotubesand other small beam-like members/devices. All of these modelsare based on a C0 theory in which the rotationdisplacement asdependent variables. For example, the model for pure bendingproposed by Anthoine [12] is based on the classical C0 couplestress elasticity theory, which includes two additional internalmaterial length scale parameters. The higher-order BernoulliEu-ler beam model developed by Papargyri-Beskou et al. [13] isbased on the C0 gradient elasticity theory, which involves twointernal material length scale parameters. The non-local Ber-noulliEuler beam model by Peddieson et al. [14], in the formula-tion the constitutive equation suggested by Eringen [15] containstwo additional material constants. More background related tothe couple stress beam based on the C0 couple stress theory,especially Cosserat-type theories which contain more than twoadditional material constants, can be found in the review byAltenbach al. [16].

Recently, due to the difculty of determining more than onemicrostructure-dependent length scale parameters and theapproximate nature of beam theories, C1 non-classical beam mod-els involving only one material length scale parameter are gettingmany attentions. One model, as a simpler BernoulliEuler beammodel based on modied couple stress theory with only one mate-rial length parameter, has recently been developed by Park andGao [17]. Ma et al. proposed a microstructure-dependent Timo-shenko beam model based on a modied couple stress theory withonly one material length parameter [18]. Tsiatas proposed a newKirchhoff plate model based on a modied couple stress theory[19]. Metin developed a general nonlocal beam theory based onC0 theory [20], where the nonlocal constitutive equations proposedby Eringen [15] are adopted. The nonclassical RL beam modelbased on the higher order shear deformation theory and C1 couplestress theory was developed by Ma et al. [21]. The non-classical RL model can be reduced to the existing classical elasticity-based RL model by using the material length scale parameter and Poissonsratio are both taken to be zero. The classical RL beammodel[22] isa third-order beam model satised the condition of shear stressequal zero on the upper and lower surfaces of the beam. For mod-erate thickness beam, the accuracy is higher than rst-order shearbeam model. Furthermore the RL beam model can be reduce thenon-classical BernoulliEuler beam model when the normalityassumption is introduced.

Composite laminate beam and plate are widely used in engi-neering. Due to the microscale such as ber, impurities and microcracks at micro matrix are involved in a laminated compositestructure, it results in classical laminate theory invalid in someproblems related to the miro-scale of laminate composites.

The objective of this paper is to develop a microstructure-

2724 WJ. Chen et al. / Composite Sdependent model for the laminated Timoshenko beam based ona modied couple stress theory with only one material length scaleparameter.2. Formulations for modied couple stress theories

Unlike the conventional continuum mechanics, the rotationvectorxi is introduced to kinematic relation of the classical couplestress theory, as well as the curvatures tensor vij and couple stresstensormij. Unlike the classical couple stress theory, Yang et al. [11]developed a modied couple stress theory in which the part ofrotation gradient in the strain tensor is symmetric.

2.1. Modied coupled stress theory

According to the symmetric couple stress theory proposed byYang et al., the strain tensor and curvature tensor can be denedas eij 12 ui;j uj;i, vij 12 xi;j xj;i respectively, wherex 12 curlu, u ui is the displacement vector and x(xi) is therotation vector. The main differences of modied couple stress the-ory with standard couple stress theory are that for modied couplestress theory the couple stress tensor is symmetric and only oneinternal material length scale parameter is considered [11], how-ever, for standard couple stress theory, the couple stress tensor isasymmetric and number of internal material length scale parame-ters is one not always.

The beam theory is a special plane problem of the plane elastic-ity, so the related 2-D couple stress theories can be given as fol-lows. Considering conventional representation in the engineering,the component representation for the couple stress theory isadopted.

2.2. Formulations of plane modied couple stress theory (C1 theory)

The displacements are represented by u and v, which are dis-placements along x and y directions. Consider the strain tensorand curvature tensor can be dened respectively as eij 12 ui;j uj;i, vij 12 xi;j xj;i, we introduce cxy = c12 + c21, vx =v13 + v31, vy = v23 + v32.

