A modified couple stress model for bending analysis of composite laminated beams with first order shear deformation

Composite Structures 93 (2011) 2723–2732

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A modified couple stress model for bending analysis of composite laminated beams with first order shear deformation q Wanji Chen a,b,⇑, Li Li a,c, Ma Xu a a

State Key Laboratory for Structural Analysis of Industrial Equipment, Dalian University of Technology, DaLian 116023, China Key Laboratory of Liaoning Province for Composite Structural Analysis of Aerocraft and Simulation, Shenyang Aerospace University, Shenyang, LN 110136, China c Physics and Biophysics Department, China Medical University, No. 92, The 2nd North Road, Heping District, Shenyang 110001, China b

a r t i c l e

i n f o

Article history: Available online 31 May 2011 Keywords: Composite laminated Timoshenko bending beam Modified couple stress theory Material length parameter Scale effect

a b s t r a c t Based on a modified couple stress theory, a model for composite laminated beam with first order shear deformation is developed. The characteristics of the theory are the use of rotation–displacement as dependent variable and the use of only one constant to describe the material’s micro-structural characteristics. The present model of beam can be viewed as a simplified couple stress theory in engineering mechanics. An example as a cross-ply simply supported beam subjected to cylindrical bending loads of fw = q0 sin (px/L) is adopted and explicit expression of analysis solution is obtained. Numerical results show that the present beam model can capture the scale effects of microstructure, and the deflections and stresses of the present model of couple stress beam are smaller than that by the classical beam mode. Additionally, the present model can be reduced to the classical composite laminated Timoshenko beam model, Isotropic Timoshenko beam model of couple stress theory, classical isotropic Timoshenko beam, composite laminated Bernoulli–Euler beam model of couple stress theory and isotropic Bernoulli–Euler beam of couple stress theory. Ó 2011 Elsevier Ltd. All rights reserved.

1. Introduction Since the 1960s, experiments have shown that micro-structure has scale effects due to impurities, crystal lattice mismatch and micro cracks at micro scales. With the material size scaling down to the order of micro scales, the stiffness and the strength of metal materials can increase with the size decreasing, which is called size effects. The size effects have been proved by many experiments in the recent two decades. For example, Fleck et al. [1] observed that the scaled shear strength increases by a factor of three as the wire diameter decreases from 170 lm to 12 lm in the twisting of thin copper wires; Stolken and Evan [2] reported a significant increase in the normalized bending hardening with the beam thickness decreasing in bending of ultra thin beams. Sun et al. [3] put forward a alternative view of the size effects in the nano-scale structures. As conventional continuum theory cannot explain or solve the problems of the scale effects, theories for microstructures need to be developed.

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. E-mail address: [email protected] (W. Chen).

0263-8223/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruct.2011.05.032

Theories for microstructures include couple stress theory and strain gradient theory. A series of research in the couple stress/ strain gradient theories have been made. For example, in the 1960s, Toupin [4], Koiter [5] and Mindlin proposed couple stress theory [6]. Between the 1980s and 1990s, Aifantis [7], Fleck and Hutchinson [8,9] developed the strain gradient theory in plasticity. Gao et al. [10] further improved the strain gradient theory in plasticity. A modified couple stress theory has recently been proposed by Yang et al. in which the couple stress tensor is symmetric and only one internal material length scale parameter is considered [11]. The couple stress theory can be viewed as a special format of strain gradient theory which uses rotation as a variable to describe curvature, while the strain gradient theory uses strain as variable to describe curvature. Though both theories can describe the scale defects at micro-scale, the couple stress theory contains fewer rotation variables than the strain gradient theory does for the strain variables. In the couple stress theory, the variables related to micro-scale impurities or defects are formulated into rotation equilibrium equations. In the strain gradient theory, these variables are formulated into higher order strain terms in geometric equations. In both cases, new parameters which describe the material scale characteristics are introduced as higher order term (4th order) into the partial differential governing equation. Yet, in conventional continuum mechanics, this partial differential governing

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equation is a 2nd order equation. Generally speaking, the couple stress/strain gradient theory for microstructures can be classified into two respective theories, C1 theory and C0 theory. For C1 theory the displacements and rotations/strains are dependent variables. For C0 theory, the displacements and rotations/strains are independent variables. In the application in the engineering, the microstructures such as sensors and actuators in micro-electromechanical systems (MEMS) and nano-electromechanical systems (NEMS) are often consist in the components of beam, plate and membrane et al. According to the application in engineering, the beam, plate and shell theories based on couple stress/strain gradient theory should be developed. The researchers have focused on the beam theory on microscale in recent years. A number of papers have been published for attempting to develop microstructure-dependent non-local Timoshenko beam models and apply them to analyze nanotubes and other small beam-like members/devices. All of these models are based on a C0 theory in which the rotation–displacement as dependent variables. For example, the model for pure bending proposed by Anthoine [12] is based on the classical C0 couple stress elasticity theory, which includes two additional internal material length scale parameters. The higher-order Bernoulli–Euler beam model developed by Papargyri-Beskou et al. [13] is based on the C0 gradient elasticity theory, which involves two internal material length scale parameters. The non-local Bernoulli–Euler beam model by Peddieson et al. [14], in the formulation the constitutive equation suggested by Eringen [15] contains two additional material constants. More background related to the couple stress beam based on the C0 couple stress theory, especially Cosserat-type theories which contain more than two additional material constants, can be found in the review by Altenbach al. [16]. Recently, due to the difficulty of determining more than one microstructure-dependent length scale parameters and the approximate nature of beam theories, C1 non-classical beam models involving only one material length scale parameter are getting many attentions. One model, as a simpler Bernoulli–Euler beam model based on modified couple stress theory with only one material length parameter, has recently been developed by Park and Gao [17]. Ma et al. proposed a microstructure-dependent Timoshenko beam model based on a modified couple stress theory with only one material length parameter [18]. Tsiatas proposed a new Kirchhoff plate model based on a modified couple stress theory [19]. Metin developed a general nonlocal beam theory based on C0 theory [20], where the nonlocal constitutive equations proposed by Eringen [15] are adopted. The nonclassical R–L beam model based on the higher order shear deformation theory and C1 couple stress theory was developed by Ma et al. [21]. The non-classical R– L model can be reduced to the existing classical elasticity-based R– L model by using the material length scale parameter and Poisson’s ratio are both taken to be zero. The classical R–L beam model[22] is a third-order beam model satisfied the condition of shear stress equal zero on the upper and lower surfaces of the beam. For moderate thickness beam, the accuracy is higher than first-order shear beam model. Furthermore the R–L beam model can be reduce the non-classical Bernoulli–Euler beam model when the normality assumption is introduced. Composite laminate beam and plate are widely used in engineering. Due to the microscale such as fiber, impurities and micro cracks at micro matrix are involved in a laminated composite structure, it results in classical laminate theory invalid in some problems related to the miro-scale of laminate composites. The objective of this paper is to develop a microstructuredependent model for the laminated Timoshenko beam based on a modified couple stress theory with only one material length scale parameter.

2. Formulations for modified couple stress theories Unlike the conventional continuum mechanics, the rotation vector xi is introduced to kinematic relation of the classical couple stress theory, as well as the curvatures tensor vij and couple stress tensor mij. Unlike the classical couple stress theory, Yang et al. [11] developed a modified couple stress theory in which the part of rotation gradient in the strain tensor is symmetric. 2.1. Modified coupled stress theory According to the symmetric couple stress theory proposed by Yang et al., the strain tensor and curvature tensor can be defined as eij ¼ 12 ðui;j þ uj;i Þ, vij ¼ 12 ðxi;j þ xj;i Þ respectively, where 1 x ¼ 2 curlu, u ¼ ðui Þ is the displacement vector and x(xi) is the rotation vector. The main differences of modified couple stress theory with standard couple stress theory are that for modified couple stress theory the couple stress tensor is symmetric and only one internal material length scale parameter is considered [11], however, for standard couple stress theory, the couple stress tensor is asymmetric and number of internal material length scale parameters is one not always. The beam theory is a special plane problem of the plane elasticity, so the related 2-D couple stress theories can be given as follows. Considering conventional representation in the engineering, the component representation for the couple stress theory is adopted. 2.2. Formulations of plane modified couple stress theory (C1 theory) The displacements are represented by u and v, which are displacements along x and y directions. Consider the strain tensor and curvature tensor can be defined respectively as eij ¼ 1 ðui;j þ uj;i Þ, vij ¼ 12 ðxi;j þ xj;i Þ, we introduce cxy = c12 + c21, vx = 2 v13 + v31, vy = v23 + v32. The geometric equations can be written as:

8 ex ¼ @u > @x > > > > ey ¼ @@yv > > > > < c ¼ @ v þ @u xy @y @x   > > 1 @2 v @2 u > > vx ¼ 2 @x2  @x@y > > >   > > : v ¼ 1 @ 2 v  @ 2 u2 y 2 @x@y @y

ð2-1Þ

Strain : fex ; ey ; cxy ; vx ; vy g: Stress : frx ; ry ; sxy ; mx ; my g: where ex, ey, andcxy are normal and shear strains in continuum mechanics. rx, ry, sxy are normal and shear stresses in continuum mechanics. vx, vy are curvatures and torsional shear strain for microstructures. mx, my are bending momentums and torsional shear stress for microstructures. The constitutive equations can be written as:

r ¼ ½rx ry sxy T ¼ De

ð2-2Þ

where

2

D1

6 D ¼ 4 lD1

3

lD1

7 5;