The geometric equations can be written as:

ex @u@xey @v@ycxy @v@x @u@yvx 12 @

2v@x2 @

2u@x@y

vy 12 @

2v@x@y @

2u@y2

8>>>>>>>>>>>>>>>>>:2-1

Strain : fex; ey; cxy;vx;vyg:Stress : frx;ry; sxy;mx;myg:where ex, ey, andcxy are normal and shear strains in continuummechanics. rx, ry, sxy are normal and shear stresses in continuummechanics. vx, vy are curvatures and torsional shear strain formicrostructures. mx, my are bending momentums and torsionalshear stress for microstructures.

The constitutive equations can be written as:

r rx ry sxyT De 2-2where

D D1 lD1lD1 D1

G

264375;

and " #( )

tures 93 (2011) 27232732m mxmy

2 G22G

vxvy

2-3

are an internal material length scale parameter, and E, l are con-stants of elasticity, D1 E1l2, G E21l.

The strain energy is expressed as

U ZVrxex ryey sxycxy mxvx myvydxdy 2-4-1

Substituting (2-1) into (2-4), we have,

U ZVrx

@u@x

ry @v@y

sxy @u@y

@v@x

12mx

@2v@x2

@2u

@x@y

!"

12my

@2v@x@y

@2u

@y2

!#dxdy 2-4-2

Integrating by parts of Eq. (2-4-2), we obtain

U ZS

rxnx sxyny

u ryny sxynxv

12mxnx myny @v

@x @u

@y

ds

Z

@rx@x

@sxy@y

12

@2mx@x@y

@2my@y2

! !u

"

@rx@x @sxy@y 12 @

2mx@x@y @

2my@y2

0

@sxy@x @ry@y 12 @

2mx@x2

@2my@x@y

0

8>: 2-6The boundary forces are

T TmxTmyTx

8:9>=>; 2-8

Substituting (2-1), (2-2), (2-3) into (2-6),The couple stress equilibrium equation in terms of

displacements:

E1l2

@2u@x2 1l2 @

2u@y2 1l2 @

2v@x@y

12 2G @

4u@x2@y2 @

4v@x3@y @

4u@y4 @

4v@x@y3

0

E1l2

1l2

@2u@x@y 1l2 @

2v@x2 @

2v@y2

8>>>>>>>>>>> 2-9

the plane is replaced by xz coordinate shown in Fig. 1.

WJ. Chen et al. / Composite Structures 93 (2011) 27232732 2725V

@ry@y

@sxy@x

12

@2mx@x2

@2my

@x@y

! !v#dxdy

ZS

rxnx sxyny 12@mx@x

@my@y

ny

u

ryny sxynx 12@mx@x

@my@y

nx

vds

ZS

12mxnx myny @v

@x @u

@y

ds 2-5

From (2-5), following governing equations can be obtained.The equilibrium equations (no body forces) areFig. 1. Schematic diagramBased on the couple stress theory, only xy is included amongthe rotations are xx = 0 and xz = 0. 12 2G @4u

@x3@y @4v

@x4 @4u

@x@y3 @4v

@x2@y2

0

>>>>:3. Basic equations of composite laminated beam of modiedcouple stress theory

In the point of view of theory of elasticity, the beam theory canbe described by introducing the hypothesis of the cross-sectioninto the plane elasticity. It is also true for the composite laminatedbeam for the couple stress theory. Considering conventional repre-sentation of beam theory in the engineering, the xy coordinate ofof Timoshenko beam.

Assumed displacements in a section of composite laminated

The strain can be written as

truce

excxzczxvxyvyx

8>>>>>>>>>>>:

9>>>>>>=>>>>>>;3-3

Substituting (3-1), (3-2) into (3-3), we have

ex @u@x du0dx z dhdxcxz czx 12 @u@z dwdx

12 dwdx h vyx vxy 12 dxydx dxxdy

12 dxydx 14 dhdx d

2wdx2

8>>>: 3-4

3.3. Constitutive relations of composite laminated beam of modiedcouple stress theory

The constitutive relations of composite laminated beam are de-ned in layer-by-layer.