D1 G

and

 m¼

mx my

"

 ¼

#(

2‘2 G 2‘2 G

vx vy

) ð2-3Þ

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‘ are an internal material length scale parameter, and E, l are conE stants of elasticity, D1 ¼ ð1El2 Þ, G ¼ 2ð1þ lÞ. The strain energy is expressed as

U¼

Z V

ððrx ex þ ry ey þ sxy cxy Þ þ ðmx vx þ my vy ÞÞdx dy

ð2-4-1Þ

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

U¼

Z " V

!   @u @v @u @ v 1 @2v @2u þ mx þ rx þ ry þ sxy  @x @y @x 2 @y @x2 @x@y

1 þ my 2

!# @ v @ u dxdy  @x@y @y2 2

Z



ð2-4-2Þ

  1 @ v @u ds  þ ðmx nx þ my ny Þ 2 @x @y !! Z " @ rx @ sxy 1 @ 2 mx @ 2 my  u þ  þ 2 @x@y @x @y @y2 V dx dy

   1 @mx @my þ ny u 2 @x @y S      1 @mx @my þ ry ny þ sxy nx þ þ nx v ds 2 @x @y   Z 1 @ v @u ðmx nx þ my ny Þ ds þ  @x @y S 2 ¼

ð2-7Þ

8 9 u > > < = ~ ¼ v u  > : 1 @ v  @u > ; @x

Z 

ð2-8Þ

@y

2  8 2 E @ u @2 v > þ ð12lÞ @@yu2 þ ð1þ2lÞ @x@y > 1l2 @x2 > > >  4  > > u @4 v @4 u @4 v > <  12 ‘2 G @x@2 @y ¼0 2  @x3 @y þ @y4  @x@y3   2 2 2 ð1þ l Þ ð1 l Þ > E @ u @ v @ v > þ 2 @x2 þ @y2 > 1l2 @x@y 2 > > >  4  > 4 > u @4 u @4 v : þ 12 ‘2 G @x@3 @y ¼0  @@xv4 þ @x@y 3  @x2 @y2

!! #

v

  3 2 @my 8 9 1 @mx < T mx = 6 rx nx þ ss ny  2  @x þ @y ny 7 7 @my T ¼ T my ¼ 6 1 @m : ; 4 ry ny þ ss nx þ 2 @xx þ @y nx 5 Tx mx nx þ my ny

Substituting (2-1), (2-2), (2-3) into (2-6), The couple stress equilibrium equation displacements:



rx nx þ sxy ny u þ ðry ny þ sxy nx Þv

@ ry @ sxy 1 @ 2 mx @ 2 my þ þ þ 2 @x2 @y @x @x@y

The boundary forces are

2

S

þ

ð2-6Þ

The boundary displacements are

2

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

U¼

8 2  @s @2 m rx mx > < @@x þ @yxy  12 @@x@y þ @y2y ¼ 0 2  2 > : @ sxy þ @ ry þ 1 @ m2x þ @ my ¼ 0 2 @x @y @x@y @x

in

terms

of

ð2-9Þ

3. Basic equations of composite laminated beam of modified couple stress theory

rx nx þ sxy ny 

ð2-5Þ

From (2-5), following governing equations can be obtained. The equilibrium equations (no body forces) are

In the point of view of theory of elasticity, the beam theory can be described by introducing the hypothesis of the cross-section into the plane elasticity. It is also true for the composite laminated beam for the couple stress theory. Considering conventional representation of beam theory in the engineering, the x–y coordinate of the plane is replaced by x–z coordinate shown in Fig. 1. Based on the couple stress theory, only xy is included among the rotations are xx = 0 and xz = 0.

Fig. 1. Schematic diagram of Timoshenko beam.

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WJ. Chen et al. / Composite Structures 93 (2011) 2723–2732

3.1. hypothesis of composite laminated beam of modified couple stress theory The hypothesis of the cross-section of the classical beam can be adopted in the couple stress theory of the Timoshiko beam [18]. In order to avoid distortion and warping of beam section under pure bending, the fiber orientation of the composite laminated beam should be orthogonal. Assumed displacements in a section of composite laminated beam can be described by

8 > < uðx; zÞ ¼ u0 ðxÞ  zhðxÞ v ¼0 > : w ¼ wðxÞ

ð3-1Þ

h

rk ¼ rkx0 skx0 z0 skz0 x0 mkx0 y0 mky0 x0 h

e ¼ ex0 cx0 z0 cz0 x0 vx0 y0 vy0 x0 2 6 6 6 Ck ¼ 6 6 6 4

8 1 > < xx ¼ 2 ðw;y  v ;z Þ ¼ 0 xy ¼ 12 ðu;z  w;x Þ ¼  12 ðh þ w;x Þ > : xz ¼ 12 ðv ;x  u;y Þ ¼ 0

ð3-2Þ

ð3 - 6Þ

iT

ð3-7Þ 3

ck11

7 7 7 7 7 7 5

ck44 ck44 2‘2 ck44

ð3-8Þ

2‘2 ck44 where x0 aligns with the direction of the fiber in kth layer, Ek1

C k11 ¼ 

1ðv k12 Þ

where h is the angle of rotation around the y-axis of the cross-section (see Fig. 1). Substituting (3-1) into the expression of the rotation as x ¼ 12 curl u, we have,

iT

2

, C k44 ¼ Gk12 ;

k k 2 12 k 1

v 21 ¼ E Ev

; Ek1 is elastic constant of kth

layer, Gk12 is shear elastic constant of kth layer, v k12 is Poisson ratio of kth layer, in which subscripts 1 and 2 represent the direction of fiber and matrix, respectively. After coordinate transformation for the stress-strain relations of the plate, kth layer in the global coordinate (x, z) of the beam can be written as follows

rk ¼ Q k e

ð3-9Þ

where 3.2. Strain of composite laminated beam of modified couple stress theory Consider the strain tensor and curvature tensor can be defined respectively as

1 2

1 2

eij ¼ ðui;j þ uj;i Þ; vij ¼ ðxi;j þ xj;i Þ According to the engineering conventional representation, the strain tensor and curvature tensor for the beam can be expressed in the vector form as follows:

e ¼ ex cxz czx vxy vyx 2

ð3 - 10Þ

iT

ð3-11Þ 3

Q k11

6 6 6 6 Qk ¼ 6 6 6 4

(

Q k44 Q k44 ek 2‘ Q 44 2

ek 2‘2 Q 44

7 7 7 7 7: 7 7 5

Q k11 ¼ m4 C k11 þ n4 C k22

9 > > > > > > =

e k ¼ C k Hð/k Þ Q 44 44 ð3-3Þ

ð3-14Þ k

where

czx > > > > > > > > vxy > > > > : vyx ;

ð3-12Þ

ð3-13Þ

Q k44 ¼ C k44 m2 þ C k44 n2 ¼ C k44

The strain can be written as

e¼

h

iT

The components of Qk are expressed as

ex ¼ u;x ; cxz ¼ czx ¼ 2c13 ; vxy ¼ v12 ; vyx ¼ v21 ; and ey ¼ cxy ¼ cyz ¼ vx ¼ vy ¼ vxz ¼ vyz ¼ 0: 8 ex > > > > > > < cxz

h

rk ¼ rkx skxz skzx mkxy mkyx

k

k

m = con/ , n = sin/ and / is angle of ply,  k  ¼ 0 0 when / Hð/ Þ ¼ . In order to avoid distortion and warp1 when /k ¼ 90 ing of beam section under pure bending, the effect of couple stress can be ignored when angle of ply is /k = 0. k

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

8 0 ex ¼ @u ¼ du  z dh > @x dx > <  dx dw 1 dw cxz ¼ czx ¼ 12 @u þ dx ¼ 2 dx  h @z     > > : v ¼ v ¼ 1 dxy þ dxx ¼ 1 dxy ¼  1 dh þ d2 w yx

xy

2

dx

dy

2 dx

4

dx

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

ð3-4Þ

2

dx

3.3. Constitutive relations of composite laminated beam of modified couple stress theory

It is well known that the principle of virtual work can be used to derive the equilibrium equation and the boundary condition. The principle of virtual work for composite laminated beam of couple stress theory ca be given by

dU  dW ¼ 0

ð3-16Þ

where The constitutive relations of composite laminated beam are defined in layer-by-layer. The stress–strain relations of kth layer in the local coordinate (x0 , z0 ) can be expressed as follows k

k

r ¼C e where

ð3-5Þ

dU ¼

n X k1

dW ¼

Z X

dU k ¼

n Z X

Xk

k1

f T du dv þ

Z

eT Q k de dv

T T du ds

 ð3-17Þ

ð3-18Þ

dX

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

WJ. Chen et al. / Composite Structures 93 (2011) 2723–2732

Substituting equation (3-4) and (3-5) and (3-12) into the equation (3-17), by the integration of the y and z coordinates in the section of beam, the equation of beam becomes

dU ¼

n X

n Z X

dU k ¼

k¼1

¼

n Z X

Z

k¼1



rkx dex þ skxz dcxz þ skyz dcyz þ mkxy dvxy þ mkyx dvyx dv

Xk



Xk



rkx dex þ 2skxz dcxz þ 2mkxy dvxy dv





!   # 2 dN dQ 1 d Y dM 1 dY du0  þ Q þ dh dx  ¼ dw þ dx dx 2 dx2 dx 2 dx 0        x¼L