The stressstrain relations of kth layer in the local coordinate(x0,z0) can be expressed as followsbeam can be described by

ux; z u0x zhxv 0w wx

8>: 3-1where h is the angle of rotation around the y-axis of the cross-sec-tion (see Fig. 1).

Substituting (3-1) into the expression of the rotation asx 12 curl u, we have,xx 12 w;y v ;z 0xy 12 u;z w;x 12 hw;xxz 12 v ;x u;y 0

8>: 3-2

3.2. Strain of composite laminated beam of modied couple stresstheory

Consider the strain tensor and curvature tensor can be denedrespectively as

eij 12 ui;j uj;i; vij 12xi;j xj;i

According to the engineering conventional representation, thestrain tensor and curvature tensor for the beam can be expressedin the vector form as follows:

ex u;x; cxz czx 2c13; vxy v12; vyx v21;and ey cxy cyz vx vy vxz vyz 0:3.1. hypothesis of composite laminated beam of modied couple stresstheory

The hypothesis of the cross-section of the classical beam can beadopted in the couple stress theory of the Timoshiko beam [18]. Inorder to avoid distortion and warping of beam section under purebending, the ber orientation of the composite laminated beamshould be orthogonal.

2726 WJ. Chen et al. / Composite Srk Cke 3-5whererk rkx0 skx0z0 skz0x0 mkx0y0 mky0x0h iT

3 - 6

e ex0 cx0z0 cz0x0 vx0y0 vy0x0h iT

3-7

Ck

ck11ck44

ck4422ck44

22ck44

26666664

37777775 3-8

where x0 aligns with the direction of the ber in kth layer,

Ck11 Ek1

1 vk12 2 , Ck44 Gk12; v21 Ek2vk12Ek1 ; Ek1 is elastic constant of kth

layer, Gk12 is shear elastic constant of kth layer, vk12 is Poisson ratioof kth layer, in which subscripts 1 and 2 represent the direction ofber and matrix, respectively.

After coordinate transformation for the stress-strain relations ofthe plate, kth layer in the global coordinate (x,z) of the beam can bewritten as follows

rk Q ke 3-9where

rk rkx skxz skzx mkxy mkyxh iT

3 - 10

e ex cxz czx vxy vyxh iT

3-11

Q k

Qk11Qk44

Qk4422 eQk44

22 eQk44

266666664

377777775: 3-12

The components of Qk are expressed as

Qk11 m4Ck11 n4Ck22Qk44 Ck44m2 Ck44n2 Ck44

(3-13

eQk44 Ck44H/k 3-14where m = con/k, n = sin/k and /k is angle of ply,

H/k 0 when /k 0

1 when /k 90

. In order to avoid distortion and warp-

ing of beam section under pure bending, the effect of couple stresscan be ignored when angle of ply is /k = 0.

3.4. principle of virtual work for composite laminated beam ofmodied couple stress theory

It is well known that the principle of virtual work can be used toderive the equilibrium equation and the boundary condition.

The principle of virtual work for composite laminated beam ofcouple stress theory ca be given by

dU dW 0 3-16where

dU Xnk1

dUk Xnk1

ZXkeTQ kdedv

3-17

ZT

ZT

tures 93 (2011) 27232732dW Xf dudv

dXT duds 3-18

where f T and TT are body force and boundary force respectively.

h h

k13

>

trucSubstituting equation (3-4) and (3-5) and (3-12) into the equa-tion (3-17), by the integration of the y and z coordinates in the sec-tion of beam, the equation of beam becomes

dU Xnk1

dUk Xnk1

ZVkrkTdedxdydz

ZX

Xnk1

Z hk2

hk2rkTdedz

!dxdy

Xnk1

ZXk

rkxdex skxzdcxz skyzdcyz mkxydvxy mkyxdvyx

dv

Xnk1

ZXk

rkxdex 2skxzdcxz 2mkxydvxy

dv

Z L0

dNdx

du0 dQdx 12d2Y

dx2

!dw dM

dx Q 1

2dYdx

dh

" #dx

Ndu0 Q 12dYdx

dw Y

2d

dwdx

M Y

2

dh

xLx03-19

where N, M, Q are the classical tractions of the beam, Y is the trac-tion of couple stress moment of the beam. They are