1 dY Y dw Y dw  d  Mþ dh

þ Ndu0 þ Q þ 2 dx 2 dx 2 x¼0 Z

L

"

rkT de dz dx dy



n Z X

ð3-19Þ where N, M, Q are the classical tractions of the beam, Y is the traction of couple stress moment of the beam. They are

8 n R P > > rk dA ; > > > :Q ¼

k¼1 n P

k¼1

R

Ak

M¼



skxz dA ; Y ¼

n R P

Ak

ðrkx zÞdA

Ak

 mkxy dA

k¼1 n R P k¼1

ð3-20Þ

The expression of the work by the external forces on the beam in the modified couple stress theory can be expressed as

dW ¼

Z

ðfu du0 þ fw dw þ fc dxÞdx þ ½Ndu0 þ Vdw

l

þ Mdhjx¼L x¼0

ð3-21Þ

where fu and fw are, respectively, the x- and z-components of the body force per unit length along the x-axis, fc is the y-component of body force per unit length along the x-axis, and N; V and M are the applied axial force, transverse force, and bending moment at the two ends of the beam respectively, and

Z

fc dx dx ¼ 

‘

1 2

1 ¼ 2

Z

fc dðh þ w;x Þdx ‘

Z ‘

!

x¼L Z

dfc

fc dh dx þ dw  fc dh dx dx ‘ x¼0

ð3-22Þ

Substituting Eqs. (3-19) and (3-21) into the equation (3-16) we have

"  !  2 dN dQ 1 d Y 1 dfc þ fu du0  þ  þ þ fw dw dx dx 2 dx2 2 dx 0   

dM 1 dY 1 þ Q þ þ fc dh dx þ ðN  NÞdu0 dx 2 dx 2       1 dY fc Y dw þ Qþ þ  V dw þ   Y d 2 dx 2 2 dx    x¼L

Y ¼0 þ M   M dh

2

Z

2727

ð3-25Þ

and displacement boundary conditions are

!

h2

k¼1

n Z X k¼1

¼

hk 2 k

X

¼

Vk

k¼1

 rkT dedx dy dz

8 N¼N > > > > < Q þ 1 dY þ 1 f ¼ V 2 dx 2 c > Y ¼ 0 > > > : M x þ Y2 ¼ M

L

4. Composite laminated beam of modified couple stress theory 4.1. Equilibrium equations in terms of displacements for the composite laminated beam of modified couple stress theory with first order shear deformation Substituting geometric (3-4) and stress–strain relation (3-9) into (3-20), we have

8 0 > N ¼ Q 11 du  J 11 dh > dx dx > > > > 0 < M ¼ J 11 du  I11 dh dx dx   ‘2 Q 44 d2 w > > þ dh >Y ¼  2 dx dx2 > > >  : Q ¼ ks Q 44 dw h dx

ð4-1Þ

the equilibrium equations in terms of displacements of Timoshenko’s beam of couple stress theory can be obtained as follows

8 d2 u 0 d2 h > > > Q 11 dx2  J 11 dx2 þ fu ¼ 0 > > > 2  2 4  < d3 h ks Q 44 ddxw2  dh  ‘ Q4 44 ddxw4 þ dx þ 12 dfdxc þ fw ¼ 0 3 dx > > > >   > > : J d2 u0  I d2 h  k Q dw  h  ‘2 Q 44 d3 w þ d2 h þ 1 f ¼ 0 11 11 2 s 44 dx 4 2 c dx dx3 dx2 dx2 ð4-2Þ where ks is the Timoshenko shear coefficient, which depends on the geometry of beam cross-section, and

8 i n h P > e k bðzkþ1  zk Þ > Q Q 44 ¼ > 44 > > k¼1 > > > > > i n h > P > > Q ¼ Q k bðz  zk Þ ðj ¼ 4; 5Þ > > < jj k¼1 jj kþ1 k  n > P Q ii bðz2kþ1 z2k Þ > > > J ii ¼ ði ¼ 1Þ > 2 > > k¼1 > > > k  > > n > P Q ii bðz3kþ1 z3k Þ > > ði ¼ 1Þ : Iii ¼ 3

ð4-3Þ

4.2. Degradation of the composite laminated beam of modified couple stress theory

ð3-23Þ

4.2.1. Classical composite laminated Timoshenko beam Substituting ‘ = 0 into the Eq. (4-1), the equilibrium equations in terms of displacement of classical composite laminated Timoshenko beam can be given as

ð3-24Þ

8 2 d2 h > Q 11 ddxu20  J 11 dx 2 þ fu ¼ 0 > > > > < 2  ks Q 44 ddxw2  dh þ 12 dfdxc þ fw ¼ 0 dx > > > > > dw 1 : d2 u0 d2 h J 11 dx2  I11 dx 2  ks Q 44 dx  h þ 2 fc ¼ 0

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

and traction boundary conditions at x = 0 and x = L are

ð3-26Þ

k¼1

x¼0

8 dN þ fu ¼ 0 > > < dx 2 dQ þ 12 ddxY2 þ 12 dfdxc þ fw ¼ 0 dx > > : dM  Q þ 12 dY þ 12 fc ¼ 0 dx dx

8 0 u0 ¼ u > > > ¼ dw > > : dx dx h¼h

ð4-4Þ

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WJ. Chen et al. / Composite Structures 93 (2011) 2723–2732

4.2.2. Isotropic Timoshenko beam of modified couple stress theory For the isotropic Timoshenko beam (H(/k)  1), the elastic constants become Q 11 ¼ EA; Q 44 ¼ GA; Q 55 ¼ GA=2; Q 66 ¼ 0; J 11 ¼ 3 0; I11 ¼ EI and I ¼ bh . Substituting these constants into the (4-4) 12 the equilibrium equations in terms of displacements classical isotropic Timoshenko beam of couple stress theory can be obtained as

8 2 > EA ddxu20 þ fu ¼ 0 > > > 2  2 4  < 3  ‘ 4GA ddxw4 þ ddx3h þ 12 dfdxc þ fw ¼ 0 ks GA ddxw2  dh dx > >   >  > : EI d2 2h þ ks GA dw  h þ ‘2 GA d3 w3 þ d22h  1 fc ¼ 0 dx 4 2 dx dx dx

 fw ¼ q0 sin pLx , i.e., fu = fc = 0. The thickness and the material properties of each layer are uniform. Boundary conditions:

ujx¼0 ¼ ujx¼L ¼ 0;

wjx¼0 ¼ wjx¼L ¼ 0;

Yjx¼0 ¼ Yjx¼L ¼ 0;

Mjx¼0 ¼ Mjx¼L ¼ 0:

ð5 - 1Þ ð5-2Þ

Substituting (4-1) into the (4-2), in terms of h and w, the (5-2) become

ð4-5Þ

2 d w

2 dx

x¼0

2 d w

¼ 2 dx

dh

dh

¼ ¼0 dx x¼0 dx x¼L

¼ 0; x¼L

ð5-3Þ

These are identical to results in the reference [18]. 4.2.3. Classical isotropic Timoshenko beam Substituting ‘ = 0 and fc = 0 into the (4-5), the equilibrium equations of the Classical isotropic Timoshenko beam in terms of displacements can be obtained as follows

8 d2 u 0 > > > EA dx2 þ fu ¼ 0 > < 2  ks GA ddxw2  dh þ fw ¼ 0 dx > > >  2 > : EI d h þ ks GA dw  h ¼ 0 dx dx2

p2 L2

ð4-7Þ

4

4

dx

¼ fw

ð4-8Þ

This is identical to result in the reference [17]. 5. Numerical example for scale effect: simply supported beam subjected to cylindrical bending A cross-ply simply supported beam shown in Fig. 2 is analyzed here. The beam is only subjected to cylindrical bending loads of

Fig. 2. Schematic diagram of simply supported beam.

wðxÞ ¼ w0 sin

ks Q 44 þ

p p2 ‘2 Q 44

4.2.5. Isotropic Bernoulli–Euler beam of couple stress theory Substituting equations h ¼ dw ; u0 ¼ 0, and H(/k) = 1 into the (4dx 7), considering only bending deformation of beam, the equilibrium equation in terms of displacements of classical isotropic Bernoulli– Euler beam of couple stress theory can be obtained as

ðI11 þ ‘2 Q 44 Þ

u0 ðxÞ ¼ 0;

px ; L

hðxÞ ¼ h0 cos

px L

ð5-4Þ

Substituting (5-4) into (4-2), we have

4.2.4. Composite laminated Bernoulli–Euler beam of modified couple stress theory Substituting h ¼ dw into the (4-5), the equilibrium equations in dx terms of displacements of composite laminated Bernoulli–Euler beam of couple stress theory can be obtained as follows

d w

The trial function is assumed as

ð4-6Þ

These are identical to results in the reference [18].

8 d2 u0 d3 w > > > Q 11 dx2  J 11 dx3 þ fu ¼ 0 > < 2 4   ‘ Q2 44 ddxw4 þ dfdxc ¼ 0 > >   4 > > : I þ ‘2 Q d w ¼ fw 11 44 2 dx4

5.1. Solution of the composite laminated beam of modified couple stress theory

L

4L2

p2 ‘2 Q 44 4L2

! w0 þ

!

p p2 ‘2 Q 44 L

4L2

 ks Q 44 w0 þ ks Q 44 þ

!  ks Q 44 h0  q0 ¼ 0

p2 Q 44 ‘2 p2 I11 4L2

þ

L2

ð5 - 5Þ

! h0 ¼ 0

ð5-6Þ

The solution of the displacements in the center of beam is obtained as follows

  q0 L4 4ks Q 44 L2 þ 4p2 I11 þ p2 Q 44 ‘2  i w0 ¼ h p4 4ks I11 Q 44 L2 þ ‘2 Q 44 4ks Q 44 L2 þ p2 I11   q0 L3 4ks Q 44 L2  p2 ‘2 Q 44  i h0 ¼ h p3 4ks I11 Q 44 L2 þ ‘2 Q 44 4ks Q 44 L2 þ p2 I11