N Pnk1

RAk r

kxdA

; M Pn

k1

RAk rkxzdA

Q Pn

k1

RAk s

kxzdA

; Y Pn

k1

RAk m

kxydA

8>>>>>: 3-20The expression of the work by the external forces on the beam inthe modied couple stress theory can be expressed as

dW Zlfudu0 fwdw fcdxdx Ndu0 Vdw

MdhjxLx0 3-21where fu and fw are, respectively, the x- and z-components of thebody force per unit length along the x-axis, fc is the y-componentof body force per unit length along the x-axis, and N; V and M arethe applied axial force, transverse force, and bending moment atthe two ends of the beam respectively, andZ

fcdxdx 12Z

fcdhw;xdx

12

Z

fcdhdx dfcdx dw

xLx0

Z

fcdhdx

!3-22

Substituting Eqs. (3-19) and (3-21) into the equation (3-16) wehaveZ L

0 dN

dx fu

du0 dQdx

12d2Y

dx2 12dfcdx

fw !

dw

"

dMdx

Q 12dYdx

12fc

dh

dx N Ndu0

Q 1

2dYdx

fc2 V

dw Y

2 Y

d

dwdx

M Y

2M

dh

xLx0

0 3-23

From (4-8), the equilibrium equations are obtained as

dNdx fu 0dQdx 12 d

2Ydx2

12 dfcdx fw 0

8>>> 3-24

WJ. Chen et al. / Composite SdMdx Q 12 dYdx 12 fc 0

:and traction boundary conditions at x = 0 and x = L are4.2. Degradation of the composite laminated beam of modied couplestress theory

4.2.1. Classical composite laminated Timoshenko beamSubstituting = 0 into the Eq. (4-1), the equilibrium equations

in terms of displacement of classical composite laminated Timo-shenko beam can be given as

Q11d2u0dx2

J11 d2hdx2 fu 0

ksQ44 d2wdx2

dhdx

12 dfcdx fw 0

8>>>>>>>>>>>>>>>:4-1

the equilibrium equations in terms of displacements of Timo-shenkos beam of couple stress theory can be obtained as follows

Q11d2u0dx2

J11 d2hdx2 fu 0

ksQ44 d2wdx2

dhdx

2Q444 d4wdx4

d3hdx3

12 dfcdx fw 0

J11d2u0dx2

I11 d2hdx2 ksQ44 dwdx h 2Q444 d3wdx3 d2hdx2 12 fc 0

8>>>>>>>>>>>:4-2

where ks is the Timoshenko shear coefcient, which depends on thegeometry of beam cross-section, and

Q44 Pnk1

eQk44bzk1 zkh iQjj

Pnk1

Qkjjbzk1 zkh i

j 4;5

Jii Pnk1

Qkiib z2k1z2k 2

i 1

Iii Pn Qkiib z3k1z3k i 1

8>>>>>>>>>>>>>>>>>>>>>>>>>>>>>:

4-3N NQ 12 dYdx 12 fc VY 0Mx Y2 M

8>>>>>>>: 3-25and displacement boundary conditions are

u0 u0w wdwdx dwdx

8>>>>>: 3-26

tures 93 (2011) 27232732 2727J11d2u0dx2

I11 d2hdx2 ksQ44 dwdx h 12 fc 0

>>>>:

4.2.2. Isotropic Timoshenko beam of modied couple stress theoryFor the isotropic Timoshenko beam (H(/k) 1), the elastic con-

stants become Q11 EA; Q44 GA; Q55 GA=2; Q66 0; J11 0; I11 EI and I bh312 . Substituting these constants into the (4-4)the equilibrium equations in terms of displacements classical iso-tropic Timoshenko beam of couple stress theory can be obtained as