ð5 - 7Þ

ð5-8Þ

m12 where ks ¼ 5þ5 for the rectangular section. 6þ5m12 The stress in the beam is

rkx ¼

pQ k11 h0 z L

sin

px L

ð5-9Þ

5.2. Solution of Bernoulli–Euler beam of modified couple stress theory  By using the trial functions of wðxÞ ¼ w0 sin pLx , the solution of the displacement in the center of beam can be obtained

8 4 < w0 ¼ 4 q0 L 2 p ðI11 þ‘ Q 44 Þ   : k rx ¼ zQ k11 pL 2 w0 sin pLx

ð5-10Þ

5.2.1. Numerical examples for the scale effects of microstructure In order to test characteristics of the scale effects of microstructure, models of simply supported laminated cross-ply beam are adopted. The sizes of the beam model are width b = 25 lm, thickness h = 25 lm, length L = 200 lm. Cylindrical bending load is q0 = 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 1 and 2 represent the direction of fiber and matrix, respectively. We choose the next two types of cross-ply laminated beam with three-layer as follows. The parameters of the first one [0°/90°/0°] are identical toEQ244 ¼ 0:4, EQ255 ¼ 0:2, E12Ibh113 ¼ 2:014, Q 44 ½0 =90 =0  = bh bh 2 (0/ 0.4/0). The parameters of second one [90°/0°/90°] are identical to Q 44 ¼ 0:3, EQ255 ¼ 0:15, E12Ibh113 ¼ 0:158, Q 44 ½90 =0 =90  ¼ ð0:3=0=0:3Þ. E2 bh bh 2

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0.2

1.6

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

0.18 0.16 0.14

1.2 1

w/h

0.12

w/h

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

1.4

0.1 0.08

0.8 0.6

0.06

0.4

0.04

0.2

0.02 0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0

1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

x/L

x/L

(a) [0, 90, 0]

(b) [90, 0, 90]

0.8

0.9

1

Fig. 3. The deflection of the beam.

0.6

0.05

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

0.04 0.03 0.02

0.2

0.01

-0.2

-0.01 -0.02

-0.4

-0.03

-0.6

-0.04 -0.05 0

0

θ

0

θ

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

0.4

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-0.8

0

0.1

0.2

0.3

0.4

0.5

0.6

x/L

x/L

(a) [0, 90, 0]

(b) [90, 0, 90]

0.7

0.8

0.9

1

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

0.5

0.5 0.4 0.3 0.2

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

0.4 0.3 0.2 0.1

0

z/h

z/h

0.1

0

-0.1

-0.1

-0.2

-0.2

-0.3

-0.3

-0.4

-0.4

-0.5 -0.08

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

-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

σx (GPa)

σx (GPa)

(a) [0, 90, 0]

(b) [90, 0,90] Fig. 5. The stress rx in section of the beam at x = L/2.

0.1

0.2

0.3

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WJ. Chen et al. / Composite Structures 93 (2011) 2723–2732

0.2

1200

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

0.18 0.16 0.14

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

1000 800

w/h

w/h

0.12 0.1

600

0.08 400

0.06 0.04

200

0.02 0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0

1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x/L

x/L

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

(b) thin beam with h / L = 0.0125

blue line

Euler-Bernoulli beam of couple stress theory

black line

Timoshenko beam of couple stress theory

Fig. 6. The deflection of the beam to compare Euler–Bernoulli and Timoshenko beam couple stress theories [0°/90°/0°]. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

1.4 1.2

w/h

1

15000

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

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

10000

w/h

1.6

0.8 0.6

5000

0.4 0.2 0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

x/L

x/L

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

(b) thin beam with h / L = 0.0125

Note

blue line black line

0.9

1

Euler-Bernoulli beam of couple stress theory Timoshenko beam of couple stress theory

Fig. 7. The deflection of the beam to compare Euler–Bernoulli and Timoshenko beam couple stress theories [90°/0°/90°].

Next, keep the thickness of the beam constant and change the material constantl to examine the scale effect. Numerical results of the deflection of the beam are given in Fig. 3, which show that the deflection of the beam in couple stress theory is smaller than that in the classical elasticity as the material constant l increases. Numerical results of the deflection of the beam are given in Fig. 3 which show that the deflection of the beam in couple stress theory is smaller than that in the classical elasticity as the material constant l increases. Numerical results of the angle of rotation of the beam are given in Fig. 4, which show that the angle of rotation of the beam in couple stress theory is smaller than that in the classical elasticity as the material constant l increases.