EA d2u0dx2

fu 0ksGA d

2wdx2

dhdx

2GA4 d4wdx4

d3hdx3

12 dfcdx fw 0

8>>>>>>>>>>: 4-7

4.2.5. Isotropic BernoulliEuler beam of couple stress theorySubstituting equations h dwdx ; u0 0, and H(/k) = 1 into the (4-

7), considering only bending deformation of beam, the equilibriumequation in terms of displacements of classical isotropic BernoulliEuler beam of couple stress theory can be obtained as

I11 2Q44d4w

dx4 fw 4-8

This is identical to result in the reference [17].

5. Numerical example for scale effect: simply supported beamsubjected to cylindrical bending

A cross-ply simply supported beam shown in Fig. 2 is analyzedhere. The beam is only subjected to cylindrical bending loads ofEI d2h

dx2 ksGA dwdx h

2GA4 d3wdx3 d2hdx2 12 fc 0>>>>:These are identical to results in the reference [18].

4.2.3. Classical isotropic Timoshenko beamSubstituting = 0 and fc = 0 into the (4-5), the equilibrium equa-

tions of the Classical isotropic Timoshenko beam in terms of dis-placements can be obtained as follows

EA d2u0dx2

fu 0ksGA d

2wdx2

dhdx

fw 0EI d

2hdx2

ksGA dwdx h 0

8>>>>>>>: 4-6These are identical to results in the reference [18].Fig. 2. Schematic diagram of simply supported beam.5.2.1. Numerical examples for the scale effects of microstructureIn order to test characteristics of the scale effects of microstruc-

ture, models of simply supported laminated cross-ply beam areadopted. The sizes of the beam model are width b = 25 lm, thick-ness h = 25 lm, length L = 200 lm. Cylindrical bending load isq0 = 1 N mm. The material constants[23]: E2 = 6.98 GPa, E1 = 25E2,G12 = 0.5E2, G22 = 0.25E2, m12 = m22 = 0.25, in which subscripts 1and 2 represent the direction of ber and matrix, respectively.We choose the next two types of cross-ply laminated beam withthree-layer as follows. The parameters of the rst one [0/90/0]are identical to Q44E2bh 0:4,

Q55E2bh

0:2, 12I11E2bh

3 2:014, Q44 0=90=0 =(0/ 0.4/0).pL2

ksQ44p Q444L2

w0pLp Q444L2

ksQ44 h0q0 0 5 - 5

pL

p22Q444L2

ksQ44 !

w0 ksQ44p2Q44

2

4L2p

2I11L2

!h0 0 5-6

The solution of the displacements in the center of beam is obtainedas follows

w0 q0L

4 4ksQ44L2 4p2I11 p2Q442

p4 4ksI11Q44L2 2Q44 4ksQ44L2 p2I11

h i 5 - 7h0

q0L3 4ksQ44L

2 p22Q44

p3 4ksI11Q44L2 2Q44 4ksQ44L2 p2I11 h i 5-8

where ks 55m1265m12 for the rectangular section.The stress in the beam is

rkx pQk11h0z

Lsin

pxL

5-9

5.2. Solution of BernoulliEuler beam of modied couple stress theory

By using the trial functions of wx w0 sin pxL

, the solution ofthe displacement in the center of beam can be obtained

w0 q0L4

p4 I112Q44 rkx zQk11 pL

2w0 sin pxL 8

(a) [0, 90, 0] (b) [90, 0, 90]

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

x/L

w/h

l=0l=h/4l=h/2l=h

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

x/L

w/h

l=0l=h/4l=h/2l=h

Fig. 3. The deection of the beam.

(a) [0, 90, 0] (b) [90, 0, 90]

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05

x/L

l=0l=h/4l=h/2l=h

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

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

x/L

l=0l=h/4l=h/2l=h

Fig. 4. The angle of rotation of the beam.