Numerical results of the stress in section of the beam are given in Fig. 5, which show that the stress in the section of the beam in couple stress theory is smaller than that in the classical elasticity as the material constant l increases. 5.2.2. Numerical examples to compare Euler–Bernoulli and Timoshenko beam couple stress theories for microstructures In order to compare Euler–Bernoulli and Timoshenko beam couple stress theories for microstructures, aforementioned models of simply supported laminated cross-ply beam are adopted. However, various sizes of the beam are chosen firstly as length L = 200 lm and thickness h = 25 lm, and secondly for a slender beam with a large aspect ratio, length L = 2000 lm and thickness

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0.5

0.5

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

0.4 0.3 0.2

0.3 0.2 0.1

0

z/h

0

z/h

0.1

-0.1

-0.1

-0.2

-0.2

-0.3

-0.3

-0.4

-0.4

-0.5 -0.08

-0.06

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

0.4

-0.04

-0.02

0

0.02

0.04

0.06

0.08

-0.5 -8

-6

-4

-2

0

2

4

6

8

σx (GPa)

σx (GPa)

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

(b) thin beam with h / L = 0.0125

Euler-Bernoulli beam of couple stress theory Timoshenko beam of couple stress theory

black line

Fig. 8. The stress rx in section of the beam at x = L/2 to compare Euler–Bernoulli and Timoshenko beam couple stress theories [0°/90°/0°].

0.5

0.5

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

0.4 0.3 0.2

0.3 0.2 0.1

0

z/h

z/h

0.1

0.4

0

-0.1

-0.1

-0.2

-0.2

-0.3

-0.3

-0.4

-0.4

-0.5 -0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

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

-0.5 -30

-20

-10

0

10

20

σx (GPa)

σx (GPa)

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

(b) thin beam with h / L = 0.0125

Note blue line black line

30

Euler-Bernoulli beam of couple stress theory Timoshenko beam of couple stress theory

Fig. 9. The stress rx in section of the beam at x = L/2 to compare Euler–Bernoulli and Timoshenko beam couple stress theories [90°/0°/90°].

h = 25 lm. We choose the cross-ply laminated beam with threelayer of [0°/90°/0°] and [90°/0°/90°], respectively, change the material constant as l = (0, h/2, h), respectively, to examine the scale effect. Numerical results of the deflection of the beam are given in Figs. 6 and 7 which show that the deference of the Timoshenko beam in couple stress theory is less than Euler–Bernoulli beam of couple stress theory for Medium thickness beam with h/L = 0.125 and for thin beam with h/L = 0.0125 under the material constant l as the same. Numerical results of the stress in section of the beam are given in Figs. 8 and 9, which show that the stress rx of the Timoshenko beam in couple stress theory is smaller than Euler–Bernoulli beam of couple stress theory for Medium thickness beam with h/

L = 0.125 and for thin beam with h/L = 0.0125 under the material constant l as the same. 6. Conclusion A new model for composite laminated beam with first order shear deformation on the couple stress theory is developed. The characteristics of the couple stress theory are the use of rotation–displacement as dependent variables and the use of only one constant to describe the material’s micro-structural characteristics. By introducing the hypothesis of the cross-section of beam, the governing equations of the composite laminated beam of couple stress theory are established by the principle of virtual work. In order to avoid distortion and warping of beam section under pure

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bending, the fiber orientation of the composite laminated beam should be orthogonal. The present model of beam can be viewed as a simplified couple stress theory in engineering mechanics. A cross-ply simply supported beam subjected to cylindrical bending loads of fw = q0sin (px/L) is solved by directly applying the newly developed beam model. Numerical results show that the present beam model can capture the scale effects of microstructure. The deflections and stresses of the present model of beam of couple stress theory are always smaller than that by the classical beam model. Additionally, the present model can be reduced directly to the classical composite laminated Timoshenko beam, Isotropic Timoshenko beam of couple stress theory, classical isotropic Timoshenko beam, composite laminated Bernoulli–Euler beam of couple stress theory and isotropic Bernoulli–Euler beam of couple stress theory. Conflict of interest None declare. Acknowledgement The work in this paper was supported by the National Natural Sciences Foundation of China (No. 11072156). This support is gratefully acknowledged. References [1] Fleck NA, Muller GM, Ashby MF, Ashby JW. Strain gradient plasticity: theory and experiment. Acta Metall Mater 1994;42(2):475–87. [2] Stolken JS, Evans AG. A microbend test method for measuring the plasticity length-scale. Acta Mater 1998;46(14):5109–15. [3] Sun ZH, Wang XX, Soh AK, Wu HA, Wang Y. Bending of nanoscale structures: Inconsistency between atomistic simulation and strain gradient elasticity solution. Comput Mater Sci 2007;40(1):108–13.

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