(a) [0, 90, 0] (b) [90, 0,90]

-0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

x (GPa)

z/h

l=0l=h/4l=h/2l=h

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

x (GPa)

z/h

l=0l=h/4l=h/2l=h

Fig. 5. The stress rx in section of the beam at x = L/2.

WJ. Chen et al. / Composite Structures 93 (2011) 27232732 2729

truc0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2w

/hl=0l=h/2l=hl=0l=h/2l=h

2730 WJ. Chen et al. / Composite SNext, keep the thickness of the beam constant and change thematerial constantl to examine the scale effect. Numerical resultsof the deection of the beam are given in Fig. 3, which show thatthe deection of the beam in couple stress theory is smaller thanthat in the classical elasticity as the material constant l increases.

Numerical results of the deection of the beam are given inFig. 3 which show that the deection of the beam in couple stresstheory is smaller than that in the classical elasticity as the materialconstant l increases.

Numerical results of the angle of rotation of the beam are givenin Fig. 4, which show that the angle of rotation of the beam in cou-ple stress theory is smaller than that in the classical elasticity asthe material constant l increases.

(a) Medium thickness beam with / 0.125h L =

Note blue line Euler-Ber

black line Timoshen

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.02

0.04

x/L

Fig. 6. The deection of the beam to compare EulerBernoulli and Timoshenko beam cougure legend, the reader is referred to the web version of this article.)

(a) Medium thickness beam with / 0.125h L =Note blue line Euler-Bern

black line Timoshenk

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

x/L

w/h

l=0l=h/2l=hl=0l=h/2l=h

Fig. 7. The deection of the beam to compare EulerBernoulli400

600

800

1000

1200

w/h

l=0l=h/2l=hl=0l=h/2l=h

tures 93 (2011) 27232732Numerical results of the stress in section of the beam are givenin Fig. 5, which show that the stress in the section of the beam incouple stress theory is smaller than that in the classical elasticityas the material constant l increases.

5.2.2. Numerical examples to compare EulerBernoulli andTimoshenko beam couple stress theories for microstructures

In order to compare EulerBernoulli and Timoshenko beamcouple stress theories for microstructures, aforementioned modelsof simply supported laminated cross-ply beam are adopted. How-ever, various sizes of the beam are chosen rstly as lengthL = 200 lm and thickness h = 25 lm, and secondly for a slenderbeam with a large aspect ratio, length L = 2000 lm and thickness

(b) thin beam with / 0.0125h L =

noulli beam of couple stress theory

ko beam of couple stress theory

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

200

x/L

ple stress theories [0/90/0]. (For interpretation of the references to colour in this

(b) thin beam with / 0.0125h L =oulli beam of couple stress theory

o beam of couple stress theory

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

5000

10000

15000

x/L

w/h

l=0l=h/2l=hl=0l=h/2l=h

and Timoshenko beam couple stress theories [90/0/90].

truc-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5z/h

l=0l=h/2l=hl=0l=h/2l=h

WJ. Chen et al. / Composite Sh = 25 lm. We choose the cross-ply laminated beam with three-layer of [0/90/0] and [90/0/90], respectively, change the mate-rial constant as l = (0,h/2,h), respectively, to examine the scaleeffect.

Numerical results of the deection of the beam are given inFigs. 6 and 7 which show that the deference of the Timoshenkobeam in couple stress theory is less than EulerBernoulli beam ofcouple stress theory for Medium thickness beam with h/L = 0.125and for thin beam with h/L = 0.0125 under the material constantl as the same.

Numerical results of the stress in section of the beam are givenin Figs. 8 and 9, which show that the stress rx of the Timoshenkobeam in couple stress theory is smaller than EulerBernoulli beamof couple stress theory for Medium thickness beam with h/

(a) Medium thickness beam with / 0.125h L =

Note blue line Euler-Berno

black line Timoshenk

-0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08-0.5

-0.4

x (GPa)

Fig. 8. The stress rx in section of the beam at x = L/2 to compare Euler

(a) Medium thickness beam with / 0.125h L =Note blue line Euler-Bern

black line Timoshenk

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

x (GPa)

z/h

l=0l=h/2l=hl=0l=h/2l=h

Fig. 9. The stress rx in section of the beam at x = L/2 to compare EulerB-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

z/h

l=0l=h/2l=hl=0l=h/2l=h

tures 93 (2011) 27232732 2731L = 0.125 and for thin beam with h/L = 0.0125 under the materialconstant l as the same.

6. Conclusion

A new model for composite laminated beam with rst ordershear deformation on the couple stress theory is developed. Thecharacteristics of the couple stress theory are the use of rota-tiondisplacement as dependent variables and the use of onlyone constant to describe the materials micro-structural character-istics. By introducing the hypothesis of the cross-section of beam,the governing equations of the composite laminated beam of cou-ple stress theory are established by the principle of virtual work. Inorder to avoid distortion and warping of beam section under pure

(b) thin beam with / 0.0125h L =

ulli beam of couple stress theory

o beam of couple stress theory

-8 -6 -4 -2 0 2 4 6 8-0.5

-0.4

x (GPa)

Bernoulli and Timoshenko beam couple stress theories [0/90/0].

h L (b) thin beam with / 0.0125=oulli beam of couple stress theory

o beam of couple stress theory

-30 -20 -10 0 10 20 30-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

x (GPa)

z/h

l=0l=h/2l=hl=0l=h/2l=h

ernoulli and Timoshenko beam couple stress theories [90/0/90].

bending, the ber orientation of the composite laminated beamshould be orthogonal.

The present model of beam can be viewed as a simplied couplestress theory in engineering mechanics. A cross-ply simply sup-ported beam subjected to cylindrical bending loads of fw = q0sin(px/L) is solved by directly applying the newly developed beammodel. Numerical results show that the present beam model cancapture the scale effects of microstructure. The deections andstresses of the present model of beam of couple stress theory arealways smaller than that by the classical beammodel. Additionally,the present model can be reduced directly to the classical compos-ite laminated Timoshenko beam, Isotropic Timoshenko beam ofcouple stress theory, classical isotropic Timoshenko beam, com-posite laminated BernoulliEuler beam of couple stress theoryand isotropic BernoulliEuler beam of couple stress theory.

Conict of interest

None declare.

Acknowledgement

The work in this paper was supported by the National NaturalSciences Foundation of China (No. 11072156). This support isgratefully acknowledged.

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A modified couple stress model for bending analysis of composite laminated beams with first order shear deformation1 Introduction2 Formulations for modified couple stress theories2.1 Modified coupled stress theory2.2 Formulations of plane modified couple stress theory (C1 theory)

3 Basic equations of composite laminated beam of modified couple stress theory3.1 hypothesis of composite laminated beam of modified couple stress theory3.2 Strain of composite laminated beam of modified couple stress theory3.3 Constitutive relations of composite laminated beam of modified couple stress theory3.4 principle of virtual work for composite laminated beam of modified couple stress theory

4 Composite laminated beam of modified couple stress theory4.1 Equilibrium equations in terms of displacements for the composite laminated beam of modified couple stress theory with first order shear deformation4.2 Degradation of the composite laminated beam of modified couple stress theory4.2.1 Classical composite laminated Timoshenko beam4.2.2 Isotropic Timoshenko beam of modified couple stress theory4.2.3 Classical isotropic Timoshenko beam4.2.4 Composite laminated BernoulliEuler beam of modified couple stress theory4.2.5 Isotropic BernoulliEuler beam of couple stress theory

5 Numerical example for scale effect: simply supported beam subjected to cylindrical bending5.1 Solution of the composite laminated beam of modified couple stress theory5.2 Solution of BernoulliEuler beam of modified couple stress theory5.2.1 Numerical examples for the scale effects of microstructure5.2.2 Numerical examples to compare EulerBernoulli and Timoshenko beam couple stress theories for microstructures

6 ConclusionConflict of interestAcknowledgementReferences